T i jii, i—iin ••*,
CONF-8909210—
Proceedings of
The 12th Werner Brandt International Conference on the Penetration of Charged Particles in Matter
September 4-7,1989
Universidad del Pais Vasco/Euskal Herriko Unibertsitatea San Sebastian, Basque Country, Spain
eman ta zabal zazu Printed in the United States of America. Available from National Technical Information Service U.S. Department of Commerce 5285 Port Royal Road, Springfield, Virginia 22161 NTIS price codes—Printed Copy: A-99 Microfiche A01
This report was prepared as an account of work sponsored by an agency of the United StatesGovernment. Neither theUnitedStatesGovernment nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the UnitedStates Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. The 12th Werner Brandt International Universidad del Pais Vasco/Euskal Hcrriko Unibertsitatea Conference on the Penetration of Charged September 4-7, 1989 Particles in Matter San Sebastian, Basque Country, Spain i I o OD to e o w w 10 CONF-8909210
THE 12TH WERNER BRANDT INTERNATIONAL CONFERENCE ON THE PENETRATION OF CHARGED PARTICLES IN MATTER
s..§ g-1, S- * ** if- 3 K'
September 4-7, 1989 San Sebastian, Spain i||ffj||f|
Date Published - April 1990 f 111 J g f f { g
3 &§ =• 5. r=. 3 P** -?=.} 5) *? i l*tJiff
Prepared by the OAK RIDGE NATIONAL L/rORATORY Oak Ridge, Tennessee 37831 operated by MARTIN MARIETTA ENERGY SYSTEMS, INC. for the U.S. Department of Energy Under Contract No. DE-AC05-84OR21400 TABLE OF CONTENTS
Page
Introduction and Overview
R. H. Ritchie vii
Toasts and Responses ix
Conference Participants xvii
Program xxi
Localised Elastic and Inelastic Scattering Processes for Electrons in Multilayer Structures A Howie. C. A. Walsh, and Z. L. Wang 1
Electromagnetic Resonances in Self Similar Arrays of Particles or Voids Francisco Claro and Ronald Fuchs 9
Influence of the Wake Potential on Inelastic Scattering Using a Small Probe in Confined Volumes P. E. Batson and J. Bruley 15
Electron Microscopy Study of the Electronic Properties of Small Silicon Particles Daniel Ugarte. Christian Colliex, and Pierre Trebbia 25
Electron Compton Scattering in the Electron Microscope P. Schattschneider and P. Pongratz 35
Dispersion Effects and Spatial Resolution in STEM N. Zabala and P. M. Echenique 61
Stopping Power for Protons in Al for the Whole Range of Velocities F. Flores, A. Arnau, P. M. Echenique, and R. H. Ritchie 69
Stopping Power for Protons Revisited Gregory Lapicki 85
Stopping Power Theory in Inhomogeneous Many Electron System M. Kitagawa „ 101 On the Z,' Correction of the Stopping Power I. Nagv and P. M. Echcnique Ill
Variational Approach to the Scattering Theory and its Application in Stopping Phenomena B. Apagyi and I. Nagy 125
Stopping Power and Ranges in Ion Implantation J. Bausells 139
Dynamical Effects in Electron Tunneling M. §unjic and L. Marusic 153
Photon Scanning Tunneling Microscopy R. J. Warmack. R. C. Reddick, and T. L. Ferrell 175
Extended Transfer Hamiltonian Method for Resonant Tunneling S. P. Apell. E. Albertson, and D. R. Penn 183
Laser Field Effects on the Interaction of Charged Particles with Degenerate and Non-Degenerate Plasmas N. R. Arista. R.M.O. Galvao, and L.C.M. Miranda 201
Screening Effects in Nuclear Fusion of Hydrogen Isotopes in Dense Media Nestor Arista, Raul Ba.-agiola, and Alberto Gras-Marti 211
Contribution of Ripplon Excitation to Interaction Energies of Particulates Rafael Garcia-Molina. Nuria Barberan, and Alberto Gras-Marti 221
q-Dependent Excitations of Metal Spheres Nuria Barberan, Manuel Barranco. F'ancesca Garcias, Jesus Navarro, and Lloreng Serra 231
Wake-Riding Electrons Emitted by Antiprotons Traversing Solid Targets J. Burgdorfer. J. Wang, and J. Muller 239
Non Perturbative Behaviours in Excitation and Ionisation of Atoms by Fast Charged Particles R. Gavet and A. Salin 255
Target Thickness Dependent Convoy Electron Production of Aligned Molecular HeH+-Ions J. Kemmler. K. O. Groeneveld, O, Heil, K. Kroneberger, and H. Rothard 267
ii Radiative Electron Capture by Channeled Ions J. M. Pitarke and R. H. Ritchie 273
Direct Methods in Structure Determination by LEED J. B. Pendry 289
Non-Local Exchange-Correlation Potential at a Metal Surface from Many-Body Perturbation Theory A. G. Eguiluz 303
Multiphonon Energy Exchange in the Collision of an Atom with a Surface J. R. Manson and R. H. Ritchie 313
Positronium Formation and Photoemission Spectroscopy for Surface Akira Ishii 329
Inelastic Positronium Formation at Metal Surfaces Toshiaki litaka 343
Energy Loss Measurements of He and H Scattered at Grazing Incidence off Ni A. Narmann, K. Schmidt, U. Imke, and W. Heiland 351
Electronic Response of Metal Surfaces to Electron Exchange with Doubly Charged Ions A. Niehaus 361
Transient Adsorption, Energy Loss, and Momentum Transfer in Low Velocity Ion Surface Scattering K. J. Snowdon. D. J. O'Connor, M. Kato, and R. J. MacDonald 375
The Neutralization Process of Highly Charged Ions Near a Metal Surface L. Folkerts and R. Morgenstern , 389
Some Basic Phenomena and their Importance in Relation to Techniques for Sputter- and Vapor-Deposition of High Temperature Superconducting Thin Films Orlando Auciello 397
Effective Mass of Bulk and Surface Positrons Due to Plasmon Excitation Shigeru Shindo 421
Variation of DOSD Moments with Bond Length in H2 and N2 Jens Oddcrshede, John R. Sabin. and Geerd H.F. Dicrcksen 435
iii De Structura Spectrorum Encrgiae Electroniorum Sccundariorum Ioniis Inductufum Prorsus Rcctorum c Corporibus Solidis Kar1-Ont(es Groeneveld. Robert Maier, and Hermann Rothard 445
Orbital Local Plasma Calculation of Mean Excitation Energies and Stopping Numbers S. B. Trickey. David E. Mcltzcr, and John R. Sabin 449
Secondary Electrons as Probe of Prccquilibrium Stopping Power of Ions Penetrating Solids K. Kroneberger. H. Rothard, P. Koschar, P. Lorenzcn, A. Clouvas, E. Veje, J. Kemmlcr, and K. O. Groeneveld 477
Secondary Electron Emission with Molecular Projectiles K. Kroneberger. H. Rothard, P. Koschar, P. Lorenzen, A. Clouvas, E. Veje, J. Kemmler, N. Keller, R. Maier, and K. O. Groeneveld 483
Direction-Dependent Stopping Power and Beam Deflection in Anisotropic Solids Oakley H. Crawford 489
Independent-Particle Model for Fusion in Cluster Impact Oakley H. Crawford 507
Bremsstrahlung Induced by 50 MeV H° Bombardment C. Stein, K. W. Habiger, B. R. Smith, Oakley H. Crawford, and R. H. Ritchie 517
Excited Substates Mixing of Fast Heavy Ions in Solids J. P. Rozet. A. Chetioui, K. Wohrer, C. Stephan, F. Ben Salah, A. Touati, M. F. Politis, and D. Vernhet 533
Radiation Mechanisms for Microscopic Relativistic Electron Beams Michael J. Moran 537
Interaction of Low-Energy Electrons and Positrons with Condensed Matter: Stopping Powers and Inelastic Mean Free Paths from Optical Data J. C. Ashley 551
Cross Sections for K-Shell Ionization by Electron Impact Ricardo Mavol and Francesc Salvat 573
The Theory of Track Formation in Insulators Due to Densely Ionizing Particles R. H. Ritchie, A. Gras-Marti. and J. C. Ashley 595
xv Track Structure: Perspectives, Progress, Problems Robert Katz 615
The Irradiation of Glycylglycine in Aqueous Solution - A Case Study of Calculations from Track-Structure to Biochemical Change J. E. Turner, Wesley E. Bolch, H. Yoshida, K. Bruce Jacobson, O. H. Crawford, R. N. Hamm, and H. A. Wright 633
Collective Aspects of Charged Particle Track Structure G. Basbas. A. Howie, and R. H. Ritchie 651
Calculations of Auger-Cascade-Induced Reactions with DNA in Aqueous Solution R. N. Hamm. H. A. Wright, J. E. Turner, R. W. Howell, D. V. Rao, and K.S.R. Sastry 673 THE 12TH WERNER BRANDT INTERNATIONAL CONFERENCE ON THE PENETRATION OF CHARGED PARTICLES IN MATTER
Introduction and Overview
R. H. Ritchie
It is in order to make a short sketch of the origin and philosophy of these workshops. The first was held at New York University in 1977, organized and inspired by our late colleague, Werner Brandt, who was Professor of Physics and Head of the Radiation Physics Laboratory there.
Werner was the driving force in the first and subsequent meetings until his untimely death, although it was my privilege to help in these and succeeding endeavors. Werner recognized the need for small and informal workshops in the community of workers concerned with the interaction of charged particles in matter. Since physical problems in this broad area often involve interdisciplinary expertise, it was expected that participants would be drawn from many different institutions and branches of knowledge and that smallness and informality would be essential for success in such gatherings. He emphasized that the main goal of such meetings should be to arrive at some sense of current problems, not merely to repeat results that have already been published.
In particular, all participants were encouraged to contemplate beforehand how their knowledge and scientific needs would relate to modern problems in penetration physics so that the workshops could address some of the outstanding questions of recent research. In a real sense, Werner here began a new mode of information exchange that involved timely and exciting discussions, required minimal effort by participants through the issuance of very informal summary reports, and included active workers with a broad spectrum of viewpoints. It has been a rare privilege for my colleagues and me to continue these workshops in Werner's honor and, in the tradition begun so ably by him, to attempt to convey through them the fascination to be found in research on charged particle penetration phenomena.
A list of past Workshops and their venues follows:
1977 - Wake Phenomena - New York University 1978 - Current Stopping Power Problems - New York University 1979 Low-Energy Particles - New York University 1980 - Matter Under Extreme Conditions - New York University 1981 - Exotic Projectiles - New York University 1982 - Dynamic Screening and Effective Charge - Honolulu 1983 - Properties of Ion-Induced Tracks in Matter • Oak Ridge National Laboratory
vii 1984 - Inelastic Near-Surface Interactions - Oak Ridge National Laboratory 1985 - Photon Emission from Irradiated Solids - Oak Ridge National Laboratory 1986/87 - Dynamic Interactions of Probes with Condensed Matter - University of Alicante, Alicante, Spain 1988 - Charged Particle Penetration Phenomena - Oak Ridge National Laboratory
The present Proceedings represents a wide spectrum of topics, but the freshness and interest of contributions to this Workshop more than compensate for the broadness of the coverage. The stimulation and excitement of exposure to the research of workers in other areas has led to interactions that may be important for the future.
Professor Pedro Elxenikc and his colleagues did an outstanding job of making arrangements for this Workshop and showed magnificent hospitality to the attendees. Pedro was ably assisted by Professor J. Colmenero, Drs. A. Rivacoba, A. Arnau, J. M. Pitarke, N. Zabala, J. Garcia, and Miriam Penalba, and by the conference secretaries M. Alvarez, A. Peon, C. Aycart, and E. Marti. Welcoming addresses by Dr. Angel Galindez, BBV; Dr. J. A Garrido, Consejero Director General de Iberduero SA; Xabier Albistur, Mayor of San Sebastian; and J. M. Aguirre, Deputy of Economy of the Province of Gipuzcoa, are given below.
Research sponsored by the Office of Health and Environmental Research, U.S. Department of Energy, under contract DE-AC05-84OR21400 with Martin Marietta Energy Systems, Inc.
viii Toast bv Dr. Angel Galindcz. BBV
Mis buenos amigos Pedro Echcniquc y José Antonio Garrido me han pedido que pronuncie unas breves palabras al término de esta cena, prólogo de la 12 Conferencia Werner Brandt que se va a celebrar aquí, en San Sebastián.
Quiero en primer lugar darles la bienvenida más cordial y afectuosa. Realmente, estoy seguro que todos Uds., que han acudido a la invitación de la Universidad del Pais Vasco, van a pasar unos días gratos, agradables y provechosos, envueltos en la hospitalaria amistad de la gente de esta tierra y dentro de este marco incomparable que es Donosti, una verdadera perla de nuestro Pais.
Deseo, también, aprovechar la ocasión que ofrecen estas jomadas científicas, en las que están implicados empresarios, para transmitir a todos Uds., inquietudes actuales de personas que han trabajado muchos años en las fronteras científicas de la tecnología, la industria y los servicios.
Estas preocupaciones se centran en la formación, la maduración y el reciclaje de las personas, principalmente los más jóvenes.
La Universidad y la Empresa son Instituciones muy vinculadas a esa tarea, exigida principalmente por la velocidad del crecimiento actual de los medios de comunicación y de los cambios tecnológicos y que, en resumen, trata de conseguir formar debidamente más gente en menos tiempo.
Tenemos la experiencia, vivida intensamente en los últimos años, de que es fácil, en el cambio de una empresa, reducir a la mitad el tiempo que históricamente se ha considerado necesario para preparar a los más jóvenes a hacer frente a ¡as mayores responsabilidades.
Igualdad y eficacia son téminos que es muy difícil conciliar, pero lo cierto es que la igualdad de oportunidades en el arranque y el estímulo en la carrera son los mobiles más adecuados para lograr la pronta maduración de las personas jóvenes.
Me parece que tanto la Universidad como la empresa deben de tener presente que hoy es esta la máxima prioridad. Formar adecuadamente más gente en menos tiempo.
Hasta ahora parecía que la acción de la Universidad y de la Empresa sobre la formación de una persona estaba saparada por una barrera infranqueable que se situaba en tomo a la edad del final de la graduación académica.
ix Creo que esta ban-era debe ser demolida. Creo que ambas, la Universidad y la Empresa, deben de colaborar desde los límites de la enseñanza secundaria, impartiendo al mismo tiempo y en el mismo tono, la ilusión, el conocimiento y el comportamiento colectivo.
Yo estoy seguro de que así será y de que todos Uds. van a trabajaren esa gran orea que es el formar y madurar a una juventud que quiere ansiosamente alcanzar, en menos tiempo, todo aquello de lo que se siente capaz.
Por último quiero levantar mi copa y brindar por esa colaboración de la Universidad y de la Empresa, por estas Jornadas, que son un ejemplo de esa colaboración, por Uds. y por sus familias.
ÁNGEL GALINDEZ
BBV Toast bv Dr. Jose Antonio Garrido. Consejero Director General de Iberduero SA
Nuestra Compañía tiene sus raices más sólidas en los grandes avances de la ciencia y en la cultura empresarial de sus recursos humanos. Queremos contribuir desde nuestro trabajo a la creación de un entorno científico tecnológico empresarial de calidad que nos permita competir en esta nueva Europa que con dificultades, pero con ilusión estamos construyendo.
En este proyecto su labor, la de los Europeos y la de nuestos amigos Americanos del norte y del sur, y Japoneses que nos acompañan es importante. No sólo para los países desarrollados. Como hemos discutido en otros foros es una responsabilidad ética de los que pertenecemos a los países desarrollados, empeñamos en desarrollar nuevos conocimientos y técnicas que permitan una solución mejor de los problemas humanos de todos los Países.
Al ofrecer esta cena de bienvenida al Duodécimo Congreso W. Brandt estamos queriendo hacer público nuestro profundo agradecimiento a todas las personas que, como Vds., hacen avanzar el conocimiento humano en aquellas cuestiones que nos permiten a nosotros mejorar la calidad de nuestro servicio a la sociedad.
Yo brindo, en nombre de mi Compañía, para que todos Vds. continúen en su esfuerzo de abrir nuevos horizontes a nuestras posibilidades.
JOSÉ ANTONIO GARRIDO
IBERDUERO
XI TOAST IN RESPONSE TO DRs. GALINDEZ AND GARRIDO
On behalf of the attendees at this 12th Werner Brandt International Conference on the Penetration of Charged Particles in Matter, I thank you for your kind remarks of welcome. In his autobiography, the famous physicist, George Gamow, tells about an interesting incident. At that time he was giving a popular lecture, open to the public, on the prospects of extracting energy from the fusion of nuclei. This was quite early in the history of nuclear physics, and the possibility of nuclear fusion had just been suggested. After his lecture, the director of the local energy district approached Gamow. He indicated great interest in the subject and made a very ingenious suggestion. He said that if Gamow would furnish him with some deuterium, presumably several decagrams in the D2O form, embedded in a copper conductor, he, the director, would make available all of the electrical power of his district to be sent through this material at an early hour on any given morning. Fortunately, Gamow was wise enough to decline to implement this fascinating suggestion.
I relate this incident to draw a parallel with the imaginative and farsighted interest in science that Dr. Garrido and Dr. Galindez have shown in their support of this Workshop and in many other instances in the past. Their remarkable vision and interest in the basic sciences are of great significance to the future of science and technology in the Basque Country and in Spain. We all thank you for this and for your magnificent hospitality.
R. H. Ritchie Oak Ridge National Laboratory and University of Tennessee
xii Opening Address of the Conference hv J. M. Apuirre. Deputy of Economy of the Province of Gipuzcoa
Buenos días Señoras y Señores.
Quiero agradecer a los organizadores de este Congreso, en particular a mi querido amigo el Profesor Echenique, la oportunidad que me brindan de hacerles esta salutación, en mi nombre y en el de todos los guipuzcoanos.
Es un orgullo poder dirigir unas palabras a un auditorio tan cualificado de profesores, investigadores y científicos de varios países del mundo. De la simple lectura de los títulos de las ponencias se deduce un eievadísimo nivel científico que espero les sea provechoso, a la vez que les deseo una agradable estancia entre nosotros, en los pocos ratos libres que les quedan para otras actividades.
sería muy presuntuoso por mi parte hablarles de la importancia de la ciencia para el progreso material de la humanidad. Cada vez más, estamos esperando de ustedes soluciones a los grandes problemas que tenemos planteados : sanitarios, de medio ambiente, energéticos, etc.... Sabemos que los resultados no se producen a corto plazo, sino con el esfuerzo continuado, con ilusión y con medios a todos los niveles: en la Administración, en la Empresas, en las Universidades y en los Centros de Investigación.
Es destacable el empuje que se ha dado en estos años a la ciencia y a la tecnología, en una sociedad como la nuestra, tradicionalmente algo reacia al conocimiento de las nuevas ideas, pero que ha comprendido perfectamente que nuestro progreso y nuestro futuro como pueblo, van unidos a otros, hasta el momento más avanzados. Multitud de problemas que tenemos, desde el déficit de la balanza tecnológica, hasta las altas tasas de paro, tienen una explicación última en el insuficiente esfuerzo en años pasados a los temas de la cultura, la formación, la investigación y el desarrollo tecnológico.
Esta tipo de Congresos tienen para mí una significación de una especie de símbolo de que nos enganchamos el tren de la modernidad y del futuro, con nuestra propia aportación, ciertamente modesta, pero que esperamos y deseamos creciente.
Esta Facultad de Químicas, reciente y joven, está alcanzando un grado de madurez en su presido científico que le permite tanto ser anfitrión de un evento tan cualificado como es éste, como de hacer sus propias aportaciones, tal como lo viene haciendo en los más prestigiosos foros y revistas científicas internacionales. Sepan Ustedes que, desde la Diputación y si bien con unas cifras modestas pero comparativarnenie con otras Administraciones con mayor competencia y responsabilidad administrativa, elevadas, apoyamos y seguimos apoyando este vivero de ideas y
xiii conocimientos que constituyen sus diversoso Departamentos.
Muchas gracias por su atención y espero que esta tarde tengamos ocasión de compartir un brindis con Ustedes, en la recepción que les ofrecemos en la Diputación, y a ia que gustosamente quedan invitados. Muchas gracias.
J. M. AGUIRRE
DEPUTY OF ECONOMY OF GIPUZCCA
xiv Toast by Xabier Albistur. Mayor of San Sebastian
I am very happy to welcome this distinguished group of scientists to the beautiful city of Donostia. Our fair town has existed since remote times, developed a prosperous maritime trade in the 14th century and is one of the loveliest and most picturesque cities in the world. You have seen some of the scenic attractions already and will understand why it is a highly popular resort area renowned for its restaurants, touristic and sporting activities.
As with most cities in the industrialized world, commercial developments in this area have grown out of the industrial revolution and the rapid increase in the resulting technology. Now we all ride on the crest of a new, high technology revolution in physics, electronics, computers, informatics, and molecular sciences. As Mayor of Donostia, I am proud to welcome a group of working scientists who arc closely coupled with these fascinating and timely developments.
The great philosopher, Seneca, wrote that "An age will come after many years when the ocean will loose the chains of things and a huge land lie revealed; when Tiphys will disclose new worlds and Thulc no more be the ultimate." Though this has been taken by many, including Columbus, as prophesying the discovery of ihe New World, it may be read as a revelation of the new world opening now in which man achieves increasingly complete control of the material world and his own genetic, physical, and intellectual nature.
Seneca also said that "a good mind possesses a kingdom." This workshop convenes many excellent minds engaged in important research. I hope that your conference will be highly profitable scientifically, that you enjoy your stay in Donostia, and that you will return in the near future.
XABIER ALBISTUR
MAYOR OF SAN SEBASTIAN THE 12TH WERNER BRANDT INTERNATIONAL CONFERENCE ON THE PENETRATION OF CHARGED PARTICLES IN MATTER San Sebastian, September 4-7, 1989
CONFERENCE PARTICIPANTS
Isabel Abril George Basbas Universidad Complutcnse Physical Review Letters Madrid, SPAIN Ridge, NY USA
J. F. Annctt Philip E. Batson University of Illinois IBM/T.J. Watson Research Center Urbana. IL USA Yorktown Heights, NY USA
Barnabas Apagyi Joan Bausells Technical University of Budapest Universidad Autonoma de Barcelona Budapest, HUNGARY Bellaterra, SPAIN
Peter Apcll G. Borste! Umea University University of Osnabriick Umca, SWEDEN Osnabriick, FRG
Nestor R. Arista Hidde H. Brongersma Universidad de Alicante Eindhoven University Alicante, SPAIN Eindhoven. THE NETHERLANDS
Andres Arnau Joachim Burgdordcr Universidad del Pais Vasco University of Tennessee San Sebastian, SPAIN Knoxville", TN USA
James C. Ashley A. Chelioui Oak Ridge National Laboratory Section de Physique el Chcmie Oak Ridge, TN USA University P. M. Curie 75231 Paris Cedex 05 Orlando Aucicllo CNRS URA 1379 Microeleclronic Center of North Carolina FRANCE Research Triangle Park, NC USA Francisco Claro Raul Baragiola Pontificia Universidad Calolica de Chile Rutgers University Santiago, CHILE Piscataway, NJ USA Vincent Cobut Manuel Barranco Universite de Sherbrooke Universidad dc Balearcs Quebec, CANADA Palma de Mallorca, SPAIN
:-:vn Juan Colmencro R. N. Hamm Universidad del Pais Vasco Oak Ridge National Laboratory San Sebastian, SPAIN Oak Ridge, TN USA
Oakley Crawford W. Hciland Oak Ridge National Laboratory Univcrsitat Osnabriick Oak Ridge, TN USA Osnabriick, FRG
Adolfo G. Eguiluz Archibald Howie Montana State University Cavendish Laboratory Bozeman, MT USA Cambridge, UK
P. M. Etxenike Toshiaki litaka Universidad del Pais Vasco Waseda University San Sebastian, SPAIN Tokyo, JAPAN
Fernando Eores Akira Ishii Universidad Autonoma de Madrid Tottori University Madrid, SPAIN Tottori, JAPAN
L. Folkerts Robert Katz Kernfysisch Versmallcr Instituut University of Nebraska Groningen, THE NETHERLANDS Lincoln, NE USA
Javier Garcia Jiirgcn Kcmmlcr Universidad del Pais Vasco Institut fur Kcrnphysik dcr Univcrsitat San Sebastian, SPAIN Frankfurt/Main, FRG
Rafael Garcia Molina M. Kimura Universidad de Murcia Argonne National Laboratory Murcia, SPAIN Argonne, IL USA
Robert Gayet Mitsuo Kitagawa Universite Bordeaux I North Shore College Talence-Cedex, FRANCE Atsugi, JAPAN
Alberto Gras-Marti Kurt Kroncbergcr Universidad de Alicante Institul fur Kernphysik dcr Univcrsitat Alicante, SPAIN Frankfurt/Main, FRG
Karl Ontjes Groencveld Norma J. Kwaak Institut fur Kernphysik der Universitat Oak Ridge National Laboratory Frankfurt/Main, FRG Oak Ridge, TN USA
Luis A. Guzman Gregory Lapicki IRST Materials Science Division East Carolina University Povo (Trcnto), ITALY Greenville, NC USA
xviii Janos Laszlo Miriam Pcnalba Max-Planck-Institut fur Plasmaphysik Universidad del Pais Vasco Garching, FRG San Sebastian, SPAIN
Ronald D. MacFarlane Jesus Perez Conde Texas A&M University Orsay U.P.S.-XI College Station, TX USA Ccdcx-Orsay, FRANCE
Joseph Manson Jose Maria Pitarke Clemson University Univcrsidad del Pais Vasco Clemson, SC USA Lcioa, SPAIN
Ricardo Mayol Sanchez Joseph Remillicux Universidad de Barcelona Institut Physique Nucleaire Lyon Barcelona, SPAIN Villeurbanne Cedex, FRANCE
Catherine J. McNeal Alberto Rivacoba Texas A&M University Universidad del Pais Vasco College Station, TX USA San Sebastian, SPAIN
Jose Luis Minchole R. H. Ritchie Dtor. Radiaciones Oncologico Oak Ridge National Laboratory San Sebastian, SPAIN Oak Ridge, TN USA
Rosa Monrcal J. P. Rozct Universidad Autonoma de Madrid Section de Physique ct Chemie Madrid, SPAIN P. M. Curie University 75231 Paris Cedcx 05 Michael Moran CNRS URA 1379 Lawrence Livcrmore Laboratory FRANCE Livermore, CA USA John R. Sabin Istvan Nagy Odcnse University Institut of Physics Odense, DENMARK Budapest, HUNGARY P. Schattschncider Arcnd Nichaus Technische Universitat Ryksunivcsileit Utrecht Vienna, AUSTRIA Utrecht, THE NETHERLANDS Lloreng Serra Crespi Hideo Nitta Univcrsidad dc Baleares Tokyo Gakugci University Palma de Mallorca, SPAIN Tokyo, JAPAN Shigeru Shindo John B. Pendry Tokyo Gakugci University Imperial College Tokyo, JAPAN London, UK
xix Ken Snowdon Ncrca Zabala Universitat Osnabrvick Universidad del Pais Vasco Osnabriick, FRG Leioa, SPAIN
Charles Stein Eugene Zarcmba Air Force Weapons Laboratory Queen's University Albuquerque, NM USA Kingston, CANADA
Marijan Sunjic University of Zagreb Zagreb, YUGOSLAVIA
Samuel B. Trickey University of Florida Gainesville, FL USA
J. E. Turner Oak Ridge National Laboratory Oak Ridge, TN USA
Jesus Ugalde Universidad del Pais Vasco San Sebastian, SPAIN
Daniel Ugarte Univcrsite Paris XI Orsay, FRANCE
• Michel A. Van Hove University of California Berkeley, CA USA
Matesh N. Varma U.S. Department of Energy Washington, DC USA
R. J. Warmack Oak Ridge National Laboratory Oak Ridge, TN USA
Helmut Winter Instilut fur Kcrnphysik Miinster, FRG
Yasunori Yamazaki University of Tokyo Tokyo, JAPAN
xx THE 12TH WERNER BRANDT INTERNATIONAL CONFERENCE ON THE PENETRATION OF CHARGED PARTICLES IN MATTER
San Sebastten, September 4-7, 1989
PROGRAM
Sunday. 3 September 1989 21:00 Dinner offered by Iberduero, S.A., at the Hotel Maria Cristina
Monday. 4 September 1989
08:00-09:00 Registration
09:00-09:30 Opening
09:30-11:30 Session I: Electron Microscopy. Chairman. M. Moran A. HOWIE: "Localised Elastic and Inelastic Scattering Processes for Electrons in Multi-Layer Structures"
F. CLARO: "Electromagnetic Resonances of a Self-Similar Array of Metallic Particles"
P.E. BATSON: "Near Edge Fine Structure: Differences Between Electron Energy Loss Scattering and Photon Absorption?"
D. UGARTE: "Electron Microscopy Study of the Electronic Properties of Small Silicon Particles"
P. SCHATTSCHNEIDER: "Electron Compton Scattering in the Electron Microscope"
N. ZABALA: "Dispersion Effects and Spatial Resolution in STEM"
11:30-12:00 Coffee Break
12:00-14:00 Session II: Energy Loss m. Chairman. J. R. Sabin
F. FLORES: "Stopping Power for Protons in Al"
xxi G. LAPICKI: "Formulas for Stopping and Range Revisited"
M. KTTAGAWA: "Stopping Power Theory in Inhomogcncous Many Electron System"
I. NAGY: "On the Z3 Correction of the Stopping Power"
B. APAGYI: "Variational Approach to the Scattering Theory and its Application in Slopping Phenomena"
J. BAUSELLS: "Stopping Power and Ranges in Ion Implantation"
16:00-17:00 Session HI: STM. Chairman. F. Flores
M. SUNJ1C: "Dynamical Effects in Electron Tunneling"
R. J. WARMACK: "Photon Scanning Tunneling Microscopy"
P. APELL: "Extended Transfer Hamiltonian Method for Resonant Tunneling"
17:00-17:30 Coffee Break
17:30-18:30 Session IV: Other Topics m. Chairman. M. Sunjic
N. ARISTA: "Laser Field Effects on the Interaction of Charged Particles with a Degenerate Electron Gas"
R. GARCIA MOLINA: "Contribution of Ripplon Excitation to the Interaction Energies of Particulates"
M. BARRANCO: "Multipote Response of Metal Spheres to q-Dcpendent Excitation Operators"
19:00 Welcome Reception
Excelentisima Diputacion Foral dc Guipuzcoa
Tuesday. 5 September 1989
09:00-11:00 Session V: Fast Ions. Chairman. J. Remillicux Y. YAMAZAKI: "Interaction of Anliprotons with Condensed Matter"
xx ii J. BURGD6RFER: "Forward Electron Production in Ion-Solid Collisions"
R. GAYET: "Non Pcrturbative Behaviours in Excitation and Ionization by Fast Charged Particles"
II. NTTTA: "On the Line Broadening of Okorokov Peak Profiles"
J. KEMMLER: "Electron Production and Transport in Thin Solids"
J. M. PITARKE: "Radiative Electron Capture by Channeled Ions"
11:00-11:30 Coffee Break
11:30-13:30 Session VI: Surface Physics. Chairman. A. Niehaus
J. B. PENDRY: "Direct Methods in Surface Structure Determination by Low Energy Electron Diffraction"
A. EGUILUZ: "Evaluation of the Exchange-Correlation Potential at a Metal Surface from the Knowledge of the Electron Self-Energy"
J. R. MANSON: "Multiphonon Energy Exchange in the Collision of an Atom with a Surface"
J. F. ANNETT: "New Universal Features of Surface Charge Density. Bonding and Stress"
A. ISHII: "Positronium Formation and Photoemission Spcctroscopy for Surfaces"
T. IITAKA: "Inelastic Positronium Formation at Metal Surfaces"
21:00 Basque Dinner at Sociedad Gastronomica Donostiarra
Wednesday. 6 September 1989
09:00-11:00 Session VII: Ion Surface. Chairman. J. B. Pendrv
W. HEILAND: "Energy Loss Measurements of He and H Scattered at Grazing Incidence off Ni"
R. MONREAL: "Energy Loss of Light Ions and Neutrals from Surface Scattering"
xxiii H. WINTER: "Coulomb Explosion of Fast Molecular Ions after Grazing Surface Scattering"
A. NIEHAUS: "Electronic Response of Metal Surfaces to Electron Exchange with Doubly Charged Ions"
K. SNOWDON: "Transient Adsorption, Energy Loss, and Momentum Transfer in Low Velocity Ion Surface Scattering"
H. H. BRONGERSMA: "New Possibilities of Low-Energy Ion Scattering"
11:00-11:30 Coffee Break
11:30-13:30 Session VIII: Other Topics(2). Chairman. A. Gras-Marti G. BORSTEL: "Calculated Photoemission and Many Body Effects in 3D- Ferromagnets"
K. J. McNEAL: "^Cf-Plasma Desorption Mass Spectromctry of Gold Clusters"
L. FOLKERTS: "The Neutralization Timescale of Highly Charged Ions on a Tungsten Surface"
R. BARAGIOLA: "Mechanisms for Multiply Charged Ion Emission in Electron Bombardment of Insulators"
O. AUCEELLO: "Some Basic Phenomena in Plasma and Ion and Laser Beam Interaction with Surfaces Related to the Synthesis of High-Tc Superconducting Films"
S. SHINDO: "Effective Mass of Bulk and Surface Positrons Due to Plasmon Excitation"
21:30 Conference Dinner at El Palacio de Miramar
Thursday. 7 September 1989
09:00-11:00 Session IX: Energy Lossf21. Chairman. J. C. Ashley
J. R. SABIN: 'Calculation of Slopping Powers in Some Aggregated Systems"
K. O. GROENEVELD: "Secondary Electron Emission as a Probe for the Non- Equilibrium Energy Loss"
xxiv S. B. TRICKEY: "Calculation of Mean Excitation Energy and Stopping Cross- Section in the Orbital Local Plasma Approximation"
K. KRONEBERGER: "Secondary Electron Emission with Molecular Projectiles"
O. H. CRAWFORD: "Direction-Dependent Stopping Power and Beam Deflection in Anisotropic Solids"
C. STEIN: "Brcmsstrahlung Induced by 50 MeV H° Bombardment"
11:00-11:30 Coffee Break
11:30-13:30 Session X: Other Topicsf3V Chairman. H. H. Brongersma J. P. ROZET: "Excited Substates Mixing of Fast Heavy Ions in Solids"
M. MORAN: "Radiation Mechanisms for Microscopic Rclativistic Electron Beams"
J. C ASHLEY: "Energy Loss of Electrons and Positrons in Solids"
R. MAYOL: "Cross Sections for K-Shell Ionization by Electron Impact"
A. GRAS-MARTI: The Theory of Track Formation in Insulators Due to Densely Ionizing Particles"
M. N. VARMA: "Radial Dose Distributions and their Application to Micro- and Nanodosimetry"
15:00-16:20 Session XI: Track Structure. Chairman. Y. Yamazaki
R. D. MacFARLANR "Fast Chemical Reactions in Heavy Ion Tracks"
R. KATZ: Track Structure: Perspectives, Progress, Problems"
J. E. TURNER: The Irradiation of Glycylglycine in Aqueous Solution, a Case Study of Calculations from Track Structure to Biochemical Change"
R. N. HAMM: "Calculations of Physical and Chemical Reactions with DNA in Aqueous Solution from Auger Cascades"
16:20-16:40 Coffee Break
XXV 16:40-17:25 Session XII: Cluster Fusion. Chairman. P. Echenique
G.BASBAS: "Ouster Fusion" R. BARAGIOLA: "Ouster Fusion" O. CRAWFORD: "Cluster Fusion" R. RITCHIE: "Ouster Fusion"
17:20 CLOSING REMARKS; P. M. Echenique and R. H. Ritchie
19:30 Social Party at Ritchie's Villa in Fuenterrabia
xxvi LOCALISED ELASTIC AND INELASTIC SCATTERING PROCESSES FOR ELECTRONS IN MULTILAYER STRUCTURES
A. Howie. C.A. Walsh and Z.L. Wang. Cavendish Laboratory, MadingJey Road. Cambridge CB3 OHE ABSTRACT: Characterisation of the microstructure of many modern composite materials depends on development and efficient use of new elastic and inelastic electron scattering processes which yield a high degree of spatial localisation. This is illustrated with reference 10 recent studies of semiconductor multilayer systems as well as natural layered crystals. In reflection siudies surface oxidation or damage seriously complicates the results. 1. INTRODUCTION Now that high resolution HREM and STM images showing individual atoms are commonplace in the scientific and popular literature, one couid be forgiven for thinking that the problems of materials characterisation are at an end. Rarely however do these images provide very direct information about local chemistry or electronic structure. This is well illustrated by the problem of characterising the artificial multiiayer structures which are of increasing importance both for semiconductor technology and metallurgy. Perhaps the most formidable challenge for electron microscopists, is posed by GaAs / AlAs superiattices where the small scattering difference between the two materials exacerbates the usual problems of quantitative HREM smicture imaging by phase contrast methods. The main effort so far has been in high resolution dark field imaging (Kakibayashi and Nagata, 1986: Bithell and Stobbs, 1989) based on the 200 Bragg reflection which, for structure factor reasons, maximises the scattering difference. Convergent beam diffraction methods have also been applied (Eaglesham and Humphrevs. 1986).
Scanning transmission electron microscopy CSTEM) is a potentially powerful combination of imaging and analysis for investigating many multilayer samples. Core loss speciroscopy CBuilock et ai 1986: Petford-Long and Long 1988) and annular dark field (ADF) imag3ng (Walls 198*: McGibbon et ai 1988) have both been usefully employed. The size of the focussed electron probe is a basic limitation in these methods but Pennycook f 1989) has recently demonstrated the exciting potential of the ADF mode in a high resolution STEM for Z- contrast imaging of multilayer and other structures at atomic resolution. Preparation of the thin multilayer samples needed for these edge-on siudies is non-trivia] and worth avoiding. Vincent et ai (1988) have extracted useful information from high-angle TEM diffraction patterns taken in plan view. Multilayers have also been imaged in bulk" at 2nm resolution by SEM fOgura. 1988) using both the back scaitered primaries and the secondary electron (SE) signals. An alternative possibility of reflection electron microscopy CREMi imaging at grazing angies (Eoothroyd et ai 1987: Buffat et al 1987) has been pursued here. combined with reflection energy loss spectroscopy using STEM. These methods have bee:: applied to samples of GaAs / A.'As muitiiayers cisaved in such a way (see fig. 1 > as to perm:: some transmission imaging and valence loss spectroscopy as well. We also report REM. vaience ioss spectroscopy ar.d SE imaging of cieaved surfaces of alum:na which are :-. l:mc of rarural uirrafine mu hi I aver sample. TEM REM
Fig. 1. 2. ELASTIC SCATTERING AND IMAGING Since AlAs surfaces oxidise much faster than those of GaAs and REM images are rather sensitive to surface disorder, it was found necessary to transfer the samples to the STEM as quickly as possible after cleaving them. Figs. 2 (a) and (b) show the bright field and ADF image of an AlAs / GaAs multilayer sample where, in a sequence of equally thick 2.8nm layers, an anomalous 5.6nm thick layer of Al Ga / As was accidendy introduced between iwo of the GaAs layers because the Ga shutter remained open. As expected, the ADF image which is dominated by incoherent scattering and Z contrast effects, shows the Ga rich regions as bright. The fact that the STEM bright field image fig. 2(a> is complementary to this, despite the considerable variations in thickness across the wedge sample, suggests that the intensity here is controlled mainly by absorption effects rather than by diffraction contrast
Figure 2. Bright field (a), ADF (b) and SREM (c) images of GaAl / As multilayer samples. Figure 2(c) is a SREM image of a different GaAl / As superlattice sample with lOnm thick layers of each material taken with the (10.10,0) reflection under surface resonance conditions. Here the contrast is similar to the ADF image, suggesting that large-angle incoherent scattering also makes an appreciable contribution to REM contrast. In some cases a large background was visible in the diffraction patterns, suggesting that the specimen had not been transferred quickly enough and that an oxide overlayer had built up.
Another case where surface layers exercise a profound effect in REM images is the a-alumina (0,1,1) cleaved surface which shows almost uniform bright contrast when first observed. After about three seconds of illumination however, some damage products start to appear on the surface. Figs. 3(a) .and 3(b) compare the same surface area before and after being illuminated by the electron beam for about 5 seconds. Inevitably the image shown in fig. 3(a) has already been slightly damaged. The damaged surface sometimes appears with three different contrast levels or domains labelled A, B and C and separated by some surface steps. The A domain appears with a very bright reflected intensity and keeps quite stable throughout. The B domain is also initially as bright as the A domain but is sensitive to beam irradiation and becomes dark showing some signs of damage products at the surface, see fig. 3(b). The products formed can cover the whole B domain but leave the A domain almost unchanged. The C domain, which is not present on all surfaces, may also undergo a damage process in a similar short period of time but cannot be visualized clearly due to its apparent roughness.
a-AJumina (0,-1,1) Surface
Fig. 3. REM images of cleaved Al (0,1,1) surface showing different surface termination domains (a) before and (b) after about 5 seconds of illumination.
Examination of the crystallographic structure of cc-alumina shows that the (O.T.I >• cleavage surface can have two or three different atomic terminations depending on the surface normal direction. For the surface normal f 0,1.1 ], the termination can be a single oxycen layer on the top of an aluminium ion layer or a single layer of A3+5 on the top of a O2 layer. Besides ihese two terminations, the surface can terminate with a double layer of O"2 on ihe top of an A3+3 layer if the surface normal is [0,1,1]. This is most probably the C domain. STEM studies of a-alumina by Bergcret al (1987) showed electron beam v;t 3. VALENCE LOSS TRANSMISSION SPECTROSCOPY Valence loss spectra obtained in transmission from individual layers sn ar. A;As I GaAs superlattice are shown in fig. 4. In contrast to previous results of Walls f 1^8,1 v.iio r-pon'id no detectable difference in the spectra, we find a consistent downward sh;f: of ;h'; r;-j3k piasmon loss combined with increased intensity near 30eV in one of :be layers. Although reliable independent identification of the layers was unfortunately unavailable at the time, both of these effects are consistent with the damped Lorentz oscillator mode] preoictior; for ::/- dielectric response of the lower band gap material with extra d-JeveJs associated w::r. GJ atoms. Fig. 4. Transmission valence loss spectra from individual iayers of &r. AIGa /' As :jperla:::r_e It is interesting that the transmission spectra obtained hue from the individuaj izyzri differ i>o:r: from pure GaAs as weii 2S from each other, even horn layers as narrow as 2.%r.rr. As fojr.c by Walls (1988) for the Si - S1O2 ir.:erf2c».:here car, be features of va'e-c» io',=. -pe-'j-a v,r.:cr. car. arise from sample structure finer th2rs y;s probe s:ze. As pointed ou: by fi:tch:e a::o ll'/xit • 19%%;. sensitivity to sucn fine 'tructure co-ic bt e-nar-c^: •:>• rr:ov:r." •:.-• 'I'fzv.: "• jr. -,:': ax:s posinori to emphasise n:gn-c excitations. /-.". adsqjate tneor. o: i:e.e::r;: ^•.•:::u'::.-:/: .:: 3'jperia'tices has vet 10 rs. worked out to :r.tsrcre; tr,'*st rss-.tv Hortr.;* <^ c '--'•. ",j.-- d sheets of eisctror. cas separated •:-• c:elet:tr:: .w-i~-.. ".'..' '.'z-irr.' :r ~.i r::~'-". •• ".••:•••"-• lz with orJv er.o'jsh erersv to excite cor.ajcf.Dn band txc::at;o''. 4. VALENCE LOSS REFLECTION SPECTRA Fig. 5 shows some reflection valence loss spectra taken in the (10,10,0) specuiar reflection under resonance conditions in the geometry of fig. 1 from GaAs crystals (full lines in fig. 5) as well as from a superlattice of Alo.3Gao.7As/GaAs in equal lOnm layers (broken lines). One of the GaAs crystals was transferred rapidly and gave the light full line spectrum with a rather intense surface plasmon peak at 12eV. A few minutes were used in transferring the other GaAs crystal which yielded the heavier full curve with a surface or interface loss at 10eV and much higher intensity in the bulk plasmon peak as well as at higher losses. These differences all seem consistent with what can be expected from the presence of an oxide overlayer. A transfer time of a few minutes was also needed for the superlattice sample and it is noteworthy that both of the relevant (broken) curves in fig. 5 exhibit die high intensity in the bulk plasmon and higher loss regions apparently characteristic of the oxidised sample. The dotted curve with the most intense bulk loss and a surface loss at 1 leV comes from the interface between layers; the dashed curve with a surface loss at lOeV comes from a Alo.3Gao.7As layer. • i.w- EKERGr lOSS It.-) Fig. 5. Reflection spectra of valence losses from individual layers of an AlGaAs supcrlatiice. The lower, thin solid line is from a clean GaAs sample and the upper, thick line curve from an oxidised GaAs sample. The dotted curve is from an Alo.3Gao.7As layer and the fourth, dashed curve is from the interface between the layers. Reflection valence loss spectra were also collected from the different A,B and C domains of the alumina surface. Despite the large differences in total reflected intensity noted ab^ve, it was found that the differences in the loss spectra in this case were very small though reproduceable. 5. SECONDARY ELECTRON fSE) IMAGES. Secondary electron (SE) images can be collected in the STEM and used to show surface morphology because of the influence of generation efficiency and escape depth. These images can have a resolution of lnm which is quite surprising if one considers the possible role of deiocalised processes like plasmon generation, propagation and decay in the chain of events leading to the generation of a secondary electron by the fast incident electron. Fig. 6 is a comparison of the scanning REM (SREM) image and the SE image taken under the same illumination conditions and almost simultaneously from the same area of an a-alumina cleaved surface after being slightly damaged. The bright A and the dark B domains are visible in the SREM image. A rather similar bright and dark contrast is observed in the SE image, with the bright A domain having a larger secondary electron emission rate but the dark B domain less. The origin of this effect is not entirely clear but three faaors which may affect SE emission are the step between domains, the change of surface work function and the change of surface conductivity depending on the nature of the terminating layer and the loss of oxygen ions. 400A :••'•-* Fig. 6. SREM image (a) and SE image (b) of a slightly damaged a- alumina (0,1,1) surface* 6. CONCLUSIONS Although considerable progress is being made in developing specimen, characterisation techniques with high spatial resolution, the precise degree of localisation available with different excitations is still far from clear. The present work illustrates the particular difficulties of REM or SE observations where the results can be profoundly affected cy contamination, oxidation or radiation damage effects. In particular, the compiex radiation sensitivity of cleaved alumina surfaces has been demonstrated. We thank the SERC for financial support. :he Cavendish Laboratory for a Clerk Maxv.»i: Studentship Berger S , Salisbury I, Milne R H, Imeson D and Humphreys C J 1987 PhiL Mag. B55 341 Bithell E G and Stobbs W M 1989 Phil Mag A60 39 Bleloch A L, Howie A, Milne R H and Walls M G 1989 Ultnunicroscopy 29 175 Boothroyd C B, Britton E G, Ross F M, Baxter C S, Alexander K B and Stobbs W M 1987 Microscopy of Semiconducting Materials 1987 (Inst of Physics London) pl95 Buffat P A, Stadelmann P, Ganiere J D, Martin D and Reinhan F K 1987 Microscopy of Semiconducting Materials 1987 (Inst of Physics London) p207. Bullock J F, Titchmarsh J M and Humphreys C J 1986 Scmicond Sci Technol 1 343 Eaglesham D J and Humphreys C J 1986 Proc Xlth ICEM (Ed T Imura, S Marcuse and J Suzucki) (Jap. Soc. EM Tokyo) p209 Horing N J M, Tso H C and Gumbs G 1987 Phys Rev B36 1588 McGibbon A J, Chapman J N and Cullis A G 1988 Proc EUREM 88 Conf (Inst of Physics London) p403 Pennycook S J 1989 Ultramicroscopy 30 58 Petford-Long A K and Long N J 1988 Analytical Electron Microscopy (ed G W Lorimer) {Inst of Metals London) p201. Ritchie R H and Howie A 1988 Phil Mag A 58 753 Ogura K 1988 Proc F/th Asia-Pacific EM Conf (EM Soc of Thailand Bankok) p 165 Vincent R, Wang J, Chems D, Bailey S J, Preston A R and Steeds J W 1988 Proc EMAG87 (Inst. of Physics London) p233 Walls M G 1988 PhD Thesis (University of Cambridge) Yagi K 1987 J Appl Cryst 20 147 Yao N, Wang Z L and Cowley J M 1989 Surface Science 208 533 ELECTROMAGNETIC RESONANCES IN SELF SIMILAR ARRAYS OF PARTICLES OR VOIDS Francisco Claro Facultad de F/sica Pontificia Universidad Catolica de Chile Casilla 6177, Santiago, Chile Ronald Fuchs Ames Laboratory, U.S. Department of Energy and Department of Physics, Iowa State University, Ames. Iowa 50011 Abstract. The dielectric response of a self similar cluster of metallic particles or random array of voids is considered. The response function obeys a recursive relation that yields a self similar distribution of modes. Their strength is also found to exhibit self similar properties. 10 I. INTRODUCTION. It is well known thai a variety of surface and bulk inhomogeneities can be well de- scribed by self similar models [1]. These models are constructed so that through several stages of amplification the details in the structure have the same geometrical features (2}. In this work we consider two such cases, a self similar cluster of metallic spheres and a random array of voids in a metallic background. Examples of such structures occur is DLA samples {3] and porous rocks (4). We are interested in the response of the system to the passage of a charged particle [5] or as electromagnetic wave, as described by an effective polarizability or average dielectric constant. n. SELF SIMILAR CLUSTER. Consider a nearly spherical cluster (the cell) of N metallic spheres of radii a, distributed following some geometrical rule. The radius of the cluster is ai = We have assumed the particles are not too close (center to center separation greater than 11 three particle radii {8]) so that the dipole approximation may be applied, although a multipolar treatment is equally possible. Ct and n, are the strength and depolarization factor of mode 8. These numbers depend only on geometry and are defined by the original geometrical ruie for building the original cluster, the cell For instance, the cell might be a regular octahedron with a sphere at each vertex. In such case there are only two active modes so that s = 1,2, only. The strengths obey the sum rule JT, C, = 1. The relation between the poiarizability of the effective sphere at stage t-f 1 and its dielectric susceptibility is Equating (1) and (2) we obtain Vi+i = 3 l where F ~ N/cr is the stage filling fraction, b, = ns - 1/3 and y - -\{4*x)~ + Equation (3) represents an iterative map that is to be iterated up to i = I -1. The normal modes are the poles in the effective poiarizability, or the zeroes of yj. ffl. SELF SIMILAR ARRAY OF VOIDS. Consider a continuous metallic medium where tiny holes distributed at random and covering a volume fraction F are punched Larger holes covering the same filling fraction F are punched next also at Because of disorder there is no exact theory to obtain the dielectric response of the system. In a dilute sample the effective medium theory may be used. Defining aa effective dielectric constant for the medium at each stage of iteration this theory- gives [9]. where as before, F is the filling fraction at the stage (voids punched at stage i in a 12 background assumed uniform). Some algebra yields from (4) where z = -{e — I)"1 = -(4arx)~1> Ci — 2/(3 - ^) is tne strength of the percolating mode, C2 = (1 - f )/(3 ~ F), and ft = 1 - F/3. This expression is identical in form with Eq. (3), with « = 1,2, and must be iterated up to t = / - 1. The zeroes of zj are the resonances of the system. IV. SPECTRAL DENSITY. Equations (3} and (5) have identical structure. Their physically significant solutions are the zeroes contained between the fixed points. They constitute a self similar set |10] and at stage I, number 2(2' -1). The strengths of these modes may be displayed through the spectral density function G[u) defined by jll], where u is either y or z, and analog of the strengths Cs in Eqs. (3) and (5), and for discrete modes may be understood as a distribution. It depends only on geometry and its integral equals unity. One obtains form (6) by inversion 1 Gt(u) = Ijmlmfriutiu - iS)]- . (7) For the cases discussed in Sees. II and HI this function has self similar properties {10]. Figure (1) shows the spectral density for a porous metal (Eq. (5)). It was obtained after 14 iterations with F = 0.05 { V. CONCLUSIONS. We have shown that a self similar cluster of spheres, or array of voids gives, rise to an also self similar dielectric response. This property is exhibited by the spectral density function in terms of which an effective polarizability or dielectric constant may be defined. Details of this work and further discussion of the properties of the spectral density will be published elsewhere [10]. The Ames Laboratory is operated for the U.S. Department of Energy by Iowa State University under Contract No. W-7405-Eng-82. This investigation was supportedby the Director for Energy Research, Office of Basic Energy Sciences, U.S. Department of Energy, and by the Fondo Nacional de Investigaciones Cient/ficas y Tecnologicas (Grant No. 0679). REFERENCES 1 For a review see S. H. Liu, Solid State Physics 30, 207, F Seitz, D. Turnbull editors (Academic Press, 1986); R. Orbach, Annu. Rev. Mater. Sci. 19, 497 (1989). 2 B. B. Mandebrot, "The fractal geometry of nature" (Freeman, San Francisco, 1983). 3 T. A. Witten, Jr. and L. M. Sander, Phys. Rev. Lett. 17, 1400 (1981). 4 A. J. Katz and A. E. Thompson, Phys. Rev. Lett. 54, 1325 (1985). 5 T. L. Ferrel and P. M. Echenique, Phys. Rev. Lett. 55,1526 (1985). 6 R. Rojas and F. Claro, Phys. Rev. B34, 3730 (1986). 7 R. Fuchs and F. Claro, Phys. Rev. B39, 3875 (1989). 8 F. Claro, Phys. Rev. B_25. 7875 (1982). 9 For a review see R. Landauer, AIP Conference Proceedings No. 40, C. W. Garland and D. Tanner editors (AIP, New York, 1978) p. 2. 10 R. Fuchs and F. Claro, to be published. 11 D. J. Bergman, Phys. Rev. 14, 4304 (1976). 14 o © OT CD Kti Ml US »i' C o "o 3 0.00 0.25 0.50 0.75 1.00 factor de depolarizacion 15 Influence of the Wake Potential on Inelastic Scattering Using A Small Probe in Confined Volumes P.F. Batson and J. Brulcy IBM Thomas J. Watson Research Center Vorktown Heights, New York 10598 When a fast electron penetrates a solid at a velocity, v, greater than the Fermi velocity, v,., a wake potential is confined within l-2nm in the charge density "wake" is created. The wake results lateral direction. Thus, if inelastic scattering from the hydrodynamic response of the system to occurs with an impact parameter smaller than the coulomb potential of the fast electron. In l-2nm, we might expect the wake potential to Figure 1, we reproduce results from Ritchie and influence the amount and type of the scattering. Fchcniquc1 for the potential and differential charge density which corresponds to the wake. These results are for a heavy, positive ion, but should be applicable to the light, fast electron with somewhat different length and density scales. The potential is highly anisotropic, having a wavelength of 2nvjwp behind the fast electron and parallel to its trajectory, and a wavelength of h/mv ahead and to the side of the trajectory. In effect, the wake is a real space representation of the plasmon and quasi-particlc continuum. For -as instance, at a certain distance behind the fast electron, the potential is mainly plasmon-like. Moving to the side, this plasmon potential becomes shorter in wavelength. When we reach the wake position at the wake angle 0 = sin-'vf/v, the plasmon potential has a wavelength similar to single quasi-particies, and the plasmon merges with the single particle continuum to the side of and in front of the fast electron. oca Scaling from results for fast ions, we expect that the depth of the wake potential will be of order e«>p/v ~ 0.25 eV for Silicon at lOOKcV incident electron energy, increasing to 0.5 eV for diamond. The integrated response charge which •• j. is responsible for this potential equals the fast electron charge within 10-20nm behind the incident electron. Therefore, 5-10nm behind the Figure 1. Wake potential and differentia! charge fast electron, the total potential will be highly density reproduced from Ritchie and Rchcniquc, distorted by the wake. On the other hand, the (1982). 16 Figure 2 shows typical energy loss results for silicon over the energy range 0-140 eV. The no loss electrons arc confined to about 0.3eV about zero energy-. The bulk plasmon occurs at 16.7 eV, and in this case we have a second peak due to multiple plasmon scattering. The Si 1^ 3 edge occurs at 99.84+0.02cV, and is sitting on a decaying backg-ound comprised of multiple single particle excitations. The present work is loosely aimed at using the Si l^-? edge as a probe of changes in the conduction band density of states as we shift the electron beam about the specimen. Figure 3 shows the l^j edge alone after stripping the background intensity. There I have stripped also the 1^ part away to reveal the shape which results from variations in the final states. In principle we want to interpret these in terms of final density of states in the crystal. In practice, there is a difficulty due to the presence of the charged hole left behind by the excitation. Therefore, the final states are relaxed downwards, 50 0 50 100 150 and a discrete hydrogenic series of lines may be formed below the ground state conduction band Energy Loss (eV) edge. A recent analysis of this core excitonic effect has been given by ("arson and Schnathirly2 Figure 2. Typical energy loss results for Si. 1 1 Si L2<3 Figure 3. Background stripped vi Si I.J7 absorption edge. The spin 3/2 component has been NX" Part extracted by a Fourier deconvolution of two impulse i functions having a reparation 0.61 cV and a relative weighting VB of 2:1. > . • ••• CB L, L3 i • 1 , , i 1 , 98 99 100 101 102 103 104 Energy Loss (eV) 17 for several semiconductors. In Si the excitonic Winding energy is about 5()nicV, while in diamond it is about 0 2eY. I'hus, we expect that in silicon the excitonic distortion should be small on a scale of 0.2- Silicon 100 105 110 11 Energy Loss (eV) Figure 5. Variation of the L3 part of the absorption with the local thickness of a thin wedge. The fine structure at the edge becomes less prominent in the thinner areas, but the edge docs not shift. The presence of SiOj cannot account for the differences. In Figure 5 we present results for the Si edge 98 100 102 104 106 108 as a function of specimen thickness. A curious Energy Loss (eV) result emerges. Ilie structure at the near edge, identified above as being due to the conduction Figure 4. Comparison of measured Si absorption band DOS, becomes less distinct in specimen with the s- and d-projected density of final states areas which are thinner than about 15-20 nm. from Reference 3. The differentiated spectrum The shape differences do not appear to be due to shows the criterion used to determine the the Si()2 nor, from other comparisons, from absolute position of the onset with an accuracy amorphous Si. High resolution images of the of + 20meV. specimen in this region showed good quality lattice images, as well. Certainly, a 15-20 nm Figure 4 shows calculated1 s- and d-projectcd thickness comprises many unit cells, so this is not DOS for Si compared to the measured edge. a real electronic structure confinement effect. It Quite a thorough comparison is possible. The is known that a conduction electron having an bottom part of this figure shows the criterion used energy 2-3eV above the conduction band edge for determining the edge onset position. This will have an inelastic mean free path of several differential method is not sensitive to shape 10's of nanometers, so it may be reasonable lhat variations in the no loss beam, and so may be damping by collision with the surfaces would used to obtain the position of the edge with an occur at these thicknesses. But evidence below accuracy of about ;f20meV. I he edge intensity suggests that something more complicated may can be obtained witli better than 2% be occuring. reproducibility when the probe i* not moved. 18 In a thick region of the sample, near a broken edge, a similar behavior is observed. In Figure 6, as the probe is moved closer lo the edge, the fine structure near the absorption onset is lost. fcj l< ••> 6 Si L2>3 y/AV i b=10nm ^*** - s*"VVv*' 5nm ,-y* 0 50 100 15 2nm Energy Loss (eV) /*^ Figure 7. Ixw loss plus core absorption for several thickness's. Multiple plasmon scattering i . . . i . . . is evident. 95 100 105 110 115 Energy Loss (eV) Figure 7 shows multiple scattering results for several thickness's in the energy range 0-110 cV. Figure 6. Variation of the I ^ scattering with In Figure 8, we have performed the single 3 5 impact parameter relative to a cleaved edge in a scattering analysis from Miscll and Jones to thick region. Fine structure disappears near the obtain the single scattering. This is commonly edge. referred to as a log deconvolution" today and is given in Fourier space (denoted by ~) But in this geometry, the probe must be moved very close to the edge. (I have pushed this limit down to <0.5nm recently. 1/90) How can ann\. = In rz— we account for the difference in behavior between the "thickness" in the directions parallel and h. perpendicular to the incident beam direction? Perhaps, the anisotropic wake potential where the a is the specimen thickness, n is the influences the final state of the core absorption, number density of scattercrs, where P^ is the scattering per unit length for an 0.06 infinite foil, Pb, is the bulk plasmon correction, P(,2 is the surface intensity, and -Pbi-jfc-ttfn.f-i.S7J. 0.04 - UJ PM = 2-jfc-t[Ai-j--l.57] = -2Pw. Therefore we may write 0.02 ; P|*lasmon ~ 2 0.00 40 60 80 100 120 so that Energy Loss (eV) 1 r» Figure 8. Single scattering analysis of the data in Figure 7. Thus, the true thickness, a, can be obtained from thc arithmetic mean of thc thickness's derived When the specimen area is fairly thick, these from the measurements of thc plasmon and total two methods yield the same answer. 'ITiercforc, scattering intensities. Figure 9 shows plots of r tnc we may integrate anos between 12-21cV and then aP^, PTot and Ppiasmnn f° range of multiply by 12()nm to obtain a plasmon thickness's in this experiment. Measured determined thickness of 35nm for the thickest intensities for thc total scattering (closed circles) area. Conversely, we obtain 36nm using the total and for thc plasmon scattering (open circles) arc integrated scattering. When we go to thinner included. As expected, thc plasmon intensity areas, thickness's derived from the two methods deviates below thc linearity, and thc total diverge. Thus, the bottom curve of Figure 8 intensity deviates above in a manner consistent yields 2.2nm using the plasmon intensity and with Ritchie's analysis. 9.1nm using the total intensity. 1.0 1 In Ritchie's 1957 paper" on inelastic scattering in thin films, he pointed out that the surface !B O.B- y. o - Plot plasmons on the top and bottom surfaces reduce 2 \ the intensity of the bulk plasmon below the value \ en 0.6 -, expected. This reduction has a logarithmic "Core Loss—y divergence in the film thickness given in units of the plasmon wavelength. On the other hand, the o 0.4 - V) surface scattering itself increases by twice as much 'Ptasmon as the reduction of the bulk scattering. This was I t described as a "depolarization" effect of the 5 0.2 or surfaces. A result is that the total scattering per unit thickness goes up logarithmically as the film 0.0 gets thinner, while the plasmon scattering per unit 100 200 300 400 thickness is reduced. Ritchie wrote for the total Specimen Thickness (Angstrom) scattering probability : Figure 9. Results for thc scattering probability I lot bl as a function of thickness following thc analysis in the text. 20 from the fast electron trajectory fan now be understood as well, bemuse tin1 wavelength of ihc wake potential is much '•boiler in this direction. Thus, normal dipole selection rules are totally inadequate to describe the scattering. What are some of Ilie implications of ihis picture? First, since the plasmon wake has a short range in the direction perpendicular to the fas! electron trajectory, its influence should be small lor large impart parameter scattering. Thus, small angle scattering should not be complicated by this effect. I nfnrtunatclv, the small scattering angle experiment requires large 98 100 102 104 106 108 110 areas of uniform thickness. These are Energy Loss (eV) experimentally difficult to obtain, and so a detailed check as a function of thickness would Figure 10. Si 1/j results normalized for scattering be difficult. It is also difficult to obtain a high probability at the various thicknesses derived enough angular resolution in the STFM using Figure 8. The data for 21.6 and 21.9 nm geometry. are superposed to show rcproducibilily of the results. Another possibility involves ihe core exciton. As discussed above, this is a localized state near the core hole. If the final state electron binding Jn Figure 10, we show the Si I,3 edges for the various spectra shown in Figures 7 and 8 above. energy is comparable to, or smaller than, the These have been normalized in intensity to the plasmon potential, we might expect to see some core intensity visible in Figure 8. For clarity in modification of the c.xcitonic contribution in presentation, the each result has been displaced favor of propagating collective final states. upwards. Notice the decay of the electronic structure information at thickness's less than Figure i 1 shows results for the K core about I5nm. The reproducibility shown in thai excitation for diamond. Two results arc shown, case is typical of what can now be obtained. In one from the partial photoyield experiment,'' and Figure 9, we have plotted the integrated core one from the spatially resolved energy loss scattering between 104 to 108 cV, normalized to experiment. 'The photoueld experiment, in the the total valence scattering at the 35nm thickness simplest interpretation gives the optical (closed diamonds). We find that the core loss absorption. It shows a classic excilonic line scattering closely follows the total valence which can be characterized with a binding energy scattering. Thus, it seems possible that the final of about 0.2 cV below a conduction band edi>e state for the core loss participates in the collective at 2R9.2 cV and with a lifetime broadening of behavior of the solid in the same way thai the 0.17cV. The line fit to this data in Figure 11 was valence electrons do. Therefore, the excitation generated by an Flliott"1 model for the exciton probability for the core loss is influenced by the following Altarelli and Dexter.11 The spatially top and bottom surfaces in the same way that the resolved energy loss case docs not show the sharp valence electron scattering is influenced. This is line. An Flliott model analysis yields a good fi1 precisely what we might expect if the applied to the data, but the conduction band edge and coulomb potential is modified by the presence of excitonic binding energy do not show a simple the plasmon wake within the material. correspondence to the phoioyield results. In particular, the edge is at a similar energy, but the modeled exciton binding energy is smaller, and The behavior of the core scattering when the the band edge is lower. Perhaps more significant, surface is located at some lateral distance avvav the lifetime broadening is greater in the electron 21 There is a possible problem with this analysis that should be mentioned. The Flliott theory was Photoyield written for electron hole interaction i/i the valence bands of semi-conductors. It was therefore set E= 0.189 up for a p-Hkc initial stale and an s-likc final state. r= 0.17 The diamond initial slate is the Is level. )t is not CBM= 289.2 clear if this invalidates the photoyicld analysis. At first glance, the relevant excitonic series should be the 2p, 3p, ... levels, with the n=l level missing. This may not be a problem fora weakly bound exciton in the Wannicr picture. 284 286 288 290 292 294 Energy Loss (eV) Figure 11. Comparison of partial photoyield (Reference 9) with the spatially resolved energy loss for the diamond Is absorption. An Flliott model fit has been applied to each results. The fits indicate binding energy, damping, conduction band edge position and spectrometer resolution respectively. 100 101 10 case. This behavior might be understandable if Energy (eV) the plasmon wake can "compete" with the core hole potential in the selection of final states for Figure 12. Comparison of photoyicld (Reference the excited core electron. In effect, the core 12) and energy loss results for Si. Below, each electrons may be excited into an excitonic level, result is differentiated to obtain the edge position. but then immediately scattered away into The valence band maximum (VFJM) is obtained propagating states - almost as if the plasmon from XPS data. The SRFHLS result obtains the potential has "ripped" apart the hydrogenic final conduction band edge (CI)M). state of the exciton. Alternatively, the periodic potential of the wake might force excitations to As a result of this problem it is best to return final states at higher angular or linear momentum to the Si [-2 3 edge where the initial and final state that the excitonic ground state. These long symmetries arc similar to the case that FHioll wavelength states might still be perturbed by the modeled. Figure 12 shows a comparison of the core hole, bul would not be bound. It is photoyicld results with the energy loss results. significant here that the exciton binding energy Fach of the results arc differentiated to show the here is 0.2 eV, while the plasmon potential depth precise edge locations. The photo yield results is 0.25-0.5 cV. Thus we don't expect a clear-cut were obtained by Mimpscl12 with an energy divergence between the photon and electron case. resolution of about O.I5cV. The absolute energy The intermediate result that we show in Figure was derived from Fbcrlmdt1* with an accuracy } I does seem reasonable. of ±50mcV. The energy loss results were obtained for the spatially revived case with an 22 energy resolution of ().28eV and an accuracy of be able to excite the core electron without ±20meV. interference from the piasmon wake. lx)oking at the photoyicld results first, we find We have done this and find that the piasmon an onset energy of 99.76cV. The differentiated intensity is dramatically smaller. (A factor of two results show two peaks, one corresponding to the in our particular case.) 1'igure 13 shows a typical edge onset and one 0.62 eV higher. The valence result. band maximum may be obtained from XPS [ i i i results and is placed at 98.72 cV. Adding the Plasmon Intensity Intensity Mormoiized To Incident Dose room temperature band gap of 1.12 cV to the \ valence band edge position yields a conduction band edge at 99.84 eV. Using the fvlliott model with this conduction band minimum, we can analyze the edge shape to derive an excitonic binding energy of 50meV. This is in good agreement with the results by soft x-ray emission.2 The energy loss results display similar differentiated structure, but the edge position is clearly shifted to higher energies. The reproducibility here is perhaps the best we can do, 0 20 40 60 and care is required. The average of many results Energy Loss (eV) places the edge onset at 99.85+0.fl2cV. Hgurc 13. Piasmon scattering intensity for the Now we should be convinced that the energy two-beam geometry described above. The loss results arc different, but we arc not sure of conditions arc indicated graphically with the the precise reason. We need to find a link vertical line indicating the (220) Kikuchi band, between the piasmon scattering and the shape of the x indicating the entrance beam angle and the the core edge. We have arranged an unusual o indicating the exit collection angle. Notice that scattering geometry, which, to our knowledge, interchanging the exciting and collection angles has not been tried before in probing plasmons. docs not alter the result. The peak at 17cV in the We tilt the crystal in selected area diffraction so bottom curve is due to quasi-clastic plus piasmon that the exciting electron beam is aligned just multiple scattering and should not be affected by outside of the (220) two-beam diffracting the channeling conditions. position. We place the collection aperture so that it defines a direction just inside of the two-beam Figure 14 shows the core loss intensity for the condition, the crystal wavefunctions thus have, parity flipped excitation compared to the respectively, odd and even symmetry with respect symmetric results — again indicating excitation to the atom columns in the crystal. (This is a and collection conditions graphically. The results well characterized result of dynamical diffraction arc normalized between 104-106cV on the theory.) Since the bulk piasmon is a scalar response and since the coulomb interaction is assumption that cxcilonic differences will be symmetric about the origin, we expect the matrix negligible in that range. Notice the dramatic element for piasmon excitation to be zero for enhancement of the near edge structure. There initial and final fast electron slates which have is also a minisculc shift downwards of the edge opposite parity. However, a core excitation position. A simple subtraction of the symmetric wherein the angular momentum changes by results from the parity flipped results yields a A/; = + l, will be allowed. Therefore, we should remarkable agreement with the photo-yield data. Both the position and the shape of the results agree nicely. 1 i 1 Si L2,3 Photoyield ,. . cri Y . co i J. EELS Enhancement /I During Channeling • 98 99 100 101 102 102 104 106 Energy Loss (eV) Energy Loss (eV) Figure 15. The difference between the results in Figure 14. The I^3 core loss scattering in the Figure 14 is displayed here compared with the off-two-bcam geometry described above partial photoyield. The similarity is striking and compared with the symmetric scattering results. the edge onset position is reproduced. The "flat" Notice that the edge is enhanced in the parity top of the photon case is reproduced also. flipped channeling case -- when the plasmon is depressed. There is a small shift in the edge also. In conclusion, the inelastic electron scattering appears to be qualitatively different from the In Figure 15, we show the difference plotted photon absorption, because the collective on and expanded scale compared with the response of the solid can take part in the inelastic Photo-yield results. The rcsemblcnce is striking. scattering in a way which favors transitions to Thus, it appears that the difference shown in final states which ordinarily arc not accessible in Figure 14 above, may be an isolation of only the the optical experiment. The results, at least for core excitonic part of the Si 1^ 3 edge. It seems well screened materials where the core exciton is astonishing that a 50meV exciton is able to distort weakly bound, closely resemble the ground state the spectrum by this amount. Future work will projected density of slates. 'i"hus, the energy loss aim to try to understand the precise origin of the may well be uscablc for determining conduction difference. bandstructure in very small volumes. References: 1. R.H. Rilchic, and P.M. Kchcriiquc, Phil. Mag. A 45 347 (19R2). 2. R.D. Carson and S.H. Schnatlcrly, Phys. Rev. led. 59 319 (1972). 3. X. Wcng, and P.Rez. Unpublished. 4. P.r. Balson. Ullramicroscnpy. 9. 277 (1982), and Phys. Rev. 1 .ell., 49, 936 (1982). 5. D. Misell and A.F. Jones, J.Phys. A 2 540 (1969). 6. P. Xu and J. Silcox, private communication. 7. M. Sarikaya and P.Rcz, Proc. 40lh HMSA (cd. Bailey, San Fransisco Press, 1983) p.486. 8. R.H. Ritchie, Phys. Rev. |06 874 (1957). 9. J.F. Morar. F.J. Ilimpscl. CJ. llollingcr. G. Hughes, and J.I.. Jordan, Iliys. Rev. Lett., 54,19M5 (1985). 10. RJ. F.lliolt, Phys. Rev. 108 1384 (1957). 11. M. Altnrelli and 1X1 . OcxUr. Phys. Rev. l.clt. 29 1100 (1972). 12. F.J. Ilimpscl, unpublished. 13. W. Fbcrhardl. (i. Kalkoircn, C. Kun/. I). Aspncs. atid M. Cardona, PJiys. Slat. Sol. (h) fiR 135 (I97R). 25 Presented to 12° Werner Brandt Workshop "PENETRATION OF CHARGED PARTICLES IN MATTER". 3-7 September 1989, San Sebastian, Spain. Electron microscopy study of the electronic properties of small silicon particles Daniel UGARTE, Christian COLLIEX, and Pierre TREBBIA. Lab. de Phys. Sol., Univ. Paris XI, Bat. 510,91405 ORSAY-FRANCE INTRODUCTION The problem of the energy lost by a fast electron interacting with a small particle has been studied for many years. Small objects have shown a behaviour substantially different from bulk materials. Moreover the electronic properties of small particles arc very important from a technological point of view because of their possible application in catalysis. Recently the advent of the Scanning Transmission Electron Microscope (STEM), which provides a focussed 100 kV electron beam within a diameter of less than 0.8 nm, has renewed the interest in the energy losses suffered by a fast electron travelling along a well defined trajectory in the vicinity of a small particle. Most of the theoretical work related to STEM measurements has been devoted to the excitation of plasmons in small spherical particles[l-9] or related gcometriesf 10,11]. Although many of the theoretical formalisms developed for small spheres are generally valid for any arbitrary dielectric function, the STEM experimental measurements have dealt mostly with metals, and their dielectric constants were usually modelized by the Drude expression for a free electron gas [11-13]. The aim of this work is to test the validity of the classical dielectric theory [14] for a more complex case. Here, we will be concerned with the electronic properties of spherical silicon particles. This material and its oxides arc very well characterized, because of their applications in microelectronics. The dielectric properties of Si and its oxides (SiO2, SiO) present complex structures [15], and provide a stringent test of the theory developed up to now. We study in a STEM the low energy loss spectrum of silicon particles, and also the spatial localization of the most important spectral features observed, using digitally acquired energy filtered images. The results are compared to the predictions of the classical dielectric theory. An anomalous peak at 3-4 eV is observed at the outermost surface of ihc SiOjj layer surrounding the particles. We conclude that this phenomenon is due to a beam induced process, which renders the SiO2 surface conducting. We modelize this effect by including a very thin coating of amorphous silicon, supposed to be produced by electron induced oxygen desorption. The theoretical spectra and images then fit the experimental results corrccdy. 26 MATERIALS AND METHODS The experimental work was performed on the VG HB501 dedicated STEM operating in Orsay. The microscope is fitted with a serial second order corrected spectrometer (Gatan 607). Both the EELS spectra and the energy filtered images were acquired with a 0.7 run probe of 7.5 mrad convergent semiangle, and a collection aperture providing a 6.5 mrad collection semiangle. The small semiconducting panicles were prepared by £ gas evaporation method [16]. Figure 1 shows an electron micrograph of a group of Si particles. They are nearly spherical in shape, their size range is 10-300 nm in diameter, and they are covered by a thin (2-6 nm) natural amorphous oxide layer, produced by exposure to the atmosphere during preparation for microscope observation. The particles are deposited on a holey carbon film, and the experimental work was performed only on particles over a hole, in order to eliminate the carbon substrate contribution to the spectra. EXPERIMENTAL RESULTS A typical low loss spectrum is shown in Figure 2. In order to study the spatial distribution of the main spectral features observed, we acquired the energy filtered images shown in Figure 3. For an easy comparison of the localization of the excitations, we have extracted line profile intensities from the images (Figure 4). The analysis of these data, together with our previous knowledge of Si-SiO2 plane interfaces [17]), yields the following immediate conclusions: - The 17 eV and the 23 eV peaks are due to the bulk plasmon of Si and SiO2 respectively. - The 9 eV contribution is due to the normal surface plasmon of the Si particle. As predicted in emulations applied to Al oxidized parbcles[5], the surface plasmon energy of a coated sphere should be found at an energy lower than that of an isolated free sphere in vacuum (for a Si sphere the dipole mode (£=l) frequency is - The 10.5 and the 12.0 eV minor features are due to SiO2 interband transitions. - Finally the low energy peak at 3-4 eV is an unexpected result, since a careful look at Figure 4, shows that it is localized at the external surface of the Si-Oxide layer surrounding the particle. 27 DISCUSSION A similar experiment has been reported by Batson [12]. He measured a low energy loss peak at 3-4 eV, on oxidized aluminium particles, and showed that it was the result of the coupling of the surface plasmons of neighbouring particles [10]. In our case the spatial localization of this low energy excitation does not show the dipole distribution that he observed, and it can be seen on all the particles. The localization of this energy loss at the outermost surface of the oxide layer is an anomalous phenomenon, because it is well known that insulators cannot have any absorption at energy values in the gap (8.9 eV for SiO2), and certainly cannot sustain surface plasmons in this energy range. This suggests the existence of some land of induced surface phenomenon (charge, contamination, radiation damage, etc.). Using the classical dielectric theory [14] in the non retarded case, we have extended the coated particle model [6] to include the effect of a second coating layer L2 (Figure 5). The expressions derived are rather long and complicated and they will be presented in a forthcoming publication. If we modelize the second coating as a contamination layer of amorphous carbon, no important effect is observed in the low loss range. This fact can be understood easily by considering the surface plasmon at a plane interface between two materials 0 and b. The condition for resonance is £ia(ca)+e1^((o)=O [18], i.e. if we consider the medium a as SiO2 (£i(0-7 eV) = 2.2) in order to have a plasmon at the interface in the 3-4 eV region, the medium b must be a conductor or at least a semiconductor whose dielectric permittivity presents negative values (£jfc(3-4 eV)< 0). Although the conditions of resonance in the spherical geometry are more complicated, this consideration still holds. The amorphous-C is a semiconducting material [19], and in the low energy region £j > 0, so an amorphous carbon layer on the coated silicon particle will not create a new oscillation in this energy range. The effect of oxygen desorption induced by electron irradiation of oxides is a well known phenomenon in electron microscopy work. The STEM analysis usually requires quite a high dose in order to get a good signal to noise ratio, in our work for example one can estimate doses as high as 108 e'/nm2 for the series of images in Figure 3. If we make the assumption that the SiO2 surface undergoes a beam damage process, and lose some amount of oxygen, a thin conducting film could be produced on the surface. In order to modelize this phenomenon, we assume that we get an amorphous surface silicon layer as a result of oxygen desorption from the SiO2 surface. The calculation using the experimental parameters of Figure 3 for the panicle radius, and the Si core, assuming a 0.4 nm amorphous Si surface layer is shown in Figure 6 for the spectrum, and in Figure 7 for the probability of excitation as a function of the impact parameter, at the energy losses of the experimental energy filtered images (Figure 4). The thin semiconducting film has a dramatic effect, and the theoretical results fit correctly the experimental measurements. The intensity of the 3 eV experimental profile (Figure 4) is higher than predicted in Figure 7, this difference may be accounted for, by considering the 28 important contribution of the zero-loss peak tail in this energy *ange. In all the calculations of this work many multipoles have been included (&15 for the curves of Figure 6 and &10 for that of Figure 7), and experimental values have been used [IS] to describe the dielectric permittivity. A conducting surface layer covering an insulating sphere sustains two surface plasmons [20], one at the external surface, and a second at the internal interface, like that of a void If the thickness of this film becomes small enough, the two surface excitations are coupled, as for the case of a thin film [21]. Thus two new frequencies are created co " and a>+; the co" branch goes to a>=0 when the film thickness goes to 0. The low energy peak observed here can therefore be interpreted as the co " branch of the coupled surface excitation on the amorphous-Si surface layer covering the particle. This would explain its localization on die outermost surface of the sphere. CONCLUSIONS The classical dielectric theory has been tested in the case of small semiconducting particles. The predictions of the loss probability show a close agreement with the experimental results, even for minor details, provided that we use the experimental values to describe the dielectric permittivity of the analysed materials: the Drude model must be replaced by a more complex representation taking into account real damping. The STEM is specially well suited to the study of small volumes. The energy filtered imaging method applied to surface plasmons is a very sensitive tool for interface visualization. The ability to make the correlation between spectrum and energy filtered image features is of great help for a correct understanding of the physical processes involved. The high dose needed in STEM studies may induce important damage effects, and much care must be taken in order to avoid incorrect conclusions. ACKNOWLEDGEMENTS Thanks are due to S. Iijima for providing the specimen for this work. The invaluable help of M. Tencc" for the manipulation of the digital control of the STEM is gratefully recognized. The authors wish to thank M. Walls for his critical reading of the manuscript. REFERENCES 1.- SCHMEITS M., J. Phys. C: Sol St. Phys.,14, 1203, (1981). 2.- KOHL H., UInran.,11, 53, (1983). 3.- FERREL T.L. Figure 1. Conventional electron microscopy image of a group of silicon particles. Figure 2. Typical low energy' loss spectrum of silicon particles. 31 3 eW 9 eV Figure 3. Family of energy filtered images of a small silicon particle lying on a bigger one. The image marked IS eV actually corresponds to 17 eV (miscallibratJon of the spectrometer). xO.33 (arb) > — 3eV — 9eV — 17 eV "" 23eV • Figure 4. Line profile intensities extracted from the energy filtered images of Figure 3. Figure 5. Electron moving with a velocity v at a impact parameter d of a double coated sphere of exterior radius rj, with a first coating layer LI of thickness (r2-r: -t) and a second coating layer L2 of thickness t. Figure 6. Calculated spectrum (r2=17.1 nm, r3 =31.63 nm, d=37.3 nm, electron energy 100 kV). a) Si core coated with a SiO2 layer. b) Idem, including also a 0.4 nm thick amorphous silicon layer (L2) on the surface. I .. *0.33 (arb) — 3eV — 9eV — 17 eV —' 23eV 30 [nm] Figure 7. Calculated energy loss probability as a function of impact parameter, for the case of a Si sphere covered with a layer (LI) of SiCfe, and an amorphous-Si layer (L2) on the surface. (r2=17.1 nm, rj=l 1.36 nm, t= 0.4 nm, electron energy 100 kV). 35 Electron Compton scattering in the electron microscope P. Schattschneider and P. Pongratz Inst. f. Angewandtt und Teehnitehe Physik, Tcchn. Univ. A-1040 Vienna, Auttria Abstract The distribution of electron momenta (electron density in momentum representa- tion) of gases can be probed by Compton scattering of either photons fa-rays or X-rays) or electrons. Recently it has been shown that Compton scattering of electrons is suited to the study of the electron momentum densities of solids on a microscopic scale. This technique, known as ECOSS, Electron Compton Scattering from Solids can be done in the electron microscope by Electron Energy Loss Spectroscopy (EELS). The advantages and disadvantages of ECOSS as compared to photon Compton Scattering are discussed. It is shown that ECOSS is a method suitable to obtain infor- mation on the ground state wave function in condensed matter. After a discussion of inherent approximations, especially the impulse approxima- tion, the reciprocal form factor is introduced. A method is proposed in order to cope with the main difficulty, namely multiple scattering. important applications of ECOSS are the study of anisotropy of momentum den- sities; correlation effects of conduction electrons in metals; and charge transfer in alloys. Keywords: EELS, Electron Compton scattering, impulse approximation, electron momentum density, reciprocal form factor, multiple scattering, anisotropy, electron cor- relation. 36 Introduction Electron Compton scattering is inelastic scattering of fast electrons at large an- gles (~ lOOmrad) off core or valence electrons. Like in the photon case, the energy of the scattered electron is increasingly lowered with scattering angle; the energy distri- bution has a halfwidth proportional to the mean squared momentum of the scatterer (Doppler broadening). The shape of the distribution can be shown to be an image of the momentum distribution of the electron in the ground state. The electron microscope is well suited to the study of the electron momentum densities of solids on a microscopic scale by this technique, known as ECOSS, Electron Compton Scattering from Solids. (10-12], In terms of electron microscopy, ECOSS is EELS in diffraction mode in the energy range of the Bethe ridge. Since the cross section for scattering of electrons is some five orders of magnitude larger than that of photons at the respective scattering angles, and since the intensity of electron sources is ~ 102 larger than for typical photon sources, it is evident that the sensitivity of ECOSS enormously exceeds that of photon Compton scattering, or to put it another way, the duration of an experiment can be reduced from days (for photon scattering) to hours or minutes. Another, and maybe the advantage is that, contrary to the photon case, ex- tremely high spatial resolution can be achieved in conventional Transmission electron microscopes (TEMs), which is important for the investigation of anisotropy in fine cry- stalline material. For instance, the anisotropy of the electron distribution in graphite was measured on a single graphite grain [9]. It is obvious that the electron microscope offers a number of instrumental features such as simultaneous monitoring of diffraction patterns, dark field images and variable magnification which are not available in the photon case. Modern electron optics have momentum resolutions of up to ~ 0.04a.u. which is a factor of five better than that of photon sources for Compton experiments. ECOSS is a relatively new technique. Radiation damage of the specimen and mul- 37 tiple scattering restricted application mainly to gaseous specimens. The first ECOSS experiment was reported in 1981 [10]. The experimental and theoretical work perfor- med so far concentrates on basic questions, simulations and comparison with photon Compton scattering {10-11,13, 8]. Theory In first order perturbation theory the dynamical form factor S is w relates to the energy E lost by the scattered electron as E = hut, the Ha- miltonian B = Ho + V consists of an unperturbed part Ho and the potential of the binding forces V, p, is the density operator, and \ the ground state wave function of the scatterer. It can be shown [5] that for high energy transfer, in the power series ±iHt of the exponentials e , all terms cancel except for that containing Ho. Physically, this meaqns thatthe potential in which the electron moves can be considered constant for the short time of interaction in a high energy transfer, and eq. (1) becomes S = ~ / t~lut < By use of the time dependent Heisenberg operator this is The momentum density operator p, (t) is the Fourier transform of the ordinary density operator n(r) n(r) = v>(r)v>+{r) (5) where xj){r) are field operators. From the Wiener-Khintchin theorem, we know that the Fourier transform of the product in eq. (5) is the autocorrelation of the fieltl operators in momentum representation, so < *(«)P; (o) >= £ < < (*K-M(*K-+«V > (•) p.r1 where the sum is over p-states within the Fermi sphere, a*, o are creation and an- nihilation operators. For large momentum transfer q, which is assumed henceforth, the final state electron can be considered to be free. For free electrons, a*{t) = a* (6)exp(—ip2/2m)t, and we may write (7} The second aspect of large momentum transfer is that the operator «£«+, creates a par- ticle in the previously empty state p' + q far beyond the Fermi momentum. Subsequent annihilation of a particle in state p + q is only possible when there is a particle, i. e. only for p = p' otherwise the probability amplitude for this process vanishes. See fig, 1. Hence J° Eventually, (K (o) >= Replacing the sum in eq. (9) by an integral (j£p -»/ ^fr) «»d the time integral in eq. (2) by an appropriate ^-function yields where we have introduced the momentum distribution 39 The momentum distribution in the many-particle system is defined as 3 ...d pnx{Pi, -Pn )x' (Pl» -A) ( . x{p) is the one electron wave function of the ground state in momentum repre- sentation. It should be noticed here that the last equality does not strictly hold in an interacting n-particle system where a one-electron wave function does not exist in a strict sense. Rather, one should use a density matrix formulation. Though, we identify p(p) with the square of a one-electron wave function henceforth. In the last step of the derivation, we integrate out the ^-function and use the relationship [11] between 5 and the differential cross section d7a/ddEil, d'o _\2mt>]2 k fd?p m kb f xy Em Here, ka kb are the electron wave numbers of the incident and scattered electron respecti- vely, E is the energy loss, and q is the momentum transfer. The variable pt is the momentum component in the direction of q. The scattering angle 0 relates to the quantities on the right by the scattering geometry (fig. l.b). The quantity J{px) in eq. (13), called Compton profile np,) (14) can be derived directly from experiment. It is a projection of the 3-dimensional mo- mentum density of the scatterer onto the direction of the scattering vector g. It is, in principle, possible to obtain the complete 3-dimensional distribution p{p) from a series of Compton experiments [6]. For fast probe electrons, then, and when the energy and momentum transfer in the interaction is large, the target electrons in the ground state can be treated as if they were free, but having a momentum distribution as if they were bound. This is the essence of the impulse approximation (IA), valid for large energy and momentum transfer in the Compton event. The reciprocal form factor Compton scattering provides information on the momentum density of electrons in the specimen. However, the same is true for elastic scattering which yields the modulus of the static form factor (the Fourier transform of the particle density): (15) Taken for granted that the phase problem can be solved (determination of the phase of F) the question is legitimate whether F[p) and p{p) contain different physical informa- tion. Eq. (15) says that F is the Fourier transform of p. Identifying p with the particle density in terms of one-electron wave functions, P(r)=¥>(r>-(r), (16) (see the comment given after eq. (12)) , we can apply the Wiener-Khintchin theorem stating that F is the autocorrelation function of the wave function x in momentum representation I (17) On the other hand, the quantity derived from Compton scattering is i.e. the diagonal element of the density matrix in momentum representation. F and p contain cemplementary information. The latter is the probability of finding an electron with momentum p, and doesn't give any clue to the phase of x- The static structure factor F on the other hand contains information on the phases of wave functions, via the autocorrelation integral, but x cannot be derived uniquely from F. The Fourier transforms of eqs. (17) and (18) are ' (f) B(f) = J Note the formal symmetry of eqs. (18, 19) and eqs. (17, 20). B is called reciprocal form factor because of its similarity to F. In complete ana- logy to the statement on the momentum space quantities F, p it can be said that elastic scattering yields the diagonal element of the density matrix in real space (probability of finding an electron at position r) without any clue to the phases of wave functions. Compton scattering yields, via the reciprocal form factor B, information on the phase relations of the wave function Contributions to the Compton Profile Core electrons The simple relationship between the momentum density and the differential cross section does no longer hold when the impulse approximation breaks down . One has to use a more accurate wave function for the final state in the matrix element. The exact hydrogenic (EH) approximation takes both <£>, and Conduction electrons For the conduction electrons the simple free electron model can often be used. In the following example, this model was applied to polycristalline aluminum. From eq. (14) we get for the CP, by projecting the states within the Fermi sphere onto the direction p,, the characteristic free electron parabola falling to zero at the Fermi mo- mentum ±pr. See fig. 3. In contrast to the L-electrons, the IA is valid for conduction electrons. The Problem of Multiple Scattering Aside from the Compton effect, other electronic excitations within the solid must be considered. Low-angle scattering off valence electrons gives rise to plasmon excita- tions peaking in the low energy loss region. Such a plasmon excitation can be followed by a Compton event. An equally dominant contribution to multiple scattering in the energy and momentum range of typical Compton experiments arises from combined Bragg-Compton scattering. The Bragg spots or rings and plasmon peaks can be consi- dered as new sources for Compton events. In the typical range of Compton scattering, multiple scattering is caused by elastic scattering into high angles followed by a Compton event. Since these Compton events correspond to scattering angles different from the single Compton scattering angle new Compton profiles are generated with different maxima and width. They overlap with the single Compton profile and alter the position of the maximum and its width. A third coupling mechanism which may be important involves diffuse quasi-elastic scattering which produces a slowiy varying intensity distribution in the diffraction plane. All these scattering events give rise to a background of 50% to 100% of the Compton peak value, and to serious masking of the line profile. The measured spectrum may 43 look completely erratic as was shown by a numerical simulation of CPs by Williams et al [13]. See fig.ll. At present, a satisfactory solution of the multiple scattering problem is lacking. However, there are some promising approaches which will be discussed in the following. Bragg scattering To determine the Bragg scattering intensity, a density distribution in the dif- fraction pattern can be used. When there is a continuum of scattering angles for the coupling of Bragg and Compton events—as is the case in polycrystalline specimens with well defined Bragg rings—the double scattering contribution forms a smoothly decre- asing background dominated by the ionisation event with the smallest scattering angle. This is why the background approximately follows a power law dependence in energy A • E~', well known from inner-shell losses. See fig. 4 The coefficient A takes on a wide range of values, but s is generally in the range 2-6 [12]. Increasing specimen thickness lowers the value of s due to plural scattering contributions, and increasing energy loss increases this value. Due to this dependence of the coefficients A and s, they should be determined at each ionization edge. Usually the energy dependence of the background is measured over a fitting region immediately preceding the edge. This procedure is only approximately valid in practice since the io- nization threshold cannot be identified exactly due to multiple scattering contributions. One method, although not rigorous, to investigate the background behaviour is to fit a model calculation of the Compton profile, including multiple scattering contri- butions and a power law background, to the measured profile. (See the demonstration experiment). In most cases, single crystals are of interest for ECOSS. So, one has to face the difficulty of extracting a single Compton profile from a measured superposition like fig. 11. Note also the difference to fig. 4. This problem could be tackled as described in the following: The intensity M measured at a particular energy loss E is a linear combination of coupled Bragg-Compton events E), (21) where P3 is the intensity of the j-th Bragg (elastically scattered) beam, and / is the inelastic intensity which would be measured if only the (000) beam were excited. P, can be calculated with standard methods of dynamical diffraction theory, and they can be measured experimentally, ft, is the angle for inelastic scattering from the j-th Bragg beam. (See fig. 22). It is always possible to choose n angles H'k such that the system of linear equations (22) can be uniquely solved with respect to I. For instance, in the two-beam case (only two Bragg beams excited) we have [, E) = Po J(na, E) + PG I(n2, E) (23) = P0I(-n3,E) + Peli-n^E) Because (at least in centrosymmetric crystals) J(fi) = /(—ft) we can solve for /(fti.2, E) in the unsymmetric case (Po ^ Pa)- By choosing the geometry such that the scattering angles i?a = jn\ |, t?2 = |H31, are equal, one obtains immediately the anlsotropy of the Compton profile /. & HE) = MpZ™*- (24) For many applications, this is the relevant .outcome which can be directly compared with theoretical predictions (See fig. 9b). 45 Plaamons The intensities of successive plasmon events can be found by integrating plasmon spectra in the bright field mode. Alternatively, a calculation based upon independent scattering events results in some Poissonian distribution for the plasmon spectrum. The channel coupling between plasmon and Compton events can be simulated by a weighted superposition of Compton profiles shifted by multiples of the plasmon energy. This was done in the following demonstration experiment on aluminum. A direct method for removal af the plural plasmon contribution is in principle possible since the convolution integral for the measured intensity can be solved for the single Compton scattering intensity by any deconvolution method (e.g. [4,14]). To our knowledge, this has not yet been tried. Quasielastic background Quasielastic scattering followed by a Compton event produces a smooth and slowly varying background in the angular and energy range of the Compton profile. When a fitting procedure is used to extract the Compton profile from measurements (as was done in the following demonstration experiment) the quasielastic contribution need not be considered separately since the smooth background shows up in the power law dependence (A • E~')., Removal of the quasielastic background should be possible using a method originally proposed by Batson and Silcox [15]. The coupling of the quasi-elastic background to the Compton event is given as , E) = I tfd'Qiti - &)![&, E) (25) where M is the result of the coupling, Q is the quasielastic intensity and / is the purely inelastic intensity. For slowly varying Q, we have approximately , E) *Q{n)f tPn'un'^E). (26) The integral is available experimentally as an energy loss spectrum in image mode, and Q can also be measured. So, the background due to quasielastic coupling could be removed. It should be noted that this procedure is only valid for radially symmetric intensities. Applications Polycrystalline aluminum In ~ demonstration experiment [8] CPs of polycrystalline aluminum were taken at a scattering angle of 5.2 degrees. Measurements were done on a cylindricai mirror analyzer (CMA) attached to a Siemens Elmiskop IA at 40keV.The angular resolution was ±2Amrad, corresponding to a wavenumber resolution of ±0.25 A l. The finite wavenumber resolution broadens the profiles by ±15eV which amounts to & momentum uncertainty of 0.23a.u. This is considerably higher than accuracies obtained from photon CS experiments (~ OAa.u. typically). The film thickness was 240 A. Measured CPs were fitted to calculations including L-shell- and conduction elec- trons as well as a background ~ AE~'. From figs. 4, 5 it can be seen that the quality of the fitting depends on the fitting range. If the region of valence electron contribution is chosen, a deviation of Z% — 4% is found. In cases of a wider energy loss range for the fitting procedure the deviations increase up to almost 100%. These results clearly show that background subtraction (i.e. choice of an appropriate fitting region) is a serious problem. From our calcuiational results it is clear that, although the IA is valid for the conduction electrons, it is not for L-shell electrons. The maximum of the L-shell CP is ~ 50eV higher than predicted by the IA. This effect causes the maximum of the measured profile - which is in essence a superposition of both contributions - to He between the IA and the exact maximum. Consequently, it does not make much sense to interpret the total CP as the electron momentum density. The asymmetry of the profile brings a commonly used practice in CP data pro- cessing into question, viz. to split the measured profile at its maximum and substitute the low energy part by the symmetrically extended high energy part. The reasoning is that multiple scattering contributes mainly in the low energy region of the profile. Given the experimental conditions encountered in ECOSS this procedure is not gene- rally valid, both because the asymmetry caused by the core contributions is not taken into account, and because multiple scattering is not negligible even in the high energy part of the spectra. In practical applications the situation is rather better, since multi- ple scattering and core contributions—both flat and broad distributions— don't much influence the widely used reciprocal form factor B{r). It is evident from fig. 6 that the L-shell contribution alters B{r) only at small wave number. Anisotropy in graphite In an electron Compton scattering study of graphite, Vasudevan et al. |9] found strong anisotropy of the momentum distribution within the basal plane, (fig. 7). Photon CS cannot give these results because the beam cannot be focused onto a single platelet of graphite the orientation of which is random in the basal plane. For comparison, fig. 8 shows predictions of various theories. The disagreement is striking. The authors speculate that antibonding sp2 ir' orbitals—which should be empty theoretically—may cause the large negative value of B at the lattice vector L100- One fact is obvious: Even todays most refined model calculations are too poor for the prediction of CS data. This may well be since these calculational methods have in general not been used to predict wave functions but energy levels and band structure rather. In any case, one should be cautious about the results of these first ECOSS exoeriments. Electron correlation in copper In order to demonstrate what can be expected for future ECOSS experiments we quote some results from photon Compton scattering: Fig. 9.a shows a Compton profile of the valence electrons in Cu, along the < 110 > direction [7]. The theory which is an SCF local density approximation (LDA) with exchange/correlation correction using linearly combined Gaussian orbitals predicts the experiment quite well. However, when the anisotropy of the profile is plotted, the situation is different: fig. 9.b shows that theory overestimates the amplitude of oscillations. In later papers fl, 2] the local density approximation was re-investigated thoroughly. The periodic deviations of LDA predictions from measurement, given in fig. 10 were traced back to electron correlation effects in the inhomogeneous electron gas. They act so as to reduce the occupation of the Fermi sphere relative to any model calculation based on a single particle concept. In an extended zone scheme, this effect gives rise to the oscillations in fig. 10. Again it can be said that present theories of the electronic structure in the solid are too poor to predict Compton profiles accurately. Conclusion After a presentation of the theory of electron Compton scattering, the advantages and disadvantages with respect to established photon Compton scattering experiments are discussed. Examples show that a) the main obstacle of strong multiple scattering can be overcome by a careful analysis of the various contributions to the Compton pro- file; b) anisotropies in the momentum distribution of valence electrons can be measured in microscopic samples, thus opening the way to electron Compton experiments in poly- crystalline specimens or microscopic segregates; c) Compton scattering is an extremely precise method for the investigation of otherwise undetectable solid state effects of the ground state, such as electron correlation. 49 References [1] Bauer G£, Schneider JR. (1984). Nonlocal Exchange-Correlation Effects in the Total Compton Profile of Copper Metal. Phys. Rev. Lett. Vol.52,23, 2061-2064. [2] Bauer GW, Schneider JR. (1965). Electron correlation effect in the momentum density of copper metal. Phys. Rev. B Vol.31,2, 681-692. [3] Bloch BJ, Mendelsohn LB. (1974). Atomic L-shell Compton profiles and incohernt scattering factors: Theory. Phys.Rev.A,Vol.9,No.l, 129-154. [4] Egerton R F. (1986). EELS in the Electron Microscope. Plenum Press, New York, 229-241 and 357-361. [5] Eisenberger P, Platzman PM. (1970). Compton Scattering of X Ray from Bound Electrons. Phys. Rev. A Vol2.2, 415-423. [6] Eansen NK, Pattison P, Schneider JR. (1987). Analysis of the 3-Dimensional Electron Distribution in Silicon Using Directional Compton Profile Measurements. Z.Phys.-Conden.Mat.66, 305-315. [7] Pattison P, Hansen NK, Schneider JR. (1982). Anisotropy in the Compton Profile of Copper. Z.Phys. B Cond.Matter 46, 285-294. [8] Schattschneider P, Hohenegger H. (1987). Electron Compton Scattering from Po- lycrystalline Aluminum. Analytical Electron Microscopy. San Francisco Press, 270-274. [9] Vasudevan S, Rayment T, Williams BG. (1984). The electronic structure of gra- phite from Compton profile measurements. Proc. R. Soc. Lond. A3S1,109-124. [10] Williams BG, Parkinson **1P, Eckhardt CJ, Thomas JM. (1981). A new approach to the measurement of the momentum densities in solids using an electron microscope. Chem. Phys. Lett. Vol.78,3, 434-438. [11] Williams BG, Sparrow TG, Thomas JM. (1983). Probing the structure of an amor- phous solid: Proof from Compton scattering that amorphous carbon is predomi- nantly graphitic. J. Chem. Soc. Chem. Commun., 1434-1435. [12] Williams BG, Sparrow TG, Egerton RF. (1984). Electron Compton scattering from 50 solids. Proc.R. Soc. Lond. A393, 409-422. [13] Williams BG, Uppal MK, Brydson RD. (1987). Dynamical scattering effects in elec- tron scattering measurements of the Compton profiles of solids. Proc.R.Soc.Lond.A 409, 161-176. [14] Schattschneider P, Zapfi M, Slcalicky P. (1985). Hybrid deconvolution for small angle inelastic multiple scattering. Inverse Problems 1, 381-391. [15] Batson P, Silcox J. (1983). Experimental energy-ioss function for aluminum. Phys. Rev. B 27, 5224-5239. 51 Tigure Captions Fig. l.a: Upper part: Geometry of Compton {scattering, p is the initial -wave -vector of the target -electron, A5*is the momentm: transferred in the interaction, pj is the final -wave -vector of the target electron. The quantity measured in Compton scattering is the distribution of-wave -vectors projected (Onto the q direction, p,. The dotted area schematicaliy indicates the spatiaJ distribution of valence electrons. .Lower parr: Sketch of the Term: sphere with allowed creation And .annihilation processes (ful3 Jines). The process indicated by z. dashed line is forbidden. Fig- 2 : Scattering geometry .and relation between momentum density and Compton •profile. Tig. S: Single scattering-profile for conduction electrons (Al, 5.2 deg) Fig. 4: .Measured profile as compared with ieasz squares nt over entire energy range. (24nm Al, 5.2 deg) Fig. 5: ieast squares it excluding background as compared with measurements for 24nm Al-at SJideg. Fitting region USeV — 6S0eV (a), 290cF — 470eV (b). Fig. 6: JReciprocal form factor JB(r) of measured Compton profile incJuding and exciuding (dotted) ir-sheU contribution. Fig. 7: HeciprocaJ form factor in < 100 > direction in the basal piane of graphite, derived 52 from ECOSS. Lattice points are denoted Lijk. From {9j. Fig. 8: Calculated reciprocal form factors in < 100 > direction in the basal plane of graphite. Full line: Tight binding, dashed line: pseudopotential. From J9j. Fig. 9a: Compton profile for the valence electrons of Cu. Dots: experiment; full line: theory. From [7]. Fig. 9b: Experimental anisotropy in the Compton profile of Cu in directions < 110 > - < 100 >. Dotted and dashed lines: experiment; full line: theory (SCF-LDA) in the Hohenberg-Kohn-Sham formalism. From [7]. Fig. 10: Difference in the Compton profile betweeen SCF-LDA theory and experiment. In the lower part, a projection of the reciprocal lattice onto the < 100 >, < 010 > plane with the Fermi surface of Cu is depicted. The oscillatory difference can be explained by the removal of states within the Fermi body to outside. From [2]. Fig. 11: Simulation of Bragg-Compton channel coupling in Si. From [13]. Fig. 12: Scattering geometry for Bragg-Compton coupling in the two-beam case. 53 core OL. cs CD cs = r\i~ a ( \ u^ t — CM o> \ / .6 8 1 6 / \ 3 8 0.8 8 . 1e.ee ise.ee 2se.ee 35e.ee Mse.ee sse.ee 6S8.ee 75 .ee es .ee $s .ee te B.ee Encrov Ios6 toV] Scoltcring profile from conduction electrons I PI 1 5.2de9l \ 1 i CD i i 1 «O \ rb.un i t o • V c N • to . 5 as h ' — •/->- <• ..« 3 // \ ^ o . > /; 2« c / \ s A CD i r-4 H •—* \ \ V\ \\ IS ! t v 0 2 8 e . s c \ CB ca « Enerev loss (eV] Profile of conduction electrons conoored with calcuioicd orof«le( (fl( 2tl0fl..5.2ae9l f i 11 ins iniervol 1 = 1190ev-478evi 56 -e.2 J2 16 28 2H 28 32 36 V« 18 Torn Todor froa ncosured orofiie including ond excluding (dotted) L-sheil coniribuiion (fll 218 A. S.2 dc9) 010 0.1 0.05 - \ B{r) 0 H^W- Blr) 0 .ll2 ' -0.05 k I 1 I ' i *. : -cnop : : 57 Cu valence profile < 110 > n Pz(a.u.) -0.04- 58 AE/keV ff g (*olid line), multiple (dotted line) and total (broken line) scattering from a silicon sample 136 A thick oriented in the (100) direction with 100 leV incident electrons. In (a), (b). (d) and (e) the scattering vector is in the {020) direction with a scattering angk of 5.5* {(a) and id)) and 3.9* ((6) and <«)). In (c) and (/) the scattering vector is in the (022} direction and the scattering angle is 5,5*. The data are plotted against both energy loss (ai-ic) and the momentum of the electrons in the sample ( -Q 000 00G 61 DISPERSION EFFECTS AND SPATIAL RESOLUTION IN STEM :s, Zabala and P.M. Echenique* Dpto. Electricidad y Electronica , Facultad de Ciencias, Euskal Herriko UnibertsiuueaApdoM*\48080 Bilbao Spain. * Dpto. FlsicadeMateriales, Facultad de Quimica, Euskal Herriko Unibensiuuea, Apdo. 1072,20080 San Sebastian. Spain. Abstract The excitation probability of a STEM electron moving paralell to a vacuum-metal interface is studied. Expresions for the energy loss probability are obtained character/zing the solid by a spatial dispersive dielectric function e(ct)Jc), within the well known specular reflection model and computations are performed for 100 keV electrons, using the Mermin dielectric function which includes both single particle and damped collective excitations in the medium. The study is focussed on the variation of surface and bulk piasmon excitation probabilities with impact parameter, specially around the interface, at distances shorter than 10 A. INTRODUCTION In the last few years, developments in the field of Scanning Transmission Electron Microscopy (STEM) have renewed interest in the interaction of high energy electron beams with surfaces at glancing incidence <-h9\ The surface piasmon excitation probability is determined by analyzing the inelastic scattering of fast electrons aligned paralell to the surface, at a given impact parameter, which can be systematically varied. Probe sizes of 0.5 - lnm and 100 keV electrons are usually employed in this kind of experiments. At such energies, the classical dielectric theory in which the target response is described by a local dielectric function e(co), has shown qualitative agreement with many experimental data. Anyway, when STEM experiments are performed to study the nature of the surface, a high spatial resolution is required.I n these cases, an accurate knowledge of the dependence of die energy loss probability on the position of the probe is needed in order to interprcte the experimental data. Howie & Milne <5> and Krivanck et ai. C7) pointed out the insensittviry of high energy inelastic scattering to surface reconstructions. In contrast to this, Ichinokawa et al. W observed, with 100 eV electrons, changes in the spectrum when oxidation or reconstruction of Si surfaces occurred. Sheinfein et al. W pointed out the detection of changes in the loss 62 spectrum with 100 keV electrons at a spatial resolution better than 8 A as the probe was moved across an AI-AIF3 interface. This last result was justified by Cheng <10), who calculated the spatial resolution fcr 15 eV energy loss in Al as the plasmon localization distance. The spatial resolution can be analyzed by studing the variations of the EELS spectrum with impact parameter, using dielectric theory, which also includes plasmon lifetime. In the local approach, when the medium is characterized by a non-dispersive dielectric function £{co), rapid variations of the spectrum near the the surface occur due to the logarythmic divergence of the loss probability as the impact parameter goes to zero, which has no physical sense. A more realistic srudy of the problem can be done by considering spatial dispersive response function £(oojc) to characterize the target ^1>12). DISPERSIVE ENERGY LOSS PROBABILITY V Figure 1 When the wave vector dependence of the dielectric response function of the medium is considered, the induced potential can be calculated making use of the specular reflection model 03,14) The p]ane boundary problem to bs solved, presented in fig. 1, can be replaced by that of a surface charge density a and two punctual charges simetrically placed at ZQ and - ZQ , in vacuum, to calculate the potential at z>0 (fig. 2a). For z<0 the potential is due, as it is shown in fig. 2b, to a negative charge density - -a <») (b) Figure 2 The energy loss by unit length is obtained from the induced potential as: dW (1) dy •x-0 ; y«vt ; and is expressed as a double integral in energy and momentum kx, paralell to the surface and perpendicular to the particle's trajectory: dW = f d(o to f dk P(CD,k ) - J d» o P((o) (2) dy s ]| 0 0 0 , where in atomic units, the differential probability to loss energy co and transfer momentum kx is given by: [e(Q,a,)- P(C0,k) (3) , where Q is the momentum paralell to the interface (4) v and £(Q,co), the surface response function defined as an integral in perpendicular momentum (5) , with (6) When dispersion effects arc neglected in e(k,co), after integration in (3) the surface response function gives the local function E(0)>, so equation (2) becomes 2 e22"0 J e(a»-l P(co,k ) = —- —pr- InJ — -I (7) ~"J^ Q l e(o)) + l TIV2 Q I e(0)) + !J In this case the integration in kx can be calculated analytically, giving the well known expression <15) 2 2cozo I £(co) -11 P((0) = __ Ko(—21) InJ t t I <8) , where KQ(X) is the zero order modified Bessei function, which presents the asymptotic behaviour: Kn(x) ~ -ln(x) , x«1 (9) To evaluate the dispersive loss probability we have considered the Mermin dielectric function (l6), which is an extension of the Lindhard function , with z = k / 2kp, to = x + iy, x = co/cop and e^ the Lindhard dielectric function or random-phase approximation for the longitudinal dielectric function of a zero temperature electron gas. To compute the locai j robability we have considered the Drude dielectric function which also includes damping: e(co) = l 2 (12) d) (a) + i y) In fig. 3 we represent the differential probability P(co,kx) given by equation (3) for the surface plasmon energy ws and different values of the momentum kx, to show how the high momentum (associated to high deflection angles) give better spatial resolution of the surface but, on the other hand, their intensity is decreased and the contribution to the total probability losses importance. In fig. 4, an energy loss spectrum is presented for Al (rs=2, y=l eV), produced by a 100 keV exterior electron at a distance of 1 A from the surface. The continuous line represents the dispersive spectrum and the dashed one .he local probability. The effect of 65 dispersion is to decrease the intensity of the maximum and to displace it slightly towards higher energies. In fig. 5 the probability to excite surface plasmons as a function of the reduced impact parameter B«2to ZQ/ v is represented until 100 A. The continuous plot corresponds to the dispersive probability and the dashed line to the local one. In the upper side we show both probabilities in the first 5 A. It is shown that the rapid variations near the interface can also be explained using dielectric theory. Near the interface those rapid variations of the spectrum occur at small distance variations but ,may be, not as small as the 4A quoted by Cheng, as it can be seen in the upper plot. Furthermore, the dispersion corrects the divergence at the boundary and the dispersive probability becomes soon the local one , before - 20 A. The rapid variations for B<1 are due to the logarythmic behaviour, but for B>1 the behaviour is exponential and the spectrum changes in units of B. For uy= ACKNOWLEDGEMENTS The authors wish to thak Prof. A. Howie and Prof. R.H. Ritchie for many stimulating discussions and Eusko Jaurlaritza for financial support. REFERENCES 1.- BatsonRE., Ultramicrosc., 2 (1982) 277; Ibid, 11 (1983) 299. 2.- Cowley J. M., Surf. Sci., 144 (1982) 587. 3.- Fan Cheng-gao, Howie A., Walsh C.A. and Yuan Yun, (to be published). 4.- Howie A., Ultramicrosc., H (1983) 299. 5.- Howie A. and Milne R. H., J. Microsc., 12fi (1984) 279. 6.- Ichinokawa T., Ishikawa T., Awaya N. and Onoguchi A., Scanning Electron Microsc., Part 1, (Chicago: SEM Inc.) (1981) 271. 66 7.- Krivanek O. L., Tanishiro Y., Takayanagi K. and Yagi K., Ultramicrosc., 11 (1983)215. 8.- Marks L. D., Solid St. Commun., 42(1981) 727. 9.- Schcinfcin M., Muray A. and Isaacson M., Ultramicrosc., 1£ (1985) 233. 10.- Cheng S. C. Ultramicrosc., 21 (1987) 291. 11.- Echcnique P. M, Phil. Mag. B, 52 (1985) 9. 12.- Núñcz R., Echcniquc P. M. and Ritchie R. H., J. Phys. C, 12 (1980) 4229. 13.- Ritchie R. H. and Marusak A. L., Surf. Sci., 4 (1966) 234 . 14.- Wagner D., Z. Naturf. (c), H (1966) 634. 15.- Echcnique P. M. and Pendry J. B., J. Phys. C, £ (1975) 2936. 16.- Mermin N. D., Phys. Rev. B, 1 (1970) 2362. 17.- Lindhard, K. Dan. Vidcnsk. Selsk. Mat. Fys. Medd., 2S, No. 8 (1954). 67 Eo=l00KeV} rs = 2 - y= 1 eV :. 3r a. Sir 0 5 impact pcrcmeter 2a(A) Figure 3 O. 045 r 0. 005 r r,, ; 0.5 2. S C. 7 0.8 Figure 4 68 0.02 0.04 3.06 0.0S 0.1- Figures 0. lp 0.08 0.2 Rfure6 69 STOPPING POWER FOR PROTONS IN Al FOR THE WHOLE RANGE OF VELOCITIES by F. Flores Departamento de Fisica de la Materia Condensada Universidad Autonoma de Madrid Cantoblanco, Madrid 28049, SPAIN A. Aruau, P. M. Echenique Departamento de Fisica de Materiales, Universidad del Pais Vasco Facultad de Quimica Apartado 1072, San Sebastian 20U80, SPAIN R. H. Ritchie Health and Safety Research Division Oak Ridge National Laboratory Oak Ridge, Tennessee 37831 USA 70 ABSTRACT The electronic stopping power of aluminium metal for protons has been calculated with explicit inclusion of the different proton charge states inside the medium. The total stopping power is obtained by adding to the calculated stopping power of each charge state, the energy loss associated with the electronic exchange processes. Good agreement is obtained between our first principles calculation and the available experimental data. 71 1. INTRODUCTION The problem of the energy loss of ions moving through solids has attracted the attention of physicists since the beginning of the century1. Fig. 1 shows schematically the different regimes for the ion stopping power as a function of its velocity. In the high velocity regime, the ions is stripped of its electronic charge and the theories of Bethe^ and Bloch3 of the electric stopping for a charge Z give a good description of the energy loss process. In the low veloci- ty regime, nuclear processes between the projectile and the target give an important contribution to the total stopping; the electronic contribution shows, however, a linear behaviour in this regime which is the result of the ion being surrounded by a cloud of polarization charge. The density functional theory allows one to calculate this cloud of polarization and the energy loss at those low velocities in a selfconsistent way. THe first explicit calculations of the stopping power in this formalism were perfomed by Echenique et al4 for hydro- gen and helium, and later were extended to higher ionic char- ges5. At intermediate velocities (v ^ z2/3 i=. ) there is no rigorous theory of the stopping power for ions mov?.ng through solids, even in the case of light ions. Effective charge theories have been proposed** to explain the stopping data. The processes by which a partly stripped ion interacts with a target are numerous and complex7; in particular, in the intermediate velocity regime electronic exchange processes between the projectile and the target must play an important role in stopping. For this reason, the development of first principles methods to calculate electronic capture and loss cross sections^ should also be introduced in the calculation of the total energy loss. 72 In this work we have calculated the stopping power for protons in aluminium as a function of the intruder velocity, with explicit inclusion of such solid state effects as the electronic capture and loss cross sections of the moving ion. 2. THE MODEL The different mechanisms included in the calculation of electronic exchange processes are shown in fig. 2. In the Auger process illustrated in fig. 2a, an electron is captured (or lost) by the ion to (or from)a bound state assis- ted by a third body that may be a plasmon or an electron- hole pair. Condensed matter effects are important here since electrons in valence band states are involved. A second mechanism leading to capture and loss is the coherent resonant interaction. Electronic exchange processes are induced by the time dependent crystal potential a* seen by the fixed ion (fig. 2b). The coherent resonant interaction resembles a close atom-atom collision at high velocities. Al low velocities condensed matter effects come into play. The third mechanism included in the electronic exchange processes is due to the shell interaction (fig. 2c). In this case, the inner electron of a target atom can be capture by the moving ion. A main assumption in the calculation of all these exchange processes is that the proton has a well defined bound lovel over the whole range of proton velocities. Vinter and Guinea and Flores9 have shown, using a selfenergy approach, the reality of the H~ state in an electron gas. proving in this way that a proton can bind two electrons inside aluminium. 73 Moreover, Norskov^O, using the local density formalism, has concluded that the H~ configuration in aluminium has an energy that is 9eVs. lower that the one for H°, with a single electron bound to the proton. This result suggests that the H" and H° configurations would be mixed for an ion velocity, v, such that ,2 a atomic units, namely, around V= 0.8 a.u. The equilibrium charge fractions of bare protons ( neutral atoms ( 4>° ) and negative ions ( $~) as determined by the different mechanisms discussed above are given in terms of the capture and loss cross sections** as: + 1 • =oioss where: D= "loss In these equs. oioss (H~) and aioss (H° ) are the loss cross sections of an electron by the negative ion and the + neutral atom respectively, while oCapt (H ) and <*capt "ft are the corresponding capture cross sections of an electron by the bare proton and the neutral atom. Fig. 3 shows the results of our calculations for •+.#° and $~ as a function of the ion velocity. For very low veloci- ties, v « 1 a.u., only negative ions survive in the Metal; for high velocities, v »1 a.u., only positive ions survive. 74 while for intermediate velocities v = 1 a.u., *° reaches its maximum. Notice that this intermediate velocity is in good agreement with our rough estimation as discussed above using Norkovs calculations. Finally, the stopping power is calculated by sunning the stopping power for each charge state weighted by the respective charge state fractions, and adding the energy lost per unit length in capture and loss processes. Fig. 4 shows how two succesive loss and capture processes leave the ion in the same state, creating an electron-hole pair excitation in the electron gas. Thus, we write for the stopping power: £ In equ.O) ^£ an while it understimates at the sane time -he stopping power of H at intermediate velocities, 0.9 sv £1.3. He can expect* however, --hat both effects -e:::: to ccrr.r er.sate each other as the results of the total s-t-cpr-r-7 power '=« below) s-jegest. 3. RESULTS FOR THE STOPPIKG POVrSP. The results obtained form ecu.(3) for the stopping power of protons moving with velocity v in hi ire shown in fig. 5 as a thick solid line (TOTAL). The curve labeled LT is the stopping power for a bare proton calculated in linear theory using an RFA dielectric function11, tc represent valence electron excitations in aluminium with r£*2. Inner shell corrections from the 2 s and 2p electrons of the Al ions have also been included. The different contributions to til* curve labeled TOTAL have been separated to show the relevance of the various terms as a function of the ion speed. Mben V* 1.2 a.u. the contributions from the negative ions In conclusion, our first principles calculation shows the relative contribution to the stopping power of the diffe- rent charge states of a proton moving with velocity v in aluminium. Combining dielectric and density functional results in the appropriate velocity range with the energy lost in the capture and loss processes, we have obtained good agreement with experiments. 76 ACKNOWLEDGMENTS One of us (AA) is grateful for careful and stimulating conversation with J Nagy. The authors gratefully acknowledge help and support by Eusko Jauralaritza, Gipuzkoako Foru Alcheudk, the Spanish Comision Asesora Cientifica y Tecnica (CAICYT) and the NATO Research Grant 0142187. The authors would also like to thank Iberduero S.A. for this help and support. Support is also acknowledged from the Office of Health and Environmental Research, U.S. Department of Energy, under contract DE-AC05-84OR21400 with Martin Marietta Energy Systems, Inc. 77 IBrBKBHU 1.- N.Bohr, Philos. Mag. 26, 1, <1913) 2.- H.A.Bethe, Ann. d. Phys. 5, 325 (1930) 3.- F.Bloch, Ann. d. Phys. 16, 285 (1933). 4.- P.M.Echenique, R.M. Nieninen and R.H.Ritchie, Solid State Comm. 37, 779 (1981). 5.- M.J.Puske and R.H.Nieminen and R.H.Ritchie, Phys. Rev. B 27, 6121 (1983); A.Arnau, P.M.Echenique and R.H.Ritchie, Nucl. Instrum. Methods B 33, 138 (1988). 6.- S..Kreussler, C.Varelas and W.Brandt, Phys. Rev. B. 23, 82 (1981); W.Brandt and M.Kitagawa, Phys. Rev. B 25, 5631 (1982). 7.- M.A.Kumakhov and F.F.Kumarov "Energy Loss and Ranges in Solids" (Gordon and Breach Science Publishers, New York, 1981); Y.H. Ohtsuki "charged Beam Interactions with Solids" (Taylor, London, 1983) 8.- S.K.Allison, Rev. Mod. Phys. 30, 1137 (1958) 9.- B.Vinter, Phys. Rev. B. 17, 2729 (1978); F.Guinea and F.Flores, J.Phys. C 13, 4137 (1980). lO.-J.X.Norskov, Phys. Rev. B 20, 446 (1978). ll.-J.Lindhard, K.Dan.Vidensk, Selsk. Mat. Fys. Medd. 2«, no 8 (1954) ; R.H.Ritchie, Phys. Rev. 114, 644 (1959). 12.-H.H.Andersen and J.F.Ziegler, in "Stopping Powers and P.anges in all Elements: Hydrogen" (Pargamon, J.'ev.- York, 1977). 73 FIGURE CAPTIONS Fig. 1 Shows schematically the stopping pcv.*er of an icn of charge Z as a function of its velocity. The intermediate regime is defined by the statistical velocity Z2/3 — Fig. 2 i.) Shows a:; Auger process for an ion moving ir. a uniforr. electron gas. (b) £hc-..£ the coherent resonant process for an ion moving in a crys-f.l. (c) Shcv.s a shell process whereby an ion moving in a crystal captures cne electron from an inner level of the target. Pig. 3 Different, charge fractions for H as a function its velocity. Fig. 4 Shows schematically how loss and capture processes create an electron-hole pair excitation in the electron gas. Fig. 5 Stopping power in atomic units of Al for protons as a function of the ion speed. The circles are the experimen- tal data. The different contributions to the curva labeled TOTAL from the fractions of bare protons (H+), neutral atoas (H°), negative ions (H~) and capture and loss processes Fig. 1 80 ORNL-DWG 87-11006 CAPTURE LOSS V EF Fig. 2 C ' Equilibrium Charge Fractions o o o * * • •f^ O> 00 < 15* T8 ORNL-DWG 9M584 + ( v/ 1 1 11 H° H" loss capture electron-hole pair excitation OflNLOMBMMBH 0.3 . 1 . 1 • TOTAL 7 /^ /o 0.2 S/ A"r / LUl X TJ|"O Jrv\ 0.1 -/A in 0.0 0.4 0.8 1.2 1.6 2.0 v(au) Fig. 5 85 stopping pomp for protons revisited Gregory Lapicki Department of Physics, East Carolina University Greenville, North Carolina 27858, USA 1. Introduction For a projectile of atomic number Zj (I.e., in effect of charge Z^«) and velocity v^, the stopping power in a target characterized by the electron density n ( n = number of electrons in an atom of atomic number Z2 per unit volume), the stopping power can be written in atomic units (aee Fig.l) as 2 S = (4» Z\ n e aQ/ vj )L , (1) where the dimensionless, reduced stopping power L is known to be proportional to v^ at low velocities and it converges to Bethc's (1930) L» to(2»{/«) - B2 - In (1 -62) , (2) with 8= Vj/c and n> = o>(Z2), in the high velocity limit. The L func- tion for protons (Z,=l) will be revisited in the low and high velocity regimes; the proposed formulas will be tested against empirical fits and fused into one function that reproduces the data at moderate velocities. The velocity ranges of 0.1-1.0, 1.0-10, and 10-100 are designated as low, moderate, and high velocity regimes. The low velocities mean proton beams of energy 25 keV or less; they are comparable or lets-than the Fermi velocity of the target electrons. The high velocities of v1>10 represent the protons with energy above 2.5 HeV. The stopping power peaks around v.= 2 i.e., for about 100-keV protons. 86 Figure 2 shows the L for stopping of protons in hydrogen, aluminum, silver, and bismuth (from the highest to the lowest curve) according to the Andersen and Ziegler (1977) fits to experiments. Since these fits were constructed to mimic the S Lindhard and Scharff (1961) found in a Thomas-Fermi treatment (see Fig. 3; note the a of Lindhard-Scharff was multiplied by the square root of 2 to ensure that the TF unit of length has the saae magnitude in a homonudear quasimoleclue as in the TF atom) L . 2 vj / (Zf3 • 4'3 )3'2 . (3) if a strange factor i^ is deleted. Subsequent publications of Lindhard et al. -- known as Notes on Atonic Collisions: III (1963), 11(1963), and 1(1968) -- refer to Eq.(3) as a "result quoted and uti- lized in a summarizing article" i.e., in their 1961 Fhys. Hev. paper. The derivation of Eq.(3) has never been explicitly presented. It would appear that to obtain SCv, , Lindhard and Scharff had to choose an 1/r interaction in their power law description of the scattering poten- tial. Such a selection, however, results in S being proportional to a instead of a that is required to derive Eq.(3). Bonderup's (1978) remarks are worth quoting here: " we are faced with the problem of possessing only fragmentary information on thm function -dE/dx. A more detailed function has been undertaken by Lindhard and Scharff many years ago ... The details of the derivation. 87 however, remained a secret and we can provide no more than the following reference: J. Lindhard and M. Scharff: Notes on Atomic Collisions, IV, Hat. Fys. Hedd. Dan. Vid. Selsk., to be published." The details of the Lindhard and Scharff derivation are still se- cret. As their Notes-Part IV have remained to be published for the last three decades, the enigmatic L of .Lindhard and Scharff -- baring an unaccounted'(albeit irrelevant for protons of Z^=l) factor of Z^11'6 -- is "rediscovered" here (see Fig.3) with a scaling of Eq.(l) as follows. Two assumptions are made: 1) with L being proportional to v., the constant of proportionality simply set to 1 i.e., L = v., 2) the n of Eq.(l) is scaled to a , the unit volume in a Thomas-Fermi treatment, rather than a , the cube of the Bohr radius as the unit volume assigned conventionally to all atoms. This gives L =(v^a/a ) which, with the specific definition of a (see Fig.3), is identical with Eq.(3). The points added to Fig.7 from Echenique et al. (1986) represent (see Fig.4) the ratios of the experimental L from Andersen and Ziegler to the L of Eq.(3). The Lindhard-Scharff low-velocity L clearly uttfe-res- timates the data, increasingly so with the increasing Z-. By contrast, the sophisticated density-functional approach of Echenique et al. (1981) results in satisfactory agreement for most of the target elements. In this work, the Lindhard-Scharff formula is modified (see the top of Fig.5). Using the method by which Eq.(3) was "re-discovered" but with a different expression for the TF unit of length in a quasimolecule of /3+ /3 the stopping medium: a =2*0.8852-*O/(Z^ z\ ) instead of a /3 /3 1/2 0.8853-ao/(zJ + z| ) , it is found that Liow = 555 vi 88 As seen in Fig.*, Eq.(4) is as good as the predictions of Echenique at al. (If«1). Z£ oscillations ar« th« defect of a statistical description of an a ton; the nagnitude of this oscillations in the experiaent-to-the- ory ratios, however, could be partly in error because the stopping power data happen to be scarce for the targets where oscillatory peaks are seen. As illustrated in Fig.7, Eq.(4) agrees with the Andersen-Ziegler fits up to v, * 1, and it overshoots the neasured stopping powers as they decline fro* the linear-in-velocity dependence and reach ns«iess around v, * 2. 5. Hlsjh velocities The Bethe-Bloch formula with "few = Kg* Z2 converges to the data remarkably well for large Z« (see Fig.7) provided the Bloch constant is tailored to then by choosing K^ - 10 eV; it fails, however, as Z2 —> 1 since it employs "few * 10 eV for Z.*l whereas the exact calculations of Dalgarno (19(2) for hydrogen give 15 eV. A significant improvement is obtained at high velocities when « * 0.275** Z2*(l * l/Zt ) or, in •», 3 -L * 15 eV-Z2(l • l/z}' )/2 (5) so that Hm * 15 eV for Z2*l and the effective Bloch constant is about 9 eV in heavy target elenents. With Eq.(5), th« Bethe's L reads (see the middle of Fig.5) Hiigh * ln I726 vl' Z2(1 * 1/Z2/3 M " *2 " ln (l It should be noted that the last two terns are insignificant in their contribution to L (see Fig.2). Equation (6) gives the stopping power in hydrogen within 2.51 of the data (see solid curves in Fig.t), while eh* Bethe-Bloch formula (see Fig.7) overestimates stopping power measure- nents in hydrogen by an unacceptable it. With the increasing Zg, the L 89 of Eq.(6) and the Bethe-Bloch become mutualy indistinguishable and fall within 11 the experiment i.e., within uncertainties of high-velocity data. At this level of experimental errors, a few percent overestimatiom of the stopping power measured in the lightest targets is significant. This could not be a symptom of an ill-description of the hydrogen atom in statistical treatment since Cq..(5) adjusts *&> to the exact value of 15 eV in hydrogen. However, the Andersen-Ziegler fits for stopping power in hydrogen were aade up to the highest available energy of 3S-MeV protons. Only truly high-velocity experiments, in which the stopping power of hundreds HeV if not GeV-protons would be measured, could yield clues as to the optimal selection of w = w (2g) for the "best" applica- tion of the Bethe formula in the high-velocity limit. 4. Moderate velocities At moderate velocities, the Lindhard and Scharff prescription of 1953, 1/2 3 2 L = i.36 x - 0.01« x ' , whore x s. vJ/Z2 , (7) seems to fill the gap between the low- and high-velocity formulas pre- sented in this work (see dashed curves in Fig.6). It links with Boa.(4) and (6), however, somewhat abruptly. Hence, a formula is proposed that is forced to transform smoothly into Eqs.(4) and (6) and that approxi- mates empirical stopping power, with deviations which are comparable with experimental uncertainties in the 1 < v,< 10 measurements. The low- and high-velocity formulas are bridged using the natural logarithms with a relatively simple argument (see the bottom of Fig.S and the dashed curves in Fig.9) : 3 2 3fl VS L = In |1 + Llo||/[1 -0.3 v j[ /(l • 2£ ) * 0.447 v|/( 1*1/^*1 | - 8* - In (1 - 82) , (•) 90 where Llo|# is given by Eq.(4) with Z^* 1. 5. conclusions and plans Tha low-velocity stopping power foraula of Lindhsrd and Scharff (1961) waa radarivad; whan sealed with a different definition of the TF unit of length for the projectlie"target quasiaolecule, it agreed with the senienpirical fits of Andersen and Ziagler (197?) up to Vj« 2/3 — the agreement was no worse than seen in prerious comparisons of these fits with the calculations of Echenique et al. (19S1.1986). Bathe's high-velocity stopping power was employed with a seaitheoretical func- tion for the average ionization potential -- this function was con- structed to equal 15 eV for hydrogen. The Andersen-Ziegler fits were found to converge to this high-velocity foraula; measurements with GeV-protons stopped in the lightest targets are needed to verify the universal utility of Bathe' stopping power in the high-velocity limit. Further work in progress will search for an expedient but store justified (if not siapler) L for all proton velocities. Extensions of Eqs.(4) and (6) to stopping power for other projectiles (Z^^l) will be investigated. Andersen, H. H. and Ziegler, J. F., Hydrogen Stopping Powers and Samges in All Elements (Pergaaon Press, New York, 1977). Bethe, H. A., Ann. Phys. 5. , 325 (1930). 91 Bonderup, E., Lecture Motes on PMMtratien of Chargad Particle* through Matter (Institute of Physics, Aarhus University, 197S), p.142. Dalgarno, A., in Atonic and Molecular Proceses, edlteo oy D. t. Bates (Aeadenic Press, New Tork, Iff2), Ch.15, p. Me*. Echenique, P. M., Vieninen, 1. M., and Ritchie, ft. «., Solid State CoasKin. 12, 779 (1M1). Echenique, P. M., Kieninen, R. H., Ashley, J. C, end Ritchie, ft. ft., Phys. Xev. A 31, 897 (1986). Lindhard, J. and Scharff, M., Hat. Fys. Medd. Dan. Vid. Selsk. 21, #13 (1953). Lindhard, J. and Scharff, M., Phy«. Rev. J2±, 121 (ltel). Lindhard, J., Nielsen, V., Scharff, H., Thonsen, P. V., Nat. Pja. Dan. Vid. Selsk. U, #10 (19f3). Lindhard, J., Scharff, M., and Schiott, H. E., Hat. Pys. Medd. Ban. fid. Selsk. 11, #14 (19C3). Lindhard, J., Kielsen, V., and Scharff, M., Mat. Pys. Medd. Ban. Tid. Selsk. 14, #10 (19«t). 92 V,* Z, ««J *, d»>»c^/•»'•>»Itxj lor 0.1 — f o -: io : 2.5 rvv O - j LAPICKI FIG. 1 93 cqUr fun) 190 FIG. 2 94 Zf r> e»g. jE ' " a« FIG-3 95 LAPICM FIG. 4 FIG. 7. Rmtioi of stopping powers for from Anderson and Ziegkr (AZ) OUf. 26), idW/4*)** to the density-functioiul predictions of Rcf. S, (rf*Yrf*W The solid circles represent experimenul diu for 16 eknealal solids, while the crosses are interpolations to materials for which no ex- perimental data are available. 96 -A (MS.) L* » IO«V- (\ 0. L h,-.,k s In 27?7 b n 51 5*5" x? LAPICKI - FIG. 5 97 LAPICKI . 6 L 14 1 1 1 1 1 «•• 4 > V .• ". • • I \ • 1i a • • • .^^B^— —^ s tf~- •V a • Out - I • II 04 1 1 1 1 | 1 1 10 FIG. 7. lUtki of stopping pd«cn lor from Andenoa and Zkfkr IA2) OUT. M. density-functiootl predictioot of Rcf. 5V while the cronci are perimenta) diu are available. 98 LAPICKI FIG. 7 99 LAPICKI •*• « v " - * "^ -J 100 LAPICKI FIG. 9 101 Stopping Power Theory in Inhomogeneous Many Eieetron System M. KITAGAWA Department of Electronics, North Shore College Atugi 243, Japan A dielectric function £ for an inhomogeneous aany-electroii gas was obtained analytically as the first order formula under the condition of the high frequency response (1]. Using the lowest order of-approxiaation in an iteration for the integral equation of £ , we also showed that the lowest order formula of stopping power exactly corresponds to one called as the "local density approximation" (LDA). In this paper, we derive the integral for- mula of stopping power for the system, in which the electron density changes one—deaensionally. Such a kind of formula is applied to the problem of the solid surface. 1. Introduction Theoretical research concerning dielectric function £ has taken the position of the fundamental theae in the field of elementary excitations in solid for a long time. Under the system of free electron gas, various kinds of theoretical works on £ were per- formed since the work by Lindhard [2], and applied to the calcu- lation of stopping power for ion passing through solid [3-6]. Comparisons between theoretical and experimental works were also performed, and LEA theory, which was introduced phenomenologica- lly in the theory of stopping power, gives a relatively good agreement with the experimental data [5]. As far as the general theory of £ , which should be extended to an inhomog/eneous many- electron gas, concerns, a basic progress has been done recently. Lundqvist reported the general framework of theory on £ for the 102 lnhomogeneous many-electron gas in 1963 f7J. By us* of the Green function's met bod, we obtained the first order of analytical fo- rmula for e under the condition of the nigh fraqiwncy r«spons«. and also showed the theoretical position of LD* theory in ISM 11]. Zn this paper, we confirm the basic progress of the theory on e In inbomogeneous aany-electroa gas by use of the density matrix formalism, and derive the theoretical formula of stopping power for the system in which the electron density is treated to change one-denensionally. Zn the following discussions, we use the notations m, -e and Z^m for the mass of the electron and the charges of the electron and the projectile, respectively. 2. Dielectric function in inhomogeneous many-electron gas The first order of analytical formula for the dielectric func- tion 6 lnr-ffl space, which is proportional to 1/tf* (« :angular frequency of electric field), was represented as follows [1J 1 e2 i 2 - 12 ^ rx *|*V^| (1) «xezn(F?) Vcft-rt —i'-| In the above, r* and r* refer to the position vectors, and n(r^) Indicates the electron density at r*. The important property of the above formula is that not oaly the term Indicating the role of the contribution due to the localised mode e)_ (r^)t*(4Xe2m{r*) /ml1'2) but also the term denoting the effect of boundary condi- tion determined from the many-electron system are included 1m eq.(l). in deriving eq.(l), the Irreducible polarisation propa- gator K makes an important role through the following equation 103 Zn eq.(2), an abbreviated notation (1) indicates the space-time coordinates which includes spin, and v means the Coulomb intera- ction between two charged particles. The detaied explanations of eq.(2) are refered to ref.(l). By use of the representation of density matrix formalism, It is given by and J where n(rf,r,) and f.(r? ,r2) mean the density matrix and the density matrix element. 0 , % and Er are the Heaviside step function, the positive infinitesimal and the last filled energy of the electron in the ground state of many-electron gas. f>. and E. are the wave function and the eigenenergy of the j-th elect- ron, which satisfiles the following Schrodinger equation From eqs.(4) and (5), and using the notation that the operator of Bamiltonian H operates r^, we have Expanding TT by I/a) in the complex energy plane under the high frequency approximation, we have 4 Z r*.r7) f j/r*.^) t l 2 [*~ (EirE.)l Using eqs.(4) and {«}, the first term in eq.(T) reduces to 104 and th* second ter» becoaes I7Hr»n(r.?t ; a«> 2 2 -(n(r».f*)+n(f|.r*))CV S"C^-^>1I Above are exactly same with the results obtained in ref.(l). in which the sum-rule of the wave function is used in calculating 7J- . If we use the such a kind of sua-rule. there appears a eff- ort ing calculation, which aeans the usefulness of the density aatrix foraalisa in solving the problem of the inhoaogeneous aany-electron gas. The result of eq.(10) in ref.(l) is recoa- firaed by eq.(9) by use of the density aatrix foraalisa, and we also obtain eq.(l) for the first order £• in inhoaogeneous aany- electron gas froa eqs,(2) and (?). It is wirthy to be noted that the tera corresponding to the second term in eq.(l) doesn't app- ear in the theory of t under the usual free electron gas. It is »ell known that the dispersion relation of plaeaon and the contribution of the single-electron excitation are derived froa the fourth tera in eq.(7), if we apply eg,. (7) to the free electron gas under the proper approximation shown la refs.(3} and (4). The aethod used in the calculation of 6 in free elec- tron gas gives a theoretical guide in deriving the higher 105 formula of £. in inhoaogeneous many-electron system. However, there appears the mathematical difficulty in obtaining such a kind of higher order term exactly. The derivation of the higher order formula is now under construction. The orthogonality between £, and £ (inverse dielectric function) is given by eq.(5) in ref.(l), and it becomes as follows if we use the "r*-tU representation Generally speaking, £"' is derived from the integral equation, which is obtained by substituting eq. (1) to eq. {10}. Applying an iteration to the above integral equation, we obtain the first order of analytical formula for i- as follows (11) |r>r?l 3. Stopping power for one dimentional case By use of the electro-magnetic theory, the stopping power S is given as follows eXt f (2) «5"(r^~^) t^-(t.Vt2)] (14) In eqs. (12)-(14). 4ff{%) means the potential from the polarisation cloud, in which the second term denotes the subtraction of the pure Coulomb part because such a part doesn't contribute to the energy loss process. j>*xt(2) indicates the number density of ternal charge if we consider a classically straight trajectory 106 of the projectile, "v* and T? denote tlw velocity and tlw iapaet parameter of the projectile. Bow we consider the systea, in which the electron density is treated to change one-dlaentionally in order to derive the sto- pping power formula for the cases of the surface and planar cha- nneling. Me take the electron density n in Inhomogeneous systea aa a function of X, and s^ la eq.(l4) as . (15) la the above, v is the two-dlmentionally Fourier component for the Coulomb interaction, which is gi In eq.(15), it is noted that the traaslatlonal lnvarlance bet- ween X and b in £"' is not satisfied, although such a kind of lavarlance is satisfied in y and s coordinates, q and M)/V corre- spond to the Fourier components over y and z. In £ given by eq. (11), (0 depends only on x, and (i£ -7f ).^ua(r^) has only the x-coaponent. Substituting ve and £ discussed here, and using the following two dlaentlonnal Fourier transformation we obtain the integral foraula of stopping "• 107 (u) U)_(X) -*-*• |X-b|cosh» e v J) l where 0J|1|X»sinh"' (q|MBeV/Wp(x)J and q^^ga*. The first tarn is corresponding to tbs usual Bathe formula, and other terse repre- sent the density-changing effect. (X-b)/lX-b| denotes the step function. It is wirthy to be noted that eq.(ls) Includes aa ar- bitrary density function. Therefore, It Is possible to apply the above foraula to one-dlsentlonally lnboaogeneous systea, for cxaaple, the surface and planar channeling problems. If we con- sider the case of 2aV2/nu_ In the above, X0(2»p{x> lX-*l/V) denotes the nodi fled Beeeel fun- ction of second kind, which Is the typical function appearing in the theory of sealclasalcal aodel. If we use eq.(lf) in eq.(l«) within the fraaework avoiding the singular point at X-b, eq. (is) is reduced to one-diaentional problea. 4. Concluding remarks The theory of the Inhnaagenecms aany-electron system la the new and charenging field. In which there are many interesting prob- leas relating with the inelaetlc scattering. In 1M». Luadeylst reported the general fraaework of the theory oa fc for each a kind of system [7]. In IMS. we derived the first order of ana- lytical foraula for 6 in the inboaogeaeous syatas under the COB- ^^B^T ^taa^sT -la^Las^aa A>a^ a^^^aTavmv^^PBjaave^F £ ^Bj^B9as^ evs^s^^Ps/w^areBi ^Cmeles^ [1] M. Kitagawa, Mucl. Instr. and Math. B13 (19SS) 133. [2] J. Lindhard and M. Scharff, K. Dan. Vld. Salak. Hat. Pya. Madd. 27,.NO.IS (1*83). [3] J. Lindhard and A. fflntar, K. Dan. Vld. Salsk. Nat. Pya. Madd. 34, No.4 <1««4). (4] L. Radln and S. Lundqvlat, Solid Stata Phya. 23 (lMt| 2. (5] ff.k. Chu and D. Powara, Pnya. Latt. 40A (1972) 23. [6] T.L. Parrall and X.H. Ritchla, Phys. Rav. BIS (1977) US. [7] S. Lundqvlst and N.H. March. Thaory of tha Electron Gas (Plemu, Naw York, 1983), Ill ON THE Z]3 CORRECTION OF THE STOPPING POWER L Nigy* ml P. M. Edtemque DepansmattD de Hsica. Facukad de Quunica UniversidaddelPafe Vasco.Apdo. 1072.20010 S«i Sebastian. Spsm ABSTRACT Higher-order correction to the stopping power proportional to Zj* is evaluated. The screening phenomenon of the bare projectile in an electron gas is characterized by simple Yukawa potentials. Second order Bom approximation for the scattering amplitude is employed to describe the elastic collision cross section. For low velocity intruder a comparison of the analytical remits with those obtained from a phase-shift calculation is made. For high velocity of the bombarding particle oar final expression for the Zj3 correction corresponds to the classical resnlt the polarization effect of a harmonic oscillator. 'Permanent address: Quantum Theory Croup. Institute of Physics. Technical University. Budapest. Budafbki ut 8. 1111 HUNGARY. 112 1. INTRODUCTION Since the original paper of Bcthe1 the energy Iocs of a charged panicle in matter has been studied thoroughly for many yean because of its practical implications. The intrinsic difficulty of the problem resides in the dynamic many-body character of the interaction of the projectile with the electrons and nuclei of the material through which the projectile passes. In the theoretical description of the electronic stopping there are two. in principle essentially, different models for the target. The atomistic description treats the stopping medium as a dense atomic fas and uses the standard method for excitations with an implicit cumulative assumption. In the homogeneous electron-gas model the stopping can be studied by response function formalism.2 In these classic first-order perturbation theories the stopping is proportional to the projectile charge squared. Zj2. because of the applied first Born approximation for the transition matrix elements. The possible (initial and final) scattering states are determined by kinematicai constraints. The problem of the divergency of the Coulomb-logarithm is avoided in different manner in the two above mentioned model. In the atomistic description, where the electrons are not free, the minimum value of the transferred impulse is related to the binding energy of the electron and the perturbing potential is pare Coulombic one. The free electron-gas model "eludes" die difficulty of the zero impulse transfer for pure Coulomb potential by the screened potential picture. It should be emphasized, that these pictures have physical grounds. The final forms of the electronic stopping obtained in the two models show significant similarity and thus there is a broad overlapping region between the dielectric and the atomic treatments of the problem. On the other hand it is know from measurements3 that the stopping power exhibits a dependence on the sign of the charge in contradiction of the "rigorous" Zj2 proportionality. The explanations for the so-called Barkas effect are different, depending on the model of the target. In the atomistic description the first Born approxiaution for the particular scattering amplitude is justified because of the pure Coulomb potential of the bare projectile (Rutherford scattering). Consequently one needs to modify the kinematicai constraint. In this model the Barkas effect has been interpreted as a polarization effect.4 In the standard (simplified atomic model} harmonic oscillator description the effect originates in the nonnegligible displacement of the atomic electron during the collision. This calculation fives a term proportional to Z|3 for the Barkas effect in the expression of the stopping power. In the electron-gas model the validity of a perturbation approach (first Bom approximation) is not obvious a priori. Here the kinematicai constraint represents the well-described pan of the 113 theory. The higher order Born terms in the transition matrix elements are related to the screening nonlinearities in a self-consistent description. The role of this nonlinearity was demonstrated is the scattering theory approach of the stopping power, for positive io». This theory uses density-functional method for the determination of the screened scattering potential and correct phase shift analysis at low velocity of the projectile.5 This approach is inherently correct to all orders in Z[. The nonlinear stopping power differs markedly from the standard dielectric (linear) result for the interesting (metallic) density of the electron gas. Note, however, that the dominant pan of this is simply due to the different scattering description. For "linear" potentials ike phase shift calculation gives results which are in acceptable agreement with the nonlinear ones.Funhermore the deviation is partly related to the different schemes for treating the screening problem itself (exchange-correlation and static local field problems). The dielectric formalism retains the momentum-space representation of the (static) scattering potential. Consequently, the transition matrix elements are calculated in first Born approximation for the corresponding (immediately given) screened potential. For arbitrary velocity the quadratic response function theory gives a possibility for the determination of the Z|3 term. This theory is aa alternative approach to nonlinearities expanding the stopping power (the induced density fluctuation) in powers of the projectile charge. Unfortunately this expansion technique rapidly becomes unmanageable, even in piasmoa pole6 or random phase approximation (RPA) of the system7. In this work, as an intermediate approach, we investigate the efficiency of the second Bom approximation for simple Yukawa type potential within the framework of the scattering theory for the electron-gas model. For low velocity projectile an additive "reaormalizMion" of the screening parameter is employed to get an approximate self-consistency. In Sec.II we present the derivation of the stopping power within, a second order Born approximation. Our numerical results for proton and aatiprotou ate summarized in Sec.ni . Finally Sec.IV, is devoted to the II. SCATTERING DESCRIPTION IN SECOND BORN APPROXIMATION The energy loss per unit length dE/dR for an ion moving with constant velocity (v) in a homogeneous electron gas is given by* 114 an .... (2.1) and for snail [v « vF, Eq. (2.1)] and high [v » vF. Eq. (2.2)] velocity of the projectile, 2 1 3 respectively. Here n i« the density of the system. vF * (3 K a) ' the Fermi velocity and 0tr the momentum transfer (transport) cross section. Atomic units ate used throufhout this paper. The "remaining" problem is the determination of the screened scattering potential for which we calculate the phase shift factors. Toe density functional method solves this problem in a self-consistent way. In order to establish the improvement we retain the momentum-space representation for the scattering amplitude f(q). Thus our method is directly comparable to the dielectric (linear and quadratic) formalism. This latter makes use the RPA. Our simple Yukawa potential V(r) — h-t'at (2.3) represents the real-space equivalent of the RPA within its Thomas-Fermi limit. The transport cross section in momentum representation is given by 2 V, °w where q is the transferred impulse and vr • vF for v « vF and vr « v for v » vF, (see Eqs. (2.1) and (2.2)]. Fortunately, the second Bora expansion of the scattering amplitude for the Yukawa potential is a standard result9 42? 2 4 2 2 2 2 where A « a + 4vr a + vr q . The first term is the Fourier transform of the potential and it is the scattering amplitude in the first Bom approximation. Close to 115 this first Born limit the arctg (x) may be replaced by it* argument x and after simple integration in Eq. (2.4) we obtain (up to the Z\ 3 order) In this equation the corresponding expressions are a| 2 where g = ( a / 2 vr ) . It is easy to show thai in the g-*0 limit (vj-li Z?(l+^.) in 2V* is the asymptotic formula for the momentum transfer cross section in which the so-called Zj3 (Barkas) factor is B=li2? 2(t In 2VT tr v4 J v2 a In the high velocity (v » vF , vr • v) limit and using a « att./v we obtain for ike stopping power with Barkas correction (an approximation for 'a' is X/2, see Ref. 10) This corresponds to the harmonic oscillator (atomic model for the target) description, where the Zj3 term originates in polarization effects (Me the Introduction). Urns for a positive charge (Zj positive) the stopping is bigger than for a negative charge (Z| negative). 116 It is very inteicninf to investigate the relevant rcnonnalizatiea of a in the static limit. We know that in this limit the Friedel. SUB rale is a strong self-consistency *1 requirement for the regular scattering potential. In order to obtain (at least approximate) self-consistency we calculate the a valve in Bom approximation by using the *2 (2.9) Friedel sum rule. We suppose that the phase shifts are small. Then the expression for the forward scattering amplitude is given by (2.10) VF for elastic scattering of an electron with impulse vF on the screened potential. Note that 8*0 scattering angle corresponds to q»0 impulse transfer. Comparing these last two expressions we obtain for the sum rule in Born approximation. In the appropriate (close to the first Bon) limit, after sobthuting [ from Eq. (25)] the relevant formula for the forward scattering amplitude, we fieri I/2 for the determination of a. In this equation OTF * ( *vF/it ) is the (approximate) RPA result. The solution of Eq. (2.11) is not difficult. A simple relation (2.12) 117 fives a very good approximation even for vF«l. Note that this expression exact in ike vF -» •• ; high density limit lor the Yukawa potential. In the following section our numerical results for proton aad aatipiwoa are summarized. HI. RESULTS FOR PROTON AND ANTIPROTON First, we sbow in Fig. 2 the screening parameter (a) as a fuactkw of the density parameter r, • 1.92 / vF. The first Bora value (Thomas-Fcrni expiesskM) is denoted by aTF(3). For proton and aatiprotoa the second Bon results (From Eq. 2nd 2nd (2.11)] are ap (2) and alp (5). respectively. For completeness this figure includes the "exact" values of a for these panicles; obtained by partial wave analysis and numerical integration requiring that the phase shifts satisfy Eq.(2.9). (These Fr Fr results are denoted by ap (l) and alp (4)] For the practically interesting density range (1.5 £ rs £ 2.5) our momentum-space calculation gives fairly good results for the screening parameters. Our numerical results for the stopping power in the static limit (dE/dR * vQ) are summarized in Fig.2. The curve 'a' is the first Bom result with aTF. (See Bos. (2.1) and (2.6) with XQ (g)]. For proton and antiprotoa there are two-two curves in secoad Bom approximation. Curve b\ is obtained with unmodified a * Ctjf valve; while bj 2n 2 otJ -il£ 0 +1) sin [ 6\ (vp) - SM (Vp)] (3.1) VF Fr Fr and numerical phase shifts (with ap and a«p )• The deviations from curve V are notable for the density range of Funhennoie, for the case of proton the analytical (02) aad pure — erica! (03) results are in surprisingly good agreement. The carves b\ and b% differ nurfcedly. For the case of antiproton there is a big deviation from curve C3 at awtallic ihmitifi 2a- Fr of the electron gas. (Remember the parameters aap and aip were in better agreement!). The curves cj aad C2 remain very close. These latter curves "break down" at about r, « 2.3. This is uaphyskat. The poteatial is mote aad more "Coulombic", therefore a simple secoad Bon approximation for the scattering amplitude is not adequate in the case of the aaUanwoa. Now. however, that for r, « 23 118 the screening parameter of the proton potential becomes smaller tkan the well-known critical vaiue Ocr « 1.19. In principle (because of the appearance of a possible bound state) a penurhative approach is not valid for higher r( values. Finally, the curves 03 and c3 are nearly symmetric with respect to the 'a' curve. The last section is devoted to the comments. IV. COMMENTS In this paper we have investigated the efficiency of the second Bom approximation for simple Yukawa potentials to distinguish the proton and antiproton stopping power. We have found that the validity and applicability of this approximation is bounded. Only for very high velocity of the projectile or very high density of the system (in the static limit) gives the perturbative method meaningful results (g->0 in Eq.2.6). In the low velocity range the nearly symmetric deviations for the proton and aatiproton (curves 03 and C3) with respect to the first Bon result (curve "a") ace somehow "extremely regular". In the light of the above a recent result for the nearly symmetric and big («40%) deviations close to the stopping maximum (typical velocity is 2-3 a.u.) needs further justification. Without physical constraint between the first order (Zf2) aid higher order (Z\*) terms, the results and statements are only acadsnuc for intermediate and smaller velocities. At this state of the theoretical description we can say that the deviations from the "onodox" result* are and might be notable. The range of validity of the lymmnrir deviations must be restricted. The choice of any model potential requires a special care in the case of the autiparticle. because of the "growing'' Coulomhk name of the scattering potential. With decreasing density the source of this is the eflickat charge density depletion, at the static limit. The detailed self-consistent investigation for antiproton. using density functional method, is in progress.14 119 ACKNOWLEDGEMENTS The authors are thankful for many useful discussions to Prof. Eugene Dr. Andres Amau. The author* gratefully acknowledge help and support by Eusko Jauriaritza, Guipuzkoako Fora Aldundia and UPV/EHU. One of us (P.M.E.) would like to Iberduero S.A. for its help and support. 120 REFERENCES 1 HA. Bethe. Ann. Phy«. 5, 325 (1930). 2 J. Lindhanl, K. Da. Vidensk. Mat. Fys. Medd. 28. so. S (1954). 3 W.H. Bttfcas. N.J. Dyer, and H.H. Hecks***. Pfcys. Rev. Lett. 11, 26 (19*3); LA Andenen et aL. ibid «. 1731 (19t9). 4 J.C. Ashley, R.H. Ritchie, and W. Bnadt. pays. Rev. MS. 2393 (1972): JJ5. Jaefcwa mi ILL. McCinhy, ibid M, 4131 (1972). Ecbcniquc, R.M. Nieminea, J.C Ashley, and RJL Ritchie. Fhys. Rev. A33, S97 (1996). J.C Ashley, A. Giw-Marti, and P.M. Ecbeaiqoe; ibid A34,2495 (1986); A. Anito, PAt Lchrmpc, and RJH. Ritchie, Nucl. Lutnm. Methods B33, 138 (19M); L Nagy, A. Anua, aad PJf. Echenique, Fhys. Rev. B3t. 9191 (198t). 6 C.C. Sung Mad RJL Ritchie, Fhys. Rev. A2S. 674 (1983). 7 CD. Hu and E. Zaremba, Fhys. Rev. 137,926S (198S). 8 P. Sifmond, Phys. Rev. AU, 2497 (19S2); I. Nagy, A. Anwu. and P-M. FrbfiqT. ibid A4f, 987 (1989); L. Booij and K. .Srhwthnmmrr. ibid B39.7413 (1989). 9 R.G. Newton, Scanerint Theocy of Waves and Panicles (Sprinfer, New Yoric 1912) pp. 291-292. 10 j.LindhantNucLInstnun. Methods 132,1(1976). " C Mael, Quantum Thewy of Solids (Wiley, New Y«k, 1987) pJ43. 12 j. Fricdd, Philos. Mat> 43.153 (1952). ^^ H. Fibffnicn and P. Sisjiimiid, Ann. Phys. (to be pwbtished). 14LNaiy,A.Antaii,P.M.Ecfaenk|tte,andE.2^tipba.Phys.Rev.B(iobejirtiliifc«i0 121 For the notations: see the text 122 For the notations: see the text 123 For the notations: see the text 124 For the notations: see the text 125 Variational approach to the scattering theory and its application in stopping phenomena * B. Apagyi and I. Nagy Quantum Theory Group, Institute of Physics, Technical University of Budapest, 1581 Budapest, Hungary Abstract. The spurious singularities arising in the Kohn and Schwinger variational methods (applicable, e.g., to the calculation of quantum reactive scattering of atomic systems) can be avoided by using a particular least-squares procedure referred to as Ladanyi variational method. With the aid of the latter method, phase shifts of very high accuracy can be computed for low energy electron scattering by impurities. Some simple one- and two-parameter models are presented for proton and anti-proton stopping by an electron gas. The models fulfill K&to'E cusp condition and/or Friedel's sum rule. " Invited paper presented at the 12th Werner Brandt Workshop on the Penetration of Charged Particles in Matter held in San Sebastian, 4-7 September 1989. 126 1. Introduction In recent decade, considerable interest has been devoted to the solution of quantum mechanical scattering problems by variational methods. This in- terest is motivated by a number of important investigations such as, e.g., the establishment of reference-standard cross sections for calibration of experi- mental electron scattering apparatus |l-2j, calculation of vibrational popu- lation inversion in chemical reaction J3j forming the basis of chemical lasers, or computing electron-molecule collisions {4,0} and photoionixation processes Compared to this activity, in atomic physics,.application of variational methods in condensed matter (solid state) physics is relatively rare. It can be explained by the fact that the interaction used in solution of problems of solid state physics is seldom non-local and channel dependent. An advantage of variational methods over direct numerical integration techniques is the easier handling complicated interactions. For example, the dimension of scattering equations to be solved by variational method remains exactly the same if one replaces local potentials by non-local ones. By use of direct integration technique, however, the dimension becomes squared and this can tax even the largest computer in case of calculating complicated multichannel reaction processes. Since variational methods treat scattering wave functions on the same footing as quantum chemical calculations do bound states, another advantage of variational methods may be to facilitate the use of standard target functions determined earlier by quantum chemistry. 127 In scattering theory there are two baiic types of variational functional* providing stationary expressions for scattering K-matrix. One is the Kohn variational functional ]7] associated with scattering state Schrodinger equa- tion plus boundary condition. The other is the Schwinger variational prin- ciple |8] which is based on the Lippmann-Schwinger equation. Since, in practice, one always deals with finite set of basis functions, both methods suffer from anomalies, the so called spurious singularities [9-lOj. Because of these spurious singularities, neither of these functional becomes a maximum or minimum principle. Therefore, the stationary principle may be replaced by a stability requirement [11]. It means that simple nonvariational methods may also yield stable results in a fairly large region of the nonlinear scale parameter characterising the basis functions. Concerning detailed analysis of variational schemes in scattering theory, the reader is referred to the lit- erature [12]. In the next section, a particularly simple expansion method, the Lada- nyi least-squares variational method will be outlined along the lines discussed in 113). In section 3 the Ladinyi method will be applied to some simple scattering problems including scattering of electrons by local and non-local potentials. Calculation of stopping power of an electron gas for slow pro- ton and anti-proton is also presented here for models fulfilling Kato's cup conditions and/or friedel sum rule. Section 4 contains a short summary. 126 2. Ladanyi varational method The simplest (and most practical) form of Ladanyi functional reads for one channel scattering as where / denotes the trial scattering radial wave function which is expanded in terms of basis functions as f{r) = £ Oi « For an 5 — wave scattering the basis consists of the following two continuum functions (3) (4) and of N discrete (square-integrabie) functions which can be chosen, e.g, to be ^•si^e-"- for * = l,2t..,tf (5) with a and fi being nonlinear scale-parameters and Jt denoting the wave mimb«r which is r«lat«d to tht scattering energy in atomic units by E = 1^/2. The functional (1) minimises the deviation (6) within a test-function space {XH} involving only square-integrabie functions which are chosen for the present application to be h r Xh{r) = Bhr e~f for h = 1,2,..,A# > N +2 (7) 129 where Bh is a normalisation factor and 7 is a scaling parameter In Ladanyi functional (1) Wk* s are positive weighting factor* and tl» normalization is necessary to exclude the trivial (/ = 0) solution. The variation of (1) with respect to the coefficients a_i leads to a simple eigenvalue problem which can be converted, by renormal- ising the coefficients to have a-i = 1, into a system of linear mhomogeneous equations (the Ladanyi equations) N Li3aj = -I,,_i, 1: = 0,..., N, (8) and a simple expression for the eigenvalue A (referred to as the measure of ike error of the approximate solution) (9) where the least-squares matrix is defined as follows (10) JC^i with the matrix elements Bhi = p dr Xk(r)(H - E)^{r). (11) The tangent of the approximate phase shift of Ladanyi method is obtained after the solution of (8) as K[N) = 00. (12) In the calculation we will employ the following simplifications in the ap- plications of Ladanyi method: «% = 1 (h = 1,...,M), fi = 7 = o, and M = AT + 10. 130 Filially, it should be pointed out that Ladinyi method avoids the Kohn- type anomalies. That is because of the appearance of the square of the Hamiltonian matrix in equation (8). 3. Application of Lmdinji varsatiousd method 3.1 Scattering of an electron by nonlocal potential. Let us consider the s- wave (I = 0} scattering of an electros by the one-channel nonlocal potential (13) representing the s-wave interaction between an electron (projectile) and a hydrogen atom (target) in the static exchange approximation. The local part of the potential reads and the nonlocal part representing exchange of the two electrons can be written in the form / M («) where the nonlocal kernel is given by wiry) = 4(-lfe-'{[l + *a)/2 - l/r^t^' (16) with 5 = 0 for singlet scattering process as, and 5 = 1 for triplet scattering states. 131 In Table 1 the tangents of the triplet scattering phase shifts due to potential (13) are listed for various approximations containing N basis func- tions together with the eigenvalue A indicating the measure of the error of the approximate solution. The remarkable convergence and stability property of Ladlnyi method as well as the non-appearance of spurious singularities within this method (13] make it reasonable that this procedure may be used as a practical computational tool in solution of scattering problems. S.2 Stopping power of an electron gas for slow proton and anti-proton. Let us consider a jeilium as stopping medium with no = Q.75/(jrrJ) electrons per unit volume, and take a proton (Z = I) or anti-proton [Z = -1) inter- acting with the electrons via pure elastic collisions. If the velocity v of the intruder entering the jeilium is much smaller than the Fermi velocity (Jbp in atomic units) of the electrons, the stopping power can be written as vQ(kp) where the transport collision frequency Q is computed from the formula Q=^ where r\i is the /th partial-wave phase shift of the electrons scattered by the screened potential V of the intruder. In the remaining part of this subsection we consider two models for slow proton/aati-protou stopping in jellium. Both models fulfill Friedel sum rule 132 and the second one satisfies also Kato's cusp condition n'(r) = -tZ (19) n{r) r=0 with n(r) = no + 6n(r) being the total density around the intruders and 6n(r) denoting the induced density. Both models automatically fulfill the total screening condition = Z. (20) In both cases one assumes the Poisson equation to be valid between induced density and scattering potential AK(r) = -4*6n(r). (21) 3.&.1 One-parameter model for stopping of $low proton mnd anti-proton kg jellium. Let us consider the Yukawa potential (ft > 0) V{r) = z£^- (22) as a crude model of the potential induced by a proton/anti-proton entering the jellium with slow velocity. Although such a simple Yukawa potential model leading to infinite den* sities at r = 0 is obviously unrealistic, it is interesting to compare Q values obtained for proton and anti-proton stopping at various densities r,. The requirement that Friedel sum rule (18) be satisfied can be made by adjusting the free parameter ft at every densities. Such an adjustment has already 133 been done by Ferrell and Ritchie (14] for a proton (Z = 1), therefore the calculation will be extended only to the case of an anti-proton {Z = — 1). The results are contained in Table 2. The marked difference between the Q values for proton and anti-proton can be interpreted as a manifestation of the well known Barkas-effect in this simple model. 3.2.2 Two-parameter model for stopping of slow proton by jellium. Following ref. [15], consider the ansatz 2 6n[r) = V0(p, v; r,)^-*" + (1 - Vo^r.))^" " (23) with the obvious conditions fi^v and ft,u>0. This ansatz automatically fulfills the total screening condition (20). By invoking cusp condition (19), we get an explicit expression for the "strength" Vo as ,r,) (24) 3 with a = fi {fi -1) and b = J/3(I/ -1). The effective electron potential is derived by solving Poisson equation (21) with the above charge density. The result r V{r) = -Vue-*» {p. + \) - (1 - l'0)e-*"{* + \) (25) is the superposition of exponential and Yukawa potentials with different "range" parameters p aud v. 134 Fixing v at some reasonable value (e.g., v - 0.3), one can adjust the parameter p so that equation (18) is fulfilled. In this way one obtains a two parameter description of proton stopping in jellium which obeys the exact rules (18)-(21). Therefore it is reasonable to compare the values of 6n(0) and Q with those calculated by more exact density functional methods. This comparison is presented in Table 3. One can conclude that the results are satisfactory in the light of the simplicity of the calculation. 4. Summary Application of variational methods is scattering theory is of great importance from practical point of view. They keep the calculation at a manageable level and facilitate the use of standard target wave functions from quantum chemistry. A particularly simple expansion method, the Ladinyi method has been applied to solve some simple scattering problems including stopping of slow protons and anti-protons by homogeneous electron gas of various densities. The impurity problem has been solved within several models which fulfill Kato's cusp condition and/or friedel sum rule. 135 Referencei 11] R. K. Nesbet, Phys. Rev. 20, 58 (1979) [ 2] R. K. Nesbet, Variational Methods in Electron-Atom Scattering Theory (Plenum, New York, 1980} [ 3] Y. C. Zhang, J. Z. H. Zhang, D. J. Kouri, K. Haug, D. W. Schwenke, and D. G. Truhlar, Phys. Rev. Lettere 80, 2367 (1988) [ 4] M. A. P. Lima, K. Watari, and V. McKoy Phys. Rev. A 30,4312 (1989) [ 5] C. W. McCurdy and T. N. Resdgno, Phys. Rev. A 40, 1297 (1989) [ 6} R. R. Lucchese, K. T^katsuka, and V. McKoy, Phys. Rep. 131, 147 (1986) | 7} W. Kohn, Phys. Rev. 74, 1763 (1948) ( 8] J. Schwinger, Phys. Rev. 72, 742 (1947) { 9] C. Schwartz, Ann. Phys. NY 16, 36 (1961) [10] B. Apagyi, P. Levay, and K. Ladinyi, Phys. Rev. A 37, 4577 (1988) [11] K. Ladanyi, P. Levay, and B. Apagyi, Phys. Rev. A 38, 3365 (1988) [12] M. A. Abdel-Raouf, Phys. Rep. 108,1 (1984) [13] P. Levay and B. Apagyi, J. Phys. B 21, 3741 (1988) [14] T. L. Fen-ell and R. fl. Ritchie, Phys. Rev. B18,115 (1977) [15] B. Apagyi and I. Nagy, J. Phys. C 21,3845 (1988) [16] C. 0. Almbladh, U. von Barth, Z. D. Popovk and M. J, Stott, Phys. Rev. B 14 2250, (1976) [17] P. Jena and K. S. Siugwi, Phys. Rev. B 17, 3518 (1978) 118] J. Gondzik aud H. Stacuowiak, J. Phys. C 18, 5399 (1985) 119] M. J. Puska and R. M. Nieminen, Phys. Rev. B 27,6121 (1983) 136 TABLE 1. Triplet reactance matrix elements K — tanijo and the eigenvalues A com- puted by Ladanyi methods at several basis-set sixes JV for the nonlocal potential (13). The wave number k is fixed at 0.5 a.u. and two values of the nonlinear scale parameter a have been chosen to exhibit the stability of the method. a = i.O a = 1.5 A* K{N) X(N) K{N) A(JV) 2 2.07761368 7.4 x 10~5 2.07028445 1.1 x 10~* 4 2.06977848 2.6 x 10~7 2.07005340 1.6 x 10~10 6 2.07011720 2.2 x 10"10 2.07006643 8.5 x 10-w 8 2.07006473 2.3 x 10"l4 2.07006656 1.6 x 10"ir 10 2.07006721 5.7 x 10-" 2.07006665 1.4 x 10-" 12 2.07006646 1.3 x 10"11 2.07006663 2.1 x 10-ai 14 2.07006663 8.4 x 10~31 2.07006663 6.9 x 10-" 137 TABLE 2. Calculated values of Q for Yukawa potential (22) with parameter fi adjusted so that Priedel sum rule (18) be satisfied within an accuracy of one percent. 2 = 1 means proton, Z = -1 stands for aiiti-proton. Z=-l Z=l r, p Q p Q 0.5 0.998 0.328 1.03 0.435 1 0.718 0.173 0.800 0.309 1.5 0.585 0.107 0.688 0.248 2 0.496 0.074 0.620 0.205 3 0.390 0.042 0.566 0.136 4 0.325 0.027 0.550 0.089 5 0.280 0.019 0.550 0.061 6 0.248 0.014 0.554 0.043 10 0.175 0.006 0.580 0.016 15 0.133 0.003 0.585 0.007 138 TABLB3. potential (2$). p kinds* OJiPhafbrn adjwtod to that «qa»- tion (IS) hokk Tbe vaiset pndictod by tdf-couwtent cakuia- tioo» am lilted lor computes. 1 1.331 0.722 0.327 OJ03 — — — 1.S 1.155 0.4S9 0.264 — — — 0.301 2 1.081 0^70 0J07 0.492 — 0.461 0.269 3 1.020 0^21 0.127 OJ04 OJK 0^72 0.163 6 1.006 0.298 0.041 0J34 — 0JI9 — *AhnbUdh * JtM ud Sincwi (17) e Gcadnk and Stadwmk 139 •To*»iMa tons jua> BSJMU x* io» muu J. Bausells Trabajo presentado «n «1 "XII W«rn«r Brandt workshop on th* Penetration of Charged Particles in Matter", San Sebastian* 4-7 Septieabre 19<9. 140 STOPPING POWER AND RANGES IN ION IMPLANTATION J. Bausells Centro Nacional de Hicroelectrbnica (CSIC) Universitat Autdnoma de Barcelona, 08193 Bellaterra, Spain 1. Introduction Ion implantation is the introduction of atoms into a solid substrate by bombardment of the solid with ions. It has important applications in the fields of materials science and microelectro- nics, where it is a fundamental technique for semiconductor device fabrication. In most cases succesful use of ion implantation requires an accurate knowledge of the implanted ion range distri- bution inside the target. The ion range distribution is obtained from the stopping of many ions that follow different trajectories inside the solid, and is therefore a process of statistical nature. Two different sta- tistical approaches are widely used to calculate ion range distri- butions: the Boltzmann transport-equation1'' and the Monte Carlo method*-'. In both cases the ion stopping power inside the solid is needed to calculate the range distribution. The ions lose energy by elastic ion-ion interactions and by inelastic electronic stop- ping. The elastic ion-ion interaction is a well-known classical two-body collision where the main uncertainty comes from the calculation of the interatomic potentials. Statistical models of the atom have been widely used in calculating elastic stopping powers. These include the Thomas-Fermi', Moliere*, Lenz-Jensen* and Bohr1' potentials. Recently, improved interatomic potentials have been obtained"'" by using Hartree-Fock atomic distributions. Inelastic stopping involves complicated processes of electron capture into (and loss from) the bound states of the moving ion. These processes determine the charge state and effective stopping- power charge of the ion as it moves through the solid. At high ion velocities, v » Ztv, (where Zl is the ion atomic number and v, the 141 Bohr velocity), the ion is stripped of its electronic charge. In this case the well-known Bethe-Bloch" linear theory for the stop- ping power can be used. At very low ion velocities, v ^ v,, the ion is dressed with quasi particles. In this velocity region large nonlinear effects are important, and the inelastic stopping power can be accurately calculated within the density functional forma- lism""". The stopping power is in this case proportional to the ion velocity. At intermediate velocities (of the order of the average electronic orbital velocity in the Thomas-Fermi atom, v * Z"*v.) an equilibrium ion charge state is reached and the stopping power calculation becomes more difficult. Ion implantation usually involves relatively light ions (8,11, o, p, si, Ar, Ge, As,...). For many years, standard ion implanta- tion applications have covered the energy region from 5 to 200 KcV. At 200 KeV, the lightest (and therefore, the fastest) ion considered, B, has a velocity of v-0.857 v., and therefore all possible ion implantation calculations are restricted to the low energy region, where the inelastic stopping power can be accura- tely calculated. Good agreement can be obtained in this case between experiment and theory. In particular, the TRIM Monte-Carlo code'-", which uses the universal Hartree-Fock interatomic poten- tial of ziegler et al." and a semi-empirical low-velocity electro- nic stopping power that agrees with density functional calcula- tions, provides quite realistic predictions, with an overall accuracy of a few percent for the range of ions in solids. In recent years, however, "high energy" ion implantation (usually up to *4 MeV) has been heavily investigated and is expec- ted to play an important role in semiconductor applicationsu>>*. At 4 MeV, boron, phosphorus and arsenic ions, for example, move with velocities of 3.83v(, 2.28v, and 1.47v,, respectively. Thus most ions used in ion implantation (and specially the lightest ones) are well inside the intermediate velocity region, where ion charge Jtate and stopping power calculations are not very well defined. As will be discussed in the next section, inelastic stopping power theories in the intermediate velocity region have been using average ion charge states to calculate ion effective charges. Recently, Sols and Floras" have obtained the distribution of charge states for light ions in the upper part of the intermediate 142 velocity region. As the implanted ion range distribution is a statistical result, the use of this distributions of charge states instead of average values in the calculation of stopping powers can lead to a different shape in the ion range distributions. In this work we study this effect in order to ascertain whether the use of charge state distributions can add new features to ion range distributions in the MeV-energy range or whether its effect is snail and need not to be considered. 2. Charge states and stopping powers at intermediate velocities The stopping power Sn of an ion of atomic number Z, can be related to the proton stopping power S, through the concept of the ion stopping power effective charge1* Z' jr (l) where S,, and S, are determined in the same material and at the same velocity. The fractional effective charge % of the ion is defined as z: -lzt (2) As proton stopping powers are well known"'", knowledge of the effective charge of an ion allows the calculation of its stopping power. The effective charge of an ion moving in a solid depends on the ion velocity, and is related to the charge state of the ion, i.e., to the occupation of the ion's bound states. If an ion of atomic number Zt moves in the medium carrying N electrons, then its ionic charge is Q«Zt-K, and the degree of ionization is Q M « - — - 1 " IT <3) Brandt and Kitagawa"-" calculated the relation between the fractional effective charge X and the charge state q of the ion. They assumed that the electronic charge bound to the ion extends over a radius A. Then the electrons in the medium at impact parameters greater than A encounter a fully screened ion of charge Q, but at smaller impact parameters they penetrate into the screening cloud and find an ionic charge greater than Q. Therefore one expects that 1 >q. within linear response theory, Brandt and 143 Kitagawa found" the result K - q + C(r.) (1-q) In 1 + where kr is the Fermi momentum of the solid, a, is the Bohr radius, and C(r.)» 0.5 is a weakly dependent function of r., the electron gas density parameter. From extensive data analysis it has been found" that a good approximation is C(r.) * \ J^-j The next step needed is the evaluation of the ion's degree of ionization (or charge state) q, which depends on the ion velocity and the target material. Bohr" suggested that the ion is stripped of all electrons with classical orbital velocities lower than the ion velocity. Kreussler, Varelas and Brandt** showed that at velo- cities of the order of v, this stripping criterion should be ap- plied to the relative velocity vr between the ion and the conduc- tion electrons of the medium. This gives, in the Thomas-Fermi model of the ion, v, which can be written in an explicit font using the result on the Bohr stripping criterion obtained by Northcliffe1* and substituting v by vr '' 1 - exp - „ , „, 1 (6) This expression agrees with Brandt's calculation" of q(vr), and fits remarkably well the experimental data for several thou- sand stopping powers11. At low ionization fractions (tj<0.3), howe- ver, there is a deviation between the theoretical expression and the experimental results. Ziegler et al." fitted an expression based on the font (6) to the experimental data, which gives M q - 1 - exp[.8O3 y/-» -1.3167 yr -.38157 yr -.0089*3 y/1 (7) whera yt« vr/(v. z,'"). He will call this expression the Brandt- Ziegler, Biersack and LittaarJc (B-ZBL) formalism to the charge stat«. The above theories provide average values for the ion charge 144 states . Recently, Sols and Floras" (SF) have calculated the cross sections and charge states associated with the electronic capture and loss of B, C, N and 0 ions Moving in aluainua with velocities Z.v, ^ v .5 3Z,v0, where the is orbital of the projectile ion con- trols the capture and loss processes. Their results for the dis- tribution of charge states for a B ion as a function of its velo- city are shown in fig. 1. 1.0 o.a o.s 0.2 0.0 it V/5VO Fig. 1. Distribution of charge states for B ions noving in Al, as a function of the reduced ion velocity. The solid lines are the results of Sols and Flores". The dashed lines are an extension of these results to the low-velocity region. The charge state of an ion greatly influences its stopping power, as can be seen in fig. 2, where we show the inelastic stepping cross sections calculated froa eqs. (1) and (4) for the six possible charge states of boron and froa the B-ZBL theory. At a given velocity, two different B ions can have different charge states according to the probability distribution shown in fig. 1, although on average they would have the charge state given by the B-ZBL theory. This aeans that individual ions will have, during their slowing down, stopping powers different from each other and froa average stopping given by eqs. (1), (4) and (7), and this can lead to a different result for the ion range distri- 145 bution. This is a kind of energy loss straggling, because we get unequal energy losses for ions that travel under identical condi- tions. 6OOOr Fig. 2. Inelastic stopping cross sections for boron ions in Al. Dashed lines: cross sections for B ions with charge Q. Solid line: ions with average charge stats given by eq. (7). As the B ion is the lightest (and therefore the fastest, at a given energy) of the ions usually employed in ion inplantation, this difference in range distributions will be, at a given energy, maximum for boron. Therefore we will concentrate our study on B ions. The difference between the two formulations can be seen by calculating the energy distribution of 25 HeV boron ions (v» 9.5S v.) after traversing 24 pm of Al. This gives an average final velocity of 4.86 v,, and thus covers the velocity rang* of the SF calculation, we have used the TRIM Monte-Carlo code", which in its standard fora follows the B-ZBL formulation, and a modified ver- sion to take into account the SF distribution of charge states in which, for each ion and each velocity, the charge state is calcu- lated according to this charge state probability distribution. Mote that this kind of calculation is implemented very easily in a Monte-carlo formalism, that follows individual ion trajectories 146 in the target. The results are shown in fig. 3. It can be seen that the energy distribution from the SF charge state theory is much broader than the B-ZBL theory distribution, due to the va- rious charge state possibilities that each ion has available in the former case. Also, the Bean final ion energy is lower in the SF formalism, due to the fact that the Q«5 charge state, which is the most probable over most of the energy range covered by the ions as they traverse the target, gives a stopping power greater than the B-ZBL formalism, as can be seen in fig. 2. It is therefo- re to be expected that there will be differences in the final ion range distributions, when using the actual distribution of charge states of B instead of the B-ZBL formalism. 10000 r 1000 in O E/EO (*) Fig. 3. Energy distribution of B ions of initial energy Eo- 25 MeV after traversing 24 ft of Al. Solid line: ion charge state given by the B-ZBL average formulation. Dashed line: ion charge states given by the Sols and Flores" distribution. In order to make an estimation of the importance of this effect in ion range distribution calculations, we need to know the distribution of charge states from v-0 to v» 5v,, where the SF calculation starts. If ft(v) is the probability of the Q*i charge state am a function of the ion velocity, we know that 5 Z f,(V) - 1 (•) i-0 147 on the other hand, the average charge state at a given velo- city v must agree with the empirical result q^Jv) given by (7) 2 £ fAv) - g».(v) (9) i-l D Indeed, the SF probabilities verify eq. (9) in their velocity range of validity, between Sv, and 15v,. Taking into account these two conditions, we have extended the results of SF down to v»o with the results shown in fig. l. He do not intend that this is the true charge state distribution for boron. Our purpose is to obtain a reasonable charge state distri- bution in order to make an estimation of the differences in ion range distributions that can be obtained when using it instead of an average value for the charge state. It is clear that (8) and (9) alone do not completely determine the form of the six ft(v). But we have to note that f, and f« are almost completely determined by the SF results and, if we assume that f, (i>l) do not extend to v-o, then f, is fully determined by (8) when fx is made to agree with (9) near v=0. This leaves ft, f, and f, to be determined. But (8) and (9) are not easily fulfilled simultaneously, so that we believe that, although our solution is not the only possible, it should not be very different from the actual distribution of charge states for B ions. 3. Results We have calculated, using the TRIM Monte-Carlo code, the ion range distributions of boron ions implanted at 4 MeV (v» 3.83vJ in aluminum at normal incidence, using the B-ZBL averaged charge states and the extended SF charge state distributions discussed above. 4 MeV is the maximum energy of standard high-energy implan- tations, and therefore this calculation should give an upper limit to the differences that can be expected between the two formalisms in actual implantations. The results are shown in fig. 4. The maximum of the ion range distribution moves from about 4.8 jn (B- 2BL) to about S.O jm (SF), which is a noticeable 4.2 % shift. Ibis fact can be understood in terms of the ion stopping powers shown in fig. 2 and the ion charge state distributions shown in fig. 1. The mean total range of the ions can be estimated from 148 (10) and it is easy to verify that an ion has a greater probability of going deeper (than when using average charge states) than of going shallower. Also, the SF ion range distribution is less skew and peaked (skewness—2.3, kurtosis- 18.4) than the B-ZBL one (skew- ncss*-l2.7, kurtosis- 230). 100000 30000 40000 50000 Depth (A) Fig. 4. Range distributions of boron ions implanted at 4 MeV in Al. Solid line: ion charge states given by the B- ZBL average formulation. Dots: ion charge states given by the extended Sols and Flores charge state distribu- tion. There is no available experimental data on MeV boron implan- tations in aluminum, but there are several results for boron in silicon3*"*. In order to verify whether the effect we are discussing gives a better agreement between theoretical calculations and experiments, we have calculated the ion range distribution of B ions implanted at 2 MeV in Si. Me have used the same extended SF charge state distributions as in Al, which should be « good appro- ximation because the electron gas parmeters of both materials at* very similar, and thus their electron stripping properties from the moving ion should also be similar. 11M results are shown in 149 fig. 5 and table 1. Zt can be seen in fig. 5 that the extended SF range distribution gives a better agreement with the experiaental SIMS data of Ingram et al.M than the B-ZBL range distribution. The use of more accurate values for the charge state distribution of B ions in Si would probably give an even better agreement with the experimental data. There is a disagreement in the deeper part of the distribution due to channeling effects which are not included in our calculation. The results of table 1 for the momenta of the range distribution show that its overall shape is in much better agreement with the data of Hong et al." in the SF calculation than using the averaged B-ZBL formalism. 1000-d zbcioo 22600 ' I _ Depth (A) Fig. s. Range distributions of B ions implanted at 2 KeV in Si. Solid line: ion charge states from the B-ZBL average formulation. Dashed line: ion charge states given by the extended SF charge state distribution. Dots: experimental data". 4. Summary and Conclusions Ion implantation in the MeV-energy range is expected to play an increasing role in semiconductor applications. This energy region involves, for light ions such as B, the intermediate ion velocity range, where capture and loss charge exchange processes between the ion and the medium determine the ion charge state and stopping power. Presently there are no reliable theories to calcu- 150 Table 1. The first four momenta of the rang* distribu- tion of 2 MeV B ions implanted in Si: Mean projected range (Rp), range straggling ( ARp), skewness and Jcurto- sis. The two theories discussed in the text are compared with experinental results. exper." SF B-ZBL Rr (A) 29720 27872 27092 AH, (A) 2060 1706 2383 Skew. I -1.94 -2.68 -11.24 Kurt. 6 15.8 18.8 160 late ion range distributions in this energy range1*. We have calculated range distributions for MeV boron ions implanted in Al and Si. We have used a standard average approxima- tion to the ion charge state as a function of its velocity, and compared this result with a calculation involving an approximation to the actual B ion charge state distribution. This distribution has been obtained from Sols and Floras" for v>5 v, and has been extended to lower velocities in such a way that the average ion charge state at each velocity agrees with the semi-empirical average charge state of Ziegler et al." We expected that, by allowing different ions to have different charge states (and therefore different stopping powers) at the same velocity, the use of charge state distributions would modify the shape of the final ion range distribution, in comparison with that obtained with the standard approach. The results of our calculations show that the differences between the ion range distributions obtained with the two formula- tions are relevant for B at 4 M«*V. Furthermore, by using charge state distributions we get a better agreement with available experimental range distributions for B in Si at 2 MeV than when we use th« standard average ion charge states. We conclude that first-principles calculations of charge states for ions moving in solids between v, and *Zv« can be very helpful to the understanding of implanted ion range distributions in the MeV-energy range, a field that is becoming increasingly important in semiconductor technological applications. References 1. J.Lindbird, H.SdJirff and l.E.Schiott, K.Du.VidMsk.Stlks.Mat.rys.Mi. 33, lo.11 (1963). 2. K.B.HisUrbot, Cu.J.fbyi. 46, 2479 (1961). 3. O.littnark ud J.F.Ziogltr, Phys.lw. 123, 64 (MM). 4. J.F.CibbOM, Hucl.Instr. ud fcta. B21, 13 (1917). 5. X.T.lobiMM and IXftntM, Phys.lM. M, 5001 (1974). «. J.P.Bitruck ud L.G.IMpitk, MucLIostr. ud Httk. 174, 257 (19M). 7. l.So«erftld, Z.Physik 71, 213 (1932). I. C.ltolitrt, I.ltatutforscb. 12, 133 (1947). 9. tf.Unz, Z.Physik 77, 713 (1932); LJeasea, Z.Pbysik 77, 722 (1932). 10.1.Bobr, K.Du.VidtMk.Stlks.lat.Fys.ltodd. II, KO.I (1941). II. S.D.Bilson, L.S.lMptck ud J.P.Bitrsack, ftys.m. BIS, 2451 (1977). 12. J.F.ttigltr, J.P.Bitnack ud O.Littiirt, "Ttit Stoppia; ud tuqt of Km u Solids'. Vol. 1 of lit Stopping ud tups of ions in Kitttr", Ed.J.r.Zioql«r, ftrqm* Ms, tm York (1915). 13. I.l.Bttbt, Z.Pbyiik 7C, 293 (1932); F.Bloch, ino.Paysik 16, 217 (1933). 14. P.R.Edmiqut, E.H.»i«tiaM ud IXtitchit, Sol.St.CMM. 37, 779 (1911). 15. P.M.Ecbttiqut, l.X.Vieuwi J.C.lsUty ud I.l.Iitdiit, tbft.Uf. 133,197 (19M). IS. V.Biibtru ud P.H.Ecbttiqut, J.Phyi. B19, Lll (19M). 17. P.I.EcbMiqut, Ihiel.Iattr. ud Btth. B27, 2SS (1917). II. J.P.Bitrsack, fucl.Instr. ud Nttb. B3S, 205 (1911). 19. P.I.Siapr, Seucond.lBttmt., I, *>.9,92 (Sept. 1917). 20. H.Brndt, in 'ltotic Collitieot ii Solids', Vol. 1. p.»l, Eds. S.Dtti, B.l.lpplttoB aid CO.IOik, Kmm, tm York (1975). 21. H.I.lndtrstt ud J.F.Zitqltr, "lydroon Stopping Powrs ud lugts in All Hants', 7ol. 3 of Tbt Stopping ud lugts of ioas in lUtttr', £d.JJ.Zi*gltr, PetoHW frtss, «w lock (19IS). 22. H.Brudt ud U.liUqm, Pnys.teT. B25, 5631 (1912). 23. H.Brudt, tucl.Instr. ud fcth. 194, 13 (1912). 24. l-.Bobr, Pbys.ltT. 51, 654 (1940); P&ys.Btr. 59, 270 (1941). 25. S.KrwHltr, C.Tanlas and ff.Brudt, Pnys.Btr. B23, 12 (1911). 26. L.C.IortncliK*, Pnys.Ier. 120,1744 {I960). 27. F.Sols and F.Flocw, Pbys.Brr. 137, 14(9 (1911). 21. D-CIngm, J.l.Baktr, D.l.italsb ud E.StratlmaB, lud.Instr. and fcth. S21, 460 (1917). 29. I.Bang, E.0ug, I.H.Cbtag, P.K.Cbu, t.X.Stratmm ud I.O.Stratbnu, lael.Iistr. ud HO. Btt, 447 (1917). 30.1.Tuun, I.totsnaki, T.aada and E.Iitani, lucl.Isstr. ud Ittk. B21, 131 (1917). 153 DYNAMICAL EFFECTS IN ELECTRON TUMNEUNO and L.MaruJic Department of Physics, University of Zagreb, POB 162, ;i001 Zagreb, Croatia, Yugoslavia Recent advances in materials science, particularly in high precision fabrication (molecular beam epitaxy) of heterostruetu- res and quanttn wells with up to atosdc •onolayer precision, have led to rapid development of aesoscopic physics - studies of phe- noaena on a subnanoaeter scale. This paper will treat one parti- cular aspect in the theory of electron tunnelinc in these structures: the role of djmaaical screening and potential* in very thin carriers (e.g. doped Al Ga As layers). The aodel is based on the exact treatment of surface/interface plaaajon excitations in layered structures and their nonlocal coupling to the electron current, and is solved in the W£B approxismtion. Analytic expression is obtained which contains the effects of screening dynamics, electron energy and decay, parallel velocity and recoil. Several cases are discussed, including self-consis- tent results and tunneling rates. 154 Recent experimental advances in the ••ainra—nt of timeline through well characterized barriers, and the observation /I/ of possible dynamical effects for tunneling tiaes coaparable to the plaaaon/screening tiaes have again emphasized the need for a theoretical description of effective barriers seen by the tunneling electrons, that would take into account the dynamics of the charge fluctuations and their coupling to the tunneling electrons. Very early, indications of deviations froa classical potential were observed experimentally, and several atteapts were made to find sesdeapirical corrections /2/. More recently it Mas realized that the origin of iaage potential is in the electron interaction with polarization aodea in the solid, in particular surface {or interface) plasaons /3,4/ which also led to the seaiclassical foraulation of the dynamical iaage potential for uniformly moving particles /5-8/. Application of this surface plasaon model to electron timeline Mas attempted by several authors /9-15/, for various experisantal situations and trying to explain different physical phenoasna, like currant oscillations in photoassisted field emission /I1,12/. Two papers /9.15/ are especially relevant for the present work; Jonson /*/ formulated the iaage potential in teraa of a non-local selfenergy which he approximately calculated for dispersionlcsa surface plasaons (SP) in a barrier outside the seaiinfinite aatal. In the first (nonselfconsistent) version his results contain soaa difficulties, probably arising froa the problaa of treating nonlocality in different approximations at various stages of the calculation, as we shall discuss later. Persson end Baratoff /15/ avoided these traps by choosing carefully their raraatfrs and calculating the tunneling rates directly, without discussing the iaage potentials and barrier shapes. This is probably sore correct physically, because the role of fluctuating potentiala in this situation cannot be fully represented by their (static) averages, but nevertheless we feel that it should be also useful 155 to calculate anl discuss the shape of the tunneling barriers. Therefore in this paper /W we set up • dynamical image potential aodel haeed on the electron coupling to two eurfaoe (or interface) plaamons, in the metal-insulator metal geoastrjr, appropriate to the GaAs-AlGaAe-aaA* systaa /I/, taking into account their dispersion and exact coupling functions. Ihe potential is calmlatort in the seoond order perturbation theory* and then eade aelfooosistant {e.g. bjr iteration). He systematically apply MB approximation, first bsonuae of its simplicity which enables us to evaluate analytically all results, and also in order to connect with the recant work on tunneling, especially related to the concept of tunneling (or dwell, or phase, or delay) tiaes /16.17/. The analytic results for the dynsnical potential barriers are presented in Sacs. II. and III., and discussed in Sees. IV. and V. in Sec. VI. we briefly coapare them to the nonlocal results of refs. /9/. In Sec. VII. we study the Modification of tunneling rates due to the dynamical effects, again in the MB approximation. The hamiltonian of the system H x H# + H>f> • The electron kinetic and potential energies are H9 s K • V Is) «> where v includes the tunneling barrier. The energy of surface/ /interface plammons of the mstml-liiiJmtnr ms*sl symtsm (Fie.II is 156 where i » (Q,p) Q is the parallel wavevactor and p::l is the SP mode parity (in a syaoetric configuration). Fie2A shows the frequencies of the syanetric ( where, inside the barrier ,'18,4,8/ The shape of the coupling functions T as shown in Fii.lb. As the problem is essentially nonlocal, we define an effective local dynamical potential as «yz,E> = J2k(z,z';E) «k(z')dz7«k where k is the parallel electron noMntua and «k is the solution of the nonlinear self consistent equation »a *' (7) The effective potential V«ff(z> = V(z) + VZtE) (8) contains the external potential barrier V(z) and the selfconsistent induced potential W. •* is the effective electron MUM, and E is its "perpendicular" energy. 157 The self energy in <6) is (Fig.2a) 1 Zk i where the total energy E» = E ••hV/a/ (11) includes the parallel kinetic energy, and In the WKB approxissition the Green's fraction of an electron v z with energy E iifn a slowly varying potential ,ff ( ' can calculated to be: * G(z,z';E) = e \ (121 / M{Z)MiZ') where * 1 « For a constant potential VQ this reduces to the version used by Jonson /9/. It is worth stressing again that this WKB Green's function becoaes less accurate near the turning points, i.e. near the edges of the potential barrier, and for electrons tunneling with energies close to the top of the barrier. However, as we shall be ultimately interested in the tvameling, i.e. electron attenuation, where the boundary regions give —11 ir contribution, we feel that WKB, as usual, provides a flood approximation. 158 III. Mow we shall aaltc two crucial stapa, again justified by the assuaptions underlying the HKB acthod. First, for slowly varying x(x) the exponent in (12) can be approxiaated by Now one could generalize the jcethod of Jonaon /9/ and study non- local potentials for the case of slowly varyinc potentials, becauae his constant potential aodel cannot be aid* self consistent. Instead, we shall here notice that the decay constant ^•j- fv(z) - E + *». + 4^(0)] >*ere iE^Q) = fi2 (Qa + 2k-Q)/2B*, corresponding to the Graen's function in the self energy (10), is large, and therefor* aakc a local approxiaation G(z,z';qV/2«*) = - -^-S(z-z') (15) q -» « this Mounts to averaging (6) over z' on a scale "q. In tens of ^i•glass this aeans that we have appradsated the diagrm (a) in Fig.3 by the diagram (b), by contracting z'=z, in the usual saaiclassical approxia»tion. Doing so, we have given up the possibility to treat the fluctuations in the tunneling current due to quantua aacfesaical interferences and siadlar |fiaiiiB»Ma which are outside the MB spproxiaation anyway. 159 Now the effective local potential (6) d* , e"2O-(ch2Q* + p) E) =-r-fiu* Y (cW f 2=- 2 P £ J J Z» where £ is given in (13) and (8). and :v(-«t ;• (17) Before proceeding with the discussion of self-consistent results, let us briefly discuss (16), which is the central result of our theory. (It can be generalized to any tunneling geometry and potential V(z) by changing the coupling functions I\ and SP frequencies « ). Hie character of the dynamical corrections is shown explicitly in the denominator in (16). the first tens defines the dynamics of the screening machaniaa. The sacoad tarn can be written in terms of the local energy dependent decay constant x(z), given by (13), and is thus related to the local electron decay. The third term AE^(Q) contains the influence of parallel electron motion (v, ) and the recoil term /5-8/. It is obvious that « here does not enter as the "timeline velocity": in this situation * plays a role of the decay To emphasize this point we should rgsmmfair that in the came of an electron moving freely with the normal velocity vx the corresponding term was va-Q * iQvx (IS) i.e. the normal velocity was also coupled to the SP Instead, * in (16) should be coneiderad as a spatial parameter, leading to the time decay oonatant T^ S 4k/c(z). In this way the dynaucs in (16) enters via the ratio n = T./T. of the two characteristic times, T and r , what* r^ depend! an the local decay of the tunneling electron. He specifically notice that thia n differs from the ratio T#/rt, usually tafcan /I.9,15/ to describe the 160 corrections to the image potential in tunneling. IV. Xonaelfconaistenfr result* / first iteration In the following we shall study the k = 0 case, which is simpler, but contains all essential physic*. In this cue the potential (16) depends on two ratios: n(z) = c(z\/fum and < = E,,/hu , and scales with e*/4a. Here a * ft3 E = —; (191 " 2m*(2m)2 is a characteristic energy of electrons in a barrier of width 2a. It is instructive to discuss the dynamical tmafc potential (16) by treating n and K as parameter*, and thus ignoring the fact that r?(z) has to be determined self consistently. This in fact corresponds to the first iteration of the potential (16), with c (z) = Vo-E. Fig.4 shows how the image potential reduces a constant tunneling barrier of height VQ. For n - 0 one recovers the classical result, modified at the interfaces by the introduction of a short wavelength cutoff which prevents divergencies /4/.With increasing n, i.e. the ratio of screening vs. decay times, the screening becomes less effective, the potential is reduced and the barrier become! less penetrable. The recoil tern £ is usually very saall, except for very thin barriers and low SP frequencies, and in particular saall effective Masses. In any case, Fig.4, shows that the dominant reduction coaes froa the dynamics, described by n. Fig.5 shows how the top of the barrier increases with increasing Q and. If. However, for low i? this is the case uhere it becosee essential to include selfconaistency. 161 V' Self-consistent results Obviously 77(z) is not a free parameter but chances inside the barrier, and should be treated selfconsistently. Already in the first iteration the electron builds up the image potential, so the energy E locally appears above the potential barrier, as the turning points tz move inwards from z - *a. In Fig,6 we show both the self consistent and first iteration solutions for the top of the barrier (2=0) as functions of e0 = = VQ- E, for two SP frequencies. In Figs. 7a and 7b we show a series of potentials for two SP frequencies and different electronenergies, evaluated selfcon- sistently. Selfconsistency always lowers the barrier in comparison with the nonselfconsistent results. Fig.4. Instead of making a local approximation (15). Jonson /9/ proceeded to calculate the self-energy (10), integrating over z" and using constant «. For a charge tunneling out of a metal into an insulator with the dielectric constant cQ he finds the modification of the classical image potential at distance z froa the surface in the fora V(z,E) = f- I dQ — <20> 0 « *• or, after integration: a f 2«'« V(E,E where the function g is 162 g(x) = x e~* Ei(x) (21) For larce x g(x) -* X, as expected, reducing to the classical result, but for finite x it shows two strange features. For 0 Neglect of surface sensitive corrections in Jonson's approach cannot be responsible for this. We have generalized the nonlocal theory to the tunneling in a symmetric M-I-M structure, and the same unphysical behaviour can be observed even in the middle of the barrier where the surface cannot influence the T T results, now as a function of v = «*/2a» *>t, which is the #/ t ratio. For n?-i the barrier is not lowered due to screening as one would expect, but increases above its VQ value. All these features do not agree with our intuitive picture of dynamical screening of tunneling electrons, and one suspect that some uncontrolled approximation produced these unphysical results. VII. Tunneling rates The shape of the potential barrier - its width and height, determines the conductance of the barrier, and this i*s one of the reasons for this study (as well as the justification for the use of VKB approximation). The tunneling rate in MJB is T(E) a e"2B B(E) = | J« where « is given by (13), and ±zQ arc the turning points. In the 163 absence of the screening the exponent in (22) would have the fora BQ(E) = 2 i/z where *>= i2m*lVg-E)/h ) . Both B(E), where the potential W in *(z) was evaluated self consistently! and B0(E) are shown in Fig.8 for the barrier width 2a = 200 aQ, and for two SP frequencies. Me also show the tunneling rate B (E) calculated for the classical (i.e. static) image potential. At this point we have to emphasize that WKB theory applies to the tunneling through a static barrier, and it would be easy to find arguments against this treatment for the case of a fluctuating potential. In Fig-2b we illustrate it by showing relative contributions of various Q components of the symmetric/antisymmetric surface plasaons to the iaage potential (albeit in the static case) for electrons at three different positions in the barrier. It can be seen that the dominant contributions to the barrier cose froa SP nodes which oscillate with frequencies often comparable or even larger than other characteristic frequencies in the problem. The use of the static barrier provides only an estimate of the dynamical effects in electron tunneling, and for further work one probably has to adopt a store sophisticated path integral/instanton approach, as in ref ./15/. On the other hand, in the latter approach one cannot study explicit fora of the iaage potentials in the barrier, as presented in this paper. VIII. Conclusions In conclusion, in this work we have calculated analytic expressions for the dynamical iaaae potentials for tunneling electrons in the MSB approxiaation, taking into account SP dispersion and coupling in a particular aetal-insulator-aetal geometry. We have shown how finite SP frequencies, wbsa comparable to other characteristic energies, prevent formation of 164 a full (static) img* potential, and thus scdify the barrier conductance. This work Has partially supported by the US-Yugoslav Joint Board on Scientific and Technological Cooperation. Grant PS 85KNIST). References 1. P.Gueret, E.Marclay, and H.Meier. Appl.Phys.Lett. 52, 1617 (1988); 2. J.G.SinBons, J.Appl.Fhys. 31, 2581 (1963); Z.A.MeinberS and A.Hartstein, Solid State Coaaun. 22, 179 (1976); A.Hartstein, Z.A.Keinber*. and D.J.OiHaria, Ftoys.Rev.B 25» 7174 (1982) 3. A.A.Lucas. Phys.Rev.g£, 2939 (1971) ;A.A.Lucas and M.Sunjw*. Phys.Letters 3S&, 413 (1972), J.Vac.Sci.Technol. 2* ?25 (1972) and Surface Sci. 3J. 439 (1972); R.H.Ritchie, Physics Letters 38A.. 189 (1972). 4. Z.Lenac and M.Sunjid, N.Ciawnto 23, 681 (1976) 5. M.Sunjid, G.Toulouse, and A.A.Lucas. Solid State Coaaun. U. 1629 (1972) 6. R.Ray and G.O.Hahan, Phys.Letters 42o, 301 (1972); see also G.D.Mahan, in Collective Properti,— of Hursical Systeas. Nobel Syaposiui 24. ed.by B.I.Lundqvist and S.Lundqyist (Acadeauc. New York. 1974), p.164. 7. P.M.Echenique, R.H.Ritchie, N.Barbetan. and John Inkaon. Phys.Rev. S22, 6486 {1981); J.R.Manson and R.H.Ritchie, Phys .Rev. BJi, 4867 (1981) 8. F.Sols and H.H.Ritchie, Solid State Cc—w. §2, 245 (1987) and Phys.Rev. Bj£, 9314 (1987) 9. M. Jonson. Solid St*te Cbwun. 22, 743 (1980) 10. Ashok Puri and M.L.Scteich, Phys.Rev. BZt, 1781 (!St3) 165 11. R.A.Your*. Solid State Consul. 4£, 263 (1983) 12. J.W.Wu and G.D.Mahan, Phys.fiev. Jgg, 4839 (1983) 13. M.C.Payne and J.C.Inkaon, Surface Sci. 152. 4C5 (1985) 14. P.M.Echenique, A.Gras-Marti, J.R.Manson and R.H.Ritchie, Phys.Bev. §22, 7357 (1987) !5. B.V.J.Persson and A.Baratoff, Phya.Rev. B2g, 9616 (1988) 16. M.BUttiker and R.Landauer, Phya.Rev.Lett. &, 1739 (1982); Phys.Scr. 12, 429 (198S); IBM J.Res.Develop. 22. 451 (1986). 17. E.H.Hauge, J.P.Falck and T.A.Fjeldly, Phys.Rev. 82fi, 4203 (1987); C.R.Leavena and G.C.Aers, Phya.Rev. §29, 1202 (1989) 18. M.Sunjitf and A.A.Lucas, Fhys.Rev. B2, 719 <1971) 19. See also M.Sunjid and L.Muruiiti, subaitted for publication 166 Fisury ctcticra Fig.l (a) Ceoaetry of the aetal-insulator-aetal systaa. (b) Coupling functions Tt (Q,z) for the syaaetric Fig.2 (a) Frequencies of syaattric <»t) and antisyaaetric («_ ) surface/interface plasaom of the H-I-M systaa. (b) Relative contributions of syaastric («. ) and antisia- eetric (w_) surface plasaons to the (static) iaage potential, for several positions of the electron in the barrier: z/a = 0, 0.5, 0.8 and 1. Fig. 3 (a) Second order nonlocal electron self-energy diagrea, and (b) its local {or seaiclasaical) approxiaation. Fig.4 Dynamical iaage potent .als in the barrier of width 2a = = 200 a0, in the first iteration. The SP energies are (a) fc>t= 0.1 By and (b> fiu^ = 0.01 By, a = 0.07 a, and •lactron energies are given on the figure. Fig. 5 Top of the tunneling barrier, in the first iteration, as a function of vo= eo/Kus, for several values of? (denoted on the figure). W is given in units of ea/4e. Fig.6 Top of the tunneling barrier - selfconsistent cal- culations. The barrier is 2a = 200 atf wide, a = 0.07a, and the surface plasson energies are (a) *«,= 0.01 By and (b) fii^s 0.1 By. Dotted lines denote the first iteration. Fig.7 Selfconaistently *^ Fig.8 Tunneling rates for the barrier with 2as 200 aQ, a*= 0.07 a, ~ir».i«»»i with no iaage potential (lo). classical isage potential (B^) and the dynaaical potential (B) for tuo SP frequencies: f*>m= 0.1 Rr 0.01 By. The ratea mn given in unita (2a/a0 )/a /a . x67 M -*• z (*) 168 ^. 2 170 Q. Z 2' -*.» 0.0 169 -o.io -10 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0-» 1.0 (a) 0.00 -0.09 - -0.10 -10 -O.I -O.« -0.4 -0.2 0.0 02 0.4 0.* 04 10 t/a 171 0.000 -0.001 -0.002' -0.003 -0.004 -0.005 -0.MK -040? 0.00 0.02 OM 0M 0.M 172 -0.00 -0.02 -0.04 I -0.06 A -0.08 A -0.10 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.+ 0.6 0.8 1.0 («•) -0.00 -0.02 .i -0.04 4 -0.06 1 -0.08 H -0.10 I i. i i i i ' ' _' -' , - -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 10 I/O 173 0.35 0.30 0.2* 0.20 0.19 0.10 COS - 0.00 0.1* 0.16 0.1* 0.12 0.10 0.M 0.M 0.04 • 0.02 - 0.00 0.000 0.004 0.00* 0.012 0.01* <="•)•« 175 Photon Scanning Tunneling Microscopy R.J. Warmack. R.C. Reddick. and T.L FerreH Oak Ridge National Laboratory P.O. Box 2006, Oak Ridge, Tennessee 37831 The PSTM is shown to be the photon analogue of the electron STM. In the PSTM, externally-supplied plane waves undergo total internal reflection at a dielectric interface. This creates an exponentially-decaying evanescent field adjacent to the interface. Photons can be made to tunnel to a nearby optically conducting fiber optic tip. The transmitted light signal is strongly dependent upon the tip to interface distance and can be regulated by feedback control to stabilize this distance. Perturbations caused by irregularities in the interface can therefore be imaged by scanning the tip laterally in the same way as STM. Subwavelength lateral and vertical resolution is demonstrated. Lateral resolution is governed by the decay of the evanescent field and the probe shape. Vertical resolution is governed by electrical and mechanical noise. Controllable parameters include the angle of incidence, wavelength, polarization, index of refraction, and probe shape. Direct imaging of standing waves caused by the interference of two coherent surface waves at the interface wiN also be shown. JCamttnn rfobe I FORCE. (0) w MAGNETIC CAPACITANCE TON )L M. Si Photon Scanning Tunneling Microscope Photo- multiplier Tube Piezoelectric Computer Control Translator and Image Processing XYZ Motion and Fiber Optic Feedback Tip Prism Fiber Optic Tip Evanescent* Totally Internally ... Field Reflected light . Sample 178 t, n - Hi - hi. «A\J Uw NP048703.C01 20 Photomultiplier Current vs. Increasing Distance from Surface. ^ 15 10 v 200 400 600 800 1000 AZ(nm) 180 , k ' 1? W n* 1 % iE* 181 Direct Observation of Interference Fringes by PSTM An interference pattern caused by the intersection of two laser beams is not ordinarily observable since the fringe spacing is so small. The photon scanning tunneling microscope (PSTM) is used here to image the interference pattern at sub-wavelength resolution. The fringe spacing displayed is 307 nm or about half the wavelength of the light used. These images ciso demonstrate that the PSTM rcsoJuiion is an even smaller fraction of a wavelength. ORNL Submicron Physics Croup 182 Holographic Grating Profile (PSTM) c -120 PSTM ADVANTAGES Sub-Wavelength Resolution Three-Dimensional Mapping Images Insulators Non-contacting/Nondestructive Samples under Ambient Conditions Wavelength Selection: Spectroscopy 183 Extended transfer Hamiltonian method for resonant tunneling S.P. Apcll Department of Theoretical Physics Umea University, S-90187 Umea, Sweden E. Albertson Institute of Theoretical Physics Chalmers University of Technology S-41296, Goteborg, Sweden and D.R. Penn Radiation Physics Group National Institute of Standards and Technology Gaithersburg, MD-20899, USA August 24, 1989 184 Abstract A scheme originally due to Heitler is utilized in extending the transfer Hamil- tonian description to resonant tunneling, for calculating transition probabil- ities and general frequency characteristics of tunneling coupled systems. We give examples for a single barrier in the context of the Scanning Tunneling Microscope and for a double barrier in connection with an irradiated quan- tum well. In this way we can very conveniently account for the saturation of the contact resistance in the STM and give a simple explanation for the high-frequency response in an irradiated double junction. 185 Introduction One of the most used schemes for calculating tunneling probabilities is based on the so called transfer Hamiltonian formalism [1]. Recently this formalism has been discussed from the point of view of extending it to situations where there are real intermediate states, like in double barriers [2,3]. This prob- lem is of current interest in the resonant tunneling through double barrier structures, with or without applied electromagnetic fields, and in the treat- ment of the Scanning Tunneling Microscope (STM) [4]. In the present paper we wiil give a description of theses processes within an extended transfer Hamiltonian formalism which is based on a scheme originally developed by Heitler [5], many years ago. We illustrate our formalism with two systems of interest in device applications: the single barrier and the double barrier shown schematically in Figure 1. Basically the single barrier has two states, left (L) and right (R), and the double barrier has also an intermediate state (I). We will treat situations having both discrete energy levels as well as a continuous range of energies since they will exhibit very different response characteristics. Let us first discuss a Fourier-space representation of transition probabili- ties in a quantum mechanical system to get the spirit of Heitlers treatment. The virtue of going over to a Fourier space representation of the physics is that we can utilize schemes developed for calculating Green's functions to infinite order in the coupling between subsystems. Those can be different electronic parts of a system but they can also be the interaction with ex- ternal radiation or an applied potential. Furthermore working in Fourier space characteristic response frequencies, such as Rabi oscillations and their strength, come out directly. We start from the time-dependent Schrodinger equation for the state vector = HT* (1) wnere Hy is the total Hamiltonian. Let /fr=Ho+H,-Bt Ho describes the un- perturbed system and H,nt the interaction. We denote an eigenstate of Ho by #n for short. The exact solution of Eq.(l) can then be expanded (2) 186 2 6n | is the probability for the system to be in the state n at time t, and 3 Dn(t) = jt\K\ (3) defines the transition probability per unit time. Notice that Um«^«, 0n(t) gives the standard Golden Rule formula for the system. We have a general constraint on our system in that: £A.(O = o (4) n since the normalization of the b's is conserved; the particle we are describing has to be somewhere. Making a Fourier expansion of bn(t): we can finally express Dn(u>) as the following convolution + u/) (6) In what follows we will be mainly concerned with D/j(o>) which we will treat as a response function of our system. We will study a system which is in equilibrium at time t=0~ in which we inject an electron into a particular state o at time t=0+. The injection procedure can be included in Eq.(4) by letting £„ Dn(t) = 6{t). We therefore have a sum-rule £„ Dn(u>) — 1 in Fourier space. From equation (3) we find the interesting static limit: a ZJn(« = O) = /*/>»(<)=! **(*>) | (7) using the fact that bn(0~) = 0. The smalt energy limit of our response function Dn(u) is thus a measure of the stationary occupancy of the state n. In the next section we first introduce Heitlen scheme for calculating the amplitudes (bn) and we apply the formalism to a single barrier, illustrating the Rabi frequency concept and the saturation of the tunneling for strong coupling. Then we go on to study the double barrier both with and without an external electromagnetic field. We end with a small commentary to the general response properties of real irradiated double barrier devices based on the physical principles of single and double barriers we have illustrated earlier. 187 Single barrier In the previous section we set up the formalism for calculating the occupation amplitudes bn(w) and here we will use the single barrier a* an illustrative example. From Heitler [5] we can write down the followingrepresentatio n for b,(t): 6w(t) = y- f~ dEGn^Eyl**-1*/* (8) £X J-co yielding (c.f. Eq.(5)) 6.(u;)«iGW + £.) W where the Green's function G*» can be written as: Gno(u>) = CU~)G~(«K(w - En) (10) with and the " injection" state Green's function (?„(«)« (a,-£.-£M)-1 (12) with its self-energy expressed in terms of a renormalized coupling strength ) - ff^ + £ ffw»{(u» - £,)£/«.(«) (14) HmB is the matrix element of the interaction part of the Hamiltonian be- tween the states m and n and can be calculated e.g. within the trans- fer Hamiltonian formalism. Now specifying to a two-level system in the form of a single barrier having only one discrete level on either side there is only one coupling matrix element Hmll«Hj|» s H. Therefore UJW«H. S(w) =H3/("+ic - £A) with e * 0* and we get: 188 defining wl>2 = -ERO - -c ± -WH (16) where 2 WH = v1^ + 4i/ ] (17) is the so called Rabi frequency [6]; the characteristic frequency for oscillating back and forth between the states o and R. Inserting bn(w) into Eq.{6), picking up the two poles in the upper half plane, we get: (18) Dfl vanishes with the coupling Hz as it should. D/}(0)=2^j- and it reduces of course to ^ when En — EQ. Comparing to dielectric functions the structure of DR(U), Eq.(lS), is char- acteristic of a bound electron system having a transition energy w^. Going into real time space we see that during a time proportional to w^1 the elec- tron has left the o-state and come back. Clearly u>^1 can be viewed as a typical time of " tunneling". Furthermore letting t—• oo DR{t^ oo) - j- | H \*6(En.) (19) to lowest order in H2, i.e. we retrieve the Golden Rule result for the transition rate between the states o and R. In a real situation we do not have the very sharp energy levels of the example above, instead we have a continuous distribution of states to the left and right of the barrier which we can characterize with their density of states PL and pn respectively which we take to be the same in what follows; PL — PR — P- Going through the same steps leading to Eq.(18) and letting all H/j=H we now find that the Rabi-oscillations are masked because the self-energy T.(u) acquires a finite imaginary part F. We also find that Dfl(O), measuring the occupancy of the right hand side state R after very long time, is now: £^T5 (20) where T = rpHTI (21) 189 and v <22> is a new renormalized coupling strength (tunneling probability) instead of H. To arrive at Eq.(22) we have approximated the energy-integrated (-function defined in Eq.(ll) by its imaginary part, to give the main physics of the problem. For thin barriers, where irpH Double barrier As our second example we consider the double barrier as shown in Figure lb. A crucial feature of the double barrier is the broadening of the intermediate state (I) due to its coupling with the continuum at either side. This broad- ening makes it qualitatively different from an atomic three-level system with well-denned Rabi frequencies which we will see in what follows. As before we focus on i>n(u>): bR(u) = iGRo(u> + ER) (23) where GUu + ER) = Utoiu + ER)GM((J + ER)t («) (24) and URO{U> + ER) = HRIt(u> + EmWtciu + ER) (25) with 2 1(" - ER)UIo + £ | Hu \ e(u; - EL)UIo (26) H L using ULO{") = Hu&u - Ei)Uj0 (27) in Eq. (26), c.f. Eq.(14). The summations over L and R represent the coupling of I to the contin- uum we have to the left and to the right (EL —* fdELpl) and E/j = E/-Ej in Eq.(25) and what follows. Neglecting the shift of the energy levels we will replace the summations with a constant life-time for the intermediate state: 2 T/ = T1R + Tu with r/t = xfdEiPi \ Hu \ . This gives finally for U/(((w): (28) As usual letting all H/j be the same (=H) and inserting Eq.(28) into Eqs.(23- 27) we get when E0=E/i: *«(«) = / ^ • w ^ M T (29) (w + tc)(w w)(w w) where \ 2 2 (30) 191 Notice that comparing to Eq.(16) we can identify in Eq. (30) an effective energy Ej — iT/ for the intermediate state. For T/ <\ E{o |, b«(o>) shows that we have a high frequency mode u>\ ~ Ej0 when E/o > 0 or a broad 2 low-frequency structure around Re[u/2] a H IEoj when E/o < 0. From Eq.(29) we now get for ; - ) + __- :_ ] (3i) When 0/ = 0 we have the limit: DM0) =1 — I2 (32) WW2 This is the same as saying that the long time response of the junction will yield a non-vanishing electron density to the right. This is clearly what one associates with a tunneling process through a junction where .depending on the coupling a certain probability exists for finding the particle to the right 2 of the barrier. For E/o > H and using T/ « 2*pH we get: In Figures 2a-c we show DA(W) calculated from Eq. (31) in three different situations: E/ < £„, E; ~ Eo and E/ > Eo for Tf/ff si. We see thai Dfl(w) vanishes for large frequencies, as it should since this corresponds to short times and there is not yet any probability amplitude to the right then. In general DR(U) exhibits a broadened oscillator behaviour around energies ~j 7 I, 7 = EiojTi. Furthermore DR(UJ) quickly vanishes for increasing J 7 | as a comparison between Figures 2a) and c) together with b) shows. Notice also that there is of course very little left of the Rabi frequency concept when we have continuous energy states both to the left and right of the barrier as well as a broadened state inside the barrier. In the following section we will extend our treatment of the double barrier to include an external potential. 192 Irradiated double barrier We have illustrated the use of a Fourier transform scheme to appreciate the characteristic energy-scales involved in a single and a double barrier respec- tively. To make contact with real devices we have furthermore to include a static bias and/or an applied microwave field. In this section we will give a brief outline of the main results for the case of an external electromagnetic field of the form E0cosut [10]. First we make an important observation. In semiconductors with a typ- ical doping, the energy scale is in the 10-100 meV range. This corresponds to very long wavelengths or penetration depths, being much larger than the tiny extension of the well region. It is therefore possible to neglect the part of the absorption which has to do with the well. This picture is in contrast to traditional approaches in this area [2,11-19] which tend to stress the im- portance of the coupling of the well with the incoming light. In our view we have absorption in the bulk electrodes outside of the well, creating electron- hole pairs which in a bias will be swept through the system and collected in the form of an I-V characteristic. The only influence is in giving an energy- selective constriction to electron transport. We thus conclude immediately that any intrinsic cut-off of such a system is determined by the optical prop- erties of the electrodes, which in reality often means the optical properties of bulk n-doped GaAs. Within the transfer Hamiltonian scheme we have to do the same steps as in the previous section, but now with an added perturbation in terms of an incoming electromagnetic field £Acosu>t. We call the coupling term in the Hamiltonian for W which we take at a constant for simplicity. Based on the arguments given above it only couples states on the left hand side of the junction with each other or on the right hand side with each other. After some straightforward algebraic manipulations we can write down a Golden Rule formula going from the state Eo to E* [10]: 3 2 oo) = y[| /(/?„) | 6(E,u)+ \ g{ER) \ 6{E,u ± M] (34) having two contributions being one the static part: /(£„)= *** <35) ERI + *!/ 193 corresponding to two tunneling events enhanced close to resonance (E/i/=0) and the other one being the dynamical part: corresponding to light absorption and two tunneling events or vice versa. In Eq. (36) we have set pi =s pn = p lot simplicity and again used the imaginary part of ( to exhibit the general behaviour. In what follows we neglect the interference between the two terms in g(E/i) and use the fact that for the situation we are investigating it is a good approximation to replace the energy-denominators containing the intermediate state with delta functions, since F/ is smaller than the other energy-scales involved [11]. This gives: 2 | g(ER) | * -Jr/WW&t,) + S(EM ± M] (37) 4 and ^ (38) yielding a general ratio between the dynamical and the static parts (i 1 for ordinary electromagnetic fields. In device physics it is appropriate to calculate the current I which flows through the junction. Since I is an energy-integration over DR{1 —• oo) the first part of Eq.(34) gives the static I-V characteristic of the junction [20], which we call IO(V). It is then easy to see that the dynamic part of the current can formally be written as: 2 I4{V,w) at (xpW) [ro(V + u;) + I0(V - «)] (39) in terms of the static I-V characteristic Ie(V). Not surprisingly this is * result which is very similar to the general treatment for irradiated Josephson junctions by Tien and Gordon [21]. The pref&ctor in Eq.(39) contains W3 which is a measure of the optical aborption in the electrodes. If we had treated this part more carefully if would of course contain the absorption frequency characteristics of the bulk electrodes. This has a characteristic cut-off frequency being the plasma fre- quency of the doping electron gas or for that matter roughly its Fermi level [22]. We therefore anticipate an upper cut-off in the response of an irradiated 194 double barrier around 40 meV (or 10 THz) based on a doping of This result therefore confirms the experimental findings in [11] of the possi- bility of observing a THz response from an irradiated double barrier. Notice also that since F/ is typically 3-5 meV it plays almost no role whatsoever for the validity of our conclusions because it is much less than 50 meV. At the end we should finally acknowledge the situation that whereas there is a response time-scale giving a cut-off around 10 THz for the isolated de- vice most experiments on double junctions are in oscillator configurations which give a cut-off in the GHz range [23-26] determined primarily by the capacitance and the series resistance of the device and not directly by its intrinsic response properties. 195 Acknowledgement This project was supported by a grant from the Swedish Natural Science Research Council. 196 References 1. See e.g. C.B. Duke, Solid State Physics, Suppl. 10, Academic Ptess N.Y., 1969. 2. M.C. Payne, J. Phyt. C19,1145 (1986). 3. L. Brey, G. Platero and C. Tejedor, Phy». Rev. B38,10507 (1988). 4. T.E. Feuchtwang and P.H. Cutler, Physic* Script* 35,132 (1987). 5. W. Heitier, The Quantum Theory of Radiation, Ch.IV:16, Dover, N.Y. 3" edition (1984). 6. See e.g. P. Stehle, Physics Reports 156, 67 (1987). 7. J.K. Gimzewski and R. Moller, Phys. Rev. B3«, 1284 (1987). 8. J. Ferrer, A. Martin-Rodero and F. Floras, Phys. Rev. B38, 10113 (1988). 9. L.V. Keldysh, Sov. Phys. JETP 20,1018 (1965). 10. D.R. Penn and S.P. Apell, to be published. 11. T.C.L.G. Sollner, W.D. Goodhue, P.E. Tannenwaid, CD. Parker and D.D. Peck, Appl. Phys. Lett. 43, 588 (1983). 12. K.W.H. Stevens, J. Phys. C17, 5735 (1984). 13. S. Luryi, Appl. Phys. Lett. 47, 490 (1985). 14. J. Kundrotas and A. Dargys, Phys. Stat. Sol. bl34, 267 (1986). 15. W. Frensley, Phys. Rev. B36,1570 (1987). 16. J.F. Young, B.M. Wood, G.C. Aers, R.L.S. Devine, H.C. Liu, D. Land- heer, M. Buchanan, A.J. SpringThorpe and P. Mandeville, Phys. Rev. Lett. 00, 2085 (1988). 17. D. Sokolovski, Phys. Rev. B37, 4201 (1988) and Phys. Lett. A132, 381 (1988). 197 18. Y. Nomura, S. Nut, S. Maruno, M. Gotoda, Y. Morithita and H. Ogata, Superlattices and Microstructures 8, 73 (1989). 19. M.Jonson, Phys. Rev. B30, 5924 (1989). 20. See e.g. the presentation in H.C. Liu and G.C. Aers, Solid State Com- mun. 67, 1131 (1988). 21. P.K. Tien and J.P. Gordon, Phys. Rev. 129, 647 (1963). 22. W. Szuszkiewicz, K. Karpierz and Vu Hai Son, Physica Scripta 37, 836 (1988). 23. T.C.L.G. Sollner, P.E. Tannenwald, D.D. Peck and W.D. Goodhue, Appl. Phys. Lett. 45, 1319 (1984) 24. T.C.L.G. Sollner, E.R. Brown, W.D. Goodhue and H.Q. Le, Appl. Phys. Lett. 50, 332 (1987). 25. J.F. Whitaker, G.A. Mourou, T.C.L.G. Sollner and W..D. Goodhue, Appl. Phys. Lett. 53, 385 (1988). 26. E.R. Brown, W.D. Goodhue and T.C.L.G. Sollner, J. Appl. Phys. 64, 1519 (1988). 198 Figure Captions 1. a) Single barrier with states L,R and o. o is the injection state. b) Double barrier with states L,R,I and o. I is the intermediate state with broadening F/. 2. Real (full line) and imaginary (dashed line) parts of the general trans- mission function DR(UJ) is plotted for the double barrier as a function of detuning 7 = (Ej — E0)fTi. w is measured in units of H. The cal- culation is done for Fr/H = 1.0 and a) 7 = -10, b) 7 = 1.1 and c) 7 = 10. Notice that a) and c) are magnified a factor of 1000 as compared tob). 199 FIGURE 1 b) H 200 Figure 2 D x 103 0 DxlO3 -10 201 LASER FIELD EFFECTS ON THE INTERACTION OF CHARGED PARTICLES WITH DEGENERATE AND NON-DEGENERATE PLASMAS N.R. ARISTA*. R.M.O. GALVAO**. AND L.GM. MIRANDA** Universitat d' Vacant, Depanamento de Fisica Aplkada. E-03080 Alicante. Espafia A general description of the effects of a strong laser field on the inelastic inieractkm between charged panicles and a plasma is presented, both for classical (dilute) or quantum mechanical (dense) plasmas. The dynamical response of the medium is recalculated, including the effects of the laser field, using a Dawson-Obennan transformation for classical systems. and the RPA formulation for quantum plasmas. We find that the energy exchange and the particle scattering rate are modified by mold- photon processes, and become anisotropic with respect to the direction of panicle motion relative to the direction of laser polarization. The RPA formalism describes the excitation of ptosmons and electron-hole pain, with simultaneous emission or absorption of photons. We calculate the contribution of these processes to the energy exchange and »the meao- freepath of the panicle, in the range of a typical low density plasma, and in me range of solid-state electron densities. New effects due to the laser field become particularly important when the laser frequency becomes close to the plasma frequency. In this case. pUsmon excita- tions can be produced below the normal velocity threshold through photon-assisted processes. This gives place to an anomalous low-velocity behaviour of the stopping power. * Centro Atomico Bariloche, Division Colisiones Atomicas. 8400 Barilocbe, Argentina **Insntuto de Pesquisas Espaciais (INPE), 12201 Sao Jose dot Campos. 5.P.. Brazil 202 VI K I •M 1 * * -w 4 - 8 i u i x H W 4: a H Ui o 41 I i 11 •N I 5 I o ^ i s J 2 w i 3 « 1 * i rx i 1 » 1 i i; ii j F X- CLASSICAL 3. Fsurier pUt»* l',3,-t) (cU»iu( J.siaV. function) ^e•. j>0 = Z« 6(f-* t) <«<» Vy : Source : s R«..tt: ire 204 4 s ?• £ V) It IM II H at | -» 9 •+ IE -T III )l It (3 2 5 X "3" 2 -I OJ n V*) ut u 1b •HI it 205 eV VOAT 206 5 Jta, 3. wf ST u« 1i j- /— '*• 1 «•» j* jt t *- *+* n •A "i** ) I -»• y _j •» So k ' >- 6 1W 1 u a. O -2 (J o J II E t'- KJ1 I o s Ul x I 4 < I ? O. ct It 207 ^v^ Ml j a. 3 f I I» pi <|1 3 c H i OIIJC f -oJ-» m 1——1 3* X I «^ i> 7 Ul H 5 H Ul z *c UJ 208 VJ s 0 «t a $ He •/ — k T i • < \ 1 -|K' in 209 I oi c 210 =frT)tscri|>t»oH (classical or a.m.) df p*tidt-bias**- laser inierjctioM processes i» tferwo of £ Processes : —i«eU«ti'c suttee M.t M pUsMA, / " K or las&r I$I$ lt IK iU ^e. ^ IO - IIOO 3. flugubr ie^evi J«wtcc ike ^ (on^ if (^ ^ 211 Screening Effects in Nuclear Fusion of Hydrogen Isotopes in Dense Media - Nestor Arista, Centro Atomico Bariioche, R.A. - Raul Baragiola, Rutgers State University, USA - Alberto Gras-Martft Universitat d'AIacant Spain. ABSTRACT Nuclear fusion rate* of isotopic hydrogen nuclei embedded in dense screening media are calculated. Ue consider the case* of a uniform degenerate electron gas and the inhomogeneou* electron density in solids. Ue derive an exact wave function for the screened nuclear interaction, and an analytical expression for the barriei—penetration factor, in the case of homogeneous screening. Tor qualitative estimates of the screening in solids, we use a Thomas-Fermi description of the electron density. A cross-over of the fusion rates of the various isotspic pairs analyzed (p-d, d-d, p-t and d-t) is predicted for increasing screening length, velocity or effective temperature of the medium. The effects of variable screening length and local effective temperatures are considered for electron gases in the range of metallic densities and compared with previous experimental and theoretical studies. 212 Present Scattering Model: * exact wave function > repulsive potential * screening in electron gas > homogeneous > inhomogeneous * TF model * statistical isotope distribution Screening Effects in Nuclear Fusion of Hydrogen Isotopes Present Scattering Model: in Dense Media - Nestor Arista, Centra Ato'mico Bariloche, R.A. - Rat' Baragiola, Rutgers State University, USA - Alberto Gras-Marti, Universitat d'Alacant, Spain. AMTIMCT * exact wave function N3 Nuclear fusion rate* of tutopic hydrogen nuclei embedded in > repulsive potential screening aedta are calculated. We consider the cases of 4 uniform degenerate electron gas and the 1nhomogeneous electron density In solids. Ue derive an exact nave function for the screened nuclear * screening in electron gas Interaction, and an analytical t-priiimn far the barrier-penetration factor, in the cave of homogeneous screening. For qualitativ* > homogeneous efttimate* of the ecreeninQ in ftolld*, w» u»e a Thdmat-Fertti description of the electron dennty. > inhomogeneous A cro*»-Dver of the fusion rates of the various isotopit pai?s »TF model anaJyted (p-d( d-d, p-t and d-t> is predicted for increasing screening length, velocity or effective temperature of the medium. * statisticai isotope distribution The effects of variable screening length and local effective temperature* are considered for electron gases in the range «f Metallic densities and compared with previous experimental aria theoretical studies. Degenerate Electron Gas: Repulsive potential a.u.: e = m - ti - 1 a » 0.529 A 18 v0 - ?.«8.|,> cm /s kT = 27.2 eV ~ 300.000 K i * relativistic: Corrections to q#: • partial degeneracy: * K, n« ~* rf: o"e-electron radius. SC -" 2 0.6 0.3 0.01 metals Jupiter Sun W. dwarf FUSION RATES IN A HOMOGENEOUS ELECTRON GAS * Attenuation Factor: T ( 1 = A s* Hulthen Potential: Strong function of q. ,v Fusion rate / scattering center: Fusion rate / volume: * Attenuation Factor: Fusion rate / pair: A" r = Fusion rate = n. A E = CM energy of relative motion fusion rates depend very strongly on the screening conditions and on the collision energy Fusion rates for p-d are larger at low screening and low-energy - fusion rales for d-d take over in the opposite limit - Obviously; screening of nuclear charge by the surrounding electron gas leads to orders of magnitude change In rates. O\ • highest q« values: strongly screened interactions as they would apply, for instance, for bound p-mu or d mu muonic systems studied in muon-catalyzed fusion reactions. - the large variations of the attenuation factor as a function of the screening parameter are similar to those found by Koonin t Nauenberg, or Van Stolen and Jones' predictions for bound d-molectHes (variable elfectlve mass for tht electrons). •200 i / *atoms / (screening length) Fusion rates versus the relative velocity of the Fusion rates for p-d and d-d reactions in a isotopes, for reacting protons, deuterons, and uniform electron gas, as a function of the tritons (p,d,t) in an electron gas with'qfl-1.27 23 -3 (corresponding to solid Pd). Light curve: p-d temperature T, for 'VlO cm and for seve- fusion rate without screening. ral values of q... Jones et al NJ -100 to FUSION RATE: FUSION RATES IN NON-HOMOGENEOUS ELECTRON GAS * Inhomogeneities > isotopes Local density approximation > electrons * Thomas Fermi for confined atomic systems WS cell r, (~ 2.88 a.u. for Pd) to 00 Isotope • metal atom interaction: • WS cell V(r) = &¥(£ Average fusion rate / pair: Tx* f»A fj (a) Radial profiles of: TF potential V, electron den- Velocity-dependent fusion rate (v), for a thermal sity ne, and screening parameter qQ, for a TF model distribution of hydrogen isotopes in the atomic cell of Pd atoms, confined within a Wigner-Seitz cell of Pd, for various local effective temperatures T ,, of radius rQ=2.88 a.u. calculated according to the present model, for p-d (b)Spatial profiles of hydrogen isotopes. and d-d reactions. ~o.l? N> i to1 v/v,h CONCLUSIONS: T*»U ru.l.n t«.. (in .-»> {ro „ . 21 * analytical model of screening in a scattering for process r««tion .c 10 * cross-over of fusion rates N5 N3 O * screening important below ~ 200 eV dramatic T - dependence * cold fusion rates very low * model of interest in other situations ? 221 CONTRIBUTION OF RIPPLON EXCITATION TO INTERACTION ENERGIES OF PARTICIPATES • 6+ 1 Rafael Garcia-Molina , Nuria Barberan* and Alberto Cras-Marti • Departamento de Física. Aplicad*, Facultad de Ciencias, Universidad de Murcia. E-30071 Murcia, Spain. § School of Physics. University of East Anglia. Norwich NR4 7TJ. United Kingdom. t Department de Física Aplicada. Facultat de Ciéncies. Universitat d'Alacant, E-03080 Alacant. Spain. Abstract Ripplons are quantized surface-tension waves in liquids. We have studied the contribution of virtual ripplon excitation: (0 to the image potential felt by a charge located in front of a liquid surface, and (ii) to the van der Waals attraction between two liquid surfaces. These interactions have been investigated as a function of the distance between the interacting systems and for planar or spherical particulate geometry. The polar or non polar nature of the medium is also considered. The range of validity of the formulae derived is discussed. t Permanent «ddrast: Departamant* da Estructura y Carntltuyanus <»• •• MtUrla, Faculta* do Física, Unlvareléa* d< Baralefie. Diagonal 6*7. E-OMZt Barcelona. Spain. 222 The image potential felt by a charge placed in front or a liquid surface piays a central role in many physical problems. For interacting systems consisting of neutral particles the van der Waals potential describes the interaction. Interactions between charged or neutral particles and liquids are of current interest in research areas of physics, such as: - condensed matter physics: formation of two-dimensional electron Wigner lattices on liquid surfaces, helium atoms scattering from liquid helium, - atmospheric physics: charge distribution in clouds. - biophysics: charges and neutral atoms in the presence of biological membranes. - aerosol physics: aggregation, coagulation and deposition of liquid drops. therefore it appears interesting to improve our knowledge of the interaction potentials previously mentioned in order to provide a better description of the phenomena where these interactions are basic ingredients. Our aim in this work is to calculate the contribution of virtual ripplon excitation: (i) to the image potential felt by a point charge located in front of a planar or spherical liquid surface (Figs, la and lb), and (ii) to the van der Waals attraction between two liquid surfaces (Fig. lc). By ripplons we mean the quanta of capillary waves at the surface of the liquid, whose dispersion relation for a planar liquid surface is (Landau and Lifshitz 1959] V * *q * Sq3 and that corresponding to a spherical liquid surface of radius R is [Landau and Lifshitz 19591 z m-im+2)» "< '—^ where g is the gravity acceleration, a- is the liquid surface tension and d is the mass density of the liquid. 223 In this work we have considered the liquid deviation from flatness (<; in Figs. 1) as a small perturbation, therefore only terms up to first order in the surface shape oscillations corresponding to the rippions have been retained. In brief, the procedure used in our study is the following. First we have calculated the interaction potential entering in the corresponding problem when a rough surface is considered; from this interaction potential, the Hamiltonian that describes the system is obtained. Secondly the ripplon field is introduced in the resulting expressions according to the second quantization formalism. And finally the self-energy due to the presence of the ripplon field is given by the shift in the zero-point energy of the system. A detailed account of this procedure may be found in Gras-Marti and Ritchie (198S1. Barberan et at. (19891 and Gras-Marti et at. (19891. Two methods may be used to obtain the interaction potential: the first one consists in solving the Poisson equation with the appropriate boundary conditions at the liquid surface. This method includes the full interaction between a charge and the liquid within a dielectric response model and we call this method the full dielectric calculation. The second one gives the interaction potential by adding the interactions taking place among the interacting particles through the whole liquid volume; this constitute* the pairwise summation method and it is a good approximation for nonpolar liquids: it does not take into account screening effects. In Figs. 2 and 3 we show the self energy corresponding to a static electron in front of a planar and a spherical liquid surface, respectively, as a function of the charge liquid-surface distance. Two liquids have been considered: helium (nonpolar) and water (polar). In Fig. 4 we show the stopping power for an electron traveling parallel and outside to a planar liquid surface. Fig. 4a corresponds to helium and Fig. 4b to water. Finally, in Fig. 5 we show the ratio between the rippton contribution to the interaction energy and the van der Waals potential energy corresponding to the interaction between two scmiinfinite liquid media whose surfaces are separated by an average distance C. A 224 more detailed description of thic research is given in Gras-Marti and Ritchie (19SS). Barberan *t at. (1969) and Cras-Marti et at. (19691. The following conclusions are derived from this work: In the case of an electron in front of a liquid surface, (i) the ripplon contribution to the interaction energy is . a perturbation to the classical image potential, but for polar liquids and short distances th» applicability of our first order approximation treatment is uncertain; in the case of a small spherical water drop (R • 5 A) the ripplon contribution to the potential energy of a bound electron is about half the binding energy calculated without considering that contribution {Ballester and Antoniewicz 1986). (ii) The full dielectric calculation and the pairwise summation method give similar results for nonpolar liquids in the case of planar surfaces but not for spherical surfaces. This seems to indicate that there are screening effects associated with the spherical geometry which are not included in the pairwise calculation and that these screening effects seem to be far more important than the effect of the permanent dipole moment of polar liquids, (iii) The ripplon contribution to the stopping power is negligible for high electron energies (v » i a.u.i. In the case of two nonpolar liquids separated by an average distance t. the ripplon contribution to the potential energy seems to be significant for distances (AS a.u. Acknowledgements Financial support from the DCICyT (project number PS88-0066). and the program of Acetones Integradas with Germany and Italy is acknowledged. 225 Reference* • Ballester, J. L. and Antoniewicz, P. R., 1966, J. Chem. Phys. §5, 5204. Barberan, N., Carcia-Molina, R., and Graa-Marti, A.. 19f9, Pltys. Rev. B 12, 10. Cras-Marti. A., Barberan, N., and Carcia-Molina, R.. 1989, "Rlpplon contribution to tht Interaction between two liquid swfMces", preprint. Gras-Marti, A. and Ritchie. R. H.. 1985, Phys. Rev. B 2L 2649. Landau. L. D. and Lifshitz. E. M.. 1959. Quid Mechanics (Pergamon. Oxford), pp. 238 and 240. 226 (a) rta. 1 vacuum •k. diXtctric liquid Jlfx, y) surface displacenar (e) vacuum 227 (at aallaa m. a to* elMalaal laafa M—till ' KT*- fall 41al««trlc 20 (0 M 100 z(eut) Ik) wut ^>k elaaalaal iaaaa aa«a««tal KT1 ' «r* tall olalctrlc —I—Itlaa 20 M N «0 228 «. s la) —* iOO *.*.. (ki »•» - ^r (•I 229 (a) Mi« wt*. t 10* W* W* -? 20 40 CO M WO j (M 10* *r* 10* 20 40 CO CO WO 230 |0» , «•"- no. s I*' uri. 0 5 10 IS ZO 231 Nuria Barberan1, Manuel Barranco2, Francesca Garcias2, Jesus Navarro3 and Llorenc Serra2 1 Departaaent d'Estructura i Constituent* de la Materia, Univeraitat de Barcelona, E-0S02I Barcelona, Spain. 2 DepartaMnt de rlcica, Universitat de sea Zllea Baleara, E-07071 ?I1M de Mallorca, Spain. 3 Departamnt da Flaica Atoaica, Molecular i Nuclear and iric, Univeraitat de Valencia - CSZC, £-46100 Valencia, Spain. 1. Introduction The increasing interest in the properties of microstructures during the last few years motivated by several problems, like the development of new reduced components in microelectronics or the fundamental mechanism of catalysis in many important chemical reactions, has generated a big amount of related literature [1]. Among the available experimental techniques, the scanning transmission electron microscope (STEM) allows the study of the response properties of targets with high resolution [2]. For small clusters, the increasing surface to volume ratio has suggested that the response would be dominated by surface type excitations. However, recent electron energy loss spectroscopy experiments on metallic spheres [3] have shown that, contrarily to what was expected, significant bulk scattering occurs even on less than 50 A diameter spheres. Bulk collective excitations on metallic spheres have been theoretically described using a non-local dielectric function [4], a hydrodynamical model (5,6] and a selfconsistent time-dependent local density approximation [7]. Here, we carry out a Random-Phase Approximation (RPA) sum rules (SR) calculation of the response (strength) function to multipole q-dependent operators of the type JL ( electron cloud of the unperturbed cluster is described either in the Kohn-Sham (KS) or an Improved Thomas-Fermi-Weizsacker (ITFW) plus a local density approximation (Dirac+Wigner) for the exchange and correlation effects. 2. RPX SUB rulas Sum rules m^ are moments of the strength function S(E) - 2 6(E - En) f k mk - J EE* S(E) dE - Li En* ! ! £ B S E3 2 2 2 a ^ (E3 -Ei )^ (4) In this calculation we have used the following electron energy density functional: E{n(r)) - J t(n) dr - 3/4 (3/x)1'3 J (n(r>)4/3 dr - a J n(r)/ r*(r) n kinetic energy densities, ra(n) - [3/4acn(r) ji/3 iS the local radius per electron, Vj(r) is the jellium Coulomb potential, R the radius of the jellium sphere, N the number of monovalent atoms and a-0.44 and b-7.8. We have used atomic units (au). 3. Results A. rinit* Spheres We have applied the SR method to Na spheres for which we have taken ra«4 au. Fig. 1 shows for Nag2 and L«l the contribution to m3(RPA) of the different terms entering its definition, that is, the kinetic term (T), exchange (ex) and correlation (cor) tenu and the direct (e-e) and jellium electron (j-e) Coulomb tens. Each contribution has been normalized dividing it by the total 013. Curve labeled C is the sum of the Coulomb e-e and j-e terms. This figure shows that for small q-values, m3 is dominated by the Couloab terms, and by the kinetic term for large q-values. Both contributions equalize at a value of q close to q,. - «fc>/vF • 0.45 1/2 au where OJp - (4«n) - 0.217 au is the plasma frequency and vr - 234 (3JC2 n)1'3 - 0.48 au is the Fermi velocity. For larger L-values, the relative importance or the kinetic term increases for small q-values, in agreement with the results obtained for surface excitations in ref. 9. The local Coulomb exchange and correlation contributions are negative, the later contribution being always negligible. This cut-off qc is a lower bound estimate of the Landau damping onset momentum. The transition from a collective to a single electron-hole excitation is related to the increasing contribution of the kinetic energy to m3 for large values of q and independently, for large values of L. 051 0.1 In Fig. 2 we represent the ITFW Ex and E3 energies corresponding to Na92 L«0. One can observe that the difference between El and E3 increases when q increases; the electron-hole (non collective) contribution becomes the main excitation mechanism for values of q £ qc. Apart from L-0, all the other modes (see figure 3 for L-5) 235 present a region at small values of q for which £x and E3 are rather q-independent, tending to the q-0 surface mode value. This means first, that the excitation is mainly of surface type and second, that the energy of the «lngl« (one for each L-value) surface mode obtained in ref. 9 for the step electron density model: 2 2 2 2 W - Op L/(2L+1) • 2/3 (2L+1)(L-l) BF /R (6) m 2 a where fir (3/5) I/ v», hjLS_IAO__ej(£xJLJt!dflfiSAdSflJa__£fiC£fiCtii£t0_-£fi£B After this flat region (which is absent in the L-0 mode), the response starts being dominated by bulk modes and both energies increase rapidly. For q > qc the spectrum is dominated by electron-hole excitations. The energy and momentum are so large that the excited electron is quasifree and its dispersion relation is - q2/2. To give an idea of how these results depend on the ITFff approximation, we show in Fig. 3 the TF and KS results for E3 L-5. One can see that the agreement between both calculations is excellent. 236 It is also worth noting in Fig. 2 the structures along the El curve which are also present in the E3 curve, although less marked. The maxima of these structures correspond to the q-quantized bulk modes in the hydrodynamical approximation. Indeed, for a constant gs electron density these modes are excited at discrete values of q such that [5] (Actually, eq. 7 is exact only for L«0 modes; for L»l it is a good approximation) . In Fig. 2 we have indicated by arrows the first four roots of eq. 7 corresponding to R«18.06 which is the jellium radius for Na92. B. Infinite systras If we calculate the m-^ and m3 RPA SR for a step electron density n of radius R and take the q and/or R —» «» limit, we get the following asymptotic expression for the excitation energy: 2 2 2 cor 2 4 CO *m3/m1 - where the term -vF/3x is the local exchange contribution, and cor £2 (n) - -2/9 a r3(n) [b+ 2 r3(n)]/[b + rs rs«4, the kinetic, exchange and correlation energy contributions are 0.138, -0.051 and -0.004 au. Thus, altogether exchange and correlation are a 40 % of the q2 kinetic energy contribution. For Al 4. Concluding raurks We have studied the RPA response of metallic spheres to q- and L-dependent operators lz,lqr) YL0 [12]. One of the main advantages of our method is that only electron gs quantities are needed to apply it. This allows one to use models of different complexity for describing the gs electron density, like the crude step density model, or Thomas-Fermi and Kohn-Sham models. Consequently, we can compare them and establish their influence on the quantitative results. In particular, we conclude that the ITFW gs electron densities yield results in good agreement with the full KS-RPA calculations. This is of great importance for the applicability of the method, since for spheres with N 2 200, selfconsistent KS-RPA calculations are not technically feasible due to the increasing washing out of the electron shell structure of the sphere. We have separately analyzed the q and L dependence of the collective excitation energies. We conclude that surface mode frequencies C0L are q-independent and that for each angular momentum value different from zero there is only one surface mode. The volume modes are q-quantized. For values of q larger than qc, i.e., short excitations wavelengths, no collective modes are possible and the excitation energy tends to that of a quasifree electron. We have obtained a simple RPA dispersion relation for the infinite electron gas which includes exchange and correlation effects, generalizing the dispersion relation obtained by Lundqvist in the plasmon pole approximation [11]. Finally, we conclude that the q-quantization is the origin of the oscillations in the scattering spectrum found by Batson [3] (see Fig. 7 of this ref.). In his experiment, he finds an oscillatory behaviour of the scattering amplitude as a function of the radius of the sphere. It seems clear from our results that for a fixed momentum q, the scattering will be resonant for all the radii R such that qR corresponds to a peak in the response function. This already stems from the step density hydrodynamical model. Here, we have shown that more realistic models also predict the same behaviour. 238 * Supported by the CAICYT (Spain), grants Nos. PB85-0072-C02-00, AE87-0027 and PB84-0388-C04-03 flj.- T. Halicioglu and C. W. Bauschlicher Jr., Rep. Prog- Phys. 51, 883 (1988) [2].- J. M. Cowley, Surf. Sci. HI, 1598 (1982) [3].- P. E. Batson, Surf. Sci. X5£, 720 (1985) [4].- R. Fuchs and F. Claro, Phys. Rev. B 25., 3722 (1987) [5].- N. Barberan and J. Bausells, Phys. Rev. B 21, 6354 (1985) [6].- R. Ruppin, Phys. Rev. B H, 2871 (1975) [7].- W. Ekardt, Phys. Rev. B 2&, 4483 (1987) [8].- LI. Serra, F. Garcias, M. Barranco, J. Navarro, C. Balbas and A. Maftanes, Phys. Rev. B 21, 8247 (1989) [9].- LI. Serra, F. Garcias, M. Barranco, J. Navarro, C. Balbas, A. Rubio and A. Mafianes, J. Phys. C (1989), to be published. [10J.- O. Bohigas, A. M. Lane and J. Martorell, Phys. Rep. 51, 267 (1979) tllj.- B. I. Lundqvist, Phys. Stat. Sol. 21, 273 (1969) [12].- LI. Serra, F. Garcias, M. Barranco, N. Barberan and J. Navarro, submitted to Phys. Rev. B (July 1989) 239 Wake-Riding Electrons Emitted By Antiprotons Traversing Solid Targets J. Burgddrfer, .7. Wang, and J. Muller Department of Physics University of Tennessee, Knoxville, TN 37996-1200 and Oak Ridge National Laboratory, Oak Ridge, TN 37831-6377 1. Introduction The dielectric response of the medium to a swift ion induces collective charge-density fluctuations which result in an oscillatory polarization potential trailing the ion ("wake"). The concept of such a "wake" dates back to Bohr1. The first quantitative treatment of the dynamical screening potential around an ion was pioneered by Neufeld and Ritchie2. Meanwhile, a large number of investigations implementing approximations at various levels 3 of sophistication have been performed . A typical example of the wake potential V.mmkt is shown in Fig. 1 for Slt+ in Aluminum at v, =* la.u. calculated in plasmon-pole approximation to the dielectric function. The plasmon-pole approximation is one of the simplest approximation which still accounts qualitatively for most of the features of the wake. Clearly, more subtle effects like bow waves and other dispersion effects are neglected. We will restrict ourselves in the following to the plasmon-pole approximation including a phenomenological damping. We also neglect in the following effects of the self-wake which can affect the shape of the wake potential4-*. In 1974, Neelavathi, Ritchie and Brandt* put forward the intriguing proposal that the additional minima of the wake potential trailing the Coulomb wdl should support bound states. Electrons trapped in these wells have been named "wake-riding" electrons. These states would correspond to a new type of bound states which is non-atomic but nevertheless strongly correlated with the atomic (or ionic) projectile. A simple variataooal calculation using Gaussian orbitals showed that the ground state is bound over a wide range of parameters*''. Wake-riding states are, in fact, only transient quasi-bound sti 240 pUu) w+ Fig. 1 Wake potential of a S projectile with * speed of vp = la.u. in an Al- foil (n, = 0.0088,7 = 0.03) in the frame of the projectile. The projectile moves in positive z-direction. which decay into the low-energy Coulomb continuum of the projectile upon exit of the leading charged particle from the solid. Consequently, wake-riding electrons are expected to form a peak in the forward electron spectrum with lab velocities new v, s: v,. The ejection of isotachic electrons in forward direction by ions penetrating solids is therefore a signature of wake-riding electrons. Unfortunately, the real work! is much more complex Ejection of electrons in forward direction in general, and in the velocity regime vt ~ 5P in particular, is induced by several competing mechanisms which can easily overshadow the wake-riding electrons. Most prominent are the "cusp* electrons which correspond to the low energy continuum states in the field of a positively charged ion formed by direct excitation or electron capture10. According to Wigner's threshold law for an attractive Coulomb field the cross section for elastic scattering approaches constant at threshold in the frame of the projectile11, 241 da a when lit, are multipole moments (anisotropy coefficients) and 0 is the polar angle of emis- sion. Both direct excitation of bound states of the projectile as well as electron capture from the valence or core levels of the target can populate those final states in the eon- tinuum. In fact, any smooth excitation function across threshold leads to (1). Upon a kinematic transformation this corresponds to a singular doubly differential cross ssction in the lab frame, The enhancement of the cross section near vt £s vp according to (2) renders the observation of wake-riding electrons very difficult if not impo—ible.In addition, multiple scattering of binary encounter electrons in the solid provides an additional source of dectrons populating continuum states with vt ~ vr. Recently, the experimental study of wake-riding electrons accompanying aatiproton transmission through carbon foils using the Low Energy Antiproton Ring facility at CERN has been proposed13. First experiments are presently underway. We have performad a theoretical study to explore the possible existence of a peak ofwake-riding" electrons accompanying antiprotons". Two important features to be discussod in detail below make their observation in the forward-electron spectrum for antiprotoas more likely than in the spectrum for positively charged ions: (a) The well-known cuspGke tnhanoameat in the forward spectrum of positively charged particles is absent, thereby facilitating the observation of wake-riding electrons which appear in the same region of the spectrum, and (b) the wake-riding states are localized a factor of - 3 closer to an antiproton than to a proton of the same speed. Electron capture probabilities into wake-riding states are therefore dramatically enhanced. 242 The threshold law (2) appears in the projectile centered continuum states as a final- state normalization factor ^) IT where v = v, - vf is the velocity vector in the rest frame of the projectile with charge Zf. An attractive final-state interaction between the electron and the ion (Zp > 0) leads to a cuspiike enhancement ~ 2irZp/v of the cross section near v« « vp, while for Z, < 0 a pro- nounced dip ("anticusp") ~ (2x\Zr\/v)txp(-2x\Zr\/v) occurs". The repulsive final-state interaction strongly inhibits forward scattering with small relative velocities v, thereby "burning'' a hole into the forward spectrum and suppressing ECC. Clearly, ELC cannot occur because an antiproton (p) does not support bound states. The presence of this "hole" for antiprotons may aiFord the opportunity to study the existence of wake-riding electrons. 2. Bound states in the wake We have calculated the bound-state spectrum in the wake for antiprotons at vr = 6 a.u.. For an estimate of the yield of wake-riding electrons it is crucial to take into account contributions from excited states in the wake. For excited states the anisotropic harmonic- oscillator approximation underlying calculations for the ground state breaks down. Figs. 2 a and b shows the potential along the beam (£) axis (i.e., p = Q) and perpendicular to the beam axis (z = zmin) where zmtn is the coordinate of the center of the first wake trough. A crucial point for the likelihood to observe wake-riding electrons is the fact that Zmin 2 (2 + s9n(Zr))^k€/4 (4) where A = 2xvr/ur is the wavelength of the wake oscillations and us, is the plasma fre- quency. Therefore, zmtm is approximately a factor 3 closer to an antiproton than to a proton. This increases the anharmonicity of the potential as clearly visible in Fig. 2a. The 243 -.1 -20 4) zfeu) Fig. 2a Wake potential near an antipcotoo with v, = 6 a.u. in carboa along the beam axis (p = 0). 0h -.1 -.2 10 20 Fig. 2b M s perpendicular to the beam iu(t* *„*, « -10.5 «.«.) 244 shape of the potential becomes important in the tail of the wavefunction which, in turn, enters the capture cross section for wake-riding states. In calculating approximate wave- functions we have retained the notion of approximate separability of the potential while incorporating anharznonic effects by writing13. 8.41 8.3- /^c~K . /—NJ~2 x=a.2- //7\ \ •-/ \ ' 8.1 - a a. CD -B.I- \\ / y\— -a.2- -B.3- -B.4- -8.5- -8.6: -8.7- -18 -28 -38 -4B Fig. 3 Wavefunction Xi(z) in the wake of an antiproton in carbon at vf = 6a.u. where S(z) and T(p) are the potential curves taken from the cuts through the potential surface (Fig. 2). The wavefunction can be written in factorized form as ,«m# (6) The resulting one-dimensional Schrddinger equations can be easily solved numerically. The solutions for the three lowest-lying states *,(r)(0 < * < 2) are shown in Fig. 3. Their binding energies (including the ground state energy in the p motion which closely rcwmblei a harmonic oscillator) are ««, =£ -0.1, ej S -0.03, and «j S 0.04. The second excited state is already a resonance in the continuum describing an electron bouncing forth and back 245 between the adjacent humps of the wake potential. Resonances in the continuum can contribute, however, when their lifetime due to autoionization is larger than the lifetime due to collisional destruction. The latter affects true bound states as well and can be taken into account within the framework of the transport theory discussed below. The numerically determined wavefunctions Xi(z) were fitted to a basis expansion of harmonic oscillator eigenstates. The latter are particularly convenient for calculating capture cross sections. It should be noticed that the accuracy of the resulting wavefunction is limited for two reasons: For highly excited states near the ionization (in our case, X1.2J the separability assumption(5) is no longer valid. We have investigated the energy spectrum in the near threshold regime of the wake using semiclassical methods16 and find significant deviations from a separable behavior characterized by ordered sequences of energy levels. Secondly, even though the fit to the harmonic oscillations basis converges rapidly in the Li norm (i.e. on the average) it can be quite inaccurate in the tail of the wavefunction which contributes substantially to the wake cross section. 3. Electron capture into the wake The transition amplitude for electron capture into wake-riding states is given in second- order Born (2) approximation by *./ = <*..*.|V, + VfCoV,|«,->, (7) where Go is the free-particle Green's function and Vrt are the (effective) interaction poten- tials of the projectile and target. At high speeds, capture from the K shell of the target (in the following carbon) dominates (i.e., $,- = $1,). Accordingly, Vt can be taken as the bare Coulomb potential of the carbon nucleus with an effective value Z, = 5.7, where we have taken into account screening effects by the passive la electron. The projectile potential Vp contains both the bare Coulomb potential and the dynamical screening potential. However, since capture requires a large momentum transfer in a hard collision and the dynamical screening potential is "soft,"1 only the Coulomb part is important. The evaluation of the 246 17 capture cross section ac using Eq (7) and standard techniques reveals the remarkable result that the first-order Born term (Bl) is negligibly small compared to the second-order Born term. This is a simple consequence of the fact that in the Bl approximation capture is mediated by high-momentum components of the initial- and final-state wavefunctiona. However, the "soft" wake potential exponentially suppresses high-momentum components •n *w«*c leading to an exceedingly small cross section11. The dominant contribution it therefore provided by the second-order Born term in Eq.{7) which closely resembles the well-known Thomas double-scattering mechanism19 for ion-atom collisions. Here an elec- tron is first scattered off the projectile by = 60°, followed by a second deflection at the target by about 60°, such that the electron finally propagates in approximately the forward direction at zero speed relative to the projectile. The fact that Vp is repulsive for antipro- tons rather than attractive for positively charged ions is immaterial since pure Coulomb scattering is invariant under charge conjugation (Zr -* -Zr) and resulting differences in the phase factor in the B2 term cancel because the Bl term is negligible. The velocity dependence of 2t 3 At v, ~ 6a.u. the cross section per carbon atom is of the order of ae » \0~ em . Using the numerical wavefunctions (Eq.6) instead of the harmonic oscillator approximation and including the first excited state increases the cross section by a factor ~ 1.3 at vr = 6a.u. Inclusion of the second excited state results in a increase by a factor 3. Since, on one hand, excited states contribute significantly to the total cross section while, on the other hand, they are rather inaccurate, there is considerably uncertainty as to the resulting emission spectrum. Their influence on the emission spectrum will be illustrated below. Considering the high solid target density and the fact that additional contributions from outer target shells have been neglected the experimental observation of a peak of wake- riding electrons should be within reach. For protons, on the other hand, the cross section is several orders of magnitude (> 6) smaller in the velocity range under consideration 247 because of the rapid decay of spatial overlap. This very likely accounts for the fact that wake-riding electrons have not yet been found10. Fig. 4 Cross sections as a function of the velocity i> of p used as input for the classical transport simulation. —, cross section for capture of a carbon /("-shell electron into the ground state (harmonic-oscillator approximation) of the first binding well of the wake dp,---, cross section for electron ejection with laboratory velocities t>, > y, in » binary collision with p is first Born approximation. 4. Transport of wake-riding electrons The wake-riding electrons, as well as electrons generated in binary-encounter events, suffer multiple scattering before exiting the foil. The determination of the observable emission spectrum requires the study of the electron transport in the presence of the field of the nearby projectile. We employ a microscopic Langevin equation,30 248 ^ (8) describing classical trajectories of an electron under the influence of the field of the pro- jectile as modified by the wake field. The electrons are subject to random forces F(t) representing stochastic collisions inside the solid. The complete solution of the transport problem is given by a Monte Carlo sampling of an ensemble of initial conditions for the phase-space coordinates which are propagated according to Eq. 8. The phase-space distribution of initial conditions consists of both binary-encounter (BE) electrons and wake-bound electrons. For the simulation of the initial velocity distri- bution of BE electrons, a first-order Born approximation for ionization has been employed. Only energetic electrons with vc > 0.8vr have been included in the transport calculation. The integrated BE cross section a BE for energetic electrons ve £ vr (Fig. 4) is several orders of magnitude larger than vc. In order to improve the statistics we have calculated the stochastic evolution of wake-bound electrons separately and added their distribution function, weighted by their relative cross section, to the binary distribution. The initial classical distribution of the wake-bound electrons is determined by the spatial probabil- 3 ity density distribution |4watt(r)| restricted to the classically allowed region and by a uniform distribution over all negative energies larger than the value of the wake minimum. In order to relate the dynamical evolution in the bulk to the post-foil experimental observation, modifications due to the penetration of the exit surface must be taken into account. The sudden breakdown of the dynamical screening near the projectile at the sur- face leads to a redistribution of the final-state population. This has particularly dramatic effects for wake-riding electrons that are in the close proximity of a repulsive Coulomb field of the.antiproton. Figure 5 shows the velocity distribution of wake electrons after the sud- den switch-on of the Coulomb field which can be interpreted as a Coulomb half scattering at the antiproton near the exit surface. The initial distribution prior to the break down of screening was an isotropic velocity distribution of the classical wake-bound states centered 249 about the projectile (v, = vr = 6 a.u.). The defocussisg by Coulomb scattering is clearly visible. Fig. 5 Velocity distribution of electrons originally bound to the wake inside the foil after (half) scattering in a pure Coulomb potential oip{vf = 6a.u.) upon exit from the solid. The effect of transport on the peak of wake-riding electron is illustrated in Fig. 6. Here we display the shape of the resulting peak of wake-riding electron* with and without modification by transport effects. The peak without multiple scattering corresponds to the singly-differential spectrum derived from the two-dimensional distribution of Fig. S integrated over the forward cone with cone half-angle 6 = 5*. Note that the peak position is shifted to lower velocities due to Coulomb defocussing, or more precisely, due to Coulomb half scattering. The shift in velocity is approximately given by where Ee is the Coulomb energy near the wake minimum 1 (10) 250 This is the amount of energy (or velocity) gained by the election in a Franck-Condon type transition from the wake potential curve to the Coulomb potential curve at the eat surface when the dynamical screening breaks down. The steady-state (or "equilibrium") diitribution resulting from steady-state production and decay by multiple scattering of '.V'alee-ridini g electrons as described by the Langevin equation (8) does not display a peak o.w o/a I 0.08 0.04 0.00 Fig. 6 Shape of the peak of wake-riding electrons emitted by an antiproton in carbon (vr s 6a.u.) into a cone with —«ni»nifr of Q - s*_ riding electrons without multiple scattering, - - - steady-state solution of the transport equation (8) for the distribution under the influence of multiple scattering. shift but an enhanced tail of low-energy electron due to slowing down and a tincrtsai of intensity near the peak. Note that the distributions of fig. 6 do not include the batkgnwad contribution due to multiply scattered binr T electrons. lairing the latter into ~*~nit the resulting distribution at electron velocities near the projectile velocity (3 < », < 7) » showa in Figs. ? and 8 for v, » 6c*. and cone hatf-angks 8§ « 5* and 2&*. In Fig. 7 251 we have ?—'">*'* that only the ground state and the first excited state contribute to the capture while in Fig. 8 we have included the second excited state in the continuum as well. In the absence of a usual cusp, three features are clearly -risible: a steep rise at the upper end of the spectrum which signifies the mrnants of the anticusp valley in the single-collision spectrum, a background due to multiply scattered binary electrons inside the valley, and a broad peak due to emission of wake-riding electrons near vt s: 5.6 on top of it. Since the wake-riding electrons give rise to a well-localised peak while the binary spectrum shows locally little angular dependence, the peak due to wake-riding electrons should become the dominant feature for sufficiently small 6s. The contribution of slowed- down binary electrons to the customary cusp peaks has also been found to be small31 for positively charged ions. 0.046 0.036 -' 0.02S 0.0« Fig. 7 Normalised convoy electron spectrum for antiprotons {t> = 6a.u.) emitted into a forward cone of half-angle 6s * 5#(—) nd 2.5* ( ), only ground and first excited state (x«,i) included. It should be stressed that the accuracy of the present calculation is limited due to the uncertainty in both the calculation of o, as illustrated in Figs. 7 and 8 and the treatment of the transport of binary electrons. Since the volume of velocity space of the observed 252 0045 0 035 to i 0025 0015 Fig. 8 As Fig. 7, but including the second excited state \2 (see »«**)• spectrum is small compared to the volume of velocity space of all initial conditions for bi- nary electrons which can contribute to the spectrum after multiple scattering, even modest statistical accuracy requires a large number of trajectories. We used a total of 1.8 x 10* initial conditions which resulted in 2s 600 events in the forward spectrum for a cone angle 0 s 5# and which required smoothing using large bin sizes {At- > 0.40o.u.)- The validity of the second Born approximation at only moderately high velocities (v, £ 6a.u.) may be questionable. Furthermore, we have observed that the cross section depends sensitively on the shape of the wavefunctions in the exponential tail which, in turn, may be affected by the separable form of the wake-bound state and by the plasmon pole approximation to a free-electron-gas model employed in the present calculation. In summary, while the observability of wake-riding electrons is not yet unambiguously established, our calculations show that chances for success are considerably higher for antiprotons that for protons. One of us (J.B.) should liks to thank Pedro Echenique, Rufus Ritchie, and Yasu Ya- mazaki for stimulating discussions. This was work was supported in part by the National 253 Science Foundation and by the U. S. Department of Energy under Contract No. DE AC05-84OR21400 with Martin Marietta Energy Systems Inc. References 1. N. Bohr, K. Dan. Vidensk. Mat.-Fys. Medd. 18, No. 8 (1948). 2. J. Neufeld and R.H. Ritchie, Phys. Rev. 98,1632 (1955), 99,1125 (1955). 3. P.M. Echenique, R.H. Ritchie, and W. Brandt, Phyt. Rev. B.20, 2567 (1979). 4. P.M. Echenique, W. Brandt, and R. Ritchie, Phys. Rev. B33, 43 (1986). 5. J. Lindhard, private communication, (1989). 6. N.V. Neelavathi, R.H. Ritchie, and W. Brandt, Phys. Rev. Lett. 33, 370, 640 £ (1974). W. Brandt and R. Ritchie, Phys. Lett 62A, 374 (1977). 7. M. Day, Phys. Rev. Lett.44, 752 (1980). 8. A. Mazarro, P. Echenique, and R. Ritchie, Phys. Rev. B31,4655 (1985). 9. A. Rivacoba and P. Echenique, Phys. Rev. B36, 2271 (1987). 10. M. Breinig et al., Phys. Rev. A25, 3015 (1982). 11. J. Burgdorfer, Pbya. Rev. A33, 1578 (1986). 12. Y. Yamazald et al., CERN Report No. CERN-PSCC 87-39 (unpublished), p. 108; CERN Report No. CERN PSCC 87 40 (unpublished), p. 108, and private commu- nication. 13. J. Burgdorfer, J. Wang, and J. Muller, Phys. Rev. Lett. 62,1599 (1989). 14. C. Garibotti and J. Miraglia, Phys. Rev. A21, 572 (1980); M. Brauner and J. Briggs, J. Phys. B19, L325 (1986). 15. J. Muller, Diploma thesis, Universitat Frankfurt, 1989 (unpublished). 16. J. Muller, J. Burgdorfer, and D. Noid, to be published. 254 17. J. Briggs and L. Dube, J. Phys. B13, 771 (2980). 18. Y. Yamaialri and N, Oda, Nud. Znstr. Meth. 194,415 (1982); P. Echenique and R. Ritchie, Phys. Lett. Ill A, 310 (1985). 19. R. Shakeahaft and L. Spruch, Rer. Mod. Phys. 51, 369 (1979). 20. J. Burgdocfer, in XVIICPEAG New York, Book of invited papers, Ed. A. Dalsr garno, (American Institute of Physics, 1989), is press; J. Burgdorfer, Transport Theory for Convoy Electrons and Rydberg Electrons in Solids, in "Lecture Notes in Physics", Vol. 294, p. 344, Springer-VerUg, Berlin (1988). 21. H. Schroder, Z. Phys. DT, 65 (1987). 255 NOH ranuiMTXvx SZEAVXOOIS ZH HCITATHW AR> IOHISATIOM or ATOMS IT FAST CHAISES MlXXCXZfl K. GAT1T and A. SALXV Laboratoire das Collisions Atoalquea, Universite Bordeaux I. 351, cours da la Liberation, 334OS Talanca Cadax, Franca ABSTRACT Non parturbaciva bahavioura in collisiona at rathar high impact valocicias ara axhibicad by means of apodal theoretical treatments. In tba casas of excitation and loniaaelon procaaaaa, it la shown that saturation effaces nay appear in contradiction with tha pradiceiona of the firat Born approximation . 1 - INTRODUCTION Tha elaceronic stopping power for faat particles of charge ZL travailing in aattsr, is generally thought to ba proportional te zj. Indeed, the description of elementary procaaaaa arlaing in atomic colli- siona at high inpact velocities, ia often made by means of perturbation treatcents like the first Born approximation (BI) for excitation and ioni* sation [1] or the Continuum Distorted Wave (CStf) theory for charge transfer [2-4] . For tha first two processes, BI predicts that total crosa sections are proportional to Z* . However, non-perturbative bahavioura cf chaaa processes have bean pointed out recently at moderately high impact velocities. An cxperiaantal study and a special theoretlcel analyais of tha excitation of tha levels Furthermore, the investigation of tha simultaneous capture and ionisation by impact of various bara ions (H*. Ha2*, 11s*. 0**), impinging at 1 MeV/nucleon on helium, showed tha standard BI to ba inadequate to describe the ionisation process at small impact parameters [8]. It In shown that the ionisation probability is also satureceo. in tha later case. A similar behaviour la also exhibited far double ionisation of helium by bara ions [9]. Whether or not tha predlctiona for tha stopping power might b* changed in view of these new results is in open question. 256 2 - T^E SATOBATIOH OF THE ELECTBOMIC EXCITATION OF HIGHLY H1A&CSP gfg rnT.T.TftTWC UITH ATOMS OF CAS The model, that has been adopted to investigate the excitation process, is strongly influenced by the actual collisions under considers* cion. The abovementioned excitation of the ion Fe2** impinging *t 400 HeV in various dilute gases is a typical instance. Therefore, the conditions of such collisions will be examined in order to introduce a simplified Modal. Since one looks after the excitation of the impinging ions Fe2**(ls2), the letters are called the "targets* in whet follows. In the same way, the atoms of the gas are named the "projectiles". A 400 HeV laboratory energy of iron ions corresponds to an impact velocity of 16.91 a.u.. To a first approximation, the electrons of the projectiles may be considered as free electrons impinging on targets Fe1** with an Impact velocity of 16.91 a.u.. Since a velocity of about 22 a.u. is required for free electrons to excite the Lyac helium-like line of Fe2t* , projectile electrons are unlikely to excite it directly. Indeed, high momentum components may be found in the orbital momentum distribution of projectile electrons. However, high aomen- turn electrons are located close to the projectile nucleus. Since impact paraaeters ouch less than the radius of the K-shell In the projectile contribute appreciably to the excitation, the nucleus of the projectile appears much aore likely to excite directly Fe2** than electrons bound to it. Thus, a reasonable model for direct excitation considers this latter to be due mainly to the Coulomb interactions between the target Fe2**and the projectile nucleus. Multiple electron processes, such a» simultaneous ioni- sation of Fe2** and charge transfer to Fe2** might also be invoked. However, they have been estimated not to be a dominant process [6]. Furthermore, in a collision where the projectile is a neutral atom, the process of electron transfer from the target into the projectile is strongly inhibited. Hence, the present model ignores the coupling between excitation and electron capture. Finally, a highly charged helium-like ion may be considered, to a good approximation, *» an ion with two hydrogen- like orbltals, which are defined with a screened nuclear charge. Thus, our model for direct excitation consists in a projectile made of a bare nucleus of charge Zp Impinging on a target made of a hydrogen- like ion. the nuclear charge of which is Zj. The projectile may excite or ionise the target but is prevented from capturing the electron. The theoretical investigation of the excitation process starts from the fractional form of the Schwinger variational principle derived in the eikonal approximation (5,10}. In this treatment, the amplitude to excite the target from the state lct> to the state 10> is. ( 4£ IVI a ) ( e ivl < ) (1) v - v c; v 257 where u£ / and i|£ ^ «ra cha standard scattering wava funcions, V Is tha targec projeccile ir.caracclon and Gj is cha target propagator. Tha expression (1) is stationary with raspaet to slight variations of j T,,- I( 31 Vli )(0'')n( J IVlB) (2) l.J DJf-( J | V - VC; V | l) (3) vhara li> and I j > balong to B, and Bj raspactivaly. Tha axprassion (2} of T3a is stationary whan B, and S^ ara enlarged. In tha last iapleaentation of cha fomalisa. Cha sacond Bora-like Matrix elements ^ j |v G* V I 1 ^ have baan avaluatad through an axpansion of Gj on th« whole discraca sptccrua of cha targec [7,11,13] . Furthermore, cha convergence co cha stationary result has baan tasted through a stretching of tha sizes of A, and Bj . To achieve chis goal, A, and Bj have been chosen in tha complete sec of trsvlling eiganscatas of tha target. One has considered the excita- tion of che levels (Is, 21) and (Is. It) of fa2** by impact of various nuclei [7,11] and che excitation of tha levels 21 and 3JS of hydrogen atom* by proton iapaccs [11,13], where t - 0.1. Sines we are dealing with target excitation froa the ground state, the states ID and IJ> are restricted to a subset which contains the lowest target states, including la) and 10) [7.11.131. Thus two kinds of avitrix elaaencs have eo be calculated : (1} First Born-like eleaents (j I V I i) - BJ^1 (Z,,. Zy, v ) (4) (ii) Second Born-like elements where v is the iapact velocity. The actual perturbing potential is obtained by omitting the long range Couloab Interaction V between the projectile and the whole target i.a. : V -i {6) 258 whara R is the incernuc', .ar distance while Z, and ZT are the charges of the projectile and of the target nucleus respectively. It is easy to show that V may be ignored in the deteraination of the transition probability J3]. thus, the potential responsible for the transition is : V - Z, [i - -~-T-\ (7) (R IR - xl where x is the target nucleus-electron distance.Then, for each pair (i.j). the following scaling law may be established for the k-th Born-like matrix element : 3 if* (7,. 2T, v) - Z% if** (I, ZT. v) (B) Thus, the matrix elements D; ( defined in the expression (3) may be written: Dn - Z,, Bjl (1. ZT. v) - Z\ RJP (I. ZT. v) (9) while one has : (j I V 1 i) - Zp Bj?" (1, 2T, v) (10) When Zp is high enough, the matrix D is doninated by the second 3orn teras 2 and D'' becoaes proportional to Z^ . Thus. T(ja tends to a finite value independent of Zp. It is known as the "s*cur*cion effect" of excitation cross sections when the charge of the projectile is increased [5,10]. For, che excitation of Fe2**(lsz) to the levels (Is, nl) where n - 2, 3 and € - 0, 1, the largest basis set is made of the 5 orbitals Is. ns , np0 , np., and np., . However three stages of approximation have been considered for and j*p^. in order to the test the convergence. \ (i) < > and are replaced by la> and 10) respectively i:i expression (1). The procedure is called the Schwinger-Born approximation (SB). (ii) |u£^ and W'g// are expanded on a two-state basis set made of la) and l@>. The procedure is refered to as Schwinger 2-2 (S22). (iii) Finally, Ar 1 I Figure 1 - Excitation of the level (ls.2p) of Fez**(ls2). impinging at an energy of -CO MeV, in che laboratory fraae. on various atons of nuclear charges Z^. The total excitation cross sections per electron axe. indicated in co2. They are plotted as functions of Zp. Experimental data [5,6] : £ population cross sections ; • direct excitation cross sectioncti s obtained after substraction of the cascade contribution (10 X for He and N, 20 X for Ar) and an estimation of the double process contribution (30 Z for N, 50 Z for Ar). These contributions have not been estiaated far Xr [6]. Theoretical results : — • — BI ; — • » — BII ; — + — SB : S22 ; S55 ; —x—FSS5 .'5,6j (see text) ; A Reading's coupled state calculations 112]. Ar _. L. '5 us a » a is 2, Figure 2 - sane legend as figure 1, but for the excitation of the level (ls,3p) 260 The coca! cross section for che excitation of Fe24* by various aeons (by various nuclei in our model) arc indicated on figures 1-2. Also plotted are che first (Bl) and second (BII) Bern approximations, as well as che first S55 results (FS55) obtained by Brendl* and Gayet [5,6.10] with a Halted expansion of che propagator fl*.. The present SSS results are close Co che FSSS ones. Details about numerical and analytical calculations may be found elsewhere [10,11]. Ic Is interesting to notice that S22 and S55 are rather close to each ocher. Except for che excitation of the level 3p, SB lies far from S55. Furcheraore, its behaviour is often different from the ones of S22 and S55 (7,11]. The new SSS results agree well with experimental data for the exci- tation of che levels 2p and 3p. A discrepancy with che results of Reading's coupled state calculations [12] appears for the excitation to 2p by argon impact. Ic is more pronounced with our new results than with the previous FS55 ones. However, SSS values for the excitation to 3p agree well with Reading's ones. Ic is worth noting chat the saturation of total cross sections that is observed experiaentally is the one predicted by our varisxlonal treatment. Thus, we have siiown, chat non perturbative behaviours may occur ac rather high iapact velocities for processes «s simple as the excitation process. In any case, El lies far above both experimental and SSS theore- tical results. Values provided by BIX are even worse except for low Z,(Zp< 3) where perturbation conditions are fulfilled. In the lacer case, all theories are in good agreement. Thus, the effort required to make a BIZ prediction appears not Co be worth che trouble. 3 - SIMULTANEOUS CAPTURE AND IOWISATIOH OF HELIUM Recent coincidence experiments (14,15,16} have been performed to study the reactions : Iq* + H«s(ls2) — i<"--"* + He* (11) — !<«•"•+ He** + e' (12) where Iq* is an ion of charge q. Since the helium target is in a singlet state, che application of an Independent Electron Model (IEM) is particularly simple. Let us denote by P£(P) and Pi(P) the single electron probabilities of capture and ionisation, respectively, in a collision with iapact parameter p.- In the present case, to be consistent with the indepen- dent electron model, Pe>j(P) has been calculated using a Harcree-Fock descripcion of che initial He(ls z) state ; further, one assumes that the second electron is not affected while the first one is ejected or captured. The single capture cross sections without or with simultaneous ionisation are 261 J^ 2Pe(P)(l • P, (P)) <*. , " 2» £ PdP 2Pe (p> P, (p) (14) respectively, if w« n«gl«cc any other channel (e.g. double capture). Tha avaluaclon of Pfi(p) haa baan made using the Continuum Distorted Wave fCDUA) codes of Belkic et «1 [17] to calculate the amplitudes of single capture. Then, the Hankel tranafonu of the latter ones penlt to evaluate Pe(P) [18]. Capcure into excited states nf, which plays an appreciable role in most cases considered here, is accounted for through tha n^3 lav (see e.g. [3]) : 3 Pe(P) a 2- pe (nf.P) +pe(n0.p) nj £. nf (15) nf-l nf- nj+1 where pc(nf,p) is the probability of capture into the final state nf. In 3 any case, ve have checked that i^ is large enough to ensure an n'f behaviour of pe(nf-,p) . For instance, a value of Kg as large as ten is s required for 0 * projectiles. The accuracy in tha calculations of pe(nf,P) has been verified by an integration over P in order to get tha total capture cross sections. Then, the latter is compared wieh that given by tha standard CDV2 code of Belkic et al [17]. The accuracy is found to be better Chan 1 Z in each case. The ionisation probability has baan calculated using two different approaches : che Born approximation and the Multlpole Expansion Defined on One Centre (MEDOC), a procedure introduced by Cheshire and Sullivan [19]. In che first Sorn calculations, the Initial (bound) and final (continuum) states of the ejected electron ar« known from a numerical integration of the Schrddlnger equation for an electron in the field of a potential VT(r), where r(r,9, For large projectile charges 2 , P, (P) as calculated with the Born approximation, aay be larger than one at small Impact parameters. The perturbation approach is clearly unsuitable to this case since the Sorn series is equivalent to a series in powers of Z,. Therefore, us made use of che MEDOC non-perturbative numerical method, which consists in expanding che one-electron wavefunction in partial waves around tha targee : * (r.c) - £* V5(8. The tiM-dependent Schrodlnger equation describing the aoclon of Che electron in both the potential VT (r) and che daw-dependant field of the projectile can be put in the fora of a set of coupled differencial equations in both r and t of Che type : - Z R&,,,(r.t) Lt,m,iz.z) (17) { 2 dt2 flr ) I'm' whero R'?>( are aatrix oleaent* of che potential between spherical harao- nics Y]J and Yj! . Thes« aquations are integrated by a finite difference procedure, in a code Bade by Salln (unpublished yet). Two points should be stressed. (I) The aethod cannot be expected to be adequate when the capture probabi- lity is large. This does not occur in Che cases considered here. (II) The limitation, therefore, is on the number of partial wrm* included in the expansion (not on the strenghc of the potential). In the present case we have included € - 0,1. A discussion on the accuracy of this approximation is given below together with the analysis of the results. The nost Important fact is illustrated in figure 3 : che capture probability decreases very rapidly for all che cases considered here when P > 0.5 au. This is in sharp contrast with the ionisation probability which decreases very slowly with impact parameter due to the dominant contribu- tion of the dlpole tera in the ionisaclon amplitude (At - 1). To evaluate che total ionisation cross section in the saae energy range, one aust inte- gr&re up to iapacc paraaecers larger than 10 au. Furthermore, che monopol* eera (A£ - 0), which plays a weak role in che total cross section, is doal- nant at saall iapact paraaecers. Therefore, no conclusion on siaultaneoua capture and ionisation can be drawn by extrapolating qualitative ideas on total ioaisation processes. A second important fact is chac the saturation effects aentioned above are auch stronger for saall lapact paraaeters (figure 3). For 0** iapacc, che Born approximation to P,(p) gives probabilities larger than one at saall iapact paraaeters. Thus, deviations froa che Born approximation are auch aore visible in siaultaneous capture and ionisation processes than in single ionisacion collisions. Hence, aeasureaencs on double active- electron processes in heavy-ion-atoa collisions at high iapacc velocities are very welcoae. a Our results for <*t\/ t are coapared with experimental data in figure 4 and in cable 2. Besides che use of che independent-electron uodel. their aain limitation lies in the introduction of t « 0, 1 only in the MEDOC calculations. We can evaluate the accuracy of this approximation by looking at Che convergence of the Born expansion with I. If we include only I - 0, 1 in che Born approximation, then che error in P,(P) is of the order of 10 Z. Of course ve cannot completely trust this argument for 0** iapact since the saturation effect is so strong that che Born results are unrealistic. However, even in this case, che contribution of t values larger Chan one decreases as p decreases. 263 peroMttr lou) Figure 3 - Probabilities of capture Pc(p; and ionisation P,(p) of one electron of helium by s 1 0 * impact at 0.7 MEV amu , , P£ (p) normalised to 1 at P - 0.05 au where its value is 0.298 ; ---, P,(P) calculated by the MEDOC procedure (see text) with t - 0,1 ; —••—P,(P) calculated by the first Bora approximation with I • 0, ...,4. 1 • OS 1.0 lapoct Mtr«y Figure 4 - Ratio of the cross section of siaultaneous capture and ionisation to the single capture cross section cr. (/cr= as a function of the iapact energy in MeV asm'' for various projectiles (H*, He2*, Li3*, 08*). Theoretical predictions [20] : evaluation of capture by che CDW approximation and of ionisation by the Born approximation C5) and/or by the MEDOC treataent (•). Experiaental data : 0. [14] ; A. [15] : o, [l«]. 264 Table 1 - Cross sections for single capture and for capture with lonisation of heliua by ion impact (in ca2)+ab stand* for a x 106 CDW+Born CDU+MEDOC Experiment Energy Ion (MeV anu'1 H* 0.4 1.9"'9 1.2-2° 6.3-2 1.9-1' 9.7'21 5.0-2 7.r21 3.2-2 0.75 1.2-20 4.1-22 3.5-2 1.0 3.O"21 7.9-23 2.7-2 3.0-2 1.5 7.3-2* 1.8"2 He2* 0.4 2.1"'8 7.1"19 3.3"1 2.3'18 4.9"19 2.V1 3.r'» 7-1? 1.4-' 0.75 1.8-'9 2.9-2° x.64-1 1.8'19 2.6-2° 1.42"1 1.1-' 1.0 5.O-20 6.1'21 1.21'1 1.0*' 1.5 7.7"21 6.1"22 7.9-2 8.5-2 Li3* 0.4 3.9•18 5.7-18 1.3 6.3'18 2.7'18 4.3'1 7.7'18 3.0''8 3.9'1 8< 0 0.75 5.I'18 6.9'18 1.37 1.1 1.0 1.8'18 1.9'18 1.05 0.9 1.5 3.7"19 2.8"19 7.7'1 0.5 Table 2 - Ionization cross sections of heliua by impact of ions with charge 2 Zp at 1.4 Mev/aau in en ) Single ionization Double ionizacion Born MEDOC Born MEDCC+ Experiment Born MEOOC Experiment (t-O.l) 1 1.6'17 1.78"18 5.7-2° 15 3.6"15 1.3'ts 3.96"1' 1.69'15 1.79-15 2.9-15 3.0-16 2. 20 6.4"15 2.03'15 6.96-'* 2.7"" 2.60'15 9.12-15 4.8-'• 36 2.07-'* 3.9'" 2.26*15 6.16-15 5.72-15 9.6"1* 1.4-;5 1.7" 13 44 3.1"1* 4.8-'5 3.37-'s 8.17"15 7.21-15 2.r 2.0"15 265 Tha agreement bacwaan our calculations and experiment is vary good. Strong deviations from cha Born approximation for P, can be obsarvad. It is particularly striking in tha casa of Li3* impact. Evan for H* impact,we gat a 25 X diffaranca at 400 IcaV, although for tha sasw anargy, tha total ioni- sation cross saction Is in full agreeaent with tha Born approximation [14]. We nota that our calculations with tha MEOOC approach for H* and Ha* at 0.4 MeV amu'1 show a discrapancy of 35 Z with tha experimental data. Excopt In thasa last casas, our rasults show that tha experimental data in coapa> tibia with an indapandant alactron model if an accurate «valuation is made within this modal. Consequently, capcura with ionlsation can ba usad to proba tha small impact parameter ionlsation probability. It is worth noting that Salin [21] showad that siailar conclusions can ba drawn in studying singla and douhla ionlsation hallua by multi- chargad ion impact at 1.4 MsV aau*' . His fraaaworlc is axactly tha saaa as cha ona quotad abova, i.a. : I EM and either tha Born approximation or tha HEDOC traatnant with < - 0,1 to gat tha lonisatlon probability. Tha singla (S) and double (D) lonisation cross sections aay ba written as : too " 2 JJoPd P 2 PfCp) [1 'Pi(p) 1 cl8) This work is particularly interesting since it is shown that saturation effects (due to non perturbativa contributions) can show up even for single ionisation. Salin's rasults are quoted in table 2. Because of the limitations of the MEDOC procedure already Mentioned , ona cannot expect MEDOC predictions to be reliable when capture processes are 5*por- tant. In the present application, CDWA calculations indicate that it might be the case for Zp£ 36 [21]. As co contributions of partial waves t>2. they have been evaluated in the Born approximation for single ionlsation. For Zpi 36, they appaar to be comparable to MEDOC results for £-0,1 (see table 2). Although Salin's results are not fully reliable for Zpi 36, they compare favourably to experimental data in any case. The difference betwaan MEDOC and Born predictions for double ionisation is impressive, indicating that perturbative treatments are irrelevant to double (and even single) ionisation when the energy of the projectile is not high enough. Thus, an independent electron model can be a powerful tool to investigate non perturbative behaviours of the lonisation process in simultaneous cap cure and ionisacion as well *s in double ionisation by fast ion impacts. COMCLPSIOH Ue have shown that non perturbativa behaviours show up even at high impact velocities. Therefore, care mist be exercised when dealing with high energy collisions. Indeed, the widely used Z| behaviour of excitation and ionisation processes cannot always be trusted. 266 [ 1] I. Laatavan-Vaiaaa, D. Hannacare end R. Gayat, J. Phya. Francs, 4f (1988) 1528 [ 2] R. Gayat. J. Phya. B 96 (1972) 483 [ 3] Dr. Balktc, R. Cayat and A. Salin, Phya. Rap. < (1979) 279 { 4] Dz. Balklc, R. Gayat and A. Saiin. Coaput. Pbyx. Coaaun. 32 (1984) 385 [5] B. Brandla. R. Gayat, J.P. Rozat and K. tfohrar. Phys. Rav. Latt. 24 (1985) 2007 [ 6] K.tfohrar, A . Chatloui, J.F. Roxat, A. Jolly, F. Faraandez, C.Staphan. B. Brandla and R. Gayat, J. Phya. B 19 (1986) 1997 [7] R. Cayat and M. Eouaaoud, (fuel. Inatrua. and Math. B 42 (1989) 515 [ 8) R. Gayat and A. Salin, J. Fhya. B 20 (1987) L571 [ 9] A. Salln, Fhya. Rav. A 3« (198?) 5471 [10] B. Brandla, Thaaa da Trolsiaaa Cyela. Wnivaraita Bordaaux 1. (1984), unpubllshad (availabla on raquaat ; taxt in franch) [11] M. Bouaaoud, Thaaa d'Etat, Univarsica Bordaaux I, (1988), unpublished (availabla on raquaat ; taxe in franch) [12] J.F. Raading, 1983, privata coaounlcaticn [13] H. Bouaaoud and R Gayac, Third Europaan ConCaranca on Atoale and Holacular Physics, Book of Abatract p 491, Bordaaux 1989, A. Salln Editor [U] H.B. Shah and H.B. Gilbody, J. Fhya. B 18 (1983) 899 115] H. Knudtan, L.K. Andaraan, P. Hvalplund, J. Soranaan and D. Ciric, J. Fhya. B 20 (1987) L253 [16] J.A. Tanix, M.W. Clark, R. Prica, S.M. Farguson and R.E. Olson, Mucl. Inatrua. Hath. B 23 (1987) 167 [17] Oz. Balkic, R. Gayat and A. Salin, Coaput. Fhya. Coaaun. 21 (1981) 153 ; Couvut. Phys. Coaaun. 22. (1983) 193 ; Coaput. Fhya. Coaaun. 22 (1984) 385 [18] R. HcCavroil and A. Salin. J. Phya. B 1 (1968) 163-71 ; J. Fhya. B U (1978) L693 [19] J.M. Chashira and B.C. Sullivan. Phya. Rav. J£2 C1967) 4 [20] R. Gayat and A. Salin, J. Fhya. B 20 (1987) LS71 [21] A. Salin. Fhya. Rav. A ?a (1987) 5471 267 Target thickness dependent convoy electron production of aligned molecular HeH*-ions' J.Kemnler*, K.O.Groeneveld, O.Heil, K.Kroneberger, H.Bothard Institut fur Kernphysik dor J.W.Goethe Universitat D-6000 Frankfurt am Main, Germany Many experiments have shown that the collective excitation of valence elec- trons effects the alignment [1] and the stopping power [2] of fast diatomic molecular ions penetrating thin solids. Collective excitations also seem to play an important role for the total electron emission induced by fast ate ic and molecular ions [3]. For a specific part of the overall electron distribution the convoy electrons, accompanying the projectile ion with equal speed and direction (v.*vp), production mechanism had been proposed which attach their origin to the wake-potential [4],[5]. Because of the long range effects caused by the wake-structure molecular projectile ions are here a sensitive tool to investigate the few Angstrom environment of a fast moving ion in a solid. When the molecular ions enters the foil the binding electrons are stripped of and both fragments are mutually repelled by the Coulomb explosion: Measuring a physical signal as a function of target thickness means here to observe it for different lnternuclear distances r*. The wake-potential is most pronounced in the direction parallel to the beam. Therefore an experiment deals with the problem that the observed effects are smeared out because of the randomly oriented axes of the mole- cular ions. Until now, no simple method is known to deliver molecular projectile ions with the lnternuclear axis aligned to the beam direction, so a selection of the corresponding events after the target is necessary. The detection of the Coulomb exploding fragments with isotropic orienta- tions requires a complex experimental setup [6]. In a recent experuMnt there the convoy electron yield Yc{-*) for ffc* penetrating a thin carbon foil has been iwasured as a function of orientation of the die luster axis. It has been found that the yield Yc{•*=&) tor fragment ions aligned to the This work has been funded by the German Federal Minister for Re- search and Technology (OffT) under the contract Number 060F173/2 Ti 476, DFG/Bonn. present address: IPN Lyon-1, Universite Claude-Bernard, F-69622 Lyon, France 268 EXPERIMENT He HeH -> C - FOIL rw PPArm O DIAPHRAGM AS HV TARGET ELECTRON TAC SPECTROMETER TIME 1: Experunentai set up for the detection of aligned fragments beam direction is smaller than for fragment ions with their axis perpendi- cular to the beam ( Yc(^=O°) < Yc(-*=90°) ). A result which does not fit into the proposed mechanism for wake-riding of convoy electrons. But also the dependence of the stopping power for N2* icns compared to the stopping for the atonic constituents cannot not yet be explained with the wake mechanism [7]. One important problem here is the effective charge or better the strength of the coupling between the fast projectile ion and the valence electrons in the solid. To get a better understanding for the contribution of the wake potential to convoy electron production it is therefore better to choose a sampler collision system. Because a complex device as described in ref. [6] was not available we used a simpler experimental set up which was able to detect the fraction w;nch is aligned with respect to the beam direction. The molecular ion beam penetrates a thin carbon target vhich is placed m the focus point of a magnetic sector spectrometer fig. 1). The emitted electrons are deflected by 90° and registered with a channeitron, whereas the Coulcmb exploding molecular fragments remain undisturbed. After a 269 MeV/u) -> 8O -. HeH* (O.3 c - FOIL «O ' f'"\ At 5.8/ 4O • • leading 1trailing H* CO 2O Z 3 O u 40 5O eo 7O TIME Fig. 2: Time spectrum of a coincidence between convoy electrons and H* fragments from HeH* (0.3 MeV/u). Trailing and leading H*-ions are clearly distinguishable. flight path f of 3.41 m the internuciear separation for HeH* of 0.3 MeV/u reaches 41 im so that only correlated fragments inclined by 5° can pass the diaphragm. With an electric field the fragments of different charge state and mass are separated before the reach the position sensitive parallel plate avalanche detector (FPAD). The separation between leading and trai- ling H+-ions is achieved with a time of flight technique which requires a start signal delivered here by convoy electrons produced during the pene- tration of the molecular ion through the solid. Because uncorrelated frag- ments, i.e. single He- and H-fragment which are strongly inclined with respect to the beam axis, can also pass the diaphragm with a second coinci- dence circuit it is ensured that only those events are counted when both, the He- and the H-ion reach the FPAD. It has been found that this condition is satisfied only by 5 % of all ions entering the detector. In a feasibility experiment [8] we studied the target thickness dependant convoy electron yield for aligned molecular projectile ions and compared it 270 1.75 HeH (O.3 MeV/u) -> C - FOIL • ALIGNED 0 > 1.5O • O RANDOM I u 1.25 (He H ; * u 1.OO > 0 Ra t 5 1O 15 2O TARGET THICKNESS ex Cpg/cm23 Fig. 3: Ratio of the molecular convoy electron yield to the sun of the yields for the atomic constituents for aligned and randomly oriented mole- cular fragments. with the yield registered with randomly oriented molecular ions. In fig. 2 a time spectrun which was started with convoy electrons and stopped with the H* fragments from HeH* {0.3 MeV/u) on carbon foil is shown. Although the time resolution of FWH = 4.7 ns was not very accurate the fraction of trailing and leading H* ions is clearly distinguishable. In table 1 the ration of both fractions Htr«ii/i««d is plotted as a func- tion of specific target thickness ex. This ratio increases with increasing target thickness signifying that loosing the correlation to the He-ion is less probable for trailing fragments. This result can be interpreted as an effect of the wake force behind the fast He-ion. In contrast to these findings the yield of convoy electron shows for alig- ned and randan oriented molecular axis the same target thickness dependence and therefore presented no hints for a wake potential influenced ccnvoy electron production. However, although the convoy electron yield is inde- pendent from the orientation of the dicluster axis, it is still a function of the inteniuclear separatico rx which demonstrates the influence of the correlated action of the charges of He and H. Concluding, we presented a simple method for the detection of aligned dia- tomic molecular ions penetrating thin foils. Our first experimental results gave hints that the molecular enhancement for the convoy electron produc- tion of molecular ions is not swply a direct effect of the wake potential, but more contributed to the correlated action of the fast projectile ions m the solid. Tab. 1: Ratio of trailing and leading H* ions Htraii/iaad for HeH* (0.3 Hev/u) in coincidence with convoy electrons for different specific target thicknesses ex. Also the non coincident charge state fractions F(q) for He* and He** are given. Target thickness Htrai 1/lacd charge state fraction [ng/cm2 3 He* He** [%] 5 :t 2 2.05 ± 0.21 32 68 6.5 ii 1.5 2.35 ± 0.25 26 74 8 ii 2 3.05 ± 0.32 22 78 20 it 2 3.4 ± 0.35 22 78 atomic value [9] 16 84 References [1] Z.Vager, D.S.Genmell, B.J.Zabransky, Phys. Rev. A 14 (1976) 638 [2] J.W.Tape, W.M.Gibson, J.Remillieux, R.Lauberti, H.F.Wegner, Itel. Instr. Meth. 132 (1976) 75 [3] K.Kroneberger, A.Clouvas, G.Scnlussler, P.Koscnar, J.Keanler, H.Rothard, C.Biedermann, O.Heil, M.Burkhard, K.O.Groeneveld, NUcl. Instr. Meth. B29 (1988) 621 [4] V.N.Neelavathie, R.H.Ritchie, W.Brandt, Phys. Rev. Lett. 33 (1974) 302 [5] V.H.Ponce, W.Meckbach, Can. At. Mol. Phys. 10 (1981) 231 [6] Z.Vaager, E.Kanter, Nucl. Instr. Meth. B33 (1988) 98 [7] M.Steuer. D.S.Geranell, E.Kanter, E.Johnson, B.Zabransky, IEEE KS-30 No. 2 (1983) 1069 [8] J.Kemnler, Dissertation 1988, Universitat Frankfurt am Main, Germany [9] J.B. Marion, F.C. Young, Nuclear Research Analysis North Holland Publ. Corp. (1968) Amsterdam 273 RADIATIVE ELECTRON CAPTURE BY CHANNELED IONS J. M. PITARKE Fisika Teorikoa Saila, Zicntzi Fakultatea, Euskal Herriko'Jnibertsitatea, 644 Posta kutxatila 48080 Bilbo, Basque Country, Spain and R.H. RITCHIE Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge. TN 37831-6123 USA and The University of Tennessee, Department of Physics, Knoxville, TN 37996 USA Abstract Considerable experimental data have been accumulated relative to the emission of photons accompanying electron capture by swift, highly stripped atoms penetrating crystalline matter under channeling conditions. Recent data suggest that the photon energies may be less than that expected from simple considerations of transitions from the valence band of the solid to hydrogenic states on the moving ion. We have studied theoretically the impact parameter dependence of the radiative electron capture (REC) process, the effect of the ion's wake and the effect of capture from inner shells of the solid on the photon emission probability, using a statistical approach. Numerical comparisons of our results with experiment are made. 274 1. Introduction Many processes can give rise to X-ray production when energetic heavy ions interact with solids. Radiative electron capture (REC) is said to occur when an ion captures an electron into one of its shells and emits a photon1. This process contributes only in a very small way to the total production of radiation; however, it becomes more important and amenable to interpretation when the bombarding ions are channeled through the solid, since in these conditions the ions never approach closer than 0.1 or 0.2A to target atoms, so that the yield of characteristic X-rays, one of the most important contributions to the total production of radiation, is very much smaller than that observed when the beam is incident on the crystal in a random direction. Furthermore, it has been shown that stripped, incident channeled ions have "frozen" charge states, and although nonradiative electron capture is the dominant mechanism for charge transfer, in ion channeling the inhibition of close collisions makes the radiative process relatively more important2. The REC process has the possibility of becoming a significant tool in the study of the momentum distribution of target electrons, so that some experimental data3"7 have been accumulated. Indeed, recent data7 suggest that the photon energies may be less than that expected from simple considerations of transitions from the valence band of the solid to hydrogenic states on the moving ion. The purpose of this paper is to extend previous5-8-9 theoretical approaches, and to study theoretically the impact parameter dependence of the REC process and the effect of capture from inner shells of the solid. 2.- Theory We consider a solid, through which a swift, heavy, stripped ion of charge Zj and velocity v passes and captures an electron accompanied by the emission of a single photon. The 275 nonrelativistic Hamiltonian of the system may be written as H=H0+Hrad+H< • <» where H^ and HTad are the hamiltonians of the electron-ion system and the free radiation field, respectively, and H1 = - - A . V (2) c is the interaction hamiltonian between them, with C \ ik.r / + \ .-. e e (3) -) k, ("fc^.J- kX ^ V ak x and ek x are, respectively, the annihilation operator and the polarization vector of a photon with wave vector k and polarization index X(K= 1,2), which in free space is perpendicular to the direction of propagation, coj^k is the frequency of a photon of wave vector k, and V is the normalization volume of the radiation field. The movement of the ion relative to the center of mass of the system has been neglected, as well as a term in H' which is quadratic in the potential vector. Unless otherwise stated, atomic units arc used throughout The transition probability for a target electron to be captured into a k-shell orbital of the moving ion, emitting a photon, may be written in first order perturbation theory as follows: iov t [dte (4) where < H'fj ( t) > represents the time-dependent matrix element of the matter-radiation interaction: < H'fi (t) > = < 01 afc < -iv.r Here (pj (r, t) is the wave function of the electron in the initial state, and JO > is the vacuum 276 state of the radiation field. In eq. (6) b represents the impact parameter. Eg represents the binding energy of the captured electron in the K orbital: 1 , 7t Z, CO E^=2Zr-^r ' (7) and E, = jv2 , (8) with v the velocity of the ion. The effect of the ion's wake has been considered including in eq. (7) the potential of the polarized medium at the site of the ion10. We should mention, however, that for typical REC experiments this potential is approximately 15eV, in such a way that this correction turns out to be not greater than a 1 or 2% of the total REC energy. REC processes can proceed by capture of free, as well as bound electrons. In particular, with the aim of understanding how the emission probability corresponding to capture of bound electrons depends on the impact parameter, one can write, on very general grounds, the initial wave function of the target electron as follows: • io>,i (10) p = q + k +v , (11) cof= cof - a , (12) and d£2 is the infinitesimal element of solid angle in the direction of emission. The spectrum obtained in this way has been found to be very much wider than that for capture from the 277 electron gas; moreover, it has been found to be wider as the impact parameter increases, as is obvious from figure 1, where the REC peak widths obtained with eq. (10) are plotted against the impact parameter when photons are emitted parallel to the beam direction. In order to study in detail the effect of capture from inner shells of the solid on the photon emission probability, a knowledge of the many-electron wave function of the solid is required; however, in order to simplify we adopt a statistical approximation5, assuming that the response of an infinitesimal element of volume in the solid at position r, where the total density is n(r), is the same as that of an electron gas at that density. Thus, we define the local Fermi energy to be 2/3 — . (13) 2 when capture occurs from the vicinity of r. Assuming in this approximation that the wave functions of electrons in a given part of the solid are plane waves: where qj represents the initial momentum of the electron and qF is the Fermi momentum (qF=1.92/rs), the REC probability per unit energy and solid angle of the emitted radiation is" '(zj+qf)4 where q'i = qj - k - v . (16) and 2 ky + 17 The integral over qt in (15) covers a Fermi sphere of electrons, so that we obtain, after some 278 algebra, the following expression: .2 ,,2 d3V ^"k f\ f2 . 3AZ+B' n (18) = TT" dq) dq q — , J dco. dQ 2 JI c v Q f* when and zero otherwise. In these expressions, A =ii sin 9 - (1 -}j.2)1/2cos 9 cos ip B = (l-| (20) 1- - (H cos 6+ (1 -p2)1/2 sinG coscp)+ —^ c 2c2 (21) (22) (23) and v \ I--cos 9 J , (24) with 9 being the angle formed by the direction of emission and the beam direction. To obtain (18) summation over the directions of ek in a plane perpendicular to the given direction of the emitted radiation has been performed. If we define the local Fermi energy to be given by (13) we may calculate, in our approximation, the REC probability per unit energy and solid angle when capture occurs from the vicinity of r, once the total elctronic density there is known. We have employed the electronic density values shown in fig. 2 for silicon, as computed 279 from a relativistic Hartree-Fock program12 using the Wigner-Seitz boundary condition, and we have found that the peak energy of the emitted radiation decreases strongly when electrons are captured from inner shells ot the solid, as is obvious from fig. 3, where REC peak energies for 160 MeV S16+ stripped ions channeled through silicon are plotted against r, that is, the position where capture occurs, when photons are emitted at 46,5° to the incident beam. Notice that the REC peak energy of photons resulting from capture of target electrons at rws, the Wigner-Seitz radius, is almost equal to EREC= E^E, (25) as may be inferred from considerations of transitions from the valence band of the solid to hidrogenic states on the moving ion, but it deviates when capture occurs from regions of relatively high electron density. This variation may explain the REC "deficit" observed in recent data, as we will see below. Finally, if we assume that the bombarding channeled ions sample on straight-line trajectories all portions of the WS sphere corresponding to impact parameters greater than a given bmjn, the statistical average of the REC probabilities per unit energy and solid angle may be written as d\ . 3 <-^i->=3 fr(r.bmin)JA.[EF(D]> (26) HO HO J J HIYI HO where A, is given by (18), and EF (r )by (13), as discussed above. On the other hand, if one assume that each electron undergoing capture from the solid experiences the full Coulomb field of the projectile, neglecting screening by other electrons of the solid, essentially the same dependence of the REC probabilities on projectile velocity is predicted5-13. 280 3. Results Figure 4 exhibits plots of the REC probabilities per unit energy and unit solid angle for a target electron to be captured accompanied by emission of radiation at 8 as obtained from eq. 16+ (26) for 160 MeV S (8 = 46.5°) ions channeled through silicon with bmin=0.3A . Figure 5 shows the dependence of the peak energy positions so obtained on the minimum impact parameter, together with the REC energies calculated from (25), and the experimentally measured REC energies 7. Notice that the agreement with experiment is good when the minimun impact parameter is 0.3 A, showing that capture from regions of relatively high electron density in the channel explains the experimentally observed REC deficit. The origin of the REC deficit is obvious, too, from figure 6, v/here measured and calculated REC peak positions for bare sulfur ions channeled through silicon are shown as a function of incident ion energy, for different values of the minimun impact parameter. This work has been sponsored by the Universidad del Pais Vasco / Euskal Herriko Unibertsitatea, by the office of Health and Environmental Research, U.S. Department of Energy, under contract DE- AC05-84QR21400 with Martin Marietta Energy Systems Inc., and by the US-Japan Cooperative Science Program of the National Science Foundation, Joint Research Project No. 87-1631 l/MPCR-168. Note: The authors are preparing a revised and more accurate version of the theory described above in which the REC probability is calculated us.'ng wave functions that describe the electron and the ion in relative and centcr-of-mass coordinates. References 1. H.W.Schnopper, H.D. Betz, J.P. Dclvaille, K. Kalata, A. R. Sohual, K. Kaiata, A. R. Sohual, K.W. Jones swd M.E. Wesner, Phys. Rew. Lett. 29, 898 (1972). 281 2. B.R. Appleton, R.H. Ritchie, J.A. Biggerstaff, T.S. Noggle, S. Datz, C.D.Moak and H.Verbeek, J. Nucl. Mater. 63, 513 (1976). 3. H.W. Schnopper, H.D. Betz and J.P. Delvaille, in Atomic Collisions in Solids, edited by S. Datz, B.R. Appleton and CD. Moak (Plenum, New York, 1975) Vol. II, p. 481. 4. B.R. Appleton, T.S. Noggle, CD. Moak, J.A. Biggerstaff, S. Datz, H.F. Krause, and M.D. Brown, ibd., Ref.l.p. 499. 5. B.R. Appleton, R.H. Ritchie, J.A. Biggerstaff, T.S. Noggle, S. Datz, CD. Moak. H. Verbeek, and V.N. Neelavathi, Phys. Rev. B 19, 4347, 1979. 6. Andriamonje S., Chevallier M., Cohen C, Dural J., Gailiard M.J., Genre R., Hage Ali M., Kirsch R., L'Hoir A., Mazuy B., Mory J., Moulin J., Poizat J.C, Remillieux J., Schamaus D. and Toulemonde M., Phys. Rev. Lett., 59, 2271 (1987). 7. CR. Vane, S. Datz, P. Dittner, J. Giese, J. Gomez del Campo, NJones, H. Krause, P.D. Miker, H. Schone and M. Schulz. Proceedings of the 11th Werner Brandt Workshop on Penetration Phenomena of Charged Panicles in Matter, Oak Ridge, Tennessee, April 14-15, 1988 8. J.S. Briggs and K. Dettman , Phys. Rev. Lett., 33, 1123 (1974). 9. J.E. Miraglia, R. Gayet and A. Salin, Europhys. Lett., 6, 397 (1988) 10. R.H. Ritchie, W. Brandt and P.M. Echenique, Phys.Rev. B 14, 4808 (1976), P.M. Echenique , R.H. Ritchie and W. Brandt, Phys. Rev. B 20, 2567 (1979). 11. This result has been obtained in the laboratory frame, and it differs from the first Born result found in the cm. system. The reason for this was not understood at the time of the conference, but, subsequendy, this question has been resolved and we have found that the correct result is only obtained in the cm. frame. This error resulted in a bigger REC deficit; the new results will be shown in a future work. 12. T.C. Tucker, L.D. Roberts, C.W. Nestor, T.A. Carlson and F.B. Malik., Phys. Rev. 178, 998 (1969). 13. H.A. Bethe and E.E. Salpeter, Quantum Mechanics of One-and Two-Electron Atoms (Academic, New York, 1957), p. 320. 282 Figure Captions Fig. 1 REC lincwidths calculated from eq. (10), as a function of impact parameter. Fig. 2 Relativistic Hartree-Fock calculations of the total electronic density in the Wigner-Seitz sphere, for silicon12. Fig. 3 REC positions for 160 MeV bare sulfur ions channeled through silicon, as obtained from eq. (18), as a function of the distance r from the captured target electron to the nearest atom in the crystal. Fig. 4 REC probability per unit energy and solid angle, as obtained from eq. (26) when photons are emitted at 0 = 46,5° with respect to the beam direction, for 160 MeV S16+ ions channeled through silicon. Fig. 5 Calculated REC peak positions for 160 MeV bare sulfur channeled through Si, with 6 = 46.5°, as a function of minimun impact parameter. Also shown are the REC energies calculated from eq. (25) and the experimentaly measured ones of ref. 7. Fig. 6 Measured7 and calculated REC peak positions for bare sulfur ions channeled through silicon, as a function of the incident energy of the ions, for b^s 0.1 and 0.3 A. Also shown are the REC energies obtained from eq. (25). 283 4 6 impact parameter (a.u.) Fig. I 284 7 100- 1 Fig. 2 285 REC energy, as obtained from eq. (25) Kj.3 286 6 7 pbotoa energy (keV) 287 7,0- 6,8- REC energy, is obtained from eq. (25) 16,6- measured REC energy 8.6.4- u 6,2- 6.0" 1 2 3 minimum Impact parameter (a.u.) Fij.5 288 100 120 140 160 180 220 EI(MeV) Kg. 6 289 Direct Methods in Structure Determination by LEED JB Pendry The Blackett Laboratory Imperial College London SW7 2BZ UK Abstract Most techniques for determination of surface structure interpret the raw experimental data by trial and error methods. For complex structures this is a very time consuming process because of the exponential growth of possible structures with degrees of freedom. Direct Methods sidestep this difficulty. They have been studied in the context of Xray diffraction and it is the heavy atom technique of Xray crystallography that inspires our approach to the surface problem. The idea is that we know the bulk structure lying behind the surface, and we can define a "reference surface" as near to the true surface structure as possible. It is subsequently treated as the "heavy atom" of the surface problem and all phases referred to the reference surface phases. The method works well for some simple examples and future applications of greater sophistication are discussed. 290 DIRECT METHODS IN SURFACE STRUCTURE DETERMINATION BY LEED 1. Introduction Surface crystallography is now firmly established as a discipline in its own right, with several hundred structures to its credit [1]. A variety of techniques contribute to the success of the subject: Rutherford back scattering, surface Xray diffraction, and low energy electron diffraction (LEED). All have so far resisted attempts to use Direct Methods. LEED is arguably the most successful of the techniques [2,3,4,5] with the majority of structures determined by this method. It owes this power to the very high density of information that the diffraction process produces: the coordinates of several independent surface atoms can be found using the information contained in diffraction data acquired over a range of a few hundred eV. Indeed it is the very success of the technique that is forcing our attention back to the Direct Methods [6,7]. Suppose that two atoms are adsorbed on a surface and we have no knowledge of their positions or relative configurations. Each atom has 3 coordinates, and we may wish to try 10 values for each coordinate. That works out to be 10 trials, each of which may involve a considerable computational effort. Even if each trial were to take only 1 second of computer time, the complete task would last for nearly two weeks of solid computing. In general for a system with N coordinates to be found and M trials per coordinate, we have, number of trials = M^. {1} 291 J. B. PENDRY There is a non polynomial dependence on the parameters, hence the problem falls into the category of NP problems which are legendary for defeating the attempts of computers to solve them. In practice we can do better than this because surface scientists perform all sorts of ancillary experiments on their surfaces: electron energy loss spectroscopy will, for example, tell us a lot about the bonding, and will certainly indicate whether the two atoms in the example above are bound into a molecule. The vast number of unrestricted trials possible illustrates just how vital this ancillary information is. Even so it is all too possible to take some data on a surface structure, to run a small subset of trials and be lucky enough to obtain reasonable agreement with experiment, and yet be badly in error because an even better tut untried structure lurks in another part of parameter space. The problem is especially serious because it discriminates against discovery of a really startling structure: anything too improbable will be dismissed by the operator as wasting valuable searching time and will remain undiscovered. Thus our capacity to add to our understanding is limited by our method of data analysis. New things are discriminated against by our inherent conservatism. The history of surface structure determination is littered with instances of false determinations. We need not feel too bad about that: even our distinguished colleagues in radio astronomy have the same problem. At one period the 292 DIRECT METHODS IN SURFACE STRUCTURE DETERMINATION BY LEED number of radio sources actually diminished with time as scientists got better at interpreting the data. But we have no room for complacency. Direct Methods are not a universal solution, but for a class of problems to which they are suited, they offer a way forward that is relatively free cf the pitfalls of the trial and error procedure. Direct Methods have been studied for many years in the context of Xray diffraction. The simple nature of Xray diffraction theory means that a Fourier transform of experimental diffracted amplitudes will give the atomic coordinates directly. Only the intensities are known so the problem reduces to one of finding the phases. This can be done either by using sophisticated theorems which limit the volume of phase space which needs to be searched, or more traditionally by the heavy atom method: a heavy atom is substituted into a complex unit cell and dominates the A diffracted amplitudes, oG- Subsequently coordinates of all other atoms are to be referred to the position of the heavy atom so that A _ is chosen to be real. Since the other atoms OG produce a relatively small change in diffracted amplitude. IAG'2 = 'AoGi2 + 2 'AOG' '6AQJ COS where 8 is the phase of 6AQ. Hence, apart from the usual ambiguity about centro- symmetric crystals, we know the amplitudes and can Fourier transform to find the coordinates. 293 J. B. PENDRY It is this heavy atom technique which inspires our approach to the surface problem, The idea is that we know the bulk structure lying behind the surface, and we can define a "reference surface" based on the same bulk structure, and as near to the true surface as our chemical intuition and other information enables us to guess. This reference surface is subsequently treated as the "heavy atom" of the surface problem and all phases referred to the reference surface phases. What use are the phases? LEED is recognised as a multiple scattering problem and Fourier transformation, even of LEED amplitudes, generally gives a jumble of nonsense. Here we introduce the second important concept in our approach [8,9,10]: provided that the reference surface is reasonably close to the real structure the diffracted amplitudes will change by an amount that is first order in the atomic displacements. Suppose that we have a surface in which one of the top layer atoms in the unit cell is displaced by an amount 5r from the reference surface. The changes in amplitudes and the displacements will be connected by a tensor, 5A(kp) = Z. T(kp,j) 6rr (3) so that the change of intensity is, A(k 2 + s T(k j) 6r 2 l pH = IVV j P' ji 'W|a + M(lcp'j) 6rj where, 294 DIRECT METHODS IN SURFACE STRUCTURE DETERMINATION BY LEED M(kp,j) = A*(kp) Sj T(kp,j) + AQ(kp) E. T* 2. Realisations of Direct Methods So far we have been deliberately vague on how a Direct Method might be implemented, because there are several realisations that can be made, all having the same underlying theory, but which look very different in their details, here are some possible implementations, starting with the most simple. linear expansion of 6t in the atomic coordinates Starting from equation {11) we expand 6t(6r.) to first order in 6t(6rj) = 6ryV ti'r^), (6) and hence. » Z. M(kp,x..} 6Xj + M(kp,Vj) 6yj + Mlk^Zj) 6zj# (7) where, M(k ,x.) = + AQ(kp) We have a matrix equation that can be inverted, provided that we have enough data points. In this very simple 295 J. B. PENDRY realisation the coordinates are given directly by the inversion process. This most simple realisation shows how the method could work. It has been implemented for systems with very small displacements [6], but suffers from the major disadvantage that equation (6) is a very bad approximation for all but the smallest values of 6r.. We stress that this is not a fundamental limitation on Direct Methods as a whole which have much wider validity in other implementations power series expansion of 6t in the atomic coordinates The next logical step is to expand 6t(6r.) to higher orders in &Zy 6t(6r.) = Al 6r. + A2 idz^)2 + A3 (6r.)3 + ... , (9) where we have used shorthand notation for the powers of a vector, A3 (6r,)3 = + A3(x,x,y) fix. 8x. 6y. + .. (10) This power series can be derived analytically from the translation operators given in the appendix of ref [2]. It converges for all values of 6r.. To take a simple example: suppose that we are sure of the x and y coordinates of an adsorbate atom, then we need only make an expansion in powers of 6z, n 6t(6z) = Sn An (6z) . (11) Substituting into (11) gives. " |Ao(ltp)|2 * Sn=0 M(kp'n) where, 296 DIRECT METHODS IN SURFACE STRUCTURE DETERMINATION BY LEED M(k ,n) = (A*(k ) J. B. PENDRY 2 5A< + A ( 6A ( 3 iA + Ao(kp) £g M(kp,g) exp(ig.6rp), (17) where g is a reciprocal lattice vector of the reference structure. In the most general case the adsorbate atom will be vibrating about within the unit cell and will be described by a probability distribution, P(6r ), and the measured changes in intensity will reflect an average over this distribution, |A(kp)|* - |Ao(kp)|> = 2g M(kp,g) P(6r } DIRECT METHODS IN SURFACE STRUCTURE DETERMINATION BY LEED more ambitious schemes The methodology outlined above requires a knowledge of the relative intensities of the beams, and preferable of their absolute intensities. LEED users have got used to the idea that absolute intensities do not matter and Direct LEED would be an even more attractive proposition if they were not required. Certainly it is true that it is possible to extract the same information by conventional analysis from, for example, the Y- function [11] for the hk beam, defined by, 1 2 Yhk(E) = L" /(L" + V^) (20) L d ln(Ink)/dE 121) where VQi is the imaginary part of the electron self energy. Direct Methods can be made to work with these more complex functions of the intensity, but certain complications develop. Consider first the case of a single adsorbate atom per unit cell whose position we are trying to find. For simplicity assume that only the z- coordinate is unknown. Using the same trick as before we can write the Y-function as a power series in the height, z, of the adsorbate atom, n Yhk(Efz) = En M{E,h,k;n) z (22) which again has the form of an invertible matrix equation. Immediately we can see one problem: the definition of Y contains poles as a function of L, and therefore the power series expansion will have a finite radius of convergence, in contrast to the power series expansion for the 299 J. B. PENDRY intensities which was absolutely convergent. Another more subtle problem occurs when we have a distribution of atoms over several values of z. We write, n I(z) = En an n< n n Y(z) = En,bn,I = Sn.bn.(En an More complications arise when there is more than one adsorbate atom whose position we are trying to find. In this instance the power series expansion for Y becomes quite a mess involving terms of the form, Y(z) = ... + c(nl,nl\n2,n2') DIRECT METHODS IN SURFACE STRUCTURE DETERMINATION BY LEED coordinates there is no reason why the Y- function should not be used. 3. Limitations to Direct Methods Firstly we do have to assume that the displacement from the reference structure produce small changes in scattering. In practice we have found through experience of the tensor LEED method that displacements of up to 0.4A can be tolerated without producing large errors. It is possible that some sort of iterative procedure could be used for larger displacements whereby an inaccurate estimate of the r•'s was used to define a new reference surface closer to the true structure. We are always free to check the structure we obtain by a conventional dynamical calculation and direct comparison with the data. The great virtue of the Direct Method is that it goes straight to an estimate of the structural coordinates without having to plod around parameter space. Secondly our method relies on differences in intensity between experimental spectra and those for a theoretical reference surface. Theory is good at predicting peak positions in LEED, but bad at absolute intensities. Ideally we should make two experiments: one on a known reference structure so that the differences could be found purely from experiment. Dynamical inputs such as the Debye Waller factor, inelastic damping, etc, have non- linear effects on the calculated intensities and some thought has to be given 301 J. B. PENDRY for each system to how the effect of intensity errors can be minimised. Despite these limitations we are cautiously optimistic that Direct Methods have a valuable role to play in the future of surface crystallography, and that they will greatly extend the range of structures that can be studied. Acknowledgements: I wish to thank my collaborators, Klaus Heinz, Klaus Miiller, Wolfgang Oed, and Philip Rous, for the contributions they have made to the ideas reported in this manuscript. References [1] JM MacLaren, JB Pendry, PJ Rous, DK Saldin, GA Somorjai, MA Van Hove, DD Vvedensky "Surface Crystallographic Information Service" (Reidel, Dordrecht, 1987), [2] JB Pendry "Low Energy Electron Diffraction" (Academic, London, 1974). [ 3 ] MA Van Hove amd SY Tong "Surface Crystallography by LEED" (Springer, Berlin, 1979). [4] MA Van Hove, WH Weinberg,and CM Chan, "Low Energy Electron Diffraction" (Springer, Berlin, 1986). [5] K Heinz and K Miiller in "Structural Studies of Surfaces" (Springer, Berlin, 1982). [6] JB Pendry, K Heinz, and W Oed, Phys Rev Lett, 61 (1988) 2953. 302 DIRECT METHODS IN SURFACE STRUCTURE DETERMINATION BY LEED [7] PJ Rous, MA Van Hove, and GA Somorjai, submitted to Surf Sci. (1989). [8] PJ Rous, JB Pendry, DK Daldin, K Heinz, K. MUller, and N Bickel, Phys Rev Lett 57 (1986) 2951. [9] PJ Rous and JB Pendry Comp Phys Ccmm 54 (1989) 137. [10] PJ Rous and JB Pendry Comp Phys Comm 54 (1989) 157. [11] JB Pendry, J Phys C 13 (1980) 937. 303 Non-local Exchange-Correlation Potential at a Metal Surface from Many-Body Perturbation Theory A. G. Eguiluz Department of Physics, Montana State University, Bozeman, MT 59717, USA 304 Abstract We report a first-principles evaluation of the exchange-correlation poten- tial (Vxc) at a metal surface. An integral equation relating (V^£) and the non-local electron self-energy (£xc) is solved numerically for a free-electron metal surface, with use of a static approximation for Src. The strongly- inhomogeneous nature of the electron density profile at the surface is treated exactly, i.e., without invoking the usual local density approximation. Our result for Vzc has the correct image-like asymptotic behavior; it derives im- plicitly from a non-local exchange-correlation energy functional. We study the effect of non-locality on the position of the effective image plane (z0) from an analysis of the image tail of Vxe and also from linear-response the- ory. The difference in the values of z0 obtained by both methods for low metallic densities is attributed to electron overlap effects. We als discuss the effects of Vzc on image-potential induced surface states. 305 Density functional theory (DFT) has proved to be an extremely suc- cessful scheme for electronic structure calculations, in which all many-body effects can be described through a local exchange-correlation (XC) potential Vxc-1 In the great majority of the applications the local density approxi- mation (LDA) is made, in which the XC potential is evaluated in a locally uniform electron gas of density n(x).1 The LDA has proved to be quite successful for many applications, such as calculations of total energies and work functions.2 However, recent variational calculations, which do not re- sort to the limit of slow density variations, yield surface energies which are substantially larger than the LDA values.3 Another drawback of the LDA is that it gives an XC potential which decays exponentially outside the surface, wheras the correct asymptotic behavior of Vxc is that it should be imagelike. Thus, an LDA based calculation failes to describe image- like surface states, which have been observed at various metal surfaces. Former attempts to overcome this shortcoming within a weighted density approximation* were successful, to the extend that an empirical form of the XC hole was used, which revealed an artificial value of the image plane position.5 Here we address the nonlocality of the XC energy functional from a more fundamental point of view. We present results of an a prior: evaluation of 6 Vxc(x) at a metal surface, in which the nonlocal electronic Green's func- tion is calculated directly, thus incorporating the strongly inhomogeneous nature of the electron density near the surface at the outset. In addition, we establish a relation between VXc and the electron density that enables 306 us to carry out linear response calculations with the same ease as LDA- based calculations. The jellium model for free-electron metals is used, in which the ionic charges are approximated by a homogeneous background.3 Our procedure is an adaption to the metal surface problem of work carried out earlier in the study of the fundamental band gap of semiconductors.7 It involves an interplay between many-body perturbation theory and density functional theory. The starting point of our calculation is the following exact integral equa- tion for 8 3 fd x' Vxc(x')fdEgo(x,x' | E) g(x',x | E) 3 3 = fd xx fd x2 fdE ffo(x,xi | E) £Xc(xi,X2 | E) g(x2,x j E), (1) where Exc is the XC part of the electronic self-energy operator, g is the exact one-electron Green's function, and g0 is the corresponding Green's function for the DFT hamiltonian.6 This equation must be solved itera- tively, since the Green's function go is implicitely determined by Vxc- As a reasonable first approximation for SA'C, which includes the impor- tant physical effect of it's nonlocality, we use the static Coulomb-hole plus 9 screened exchange (COHSEX) approximation. The approximation g = gQ is made throughout. Eq. (1) then turns into a one-dimensional integral equation, depending on the electronic polarizability x° f°r zero wave vec- tor parallel to the surface,10 which can be solved by standard iteration: Starting from Vxc in LDA, the wave functions and energy eigenvalues ob- tained self-consistently with Vxc ars used in the computation of go, X*°\ 307 and SA'CI and the integral equation is solved, yielding a new potential Vxc- The procedure is repeated until convergence is achieved. Results for an Aluminum surface (r,=2.07) are given in the figure. Both the converged solution and it's LDA-COHSEX counterpart are shown. The crucial difference between the "local" (i.e. LDA) and nonlocal results for Vxc(z) is that the latter potential can be approximated by an imagelike potential sufficiently far in front of the surface. This represents an impor- tant improvement for studying, e.g., image-potential bound surface states or tunnel barriers in a scanning-tunneling microscope. Of great importance in surface physics is the effective image plane po- sition z0. It can be defined either by the asymptotic behavior of the XC potential (see figure caption), or by linear response (i.e., by calculating the density induced by a weak static electric field applied normal to the sur- face). The centroid of the induced density equals the linear-response value s of z0. Here we address the question wether these two values are the same, in other words, are there electron-overlap effects built into the equation that controls the image tail? We determine a value for z0 both from the image tail and in linear response. The latter is done by solving the DFT equations in the presence of a dc electric field, using the XC potential for the undisturbed density n.6 For different bulk metallic densities, the results are as follows: From the image tail of Vxc(z), we found that z0 (measured from the jellium edge) ranges from 1.19 a.u. for r, = 2.07 (Al) to -0.11 a.u. for r, = 5.63 (Cs). For the nonlocal linear response calculation, the results for z0 axe 1.16 a.u. for 308 r, = 2.07 and 0.68 a.u. for rt = 5.63, whereas for the local linear response calculation, the corresponding values are 1.25 a.u. and 0.795 a.u., respec- tively. The values agree basically for high-density metals (r, = 2.07), but differ substantially for low metallic densities. This interesting result can be understood from the fact that the work function decreases as r, increases. A smaller value of the work function leads to a more pronounced tailing out of the electrons into the vacuum. One then expects that any electron-overlap effects built into the image tail of Vxc would become more significant as rt increases, in agreement with our observation. That overlap effects exist at all is a consequence of the fact that the asymptotic limit is reached rather quickly, as seen in the figure. This behavior is in disagreement with pre- vious studies by Ossicini cf al.,A who found, that the asymptotic limit is reached slowly, due to an artefact of the empirical choise of the XC hole.5 However, in agreement with these studies4 we have that the nonlocality leads to a small (as 10% ) reduction of the linear-response value of ZQ. The results of the present paper, based on the use of the COHSEX approximation for Exc, establish only a qualitative test for Vxc- Quanti- tatively, our LDA linear-response values of zo differ substantially from the result obtained in LDA using Wigner's interpolation for XC,2 i.e., 1.55 a.u. for rt = 2.07 and 1.17 a.u. for r, = 5.63. Improved results are expected from an evaluation of Lxc in the "GW approximation,9 which is currently in progress. 309 References 1 P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964); W. Kohn and L.J. Sham, ibid. 140, A1133 (1965). 2 N.D. Lang and W. Kohn, Phys. Rev. B 7,354 (1973). 3 E. Krotschek and W Kohn, Phys. Rev. Lett. 57, 862 (1986). * S. Ossicini, CM. Bertoni, and P. Gies, Europhys. Lett. 1, 661 (1986); S. Ossicini, F. Finocchi, and CM. Bertoni, Surf. Sci. 189/190, 776 (1987) and Vacuum (this volume). s E. Chacon and P. Tarazona, Phys. Rev. B 37, 4020 (1988). 6 A.G. Eguiluz and W. Hanke, Phys. Rev. 39, 10433 (1989). 7 L.J. Sham and M. Schliiter, Phys. Rev. Lett. 51, 1888 (1983); R.W. Godby, M. Schliiter, and L.J. Sham, ibid. 56, 2415 (1986); Phys. Rev. B 35, 4170 (1987). 8 L.J. Sham, Phys. Rev. B 32, 3876 (1985). 9 L. Hedin and S. Lundqvist, in Solid State Physics, edited by H. Ehrenreich, F. Seitz, and D. Turnbull (Academic, New York, 1969), Vol. 23, p. 1. 10 A.G. Eguiluz, Phys. Rev, B 31, 3303 (1985); Phys. Scr. 36, 651 (1987). 310 Fig. 1. Nonlocal XC potential(solid line) for r, = 2.07 (Aluminium). Also shown is the XC potential in LDA (dashed line) and the classical image potential V{m — — 4.*_ . (dashed-dotted line), z is measured from the jellium edge in units of Ap = 4.05A, z0 = 0.63k. 01- ) A0H3N3 313 Multiphonon Energy Exchange in the Collision of an Atom with a Surface J . R. Mansor. Department of Physics and Astronomy Clerason University Clemson, South Carolina 29634 U. S. A. and R. H. Ritchie Department of Physics University of Tennessee Knoxville, Tennessee 37996 U. S. A. and Health and Safety Research Division Oak Ridge National Laboratory Oak Ridge, Tennessee 37831 U. S. A. Abstract We consider the collision of an atom with a surface under conditions where the energy transfer involves multiple phonons. Such conditions occur with either relatively high incident energies or hot surfaces. Several simple models for the energy exchange to the surface by a projectile with translational energy in the eV range show that such a particle should lose a large fraction of its energy upon collision. A more elaborate model, based on quantum mechanical transition matrix theory, agrees well with new experimental data for the scattering of low energy helium atoms with alkali halide surfaces. 314 I. Introduceion There is considerable interest in the problen of energy exchange when an incident, low-energy atom or Molecule is reflected from a solid surface. ' Such studies have provided invaluable information on surface phonons as veil as knowledge of the particle-surface interaction. Recently, there have been some interesting experiments in which the energy exchange with the surface is measured to be a large fraction of the incident translational energy. •* Typically, such conditions occur when the incident aton or molecule has energy in the electron volt range , or when the surface temperature is large compared to its Debye temperature. The primary candidates for energy exchange mechanisms in such collisions are electron-hole pair creation and phonon excitation. It has been understood for some time that electron-hole pair creation is a process with very low probability in the case of helium atoms as projectiles. More recent examinations confirm this conclusion and extend it to a much larger class of incident atoms and molecules. In particular, for rare gas atom collisions, direct excitation of electron-hole pairs appears unlikely . Thus recent observations of electron-hole pair creation by heavy rare gas atoms incident on semiconductor surfaces may be due to secondary effects, such as local heating ** a result of energy exchange on collision. The object of this work is to examine the phonon mechanism for the energy transfer. We first consider two straightforward mechanisms, the direct mechanical excitation of acoustic phonons, and excitation of surface optical phonons via the interaction with the polarization field. In the case of moderately energetic rare gas atoms both of these mechanisms give a significant transfer of energy to the surface, in qualitative agreement «rith experiment. Ue then move on to a more elaborate model for the energy exchange based on a quantum mechanical transition matrix formalism. This model is used to calculate the temperature dependence of the diffuse inelastic scattered intensity and its distribution in energy. When compared with the detailed energy-resolved experiments of Safron et al. for the scattering of helium by alkali halide surfaces, good quantitative agreement is achieved. An important result of this agreement is that measurements of the diffuse inelastic background signal show directly the numbers of phonons exchanged as a function of incident energy and surface temperature. 315 II. Phonon Excitation Models We will take a fully quantum mechanical approach to the problem. There are two major reasons for using quantum rather than classical mechanics. First, there is substantial evidence that quantum effects such as diffraction are important in the surface scattering of heavy atoms7' , and secondly, a classical approach would preclude obtaining information on microscopic details of the interaction such as numbers and sizes of phonon quanta exchanged. The total probability per collision for exchanging a phonon with the surface is given by the generalized golden rule 2* 2 Rt 2 <|Tfi| «(Ef - Ei)> , (1) where the sum is over all final states of the particle and crystal, the brackets signify an average over the initial states of the crystal, and the incident current is ji - *k^z/m with m the particle mass and k^z the component of incident wave vector normal to the surface. The transition matrix is related to the interaction potential V through the operator relation T - V + VGT (2) where G is the Green function of the particle. In atom-surface scattering the dominant contribution to the inelastic transfers come from vertical motions of the surface atoms , and it is a reasonable approximation to consider a one dimensional potential depending only on the dimension z perpendicular to the surface. Expanding the potential as a series in the normal displacements u gives V(z,u) - V(z) + V'(z)u + V" ( )/ 1 o2 f f i -y- E|V'fi | | dt e where <^ and where H is the mass of a surface atom, u^ is the maxiaum crystal frequency. n(w) is the Bose-Einstein function, and p(u) is the frequency distribution function. Ue will, for actual calculations, further approximate p(u) by a Debye distribution p(w) - 3u /u^ . For the interaction potential we will take an infinitely repulsive hard wall, a model which has been widely used and often gives excellent results in atom-surface scattering. The matrix elements are well known and are given by fi fz iz * Combining Eqs. (6) and (5) with (4) and carrying out the trivial integration involving Che f-function leads to 4«kiz rf p(«> f R< dw jk. n(u) + k_ |n k with k+ - iz ~ 2«u/rt. If the incident normal energy is sufficiently high we have k+ — k^z and Eq. (7) becoaes siaply (n<«) +1/2) . At low temperatures only phonon creation processes occur and writing the energy of normal motion as E^ - ft kj2 /2a we have t i«I)M . (9) while at high temperatures n(u) » k^T/ftu; and the probability of energy exchange , is /ftwDM) (iksT/*wD) . . (10) 317 Of more immediate interest to che quescion ac hand is che total energy exchange. This is readily obtained from Eq. (7) by weighting the integrand wich flu. The result, valid for either low or high temperatures is E - SmE^M . (11) This staces that che fractional energy exchange for a parcicle incident normally on the surface is 8m/M, clearly indicating ChaC a heavy, energetic rare gas atom should lose a substantial fraction, If not virtually all of its energy upon collision. In spice of the fact that the results expressed in Eq. (11) agree qualitatively with the results measured in Ref. (3), there is ample evidence to suspect that it is inadequate. In its simplest terms, Eq. (11) gives the results for the Born approximation with the exchange of only a single phonon. A more complete calculation would include multiple phonon contributions coming from higher orders in the Born series, and would show that each contribution in attenuated by a Debye-Waller factor. The multiphonon corrections and the Debye-Waller factor tend to cancel each other as we show in the more detailed calculation in the section below. In fact, we note in passing chat this cancellation is complete in the special case of a collision wich a surface having only a single vibration frequency, i. e., for Che inelastic scattering of a poinc parcicle wich an Einstein oscillator. In this case, if R is che probability of single quantum exchange in the Born approximation, then the probability PN of N quantum exchange is given by the Poisson distribution PN - exp(-R)R /N!. The cotal energy exchanged is obtained upon multiplying PN by Nfiw and then summing over all N. The result is E - rtuR, which is identical to che Born approximation result. This simple result, although not a justificacion of che use of Eq. (11), does indicate how higher order effects tend to cancel. Before leaving this section it is of interest to consider another contribution Co che phonon energy loss, che exchange of surface optical phonons, which as our calculations indicate, can also be a significant effect. Surface optical phonons in insulacors and semiconductors can be excited by the polarization field of the incident particle and surface, the sane field which has been shown Co be very inefficient for the excitation of electron-hole pairs. The mutual polarization requires transitions of the incident atom to virtual excited states, hence the first nonzero contribution comes from the second order Born approximation for the cransition matrix <2), 318 Tfi - i fe " ii • Ei • Ei where the polarization potential is* [. ] iK-E -K|z| V(X) - S TJJ i . J4(K) e e (&£ + aK) (13) with Ji(K> - (iKx. iKy, -K) 2 2 2 I" . - jrZe ftu>s/L K (15) where us the surface phonon frequency. In this case we take the matrix elements with respect to a plane wave basis set. Rather than calculating Rj directly it is more convenient to first calculate the self-energy for inelastic transitions , 2^(z), and then the inelastic transition probability becomes - fdz Zi(z) (16) zo where zQ is the classical turning point of the repulsive part of the potential V(z) of Eq. (3). We have carried out a number of calculations for a variety of incident rare gas atons interacting with semiconductor surfaces. For incident energies in tht; eV range this mechanism gives energy losses which are a substantial fraction of the incident energy. The conclusions of this section are that two different models describing phonon exchange of the incoming particle with the surface, direct mechanical excitation of acoustical modes and excitation of surface optical modes, both lead to the same qualitative result. A moderately fast incident atom will lose a large fraction of its energy upon collision in a. very short time. In the next section we develop the theory in a manner that allows us to examine in detail the multiphcnon nature of the energy exchange. 319 III. Multiphonon Exchange For a more detailed treatment of multiphonon scattering we go back to Eq.(l) and apply the Van Hove transformation directly, vhich when expressed In terns of the transition rate from particle state i to state £ is wfi - —* dtL e iH t/ft -iH t/fi T(t) - e T e ' (18) Without actually calculating the transition matrix, for example by assuming a model potential and using Eq.(2), we can gain a great deal of insight by noting that the T-matrix elements of interest in scattering theory are given by the coefficients of the wave function asymptotically far from the surface. (W) T im ffii In Eq.(19) the ket |n£> is a state of the noninteracting crystal and the suns over final states are carried out consistent with overall energy conservation between particle and crystal. If the crystal is rigidiy displaced through a displacement Jl parallel to the surface and uz perpendicular, the effect on the transition natrix is iK-U i(kfz izz Tfi -» Tfi e e " (20) Ue are considering here the question of energy transfers involving exchange of many phonons and in this case the overwhelming majority of the phonons will be of small energy and hence of large wavelength. The displacement of a long wavelength phonon will approximate a rigid surface displacement, and this will be even more true if the wavelength is large compared with the coherence width of the particle wave packet. Making the assumption that the only tine dependence of the transition operator is in the displacement appearing in Eq.(20) the transition rate (17) takes the form 320 w T dt e fi " S fi where we use the notation k. - ( E, kj2 + k^2 ) and u - ( U,uz ). Vithin the harmonic approxination Che average over crystal states can be readily carried out1A to give wfi--2 lTfir •"* dt< where e'2W is the of the form of a standard Debye-Waller factor with U - <(k-Ji) >/2. Equation (22) is the essential result. Although we have used a very simplified approach to arrive at (22) it can be obtained in a much more rigorous way by decomposing the transition matrix into contributions from surface unit cells. Then a generalized form of £q.(22) capable of describing all the diffraction beams is obtained after making that single assumption, that the unit cell is not deformed by the phonon. This assumption is consistent with that stated above, and implies that the results should be valid for long wavelength modes. The important features of Eq.(22) are made clearer if we rewrite it as 2W(k T) wfi - F(fc) e- -' S(k,,T) . (23) where T is the surface temperature. This is in the classic form of a fora factor (essentially the square of Tj^) multiplied by the Debye-Waller factor and a structure factor, where the structure factor is the Fourier transform appearing in (22) f i - fIdt e e (24) The structure factor contains all the dynamics of phonon transfer. In particular, if it can be measured independently of the Debye-Waller factor as a function of temperature, it gives immediately the number of phonons involved in the scattering process. To see this we note that the displacement function appearing in the exponential of (24) is proportional to T for temperatures large compared with the Debye temperature. If this 321 exponential is expanded in terms of powers of the displacement correlation function, the result is a structure factor which is a power series ir. T — d IK, L) dn ~ 9i 1 ~ 341 T ... ~ Jml T . • . • l^J j V aL £ II The n term in the series is the contribution arising from the transfer of n phonons, hence the dominant terms in the series at a given temperature tell immediately which numbers of phonon transfers are important. Note that unitarity arguments (conservation of particle flux) would imply that S(k.,T) should approach exponential behavior at large T in order to cancel the decreasing exponential behavior of the Debye-Waller factor. This behavior is precisely the general case of the cancellation of the multiphonon corrections by the Debye-Waller factor discussed in the previous section. In the next section we derive explicit expressions for a simple phonon model. IV. Comparison with Experiment Recently, a series of experiments has been carried out measuring inelastic transitions in He-surface scattering which is ideally suited to the theoretical development presented here. The experiments involved helium beams scattered from alkali halid* crystals at energies from 50 to 100 meV and surface temperatures in the range 100-700 K. The detector was placed at the specular position and the scattered intensity was energy resolved by time-of- flight measurements. This enabled a separation of the elastically reflected specular peak from the diffuse inelastic background. As a function of increasing temperature the specular peak was observed to decrease exponentially as expected according to the Debye-Waller factor, while at the same tine a diffuse inelastic "foot" rose out of the background underneath the elastic peak. We can apply the theory of the preceding section using a Debye phonon spectrum as in Sec. II above. We look only at the high-temperature limit and consider only the vibrational displacements normal to the surface. Eq.(5) is used to evaluate <\iz > from which we obtain the exponent of the Debye-Waller factor as 3 o kBT 2 The one parameter in this theory is up or the Debye temperature 6^ and this can be evaluated by comparing (26) with the measured value for the thermal attenuation of the specular elastic peak. 322 Using Eq.{22> we can calculate Che intensity of Che observed diffuse inelastic foot. The simplest quantity to check is che diffuse inelastic intensity at near zero energy exchange, or the height of the foot at the position of the specular peak in the time-of-flight spectrum. In this case the Debye-Waller factor and the form factor are the sair.e for both the elastic and diffuse, inelastic intensities, thus their ratio is given by the structure factor of Eq.(24) evaluated at (f - <£- In order to evaluate (24) vn must first determine the displacement correlation function in the exponential. Again caking normal vibration displacement and the Debye frequency distribution this is readily evaluated with the aid of (5; 2 <}c-a(O) k.-u(t)> —> (ki2 + kfz) 3kRT n k + k 2 " < iz fz> 3 (27) With the results of (27) it is possible to find closed form expressions for the Fourier transform of (24). However it was found simpler to evaluate it numerically. The results of these calculations are shown in Fig.(l) for the case of 44meV heliua acorns incident on a NaCl crystal surface. The filled circles are the experimental points for che Debye-Waller thermal attenuation of the specular beam, and the line drawn through them is our calculation with a Debye temperature of 33OK. This choice compares favorably with the bulk Debye temperature of 321K. • The remaining experimental points are the diffuse inelastic intensity close to the elastic peak, the crosses are measured on the side in which the He gains energy and the squares are on the energy-loss side. The straight line drawn through them is a fit to the data having a slope of the calculation from (24) using the same Debye temperature. In the high- temperature region the agreement is quite good. Calculating the actual shape of the diffuse, inelastic foot would require a choice of an interaction potential model in order to determine the form factor appearing in (22). However, without complete knowledge of the form factor we should be able to estimate Che temperature dependence of the full width at half maximum (FWHM) of the diffuse foot. In this case we note that the actual quantity measured is the differential reflection coefficient, which is che transition rate multiplied by che available volurae in phase space and divided by the incident particle flux: 323 2 oR m |kf| w 3 3 fi • dEfdfif (2>r*) kiz The form factor in (28) acts as a cut-off factor at large energies, but we make the assumption that it does not vary strongly in the region of the half maximum. In Fig.(2) the filled circles are the measured points for the FWHM as a function of temperature. The solid curve is the calculation using (26) in which we have used the form factor as a free parameter to scale the vertical axis. This does not allow us to make a quantitative comparison, but the qualitative trend is correct. The FWHM increases nonlinearly with temperature which indicates that multiphonon processes dominate at high T. Finally, we can address the question of how nany phonons are involved in the scattering process at a given temperature. This is given in Fig.(1) where it is seen that the structure factor is closely approximated by an increasing exponential at high T, S(T) « e . The n tern in the power series of Eq.(25) is then (cT)n/n! and the largest terms of the series are those for which cT •» n. Taking the above cited value of c - O.OO6O8K , we see that at T - 400K the typical number of phonons involved in a single scattering event is about two or three, while at 700K it is more of the order of four or five. V. Conclusions We have considered the energy exchange process in single collisions of atoms and molecules with solid surfaces. We find that two separate mechanisms of phonon transfer, mechanical excitation of acoustical modes and excitation of surface optical modes by mutual polarization, can give rise to energy exchange that is comparable to the incident kinetic energy of the particle. The implication is that such collisions can deposit significant amounts of energy in a small region of the surface in a very short time. This helps support the idea that electron-hole pairs, which are known to be created in such collisions, are caused by secondary processes related to the local heating. In order to obtain a more detailed picture of the nature of the phonon- mediated energy exchange process, we have developed a theory of nultiphonon, inelastic atom-surface scattering which is valid for long wavelength vibrational modes. Predictions of this theory for the intensity and width of the diffuse inelastic signal in atom-surface collisions agrees well with recent experiments with He beams scattering off of alkali halide surfaces. These results give important information on the nature of the collision, such 324 as the number of phonons exchanged, and they also show promise of boing useful In analyzing experimental data in which it is desired to separate out the elastic and single quantun contributions from the total scattered signal. Acknowledgments. One of us would like to thank J. Skofronick anc* S. Safron for useful discussions. This research was supported by the Office of Health and Environmental Research, U. S. Department of Energy, under Contract No. DEACOS- 84OR21600 lith Martin Marietta Energy Systems, Inc., and by the U. S.-Spain Joint Committee for Scientific and Technological Cooperation. Figure Captions 1. Elastic and diffuse inelastic scattered intensities in the specular direction for He reflected by NaCl as a function of surface temperature. The He energy is 44aeV. The filled circles are the elastic peak intensities and the line drawn through the points is the theoretical calculation. The squares are the diffuse inelastic intensity on the creation side of the elastic peak, while the crosses arc from the annihilation side . The straight line is a fit to experiment, and the theoretical prediction is the solid curve displaced below. 2. The full width at half maximum of the inelastic foot measured in the time-of-flight spectrum for the system of Fig.(1), The points are the experimental data and the solid curve is the theory of Eq.(28). 325 References 1. V. Bortolani and A. C. Levi, Rivista del Nuovo Cimento, 2.1 (1986). 2. V. Celli, in Many Body Processes at Surfaces, D. Langreth and H. Suhl, eds. (Academic Press, New York, 1984) p. 315. 3. A. Amirav and M. Cardillo. Phys. Rev. Lett. 52, 2289 (1986). 4. S. A. Safron, W. P. Brug, G. Chern, J. Duan, and J. G. Skofronicic, J. Vac. Sci. and Techn., to be published. 5. 0. Gunnarsson and K. Schonhasuer, Phys. Rev. B££, 2514 (1982). 6. J. Annett and P. Echenique, Phys. Rev. £3_£, 8986 (1987). 7. E. K. Schweitzer and C. T. Rettner. Phys. Rev. Lett. £2. 3085 (1989). 8. R. R. Hancox, Phys. Rev. 42., 864 (1932). 9. G. A. Armand and J. R. Manson, Phys. Rev. Lett. £3_, 1112 (1984). 10. J. H. Weare, J. Chem. Phys. £1, 2900 (1974). 11. J. R. Manson and G. Armand, J. Vac. Sci. Technol. AS, 448 (1987). 12. G. Boato and P. Cantini, Advances in Electronics and Electron Physics £Q, 95 (1983). 13. J. R. Manson, R. H. Ritchie and T. L. Ferrell, Phys. Rev. B21, 1080 (1984). 14. A. A. Maradudin, E. W. Montroll and G. H. Weiss, Theory of Lattice Dynamics in the Harmonic Approximation (Academic Press, Hew York, 1963). 15. J. Skofronick, S. Safron, G. Chern and J. R. Manson, to be published. 16. J. T. Lewis et al., Phys. Rev. 1£1, 877 (1967). 17. The Debye temperature would be expected to be smaller for the surface region, as is found in the case of electron scattering. However, the Debye temperature as measured by He scattering is usually found to be about that of the bulk. The reason for this is usually ascribed to the fact that the He atom, being a large particle, interacts with more than one surface atom. This point is discussed in Ref.(l). Intensity (Arbitrary Unlti) <3 00 0 \ s \ O \ \ s C* c M Ml K D \ n m 3 \ \ O / a s t n \/ tt rt M K K \ o A r* \ / \ V / \ 80 0 \ s 60 « * 40 e 1 -* — * • * 20 800 200 400 600 T (K) Surface Temperature 329 Positronium Formation and Photoemission Spectroscopy for Surface Akira Ishii Faculty of General Education, Tottori University Koyama, Tottori 680, Japan Theoretical feasibility study of positronium formation spectroscopy is presented as a complementary spectroscopy to the UV photoemission. The- ory is very similar to photoemission if we recognized that Ps corresponds to a photon, the incident positron corresponds to a time-reversed LEED state. Calculation for Al(100) and Cu(100) showed that positronium formation is quite sensitive to the surface electronic structure. The time-reversed process of positronium formation is also possible to use as a spectroscopy. 330 1. Introduction The attempt to get more complete surface electronic structure is quite im- portant, because it affects results of photoemission analysis. As well-known, determination of band structure of solid is one of the most important prob- lem in physical science. However, especially for valence band electron, the- oretically reproduced photoemission spectrum depends strongly on model of surface barrier. Thus, we should determine the surface barrier to get rig- orous band structure from experiment. Nevertheless, at the moment, it has been believed to be very difficult to obtain information of surface electronic structures around the surface barrier separately from the bulk ingredients. Today, UV photoemission would be the best method for it. However, it is well-known that UV photoemission spectrum is consisted from photoelec- trons which comes from at least several atomic layers in the surface region. Therefore, photoemission spectrum is dominated by features which origi- nate from the bulk band structures. Identification and elimination of them to isolate surface served features are usually very difficult, because no one knows what is the expected solution. Especially for clean surface, it is very difficult to distinguish surface features among bulk features. Since positronium (Ps) is formed only outside of surface for metal and semiconductor, Ps spectrum from surface has been expected to be a very good probe for the surface electronic structure of the topmost layer[L-16j. Although ion neutralization has been also expected to be a good probe for electronic structures of the topmost layer, Ps formation spec'.roscopy (PsFS) is definitely better. The reason is that, by using PsFS, we can measure momentum distribution of electron from the top to the bottom of the band. Recently, STM (scanning tunneling microscope) would be considered as a good probe not only for surface atomic structure but for surface electronic structure. However, since STM is dominated from structure of using tip, the density of state of the tip should be removed from STM spectrumfi?]. Because we could not control atomic structure of the tip in detail, PsFS is better to get detailed information of surface electronic state. However, previous works on PsFS are not enough to get the advantage of this spectroscopy. Though there are some calculation on Ps spectrum formed from thermalized positron, no calculation on Ps spectrum formed from energetic positron beam incident on solid surface. Similarly, though there are a lot of experimental work on thermal positron, very few works 331 are on energetic Ps production by using positron beam[lj. Definitely, there are a lot of advantages on energetic positron beam method. First, we can observe electronic structures not only just below the Fermi surface but also of a whole band. Second, we can easily remove smearing of the spectrum due to the level broadening of Ps near surface[13]. Recently, some intense positron beam lines have begun to work in U.S., Germany and Japanfl]. Therefore, we can expect experimental support on the present work. 2. Theory In this letter, we present the first calculation of Ps energy spectrum by using multiple scattering theory presented by Ishii and Pendry[14], where we have included the separate ingredients known to be important in Ps formation: band structure, surface effects, matrix element, diffraction of the incoming positron wave and higher order corrections. Since the theory and program we used is quite similar to photoemission calculation[19,20], we can compare calculated Ps spectrum directly with the photoemission spectrum. In the recent Green's function formalism of PsFS[14], Ps momentum distribution, /(p), can be written as follows; (1) = j J J J dr+dT-dM>+dl>- (2) - £poiiitron(q) - E (3) where $p, is the wave function of incident Ps assumed to be a plane wave for the center of mass motion and the IS orbit for the relative motion. 332 xexp(lp+ . (4) for outside of the surface, where R is reflection coefficient of Ps calcu- lated by using the step barrier model for Ps with Ps work function. For the inside, r+ - r_ (5) According to ref.[14], the imaginary part of Ps inner potential is as- sumed to be an adjustable parameter which gives non- existence of Ps in bulk. 0q in eq.(2) is the wave function of emitted positron calculated as a LEPD wave by using LEED program. The surface barrier for positron is assumed to be a step potential whose height is defined by the positron work function. Usually, positron work function is very close to zero[l], so that effect of the barrier is not important for energetic positron. The imaginary part of the positron inner potential is assumed to be similar in value to that of electron. Though positron imaginary inner potential has been reported to be larger than electron one[l], we should use it as an adjustable parameter after experimental work is reported. G is the electronic Green's function for a finite step barrier[21,22j. The reflection and the transmission coefficient in it are calculated by using the step barrier model with the correction of multiple scattering effect as mentioned in ref.[14]. [ ,| (z. *'<0). (7) 2 (27r) * I i 333 (8) As pointed out in ref.[13], the dynamical effect or effect of finite life- time of Ps near surface due to the electronic interaction with the surface is negligible if the incident positron energy is far larger than the Fermi en- ergy. In fig.l, we show higher order contribution in Ps spectrum[13]. V£ is the strength of the matrix element which is here chosen to be a simple one. We found that for strong coupling case, the higher order contribution is important. However, in the case of PsFS using positron beam of several tens electron Volts, the coupling can be treated to be the weak one.[13,18] We can calculate /(p) from the ordinary golden rule formula: 7T (10) This is just same as a formula of the photocurrent calculationflSj. We could consider the above as a picture of a propagation: Ps comes from the vacuum with the time-reversed positron wave field, annihilates by the interaction, V, into the electron (hole) described by G, and is recreated by V with the time-reversed positron. The detailed analytic form of S is presented in ref.[14] by using muffin-tin potential formalism. 06 y,=oooooi 4 2 03 2 —1— 1 1 c 02 0001 li c 1 O d 002 i A - 0.01 •f-V / 1\\\ in ,.— Z Ltl J ~\ 1 ,- 0 002 \ IN 0.1 1 _1_ -10 -5 ENERGY Cigurf 3 An e.t&mplc ofthe angle-t»olvcd phatoemifiion tpci-tium for AI(IOO) The incident photon eneigy u 45eV and the ingle it 45° The emution »ngle o{ the photo- 005 o.l elccttoti is notm&l to the nu{»ce The lohd line it theoieti<:sl|14| 4nd the duhed one it expetim«nt4i|33| Pi ENERGY ( ATOMIC UNIT) E o Figure i Pi ip*elr* tot t inflow bind ihowfd Mow the l)guir|ll) Vj it * cuupl in • iimplt liiodel m4tm «lcm«nl htving Luieiitntn foini foi *\*t numtxti 335 3. PsFS spectra First, we show some recent results of PsFS spectrum calculation.[16] Al(100) is a very good example, because there is a very sharp peak of surface state in angle-resolved photoemission spectra[23]. We present in fig.2 a reproduction of photoemission spectrum[24] by using a program code of photoelectron calculation[20], where the muffin-tin potential of aluminium atom is taken from the book of Moruzzi, Janak and Williams[25]. The reproduction seems to be well including the surface peak. Next, we present in fig.3 an angle-resolved Ps spectrum, where incident positron angle and emitted Ps angle are both assumed to be normal, for direct comparison with the photoelectron spectrum. In fig-2, we notice at least the following two points. First, the surface peak is quite clear. Second, there are no bulk transition peak in contrast to the photoemission. Both of two should be considered with incident enegy dependence of the spectrum. Ps spectra for other incident energies are shown in fig.4. According to the calculation, we notice that the height of the surface peak has a very drastic dependence on the incident positron energy. Although surface peak height of photoemission spectrum also shows incident photon energy de- pendence[23], that of Ps is stronger. Just like I-V curve of LEED, we could perform the I-V curve analysis of the surface peak height of Ps spectrum. This analysis is not for determination of surface atomic structure but for determination of the shape of the electronic surface barrier potential. For the second problem, we can confirm that there are no bulk tran- sition peak in Ps spectrum. This is not disadvantage of the spectrosopy. The clear bulk transition peak means that the whole spectrum is bulk like: the transition occurs mainly under the surface. Therefore, the no-buik- peak feature is a great advantage to detect the surface local phenomena, e.g. electronic surface potential, surface states and electronic structure of adsorbed atoms. In other words, PsFS is suitable to observe surface density of states of electron directly. To demonstrate the DOS-like shape of Ps spectrum, we calculate a Ps spectrum for Cu(100) where is d-band just below the Fermi lewl.{16] In fig.5, we show the calculated Ps spectrum for 45° incident and 45° emission. We found that the shape of the spectrum is something similar to the DOS of d-band. We found that the d-band like spectrum seems to be shifted toward the Fermi level, in contrast to the photoemission spectra[26,27,28j. It agrees with a theoretical suggestion for surface band[29]: d-band at sur- 20. e a o CO 10. c -4 -3 -2 -1 0 = BF -f. -I -It -2 -I 0 tt KF Energy (eV) Energy (eV) t'igur< 4 Figure 3 The incident energy dependence of I'iFS tpecttum of AI(IOO) The incident energy The Pits spectrum of AI(IOo) »utf»ce. the ie»i »nd nn»jin»ty ptit of pouttan inJ of pmilroft i» 42 2, 43 5, it 9eV, teipectively the othet pu^meteti tie itme •• fig 3 fi pdlHltiii tee 0 02?,-4 0,. JS» »nd .57 2«V, tetpectively the seieening ptitmelet It (cf Kf.|I4|) U I 0 »lomic uluU the im*ji«4i)f p»it of eieclton poUntiil in »olid n '0 OfteV the incident en«i(y of potiliaii i> 40 8eV The incident and the emiitien tn|lt it aoiffitl la the mrftce 0.3 r—T-1—T—T—r—— s 0.9 1 \ 0.8 : || mm O.G (A • o 4) 0.5 0.1 : m S 0.3 • 0.2 - A ] •:l -5 -UL1)4 -3 -2 -1 0 = EF Energy (eV) figure i The P.FS .pecttum of C«(100) , ,a« The ,,«1 »nd , » «y p.,i of po.u.on »d u m pn EP = 0 1 2 3 |4 5 P. p.*** «, .0.J7..I.0. - 2 J7 M- .27 2«V, r«p«,i«|y The .««„,„, p.ume.e, M (rf <«»|I4|) U 1 0 4tonUC .a,.. The im»pn.ly ptrt of eleetton pote.titl in .oUd i. -0 4.V Energy (eV) The ..ddfl «•«„ „ ,0eV The indent M(|. of po.,,,0. Md ,h, tlBIMIBi ..„, tf,p, Figure S •re 45' The invent P»FS ipecttum of Cu(100) aurhce Thf i«l knd imi|iut) ptrt of potitron tnd Pi poltntiili tte -0 27,-4 0, • 2 17 »nd -2?2cV, leipeclively The icieen- '»« pMimetei M (ef tef |I4|) U 1.0 Home tiniu. The imtfintty ptti of election potentitl is tolid ii -0.4eV The incident enei|)r of P* cettttpondt to the iiodtut e»t|y of SOeV fat PtFS 338 face will be narrowed and shifted. The effect has not yet obserbed clearly in photoemission spectrum, because photoemssion would be more bulk-like. Since PsFS is more surface-like, we can observe the clear shift of the d-band at the surface. On the other hand, we could not measure bulk band structure by us- ing PsFS. Thus, PsFS and photoemission are complementary spectroscopies to each other. PsFS is more surface-like and photoemission is more bulk- like spectroscopy. Using the both spectroscopy after the determination of surface atomic structure by LEED, we could obtain more complete experi- mental picture of surface electronic structure. 4. inverse PsFS spectrum The time-reverse process of PsFS is possible, if we have monoenergetic Ps beam. Fortunately, Brookhaven National Laboratory has it and some other laboratories plan to build it[l]. Thus, Ps beam line is available for surface studies. The original idea of the inverse PsFS was presented by Ishii[30,31]. The relation between PsFS and inverse PsFS is just like that of photoe- mission and inverse photoemission. Just like inverse photoemission, inverse PsFS is used to detect unoccupied surface electronic structure above the Fermi level. Here we show one example; inverse PsFS spectrum of Cu(100). In fig.6, we found a very sharp peak. Since the position corresponds to the surface peak found in two-photon photoemission spectroscopy[32], our peak is also surface peak. The interesting point here is that we did not use the image potential: we use just square barrier. Thus, it is quite interesting thing that we can also see surface peak without image potential. Same as PsFS, it will be a good idea to perform I-V curve analysis for inverse PsFS. Since for many metal, we have surface energy gap below or above the Fermi level. Thus, if the inverse PsFS is established, we can do the I-V curve analysis to determine surface barrier for a lot of surfaces. 339 5. Conclusion We present a theory of PsFS and calculate angle-resolved Ps spectrum by using multiple scattering formalism to include band structure. The result for Al(100) gave us two features: strong energy dependence of the sharp surface peak and non existence of bulk peak. The result for Cu(100) showed us that the d-band like shape in the Ps spectrum is shifted toward the Fermi level. The inverse PsFS is also possible as a spectroscopy to unoccupied surface state. We show a demonstration spectrum for Cu(100) where we have a sharp peak without image potential. Since PsFS seems to be a very good probe to detect the surface barrier picture, experimental work of the angle-resolved PsFS by using energetic positron beam is expected. Acknowledgement The author thanks the Computer center of the Institute for Molecular Sci- ence for the use of the HITAC M-680H AND S-820/80 computer. 340 References [1] RJ.Schultz and K.G.Lynn, Rev.Mod.Phys.60 (1988) 701 [2] A.Ishii, Surf.Sci.147 (1984) 277, 295 [3] A.Ishii, Phys.Rev.B36 (1987) 1853 [4] A.Ishii and S.Shindo, Phys.Rev.B35 (1987) 6521 [5] S.Shindo and A.Ishii, Phys.Rev.B35 (1987) 8360 [6] S.Shindo and A.Ishii, Phys.Rev.B36 (1987) 709 [7] A.Ishii and S.Shindo, Surf.Sci.189/190 (1987) 988 [8] A.Ishii and S.Shindo, Nucl.Instrum.Methods B33 (1988) 382 [9] S.Shindo and A.Ishii, Nucl.Instrurn.Methods B33 (1988) 392 [10] R.H.Howell, I.J.Rosenberg, M.J.Fluss, R.E.Goldberg and R.B.Laughlin, Phys.Rev.B35 (1987) 5303 [11] D.M.Chen, S.Berko, K.F.Canter, K.G.Lynn, A.RMiIIs,Jr., L.O.Roellig, P.Sferlazzo, M.Weinert and R.N.West, Phys.Rev.B39 (1989)3966 [12] A.Ishii, J.B.Pendry and D.K.Saldin, in Positron Annihilation, Eds. L.Dorikens, M.Dorikens and D.Segers (World Scientific, Singapore 1989) [13] A.Ishii, Surf.Sci.2O9 (1989) 1 [14] A.Ishii and J.B.Pendry, Surf.Sci.2O9 (1989) 23 [15] A.B.Walker and R.M.Nieminen, J.Phys. F16 L295 (1986) [16] A.Ishii, to be published [17] M.Tsukada and N.Shima, J.Phys.Soc.Japan, 56 2875 (1987) [18] A.Ishii and S.Shindo, in Atomic Physics with Positrons Eds. J.W.Humberstone and E.A.G.Armour (Plenum, New York, 1987) [19] J.B.Pendry, Surf.Sci.57 (1976) 679 [20] J.F.L.Hopkinson, J.B.Pendry and D.Titterington, Computer.Phys.Commun.19 (1980) 69 C.G.Larsson and J.B.Pendry, private communication [21] J.E.Inglesfield, J.Phys.C4 (1971) L14 [22] J.E.Inglesfield and E.Wirkborg, J.Phys.C6 (1973) L158 [23] H.L.Levinson, F.Greuter and E.W.Plummer, Phys.Rev. B27 727 (1983) [24] A.Ishii and T.Aisaka, to be published [25] V.L.Moruzzi, J.F.Janak and A.R.Williams, Calculated Electronic Properties of Metals (Pergamon, New York, 1978) [26] L.Ilver and P.O.Nilsson, Solid State Commun. 18 677 (1976) [27] J.B.Pendry and J.F.L.Hopkinson, J.Phys. F8 1009 (1978) [28] D.M.Lloyd, C.M.Quinn and N.V.Richardson, J.Phys. C8 L371 (1975) [29] R.Haydock, V.Heine, M.J.Kelly and J.B.Pendry, Phys.Rev.Lett.29 868 (1972) [30] A.Ishii, in Proceedings of the International Workshop of Positron Part B\x\ press [31] A.Ishii, to be published. [32] K.Giesen, F.Hage, F.J.Himpsel, H.J.Reiss and W.Steinmann, Phys.Rev.33 971 (1987) 343 Inelastic Positronium Formation at Metal Surfaces Toshiaki Iitaka Department of Physics, Waseda University 3-4-1 Ohkubo Shinjuku, Tokyo 169, Japan BITNET: PARITY@JPNWASOO Abstract We calculate the probability of inelastic positronium formation with excitation of electron hole pair. In the case of thermalized positrons, it is shown that the inelastic process is small compared with direct formation process. 1 Introduction Positrons implanted into solids are thermalized in the solids, and diffused to the surface. Some of them are emitted from the surface as energetic positroniums changing the binding energy into kinetic energy [1]. Positro- nium formation processes can be classified into two types, the direct- and inelastic-formation which correspond to resonant- and Auger- neutralization of ions [2], respectively, though many authors have been studied the theory of the direct formation process only [3, 4, 5, 6]. We sti'dy, in this paper, inelastic formation by using surface dielectric function method, which has been widely used to get a simple formalism of low energy electron inelastic scattering [7, S], stopping power and energy straggling of ions near a metal surfaces [9, 10], electron-hole pair excitation in atom-surface scattering fll] and x-ray photoemission spectrum [12]. 344 2 Model Taking the first Born approximation with respect to the coupling of positro- nium with the metal electrons and RPA approximation with respect to the interaction between the metal electrons, differential probability of inelastic formation is expressed by 1 1 <(£,-£) (1) where |or), |/2), \6), |T),|7') represent the state vectors of positron, electron, positronium and ground and excited states of metal electrons, respectively. After some manipulation [11] and introduction of the spectrum function [7, S], we get dP f 3 2 - = - jdKt f d k \M\ S(Q, (2) dKydK where Q|| = ky — K|| and and J k i K* The reduced matrix element M is defined by \M+ - M-|2 (5) where the matrix elements M+ and M~ are calculated by using the wave functions of positronium, positron and electron. ± 3 + 3 M = Jd r jd r- The surface dielectric function I(Q, u) is defined by 345 We employ the approximate form for the RPA dielectric function e(k,u)) in the small **» region, 2 2 t[k,u) = 1 + Jp/{s k ^ - iiruO(2kf - k)/2kv}] - J). (8) Expanding eq.($) in a power series in .J to the first order and introducing it into eq.(7), we get the spectrum function with the surface dielectric function [9, 13), J} 2\ B(Q) where MQ) = | kTFxj4k) - Q* + 2kj y/Q* + k\F y. log' — 2 - Q - 2k, JQ* + k\F {Q2-*•••?• In the above equations, v, and kjF = -JZuipIv, mean fermi velocity and Thomas-Fermi wave number, respectively. In eq.(2), we have assumed that the position of electron- (positron-) po- tential step and the position of metal surface are the same. This assumption is consistent with the experimental observation that positron and positron- ium emission can be described by the same transmission model [14]. The wave functions are derived from this model potential taking into account the fact that positronium does not exist in the electron gas corresponding to metals [15]. For comparison, direct formation probability is aiso calculated. We em- ploy the screened Coulomb interaction as the electron-positron coupling [3]. The differential probability becomes [2, 3] dP Cl2 ! - = 2rr jdl where k|, = Kfl and k? = 2(-^ - £ + % + (12) where A/c is the matrix element of the Coulomb interaction, c 3 + + M = Jd r y>r-#,,(r , T-)--l—4>+(r*)4r{,-). (13) 3 Results and Discussion We have calculated Ps momentum distributions with the parameters for Al(100) at 300K. Normalized contour maps of the ejected Ps momentum distribution of inelastic and direct formation eqs.(2) and (11) are shown in fig.l and fig.2 where l(mrad) = mc x 10"3. Though the contour maps are similar in shape, the total probability of direct formation is about 100 times larger than that of inelastic one. We can conclude that Ps formation proba- bility is dominated by direct process. The difference between our theoretical results and the experimental con- tour map obtained by Lynn et al. [16] in low Kz region (fig-3) cannot be explained by a simple first Born calculation. The corrections for this dif- ference are discussed by Walker and Nieminen [3] and Shindo and Ishii [5]. However we did not made any corrections because the purpose of this paper is the comparison between direct and inelastic Ps formation probabilities. We have neglected the process accompanied with plasmon excitation be- cause thermalized positrons does not have enough energy to excite a plasmon i.e. u> < Upin eq(4). For the energetic positrons, this process may have im- portant contribution for Ps formation probability as well as electron hole pair process. The probability can be calculated with our formulation by expanding exact RPA dielectric function with respect to 1/w instead of ui. We can extend our theory to multiple scattering formalism [17] by intro- ducing the spectrum function into the Green functions. It will provide the information about the imaginary part of Ps self energy in the vicinity of the surface and multiple inelastic scattering of c+ and Ps. The limit of our calculation is that it is based on the step density dielectric function. For the more accurate calculation, some improvements are needed. 5 Kt (mrad) Kx (mrad) Kz (mrad) Figure 1: Contour maps of the ejected Ps momentum distribution of (a)inelastic Ps formation (eq.'2), (b)direct Ps formation (c)experimental data obtained by Lynn et al. (16) 348 One is the improvement of spectrum function for more realistic electron den- sity distribution, which may be obtained by low u> expansion of dielectric function of nonuniform electron gas rather than l/u expansion [18]. The other is to calculate Auger matrix elements which is popular in the field of ion neutralization [19, 20, 21, 22, 23]. In this paper, we have shown that Ps formation of thermalized positrons is dominated by direct process. It will be important to study the effect of multiple scattering and the probabilities for more energetic positrons. 4 Acknowledgment The authors are grateful to Prof. Yoshi-Hiko Ohtsuki, Prof. Mitsuo Kita- gawa, Dr. Shigeru Shindo, Dr. Akira Ishii, Dr. Masahiko Kato and Mr. Hi- roshi Arimoto for helpful suggestions, discussions and encouragements during the work. References [1] Peter J. Schultz and K. G. Lynn. Interaction of positron beams with sur- faces, thin films, and interfaces. Reviews of Modern Physics, 60(3):701, 1988. [2] Akira Ishii. Theory of Positronium Formation at Solid Surfaces. PhD thesis, Waseda University, 19S5. [3] A B Walker and R M Nieminen. Theory of positronium momentum spectra at metallic surfaces. Journal of Physics, Fl6:L295, 1986. [4] Shigeru Shindo and Akira Ishii. Quantum theory of positronium forma- tion at surfaces. Physical Review, B35(16):8360, 1987. [5] Shigeru Shindo and Akira Ishii. Calculation of momentum distribution of posit.roniinn ejected from surfaces. Physical Jlrvtrw, IWfi{ 1 ):70.0, 1.987. [6] Akira Ishii. Theory of positronium formation on solid surfaces. Surface Science, 147:277, 19S4. 349 [7] Masakazu Ichikawa. Inelastic Scattering of Low-energy Electrons from a Crystal Surface. PhD thesis, Waseda University, 1974. [8] Masakazu Ichikawa. Inelastic scattering of low-energy electrons from a crystal surface. Physical Review, B10(6):2416, 1974. [9] R. Nu nez, P. M. Echenique, and R. H. Ritchie. The energy loss of energetic ions moving near a solid surface. Journal of Physics, C13:4229, 1980. [10] Masahiko Kato, Toshiaki Iitaka, and Y.H. Ohtsuki. The skipping motion of ions and their energy loss spectrum. Nuclear Instruments and Methods in Physics Research, B33:432, 198S. [11] James F. Annett and P. M. Echenique. Long-range excitation of electron-hole pairs in atom-surface scattering. Physical Review, B36(17):8986, 1987. [12] Masahiko Kato. Singularity index of the core-level x-ray photoemission spectrum from surface atoms. Physical Review, B38(15):10915, 19S8. [13] Masahiko Kato. Energy Loss of Charged Particles at Metal Surfaces. PhD thesis, Waseda University, 1986. [14] D.T. Britton, P.A. Huttunen, J. Makinen, E. Soininen, and A. Vehanen. Positron reflection from the surface potential. Physical Review Letters, 62(20):2413, 1989. [15] H. Kanazawa, Y. H. Ohtsuki, and S. Yanagawa. Positronium formation in metals. Physical Review, 13S(4A):1155, 1964. [16] K.G. Lynn, A.P. Mills, Jr., R.N. West, S. Berko, K.F. Canter, and L.O.Roellig. Positron or positroniumlike surface state on Al{100) ? Physical Review Letters, 54(15):1702, 1985. [17] Akira Ishii. Theory of positronium formation at metallic surfaces I. Surface Science, 209:1, 19S9. [18] Mitsuo Kitagawa. Calculation of the stopping power at the surface of a solid. Nuclear Instruments and Methods in Physics Research, B33:409, 1988. 350 [19] J.W. Gadzuk. Theory of atom-metal interactions ii. Surface Science, 6:159, 1966. [20] S. Horiguchi, K. Koyarr.a, and Y.H. Ohtsuki. Auger neutralization of slow protons at surfaces, phys. stat. sol., (b)87:757, 1978. [21] Y.H. Ohtsuki. Charged Beam Interaction with Solids. Taylor k Francis Ltd, London, 1983. [22] K.J. Snowdon, R. Hentschke, A. Narmann, and W. Heiland. Auger and resonant neutralization of low energy ions near metal surfaces. Surface Science, 173:581, 19S6. [23] R. Hentschke, K.J. Snowdon, P. Hertel, and VV. Heiland. Matrix el- ements and transition rate for auger neutralization of low energy ions near metal surfaces. Surface Science, 173:565, 19S6. 351 Energy Loss Measurements of He and H scattered at grazing incidence off Ni A. Narmann, K. Schmidt, U. Imke and W. Heiland University of Osnabriick, D-4500 Osnabnick, RFA Abstract The inelastic losses of a particle scattered from surfaces is determined by its charge state along the trajectory. Measurements using He and H at energies from 300 eV to 10 keV give information about the energy dependence and the peak shape of the energy spectra. The energy dependence is related to the velocity dependence of the electronic stopping power, the peak shape affords additional information about the charge state and the straggling. 352 1. Introduction In recent experimental and theoretical studies of the interaction of slow He particles with Ni surfaces we have shown that the charge state fraction of the scattered particles and the inelastic energy loss are closely related1^). The theoretical analysis of the experimental data is based on the dielectric response of the metal to the moving charge46'. It has been shown that Auger neutralization is the dominating neutralization process whereby the s-electrons are mostly contributing owing to the surface channeling at rather large distances above the surface. Also the inelastic loss is dominated by the s-electrons. An interesting result is the rather large straggling observed experimentally and theoreti- cally^). The surface channeling effects have been studied in detail^ using the code Marlowe10'. We found for particles scattered along a random surface direction planar channeling and an increase of the trajectory lengths with energy (= velocity). Since the surface friction coefficient yt° turns out to be approxi- mately proportional to the velocity too, hence the inelastic loss is Qp = Ys° v L * v3, where L is the trajectory length^). We extend the experiments in the present work to the interaction of H with a Ni(lll) surface. 2. Experiment and Results The experimental system is described elsewhere11'. It is essentially a time-of- flight (TOF) system operating at a fixed angle of incidence of 5° and laboratory scattering angle of 10°. The angle of acceptance of the detector is 1.2° (full width). The crystal surface is prepared by sputtering and annealing and controlled by low energy ion scattering spectrometry (ISS). With He we covered an energy range from 250 eV to 14 keV. With H we obtained reproducible beams and results between 300 eV and 10 keV. These energy ranges correspond to velocity ranges of 2105 m/s to 9105 m/s and 3105 m/s to 13105 m/s respectively. The corresponding ranges of the "perpendicular" energy component Ej. are 2 eV to 106 eV and 2.3 eV to 76 eV respectively. Fig. 1 shows energy spectra of neutral H for 4 primary energies of the incident H+. There is an appreciable change in the width and asymmetry of the peaks. Fig. 2 summarizes the results in the form Qp (peak energy loss) vs the primary energy Eo and vs the velocity v. For a comparison we reproduce the He results^) here, i.e. Qp(He) vs Eo and vs v3 (Fig. 3). 353 3. Discussion We observe a striking difference even in the overlapping velocity region between the H and He data. For He we find Qp « v with ns3, for H it is rather Qp a vm with m = 1. This would imply i) that in case of H the surface friction coefficient is constant - as a bulk friction coefficient5^) and ii) that the average trajectory length changes very little with energy. Above about 1-106 m/s the H energy loss follows again approximately a straight line - within the scatter of the data but certainly with a steeper slope. There is the possibility to correlate the energy of 6.5 keV, i.e. where the slope of Qp vs v brakes, with the breakthrough angle ipb for channelingt2,i3). An estimate of the critical angle ips for low energies wich is closely related to the breakhrough angle is given by: with (l) where ai2 = 0.855 ao (Z\ "2 + Z2I/2)"m is the Firsov screening length and d the spacing in the rows of Ni atoms. For a [110] chain this yields tps = 6.9° at 6,5 keV for H and tps = 5° for He at 13 keV, i.e. the breakthrough is probably not observable in our range of energies for He. It should be noted that the breakthrough angles are smaller for planar channeling^,13» which is probably the case for the "random" azimuth (45° off [110]) chosen here. The reduction is however of the order of 20 % only 13). So it is plausible to relate the brake in the H- energy loss curve at 6.5 keV to the change from surface scattering to bulk scattering. The analysis of the He scattering2,9) has shown that the average trajectory length increases with increasing energy in the range of surface scattering. Since the range of Ex is the same for H we have no reason to assume a different behaviour for H. Previous TRIM calculations'•*> showed the same behaviour, i.e. an increase of the average path length with energy. That is, the two assumptions made above are probably wrong. The energy spectra of the scattered He have been successfully described3.7,8) by Q > (2) gy a -* -m o with dQ ~ v udl and, dQ = v vdl 4 5 • • 354 where ys* and ys° are the surface friction coefficients for the He* ion and the neutral respectively, tA is the Auger-neutralization lifetime, 6 the step function and 1 the length. This equation (2) is then convoluted with the proper energy loss straggling. The fact which makes the average surface friction coefficient ys° to increase with velocity is the increase of the survival length of the He* ions with increasing velocity, hence the larger \'s* contributes increasingly to the energy loss. For H we face a quite different situation. Hydrogen has a stable H" state, such that after surface scattering an appreciable yield of H* is observed1^. 16). Recent calculations have shown that in the bulk the actual charge state of H is indeed H' at low velocity. With increasing velocity the H" part decreases monotonously whereas the H° part increases until the H * part starts to take over at about 1 a.u.. The stopping power for H" (and H+) is larger than for H0'7'. Therefore we assume in analogy to the He case for H: dQ~ = v'udl and dQ" = y°a dl I3I with decreasing survival length for the H~ with increasing energy due to resonant neutralization (loss) in the case here. We expect ys' > Ys°- Our experimental result indicates that the resulting decrease of Qp is compensated by an increase of the average trajectory length. Theoretical studies and Marlowe calculations are in need to prove the open points in the discussion given here. Acknowledgement: This work is part of the D.O.M. collaboration (Donostia - Osnabruck - Madrid) partly sponsored by the DFG and the DAAD. Discussion with P.M. Echenique, F. Flores and R. Monreal are gratefully acknowledged. 355 References 1) R. Monreal, E.C. Goldberg, F. Flores, A. Nannann, H. Derks and W. Heiland, Surf. Sci. 211/212 (1989) 255 2) A. Narmann, H. Derks, W. Heiland, R. Monreal, E. Goldberg and F. Flores, Surf. Sci. 217 (1989) 255 3) R. Monreal, F. Flores, A. Narmann, W. Heiland, S. Schubert and P.M. Echenique, Radiation Effects and Defects in Solids 109 (1989) 75 4) F. Sols and F. Flores, Phys. Rev. B30 (1984) 4878 5) P.M. Echenique, R.M. Nieminen, J.C. Ashley and R.H. Ritchie, Phys. Rev. A33 (1986)897 6) P.M. Echenique, F. Flores and R.H. Ritchie, Solid State Physics, Series 1989 7) A. Narmann, R. Monreal, P.M. Echenique, F. Flores, W. Heiland and S. Schubert, to be published 8) A. Narmann, K. Schmidt, W. Heiland, R. Monreal, F. Flores and P.M. Echenique, Proc. 13th ICACS, Aarhus 1989, to be published in Nucl. Instr. Meth. B 9) H. Derks, A. Nannann and W. Heiland, Nucl. Instr. Meth. B, in press 10) M.T. Robinson and I.M. Torrens, Phys. Rev. B9 (1974) 5008 11) B. Willerding, H. Steininger, K.J. .Snowdon and W. Heiland, Nucl. Instr. Meth. B2 (1984) 453 12) J. Lindhard, Mat-fys Medd. Kong. Danske Videnskab. Selskab 34 (1965) No 14 13) M. Hou and M.T. Robinson, Appl. Phys. 17 (1978) 371 14) H. Steininger, B. Willerding, K. Snowdon, N.H. Tolk and W. Eckstein, Nucl. Instr. Meth. 2 (1984) 484 15) W. Eckstein, H. Verbeek and R.S. Bhattacharya, Surf. Sci. 99 (1980) 356 16) J.N.M. van Wunnik and J. Los, Physic* Scripta T6 (1983) 27 17) M. Penalba, A. Arnau and P.M. Echenique, Proc. of the 13th ICACS, Aarhus (1989), to be published in Nuci. Instr. Meth. B 356 Figures Captions Fig. 1: Energy spectra of H scattered from Ni(lll) (glancing angle 5°, scattering angle 10°) for four primary H+ energies. Fig. 2: Energy loss of H° scattered from Ni(lll) (glancing angle 5°, scattering angle 10°) as function of the primary H+ energy (top) and of the velocity (bottom). Fig. 3: Energy loss of He0 scattered from Ni(l 11) (glancing angle 5°, scattering angle 10°) as function of the primary He * energy (top) and of v3 (bottom, v — velocity). 357 2300 2400 1M0 2H0 1790 2M0 MOO MOO JIM KMTOCCHCftBTlaVj E,-e«ZO«V "S 0.4 ejso itao tioo 72:0 790s 7100 1100 »«oo 17M MOO 4000 *Mt 4400 4M0 XMTICCNCET[ 800 Energy loss of H+-lons scattered of f Ni{111) 700 under grazing incidence 57 S00 v ^500 o >• 400 o os £ 300 Id 200 100 f 2000 4000 6000 8000 PRIMARY ENERGY [eV] 800 Energy loss of H+-lons scattered off Ni(l 11) 700 under grazing incidence 5* 600 v J8 500 o V 400 O I 300 200 - III 100 - ..»' .-•••*' 4 6 8 10 12 VELOCITY s-1* 105] 359 500 Energy loss of He+-lons scattered 700 off Ni(i 11) under grazing incidence ST 600 v 8 500 O >- 400 o an £ 300 ui 200 - 100 - . 11 * t 2000 4000 6000 8000 10000 12000 14000 PRIMARY ENERGY [eV] 500 - Energy loss of He+-lons scottered 700 offNi(m) under grazing incidence - " eoo - - 0) to -uu - - O >• 400 o g 300 - 200 100 > I I - t i i i i 10JO 200 300 400 500 361 ELECTRONIC RESPONSE OF METEAL SURFACES TO ELECTRON EXCHANGE WITH DOUBLY CHARGED IONS A.Niehaus Fysisch Laboratoriun, Rljksuniversltelt Utrecht, 3508TA Utrecht ABSTRACT Electron spectra are discussed that arise from gazing incidence collisions of doubly charged keV-ioru with single crystal surfaces at well defined angles of observation. New results are presented for doubly charged He.Ne.Ar colliding with a Ni(110)•surface, and are compared with corresponding results for a Cu(llO)- surface reported earlier. Evaluation of the spectra in tents of Che model developed in connection with the Cu-data confirms also for Ni the earlier conclusion that the metal surface "responds" delayed Co the charge changing processes in front of it, with delay times of approximately 5x10 s. New results are also reported and discussed for the angular distribution of auto ionization electrons arising from doubly excited He- and Ne- atoms formed by capture of two surface electrons. It is found that these distributions are well described by the complex sub level population amplitudes of the corresponding decaying atomic states if a slight distortion by the charge- image charge interaction is taken into account. In the collision plane formed by the incoming beam and the surface normal the distributions are found to be oriented parallel to the surface normal. Changes of electron spectra upon controlled chemisorption of oxygen atoms are discussed. Changes of the structures due to auto lonization suggest that each auto ionization line has a low energy satellite. This satellite is tentatively attributed to the excitation of a surface plasmon. Comparison with numerical calculations using existing formulae of linear response theory support this interpretation. INTRODUCTION In collisions of keV-ions at sufficiently low angles of incidence the atomic particle is reflected by the combined repulsive interaction of the nearest surface atoms, and does not penetrate into the surface. Since in such collisions the distance of closest approach to a surface atom is large, the mechanisms of kinetic emission are inefficient. On the other hand, the various mechanisms of spontaneous emission of an electron are very efficient, due to the long residence time of the atomic particle in the region of appreciable electron density above the surface. It may therefore be expected that electron spectra arising from such grazing incidence collisions are dominated by contributions from spontaneous, or "potential"-, emission. In the independent electron picture the simplest and most probable transitions leading to spontaneous emission involve two electrons: one 362 electron becomes nor* strongly bound In Che transition,by filling a "hole" in Che ion, and Che other one gains the transition energy of the first one. It is emitted into the continuum if its binding energy before Che transition was sufficiently low. Since in addition to the two-electron "Auger-type" transitions there also occur one-electron resonant charge exchange transitions (RT) between the ion and che surface. Chare arise several spontaneous lonlzatlon mechanisms that nay be distinguished by the "types" of electrons involved, metal electrons or atonic electrons: 2+ Fig.l: Tentative decoaposition of a He spectrum obtained at an elevation angle of 2"and 2keV.Peaks are due to AU.the low energy hump to ACl.and the 'rest'(the dashed curve) to AC2. the surface,i.e.,the surface density of electronic states (SDOS).A aodsl based on the quantities (i-iv) was developed in connection with electron spectra for the Cu(llO)-surface [1,2].Only three free parameters were introduced to adapt the calculated spectra to the measured ones. The calculations were carried out assuming the following reaction paths. 2+ 2 2 A + M < E > A J*- M * AU l> A + + e <- RT A + AClJ A + M*++ 2e' Qualitative agreement with aeasured spectra was found to be achievable when on* of the three free parameters, th* so called response parameter, was given a value that allowed to Interpolate between a "diabatic" and an "adiabatic" response of the metal electrons to th* spontaneous electron emission processes. Her* "adiabatic' means that in the final state of the transition th* aetal electronic system is completely relaxed, and "diabatic" means that th* system Is frozen. Th* finding that th* metal response is intermediate between diabatic and adiabatic means that the ion- metal system is excited after th* electron is eaitted. Calling E the excitation *n*rgy,e th* energy of an ejected 364 electron, and tA the energy the ejected electron would have in case of adiabatic behavior,we have the relation <(z)- with r being the characteristic relaxation tine of the sysrea.t. depends on the velocity of the ejected electron. Ve therefore introduce as free parameter the constant C defined by the relation t. / r - CX V e /«* (4) ion' F with < being the Fermi energy of the metal electrons. All spectra obtained for Cu(llO) were best reproduced in calculations when a value close to C-0.5 was chosen.With this value, and by expressing t. by a characteristic distance L. the emitted electron must ion J ion travel at its velocity v to complete the ionization event, i.e.as t, -L, /v , one obtains from (4) for the relaxation time r: ion ion' a ' a W (V2) <5> with v being the velocity of electrons at the Fermi level. If a characteristic distance of 10a.u.is assumed, this leads for Cu to a relaxation time of approximately 7x10 s. As an example of the agreement achieved by using this value in the calculations, in contrast to this complete disagreement obtained when using an infinitely short relaxation time(adiabatic behavior),in fig.2 calculations and the measured spectrum for He / Cu(110) are shown. The identification of the mainly contributing spontaneous ionization mechanisms, and the finding that a finite relaxation tin* of the metal electronic system exists and is rather sensitively reflected in the electron spectra, are the aain results of our recent work. Regarding the complexity of the events occurring in grazing incidence ion surface scattering, it is obvious that our model is only a first step in the development of an accurate description. Further refinements, guided by 365 experimental daca, are certainly necessary. In the following we will present some new experimental data that may be useful for this purpose. Fig.2: Comparison of experimen- tal spectrum (a),with model cal- culations; (b) assuming adiabatic behavior,and (c) with intermedi- ate behavior (parameter C-l/2). 10 20 30 40 ELECTMON ENERGY [*vj COMPARISON OF DATA FOR CU(llO) AND FOR NI(llO) Electron spectra for lkeV He2+,Ne2+,and Ar2+, colliding with a Cu(llO)- and a Ni( 110)-surface are shown in fig.3 and ' fig.4. respectively. For both cases the electrons are detected in the collision plane (8-0°), and perpendicularly to the incident bean (0-90°).The angle of incidence is 8° for Cu(llO), and 2° for Ni(llO). Qualitatively the corresponding spectra ere similar for Cu and Ni, indicating that the same ionizatlon mechanisms are dominant,namely,ACl,AC2,and AU.The main differences are that AU is much less pronounced for Ni, and that the AC2-component is even more broadened with respect to the shape that would correspond to an adiabatic behavior of the metal.It should be pointed out that th« differences do not result from the different angles of incidence:at 8° elevation angle the AU-component for Ni is even less pronounced.In fig.5 a H«2+/ Ni(110)-spectrum is shown together with calculations assuming the same reaction paths as for 366 Cu(llO) (see scheme 1). Fig.5c represents the calculation in which ths response parameter C-0.5 is used, which yielded the best agreement for Cu. Obviously,the agreeaent is such less satisfactory for Ni.In fact,within the present formulation of the model, no satisfactory agreement with the experimental spectrua can be achieved. As is suggested by the indicated tentative t.tC"»OH 1*1*0* r*i Fig.3:SpectraforCu(110) Flg.4:SpectraforNi(110) -C '5 20 Ji 30 IS -Z "5 JO 25 3C 35 Fig.5:Measured Si(llO) spectrun for an elevation angle of 6°.and a collision energy of 1 keV (a).Calculated spectrua for adi- batic behavior (b),and intermediate behavior (c) decoaposition of the experimental spectrua in fig.5a, the AC2- coaponent extends to very low energies suggesting extra energy losses for this process that are not included in the model. Another interesting difference, not visible in the spectra of fig. 5 because of the discrimination against low energy electrons by our spectroaeter, arises for Cu and Hi at low energies in the He - spectra. These low energy parts can be measured if an acceleration voltage is applied between surface and detector.The 367 resulting spectra for Cu and Ni are shown In fig. 6 and fig.7,respectively. A narrow peak at an energy that corresponds to 2 eV in the emitter fraiie is observed for Cu and not for Nl.The same peak has been observed by Varga et al.[3], and has been ascribed to the following mechanism: PI Ne2++ M Ne+*(4s)+ M+ -• H«**(3«)+ H2++ e" (6) z • 2 I C.tC"•Ox £NC«O» C«] 2+ Fig.6: Measured electron spectrum for lkeV He incident on Cu(110).Electrons are accelerated by 4 eV. O JO CLCC'IW !«(•« [.»] Fig.7: Measured electron spectrum for lkeV K« ^incident on Ni(110).Electrons are accelerated by 4 eV. i.e.,Penning lonization of the metal by the excited He**(4s) ion. Since the electron spectra due to Penning lonization to a certain approximation reflect the (SDOS) directly, the observed peak would have to be ascribed to the d-band of Cu. and its absence in the case of Ni, to the fact that no peak like structure occurs due to the d-band in the (SDOS) .UPS-spectra of the two metals, which of 368 course are less sensitive to the (SDOS), sees to confirm this conclusion. INELASTIC ENERGY LOSSES OF EMITTED ELECTRONS Recently we neasured electron spectra for doubly charged He* and Ar- ions scattering froa clean and oxygen covered Cu(110)[2]. Ic was found that the auto lonlzation peaks decreased significantly when half a mono layer of oxygen v*s cheaisorbed.In addition, we observed that at energies about 3 to 4 eV below the auto ionization peaks also a significant decrease of the intensity occurred. To denonstrate this observation, ve show in fig. 8 the difference between the spectra for the clean and the oxygen covered surface in the region around the auto ionization peaks.Variation of the detection angle in the scattering plane results in an energy shift of the the two structures of both ions, indicating that the two structures belong to auto ionization of the moving particle. • e [«v] 330 360 Q [«v]——« 3 *32 I 0 ——t [tv] 10O 12.19 Q£,v] —8 5 4 1 2 i 0 Flg.6: Auto ionization peak and low energy satellite for He and Ar. 369 A plausible explanation for Che occurrence of the extra low energy satellite is that, it is due to a discrete energy loss of tho emitted auto ionization electron by excitation of a surface plasmon.The occurrence of extra energy losses due to ehe fact that in the final state of an ionization process a localized hole is suddenly created co which the aetsl electrons cannot adiabatically respond, has been extensively discussed in the literature in connection with Inner shell photo lonization [4], Froa these discussions it appears appropriate to ascribe the high energy auto ionization peak in our spectra to the so-called adiabatlc peak, which corresponds to a fully relaxed final state.however, is populated wich diminished probability due to a finite probability for the excicacion of a surface plasaon. The probability for excitation of a surface plasaon by an electron emerging froa the surface is related to che imaginary part of its coaplex self energy [5].Within linear response theory for this probability an approxinate expression has been published [5]. We have numerically evaluated this expression for the case of He auto ionization(eleccron energy 35.8 eV,vertical emergence froa che surface).The result of this evaluation is shorn in fig.9a.The low energy structure is broadened as compared to the original peak due to dispersion of che energy of che plasaon.We nocice Chat Che calculated ainiaua energy loss is auch larger than the measured one, and chat the plasaon excitation probability ?.s lower Chan measured.Fig.9b 3hows a calculation for which the electron density at the surface is assumad to be lower by a factor 6.S than che bulk density.Surface plasnon frequency as well as che excitation probability now agree rather well with Che aeasureaenc. AC this point ic is important co realize chat, despite the remarkable agreement between aeasureaenc and calculation, other contributions to the low energy satellites cannot be excluded.Especially, a larger contribution froa Penning ionization by He in combination of a peak like structure in che (SDOS) of Cu could ,in principle, also result in Che observed satellite. This will be investigated further In the future. 370 2.5 t A c 1.5 a 1 5 0.5 - -10 -5 Energy loss -15 -10 -5 Energy loss (eV) fig.9: Calculated energy loss spectrua for electrons of 35.8 eV emerging vertically from a Cu-surface.The low energy satel- lite Is due to excitation of one surface plasnon. (A)'bulk'electron density is used.(B)density lower by 6.5. ANGULAR DISTRIBUTIONS OF AUTO IONIZATION ELECTRONS It Is to be expected that to each of the mechanisms contributing to the electron spectra a characteristic angular intensity distribution exists which in general will be anisotropic. In order to obtain a physically aeaningful experinental result it is therefor* important to isolate in the spectra a special component *.hat belongs to a certain mechanism. This is possible to a good 371 approxlnation for the component belonging to auto lonlzatlon. For 2* 2* the components belonging to auto lonization of He and of He we have recently determined angular distributions for the case of a Cu(llO) surface and of grazing Incidence collisions at 1 keV and an incidence angle of 4° [6 ].Measurements war* carried out In two planes, in the collision plane, and in the plan* which is perpendicular to it and contains the surface normal,. The directly measured intensities were corrected for angle dependent detection efficiencies. In fig. 10 and fig.11 we present the results for He and Ne, respectively.In connection with the theoretical interpretation outlined below, a polar coordinate frame is chosen whose z-axis is perpendicular to the scattering plane, the x,y-plane, and whose x- axis points Into the direction of the primary beaa. The left part of both figures shows the distribution in the scattering plane (polar angle $ -90°,varying azimuthal angle 4 ). The direction of the surface normal at 4 — 94° is indicated by the dashed line. The right part of the figures shows the distributions in the other detection plane. In this plane 4 is nearly constant and close to 90°, while # varies from 0° to 180°. The direction of the surface normal is here at I - 90° ( 4 - 94°).The intensity in the various detection directions is plotted as distance from the origin. The data points are the experimental results, and the drawn lines represent calculations. The auto ionizing states to which the distributions belong are indicated in the figures. In the case of He, it is not possible to decide from the energy position of the auto ionization line whether the He2*(2s2p)XP , or ths He2*(2p2)1D is responsible. In fig.10 therefore the experimental points are compared with calculations based on the two possibilities. All distributions are strongly anisotropic, and are oriented in the direction of the surface normal. In case of the distributions of the S-states, the anisotropy can be ascribed to the refraction of the electrons by the attractive electron surface potential. In fact, the theoretical curves shown result when an isotropic initial distribution is assumed, and when the electron is retarded by approximately 3.S eV when it escapes from the surf ace. This result suggests that the observed distributions result from an undisturbed initial distribution, which is modified after the 372 emission process in the MM way for all processes. This is the basis for our theoretical calculations. The relation between the initial distribution and the complex sub level population amplitudes of the decaying atomic state is the same as in the case of auto ionlzadon of an isolated a ton, except for Che fret that only pare of the distribution can be aeasured. Since the collision plane as well as the deflection angle is well defined in grazing incidence collisions, the situation regarding the preparation of the auto ionizing atom is foraally the saw as in the case of gas phase collisions when impact parameter and collision plane are selected by measuring angular distributions in coincidence with one of the scattered particles. For the description of our initial angular distributions in terms of the population amplitudes of the decaying atom we may therefore use the same relations as for the description of coincident angular distributions in gas phase collisions.These relations are well established and have been applied in the past [7,8]. In the case of He, which auto ionizes into an S-state of the ion, in principle "complete" Information on the excited state can be obtained, i.e., all complex population amplitudes can be determined from the angular distribution. -In the 2 case of Ne, which auto ionizes into P final states, the ratio of matrix elements belonging to 1-1 and 1-3 outgoing electron waves, respectively, remains undetermined for the decaying 0 state. For the decaying P state only 1-1 outgoing waves arise due to parity conservation.. The hamiltonian relevant for the collision system has reflection symmetry with respect to the collision plane. This results in symmetry constraints for the possible values of the population amplitudes.In the coordinate frame used here, these constraints result for a P-state in the condition that the amplitude for the magnetic quantum number m—0 oust be zero, and for a D-state that the amplitudes for m-+l,-l oust be zero. By adapting the other complex amplitudes to obtain best agreement with the experimental distributions- after taking into account above mentioned refraction correction- the above mentioned formulae yield the theoretical distributions shown in fig.10 and fig.11. The population amplitudes a - /*/ x exp(ixa ) determined In this way are given in table 1. 373 Fig.10: Measured (points) and calculated (llnes)distribu- tions for Ha. Fig.11: Measured(points) and calculated (lines)distribu- tions for Ne. He(2s2p)1P He(2p2)lD /aj/ - 0.71 - 0.05 24 .61 a1 /*_2/ - o. i " ° /a j/ - 0.71 a 05 /* / — 0.42 - 0 i""0- °n /«2 / - 0.87 - 0..11 Ne(2p43s2)3P Ne(2p43s2)lD /a j/ - 0.71 a . .512 55 - 0.48 r - /a_2/ - 0. a-2 /a / - 0.71 a. - i.52 78 - 0 x /•o / - °- an /•2 / - o.28 a2 - 2.5 Table 1:Population amplitudes of the auto-ionizing states.Phases are given in radians. 374 The agreement assuming a D-state for He Is significantly better than the one obtained if a P-state Is aastaMd.lt la further remarkable that the distributions are oriented along the surface normal and not along a direction perpendicular to the incoming beam. Since all the electrons are ejected befor* the atoms are deflected, an orientation with reapect to the surface normal is not a priori expected.Finally we mention that for the given symmetry angular momentum perpendicular to the collision plane aay in principle be contained In the atomic states, from table 1 it is seen that only the 0-states contain such angular momentum. ACKNOWLEDGEMENT This work was performed as part of the prograa of the 'Stlchting voor Fundamenteel Onderzoek der Materle'(POM), with financial support from the 'Nederlands* Organisatie voor tfetenschappelijk Onderzoek'. REFERENCES [1] P.A.Zeijlmans van Eanichoven, P.A.A.F.tfouters, A.Niehaus. Surfaco. Sci. 19S (19S8) 115 [2] P.A.A.F.tfouters, P.A.Zeijlmans van Eamichoven, A.Niehaus. Rad.Eff.and Oef.in Solids 109 (1989) 111 [3] P.Varga, W.Hofer, H.Winter. Surface Sci.117 (1982) 142 [4]tf.F.Egelhoff Jr. , Surface Sci.Rep.6 Nos.6-8 (1987) [5] P.K.Echenique, R.H.Ritchie, N.Barberan, J.Inkson, Fhys.Rev.B23 (1981) 6486 [6] P.A.A.F.Wouters, A.Niehaus ,to be published [7] E.Boskaap, R.Morgenstern, P.v.d.Straten, A.Niehaus, J.Fhys.B17 (1984) 2823 [8] E.Boskamp, O.Griebllng, R.Morgenstem, G.Nienhuis, J.Phys.B15 (1982) 374S 375 TRANSIENT ADSORPTION, ENERGY LOSS, AND MOMENTUM TRANSFER IN LOW VELOCITY ION SURFACE SCATTERING K. J Snowdon*, D. J. O'Connor, M. Kato, R. J. MacDonald Department of Physics, University of Newcastle, Newcastle, NSW, 2308, Australia Abstract Experimental evidence of transient adsorption or skipping motion of 2OO-2OOOeV Si* beams from a Cu(111) surface under angles of incidence from 1-20° is presented. Detailed analysis reveals that at all incident energies and angles, the highest energy loss peak of specularly scattered Si * and Si' ions has suffered an energy loss equal to or exceeding the "perpendicular energy" of the incident beam to the surface. This would be sufficient to induce trapping by inelastic processes alone. Furthermore, we can deduce from the data that detrappgng must occur through a combination of conservative processes and momentum transfer between the perpendicular and parallel momentum components of the beam to the surface. Unambiguous evidence for such coupling of the momentum components is provided by data for 5O-2OOeV Na * and K * scattering from the same surface. The origin of this coupling is explained. Present and Permanent Address: Fachbereich Physik, Universitat Osnabruck, D-4500 Osnabruck. W. Germany 376 Introduction Experimental studies of small angle diatomic molecular ion beam scattering from carefully flattened metal surfaces have successfully illuminated some aspects of the adsorption dynamics of neutral molecules at thermal energies [i-7].The behavioural similarity, at grossly different molecular translation*! energies, seems related to the nevertheless near-adiabatic interaction Both the scattering and adsorption appear to proceed on the same or similar potential energy surface (PES), and it is tempting to assume that the fast motion of the ion beam along the surface has little influence on the component of the motion perpendicular to the surface. The scattered beams, however, suffer quite high energy losses, and although these losses convert to quite small losses per unit path length in the close interaction region, the local electronic excitation of the metal must be non-negligible. This troublesome observation demands a detailed examination of energy and momentum transfer processes between low velocity atomic and molecular ion beams and surfaces under small angle scattering conditions. Conveniently, this regime is also close to the limit often addressed theoretically, of motion parallel to the surface. This paper reports results of the scattering of Na*, K* and Si* beams from a Cu(1 11) surface. Experimental Ion beams of Si *, Na * and Kr were generated in an ion source, accelerated, and focused onto the (111) face of a Cu crystal. This crystal (99.999% purity) was cut and polished with diamond paste down to 1pm within 0.5° of the (111) plane, and following installation on a two-axis goniometer, was further cleaned and polished under UHV conditions (2x10~'°torr) by a combination of glancing incidence sputtering and annealing. Scattered positive and negative sons were energy analysed by a 180° hemispherical electrostatic energy analyser operating in AE/E = constant mode. This spectrometer was also mounted on a two-axis goniometer. Mass analysis of the primary beam, and of both scattered and recoil ions was performed by time of flight techniques. The primary beam energy and direction was measured by translating the target out of the beam and passing the beam directly into the energy analyser. This ensured very accurate determination of both the scattered particle energy ioss relative to the incident 37 beam energy, and the scattering angle. Further details of the experimental setup and procedures are provided elsewhere {8,9]. Results The energy spectrum of specularly scattered Si* ions obtained from a i744eV5i* beam incident at 4° to the target surface is reproduced in Fig. 1. The spectra at other beam energies trom 200-2000eV, incident angles from 1-20°, and of negative ions over the same energy and angular ranges are similar. The energy loss of each peak (which we define as the energy difference between the incident beam energy and the centroid of each peak) exhibits a linear dependence on the incident beam energy. Data for Si" ions are shown in Fig. 2. The curves for Si * are similar, except that in this case, the extrapolated energy loss at zero incident beam energy is 13.6±2.5eV, rather than the 3.4±0.5eV determined for Sr (based on the lower 3 curves in Fig. 2). These losses are independent of the crystal azimuth over the (carefully studied) range ±15° about the Defining the energy loss AEi as the energy difference between the maximum of the highest energy peak and the incident beam energy, and R = AEJ/EJ. (where Ex = Eosin26,, with Eo and 9, being the incident beam energy and incidence angle to the surface plane), we show in Fig. 4 the dependence of R on 8, for specular scattering and both positive and negative final charge states. The peaks of the angular distributions of Si * and Sr ions scattered from Cu{111) lie just below the specular exit direction de. closer to the surface) [8]. This is in stark contrast to the results for the scattering of Na* and K* ions from the same surface. Here a complete breakdown of the specular scattering approximation is observed (Fig. 5), in the absence of measurable (within ± 0.14eV) energy loss 19]. 378 Discussion In [8,10] we argue that the highest energy peak in the kinetic energy spectra such as Pig. 1 represents a single reflection of the beam from the surface, and that each subsequent peak represents a further reflection of the beam, from the surface, in a binding potential. The well defined "skipping trajectories'" or vibrationai motion in this binding potential are associated with well defined trajectory lengths along the surface. Since inelastic energy losses are. for small losses, proportional to the trajectory length, a weil defined series of regularly spaced peaks in the scattered particle kinetic energy distributions are resolvable. More surprising is the apparently linear dependence of the energy loss on incident beam energy, when all first order theories predict a linear dependence on velocity (10,11]. A more careful analysis, however, shows that both attractive and repulsive non-step-like potentials lead to strongly energy-dependent trajectory lengths in the strong interaction region where the energy loss occurs. Explicit calculations for a model Si/Cu(111) interaction potential showed that a linear energy loss dependence on incident beam velocity, combined with this energy dependent trajectory length effect, comes close to reproducing the observed linear dependence on incident beam energy [12]. The non-zero energy losses of both Si* and Si" ions at zero extrapolated incident beam energy was shown in [8,10] to be a simple consequence of the endothermicity of the charge changing processes responsible for production of Si* and Si" from the reaction intermediate existing at the Cu(111) surface. That these losses are non-zero and consistent with energy conservation considerations provides verification of the precision of our measurements. We have performed a binary scattering calculation of the scattering of Si * from a defect free CU(1 11) surface. This three dimensional binary collision simulation is similar in structure and operation to MARLOWE [13], but has been optimized for simulating low energy ion scattering off the first few atomic layers of a crystal, and includes inelastic energy loss and thermal vibrations. The calculation used the Biersack "universal" potential with the Ziegler screening length [14] and a "detector" angular resolution identical to that used in the experiment (0.5° FWHM). The excellent agreement between theazimuthal scan simulation, which exhibits 100% reflection from the surface layer, and the experiment provides yet further support for an interpretation of the results in terms of scattering from 4he crystal surface. Furthermore, the experimentally measured depth of the channeling dip is precisely reproduced by the simulation, indicating the high 379 quality of surface preparation achieved in this experiment A trajectory analysis identifies the origin ^f *he cusp in the < 110> direction as arising from focusing in the corrugated surface potential. We observe no dependence of the energy loss on crystal azimuth. This is in stark contrast to recently reported results for He* scattering from Ni(110) 115]. These latter results, however, were poorly reproduced by a binary scattering simulation, leading to the conclusion that a high defect density existed on the surface [16]. The observed dependence of the energy loss and straggling on azimuth in that experiment may very well be due entirely to a changing defect density (as seen by the beam) with azimuth. Our results clearly emphasize the importance of good surface preparation in fundamental studies of this type. To access a binding potential, the incident Si* beam must elastically or inelastically lose, during the first reflection, an amount of translational kinetic energy at least equal to the incident perpendicular energy E j.. That at least this amount of energy is indeed lost over the full range of incident beam energies and incidence angles is clearly shown in Fig. 4. Indeed, beyond incidence angles 9, = 10°, exactly Ex is lost through inelastic processes alone. A necessary condition for trapping of the beam, derived from total energy conservation considerations alone, is therefore fulfilled. However this condition is by no means sufficient. If the concept of perpendicular energy scaling is valid, and we have perfect decoupling of the motion and energy losses parallel and perpendicular to the surface, the result (Fig. 4) for 6; >10° hints that for 6i >1O° at least, only the perpendicular component of the projectile motion contributes significantly to the observed final energy loss. The perpendicular component of the motion indeed contributes proportionately much more to the total energy loss than the parallel motion. The ratio of the J perpendicular to parallel energy R' = Ex/Et (where E( = ^- mv( and v( is the beam velocity component parallel to the surface) at 4° incidence changes for the ingoing and outgoing beams by a factor R'in/R'out = 1-3, increasing to 1.8 at 12° incidence. The bound state we access in the Si/Cu{111) interaction is however, transient, otherwise we would not observe re-emission (ie scattering). Furthermore, the energy losses plotted in Fig. 4 were measured at the specular scattering angle (26i). Thus, although we have satisfied a necessary condition to access a bound 380 state of the system, we must also provide a means of exiting from this bound state with a non-negligible final perpendicular energy. If the initial perpendicular energy were wholly dissipated in the first reflection to permit access to the bound state, then transfer of parallel momentum p into perpendicular momentum px of the beam must occur during the subsequent interaction. If such p,-»px coupling exists, then we expect that Px-»P(coupling also exists. Clear proof of such coupling, in the absence of inelastic energy loss, is provided by the Na * and K * scattering data (Fig. 5). Such coupling is completely inconsistent with the concept of perpendicular energy scaling. That perpendicular energy scaling does not apply to the Si/Cu{111) interaction is also illustrated by the absence of a threshold for the production (scattering) of Si' at (from) the Cu(111) surface at the Ex corresponding to the endothermicity of this reaction (3.4eV (8,101). Such a threshold, however, has been observed in the production of O2" from O2* at a Ag(111) surface [6]. The data presented here implies that we have an efficient mechanism of momentum transfer between the perpendicular px and parallel p components of the projectile motion in the surface region. The physical origin of this coupling appears to reside in the fact that the perpendicular component of the momentum of metal electrons is not well defined in the surface region. This results for example in a formula for the stopping power in the z (surface normal) direction [11] 2 -dE -iZe) a „ n */-n dz nn**uu " A kl + n 2 2 2 lu; - ka ) + * ux in which the 2 component q of the metal electron momentum k = (k, q) never appears. This implies a strong coupling between p( and px in the case of surface scattering. This coupling is absent for particle motion in homogeneous media. We demonstrate in detail in a separate publication [17] that indeed = In contrast, for surface scattering, that the distribution function describing the projectile momentum component px is strongly asymmetric. The presense of strong pj.-p( coupling in the absense of significant inelastic energy loss is not necessarily contradictory, since the force acting on the particle in the near surface region contains both conservative (dynamical image) and non-conservative (dissipative) components [11]. Conclusion We have presented experimental evidence which demonstrates an independence of inelastic loss processes on crystal azimuth, despite the presence of strong surface channeling effects. We also show that observed energy losses need not, due to an energy dependent trajectory length effect, reflect the linear or other dependence on incident beam velocity expected from first or higher order theoretical considerations. We also show that the concept of perpendicular energy scaling is not always valid in small angle ion-surface scattering. On the contrary, strong coupling of the beam momentum components perpendicular and parallel to the surface can occur. Further studies are required to establish whether the observed transient adsorption of ion beams at metal surfaces is related at al! to inelastic processes, or whether p-pj. coupling is the primary mechanism. Theoretical studies to identify the physical parameters determining the strength of this coupling are in progress. Acknowledgements This work was supported by the University of Newcastle, the Australian Research Council, and the Deutsche Forschungsgemeinschaft. 382 References 1. Pan Hochang.T. C. M. Horn, A. W. Kleyn, Phys. Rev. Lett. 57 (1986) 3035 2. P. H. F. Reijnen. A. W. Kleyn, U. Imke, K. J. Snowdon, Nucl. Instrum Meth. 8 33(1988)451 3. H. Akazawa, Y. Murata, Nucl. Instrum. Meth. B 33 (1988) 442 4. H. Kang, S. Kasi, J. W. Rabalais, Nucl. Instrum. Meth. B 33 (1988) 438 5. S. Schubert, U. Imke, W. Heiland, K. J. Snowdon, R. H. F. Reijnen, A. W. Kleyn. Surf.Sci. 205 (1988) L 793 6. P. H. F. Reijnen P. J. van den Hoek, A. W. Kleyn, U. Imke, K. J. Snowdon, Surf. Sci. in press. 7. J. H. Rechtien, U. Imke, K. J. Snowdon, P. H. F. Reijnen, P. J. van den Hoek, A. W. Kleyn, Surf. Sci. submitted. 8. K. J. Snowdon, D. J. O'Connor, R. J. MacDonald, Surf. Sci., in press 9. K. J. Snowdon, D. J. O'Connor, M. Kato, R. J. MacDonald, Surf. Sci. submitted 10. K. J. Snowdon, D. J. O'Connor, R. J. MacDonald, Appl. Phys. A 47 (1988) 83; Phys. Rev. Lett. 61 (198fc) 1760 11. see for eg. R. Nunez, P. M. Echenique. R. H. Ritchie, J. Phys. C 13 (1980) 4229; T. L. Ferrel, P.M. Echenique, R. H. Ritchie, Solid State Commun. 32 (1979) 419, M. Kato, Y. H. Ohtsuki, Phys. Stat. Sol. (b) 133 (1986) 267 12. K. J. Snowdon, D. J. O'Connor, R. J. MacDonald, Rad. Solids, in press 13. M. T. Robinson, I. M. Torrens. Phys. Rev. B 9 (1974) 5008 14. J. P. Biersack, J. F. Ziegler, in Ion Implantation Techniques, edited by H. Ryssel and H. Glawisching, Springer Series in Electrophysics, Vol. 10 (Springer- Verlag, Berlin, 1982), pp. 122-156. 15. A. Narmann, W. Heiland, Rad. Solids, in press. 16. A. Narmann, Thesis, Universitat Osnabriick, 1989. 17. M. Kato, K. J. Snowdon, D. J. O'Connor, R. J. MacDonaid, Nucl. Instrum. Meth., submitted. 383 3UUU Si**Cu(111) -• Si** CuliH) A Eo= 17UeV. 8j =9r =4° 2000 • •• Yiel d : y 200 XX) E. /* "l 1 1 i j 1300 UOO 1500 1600 1700 Energy (eV) Fig. 1 Kinetic-energy distribution of Si* ions scattered from a Cu{111) surface {incident beam energy Eo = 1744eV, incidence angle to surface Oi* observation angle to surface 8r = 4°, azimuth -5° to < 110> direction). 384 300 Si*»Coi11i]*sr.[Cu(111)-2e-l 250 200 WOO 1500 2000 Energy (eV) Fig. 2 Dependence on incident energy Eo of the energy loss of each peak in the energy spectrum of Si" ions produced in the glancing incidence scattering of Si * from Cu(111). The slopes (x 102) of the least-sqares-fitted lines are indicated (crystal azimuth as for Fig. 1) 385 I— SiMCul111)-2 0.8 •o 0.6 N 0.4 0.2- 0.0 -10 0 10 Azimuth (deg) Fig. 3 Comparison of the a2imuthal dependence of the yield of scattered Si" ions (for the highest energy peak) with the results of a binary scattering calculation at the sample temperature of 300K(Eo = 1993 eV, 6, =6r = 4°, where 8r is the angle of exit of the scattered beam to the Cu(111) surface). 386 Si**Cu{11D—Sr*[Cu(111)-2e"J Si**Cu(tn)-*-Si%Cu(111) E0=200-2000eV, 6: = 0r 20 uT 10 II I 2 1 10 15 20 0,- (deg) Fig 4 Dependence of the ratio R = AEi/Ei (see text) on incidence angle of the beam to the surface for specular scattering. The ratio R is independent of the incident beam energy Eo within the error bars shown 387 10 20 30 Scattering Angle (deg) Fig. 5 Energy integrated angular distributions of K* andNa* ions scattered from Cu(111) in a plane defined by the incident beam and the surface normal. The scattering angle at which the surface piane lies is indicated by the vertical bars with the incidence angle setting error bar of 0.5°. An additional uncertainty of - 0.5° exists in the absolute position of this bar with respect to the scattering angle. The non-zero intensity at exit angles below the indicated surface positions arises from a combination of these setting errors and the finite instrumental angular resolution. (Eo = incident beam energy, 9,= incidence angle of the beam to the surface, scattering angle is the angle between the incident and scattered particle directions, lines are drawn to guide the eye). 389 The neutralization process of highly charged ions near a aetal surface L.Folkerts and R.Morgenstern KVI, Rijksuniversiteit Groningen, the Netherlands Abstract: We present high resolution energy spectra of electrons obtained from collisions of C5*, N6* and 07* on a polycrystalline tungsten surface. Vm identify tvo structures in the spectra, at lov energies the LMM Auger electrons, and at high energies the KLL Auger electrons, in which a substructure is resolved. The implications for the neutralization/ deexcitation process are discussed. Introduction: During the neutralization of multicharged ions approaching a metal surface, a large number of secondary electrons is ejected. Most of these electrons have lov energies (<30 eV), but because of the great variety of processes leading to these lov energy electrons, hardly any structure can be resolved in this part of the spectrum, and therefore it is difficult to deduce dependable information about the neutralization process. One possibility to circumvent this problem is to use multicharged ions which have an innershell hole, since neutralization of these ions results in the ejection of high energy electrons with more pronounced structures [1,2,3,A]. In this paper ve report on nev measurements of H-like ions colliding on tungsten, in which we have resolved substructures in the high energy peak, and also resolved a peak in the lov energy part of the spectrum. The neutraliztion process: We consider a multicharged ion Aq* approaching a metal surface. The electronic properties of the metal are described by the conduction band with a depth s for free electrons and a vorkfunction * from the vacuualevel to the fermilevel. It is known that already at relative large ion-surface distances (e.g. at s » 2q + 7 [a.u.] [5]) vacant high lying projectile levels can capture electrons from the conduction band by resonant neutralization (RN). This results in a inulti-excited atom. 390 In the following description of the deexcitation process, ve consider tvo deexcitation mechanisms. The first one is a pairwise decay of the excited electrons via autoionization (Al), one going dovn to a lower lying level and one being ejected to an empty state above the conduction band or to the continuum; hereafter the ejected electron is replaced via RN into the resonant level. Th« second mechanism ve want to consider, is the following: at large distances from the surface only the weakly bound atomic levels have a sufficient overlap vith the metal electron vavefunctions; vith decreasing distance also the deeper atomic levels get an adequate overlap and will be populated by resonant electron capture fron the metal. b. Fig. 1: Simplified schemes for the neutraltzation/deescitation process. In •part a) of the figure the 'ladder-picture' (set test) is illustrated. In part b) tkt 'resonant-picture' (see text) is illustrated , for two different atom-surface distances. RM'Resonant Neutralization, RI*Resonant lomzatzoi, AI=Avtoiom.zation. 391 Starting with the multi-excited atom after the first RN steps and vith the tvo given deexcitation mechanisms, one can build tvo different scheme.: for the total neutralization/deexcitation process, depending on the relative probability of the tvo mechanisms. If the autoionization steps are fast, rhese vill be dominant in the deexcitation. So after several of such steps, electrons vill accumulate in lover levels from which again Al-processes can occur (see fig la). Eventually innershell vacancies vill be filled by AI. Since the electrons in the deexcitation process have to come dovn step by step, ve vill refer to this scheme as the 'ladder-picture' (LP). In the other scheme (see fig lb) it is assumed that the AI processes are slov. In this case the next lover level is not yet populated by AI at the moment that the atom has approached the surface close enough for this level to become resonantly filled. Eventually at very short distances from the surface, the lov lying levels vill be filled directly, vhereafter finally AI steps vill fill existing innershell holes. This scheme ve vill call the 'resonant-picture' (RP), because succesive levels are filled by RN one after the other. To distinguish betveen the 'ladder-picture' and the 'resonant-picture' in an experiment, one has to realize that in both pictures autoionization steps cam occur. In particular the filling of innershell holes vill occur by AI in both schemes. However there is an important difference. In the LP the Al-step filling the innershell hole can occur as soon as tvo electrons descended part of the ladder, whereas in the RP the Al-step is most likely to occur only after the level just above the innershell has become resonant. The distribution of electrons (DOE) at the moment of AI is therefore completely different in the tvo cases, and since this DOE has an important influence on the energy of the ejected electrons, the 'ladder-picture' and the 'resonant-picture' vill result in different electron energy spectra. The high energy part of the spectra: Our experimental results from slov H-like ions (C5*,NS* and Q7*) colliding on a polycrystalline tungsten surface are shown in figure 2. The high energy peaks result from the autoionization step in vhich the ls-hole of the projectile is filled. 392 -- '*• ons an .-lOev Fig, 2: Secondary eZectron energy spectra from collisions of 100 eV C5*, 90 eV it*, and 100 eV Q1* on tungsten respectively. Note the scale factor for the high energy part of the spectra. The horizontal brackets indicate the electron energies expected for autoioniztng decay of doubly excited (is 2snl) configurations. The vertical ticks indicate the position of the KLL Auger lines in singly ionized atoms. The horizontal brackets mark the position of Auger-electrons from doubly-excited atoms, resulting from tvo electron capture in collisions of H-like ions with H^ or He [6J. Indicated are the energies of electrons from autoionization from the states 2121', 2lll' and 2IAI'. These transitions are close to what is expected in the 'ladder-picture'. As starting point for the determination of the electron energies expected in the 'resonant-picture', we use the KLL Auger electrons, which are found if ls-innershell holes are created in neutral atoms by means of electron or photon impact. The energies of these KLL Auger electrons are indicated by the ticks on the energy-axis [7J. 393 From the indicated energy ranges, one cannot determine vhich scheme gives the better agreement vith the experimental results. Both are in the correct range, but neither fits perfectly. There are however some remarks to be made, which deal vith the fact that the DOE resulting from two electron capture in H-like ions on the one hand and from innershell hole creation in neutral atoms on the other hand do not agree completely with the expected DOE in the 'ladder-picture' and in the 'resonant-picture' respectively. The first comment is that in the LP not only the autoionizing electron pair is present, but there are other excited electrons in higher states. These electrons will partly shield the core charge, leading to higher energies of the ejected electrons. This worsens the agreement of the indicated ranges (horizontal brackets) with our experiments. Secondly in the RP the AI will not occur in a singly charged ion, as it is created by innershell ionization due to electron/photon impact, but rather in a completely neutralized atom Again as a consequence of extra shielding of the core charge, this increases the electron energies (-15 eV in case of N). This energy shift is just enough to overcome the discrepancy of the indicated energy positions (ticks on energy-axis) vith the experimental results. Another important argument favoring the RP comes from the peak substructure which shows up in the spectra, the best example being the Ns*-spectrum (see fig 2). These subpeaks do agree very well with expected subdivision due to the electronic sublevels in case of KLL Auger processes. The different peaks ara refered to as K-L L (1S), K-L L (:P), K-L L (3P), K-L L (1S) and 11 1 23 1 23 23 23 K-L L (:D), where the latter two are not resolved [6J. Therefore our conclusion from the high energy part of the spectra is that the 'resonant-picture' holds the best. The low energy part of the spectra: In the low energy part of the spectra, there is also a pronounced peak (at 33 eV for C5*, AA eV for Ns* and at 60 eV for 07*). From the energy positions, ve consider these electrons to result from LMM Auger processes, but the corresponding energies are not exactly known, since these transitions can occur from quite exotic distributions of electrons. For instance, the M-shell can contain up to q electrons (q is the original 394 charge state of the primary ion), vith the L-shell still empty, and the K-shell hole still existing. , CO i T _ ' c4" A '30 'SO *«5 SCO 5S!«v Fig. 3: Secondary electron energy spectra f**om collisions of 100 eV tP (q*3,4,5,S,7] on tungsten. Note the scale factor for the high energy part of the spectra. Trie low energy peak, identified zs being due to LMH Auger electrons, is also present in the spectra of the lower primary charge states. These LMM Auger transitions hovever do not depend on the existence of an innershell hole and should therefore also be found if ions vith lower charge states are used as projectiles. This can easily be verified in the electron spectra obtained for 09* 395 (q.3,A,5,6,7) colliding on V, vhich are shovn in figure 3. Indeed the low energy peak shovs up independent of a projectile innershell hole. However the energy position of the LMM peaks does depend on the primary charge states, since additional electrons in the (K-) L-shell shield the core charge for both electrons involved in the autoionization, resulting in lover energies for the ejected electrons. In figure 3 it can be seen that also the spectrum for He-like 0** exhibits an electron structure at high energy, vhich results from KLL Auger processes. This is because the 06* beam carries about 5% metastable O**(ls2s)JS, vhich live long enough to survive transportation from the ECR-ionsource to our apparatus. From the number of electrons in the LMM peak as compared vith the number of electrons in the KLL peak, in figure 2, one can draw an important conclusion vith repect to the 'ladder-picture' and the 'resonant-picture'. At lov collision energies the integrated numbers of electrons in the KLL-peaks are close to one electron per ion, i.e. most of the K-shell holes (one per ion) are filled by a KLL process. The LMM Auger peaks contain about tvo electrons per ion, so only tvo L-shall holes are filled by LMM Auger processes. However from the energy position, and the substructure of the KLL peak, ve have concluded that the L shell contains up to q electrons, so the additional electrons have to come fro«n direct neutralization. It is therefore again the RF, vhich can explain the experimental results. Conclusion; The neutralization of multicharged ions at a metal surface starts at large ion-surface distances vith the filling of high lying electronic levels. The deexcitation of the resulting multiexcited atom is the best described by the 'resonant-picture', in vhich the lover levels are filled by resonant neutralization due to increasing overlap vith the metal electron vavefunctions at smaller ion-surface distances. Between tvo succesive level fillings autoionization processes can occur, from vhich the KLL Auger and the LMM Auger are identified in the measured electron energy spectra. 396 References; [1] S.T.de Zvart, A.G.Drentje, A.L.Boers and R.Morgenstern, Surf.Sci 217 (1989) 298.298. [2J L.Folkerts and R.Morgenstern, J.Physique 50 (1989) Cl-541. [3] M.Delaunay, M.Fehringer, R.Geller, P.Varga and H.Vinter, Burophys.Lett. 4 (1987) 377. [4] F.U.Meyer, C.C.Havener, S.H.Overbury, K.J.Snovdon, D.M.Zehner, V. Heiland and H.Herame, Nucl.Instr.and Meth. B23 (1987) 234. [51 K.J.Snowdon, Nucl.Instr.and Meth. B34 (1988) 309. [6] M.Mack, Nucl.Instr.and Meth. B23 (1987) 74. [7] F.P.Larkins, Atomic Data and Nuclear Data Tables 20 (1977) 311. 397 SOME BASIC PHENOMENA AND THEIR IMPORTANCE IN RELATION TO TECHNIQUES FOR SPUTTER- AND VAPOR-DEPOSITION OF HIGH TEMPERATURE SUPERCONDUCTING THIN FILMS Orlando Auciello Microelectronics Center of North Carolina, Research Triangle Park, NC 27909-2889 and North Carolina State University, Materials Science and Engineering, Raleigh, NC 27695-7907 ABSTRACT The processes involved in plasma and ion beam sputter-, electron evaporation-, and laser ablation-deposition of high Tc superconducting thin films are critically reviewed. Recent advances in the development of these techniques are discussed in relation to basic physical phenomena, specific to each technique, which must be understood before high quality films can be grown. Low temperature processing of films is a common goal for each technique, and efforts in this area using activated oxygen sources are discussed in view of integrating the high Tc superconducting thin films with current microelectronic technologies. 398 INTRODUCTION production of high Tc superconducting films, which include, for example, chemical The discovery of the new high vapor deposition and spin-on processes. temperature superconducting materials These methods, however, appear to be (Bednorz and Muller 1986; Wu et al. less compatible than those mentioned (1987); Maeda et al. 1988; Sheng et al. above with the requirements of 1988) initiated intense research in a wide superconducting device fabrication variety of processing techniques in an technology. For example, spin-on effort to utilize the unique properties the processes ( May et al. 1988) produce compounds possess.These materials were films with poor grain-grain contact and first synthesized in bulk form, probably must be subjected to an annealing step at because the hardware involved is less > 900 °C in order to sinter the sophisticated than that necessary for thin disconnected islands of superconducting film processing. However, research materials. indicates that thin films possess better Only those techniques which hitherto current carrying capabilities and appear more compatible with device microstructurc than their bulk counterparts. fabrication technologies will be discussed Present indications are that thin films, with here. Additionally, this review will be the potential for integration with limited to analyzing current technological microelectronics technology, will be used developments in each method and basic in the first practical applications of these phenomena related to the deposition materials. processes. A thorough examination of The development of automated these techniques, throughout the systems for the production of literature and from our own work, has superconducting films in an integrated revealed a series of advantages and deposition/processing cycle is a challenge disadvantages of each method. It is that must be met before large scale impossible to cite all the literature in this manufacturing of these materials can be rapidly growing field of research; therefore, reproducibly performed. Numerous only representative references will be specifically developed techniques for thin cited. film deposition and adaptation of A manufacturing process for techniques from more general coating producing high temperature technologies have been applied for superconducting films for technological synthesizing high temperature applications should at least include the superconducting films (Harper et al. following characteristics: (1) applicability (Eds.) 1988). However, it appears that of the processes to deposition of materials techniques based on sputtering (plasma with different physical and chemical and ion beam-induced ), laser ablation, or properties; (2) compatibility with evaporation (electron beam-induced, or integrated device processing, which molecular beam epitaxy) are emerging as includes production of as-deposited high the leading methods which are both temperature superconducting films on compatible with technologically feasible substrates at the lowest possible deposition rates and currently used temperature; (3) production of high quality, techniques for device fabrication. These epitaxial films with high critical currents; methods appear to be capable of producing (4) simple and low cost deposition with the high quality, high purity, epitaxial films capacity for high deposition rates; (5) that are necessary to optimize materials ability to produce patterned structures, properties and performance. Other superlattices and layered heterostructures, techniques have been applied for the particularly for sandwich-type Josephson 399 junctions; and (6) reproducibility of the increases. Preferential sputtering and ion deposition process. bombardment-induced topography can be severe, as shown in Fig. 2 for the case of YBa2Cu307.x (Auciello and Krauss REVIEW OF DEPOSITION/ 1988). Other problems, related TO PROCESSING TECHNIQUES obtaining layered structures and tailoring the film composition, are discussed below. 1 Plasma Sputter-Deposition I" FICLD In plasma processing techniques '•'ffllCdhaOl (Harper et al. (Eds.) 1988), deposition of Hi|tll» Hl|* Vllltf* high temperature superconducting films is achieved by sputtering of targets exposed to a dc- or rf-plasma discharge within a high vacuum chamber. This chamber is filled with an inert gas, such as argon, to pressures ranging from 1-100 mTorr . The partially ionized plasma is generated adjacent to the solid target which is constructed from either a bulk superconductor (YBa2Cu307.x for example) or elemental materials (Y, Ba, Cu or their oxides for example). The targets are typically symmetrically distributed in front of a substrate Fig.l. A magnetron sputtering system for positioned at a distance of several deposition of high T superconducting centimeters. The target (cathode) is c negatively biased so that its surface is films. bombarded by positive ions from the plasma. It is common practice to use magnetic fields to form electron traps Preferential sputtering results in which are configured such that the ExB compositional changes both in the lateral electron drift currents converge on the composition and to a certain depth below cathode surface. This concept, known as the surface of the sputtered target. There magnetron sputtering (Fig. 1), results in are corresponding compositional changes higher cathode erosion rates than other in the sputtered flux and therefore in the plasma sputtering methods (Thornton deposited films. This is a transient 1988). Magnetron sputter-deposition is condition, since a steady-state situation is the plasma related technique most widely generally achieved in which the sputtering used at the present time. rate of the components of the material is proportional to the bulk composition (Betz The plasma sputter-deposition and Wehner 1981).However, in most techniques discussed above are notcases, a relatively thick layer must be without problems, especially when using removed before the steady-state condition multicomponent oxides for targets, as is reached (Betz and Wehner 1981), would be the case for superconductor thin which can require bombardment of the films. For example, when using a target as long as 10 hours (Hong et al. multicomponent target it is difficult to 1988) before the deposition starts. This change and control film stoichiometry, extended bombardment may in turn lead to particularly as the number of components substantial surface topography 400 development (Aucicllo and Kelly 1984), depositing thin layers of each material which can also affect the sputtered fluxes (Bhushan and Strauss 1989). Practical of the target materials. considerations will limit the deposition rate and/or the layer thicknesses in these cases. Recently, other problems related to plasma-assisted deposition have been identified, which are common to methods involving both bulk oxide superconductor or elemental material (or their oxides) multi- target arrangements: For both single multicomponent and multiple single targets, impurity incorporation into the film, from the plasma container walls or the gas used- to sustain the discharge, is common with the plasma techniques. A relatively large number of O" ions Fig. 2. Micrograph of ion beam induced may be produced during the sputtering of surface topography after bombarding a oxide targets. These ions can be YBa2Cu3O7 sample with 10 keV Ar+ accelerated through the plasma-cathode ions. See Figure 8 of this review for potential fail and neutralized upon entering comparison. (Auciello and Krauss 1988). the plasma region. The neutralized O* ions travel through the plasma and impact on To overcome the problems mentioned the growing films with enough energy to above, some groups (Harper et al (Eds.) produce undesirable damage and/or 1988) have used elemental materials (Y, sputtering of the film. This phenomenon, Ba, Cu, or their oxides. Y2O3 , BaO2 .CuO designated as the "negative ion effect", for example) in multitarget arrangements, has recently been demonstrated during where each material serves as a cathode of magnetron sputter-deposition of Y-Ba-Cu- a magnetron sputtering system. While this O films from YBa2Cu3Oy.x {Rossnagel method addresses the issue of preferential and Cuomo 1988) and an Y2O3, Ba2CC>3, sputtering, other problems are raised. and CuO multitarget arrangement (Shah Simultaneous sputter-deposition from and Garcia 1988). Major consequences of elemental target materials exposed to the secondary particle bombardment of the independent magnetron sources leads to growing films that have been observed are: compositionally inhomogeneous films. (i) a dramatically reduced ion current to the Since the targets are spatially separate, target (reduced sputtering rate) and the overlapping deposition fluxes will not increased erosion of, and damage to the be identical at all points on the substrate growing film (Fig. 3a) with an associated (Kang et al. 1988). One method tc stoichiometry change (Fig. 3b); (ii) in improve composition homogeneity with extreme cases, the negative ion effect can multiple sources is to increase the actually be severe enough to produce substrate to target distance, leading to negative growth rates; (iii) in the case of very inefficient use of target materials. the single elemental-oxide targets Another alternative is to maintain a fixed arrangement mentioned above, where the substrate-target geometry by moving the substrate sequentially into position in front substrate is sequentially positioned under of each fixed source and alternately each target, film erosion has been observed only under the Y2O3 and 401 23 but not under the CuO targets, which has fueled some speculations that not only neutralized O" but also other 400 - energetic sputtered neutrals particular to each oxide may contribute to the film erosion (Shah and Garcia 1988); (iv) 30.0 - another effect observed in the bulk superconductor target case is the rather strong variation in the deposition/erosion 7 200 - rate as a function of the radial position of the substrate under the target (Fig. 4). 100 (- 1 l 1 i § 0 i o: - -•0.0 - 02 lo Target 0 ! 1 -20.0 DO "•""' I • \ ~ 0 1 -soo «0 40 20 0 20 40 SO i i 1 [)? Dnienet From Ton]ti Ccxiir I mm) PCKCEKT OXVOEN |tn *r 35 1 1 1 1 Fig. 4. Deposition/erosion rate as a function of the radial position of the 2 30 Desired Uu Levtl substrate under the target (Shah and Cu _. Carcia 1988). Desired Ba Ltvel A number of different methods have N been implemented in order to control the 1 5 - - negative ion bombardment effect. A 10 straightforward method is to increase the plasma pressure such that the energy of os _ the neutralized negative ions is reduced, — — by multiple collisions with plasma species, 00 +-— l 1 0 20 "0 60 80 '00 until it is below the energy necessary to PERCENT OXYGEN (In Ar background gn) sputter the growing film (Adachi et al. 1987). A second means of obtaining the Fig. 3. (a) Net magnetron sputter- appropriate film stoichiometry is to modify deposition rate for YBa2Cu307.x as a the target composition in order to function of O2 concentration in deposition compensate for the preferential sputtering associated with the negative ion effect chamber background gas. (b) Composition (Moriwaki et al. 1987). This method is of a magnetron sputter-deposited Y-Ba- time-consuming and somewhat unreliable Cu-O film, normalized to Y=1.0, as since any change in processing parameters determined by RBS (Rossnagel and may alter the final film stoichiometry. Cuomo 1988). Unconventional sputterine Geometries in 402 which the substrate is not subject to ion minimizing the bombardment of growing impact (Fig. 5) (Sandstrom et al. 1988) films by secondary electrons and ions (Fig. may ameliorate the negative ion effect, at 6) emitted from the target in an attempt to the cost of deposition rate and film elucidate their contributions to undesirable thickness uniformity. alterations of the film characteristics. .Substrate. so-- e e i- US GUN H 7777^ \ MAGNETRON SOURCE IE WflTER HOLOE" 360* HOTAriON- (b) CEHBMIC HEATER-^C^>-' ~*aFE" Substrate^ 77H7T "^> ^^*^ -Target-^ (c) (d) Fig 6. Methods for minimization or elimination of secondary electron and ion bombardment of growing high Tc Fig. 5. Sketch of an unconventional superconducting films in plasma sputter- geometrical arrangement of substrates out deposition. of the main stream of sputtered-neutralized negative ions to minimize the plasma- Analysis of the four sketches in Fig. 6 induced negative ion effect in plasma- indicates the following: (a) Both secondary assisted deposition of superconducting neutralized ions and electrons impact on films (Sandstrom et al. 1988). the substrate, (b) Bombardment of secondary electrons is largely eliminated In addition to the negative ion effects by trapping in a transverse magnetic field , discussed above, bombardment of high Tc although neutralized secondary ions can superconducting oxide targets by plasma still impact on the substrate, (c) ions results in a relatively high yield of Bombardment of the substrate by secondary electrons, which also bombard secondary ions is controlled by bias the growing film. These electrons alter voltage (it is not clear which secondary stoichiometry, increase substrate ions the authors are considering, see text), temperature beyond that purposely and (d) Bombardment of both secondary independently established for film electrons and ions is eliminated (again, it processing, and may possibly lead to is not clear from the discussion presented electron bombardment-induced defects. by Tcrada et al. what is the main difference These secondary electron bombardment between (c) and (d)) (Terada et al. 1988). associated phenomena may result in undesirable alteration of the film The analysis of films deposited under characteristics. Terada et al. (1988) have conditions (a) and (b) yielded recently investigated different methods of compositions as indicated in Table 1. 403 The main results, which partially confirmed Table 1. Relationship between film some of the effects discussed above, were: composition and deposition conditions. (i) A slight increase in Cu and Y and decrease in Ba contents in the films for an - increase in pressure from 3 to 20 mTorr. Press. Composition Deposited (ii) A noticeable dependence of deposition mTorr (a) (b) rate on pressure, such that a significantly higher rate was observed at 6 mTorr, 60 Yl.0Ba0.68Cu1.0 Yl.OBa1.8Cu3.O probably due to competition between ion 10 YioBao3oCuo5 Yj oBaj 2Curj.8 collection by the cathode and gas 1 scattering of the sputtered material. (iii) A slight increase in Y and Ba and Terada et al. (1988) argued that the decrease in Cu contents of the films as a data presented in TabJe 1 demonstrates f™ctiotlof increasing power in the range of that the suppression of electron 20 to 75 watts. bombardment in condition (b) resulted in focused onto spatially separated produce a completely homogeneous film elemental targets (Y, Ba, Cu for example) (Wessels 1989). located in water-cooled holders (Bunshah It should be noted, however, that the et al. 1982) (Fig. 7). use of one electron gun for each target adds complexity and cost to the hardware, because of the needed replication of the electronics necessary to control each source. Accurate adjustment of the sources to keep the evaporation rate of each material within tight tolerances is mandatory in order to produce stoichiometric films. This problem is obviously accentuated as the number of elemental material components of the films increases. Little can be discussed about basic phenomena involved in electron beam- induced evaporation, as studies on transport processes of evaporated species and dependence of film characteristics on geometrical arrangements have not been Fig. 7. Sketch of an electron beam vapor- performed, according to the authors' deposition system for production of high Tc knowledge. Evaporation with assisted or superconducting films. activated deposition (ARE) has proved to be extremely important for deposition of This technique is particularly useful crystalline films. These processes are for evaporation of refractory materials. discussed below in further detail. Precise film stoichiomctry and abrupt interlaces are difficult to reproduce and control using this method, since direct 3 Laser Ablation Deposition shuttering of individual beams near each source is difficult without disturbing the Pulsed lasers have been used to focused magnetic flux and thus quenching deposit Y-Ba-Cu-O films by ablation of the target material evaporation. The sintered YBa2Cu3O7 targets (Dijkkamp simultaneous evaporation of the elemental et al. 1987, Narayan et al. 1987, Harper et materials from different spatial locations al. (Eds.) 1988). This appears to be one of tends to produce non-uniformity in the more successful techniques used to composition and thickness across the film produce high Tc films from bulk surface. This and the shuttering problem superconductor targets on a laboratory have been partially overcome by placing scale. However, some characteristics of the substrate on a moveable holder the laser ablation process, determined in (Wessels 1989). The substrates are recent studies, indicate that laser ablation- sequentially positioned over each hearth deposition, as hitherto implemented, may for a pre-determined time to produce the present some difficult problems, desired layer thicknesses. For deposition particularly related to control ~»f film of YBa2Cu 307.x thin films, if the layer characteristics over large areas for scaled- thicknesses are of the order of SO A, the up industrial applications. subsequent annealing step suffices to The target stoichiometry is not exactly reproduced in the deposited fiims, a 405 fact which may contribute to the observed slightly lower Tc in these films. Dijkkamp et al. (1987), Auciello et al. (1988) and Sudarsan et. al. (1988) have shown that the variation in film composition may be due, at least partially, to significant topographical and associated compositional changes on the target surface induced by the laser impact during the ablation process (Fig. 8 for example). Depending on the laser power and wavelength (Auciello et al. 1988, Sudarsan et al. 1988), extreme variations in target composition and morphology have been observed at various points in the laser- induced etch pit. For a 308 nm (~ 3 J/cm^) excimer laser irradiation of Auciello et al. (1988) found a Cu depletion at the center of the laser impacted area, while Sudarsan et al. (1988) found Cu enrichment at a similar position, when irradiating a similar target with an 193 nm (0.41 J/cm^) excimer laser. These results indicate that more systematic studies are necessary to elucidate the effect that the irradiation of superconductor targets, by lasers of different wavelength and energies, may have on the characteristics of deposited films. Laser irradiation may result in the formation of cones, as seen in •S> -1 0 1 2 j Fig. 8. The presence of these features may Olflinc* From Ctnln (mm) affect results of the compositional analyses in that compositional variations are pu 'uced in each individual cone with Fig. 8. (a) SEM micrograph showing a appa. Mit differences from the top to the characteristic laser-induced surface side of the cones (Auciello, 1989). topography of a single phase YBa2Cu3C>7 Noticeable lateral non-uniform target impacted by 1200 pulses of an composition and thickness of laser-vapor excimer laser ( 308 nm, 0.28 J) during the deposited films and angular variation in the deposition of an Y-Ba-Cu-O film, (b) AES stoichiometry of the flux of ablated species line scans at different positions across the from superconductor targets has recently laser impacted surface of the same target been observed by Singh et al. (1988), as above (Auciello et al. 1988). Venkatesan et al. (1988), and Auciello (1989) (See Figs. 9 and 10). 406 IS (o) Figs. 9 and 10 clearly show that there E- 1.5 J/cm2 are two distinct regions in the deposited In films with different thicknesses and composition, which may be correlated with iO.S two regions in the angular distribution of the ablated species flux, i.e.. a narrowly and a widely dispersed components, 00 4 0 respectively, around the surface target (b) normal (Venkatesan et al. 1988). Fig. 9b 'Cu/Bo indicates that the film composition in the outer region of the film does not correspond to the stoichiometric composition of the i.O target. On the other hand, the forward- peaked component of the ablated flux appears to yield a film composition closer S 10 IS 30 » X 35 40 4S to that of the target. 6 Venkatesan et al. (1988) have Fig. 9 Angular distributions of (a) considered various possible models, thickness (the dashed line is the cos 0 fit) briefly described in this paragraph, to and (b) composition of an Y-Ba-Cu-O film explain their experimental data, which deposited by laser ablation of an mainly relates to ablation processes YBa2Cu307.x with a laser energy density produced by an excimer laser (248 nm) 2 with power densities in the range 1.5-3.5 x of 1.5 J/cm (Venkatesan et al. 1988). 107 W/cm2. Within a few nanoseconds of laser impact on the target surface, material Of (0) (O can rapidly evaporate, which may result in o a subsequent forward ejection of that material due to a secondary process. The to. £-1.1 J/cm,'S 0« 2 E« O.S J/cm possible secondary processes that were considered are: (1) surface shock waves I generated by the rapid surface evaporation, as seen in a number of laser-surface 00 I interaction studies (Bonch-Bruevich and (b) s (d) Imas 1968), (2) a subsurface explosion ACuA Ofe/Y CuA generated by a combination of a hot subsurface region, capable of building-up a & a high vapor pressure zone, and a cooler surface produced by the rapid evaporation O O O O 2 of material from the surface (Gagliano and Cu/Bo *-• » . » * Paek 1974), (3) the production of a high 0 3 10 IS JO 33 30 33 40 0 3 10 IS 20 JJ 30 JS 40 concentration density of ablated species in e e a region in front of the laser impacted surface (the so called Knudsen layer) (Kelly and Dreyfus 1988), which could iead Fig. 10 Stoichiometry and thickness of Y- to a high number of collisions among the Ba-Cu-O films as a function of laser energy ablated species resulting in a 2 density: (a) and (b) 1.1 J/cm ; (c)and (d) recondensation process and particles 0.5 J/cm2 (Venkatesan et al. 1988). emitted in a forward peaked direction. 407 With the data presently available, it 11(86 eV) (I and II designate neutral and is difficult to decide which of the ionized excited species respectively) are mechanisms described above or their relatively high (Zheng et al. 1989) and can combination may be the most appropriate in principle be explained in terms of the to describe the experimental observations, theory of supersonic molecular beams or if they generally apply to lasers of (Anderson et al. 1966), where the velocity different wavelength and power. distribution function of laser ablated atoms The precise mechanisms involved in from a solid can be expressed as: these effects are still being investigated. Their understanding may be fundamental to f(v) = Av3 exp[- m(v-vo)2 / 2 k Ts] (1) solving possible difficult problems that these effects may pose for scaling the where v is the velocity of the atoms, v0 is laser-vapor deposition technique to large the "stream" velocity, m is the mass of the areas. This coverage is necessary for the atoms, k is Boltzman's constant, Ts is a integration of high Tc superconducting films temperature parameter describing the with current semiconductor technology. velocity spread, , and A is a normalization Film thickness uniformity over large areas constant. Fig. 11 shows, as an example, could be achieved by planetary motion of typical optical emission time of flight the substrate. However, it is not clear (TOF) experimental spectra for Cu, Y, and whether this mechanical motion of the Ba atoms ablated from an YBa2 01367.x substrates will solve the problem of the target, by an ArF laser (193 nm), and the different film stoichiometry produced by the fitted curves obtained from eq.(l) (Zheng two component laser ablated plume. et al. 1989). Additionally, measurements of Whether this problem can be solved by ablated neutral atom velocities as a using a large diameter (£ 3 inches) laser function of distance along the normal to the beam also remains in doubt. Obviously, target surface indicated that Cu atoms further work is necessary to optimize the appear to initially move faster than Y, Ba, laser ablation deposition technique for and O atoms, and the velocities of all these large scale technological applications of species tend to equilibrate at about 7 cm high Tc superconducting films. from the target surface (Fig. 12, Zheng et In addition to the study of the basic al. 1989). The almost zero velocity of the phenomena discussed above, an accurate Ba II species was not explicitly discussed understanding of the evolution of the by Zheng et al. (1989) However, a ablated plume from the target and the possible explanation proposed by the transport of the ablated species towards authors of this review relates to the fact the substrate is necessary to optimize the that Ba II represent Ba ions in an excited deposition process, and thus the film state. The ablated species in an ionized characteristics. Recently, studies have state are moving against an electric field been performed on the evolution of the established between a positively biased plasma plume generated when laser beams ring and the grounded target (Zheng et al. impact upon bulk superconductor targets. 1989), which could effectively reduce the In particular, the velocity (energy) velocity of the Ba ions to thermal values distributions of ablated speciss (Y. Ba, Cu, (i.e. £ 1 eV). The equilibration of the and oxide molecules (e.g. BaO)) from neutral atom velocities at about 7 cm from YBa2C 11307.x targets have been the target surface has been correlated with measured by different groups (Venkatesan the experimental observation that the best et al. 1988, Zheng et a!. 1989). The mean quality films were obtained when kinetic energies measured for Cu I (41 positioning the substrates at that eV), Y 1(43 eV). Ba 1(42 eV), and Ba particular distance (Zheng et al. 3989). 408 On the other hand, Venkatesan et al. (Sudarsan et al. 1988). As the power (1988) used laser ionization mass density is increased, the composition of the spectrometry (LIMS) to identify ionic film more closely approaches that of the species emitted from YBa2Cu307.x taTget (Geohegan et al. 1988). At very targets impacted by a Nd:YAG laser (266 high power levels, the film composition nm, 5 ns pulses), and post ablation may therefore be less dependent on ionizarion (PAI) to determine the masses fluctuations in laser output. However, very and velocities of the ablated species, which high power levels may contribute to the were also demonstrated to be very ejection of molten matter from the target as energetic. Both the LIMS and PAI studies small (~ micron-sized) particles which are were interpreted as an indication that collected on the substrate, a process which ejection of stoichiometric clusters of could inhibit producing superconducting YBa2Cu3O7-X from the targets can be junctions. A current practice to minimize or ruled out. Instead, binary and ternary sub- avoid the influence of target laser-induced oxides were emitted from the targets, at compositional changes on the film least under the laser irradiation conditions stoichiometry involves continuously used by Venkatesan et al. (1988). rotating the target in order to permanently Different laser wavelengths, coherence, expose a fresh surface to the laser fluence, photon rates, energy, duty cycles, (Dijkkamp et al. 1987). target and vacuum conditions, etc. may More recently, Singh et al. (1989) render comparisons of data obtained in analyzed many of the experimental results separate systems unreliable. In fact, discussed above in terms of a model Deshmukh et al. (1988) observed mainly developed for simulating the laser metallic species (Bi, Sr, Ca, Cu) in the deposition process. The model is based on plume generated by laser irradiation of the hypothesis that a high pressure plasma Bi2CaSr2Cu2O9, when using an ArF is generated in front of the muiticomponent excimer laser (193 nm, 20 ns pulses), and superconductor target upon laser impact on only very small peaks were identified as the surface. This plasma is initially oxide molecules (i.e. BiO and CaO), which confined in a very small volume and appear to be minor constituents in the subsequently is allowed to suddenly laser-induced plume from these materials. expand into the surrounding vacuum and Contrary to the uncenainties still existing interact with a substrate located at a in relation to the identity of ablated species certain distance in front of the target. for different experimental conditions, a The model includes three separate general trend is emerging in that the regimes, namely: (i) the interaction of the ablated species from superconducting laser beam with the target surface, (ii) an targets appear to be very energetic, which initial isothermal expansion of the laser- could significantly contribute to the growth induced plasma, and (iii) a final adiabatic of high quality crystalline films at relatively plasma expansion leading to deposition of low ( £ 600 °C) substrate temperature. films. Singh et al.'s (1989) model accurately describes some of the An important parameter in laser experimental results discussed above: ablation-deposition of high Tc superconducting films is the laser power (1) The highly focused plasma plume (energy) density. At relatively low laser component along the target surface normal, power densities, it is found that the which has been observed by several composite target material does not groups (Singh et al. 1988, Venkatesan et vaporize congruently and the film al. 1988, and Auciello 1989) is due to composition for a given target is dependent anisotropic expansion velocities of the on both the laser power and wavelength 409 ablated species in the plume, such that the expansion velocities are related to the initial dimensions and temperature of the plasma, and the atomic weight of the respective species. (2) The energy density appears to control the maximum plasma temperature, which determines the expansion velocity of the plasma. ~l '0 20 JO I r Fig. 12 Dependence of the ablated species velocity on the distance from the target A. surface along the normal for various species: ») Cu I, (O) Y I, (A) Ba I. (•) \ O I, (A) Ba II (Zheng et al. 1989). (3) Based on simple energy balance -10 0 10 JO 30 considerations, the ion velocities follow a »> TME-OF-FLtGHT »m) cube root dependence with the energy density (i.e. v a E^^), which is in good agreement with limited expenmental data on ablation of ZrO2 by a CO2 laser beam (Sankur et al. 1987). (4) The compositional variations in superconducting thin film stoichiometry can be attributed to either non-stoichiometric evaporation of the target at low energy densities or to different expansion velocities of the species corresponding to different atomic masses. Fig. 11 Optical emission TOF spectra of There is one point of discrepancy Cu I, Y I, and Ba I taken at 7.2 cm from the between Singh et al.'s (1989) work and target surface. The solid lines are results from other groups (Venkatesan et theoretical fits using eq. (1). The initial al. 1989), in that the former observe a non- spike is due to scattering and laser- stoichiometric film component on the induced fluorescence, and can be used as a central area of the deposited Him and a time marker (Zheng et al. 1989). stoichiometric one at the periphery, while are not present or are much smaller in the Venkatesan et al. observe the opposite. ion beam sputter-deposition case. Based on the discussion presented However, initial work using this technique above, the laser ablation process may involved the utilization of ion beams under encounter serious difficulties for the rather uncontrollable conditions to sputter production of thin tailored multilayer YBa2Cu3C>7_x bulk superconductor structures, at least with the bulk targets. Improper confinement of the superconductor targets currently used. beams led to the introduction of impurities Some of the characteristics of laser which resulted in films with low Tc and ablation deposition which present poor characteristics (Kobrin et al. 1987). problems for laboratory studies may turn More recently, other researchers have out to be advantageous for production used the ion beam sputter-deposition purposes. By using large substrates at a technique to produce high quality films from correspondingly large target-substrate bulk superconductor targets (Gao et al. distance, it may be possible to reduce the 1988). Again, as in the plasma sputter- deposition rate to a controllable level while deposition method previously described, a at the same time maintaining the laser rather long pre-deposition sputtering of the power density on the target at a level multi-component targets was necessary in which produces stoichiometric deposition order to stabilize the target surface of the thin film material. However, the composition. The technique can, however, highly directional nature of the ablated yield films with Tc as high as the laser plume produced by focused lasers will ablation-deposition method and of equal or probably require a planetary substrate better quality, and yield higher Tc and motion. Expensive large beam diameter better quality films than produced with the lasers can help circumvent the need for this plasma sputter-deposition techniques. mechanical solution on large areas. Obviously, further work on this and other More recently, a new automated ion areas related to laser ablation-deposition beam sputter-deposition technique has of high T superconducting films is been developed (Krauss and Aucielto c 1989, Krauss et al. 1989, Kingon et al. necessary to develop this technique to a 1989), which has the following features: level compatible with technological (1) A high current ion beam generated by a applications. Kaufman-type ion source (Kaufman et al. 1982), which initially was directed at 45* with respect to the target surface normal 4 Ion Beam Sputter-Deposition (see Fig. 13). Fundamental studies on the ion-solid interactions have since revealed Ion beam sputter-deposition of high that normal incidence is the optimal Tc superconducting films has been much system geometry (Ameen et al. 1989). less investigated than the methods (2) A rotatable target holder driven by a previously discussed. However, this computer-controlled stepper motor, which method is well suited for deposition of serves to sequentially position elemental multi-component oxide films such as the material (oxide) targets in front of the high Tc oxide superconductors, since many sputter-beam. of the undesirable effects (substrate (3) A quartz crystal resonator (QCR) bombardment by energetic negative ions (Fig. 13), which measures the amount of and electrons, impurity introduction in films each elemental material deposited and from plasma-wall interaction in the sends a feedback signal to the computer deposition chamber, etc.) already when the preprogrammed necessary discussed for plasma sputter-deposition amount of an element, to produce a desired 411 film composition, is reached. The QCR advantages over those featuring multiple feedback signal activates the computer for ion beams, namely: (a) the use of only one shutting off the ion beam while rotating the ion beam, which simplifies hardware target holder to position the next target design and reduces cost, particularly when under the beam. producing films with more than three (4) Various computer-operated controls to components, makes computer control more regulate:(a) the introduction of processing manageable, and avoids having to gases ( oxygen for example) into the accurately control various ion beam target chamber or in a sub-eV atomic or currents simultaneously, (b) the sputtered energetic ion beam source directed at the fluxes of all elemental target materials substrate, (b) the interposition of shutters originate from the san;? spatial location in or masks between targets and substrates, the one computer-controlled ion beam (c) the substrate temperature, and (d) other processing steps. A specially designed computer program (Krauss and Aucieilo 1989) is a fundamental component of the automated ion beam sputter-deposition system. The potential of this technique has recently been demonstrated by producing a m superconducting YBa2Cu3O7-x ^ (Kingon et ai. 1989). However, much work is still needed to optimize the method. In particular, studies on basic ion beam-solid interaction phenomena, related to the elemental targets relevant to this review (Y, Ba, and Cu for example), are necessary. Comprehensive experimental ION and computer modelling, using the TRIM SOURCE code (Biersack and Haggmark 1980), studies are currently underway and first ROTATJNO TARGET results will soon be presented (Ameen et A5SEMILY al. 1989). Briefly, these studies have shown that relatively light ions such as Fig. 13 Schematic of the automated ion Ar+ impacting at 45° with respect to the beam sputter-deposition system. The target surface normal results in an computer monitors the quartz crystal undesirably high scattered ion flux directed resonator (QCR) output, turns the ion at the substrate, which leads to a beam off, and rotates the targets when the deleterious incorporation of gas into the desired amount of sputtered material is films. deposited (Kingon et al. 1989, and Krauss et al. 1989). Other groups are now developing ion beam sputter-deposition techniques involving the use of multiple beams to system, which results in more uniform sputter elemental materials from different films across the substrate surface both in spatial locations and simultaneously thickness and composition, (c) the ion deposit them onto an appropriately scattering fluxes and angular distributions situated substrate. However, the of the sputtered fluxes may be more easily automated ion beam sputter-deposition controlled in the one ion beam system. method described above has some 412 Compared with the plasma sputter- during the past eighteen months directed deposition techniques, the ion beam at modifying the techniques discussed method offers the following advantages: above to produce as-deposited crystalline (a) A much lower flux of impurities in the superconducting films at substrate target chamber during deposition, since the temperatures <, 600 °C (Moriwaki et al. focused beam can be made to mainly 1987, Terashima et al. 1988, Wasa et al. interact with the target, contrary to the 1988, Witanachchi et al. 1988. 1989, plasma sputtering case, where the plasma Margaritondo et al. 1989, Kwo et al. 1989. has a rather strong interaction with the and SPtE 1989). Production of films at £ target chamber walls, (b) Control of the ion 600 °C is critical for the application of high angle of incidence with respect to the Tc superconducting films to device target surface, and therefore, to first order, technologies, particularly the Si-based of the sputtering and ion scattering semiconductor technology. processes. Primary ion scattering can have Different methods, depending on the a major influence on the characteristics of deposition technique used, have been the films, as demonstrated by work utilized to dynamically introduce oxygen currently underway (Ameen et al. (1989)). into the growing film in order to produce as (c) No negative ion bombardment of the deposited high Tc superconducting films at growing film as in the plasma sputter- the lowest possible substrate temperature deposition case, which requires extra Witanachchi et al. (1989), for example, attention in regard to the target-substrate first demonstrated the use of an configuration. oxygenated laser-induced plasma-assisted deposition method for producing as LOW TEMPERATURE FILM deposited Y-Ba-Cu-O superconducting PROCESSING films at about 400 oC. They located a positively biased (about 300-400 V) The deposition techniques described platinum ring in front of a grounded target, above generally require a high temperature which was used to sustain an oxygenated (about 900 °C) post-deposition anneal. plasma triggered by the laser-induced YBa2Cu3O7.x, for example, is unstable for plasma plume from the target in xM, decomposing into Cu metal and combination with an oxygen jet directed at yttrium and barium oxides. In practice, the substrate and passing through the ring. therefore, deposition processes occurring (Fig. 14). The high voltage between the at low oxygen activities result in ring and the target was used to sustain the amorphous, oxygen-deficient films. The oxygenated plasma at 0.1-0.5 mTorr of high post-deposition anneal temperatures oxygen pressure in the deposition are necessary, since crystallization into chamber. The hypothesis proposed by YBa2Cu3O7_x requires both cation and Witanachchi et al. (1988) was that excited anion diffusion. The oxygen stoichiometry oxygen species created in the plasma were of x=0 is achieved by a final anneal at 400- dynamically introduced in the growing film, 600 °C or by slow cooling from 900 °C, which results in as-deposited both processes being conducted in an superconducting films at lower substrate oxygen atmosphere (Tarascon et al. 1987). temperatures. Several groups are now The high temperature post- using this laser ablation deposition deposition anneal can result in deleterious technique. However, more work is substrate-film interactions, making this necessary to better understand the particular process incompatible with Si- mechanisms responsible for the observed based semiconductor technology. results. Therefore, much work has been done 413 HEATED SUBSTRATE Kanai et al. (1989) produced as- k/wwi deposited high Tc films on substrates at 480 °C, by laser (ArF, 193 nm) ablation deposition in a N26 atmosphere. N2O was employed instead of O2 because the ArF laser, which performs the double function of ablating the target and activating the MUTATING oxygen, is one order of magnitude more TARCCT efficient in dissociating N2O than O2 to produce active oxygen species. These effects have been verified by et al. (1989), in which a Nd-YAG laser was used to ablate an YBa2Cu3O7_x target , Fig. 14 Schematic of the laser ablation- while an ArF laser beam, directed parallel deposition system used for producing as to the substrate surface, inter.ected the deposited high Tc superconducting films at ablation plume. As-deposited super- low substrate temperature (Witanachchi conducting films at low substrate et aJ. 1988). temperature were observed only when using a N2O atmosphere. The effect of Mizuno et al. (1989) used an Ar-s-02 location of the ArF laser between the plasma to sputter-deposit polycrystalline substrate and target was also studied. Y-Ba-Cu-O films on GaAs (100) with a These experiments demonstrated that the CaF2 buffer layer. The substrate best films were produced when the ArF temperature, about 450 °C during laser beam intersected the ablated plume deposition, was subsequently dropped to in a N2O atmosphere close to the 250 °C for 30 minutes followed by an slow substrate surface. The spatial location cooling down in an O2 atmosphere after where the activated oxygen is formed may deposition. This procedure produced an be critical to an effective incorporation of as-deposited high Tc superconducting film, atomic oxygen into the growing film, since although with a relatively low Tc (45 °K). atomic species created further away from In this case, the activated oxygen was the substrate may be subjected to a high produced by a plasma established between rate of recombination due to the necessary the superconductor target (cathode) and relatively high pressures in the deposition the substrate (anode). A possible reason chamber. for the relatively low Tc obtained may have An ozone jet directed at the been the bombardment of the growing film substrate produced as-deposited by negative oxygen ions and secondary YBa2Cu3O7.x superconducting films electrons from the target, which can (Berkeley et al. 1988). Deposition at a produce deleterious compositional substrate temperature of 590 °C in the changes. These deviations have been presence of this ozone jet resulted in T demonstrated by different groups c values of 40 °K, and 700 °C resulted in -80 (Rossnagel and Cuomo 1988, Terada et al. °K transitions. 1988). The results indicate that not all Electron cyclotron resonance (ECR) sources of activated oxygen are viable sources have been particularly effective in alternatives for the production of high quality as-deposited superconducting films. allowing in-situ deposition of YBa2Cu3O7.x, as demonstrated by Moriwaki et al. (1989) and Aida et al. 414 (2989). These workers utilized electron CONCLUSIONS beam evaporation in conjunction with oxygen from an ECR source directed at the A limited review has been presented substrate. The dominant species of the on what appears to be the most promising oxygen plasma near the substrate was techniques for production of high Tc identified to be O2+. and the electron superconducting films compatible with temperature estimated (by Langmuir relevant technologies such as probe) to be - 8 eV. The ECR source is microelectronics. It is clear that much work also known to yield a significant is still needed to understand many of the concentration of oxygen radicals. basic processes occurring during Crystalline YBa2Cu307.x resulted at deposition. This understanding is deposition temperatures as low as 450 °C. fundamental for achieving the necessary Aida et al. (1989) also showed that control in film synthesis, which will lead to in the case of the evaporation of copper applications in device fabrication. alone, Cu metal is deposited in a molecular Outstanding issues that need to be oxygen atmosphere (pressure 2 x 10*2 addressed are: ton) and temperatures as high as 600 °C. (1) Control on uniformity of film This is consistent with the known composition and thickness across thennodynamic stability of Cu as a function extended areas. This is critical for the of oxygen partial pressure. However, in integration of high Tc superconducting films the presence of "active" oxygen originating with the current microelectronics from the ECR plasma, CuO was deposited technology. at substrate temperatures as low as room (2) Production of as-deposited temperature. This study emphasizes the superconducting films at the lowest oxidative ability of the ECR plasma, and possible substrate temperature. This is furthermore indicates that additional also critical for the integration of these energy, probably derived from the oxygen films with the current silicon-based ions, can increase the mobility of deposited microelectronics technology. species on the film surface. (3) Integration of the deposition The ECR source has also been techniques with patterning methods for the utilized in a novel sputtering geometry by fabrication of devices based on high Tc Goto et al. (1989). This sputtering method superconducting films. In relation to this has the advantage that the substrate is not issue, much work is also needed to exposed to the plasma, with the absence develop and optimize patterning therefore of any associated damage. techniques, although some work has The studies discussed above clearly already been done on this topic. indicate that the dynamic incorporation of Given the pace at which research on activated oxygen is fundamental for high Tc superconducting films has been producing as-deposited high Tc developing, relevant advances can be superconducting films at relatively low expected in the near future. substrate temperatures However, further work is necessary to better understand the ACKNOWLEDGMENTS complex processes involved in the The author would like to activated oxygen-assisted deposition of acknowledge the productive collaboration superconducting films. with Drs. A.R. Krauss (ANL), A.I. Kingon (NCSU), M. S. Ameen (NCSU), and T. Barr (U. of Wisconsin), useful discussions with Dr. R.F. Davis (NCSU), the help of the graduate students C.S. Soble, A. Rou, and T. Graetinger, and support from DARPA (N- 00014-88-K-0525), NSF (DMC-8813502, 415 DMR-88-07367), ONR (N-00014-88-K- 0526), and DOE (Basic Energy Sciences W- 31-109-ENG-38 and DE FG05-88ER Aucieilo O and Krauss AR. (1988). A 45359). Critical Analysis of Deposition Techniques REFERENCES for Production of High Temperature Superconducting Films. Proc. 3rd Intl. Adachi H, Hirochi K, Setsune K, Symposium on Surface Science, Salzburg, Kitabatakc M, Wasa K: (1987). Low Austria, p. 197- 203. temperature process for the preparation of high Tc supercondcuting thin films. Appl. Auciello O, AthavaleS, BourhamM, Phys. Lett 51, 2263- 2265 . Hankins OE, Gras-Marti A, Vallei-Abarca JA. (1989). Transport Processes in Aida T, Tsukamoto A, Imagawa K, Plasma-Assisted Deposition of Fukazawa T, Saito S, Shindo K, Takagi K, Superconducting Thin Films. Proc. Miyauchi K. (1989). Thin film growth of European Conf. on Surface Science (in YBa2Cu 307.5 by ECR oxygen plasma press). assisted reactive evaporation. Jpn. J. Appl. Phys. 28, L635-L638 . Bednorz JG, Mullcr KA. (1986). Possible High Tc Superconductivity in the Ba-La- Ameen MS, Auciello O, Rou SH, Soble Cu-0 System. Z. Phys. B64, 189-193. CS, Graettinger TM, Krauss AR, Kingon AI. (1989). AVS National Symposium. Berkeley DD, Johnson BR, Anand N, Boston. To be published in Araer. Inst. Beauchamp KM, Conroy LE, Golman AM, Phys. Conf. Proc. Maps J, Mauersberger K, Mecartney ML, Morton J, Touminen M, Zhang Y.Y-J. (1988). Anderson JB, Andres RP, Fenn JB. (1966). In Situ Formation of Superconducting title.... Adv. Chcm. Phys. 10,275- . YBa2CU3O7.x Thin Films Using Pure Ozone Vapor Oxidation. Appl. Phys. Lett 53,1973- Auciello O and Kelly R (Eds.). (1984). 1975. "Ion Bombardment Modification of Surfaces: Fundamentals and Applications", Bctz G and Wehner GK. (1981). title, Elsevier Science Publishers. in "Sputtering by Particle Bombardment, HM, Ch.2, Berisch R (Ed.). Springer Auciello O, Krauss AR, Santiago-Aviles J, Verlag,p.ll- . Schreiner AF, and Gruen DM. (1988). Surface Compositional and Topographical Biersack JP and Haggmark LG. (1980). A Changes Resulting from Excimer Laser Monte Carlo Computer Program for the Impacting on YBa2Cu3G°7 Single Phase Transport of Energetic Ions in Amorphous Superconductors. Appl. Phys. Lett. 52, Targets. Nucl. Instr. Meth. 174, 257-269. 239- 241. Bonch-Bruevich AM and YA Imas.(1968). Auciello O. (1989). Unpublished. Recoil Momentum and Ejection of Metal Particles Under the Action of Giant Laser Auciello O and Krauss AR. (1988). Proof- Pulses. Sov. Phys. Tech. Phys. 12, 1407- of-Principle of a Novel Method for Sputter- 1409. Deposition of High Tc Superconductor Films via A Computer-Controlled Mas* Bunshah RF et al.(Eds.). Deposition Analyzed Ion Beam. Amer. Inst. of Phys. Technologies for Films and Coatings. Conf. Proc., Harper JME, Colton RJ, (1982). Noyes Publications. Feldman LC (Eds) No 165, 114- 121. 416 Bhushan M, Strauss AG. (1989). Films by ECR Plasma YBa2Cu3Ox Thin Films Prepared by Sputtering. Jpn. J. Appl. Phys. 28, L88- Sequential Sputtering of YBa2C>3, BaF2, L90. and CuO Targets . Book of Abstracts, Materials Research Society Spring Harper JME, Colton RJ, and Feldman LC, Meeting, p. 296. (Eds.). (1988). American Vacuum Society Topical Conference Proceedings on Thin Deshmukh S, Rothe EW, Reck GP. Film Processing and Characterization of Kushlda T, and Xu ZG. (1988). Emission High Temperature Superconductors. Amer. Spectra from ArF laser ablation of High Tc Inst. of Phy*. Conf. Proc. No 165. Superconductor Bi2CaSr2Cu2O7- Appl. Phys. Lett. S3, 2698-2700. Hong M, Liou SH, Bacon DD, Grader GD, Kwo J, Kortan AR, and Davidson BA. Dijkkamp D, Venkatesan T, Wu XD, (1988). Superconducting Tl-Ba-Ca-Cu-0 Shaheen SA, Jisrawi N, Min-Lee YH, Films by Sputtering. Appl. Phys. Lett S3, McLean WL, and Croft M. (1987). 2102-2104. Preparation of Y-Ba-Cu Oxide Superconductor Thin Films Using Pulsed Kang JH, Kampwirtb RT. Gray KE, Marsh S, and Huff EA. (1988). Superconductivity Laser Evaporation from High Tc Bulk Material. Appl. Phys. Lett 51, 619- 621. in Thin Films of the Bi-Ca-Sr-Cu-O System. Phys. Lett. A128, 102-104. Enomoto Y, Murakami T, Susuki M, Morwaki K. (1987). Largely Anisotropic Kang JH, Kampwirth RT. and Gray KE. Superconducting Critical Current in (1988). Superconductivity in Sputtered Thin Films of Tl-Ba-Ca-Cu-O. Phys. Epitaxially Grown Ba2YCu3O7-y Thin Lett.A131, 208- 210. Film. Jap. J. Appl. Phys. 26, L1248-50. Kang JH, Kampwirth RT, and Gray KE. Gagliano FP and Paek UC. (1974). (1988). High Critical Field Anisotropy of Observation of Laser-Induced Explosion of Superconducting Bi-Sr-Ca-Cu Oxide From Solid Materials and Correlation with Highly Oriented Thin Films. Appl. Phys. Theory. Appl. Opt. 13, 274-279. Lett. 52, 2080-2082. Gao J, Zhang YZ, Zaho BR, Out P, Yuan Kanai M. Kawai T, Kawai S, Tabata H. CW, and Li L. (1988). High Tc Y-Ba-Cu-O (1989). Low Temperature Formation of Thin Films by Ion Beam Sputtering. Appl. Muhilayered Bi(Pb)-Sr-Ca-Cu-0 Thin Phys. Lett 53, 2675-2676, Films by Successive Deposition Using Laser Ablation. Appl. Phys. Lett. 54, Geohegan DB, Mashburn DN, Culbertson 1802-1804. RJ, Pennycook SJ, Budai JD, Valiga RD, Sales BC. Lowndes DH, Boatner LA, Kaufman HR, Cuomo S3, Harper JME. Sondner E, Eres D, Christen DK, and (1982). Technology and Applications of Christie WH. (1988) .Pulsed laser Broad-Beam Ion Sources Used in Ablation of Thin Superconducting Films of Sputtering. Pan I. Ion Source Technology. Ho[Ba2Cu3O7.x and YiBa2Cu3O7.x. J. J. Vac. Sci. Technol. 21,725-736. Mater. Res. 3, 1169-1179. Kelly R and Dreyfus RW. (1988). Goto T, Masumoto M, Toshio H. (1989). Reconsidering the Mechanisms of Laser Preparation of Superconducting Sputtering with Knudsen-layer Formation 417 Taken Into Account. Nucl. Instr. Meth. May P, Jedamzik D, Boyle W, and Miller Phys. Res. B32, 341-348. P.(1988). Production of Superconducting Thick Films by a Spin-on Process, Kingon AI, Auciello O. Ameen MS, Rou Supercond. Sci. Technol. 1,1-4. SH,and Krauss AR. (1989).YBa2Cu3O7.x Films Deposited by a Novel Ion Beam Moriwaki K. Enomoto Y, Kubo S and Sputtering Technique. Appl. Phys. Lett. Murakami T. (1988). As-Deposited 55, 301-303. Superconducting Ba2YCu3O7_y Films Using ECR Ion Beam Oxidation. Jpn. J. Klein JD and Yen A. (1989). The Influence Appl. Phys. 27, L2075-L2077. of r.f. sputter variables on the composition of YBa2Cu3(>7 Films. J. Vac. Sci. Technol. Narayan J, Biunno N, Singh R, Holland A (in press). OW, and Auciello O. (1987). Formation of Thin Superconducting Films by a Laser Kobrin PH. DeNatale JF, Housley RM, Processing Method . Appl. Phys. Lett. Flintoff JF, and Harber AB. (1987). 51, 1845- 1847. Superconducting Oxide Thin Films by Ion Beam Sputtering .Advanced Ceramic Shing RK, Biunno N, Narayan J. (1988). Materials 2,430-435. Microstructural and Compositional Variations in Laser-Deposited Krauss AR and Auciello O-(1989). Patent Superconducting Thin Films. Appl. Phys. Pending (ANL-IN-87-91). Lett. 53. 1013-1015. Kiauss AR, Auciello O, Kingon AI. Proc. SPE-Int. Soc. Opt. Eng. (USA). Ameen MS, Barr T, and Liu YL. (1989). (1988). High Tc Superconductivity: Thin Rev. Sci. Instr. (submitted). Films and Devices, vol. 948. KwoJ, HongM, Trevor DJ, Fleming RM, Rossnagel SM and Cuomo JJ. (1988). White AE, Farrow RC, Kortan AR. and Negative Ion Effects During Magnetron Short KT. (1988). In-Situ Epitaxial Growth and Ion Beam Sputtering of YBa2Cu3O7 . of YBa2Cu3O7_x Films by Molecular Amer. Inst. of Phys. Conf. Proc., Harper Beam Epiuxy with an Activated Oxygen JME. Colton RJ, Feldman LC (Eds.) No Source. Appl. Phys. Lett. 53, 2683-2685. 165, 106- 113. LaibowitzRB, Koch RH, Chaudari P. and Sandstrom RL, Gallagher WJ, Dinger TR, Gambino RJ. (1987). Thin Superconducting Koch RH, Laibowitz RB, Kleinsasser AW, Oxide Films. Phys. Rev. B35 , 8821- Gambino RJ, Bumble B, and Chishclm MF. 8823. (1988). Reproducible Single Target Sputtering Process for High Temperature Maeda H, Tanaka T, Fukutomi M, Asano Superconducting Films and Devices. T. (1988). A New High Tc Oxide Superconductor Without a Rare Earth Sankur H, Denatale J. Gunning W, Nelson Element Jpn J. Appl. Phys. 27. L209-L210. JG. (1987). Dense Crystalline ZrO2Thin Films Deposited by Pulsed-Laser Margaritondo G, Joynt R, and Onellion M Evaporation. J. Vac. Sci. Technol. A5, (Eds.). (1989). title . Amer. Inst. of 2869-2874. Phys. Conf. Proc. No 182. 418 Shah SI and Carcia PF. (1988). Growth of Venkatesan T, Wu WD, Jnam A. and YBa2Cu3O7-x Thin Films by Sputtering. Wachtman JB. (1988). Observation of Two AIP Conf. Proc. 165, 50-57. Distinct Components during Pulsed Laser Deposition of High Tc Superconducting Sheng ZZ, Hermann AM. (1988). Bulk Films. Appl. Phys. Lett. 52. 1193-1195. Superconductivity at 120 K in the Tl- Ca/Ca-Cu-O System. Nature 332, 138- Venkatesan T, Wu XD, Inam A. Jeon Y, 139. Croft M, Chase EW, Chang CC. Wachtman JB. Odom RW, di Brozolo FR, Shing RK, Holland OW, Narayan J. (1989). and Magee CA. (1988). Nature of the Theoretical Model for Deposition of Pulsed Laser Process for the Deposition of Superconducting Thin films Using Pulsed High Tc Superconducting Thin Films. Appl. Laser Evaporation Technique. J. Appl. Phys. Lett. 53, 1431-1433. Phys. (in press). Sudarsan U, Cody NW, Bozack MJ, and Wasa K, Kitabatake M, Azachi H, Solanki R. (1988). Excimer Laser-Induced Setsune K. and Hirochi H. (1988). Sputtering of YBa2Cu3O7.x, J- Mater. Superconducting Y-Ba-Cu-0 and Er-Ba- Res. 3, 825-829. Cu-O Thin Films Prepared by Sputtering Deposition. AIP Conf. Proc. 165, p. 38-49. Tarascon JM, McKinnon WR, Greene LH, Hull GW, Bagley BG and Vogel VG. Wessels B.(1989). Private communication (1987). Processing and Superconducting Properties of Perovskite Oxides. Adv. Witanachchi S, Kwok HS, Wang XW. and Ceram. Materials 2, 498-505. Shaw DT. (1988). As Deposited Y-Ba-Cu- O Superconducting Films at 400 °C. Appl. Terada N, Ihara H, Jo M, Hirabayashi M, Phys. Lett. 53, 234-236. Kimura Y, Matsutani K, Hirata K, Ohno E, Sugise R, Kawashima F. (1988). Sputter Witanachchi S. Patel S, Kwok HS, and Synthesis of YBa2Cu3Oy As-Deposited Shaw DT. (1988). As Deposited Y-Ba-Cu- Superconducting Thin Films from O Superconducting Films on Silicon at 400 Stoichiometric Target: A Mechanism of C. Appl. Phys. Len. 54. 578-580. Compositional Deviation and its Control. Jap. J. Appl. Phys. 27, L639-L642. Wu MK, Ashbum LR, Torng CJ, Hor PH, Terashima T, Iijima K, Yamamoto K, Meng RL, Gao L, Huang ZJ, Wang YQ, Bando Y, and Mazaki H. (1988). Single- Chu CW. (1987). Superconductivity at 93 K in a New Mixed-Phase Y-Ba-Cu-O Crystal YBa2Cu3O7.x Thin Films by Activated Reactive Evaporation. Jap. J. Compound System at Ambient Pressure. Appl. Phys. 27, L91-L93. Phys. Rev. Lett 58, 908-910. Thornton JA. (1988). Physical Vapor Zheng JP, Huang ZQ. Shaw DT, Kwok HS. Deposition. In "Semiconductor Materials (1989). Generation of High- Energy Atomic Beams in Laser-Superconducting and Process Technology Handbook", G. E. Target Interactions. Appl. Phys. Lett. 54, McGuire (Ed.), Noyes Publications, p. 280-282. 329- 454. Zheng JP, Ying QY, Witanachchi S, Huang ZQ. Shaw DT, Kwok HS. (1989). Role of the Oxygen Atomic Beam in Low- Temperature Growth of Superconducting Films by Laser Deposition. Appl. Phys. Lett. 54, 954-956. 421 Effective Bass of bulk and surface positrons due to plasaon excitation Shlgeru SHINDO Tokyo Gakugel University, Koganei, Tokyo 184, Japan abstract We calculate the effective mass of bulk and surface positrons. We show that the mass of bulk positrons is enhanced due to plasraon excitations. We calculate also the effective mass of surface positrons due to surface plasmon excitations. 422 1. Introduction It has recently been known that a low energy positron beam provides a useful probe for studying solid surfaces, such as the LEPD (Low Energy Positron Diffraction), PELS (Positron Energy Loss spectroscopy). PsFS (Positronium Formation SpectroscopyI [1|. and so on 12). For the experimental analysis, the Investigation of the coupling of the positron with elementary excitations of solids (inelastic scattering) is very important. A recent experiment [3| suggests that the inelastic scattering cross section of positrons in solids is much enhanced In comparison with the electron beam. We have given an explanation for this experimental facts by taking account of the higher order perturbation terms [4]. In this paper, we calculate the effective mass of thermal positrons due to bulk and surface plasmon couplings. Experimentally, it has been Indicated that the effective mass of positrons in the bulk of metal (m") becomes nearly twice of the bare positron mass In alkali metals[5,6] . Many theoretical models have proposed for the explanation of the large effective mass, for example, positron band mass, coupling with phonon. and so on. However, there is no theory explaining the experimental large positron mass enhancement, except Ishii's "soliton bag" theory taking account of the plasmon couplingl9j. Recently, Hyodo et al.[8) measured the effective mass of positrons in potassium and found that the effective mass m»/m Is 1.35± 0.15. which is much less than the previously measured value(5.6]. Then, we recalculate the positron effective mass in metals due to plasmon coupling using the self-energy approach. We also take Into account the sur- face plasmon coupling In order to predict the effective mass of surface positrons. 423 2. Plasmon excitation It is widely known that optical potential formulation repre- sents a very powerful method in various scattering phenomena In nuclear physics, atomic physics and sold state physics. In order to deal with a complicated many-body problem in the scattering processes, we introduce an effective potential, called optical potential, which should at least In principle include the com- plication of the many-body problem. The term of "self-energy" is used as the expectation value of the optical potential over the initial wave function of the considered projectile. Using the first-order perturbation theory, the self-energy is represented by < k I VOm| k'> < k* I Vmo i k> 1 I (1) m k' E* - Eu- - Em • Eo • id where E* and Ek- are the initial and final energies of the positron. I k> and I k*> are the associated initial and final plane wave states, respectively. Eo and Em represent the ground and excited states of the target, respectively. When we consider the electron excitations, the Coulomb operator coupling Is used for V in eq.(1) as follows: V= - (2) 1 r« - rp I where r« and rP represent the positions of the solid electron and the positron, respectively. By introducing the dielectric function by 424 Ait e* (p m q, ai ) ftq2 (3) where p Is the density operator, we have the expressions of the self-energy as follows: (i^q An e2 1 Z = -/ dw S ( 1 ) 6 (4) In order to investigate the behavior of thermal positrons whose energy is much less than electron Fermi energy, we expand £ of eq.(4) by the initial momentum k as C L. (k = 0) * C (k = 0) (5) 2 ^k2 Since the total energy of positrons in the electron gas is given by M2 k= K2 k2 E = * C = + E (k=0), (6) 2m 2m* the effective mass.m*. is represented in terms of the self-energy as follows: 425 m ^ m" = m (1 • £ < k I Vom I k'Xk'l Vmo|k> 2 =51 I (8) 2 a a m q (Em - Eo • (Mq) /2m)) mk Then the first term of the right-hand-side of eq.(5) gives the static self-energy in the electron gas. The second term represents the effective mass m". For the plasmon excitation, we take the high-frequency limit of the dielectric function, as q.w ) = 1 - *16 (8) where w P is the plasmon frequency. Using this dielectric func- tion, the static self-energy £ (k=O)is calculated as follows: z (k=0) = e / mw P/(2M) (9) We choose the plasmon cut-off wave number as infinite, which does not affects the results seriously. The effective mass, m* , given by eq.{7) is calculated as follows: m- /m = (1 - where 426 ,,e2 q2 m e2 CO ; dq = / H/(2mw P) (10) [n fig.l. we show the ratio of the effective mass of positron to the bare mass,(m*/m) .as a function of r« parameter where An ir™aB)-'/3 = 1/n, aa Is the Bohr radius. Our calculation shows good agreement with the recent experiment by Hyodo et al.[8J for the bulk positron in potassium. We note that our present calculation gives smaller effective masses in comparison with the old experi- ments by the same authors [ 5", 6] Now, we mention Ishli"s{9j "soli ton bag" model for the effective mass of positrons in metal, which gives larger effective masses than the present calculation and shows good agreement with the experiments by Stewart et al.[6.7]. In his model, the plasmon coupling of the positron is taken into account, which is equivalent to our present calcula- tion. This means that the plasmon contribution is essential and important for the effective mass of slow positrons in metals. However, according to the "soliton bag" model, the effective mass is independent of the coupling strength (e2 in the present theory), which is in contrast to our theory. 427 3. Surface plasmon excitation In the previous section, we mentioned that the contribution of the plasmon coupling is Important to explain the experiments of the effective mass of positrons in metals. The coupling with the surface plasmon may equally contribute to the effective mass of surface positrons. The effective mass effect is very important for the behavior of surface positrons, for example positron surface state (image potential state) and the positronium (Ps) formations. Now. we assume that the surface positron moves along the sur- face, that is. surface parallel direction distant from the sur- face, z. Then the self-energy of the positron Is given byllO) d2Q 2n es 1 c = - / do; / e-aQ- ( -1)6 (o> •HkQ/m* (MQ=)/2m) (2s )2 Q e .(Q.w ) (11) where (12) E _(Q, ) ( E (q. w ) • 1) By using the dielectric function of eq.(8). the static self- energy E (k=0) due to the surface plasmon excitation becomes as follows: e2 (k = 0) = a { ci(2a z)sin(2o z)-si(2o z)cos(2a z)} (13) 2 428 where a = /• (2uiu ,)/M ci and si are cosine and sine integrals, respectively. We note that eq.(13) coincides to the familiar image potential of a charge distant from the surface. -e2/(4z) Oj l-»co. Similarly to eq.(10), the effective mass due to the surface plasmon excitation is calculated as follows: m-(z)/m = (1 - r (z))-x (14) where the effective mass depends on the distance from the surface, z. 2e2m T (z) = v K/(2mw „) f(a z) (15) where function f(x) is defined by y f(x) = /* exp(-2xy) dy * In fig.2, we show the effective mass of surface positrons due to surface plasmon excitation as a function of the distance from sur- faces. Calculations are done for r« = 2.0 (corresponding to aluminium) and for r» = 5.0 (corresponding to potassium). It is seen that the positron distant from the aliminlum surface of the Bohr radius has an effective mass about factor 1.1 times to the bare mass. 429 The effective mass of positrons trapped into the surface image potential states, m*, is calculated from eq.(14) as Jo where^f(z) is the wave function of positrons in the surface state. When the image-potential is chosen as -e /4z, the wave function corresponding to the deepest energy level is given by Then, m* is given by Fig.3 shows the effective mass of positrons in the image- potential states as a function of rs parameter. It is seen that the effective mass increases several percent to the bare mass. Since the present effective mass is calculated on the basis of the first-order perturbation theory, we may apply it to that of electrons in the image-potential state. Concerning to this problem, there are experiments using inverse photoemission and two-photon photoemissions, where it is pointed out that the effective mass of electron exceeds by several ten percent of the bare mass DO. Theoreticall, Echer.ique and Pendry£l23 and Echenique£,13j considered the coupling with electron-hole pair excitation and with surface plasmon excitation, respectively. They found that those couplings increase the effective mass by at most a few percent, which coincides to the present calculation due to surface plasmon excitation. 4. Concluding remarks We calculated the effective mass of bulk and surface positrons due to bulk and surface positron excitations. Our calculation of the bulk positron agrees well with the experiment by Hyodo et al.{8). We predicted the effective mass of surface positrons due to the surface plasmon excitation. When we apply the present calculation to the effective mass of electrons in the surface image-potential state, which is equivalent to the previous theory by Echenique and Pendry ft.2,13], we found that our calculation cannot explain the experimentally obtained large mass enhancement. The author would like to thank Drs. A.Ishii and H.Nitta for valu- able discussion. This work is partly supported by the Institute for the Molecular Science. 430 References [1] A.P.Mills,Jr. .L.Pfelffer and P.M.Platzrnan. Phys. Rev. Lett. 51 (1983) 1085 |2J B.Pendry. In "Positron Solid State Physics" edited by W.Brandt and D.Dupasquir (North-Holland, New York 1983) [3] R.Mayer. C-S.Zhang,K.G.Lynn.W.l£. Frieze.|-\Jena, and P.M.Marcus. Phys.Rev.B35.3102 (1987) I4| S.Shindo. Nucl.Inst.Meth.B (submitted) [51 A.T.Stewart, J.B.Shand and S.M.Kim. Proc.R.Soc.London 88 (1966) 1001 [61 S.Kim and A.T.Stewart, Phys.Rev.Bll (1975) 2490 (7! S.Shindo and a.lshii. Nucl.Inst.Meth.B33 (1988) 392 [8j T.Hyodo, T.Mc#1ullen and S.T. Stewart. "POSITRON ANNIHILATION" edited by P. G.Coleman. S.C.Sharma and L.M.Diana (North-Holland. New-York 1982) [9] A.lshii, Prog.Theor.Phys. 70 (1983) 664 [10| R.Kawai.N.ltoh and Y.H.Ohtsuki. Surf.Scl. 114 (1982) 137 [lllK.Giesen, H.Hag, F.J.Himpsel, H.J.Riess, W.Steinmann and N.V.Smith, Phys.Rev. B35 (1987) 975 Q23p.M.Echenique and J.B.Pendry, Surf.Sci. 166 (1986) 69 Q3JP.M.Echenique, J.Phys.C18_ (1986) L1133 431 Figure captions Figure 1. Effective mass of bulk positrons due to plasmon excitation as a function of r. parameter. Solid line represents the theoretical results and the experimental results by Stewart et al.(6] are marked by circles. O . and the experimental result by Hyodo et al.[8] for potassium is marked by a triangle.A . Figure 2. Effective mass of surface positrons due to surface plasmon excita- tion as a function of the distance from surfaces. Calculations are done for r™ = 2.0 and r. = 5.0. 432 \ 2. 433 nrf/m-1 434 O.oi - 435 Variation of DOSD Moments with Bond Length in H2 and by Jens Oddershede and John R. Sabin Odense University and University of Florida and Geerd H.F. Diercksen Max-Planck-Institute for Astrophysics Presented at the 12th Werner Brandt Workshop - 1989 Abstract Using the polarization propagtor formalism, the S(/i), Lip) and /„ (—2 < n < 2) moments of the DOSD of H2 and N2 are calculated. The dependence of these quantities on bond length is discussed, and the reliability of the results is evaluated. Finally, a comment is made on the possible connection between the bondlength variation of the mean excitation energy for stopping and the large phase effect in N*2: there isn't any. 1. Introduction There can be significant change in molecular bond lengths; both dynamic, due to vi- bration, and static, due to physical environment or phase1-2. For example, the equilibrium 3 bond length in gaseous N2 is measured to be 2.07 au , while in the van der Walls solid measurements give values from 2.04 to 2.17 au3. In both these casea, since the time scale for electronic motion is so much shorter than that for nuclei, one would expect bond length dependent changes in the dipole oscillator strength distribution (DOSD) of the system, and consequently of the properties associated with it. In particular we are interested in h, the mean excitation energy for stopping, which is the material parameter describing energy deposition by swift, charged particles in material (in this case molecular) targets. We are especially interested in whether the variation in /<> with bond length can play a role in the sample phase dependence of stopping in molecular solids, especially NV In this paper we report the variation of the moments of the DOSD for Ej and Nj at calculated in the polarization propagator formalism. 436 2. Theory and Computation The moments of the DOSD are defined as dE (2) (3) where the Schiff sum is over all discrete and continuum excitations of the system. In the range -6 < n < 2, the S(/i) can be related to many properties of the interaction 4 of radiation with matter . Io and h are the mean excitation energies for stopping and straggling, respectively. In order to calculate the moments of the OOSD, all excitation energies and oscillator strengths (transition moments) of the system are needed. We obtain these from the energy dependent polarization propagator, which in its spectral representation can be written5, for operators P and Q <0\P\m> which has first order poles at E = ±(Em — EQ). For the case of the dipole operator, P = Q = f, it follows from eq.4 that Thus calculation of the poles and residues of this polarization propagator yield the exci- tation energies and {dipole length) transition moments, and therefore oscillator strengths, directly. Similarly, P = Q = p yields the dipole velocity form of the oscillator strength. We use the matrix form of the propagator5, and evaluate the matrix elements4 through given order in the fluctuation potential V = Ho — FQ, thus including correlation in the ground and excited state to a specific order. The results reported here are calculated to first order and use the Hartree-Fock ground state as the reference state; JO >= \HF >. In this case, we have the consistent first order polarization propagator approximation (FOPPA) which is identical to the random phase approximation (RPA). The details of the calculations are published elsewhere7. The basis sets used to cal- culate the reference state consisted of 90 GTO's in the case of H2 and 102 GTO't for N2. The two molecules have 42 and 298 dipole allowed excitations, respectively. The Thomas-Reiche-Kuhn (TRK) sum rule, exact in RPA, 5(0) = Y. /o« = N W 437 provides a convenient criterion for the assessment of basis set quality. As the oscillator strength in the velocity formulation 2 / is sensitive to the inclusion of tight functions in the basis set, while in the length formulation fL = f I < °IH™ > \2£om (8) it is most sensitive to more diffuse functions, the agreement between fv and fL measures the balance of the basis. The agreement of 5(0) with the total number of electrons in the system, .V, is a measure of basis set completeness. From the TRK sum rules for H2 and Nj given in Table 1, it is clear that the basis sets used are both well balanced and reasonably complete for the properties which we consider. All calculations are reported in Hartree atomic units unless otherwise specified. 3. Results The moments S(^) and I(/i) (-2 < fj. < 2) for H2 and N2 are given in Table 2. There are few other data with which to compare, but the empirical results of Zeiss et al.g are both comprehensive and reliable, and we consider them normative. In Table 3 we compare our data at the isolated molecule equilibrium geometry to those of Zeiss et al.. which are derived from experiment, and thus represent vibrationally averaged values. We also include the MSXa(a = 1) local density functional results of Kosman and Wallace9 for N2 which are the only other complete theoretical evaluations of many DODS moment* of which we are aware. It is clear that our results agree well with those of Zeiss et al. for fi < 0. The results deteriorate for positive values of fi, and the effect is larger for N2 than for H2. This is due to the fact that we use a discrete representation of the continuum, and the higher lying excitations, precisely those which most effect the p > 0 moments, are poorly represented. As there are many more such excitations in N2 than in H2, the effect is larger in N2. The lower (bound state) excitations are much better represented, as reflected in the better values of S(fi) for p < 0. Kosman et al.9 tend to agree with the Zeiss et al. N2 results for /x > 0 and get poor results for y. < 0. This is because the lower lying excitations are poorly represented in the XlSXa method. There is no agreement except for /x = 0 between our results and Kosman's. There are no general rules for the behaviour of DOSD moments with internudear distance. In the case of H2, the low lying cr* *— ag and ir* <— xg transitions dominate the spectrum, and these transitions decrease in energy with increasing RHH- AS the oscillator strengths do not contribute significantly to the radial dependence of the sum rules (5(0) is independent of R), the E2 S{fi) increase for \i < 0 and decrease for p > 0 with increasing RHH- This behaviour is also understandable in terms of the relations of the S{fi) to 438 physical quantities. The 5(-2) moment is the molecular static polarizabiiity,4'10 which increases with increasing RRH- Similarly 5(—1) is related to the sum of the squared radii of the electrons, that is, the size of the system,4'10 which also increases in H3 as the molecule dissociates. 5(1) can be related to the mean kinetic energy of the all of the electrons, which, in conjunction with the virial theorem, would lead to a decrease with increasing bond length. The same arguments hold for N2, and an increase in S(fi) {p = — 1, —2) is seen in this case as well. For 5(1) in N2l the situation is more complicated. There is an initial decrease in 5(1) at short /J.v.v on stretching, but this reverses just before Rtq, and 5(1) begins to increase. The 5(1) sum rule can be written as a sum of the kinetic energies < £pf > a of all the electrons plus momentum correlation terms of the type < 21 pj • pj >. As the kinetic energy term must be maximum at Rtq and decrease thereafter, the increase in 5(1) as the bound stretches must be due to correlation. The effect is about 8% from Req, well within estimates of the size of the correlation term in some atoms4. If one looks at the mean excitation energy for stopping (/o) in H2, it is seen to be a strong decreasing function of RHH (Table 2a and the Figure). This suggests that vibra- tionai averaging might be useful. When carried out7, it is seen that the mean excitations energies /o and I\ are each lowered by ~ 0.3 eV over the static value at Rtg. The question then arises as to whether this same bondlength dependence is present in N2, and if so, if the differing bond lengths in Nf" and N|°'"f can account for the large reported phase effect2. From Table 2b and the Figure it is clear that there is a bondlength dependence of /o in N2. [In fact, the curve of Io vs R for N2 is nearly parallel to that for H2!] If one takes the most favorable values for gas and solid bond distance, i.e. those that differ most, 3 the difference in IQ is only a little over 1 eV. It is thus clear that the large phase effect in N2 cannot arise from Bethe stopping, even if the total change is attributed to a change in the valence part of Io- 4. Summary We have calculated the 5(/z), L(p.) and /M moments of the DOSD of H2 and Nj as a function of internuclear distance using the polarization propagator formalism. The results compare well with available data for the static cases at Req, and are consistent in their bondlength dependence with predictions from sum rules. We thus expect the moments calculated in this way to be reliable to about 5% for ft < 0. For p > 1, the high lying excitations are poorly described, and the moments deteriorate. It is clear that bond length dependence of Jo for N2 cannot lead to changes in Bethe stopping large enough to explain the observed phase effect. 439 Acknowledgements The authors would like to thank the Danish Natural Science Research Council (grants # 11-6844 and # 11-3992), the U.S. National Science Foundation (grant # INT-8609943), the U.S. Army Office of Research (grant # DAA-L03-87-K-0046) and NATO (gran. # 622/ 1984) for support of this work. References 1. D.I. Thwaites, Rad. Res. 95 495 (1983); Nucl. Instr. Meth. B 12. 84 (1985). 2. P. Bargesen, Nucl. Instr. Meth. B 12. 73 (1985). 3. T.A. Scott, Phys. Rept. 27, 89 (1986). 4. J.L. Dehmer, M. Inokuti and R.P. Saxon, Phys. Rev. A. 12, 102 (1975). 5. For a complete description of the method in both formal and computational aspects, see: J. Oddershede, Adv. Chem. Phys. 69, 201 (1987), and J. Oddershede, P. Jargensen and D.L. Yeager, Comp. Phys. Rept. 2, 33 (1984). 6. Calculations were carried out using the MUNICH system of propons. cf. G.H.F. Dier- cksen and W.P. Kraemer, MPI spcial Report, 1981. 7. H2: G. Geertsen, J. Oddershede and J.R. Sabin, Phys. Rev. A 34, 1104 (1986). N2: G.H.F. Diercksen, J. Oddershede and J.R. Sabin, to be published. 8. G.D. Zeiss, W.J. Meath, J.C.F. MacDonald and D.J. Dawson, Can. J. Phys. 55, 2080 (1977). 9. W.M. Kosman and S. Wallace, J. Chem. Phys. 82, 1385 (1982). 10. J.O. Hirschfelder, W.B. Brown and S.T. Epstein, Adv. Quantum Chem. 1, 255 (1964). Table 1: Thomas-Reiche-Kuhn Sum rule for H2 and N2 in Dipole Length and Dipole Velocity Formulations HS "2 5(0)'' 2.000 13.980 S{0)v 1.998 13.763 N 2 14 a. ft = Rt = 1.4011 au. b. R = R< = 2068 au. <>.•> (ii'iiti l;«cll at ion inuniicB for Ilifi .IXTN| and H, a; It(mi) 0.3011 0.5011 0.H01! I (Kill 1.2011 1.3011 1.3511 .4011" 1.4511 1.5011 1.6011 1.8011 S( 2) 1630 2.037 "2.859 3.541 4.330 4.765 4.993 5.228 5.470 5.718 6.237 7.358 5( I) 1.654 1.803 2.230 2 496 2.771 2912 2.983 3.054 3.125 3.197 3.341 3 631 5(0) 1.998 1.99!) 1.999 1.999 2.000 2000 2 01)0 2.000 2 000 2 000 2 000 2 001 5(1) 3.382 2.895 2.324 2.047 1.833 1 744 1.703 1 .064 1.627 1.592 1.528 1.119 5(2) 10.932 7.570 4.506 3.498 2.916 2.694 2.598 2.510 2.248 2.352 2.215 2.002 /-( 9) 0.069 0.289 0.841 1.392 2.107 2 533 2.7<>t> 3.009 3.266 3.536 4.118 5.455 L( I) 0.139 0.047 -0.428 0.743 1.103 1.298 1.399 1.503 1.609 I 712 1.940 2.409 HO) 0.638 0.375 0.016 0.255 0.473 0.575 0.624 0.672 0.719 0.765 0.853 1.017 2.661 1.735 0.814 0.446 0.201 0 1.15 0.075 0.040 0.007 0.023 0.076 0.156 L(2) 17.683 10.786 4.692 3.240 2.659 2.440 2.351 2.273 2.203 2.136 2.016 1.847 37.45 32.83 27.00 23.96 21.48 20.41 19.91 19.45 18.99 18.57 17.76 16.37 a. UFA, (lipole length Table 2b: Nj-Moments of the DOSD" S(/i) (in au) and /,, (in eV) as a Function of Internuclear Distance R(au) 1.500 1.720 1.870 2.020 2.069* 2.120 2.l70r 2.500 S(-2) 8.133 9.208 10.084 11.060 11-391 11.760 12.123 14.695 S(-l) 7.946 8.500 8.888 9.278 9.402 9.536 9.665 10.484 5(0) 13.951 13.960 13.968 13.977 13.980 13.982 13.985 13.994 S{\) 154.712 143.251 141.840 147.303 149.131 150.863 152.274 161.089 L(-2) -1.885 -1.796 -2.427 -3.231 -3.523 -3.859 4.200 - 6.883 I(-l) 1.157 0.685 0.297 -0.135 -0.281 -0.443 -0.602 -1.709 1(0) 17.199 16.482 16.061 15.691 15.582 15.468 15.362 14.764 1(1) 620.365 508.030 498.039 560.495 581.103 600.663 616.635 715.058 /. 93.36 88.61 85.92 83.62 82.95 82.26 81.fi2 78.15 a. RPA, dipole length b. R =s Rtq for isolated molecule c. distance in the solid 443 e 3: Moments" of the DOSD at R\q : 5(M), S(M) in au, In in eV. i this work Zeiss et al.d this work Zeisi et ai.d Kosman et al.c 5(-2) 5.228 5.428 11.391 11.74 20.21 5(-l) 3.054 3.100 9.402 9.484 11.59 5(0) 2.000 2.000c 13.980 14.000 13.98 5(1) 1.664 1.676 149.130 138.0 138.8 5(2) 2.510 3.771 — 12300 15210 £(-2) -3.009 -3.260 -3.523 -3.980 -14.31 £(-1) -1.503 -1.586 -0.281 -0.391 -3.993 £(0) -0.672 -0.691 15.582 15.41 14.59 £(1) 0.040 0.114 581.103 467.5 468.7 £(2) 2.273 8.825 — 76820 121000 19.45 19.26 82.95 81.84 77.27 a. RPA, dipoie length b. H2: Rtq = 1.4011 au, N2: Req = 2.068 au. c. constrained to satisfy the TRK sum rule. d. ref.8 e. ref.9 445 The 12" lernte Brtndt laleroilioail Conlrreact 00 *n» ftnttntioa ot Cfrqtt firtidti Tbroufä Httttr" Evsktl Htrnko Voibtrtimtt*. Su Seins tut. Spin, 4.-7.9.191$ DE STIUCTUM SFICTKOttm OKKIU CLECTKMJOIUIt SECUTOMIOtWI IOKIIS INDUCTOlim «OK'UI «JCTOU» t COBfOIIKJt SOLID« IT8E STRUCTVIE OF 10» ISDUCEB SECCMMH ELECTIM BTIGY SfECTM Flo» SOLID« II FOmiD BlIECTIO«) «arl-Oatjes Croeacveld. token Haler, leraaa« lotbard Institut (Or Kernphysik der Jobass-Volfgasi-Goctne-Universitlt Hujuat-Euler-Str. 6. S-6000 Frankfurt <• Mjio «0. Se P»CS: 79.20.rt. 75.2O.Hc. 34.90.*q SUMNAtlUR 3rcvi CG»prfBfnsioDe pDienoaem »»jssionn cineticie elcctrosiorus »cundtriorua « corporitu» «oliilji icon» mdictit data structuri« pnncipilei s^cctrorus entr?ue elcctronloriB lecundtrionui ex lolm tenuitus protomi« it et prorsu» rectorua. id est la ridn directiooca. deaoaitrutur. »BSTUCT After a short introduction into tbe pbenoaeaoD o( kinetic 10s laduced «econdirr electros ralinoa froa lolad». tkc basic features of proton induced secondary electron spectra troa tbin foil* ID forward-(beaa-J direction ar« ftrictlT discussed. Vtcb einer korim ElnfQbruni] zu» Phanoaei der kiottucteo loneainduzierteD SekuodlrelektroDeoeaiiiloo aus Fcstkorpcra «erden die vesentUchea Stmlctureo in Protosen-induzierten Sekundlrelektronen-Eaergic-SpcktrcB voi dficsei Toll«» n vor»irti- (Strabl-) »lcSfjDg vorbestellt. 2. fUETtLTlO 11. imooumo iisrotici Productio electromorua secundariorua praecipnl aoaentl Exaamatio accioai* »utuae :oJer radiatio coaseguentia actioais autuae radlatioms looizantis cua larua et aatcriaa initiua habuit. cua acno 115»* radii •ateria eat. Idcirco distrlbutlonea anfalarea et enerqiae cathodl a FlOcker atque radii caaalis Uli' ab E. Soli- eiectromorua secuadanorua scire la diversit doctrims stein laveati essest. 1.1. Taoasoa tria anno l(«7* radio« valet velut in biologia radiatioms. ID cbeaia, 10 phy- cathodl electroma. 1. VIM anno noo* radio« caisll« sic« plasaati* 1 nque exaaiaatlne iujionu nudearit loaia esse coaperuit /I/. 'quantua spectat ad actioaea autuaa inter plasaa et pi- rietea adque generatioDea enerqiae theraoauclearlsJ et In Etfectus electroaiorua secundarioraa «aissiom« losi}« •ultls ratiooibus superiicierua exaainaadarua velut 10 isdoctae laa anso 1«99* qjjJea a K.P. villard 4escri»tia> aicroscopta electronica. erac /6/. quea ellectaa ar.no deaua 1905* J.J. Thoaao* /I/. E. lutberford 111. c. ttcMbauer /f/ «aisaioica Blc eaisvio electroniorua secuDdariorua loans inducta electroniorua secundariorua esse recofnoveruat. ?rian* praevalet praecipueque ea protonns inducta (pro veloci- Tcsneider anno 1931* eaissiocea electromorua secunda- '.atibus loniorua aaioribus quaa v,/v.«i- . 0.1. id est rlorua prorsus rectorua tx Join« tenuibus «irperiaestii »» < 2.2-10' ca/s). quoniaa baec aulto (icilics cua tbeo- exaaioavit. rus coaparari potesl quta exeapli gratia eaissio elec- ^roniorua secundariorua electronns inducca /1/. Specia- Studiua eai**ionis electrcniDrua secucdariorua äenuo tia experiaenta de eanssione electroniorua secundanorua excitatua est. cua larriscn et Lucas acno 1*70* cacuae« prorsu« rectorua ex !->liu tenmbus studioa excitans. cua eleclroaiorua coaitaatiaa ill isveaisieat /11/. Exptri- saxiaa pars enerflae asissae loniorua ad prodactioaea »enta de iollis dispcrsioat pur^atis cua »upertltietms electroniorua secundariorua prorsus rectorua cosscaatir accuratl« priaua ex anao 19ii> :n collaborallone sostra nonnullaqne genera productionis tantua in dlrectlone rrancoJarteo»! condar! pcteraal /*. 1J-J»/. radii inveniantur /2-4/. 446 H+(1.2MeV) -*C( 8=0° 0 20 40 60 -energia electroniorumfeVl* * e'secundaria "vera" e" Auger kma KLL corbonis e~ comitontiQ {V« =Vp J e-conflictus binorii (Ve=2Vp) i i 5 10 15 velocitas electroniorum Flf. 1.: Zpectrum encrgitt electronioma ifcundariorua protooilt iaductorua (Er(l') • 1.2 Rev) prariu rector ex tolio ctrbonia (d = 1500 A). Sapra a dcxtri pan aialsat tntrgitt cmidc* ipcetrl aoaatratir. in. sunuE (3) Electronia Augerlaaa KLL caraoaia cnaractcriatle* circua enerqlaa E. > 270 ev 1». > 4-5 IV. FM»*ECTUS Mutjtionw quoqu* exaiiattu dignat suct, qua* proltctl- 11« intrj corpit jolidi» aubeHBt, 14 tjt mutiOMf Supcrliclc* corporli «olldi in aultii applicatloalbui laprini* camcatioai*, unde tunc vcrlalailita* Mtitio- aodarDii aaxiai aoatDti eft. fl*c dc ca*ia iapriai* *oa ni» carricatioai* proicctills coaptnri pot*«t, qu eHectui exaainare oj*ortet. QUI in «uperficie v«l prop* deindt III proprUtatM dettiniri poiiut. qait t< lor- eaa liunt. Ad ld priccipue tnniporUtio tt triBtainio •M ipectrl electranioruB «ecuodirloruB aaflf tx Mrt« electronioruB «ecundariorua per supcrticita corpori* proUctiliiw pertinut. velut cacuaci cltctronierM solidl itque purgttlo dispersona accuratioqut luptrli- coaitiBtlua, cuiu !orM «t iotcnsitit tt id earrlntlo- ciei tpectant /2. 15. 17/. nca «nergiMque proiectillu» /24/ et id purfitiosm »»- p*rtid*i atquc uttrin (olli rcitrrl poitut /25/. riui autM de energiit et jnjuloru» dimibutloni 4t4l( camcatlone earua quoquc ptrticularua eoaparlr* atctft* V. GMTIAE eit, quu dt suptrllcit diiptrguntur. D* f»if /20/ p«r- tiwiu* «tla» de corporibui solidii /21/ exptrlBeata, quat rro IUXIUO «t in txptrlMitii it In lsttrprttatloM hunc atptctua tractant, iaa !acta tunt . Pro Mtu ioll- iuaurnB noslt otlato ouibat colltfU collakoraCloal* di* vero talibuf experiMBtii diflicultatc* oppositat ooatra* gratlai a«tr* «elima, 4111 tuit Xarkai Sckoiilf, sunt. quoni»» superliciei Mtac seaper ccrtaa feoMtriaa Actaia Albcrt. Ttttr Lorcnicg, NDraas ttlltr. Kvrt «ro»*- experiacBto praeponit. Ea de cauta detectio lonioru» d* berfcr. Olivcr 1*11. Christopk Blcd*na». Jtrfta Ktm- supcrficie disperforiui grave problcu i»pont. cua aotl- ler. Oietcr Hofaa». Gratlai pratcipvas agiaas «tiM (la csergiae anguliqu* m»ioni> coutmtnr. Qua d* Erlcb Zanftr. lorit Ducrlij. Vcrntr Tbeiaitffr. Rarco c PUBilCATICWEt ADIIBITHE III J. Scboa 'Secondary Electros Eaiaaioa Iron Sollda by Clectroa and Protoa Boabardeaeat* Scaaaiaa Mcroacoar 1 (19»») 60? Ill K.O. Groeneveld "Xoclear Track Foraatioi lalataa Electron Productioa aad Traniport Froa Ioa renetratioa throaik Sollda" Hucl. Tracka ladiat. Xeaa. 15 (HID 51 III "Forvard Electron Eaiaaioa in loa Colllaiona* I.O. Groaneveld, V. Keckbaca and I.A. Sellia. Ha. Springer Ver- lag. Heidelberg. UP213 (1»4) HI B. Kotbard. K. Kroneberger. N. Bcrkbard, C. Biederaana. J. Keaaler. 0. Hell, (.0. Groenevela "telraclio* ol Di- rected Shock Electron! at Planar Solid Surfacea" J. Phyaique iParia) 50 (19M) C2-10S .'5/ R. Raken. 8.C. Volf "»toa- und Quantenpbyaik" Springer Verlag, Heidelberg, editio V (1M?) p.23 /(/ n.r. Villard "Sur lea Itayona Catnodiques" Journal de Payaique theorlqae et appliqaee { (H»») S HI J.J. Thoaaoa "On tba Poauiva Electrilicatioa of a Kiya and tbe Eaiaaioa ol slovlv Moving Cataote iaya by ta- dio-active Subatancea" Proc. Caabrldge Pali. Soc. Ji (1904) 41 />/ t. Rutherford "Charge Carried by the a and I laya of iadiua" Phil. Hag. 10 (1905) If] /»/ C. FQcbtbauer "Ober tin von ICanalatrafelen erzeugta fcxindaratrahliiig und Sber eint (allcxloa der Kanalatrta- lea" Phya. Zeiticariit 1 (!40i) 153 /iO/ G. Schneider "Aualoiung von Sekundlrelektronen durcb Vaaaeratoftkanalatrahlea la Netallen" Ana. d. Pbya. 5 (H31) 357 448 IWI K.G. Iirnion. U.K. Lucas "Secondary Electron Spectra froa roll* under Llgbt-ion Boabardeaent" Pliys. Leu. 3JU (1110) 142 /12/ ». Loll. K. Burkhard. P. Koackar. J. Keaaler. I. lotlard. C. Blederaana. D. Boteann. K.O. Oroeneveld "CUnhifb Vacuea Analyzer (or Convoy Electron Neasireaeats" Kiel. Inttrua. Ketb. A24S <19Bt) HO /13/ N. Burkhard. I. lothard. C. Bladcraaia, J. (easier. I. Jroeeberger. P. Itoachar, 0. flcil. K.O. Groeneveia "Hea- vy-Ioa-Indactd Shock Electron! froa Sputter-Cleaned tmlii Surttcf- Thya. Kev. Lett, it (19«7) 1173 /It/ «. «ot»ar«, n. Burkkard. C. Bicdenaiii. J. Keaalir, t. Xroatber«er. T. Koachar, 0. 8etl, S. Katmtna. (.0. Sroe- nt»tld "Deptodenc* o( Coivoy Electros Ea>g«lo( o* Sttrlace Propertu* and Target Raterwl* 1. Pfir»i<]«« /IS/ It. lurkkard, H. dothard. J. Xe*Bl*r, «:. Kroneberger. K.O. Croencveld "Surface Chartctensatioa a! Tblo Solid foil Tarftti by Ion Icpact" J. rhyi. D£l (1)11) 412 m/ H.r. BurWurd. I. (oltiard, 1.0. Croeaeveld "Slnjlt-Electroa Deexcitatlon of Voluae Plasaoni Induced by Beavy Ion* in This Solid Foils" phjdct status «olid» (b) 1£7 (l»t) %<1 IVtl I. lotbard. H. Burkbard, C. Biedcruia. J. teaalcr. T. Koscsar, I. Kronebetfer. 0. Bui. D- Bofunn. t.O. t;roe- neveld "The Inlluence of Solid Surtacti on Lev-Energy Convoy Electron Eaissios" J. fbys. C21 (13tt) 5031 III! H. lotbard, X. Eroneberger. H. ScbOfnuj. r. lorenzei. E. Veje, *. Keller, t. Haier. J. tewler. C. Siederaasa. A. Albert, 0. lell, 1.0. Croeneteld "Secondary tlectroa Velocity Spectra and Angular Distributions (torn Ions reMtratln Tkli Solids" Iwl. Iistr. Ustk. | (1»«) aiblicabitir Ml I. »ot»»r4. I. Kroieberger. f. Lortme, K. Tobiassei. C. lledtrsau, «. Naier. K.O. Croeneirtld "Ultrabocbv*- lro«M|)Mt*tir ait Targetktbluif zar Stkandlrtlektroneaspcktroskopie" Jabresbcrlcbt dex Instltuts far Kerapbyslk dtr J.V.- Ooethe-UaiveriiUt IP-41 U»ll) II not J. Ullrlck, I. Scbaldt-Bocklif, C. K«lbcb "Dcterainatlcm ot very saall Prolectlle Scattering Angles ty Pleasu- ring lecoU-ioa transverse Velocities" Xscl. Instrua. Heth. Alt! (l»l») 21* n\l r. liffi, k.H. Spiia "Mass Snectroaetrlc Iavestlgitioa ot Desorbetf loss froa a Vallne Staple by ">C! Fiiaion Fragaests" II Noovo Ciaeito BU <1»»») 101 122/ I.C. Vllcy, I.I. HcUre* "Tlae-ot-Flight Spectroaater vitk laproved lesolntion" kev. Sci. Initr. 24 (J»55) 3350 1111 M- Ulltr. 0. Reil, J. (eaaler, X. Kller. I. Hothird. t. Lorenzei, S. Lencinas. K. (roneberger, M.«. Lucas. I.A. Sellin. K.O. Croeaeveld "Oatersicbntea zar Erzeamsf von Cssp-ElectTon«n aittels Koinzidenz ait Mckstot- icnea" Janresbericht des Instituts (Or Kernphysik der J.f.-Goethe-Universitlt iff-48 (1911) 3t /HI J. Keaaler, 0. Bell. p. Kosckar. C. Biederaaaa, t, Iotk»r«. «. (roneberger. * Plecicbacn. (.0. Croeceveld "Eflui- libriua Convoy Electron Production and Charge Exchange* 1. Phya. BU H9B9). tl Hi/ t. Loreszea. R. lotkard. K. Kronebergtr. J. Keaalcr, n. Burkhard. K.O. Groeneveld "Target naracterlzation by fast loa Isaaet" Nucl. Imtr. Ketn. A (11111 pabllcabitar 449 Orbital Local Plasma Calculation of Mean Excitation Energies and Stopping Numbers S.B. Trickey, David E. Meltzer, and John R. Sabin Quantum Theory Project Depts. of Physics and of Chemistry University of Florida Gainesville, Florida 32611 T.F. 1114 12th Werner Brandt Workshop 450 ABSTRACT The extension of the Oddershede-Sabin orbitally-decomposed form of the kinetic theory of stopping from gas phase to films and crystals necessitates the generation of orbital mean excitation energies calculated within the local spin density approximation (LSDA) to density functional theory (DFT), the preeminent theoretical model for such extended systems. In LSDA, the Kohn-Sham (KS) orbitals and orbital eigenvalues are simply artifacts for the construction of the electron density p. They have no rigorous validity for estimation or calculation of excitation energies. As an alternative we present an orbital-density or local-density-of states generalization of the Lindhard-Scharff Local Plasma Approximation (LPA). We test the scheme via systematic study of mean excitation energies /„/ for each atomic central field orbital for all atoms with Z < 37. Stopping cross-sections for a representative sample (about half) of these elements have been calculated using these /* values. We report four which illustrate our findings: O, Al. Ar, and Ca. The results do not differ substantially from those of Oddershede and Sabin (who used /„/ values obtained from work of Inokuti and co-workers), with the discrepancy generally less than 15%. Comparison of the LPA and OS orbital mean excitation energies exhibits a number of striking, but difficult to interpret systematic features. Although the two methods give very different IJs for core orbitals, there is little effect on the resulting stopping cross sections because the valence orbital /„/ predominates at lower projectile energies and the inverse square velocity dependence of the stopping overwhelms moderate differences in I^'s at high velocities. 451 1. Motivation: Stopping and Quantum Size Effects Because ordered, unsupported ultra-thin films (u = 1,2,3,4... atomic planes) are the obvious limiting case of overlayers in nanoscale technologies, we have undertaken the a priori prediction of their structure and properties. Our focus is on proton stopping by a film at its calculated equilibrium geometry. The physical distinction between the ultra-thin film and its counterpart bulk crystal is evident. The film has crystalline periodicity and length scales in two dimensions but thickness of roughly molecular scale in the third. An immediate consequence of such multiple, atomic-level length scales is the existence of quantum interference effects. These occur in both a slab of homogeneous electron gas1-2 and in real films, where thickness is a discrete parameter (the layer number).3 It is easy to show that quantum size effects (QSE) will occur in the stopping number Uv) (with v = projectile velocity) of a film.4 The key parameters are (p. the Fermi energy and W, the occupied bandwidth. Both are functions of v. Their central role can be seen in the kinetic formulation of stopping.5 In it one has6 L(v)tx fdv'J(v')f(v',v) with J(v) the Compton profile andftv'.v) a somewhat intricate function which involves the Bethe logarithm 2 LB(v) = ln[(2mv )/l] /, the mean excitation energy, conventionally is taken to be a material-dependent parameter which characterizes the target in its entirety. Clearly / is dependent on the occupied-state energetics of the target electrons. These are characterized, at the least level of detail which is meaningful, by €? and W. Since those parameters are t/-dependent, in general / will be i/-dependent also. Furthermore, the atomic stopping studies of Sabin and Oddershede7 have shown that a high-quality stopping calculation must distinguish among the energies of core, semicore, and valence states. The distinction occurs first in the mean excitation energy /, which for an atom must be replaced by a set of orbital values /„/. Secondly, the distinction appears in the momentum density which defines the Compton profile. Clearly these distinctions, which are based on chemical differences, will carry over to films and solids. Matters are made more interesting by the fact that both f F and W are functions of nearest- 7 9 neighbor spacing am as well. ' For both monolayers and solids a reasonable approximation is 2 W(v.ann) -x ann(u)~ . In this expression we have made explicit the fact that ultra-thin film am's typically depend 8 10 strongly on u and differ substantially from am for their counterpart crystals. * The challenge which motivates this work can now be stated. With the exception of a few limited Hanrce-Fock calculations, the state of the an for predictive calculations of i/-layers and 452 crystal properties is density functional theory (DFT) in the local spin-density approximation (LSDA).11 This fact poses several problems. 1. The one-electron eigenvalues generated in the LSDA by solution of the Kohn-Sham (KS) equations do not have a rigorous interpretation as one-electron excitation energies.12 with one exception, the calculated e?.13 2. Similarly, the KS orbitals are not rigorously interpreuble as one-electron wavtfunctions, but only as artifacts for die construction of die electron density. 3. While LSDA generally provides a remarkably good account of am values, the pressure- dependent ordering of crystal phases, bulk moduli, and similar ground state bulk properties, 14 it is known empirically that the calculated am's suffer from systematic contraction with respect to experiment. A corresponding systematic increase in W(v#m) can be expected. These facts show that the LSDA energy bands of solids or {/-layers must be used with great caution when it comes to predicting or interpreting excitation energies. LSDA eigenvalues for insulator and semiconductor bandgaps, for example, provide estimates which are drastically undersized relative to experimental values.13 Although the Sabin-Oddershede7 calculations are a major step forward in incorporating chemical effects, they also illustrate a more complicated version of the challenge. They utilised oscillator strength moments calculated by Inolcuti et al16 from matrix elements derived in the Hartree-Fock approximation into which were inserted KS orbitals and eigenvalues from an LSDA model which, for purely historical reasons is called Hartree-Fock-Slater (HFS). By now it is well understood that the HFS model is not a reliable approximation to restricted Hartree-Fock. Rather it is a very simple LSDA model which has been artificially parameterized to give a rough approximation to atomic Hartree-Fock orbital eigenvalues. Substantial numerical exploration has shown that the parameterization fails to describe crystal and film properties and. further that the orbitals themselves are not faithful replicas of authentic Hartree-Fock orbitals. These restrictions are specific examples of the general problem that LSDA (i.e. KS) orbitals cannot be used with any rigor as if they were HF orbitals. Resolution of the dilemma by constructing a proper theory that involves direct use of die LSDA eigenvalues and orbitals seems quite difficult; we know of little progress along those lines. An alternative is to calculate the Ink (and corresponding energies in the ultra-thin film) directly from the density, die central quantity in DFT. The simplest version of this idea is the local plasma approximation, our topic for the remainder. 2. Local Plasma Approximations In the original form due to Lindhard and Scharff,17 the local plasma approximation (LPA) reads 3 In I = i jd rp{r)ln[x^r{r)} with the local plasma frequency given by 453 (The parameter \ is an ad hoc correction factor which Lindhard and Scharff argued should be \/2. Ziegler18 has studied the empirical systematics of x in elemental crystals. For reasons which we shall detail elsewhere,19 the most advisable choice in the present cue is x — 1.) The LPA has been subjected to much scrutiny and no little criticism.20 Nevertheless the LPA has two enormously appealing features with regard to the challenge of working from inside the LSDA: 1. It depends on the central quantity in the theory, die density; 2. It is a local density approximation, that is, the physical quantity is calculated from the homogeneous electron gas density equivalent at the point in question. The most obvious deficiency of the LPA is that it lumps core, semi-core, and valence slectron response into a single number. An evident response is to construct an energy-selected LPA, that is, one in which the excitation energy of all the electrons associated with a specified energy 6 (below er or the highest occupied state) is estimated by applying the LPA to the density associated with those electrons. For the simplest test case, the central field atom, energy sorting has a one-to-one correspon- dence with KS orbital labels. Thus we can write an Orbital LPA (OLPA) as In Inl = -L A, | dr r> Pnl{r) In ^ o The piefactor .4; comes from angular integrations and has the values 1.0, 1.2411, and 1.2775 for / = 0, 1, 2 respectively. With the OLPA /'s in hand, it is possible to proceed almost exactly as in Ref. 7. One distinction which requires attention is the weight factor wn/ which appears in Ref. 7's use of the rest frame Bethe stopping number 2 7 (Here an, = (7n//2m)^ is a kinematic cutoff.) Oddershede and Sabin used "ni + *nl (0) Unl = 2 with the snt (0) related to the atomic oscillator strengths /„/ by nn//nl = nl n/ with values as calculated by Inokuti and co-workers.21 Both the orbital occupation numbers and the weights must satisfy the sunn rule ""1 = z nl nl 454 with Z the atomic number of the target material. Elsewhere19 we present the argument that the most economical consistent choice of weights for the OLPA is the choice used here: The essence of the argument is that the occupation numbers are the weight factors for the KS orbitals in the density itself, hence should also be the weight factors in a density-based calculation of the mean excitation energy. In the energy-selected theory there is no intrinsic definition for a "total" mean excitation energy / in terms of orbital mean excitation energies /„/ but an obvious choice exists, namely that of Ref. 7: . . 1 v* , _ u ln In I = —2 y. nl *«l 1A Alternatively one may consider the original LPA total llFA, which uses the whole-atom p (r), as studied by Chu and Powers.22 For direct comparison with the results of Chu and Powers, we have carried through such a calculation, including the use of the X<* LSDA with a=l, as used in Ref. 22. The other subtlety in grafting the OLPA into the energy-selected version of the kinetic theory of stopping involves proper determination of the velocity distribution of the electrons for each KS orbital, p*i (v). The dominant contribution arises from the Fourier transforms of die KS eigenfunctions. However, unlike Hartree-Fock theory, there is a correction term.23 The woric of Tong and Lam24 suggests that its magnitude is quite small. In the present context the correction terms calculated in Ref. 24 for Ar and Ne would make a negligible contribution to the stopping cross sections. Hence they have been omitted. We solved the Kohn-Sham equations for the central field atom via Boettger's25 linear combination of Gaussian orbitals code. We have used the Hedin-Lundqvist exchange-correlation model with the Moruzzi, Janak, Williams parametrization26 throughout Radial integrations were done on a 401 point version of the the familiar Herman-Skillman27 doubling mesh. 3. Results Table 1 gives results for the Jn; for all atoms with Z < 37. For comparison Table I also gives /„( values from Oddershede and Sabin as they derived them from Inokuti et al. Total /'s obtained via the two methods described in the preceding section, as weU as those from Oddershede and Sabin, are shown in Table 2. For comparison, empirical values2* are also shown. In general, empirical values of / obtained from experiments with bulk solids would be expected to show significant discrepancies29 from values calculated on isolated atoms. The ratio of the OLPA IjfA to the Oddershede-Sabin (from Inokuti et al.; recall above) " as a function of Z is particularly interesting for the outermost KS orbital. Fig. 1 shows this ratio for the highest energy orbital except tor Z = 21,22,23,24. For those elements, the 4s KS orbital lies on the overall trend line, while the actual highest energy occupied orbital, the 3d, lies far above it (as shown in the dotted line in Fig. 1). The ratio for the highest energy orbitals 455 ranges mostly between 1.2S and 0.75, while for the deeper KS orbitals it ranges up to a factor of three or greater. Also evident in Fig. 1 are clear correspondences with valence shell occupation. We have tested whether the weights «„/ used by Sabin and Oddershede alter these trends. The test is to recalculate the orbital /n/'s from the oscillator strength moment data of Inokuti et al., but with u>ni = "ni. instead of the Oddershede-Sabin choice discussed above. These air reported in Table I as well, where they are denoted I'J1. There is little qualitative change in A : PA the trends just summarized, but the values of I™Jiff are significantly lower than Iff /I^{ for the inner orbitals. M Of course these /fj/ would not be under study were it not for the motivation of calculating stopping in the LSDA. An appropriate test of these OLPA mean excitation energies therefore is to calculate the atomic stopping cross sections just as did Oddershede and Sabin but with these I's. Figures 2-5 shew the proton stopping cross section as a function cf projectile velocity (Hartree au) for O, Al, Ar, and Ca. The selected proton energy range is roughly the region of the peak ( Le., « 100 XeV) to the vicinity of 10 MeV (i.e., 20.008 atomic units). Above that energy, even large variations in either the total or the orbital mean excitation energies make nearly negligible differences in the resulting stopping cross sections because of the v'2 dependence of the stopping. Those figures show results from the OLPA, from Oddershede and Sabin, and for comparison, the fits to empirical data given by Janni.28 The two calculations are, generally speaking, in good agreement. Except for the region near the peak of the curve (where the low-velocity corrections ignored in both calculations become very significant), the agreement is typically within 15%, with the OLPA stopping cross sections slightly above the Oddershede and Sabin results. Both calculations agree well with the empirically based results of Janni, again with the exception of the low-energy regime. There, however, both Barkas and Bloch corrections (omitted here) are large.30 Effects cf the target phase ("solid-state" effects) also could be significant in comparison with some of Janni's fitted data, except for those systems (such as N2, O2, and the rare gases) in which gas phase measurements are available. It is useful to consider why the OLPA and Oddershede-Sabin stopping cross sections are in such good agreement in the face of their quite different results for several of the orbital mean excitation energies. The obvious qualitative answer is that, in the energy range in question, the outer orbitals contribute far more to the stopping than do the inner orbitals. Since the two calculations agree substantially on the /„/ values for the valence orbital, the respective stopping cross sections which they produce do not differ greatly. A key aspect of this predominance is that the contribution of a particular orbital to the stopping cross section is proportional to the occupation of that orbital. For the elements exanrned, it is usually the case that the outermost orbital has an equal or greater occupation than most of the other orbitals in the atom. Elsewhere15 we treat these qualitative considerations analytically. 4. Concluding Remarks Several previous stopping calculations have used diverse decompositions into separate orbital contributions. A few of these have been based in some way upon the LPA also. Thus Pathak31 has described s scheme for finding the .-topping of channeled light ions in which the stopping 456 has contributions from valence (treating separately both single-panicle excitations, and collective "plasma-like" excitations), conduction, and core electrons. Tung and Watt32 have decomposed stopping contributions into inner-shell ones (for which the LPA is used), and valence contributions (for which a dielectric response function is employed). They used a Hartrce-Fock electron density as input to the LPA. Whether this procedure can be rationalized with respect to known results for the DFT-uSDA dielectric function (e.g. Hybensen and Louie33) is unclear but seems unlikely in view of the mixing of HF and LSDA quantities. Kwei, Lin and Tung34 have used the LPA to extract values for die generalized oscillator strength via the relation between the oscillator strength and the electronic charge density implicit in the LPA. In that work, sub-shells are treated separately, and account is taken of the binding-energy excitation threshold of the individual subshells. Although the differences between the OLPA mean excitation energies and those obtained from the matrix-element calculation of Inokuti, et al., are large for the inner orbitals, the benefits of the orbitally decomposed method of calculating die stopping cross section are retained nevertheless. The overall stopping curves whivh result from the use of the OLPA are closer to those found by Oddershede and Sabin, and to empirical fits, man stopping curves which are produced by the use of a "single-/" approximation to the atomic mean excitation energy. Therefore the OLPA appears to be a promising method of calculating stopping cross sections for films and solids, in which more rigorous approaches to stopping are difficult or impossible to implement. 5. Acknowledgements We have benefited from helpful discussions with J. Oiidershede. This work was supported in pan by the U.S. Army Office of Research under contract DAA L03-87-K-0046. 6. References 1. F.K. Schuite, Surf. Sci. 55 (1976) 427. 2. E.E. Mola and J.L. Vicente, J. Chem. Phys. 84 (1986) 2876. 3. (a)P. Feibelman and D.R. Hamann, Phys. Rev. B 29 (1984) 6463; (b)I.P.Batra, S. Ciraci, G.P. Srivastava, J.S. Nelson, and C.Y. Feng, Phys. Rev. B 34 (1986) 8246 4. S.B. Trickey, D.E. Meltzer, and J.R. Sabin, Nuci. Inst Meths. B 40/41 (1989) 321 5. P. Sigmund, Phys Rev. A 26 (1982) 2497. 6. J.R. Sabin and J. Oddershede, unpublished, see also Phys. Rev. A 35 (1987) 3283. 7. J.R. Sabin and J. Oddershede, Phys. Rev. A 26 (1982) 3209; AL Data and Nuc. Data Tab. 31 (984) 275. 8. W.A. Harrison, "Electronic Structure and the Properties of Solids" (Freeman, San Francisco. 1980), pp 149-51. 9. J.C Boettger and S.B. Trickey, J.Phys. F 16 (1986) 693; J.Phys.: Cond. Matt 1 (1989) 4323 10. LP. Batra, J. Vac. ScL Tech. A3 (1985) 1603. 11. E. Wimmer, Surf. Sci. 134 (1983) L487. 12. Reviews and references to the original literature are in "Theory of the Inhomogeneous fciectron Gas", eds., N.H. March and S. Lundqvist (Plenum, NY, 1983); "Local Density Approximations in Chemistry and Solid State Physics", eds.. J.P. Dahl and J. Avery (Plenum, NY. 1984); "Density Functional Theory of Many-Fermion Systems", S.B. Trickey ed. (Academic. San Diego, 1989) 13. J.P. Perdew, Internal J. Quantum Chem. S19 (1986) 497; W. Pickett, Comments on Solid State Phys. 1? (1986) 57; S.B. Trickey, Phys. Rev. Lett 56 (1986) 881 and references in each. 14. M. Levy, J.P. Perdew, and V. Sahni, Phys. Rev. A 30 (1984) 2745. 15. H.J.F. Jansen, K.B. Hathaway, and A.J. Freeman. Ph;-s. Rev. B 30 (1984) 6177; J.C. Boettger and S.B. Trickey, Phys. Rev. B 32 (1985) 3391. 16. J.L. Dehmer, M. Inokuti, and R.P. Saxson, Phys. Rev. A 12 (1975) 102; M. Inokuti. T. Baer, and J.L. Dehmer. Phys. Rev. A 17 (1978) 1229. 17. J. Lindhard and M. Scharff. K. Dan. Vidensk. Sels, Mat. Fys. Medd. 27 (1953) 15. 18. J.F. Ziegler, Nucl. Inst. Meths. 168 (1980) 17. 19. D.E. Meltzer, J.R. Sabin, and S.B. Trickey, Phys. Rev. A (submitted) 20. R.E. Johnson and M. Inokuti, Comments At. Mol. Phys. 14 (1983) 19; E. Kamaratos, Chem. Rev. 84 (1984) 561. 21. J. L. Dehmer, M. Inokuti, and R. P. Saxon, Phys. Rev. A 12 (1975) 102 ; M. Inokuti. T. Baer, and J. L. Dehmer, Phys. Rev. A 17 (1978) 1229 ; M. Inokuti, J. L. Dehmer, T. Baer, and J. D. Hanson, Phys. Rev. A 23 (1981) 95 22. W. K. Chu and D. Powers, Phys. Lett. 40A (1972) 23. 23. L. Lam and P. M. Platzman, Phys. Rev. B 9 (1974) 5122, Phys. Rev. B B 9 (1974) 5128; Gerrit E. W. Bauer, Phys. Rev. B 27 (1983) 5912. 24. B. Y. Tong and L. Lam, Phys. Rev. A 18 (1978) 552 . 25. J. C. Boettger (unpublished). 26. V.L. Moruzzi, J.F. Janak, and A.R. WiIliams,"Calculated Electronic Properties of Metals" (Pergamon, NY, 1978). 27. F. Herman and S. Skillman, "Atomic Structure Calculations" (Prentice-Hall, Englewood Cliffs, NJ, 1963). 28. J. F. Janni, At. Data Nucl. Data Tables 27 (1982) 341 . 29. For example, J. R. Sabin and J. Oddershede, Phys. Rev. A 39 (1989) 1033. 30. F. Bloch, Ann. Phys. 16 (1933) 285; L. H. Andersen, et al., Phys. Rev. Lett. 62 (1989) 1731 and references therein; H. H. Mikkelsen and P. Sigmund, Phys. Rev. A 40 (1989) 101 and G. Gabrielse et al., Phys. Rev. A 40 (1989) 481. 31. A. P. Pathak, Phys. Status Solidi (b) 71 (1975) K35 . 32. C J. Tung and D. E. Watt, Radiation Effects 90 (1985) 177. 33. MS. Hybertsen and S.G. Louie, Phys. Rev. B 35 (1987) 5585. 34. CM. Kwei, TX. Lin, and CJ. Tung, J.Phys. B 21 (1988) 2901. 458 7. Table Captions TABU' I. Orbital mean excitation energies for all atoms through Z * 36. Present results: llfA . Ref. ': Ps. /w are obtained in the same way as I05 except that the weights u^ are set equal to the orbital occupation numbers «„/. (see text). TABLE 2. Total Mean Excitation Energies. The values in the first column are obtained by employing the Hedin-Lundqvist LSOA total atomic charge density. The values in the second coluvin are obtained in the same way, but with the Xa (a-1) LSDA. The values in the third column are obtained from the OLPA orbital mean excitation energies (first column of Table I) as discussed in the text The values in the fourth column Uos) are the results of Ref. 7. Ths values in the Anal column are from Ref. 28; the asterisks for H, N and O indicate that ths empirical results are for diatomic gases as opposed to the calculated values which are for isolated atoms. 8. Figure Captions Figure 1. Values of the ratio l°s lfiJ>A for all atoms up through Z - 36. for the "outermost" (highest energy) orbital. (For Z = 21, 22, 23, 24, the values for the actual highest energy orbital (3d) are indicated by the dotted line and crosses; the solid line and circles correspond instead to the values for the 4s orbital.) Figure 2. The stopping cross section for Oxygen. This work: solid line (—); Ref. 7: dotted line (....); empirical fits (Rcf. 28): dashed line ( ). Note that the empirical data correspond to O2, in contrast to the calculated values, which are for monatornic O. Figure 3. As in Figure 2, for Al Figure 4. As in Figure 2, for Ar. Figure 5. As in Figure 2, for Ca. 459 TABLE 1 s 1 z Orbital | I1-** (eV) I I° (eV) I Iw (eV) j 1 Is | 11.252 | 14.99 | 14.99 2 Is | 33.661 38.83 38.83 3 Is 69.733 109.32 109.32 2s 3.1747 3.29 3.29 Is 113.91 203.78 203.78 2s 7.8089 7.32 7.32 Is 16*39 320.21 329.96 2s 12.137 16.33 16.10 2p 8.0719 11.55 11.30 Is 220.78 451.34 417.14 2s 16.926 27.57 25.95 2p 17.192 20.97 21.98 Is 283.07 590.00 512.22 2s 21.822 41.24 36.08 2p 27.590 32.68 35.84 460 Orbital (cV) (cV) (eV) Is 349.72 729.41 591.81 2s 27.331 56.86 45.62 2p 40.822 46.64 52.97 Is 421.58 861.33 648.29 2s 32.800 74.04 53.98 2p 54.778 62.86 73.48 10 Is 497.75 982.68 684.46 2s 38.347 92.22 60.71 2p 70.810 81.37 97.48 11 Is 577.82 1110.36 729.27 2s 47.119 119.24 78.34 2p 93.513 124.41 152.84 3s 2.9770 2.46 2.37 12 Is 661.87 1243.15 773.82 2s 56.769 151.05 99.24 2P 118.75 169.86 210.60 3s 6.0801 4.45 4.33 461 s Orbital I° (eV) Im(eV) 13 Is 749.74 1373.04 813.26 2s 66.869 187.14 122.55 2p 145.18 221.15 277.45 3s 8.3960 9.01 9.18 3p 4.4843 4.85 4.14 14 Is 841.22 1497.54 847.88 2s 77.390 226.08 147.07 2p 172.81 278.63 353.69 3s 10.753 14.56 14.45 3p 9.0573 8.87 8.32 15 Is 936.24 1618.33 875.75 2s 88.561 266.84 171.01 2p 201.65 342.50 439.31 3s 13.151 21.30 19.70 3p 14.244 13.71 13.73 16 Is 1034.7 1733.73 901.09 2s 100.29 308.23 193.92 2p 231.89 412.71 535.51 3s 15.569 29.40 24.63 3p 20.061 19.37 20.37 462 Orbital ILPA(eV) I°s(eV) 17 Is 1136.5 1844.43 920.79 2s 112.49 349.09 214.21 2p 263.31 489.36 640.98 3s 17.994 38.91 28.92 3p 26.461 25.76 29.16 18 Is 1241.21 1948.72 936.98 2s 125.13 388.29. 231.36 2p 295.87 572.56 756.99 3s 20.531 49.93 32.45 3p 33.167 32.95 37.18 19 Is 1349.55 2055.32 953.82 2s 138.35 429.39 249.31 2p 329.70 662.36 881.29 3s 23.945 60.43 38.23 3p 42.135 48.78 57.17 4s 2.2748 1.60 1.44 20 Is 1460.6 2161.01 972.76 2s 151.92 472.33 268.24 2p 364.80 759.01 1014.59 3s 27.665 72.49 44.84 3p 50.694 64.83 76.96 4s 4.3528 2.70 2.46 463 Orbital ILPA (eV) I°s (eV) I01 21 Is 1574.7 2262.31 993.10 2s 165.93 519.23 291.52 2p 401.10 847.49 1104.02 3s 30.843 82.66 50.19 3p 57.642 77.32 87.83 3d 13.038 42.69 57.34 4s 4.9102 3.15 2.90 22 Is 1691.7 2433.00 1036.61 2s 180.34 582.70 322.44 2p 438.50 933.14 1173.30 3s 33.669 91.65 55.02 3p 62.954 90.43 97.30 3d 19.248 56.31 81.29 4s 3.5209 3.51 3.27 23 Is 1811.5 2539.60 1060.47 2s 195.11 625.00 343.36 2p 476.86 1012.46 1221.23 3s 36.843 102.22 60.51 3p 69.681 103.49 104.69 3d 27.394 69.43 105.72 4s 3.7594 3.79 3.56 464 s Orbital lLPA(eV) I° (eV) 24 Is 1934.1 2652.59 1082.37 2s 210.25 692.84 377 50 2p 516.23 1081.66 5221.76 3s 40.056 109.28 63.91 3p 76.487 112.68 101.27 3d 36.215 65.38 96.7! 4s 4.0038 3.23 2.99 25 Is 2059.2 2811.39 1126.84 2s 225.75 746.73 400.57 2p 55650 1159.55 1265.49 3s 43.621 122.96 71.31 ^P 84.672 129.90 166.43 3d 52.516 99.38 165.54 4s 6.4937 4.57 4.35 26 Is 2187.1 2881.38 1134.73 2s 241.68 792.34 422.85 2p 598.30 1224.75 1265.84 3s 46.774 138.00 78.68 3p 91.176 140.53 114.78 3d 58.954 113.89 196.00 4s 5.8341 4.92 4.71 465 Orbital I°s(eV) 27 Is 2317.6 3023.68 1168.13 2s 258.17 875.16 461.85 2p 641.12 1263.85 1230.88 3s 49.967 148.13 83.92 3p 97.488 153.46 115.95 3d 65.997 132.33 236.21 4s 4.7590 5.17 4.97 28 Is 2450.8 3227.87 1222.69 2s 274.81 906.75 476.92 2p 684.71 1315.15 1204.83 3s 53.292 158.33 89.27 3p 104.37 163.22 114.03 3d 75.091 149.45 274.62 4s 4.6213 5.50 5.31 j 29 Is 2586.0 3382.26 1215.74 2s 291.90 1010.46 512.70 2p 729.22 1336.95 1099.27 3s 56.431 165.20 91.08 3p 111.18 165.69 99.25 3d 90.745 144.29 254.55 4s 4.5007 5.52 4.43 466 Orbital ILPA(eV) I°s(eV) 30 Is 2723.8 3449.96 1266.69 2s 309.15 1089.16 560.70 2p 774.23 1372.04 1107.12 3s 60.538 179.44 100.04 3p 120.25 181.11 107.92 3d 105.64 188.44 366.02 4s 7.7905 6.19 6.02 31 Is 2864.1 3642.55 1313.7.6 2s 326.78 1100.06 562.98 2p 819.73 1416.84 1088.06 3s 64.660 206.97 113.41 3p 129.53 199.95 116.81 3d 125.28 240.33 492.99 4s 9.3283 9.67 9.79 4p 4.7414 4.84 4.00 32 Is 3006.5 3650.25 1300.89 2s 344.69 1185.35 594.42 2p 866.93 1434.24 1053.02 3s 68.851 214.33 117.83 3p 139.49 225.58 134.24 3d 147.10 295.59 630.17 4s 10.889 13.91 13.88 4p 8.7720 8.13 7.37 467 LPA s Orbital I feV) 1 t° (eV) I" (eV) 33 Is 3151.3 3852.74 1347.98 2s 362.88 1202.15 594.83 2p 914.58 1484.64 1044.18 3s 73.568 237.50 128.61 3p 149.93 249.84 149.99 3d 163.94 353.88 776.34 4s 12.520 18.60 17.56 4p 23.096 11.79 11.47 34 Is 3298.4 4022.64 1380.38 2s 381.43 1286.55 622.73 2p 963.06 1541.54 1041.00 3s 78.238 264.87 140.74 3p 160.83 277.32 168.04 3d 183.80 415.85 932.72 4s 14.138 24.07 20.81 4p 17.429 15.21 15.54 468 Orbital TLPA (eV) l°s(eV) 35 Is 3447.6 4150.43 1403.78 2s 400.28 1282.35 612.61 2p 1012.4 1545.54 1007.89 3s 83.122 277.06 146.03 3p 172.06 31534 192.41 3d 203.67 477.69 1088.72 4s 15.722 30.77 23.63 4P 22.098 19.72 21.16 36 Is 3599.1 4229.23 1409.08 2s 419.46 1353.15 632.03 2p 1062.5 1588.54 1000.29 3s 88.121 290.12 150.96 3p 183.57 344.29 212.97 3d 224.99 564.77 1272.43 4s 17.270 38.22 25.55 4p 27.068 23.81 26.67 469 TABLE 2 TOTAL I z ILPA ILPA ILPA Ios(eV) IEMHRTCAL [calculated [calculated [calculated (eV) from total from total from charge density, using logarithmic density] (eV) Xa with sum of a=l] (eV) orbital I's] (eV) I 11.3 13.9 11.3 14.99 20.40* 2 33.7 38.4 33.7 38.83 39.10 3 25.9 29.2 24.9 34.00 57.20 4 31.4 34.4 29.8 38.62 65.20 5 38.8 43.4 31.7 50.22 70.30 6 49.7 55.7 40.0 61.95 73.80 7 61.7 68.3 50.2 76.79 97.80* 8 76.3 84.7 63.2 93.28 115.7* 9 91.2 100.1 76.9 111.31 124.8 10 107.6 116.3 92.5 130.94 1318 11 99.2 106.9 84.0 123.14 143.0 12 101.1 108.2 85.2 120.74 151.1 13 102.1 109.5 81.9 123.67 160.1 14 107.5 115.2 85.2 131.04 174.5 15 114.5 122.6 90.7 140.34 179.1 16 122.5 130.8 97.4 151.26 183.6 17 131.1 139.5 105.0 162.87 182.6 18 140.0 148.4 113.0 175.35 181.6 470 TOTAL I Z ILPA ILPA Iu>A Ios 19 133.8 140.8 106.4 168.20 186.8 20 133.6 140.6 105.4 163.52 191.9 21 141.9 149.9 105.6 171.63 214.5 22 154.6 167.6 103.5 182.05 228.0 23 164.9 174.7 108.6 191.70 237.6 24 175.9 190.8 125.9 211.19 315.8 25 183.5 194.6 130.4 215.75 268.8 26 196.4 209.6 139,6 226.42 278.2 27 210.3 224.1 151.2 239.56 295.3 28 222.2 236.0 160.8 251.64 302.3 29 237.4 247.3 173.6 268.50 323.3 30 241.8 255.4 174.4 278.84 323.1 31 243.3 256.2 172.2 283.42 301.8 32 249.1 260.1 175.8 290.81 280,6 33 254.5 266.4 179.3 300.39 290.6 34 261.2 273.1 184.4 310.38 300.9 35 268.4 280.2 190.2 319.73 325.1 36 276.1 287.6 196.7 329.59 340.8 471 v 1 i 1 15 -A % O / - 10 6 CO 5 \ n i i i ^—_i 0 5 10 15 20 . 1 ...v (atomi. 1c ...units) . 1 .... FIG. 2 Aluminum Stopping 0 5 10 15 v (atomic units) FIG. 3 Argon Stopping "I 6 CO 5 10 15 20 v (atomic units) FIG. Calcium Stopping 100 o 6 (.n CO o 5 10 15 v (atomic units) FIG. 5 477 The 12l* Werner Brandt International Conference on "The Penetration of Charged Particles Through Matter" Euskal Herriko Unibertstatea, San Sebastian, Spain, 4.-7.9.1989 Secondary Electrons as Probe of Preequilibrium Stopping Power of long Penetrating Solids K. Kroneberger, H. Rothard, P. Koschar, P. Lorenzen. A. Clouvas". E. Ve]eb, J. Kemmlerc , K.O. Groeneveld. Institut fur Kernphysik der J.w. Goethe Universitat, D6000 Frankfurt/Mam 90, Germany The passage of ions through solid media is accompanied by the emission of low energy secondary electrons Z1 /. At high ion 7 velocities vP (i.e. vf > 10 cm/s) the kinetic emission of electrons as a result of direct Coulomb interaction between the ion and the target electron is the dominant initial production mechanism /z/. The energy lost by the ion and, thus, transferred to the electrons is known as "electronic stopping power" in the solid. Elastic and ineleastic interactions of primary, liberated electrons on their way through the bulk and the surface of the solid modify strongly their original energy and angular distribu- tion /3/ and, in particular, leads to the transfer of their energy to further, i.e. secondary electrons (SE) /*/, such that the main part of the deposited energy of the ion is eventually 478 transferred to SE. It is, therefore, suggestive to assume a proportionality between the electronic stopping power S. of the ion and the total SE yield g, i.e. the number of electrons ejected per ion /I/. Indeed, it has been shown experimentally (over three orders of magnitude of S. and vp, see e.g. /'/, /*/, /T/, /•/) that this proportionality holds within a factor of about 2 if the ion charge state is close to the mean ion charge at that energy. The energy distribution of SEs, as measured outside the solid, is strongly dominated by low energy electrons, i.e. E* < 30 eV /•/, /2/. Electrons of this energy are known to have small escape depths, LSE < 20 A (inelasic mean pree path) /10/. This small escape depth has an important consequence: The SE yield g is sensitive only to events in the first few atomic layers at the entrance and the exit surface of the solid penetrated by the ion. Following Sternglass /I/ we consider schematically for kinetic- SE emission contributions from two extreem cases: a. SEs produced mostly isotropically with large impact parameter, associated with an escape depth LSE from the solid. b. SEs produced mostly unisotropically in foreward direction with small impact parameter (6-electrons), associated with a transport length Ls . On their diffusion from the point of origin to the surface these electrons suffer elastic and inelasic collisions /2/ thereby transfering energy to further electrons. According to the par- 479 tition rule (see Bohr /ll/ and Brandt et al. /"/) the fraction of process a. is given by the partition factor (1-B). The assump- tion is that always a dominant or a constant fraction of the energy lost by the ion is, transferd to kinetic energy of SEs. The model of Koschar et al. /'»/ that evolves here is a four step model: 1. The ion is prepared into a certain charge (or excited) state with a mean free path for charge (or excitation) change Lck, 2. an electron is produced in close or distant collisions, 3. the produced electrons are transported in the solid, associ- ated with characteristic lengths (Lsi, Li), thereby creating further SEs and 4. transmitted through the surface. The transmis- sion aspect is discussed in ref. Z1*/, Zi-oscillation of g in /'/, the contribution of the electrons of the ion impacting the solid is addressed in ref. /"/ and the role of collective exitation modes are inferred in /»•/. The pre-equilibrium stopping power is related to the "fractional stopping power" first introduced by Allison /*7/. For the interpretation of the data it is important to compare the above mentioned lengths Lc», LSE and Le (in most cases discussed here Let > LSE). Using this concept data from ref /I3/ have been analysed. They yield surprisingly consistent and reasonable parameters for B. LSE, La and, via the pre-equilibrium stopping power concept St (qi , .n), the effective charges qi.«tf. It is most remarfcable that the initial state shell configuration clearly structures the effective charges resp. the S* (qi , «ft )-values. 480 Thus, the measurement of SEs offer an alternative method to deduce SE-related signals, as e.g. both equilibrium electronic stopping power S« and pre-equilibrium stopping power S*(qi,«fr) near the solid surface. The near surface stopping power data are pre-equilibrium data, which will depend on the incident effective charge state q«.«ff. Stopping power values listed in tables {see e.g. /»•/) refer generally only to equilibrium charge states. However, with sufficiently thin targets or at sufficiently high ion velocities (e.g. GANIL, or GSI-SIS) we are dealing with preequilibrium conditions where stopping power values are significantly different from equilibrium data and are neither available in the literature nor easily accessible experimentally. This nowel approach gives both new insight in the fundamental processes of ion penetration through solids and gives needed heavy ion stopping power data for operation and experiment with the new generation of high energy heavy ion accelerators. This work has been supported by the Bundesministerium fur Forschung und Technologie in Bonn under contract number 060 F173 / 2 Ti 476. • Aristotelian University, GR54006 Thessaloniki, Greece » H.C. Orsted Institut, Universitet, Kopenhagen, Denmark e and IPNL de l'Universite de Lyon I, France References 1. E.J. Sternglass, Phys. Rev. 108 (1957) 1 2. D. Hasselkamp, Comm. Atomic and Mol. Physics 21 (1988) 241 3. S. Lencinas, J. Burgdorfer, J. Kemmler, O. Heil, K. Krone- berger, N. Keller H. Rothard, K.O. Groeneveld Submitted for Publication 1989 4. J. Schou, Scanning Microscopy Vol. 2 No. 2 (1988) 607 5. H. Rothard, K. Kroneberger, M. Burkhard, J. Kemmler, P. Koschar, O. Heil, Chr. Biedermann, S. Lencinas, N. Keller, P. Lorenzen, D. Hofmann, A. Clouvas, K.O. Groeneveld, E.Veje Radiation Effects and Defects in Solids (1989) 6. A. Clouvas, H. Rothard, M. Burkhard, K. Kroneberger, R. Kirsch, P. Misaelidis, A. Katsanos, Chr. Biedermann, J. Kemmler, K.O. Groeneveld Phys. Rev. B39 (1989) 6316 7. H. Rothard, K. Kroneberger, A. Clouvas, E. Veje, P. Loren- zen, H, Keller, J. Kemmler, W. Meckbach, K.O. Groeneveld. preprint (1989) 8. K. Kroneberger, H. Rothard, M. Burkhard, J. Kemmler, P. Koschar, O. Heil, Chr. Biedermann, S. Lencinas, N. Keller, P. Lorenzen, D. Hofmann, A. Clouvas, E. Veje, K.O. Groeneveld, J. Physique 50-C2 (1989) 99 and this conference 9. H. Rothard, K. Kroeneberger, M. Schosnig, P. Lorenzen, E. Veje, N. Keller, R. Maier, J. Kemmler, Chr* Biedermann, A. Albert, 0. Heil, K.O. Groeneveld, Nucl. Instr. Meth. (1990) 10. C.J. Powell, Scanning Electron Microscopy 4 (1984) 1649 and Electron Beam Interaction with Solids (1984)10 11. N. Bohr, Kgl. Dan. Vid. Sel. Math. fys. Medd. 18, #8 (1948) 12. W. Brandt, M. Kitagawa, Phys. Rev. B25 (1982) 5631 13. P. Koschar, K. Kroneberger, A. Clouvas, R. Schramm, M. Burkhard, 0. Heil, J. Kemmler, H. Rothard, H.-D. Betz, K.O. Groeneveld, Phys. Rev. A (1989) in print, and ref. quoted therein 14. H. Rothard, K. Kroneberger, M, Burkhard, Chr. Biedermann, J. Kemmler, 0. Heil, K.O. Groeneveld J. Physique 50-C2 (1989) 105 15. K. Kroneberger, A. Clouvas, G. SchluBler, P. Koschar, J. Kemmler, H. Rothard, C. Biedermann, O. Heil, M. Burk- hard, K.O. Groeneveld Nucl. Instr. Meth. B29 (1988) 621 16. K.O. Groeneveld Radiation Effects and Defects in Solids (1989) in print 17. see: S.K. Allison, Rev. Mod. Phys. 30 (1958* 1137 18. J.F. Ziegler, "Stopping Cross Sections...1 Pergamon Press, New York, (1980) 483 Secondary Electron Emission with Molecular Proiectiles* \. ivroneberger, H. Rothard. P. Koschar, P. Loren.-^n. A. Clouvas1, E. VpjeJ, J. Kemmler3, \. teller, R. ^aiec ind K.O. Croeneveld Institur fur Kernphysik der J.W. ^oethe-fnivers; * .it D-6000 Frankfurt/lain, Germany PACC: 79. 2O.Rf. 79.20.Vc •This work has been founded by BMFT, Bonn, under contract number 060F110/II T.. 439 'Aristotelian University of Thessaioniki, Greece 2H.C. 0rsted Institute, (Copenhagen, Denmark 3now at IPS" Lvon, I'niversite Cljude-Bernard, Lyon, France j.bst ract We present results (or the secondary electron emission (SEE) from thin foil targets, induced by both molecular ions and their atomic constltuents AS projectiles. The Sternglass theory for kinetic SEE states a proportionality between y and the electronic stopping power, S«, which has been verified in various experiments, with comparing secondary electron (SE) -yields induced by molecular projectiles to those induced by monoatomic projectiles, it is therefore possible to test models for the energy loss of molecular or cluster projectiles. Since the atomic constituents ot the molecule are repelled from each other due to Coulomb explosion (superimposed by multiple scattering) while traversing the solid, it is interesting to measure the residual mutual influence on SEE and S« with increasing mternuclear separation. This can only be achieved with thin foils, where •as in the pres^nr Tase) the SE-vields from the exit surface :-jn be measured separately. We measured the SE-yields from the entrance (y») and exit (yt) surfaces of thin C- and Al-foils (150 to 1000A) vith CO*, C4 and O» (15 to 85 keV/u) and Hz* and H* (0.3 to 1.2 MeV/u). The aolecular effect defined as the ratio R(y> between the yields induced by molecular projectiles and the sum of those induced by their atomic constituents was calculated. The energy dependence of R(y) can be well represented by the calculated energy loss ratio of di-proton-clusters by Brandt. This supports Brandt's model for the energy loss of clusters. Introduction The secondary electron emission (SEE) froa solids under ion bo«bardnent has been found to be a powerful tool for the examination of ion-solid- interaction, especially with regard to the stopping power (\,2.3.4/. The proportionality between the electronic stopping power S. and the secondary electron yield y is quite well established for a wide range of target materials (Zt), projectiles livides the product :on of secondary electrons (SE) into two in rlos>? •.•olli.aions fast o-eiectrons are produced, petked in forward Jirection. They transport their energy in forma direrf ;on, until 'hpy •! L jt.ribur ^ ir tn slow 5E in collision cascades within a dist-jjioe '_6 comparable with their range /1,5,6/. .-.; 2low CE are directly produced by the projectile in distant collisions. H^ie the iong range coulomb force of the projectile charge causes a :ollwt:vo escitation of the electron plasma bk winch small amounts jf energy (E-2SeV) can be transferred to single electrons. These slow si show ^ uniform angular distribution. The number of slow SE produced in each process is proportional to the fraction of the specific energy loss of the projectile, dE/dx, for each process, divided by the mean energy used up to free one SE. The SE can be emitted from the solid if their origin lies within an escape depth 1st m the order of 10A Df ^ Y Molecule) „,.,.,,., dE/dx Molecule) a M, R(y) = : y-Itomic :onst.) ' R(dE/^» = r dE/dxUionwc const.) eq" il) "' where a ratio S*l signifies the appearance of a molecular effect. The internuclear separation between the atonic constituents on their way through the solid increases due to the coulomb explosion and multiple scattering. Thus molecular effects will decrease in forward direction with increasing target thickness. This can only be studied by measuring the SE yields from both the entrance and the exit surfaces of thin target foils. Experiment Most of the experimental set-up is described in detail in /6/. In the present set-up, however, experiments were carried out in an ultra high vacuua (UHV) chaaber and with sputter cleaned target foils /ll/. Clean surfaces are essential for the aeasureaent of y, since the SC yield strongly depends on the surface conditions, and for the definition of <*» and thus k* in the last layers of the foil, where aott eaitted SE originate. Experiments were carried out with B* and Hi* (0.3 to 1.2 MeV/u) and C*, 0«, CO* (15 to 85 keV/u) as projectiles, and C-, Al- and Cu-foils (150A to 1000A thickness) as targets. 485 iiired molecular effect ratios a how no •'.ear iependenc» on the target foil vhicknejs. in contrast to earlier results /6/, so in the following we iise only the mean values, f.^rwH over r!ie farget thickness. In backward direction, fi 1.26 *« 10 & O.76 3OO GOO 9OO 12OO £ O.7O O 2O 40 ©O 8O Projectile Energy per Nucleon (keV/u) fi<7. 1: lolecular effect ratio for yt> from C- and Al-foils, induced by CO* (bottom), and from C-, A1-, Cu- and Ti-foils. induced by H»' (top). In forward direction (fig. 2), however, only for snail target thicknesses (x<=iOOA, depending on the projectile velocity /6/), the binding electron can contribute to a aolecular effect. A nolecular effect in SEE due to the correlated motion of the atonic constituents can only occur, if they have not yet separated more than the screening length, X«, vithin the last target layer correspongmg tc the escape depth of the SE, 1st. Our results show no target thickness dependence of R(yt), which indicates that the acoaic constituents have already separated too far. The observed value R(yi)*l say thus not be due to a Molecular effect but rather indicate a systematical error. Though, other experiments carried out with Sj'-projectiles /10/ also show R(dE/dx)=0.9 at target thicknesses comparable to ours. Also, •onte- carlo-calculations for r» of CO* at low velocities (v»=v») /9/ and experiments with different molecules /10.16/. indicate that the atomic constituents may stay together for rather large distances {some 100A) if one particle is captured in the first minimum of the wake potential of the leading particle. 486 f 1.20 m •01 1.0 -A • 4- - p S cT .7© t D 5 1 a 300 000 9OO 12OO O 2O 4O OO SO ProJectJJe Energy per Nucleon (keV/u) Fi ! :l- i Molecular effect ratio for yf from C- and Al-foils, induced by CCr (bottom), and from C-. A]-. Cu- and Ti-foils, induced bv Hi' ((op). For low velocities (vp 2ve) Brandt's calculations show a decrease of the energy loss ratio towards R(dE/dx)=l -it vp^:.5vo. Our results, as well as recent results for y* from thick rar^r -.-th Hi* projectiles', follow this tendency even continuing to Sfys.) 1 it P 1.5v. /:/. Tins indicates that the proportionality between energy loss sn.i Zllll is validd alsoo for molecularr projectiles and qives us the possibility to calculate energy loss data fro« 3E-y;eid measurements. Conclusion we measured the Secondary Electron yield coefficients in forward iit) and backward (yb) direction from thin solid foils 7. .;. ':r^rngiaas. Phyj. Sev. ^£8 (1957) 1 rip, .J. Liebig v.niv. Ziessen. Germany, Hdb; 1 .• *: :onsschnf * H. '. 7: l.-chki;! n, K. 0. E. JriT.MiHk»»hl, :'hvs. Ccri|JrJ 76 f 1n^3J 39 ; \. •••:juvas. S. G. Kjrsch, \. Kats-inos, P. Misaeli-l*s. ".. ?..sharri, H. ?(irl:.u«i, •"!. 3i«?defnidnn, J. Kemmlei. K. Kroneber^es, \. j. "foeneveid. P. K'isi'har. K. Kroneberqer, ^. Clouvas, R. Schramm, H. D. 3etz. 1. B;irkn.ud. 0. Sell, J. Kemmler, H. Sothard, K. 0. Groeneveli, this ci;nf erence '6/ K. Vrooeber'jer, A. Clouvas, 0. Uchlussler, P. Koschar, J. Kemmier, H. Rothard, C. 3iedermann, 0. Heil, 1. 3urkhard, K. 0. Oroeneveld, \ucl. Instr. Me'Ii. B2i (1983) 621 /?' \. Bohr, K. Dan. Vid. Selsk. ^lat. Fys. Medd. 3J N'o.8 (2948) ,'S/ W. Brandt, J. Remheimer, Phvs. Ke1.'. B2 (1970) 310-5 /9/ J. Kemmler, P. K'cschar, 1. Burkhard, K. O. Groeneveld, \ucl. Insfr. Meth. Oil :i985) 62 /10/ H. F. Jfeuer, 0. S. Genmiell, E. ?. Kanter, E. \. Johnson. 3. J. Zabransky. IEEE Trans. Vucl. Sc:., Vol. NS30 (1983) 1069 III/ M. Burkhard, H. Rothard, J. Kemmler, K. Kroneberger, K. 0. Groene\eld. J. Phys. D2i (1988) 472 and M. Burkhard, w. Lotz, K. 0. Groeneveld. J. Phys. E2J. C9B8) 759 /IT/ J. Holzl, K. Jacobi, Surf. Sci. U (1969) 351 /I3/ w. 3rar.dt, R. H. Ritchie, Xucl. Ir.str. Meth. U2 (1976, 43 /!•}/ V. Cue, V. V. ie Castro Faria, M. J. Gaillard, J. C. ?c:zz:. J. Remillipu\, Vucl. Instr. Meth 170 (1980) 67 /;5/ A. Clouvas, M. J. Gaillard, A. G. de Pinho, J. C Poizat. J. Reraillieu\, ;. Desesquelles, N'ucl. Instr. Meth. 32 r 1084) 273 .';•)' *. P. Kanter, P. J. Cooney, Phys. Rev. A20 (1979) 334 489 Direction-dependent Stopping Power and Beam Deflection in Anisotropic Solids Oakley H. Crawford Health and Safety Research Division Oak Ridge National Laboratory Oak Ridge, Tennessee 37831-6123 Abstract Directional effects on the motion of swift ions in anisotropic media are studied. The stopping power is a function of the direction of the velocity relative to the principal axes of the medium, and there is a nonzero lateral force on the ion tending to bend its trajectory. These effects arise from the anisotropy of the dielectric response, and are distinct from channeling. Simple expressions are derived for the stopping power and lateral force in the nonrelativistic, high—velocity limit, and results are presented for crystalline graphite. I. INTRODUCTION The forces acting on a swift ion in a solid are determined by the dielectric function f(w,q), when scattering by nuclei can be ignored. In a crystal, or in an 490 amorphous layered or oriented medium, e(u,q) depends on the direction of q, leading to the following directional effects on a swift ion: The stopping power depends on the direction of the velocity v, and there may be a lateral force (perpendicular to v). The total force and its components are illustrated in Figure 1. Note that the stopping power is just the negative of the V component of the total force. Channeling is of course distinct from the above effects, can occur simultaneously with them, and is likely to interfere with their observation in crystals. The differential probability of transfer of momentum oq and loss of energy hw by a swift projectile may in principle be calculated from the differential inverse mean free path P(v,q). This latter function is defined as the probability per unit path length per unit u per unit volume in q space, that the medium will gain energy hu and momentum Jtq from the projectile. It has the following well—known form, 1' 2 in the Born approximation: where e(c/,q) = q • e(w,q) • q, (2) where q is a unit vector, e(w,q) is the dielectric tensor, and where — eZp and v are the charge and velocity of the projectile ion. Eq. (1) is often used (along with dispersion relations) to investigate the dielectric tensor of crystals from measurements of energy loss and scattering of 491 electrons, using thin enough targets that multiple scattering may be either avoided or corrected for. In the case of thicker targets, where projectiles suffer a large number of inelastic collisions, statistical measures of the force are useful. In the following, the expectation value, or time average, of the force is studied, and all references to force and its components should be interpreted accordingly. The stopping power 5 and the lateral force Fx are obtainable in principle as averages over P(w,q) (see below). However, simpler expressions for S and Fx would be useful. More importantly, P(w,q) is generally known for small values of q only (at best), so direct calculation of 5 and Fx from e(w,q) or P(uj,q) is not possible in practice. The present work addresses two questions: Can 5 and Fx be described by simple expressions in the nonrelativistic high—velocity limit, and how large are the directional effects for protons in graphite. n. STOPPING POWER Consider a uniaxial solid. There is one unique axis, called the c axis, and in the principal—axis system the dielectric tensor €(w,q) has the following nonzero elements: e±(ui,q) (twice) and f||(w,q). These elements describe the response to applied fields in directions respectively perpendicular to, and parallel to, the c axis. The stopping power S(a) is given as a function of the angle a between the velocity and the c axis by the expression S{a) = \d*q\ dw P(u),q) Tlw, (3) 492 where the first integral extends over all q. We have shown that in the nonrelativistic high—velocity limit, the above expression goes asymptotically to the following: mv where /(a) is given by the following: dwlm -1 J o (ei+ Ae sin^a 2 (|argcx| < 7T, |arg[ex + Ac sin aj| < ir) , (5) where and where c and £•• stand for e (w,0) and e|j(u,0), respectively. Eq.(5) is the leading term in a high—velocity asymptotic expansion of Eq. (3). The next term is of order vA lnu Note that Eq. (5) for S(a) has the same form as in the Bethe theory for homogeneous isotropic media, except that here the mean ionization potential /(a) is a function of the direction of motion relative to the c axis. In the special case where e^ and e^ are identically equal, Eqs. (4) and (5) reduce to the Bethe 493 expression. Note also that /(a) is calculable from knowledge of the optical constants, c (w,0) and e,<(u,0), alone. HI. STOPPING POWER CALCULATIONS Calculations reported here are all based on Cazaux's model for the optical constants of graphite. This model consists of a simple form obeying f—sum rules for four electrons per atom (i.e., ignoring the K—shell electrons), which is fit to experimental values of the optical constants in the energy range of valence-shell excitations. The model was modified slightly in the present work, as will be described elsewhere. Figure 2 shows I(a) calculated for graphite, as a function of angle a. One sees that I(a) has its minimum value at a= 0, which is a reflection of the fact that the transition energies hu>rp are lower for q perpendicular to the c axis than for q parallel to it. Note also from Fig. 2 that /(a) ranges from 30.6 to 34.8 eV, which values are considerably less than that of the empirical mean ionization potential /, 77.3 eV, of carbon. This is because excitation from the K shell is not included in Cazaux's model. Previous work shows that neglect of the K—shell electrons is justified in approximating the stopping of ions of energy less than (M /m)Bjy-, where M is the projectile mass, and Br, is the K—shell ionization potential. In Figure 3, the proton stopping powers 5(0°) and 5(90°) computed from Eq. (15) are compared with the proton stopping power of graphite given by Andersen and Ziegler. This comparison suggests that Eq. (15) with the Cazaux model is unreliable at energies less than 100 keV, is of marginal accuracy between 100 and 200 keV, and is reasonably accurate for proton energies from 200 keV up 494 to at least 10 MeV. However, the applicability of Cazaux's model to the stopping power for protons of energy exceeding 1 MeV is questionable, according the above discussion, since (M/m)B^is 1.0 MeV for carbon. Figure 4 shows the angular dependence of proton stopping power in graphite for proton energies ranging from 102 to 10* keV. Each curve is normalized to unity at a = 0. The stopping power is at a maximum at a = 0, and falls a few per cent as a increases to 90°. IV. LATERAL FORCE The lateral force, Fx, is given by duP(u,q)qx, (6) o where qx is the component of q in the x direction. The X axis (perpendicular to v and coplanar with v and the c axis) is in the direction of the lateral force vector depicted in Figure 1. The asymptotic high—velocity form of Eq. (6) is found to be as follows: 4 2 iire Fx(a) = h P sina cosa G(a), (7) mv where 2 fx G(a) — 7) J du Im v 0 (ex+ Ae 495 2 (|argej < r, jarg[ex + Ac sin a]| < JT) , (8) in which Ae = en — e , and e,, [e ] stands for CI<(CJ,0) [c (a/,0)]. The first term omitted from Eq. (24) is of order v*. V. LATERAL FORCE CALCULATIONS Figure 5 shows the lateral force on 200—keV protons in graphite, as a function of angle a, as computed from Eq. (7). In this medium, G(a) is not a strong function of a, so the Fx(a) curve has approximately the shape of sina cosa. Figure 6 shows Fx again, this time for different angles a, as a function of energy. Recall from Eq. (7) that the velocity dependence is just v2. The units, keV//xm, used for Fx in Figures 5 and 6 are unusual units for force, but they are conventional for stopping power (which is just —Fv, the negative of the force component in the V direction). The same units are used here for Fx as for 5, to facilitate comparison. Perhaps the most immediately useful way to look at the magnitude of Fx(a) is to compare it with S(a). It may be seen from Figure 1 that the angle 7(a) between the net force and the negative of the velocity vector is given by just the ratio, = aictaa(f^). (9) 2 The ratio Fx(a)/S(a) is shown in Figure 7 for a = 45°, for the 10 - 10* keV range of proton energy. One sees that its value is a few per cent. Thus, the net force on swift protons under these conditions deviates from being antiparallel with the 496 velocity by an angle of a few hundredths of a radian. The effect on the proton is that (on the average) it travels a curved path, with its trajectory bent toward the caxis. VI. SUMMARY The stopping power S(a) and lateral force Fx(a) are found to be given by simple expressions in the nonrelativistic high—velocity limit. S(a) has the same form as in an isotropic medium, except that here it is a function of a. Calculated results are presented for protons of 102 to 10* keV in crystalline graphite. The stopping power varies by a few per cent with a. At a = 45°, the lateral force is a few percent as large as the stopping power. The directional effects studied here may have practical consequences in the irradiation of biological systems, which can be quite anisotropic. ACKNOWLEDGMENTS This research was sponsored jointly by the U. S. Air Force Office of Scientific Research, under Interagency Agreement DOE No. 1262—1262—Al and the Office of Health and Environmental Research, U. S. Department of Energy, under Contract No. DE-AC05-84OR21400 with Martin Marietta Energy Systems, Inc. 497 References 1. J. Hubbard, Proc. Phys. Soc. 48, 441 (1955). 2. V. M. Agranovich and A. A. Rukhadze, J. Exptl. Theoret. Phys. (U.S.S.R.) 35, 1171 (1958) [Sov. Phys. JETP 35, 819 (1959)]. 3. H. Raether, Excitation of Plasmons and Interband Transitions by Electrons, Vol. 88 of Springer Tracts in Modern Physics (Springer—Verlag, Berlin, 1980), and references therein. 4. 0. H. Crawford, to be published. 5. J. Cazaux, Opt. Comm. 2, 173 (1970). 6. H. H. Andersen and J. F. Ziegler, Hydrogen Stopping Powers and Ranges in All Elements, Vol. 3 of The Stopping and Ranges of Ions in Matter, edited by J. F. Ziegler (Pergamon Press, New York, 1977). 7. 0. H. Crawford and C. W. Nestor, Jr., Phys. Rev. A 28, 1260 (1983). 8. J. A. Bearden and A. F. Burr, Rev. Mod. Phys. 39, 125 (1967). 498 Figure Captions FIG. 1. Forces on a swift ion in a uniaxial medium, for the case Fx > 0. FIG. 2. Mean ionization potential I(a) of graphite, as a function of angle a, calculated from Eq. (5). FIG. 3. Proton stopping power of graphite at two different angles, from Eq. (4) (solid curves), compared with the experimental stopping power of carbon, from ref. 6. FIG. 4. Proton stopping power in graphite, from Eq. (4), versus angle, for different proton energies. Each curve is normalized to unity at 0°. FIG. 5. Lateral force on 200-keV protons in graphite, as a function of angle, from Eq. (7). FIG. 6. Lateral force on protons in graphite, as a function of proton energy, for different directions of the velocity, calculated from Eq. (7). FIG. 7. Ratio of Fx(a) to S(a) for protons in graphite, as a function of proton energy, at a = 45°, calculated from Eqs. (4) and (7). ORNL-DWG 89-16939 VELOCITY CAXIS LATERAL FORCE FORCE STOPPING V POWER ORNL DWG 89-16941 V 36 r \ J - GRAPHITE 34 o > o 0) 32 30 0 30 60 90 ANGLE a (deg) ORNL-DWG 89-16942 E PROTONS IN GRAPHITE - zl CD 200 -THIS CALCULATION LLJ 00 O Q. O 100 ^^ANDERSENAND z ZIEGLER Q_ Q_ CO 0 10 102 103 10' PROTON ENERGY (keV) ORNL-DWG 89-16943 1.02 • I ' I LU 5 L PROTONS IN GRAPHITE o Q. 1.00 o O LJAl l J 0.96 3 LU 0.94 0 30 60 90 ANGLE a (deg) ORNL-DWG 89-16944 10 V E PROTONS IN GRAPHITE a .=1 200 keV CD LU O occ LL < LU 0 30 60 90 ANGLE a (deg) ORNL-DWG 89-16945 15 | 1 a E V PROTONS IN GRAPHITE \45° I 10 — LJJ O CC I/oI 00 O LL, < 5 CC Hi H n 10' 103 10' PROTONS IN GRAPHITE (LATERAL FORCE) / (STOPPING POWER) 507 Independent-Particle Model for Fusion in Cluster Impact Oakley H. Crawford Health and Safety Research Division Oak Ridge National Laboratory Oak Ridge, Tennessee 37831-6123 USA Abstract Theory is presented in an independent—particle model for the fusion yields from impact of accelerated (D2O)n clusters on TiD. It is assumed that deuterons from an incident cluster penetrate the medium independently of each other. Comparison is made with the recent experiment of Beuhler, Friedlander, and Friedman. The measured fusion yields, for n values ranging from 22 to 1300, are many orders of magnitude higher than those calculated from this theory. This implies that the mechanism depends crucially on cumulative or cooperative effects involving many cluster atoms. 508 I. INTRODUCTION The first report of the discovery of cluster—impact fusion was made today at this conference. Beuhler et al. have observed d-d fusion from impact of accelerated, charged clusters of DgO molecules on TiD targets. Novel methods of achieving nuclear fusion have been of great interest since the claimed discovery early this year of cold fusion, 2 ' 3 and this latest report is certain to stimulate continued widespread interest in phenomena attending the penetration of charged particles in solids. The purpose of the present work is to calculate d—d fusion yields from cluster impact in an independent—particle model, compare with experiment, and draw conclusions concerning the mechanism. The calculations below are based on a model in which clusters break up on impact with the TiD target, whereupon the deuterons (and oxygen, which is ignored) penetrate the solid and are transported in it independently of each other. No account is made for compression or heating of the medium. Incoming deuterons lose energy to electrons and phonons, and finally come to rest. In the process, a given projectile deuteron may, with small probability, enter into a fusion reaction with a deuteron belonging to the target. This model is a reasonable one for a first look at cluster impact fusion. It is clearly valid for impact fusion of D , DgO , and sufficiently small clusters at the experimental energies of 200 keV and up. Finally, it is interesting to compare the recent experiment with this theory. On the basis of this comparison, it is argued in the final section that a successful explanation of impact fusion of clusters must include a mechanism for large increases in relative energies of some d(beam)-d(target) pairs. 509 H. THEORY + Consider a (DOO) cluster of translational energy E, incident on a deuterium—loaded target. Suppose the 2n deuterons penetrate the target, each initially with the same velocity as the cluster. At this point, two approximations are made, whose validity will be considered elsewhere. First, the target nuclei are treated as if their positions comprise a uniform random distribution. Thus, the probability dY that a d—d fusion reaction (of any of the 2n fast deuterons) occurs while one typical deuteron travels a distance dR (measured along its path) is written as dY = 2nN a(Ed) dR, (1) where N is the number of target deuterons per unit volume, and crfE^) is the reaction cross section, which depends on the energy E, of the deuteron. Next, the continuous—slowing—down approximation (CSDA) is made, whereby dR is replaced in Eq. (1) by (dE ,/dR)"kiE,. The resulting equation is immediately integrable to give the following expression for the yield, or number of reactions per cluster, Y(E): EdO dE -1 Y(E) = 2nN f (2) where E^ = (M^/nM^ Q)E is the initial energy of the deuteron, given by the product of the mass ratio and the energy of the cluster. Note that the quantity in the brackets in Eq. (2) above is just the deuteron stopping power of the solid. 510 The cross section of the D(d,p)T reaction is given ' at the energies of interest by (3) with n 2X77 = 1*h = 269170 (Ed ao/e2) where v is the velocity, -e and aQ are the charge of the electron and the Bohr radius, and S is the astrophysical constant extrapolated to zero energy, whose 6 5 2 value is 55.7 keV b = 7.31 * 10~ e aQ. The prefactor of 2 in Eq. (3) appears because the energy E, in the denominator is the projectile energy, instead of the relative energy. Eqs. (3) and (4) neglect the effect of screening on penetration of the Coulomb barrier. One may infer from the theory of Arista et al. that screening does not have a large effect at the deuteron energies considered here. The stopping power is the sum of the electronic and nuclear stopping o powers. For the electronic stopping power, values given by Andersen and Ziegler for Ti metal were used. This should be a reasonable choice for the purpose, particularly since, as can be seen in Eq. (2), it is the ratio of the stopping power to the atomic density that is required. The nuclear stopping is calculated from LSS Q theory for the Thomas—Fermi potential. The sum of the nuclear stopping by Ti and by D (assumed to be present in equal number) is used. At all the energies relevant to the present results (0.6 to 300 keV) it is found that the electronic stopping is greater than the nuclear stopping by Ti, which in turn is greater than the nuclear stopping by D. 511 If the nuclear stopping is neglected, the stopping power at the energies of concern here is proportional to the velocity, v. Upon substituting (-dE^/dR) by Av, Eq. (2) may be integrated, giving the following: } () ire A where 7)n is r), evaluated at E j = E ,0. However, the above expression is not used in the present calculations, since at the lower end of the range of deuteron energies considered the nuclear stopping is not negligible. m. RESULTS AND CONCLUSIONS The figure displays the results. The circles are the yields (number of D(d,p)T fusion reactions per cluster), computed from Eq. (2), for 300-keV clusters composed of 1 to 50 D2O molecules. The squares are yields of the same reaction, measured by Beuhler z* al., for clusters of 22 or more molecules. In each case5 the clusters are accelerated to 300 keV energy. Also calculated, but not shown in the figure, is the yield from 300-keV D+ ions, which is 1.7 * 10~7 D(d,p)T reactions per incident ion. The experimental data extend to clusters of 1300 molecules. As the figure shows, the theory falls below experiment by a huge factor. At n = 50, the two differ by 18 orders of magnitude! More surprising, the experimental yields increase with increasing n until n ~ 100, and do not begin to decrease until n ~ 500, whereas the theoretical yields decrease monotonically (and rapidly) with increasing n on account of the strong energy dependence of the fusion cross section. 512 The figure also illustrates the necessity of excluding low—mass deuterium-containing ions from the accelerated beam in such an experiment. (This problem is dealt with in detail by Beuhler, et al .) Thus, the calculated + + fusion yield from 300-keV D2O (or DgO ) is about 10 times as large as the yields measured for 300-keV clusters. Also, the calculated yield (not shown) from 300—keV D+ ions, 1.7 * 10 D(d,p)T reactions per incident D , is about 5 orders of magnitude larger than the experimental yields from 300-keV clusters In view of the immense differences between theory and experiment displayed in the figure, alternate mechanisms must be considered. It is important in this connection to note that the observed reaction is evidently d(beam)-d(target) fusion, rather than d(beam)-d(beam) or d(target)-d(target) fusion, because little or no reaction occurs when the deuterium in either the beam or the target is replaced with hydrogen. Let us suppose that a mechanism exists for increasing the energy of every deuteron in a cluster, and estimate on the basis of the above calculation how large an increase is required to explain the measured yield. For deCniteness, I in consider 300-keV (D2°HoOO ' wnose measured yield is 4*10 reactions/cluster. Reference to the figure shows that the theoretical yield from + (DgOJg is about equal to the above experimental yield from (D2O),QQQ , when expressed in terms of reactions per incident deuteron. However, the initial energy E .Q of each deuteron in the n = 6 cluster is 167 times that in the n = 1000 one. Thus, an increase by a factor of 167 in E,y (bringing it up to 5 keV) for each of the 2000 deuterons in (D2°)l000+ would brmS tneory int<> agreement with experiment. However, the above result is unphysical, requiring much more energy than the 300 keV that is available. On the other hand, a sufficiently large energy 513 increase (more than a factor of 167) in a portion of the deuterons would have the desired effect, without exceeding the total energy of the cluster. Applying the same considerations to other clusters in 600— to 1300—molecule range leads to similar conclusions, with other factors (having value 100 or more) replacing the above factor of 167. In conclusion, it appears that an explanation of impact fusion of large clusters must include a mechanism by which the energy of some cluster deuterons (or the relative energy of some d(beam)-d(target) pairs) is increased by a large factor from its value prior to impact. This factor exceeds 100 for 300—keV clusters in the 600— to 1300—molecule range. ACKNOWLEDGMENTS The author thanks L. Friedman for a helpful discussion of the experiment. This research was sponsored by the Office of Health and Environmental Research, U. S. Department of Energy, under Contract No. DS-AC05-84OR21400 with Martin Marietta Energy Systems, Inc. 524 References 1. R. J. Beuhler, G. Friedlander, and L. Friedman, Phys. Rev. Lett. 63, 1292 (1989). 2. M. Fleischmann, S. Pons and M. Hawkins, J. Electroanal. Chem. 261, 301 (1989); 263, 187 (E) (1989). 3. S. E. Jones, E. P. Palmer, J. B. Czirr, D. L. Decker, G. L. Jensen, J. M. Thome, S. F. Taylor, and J. Rafelski, Nature (London) 338, 737 (1989). 4. 0. H. Crawford, to be published. 5. J. D. Jackson, Phys. Rev. 106, 330 (1957). 6. N. Jarmie and R. E. Brown, Nucl. Instr. Meths. in Phys. Res. BIO/11, 405 (1985). 7. N. R. Arista, A. Gras-Marti, and R. A. Baragiola, This Conference: ibid, Phys. Rev. A 00, 0000 (1989). 8. H. H. Andersen and J. F. Ziegler, Hydrogen Stopping Powers and Ranges in AU Elements, Vol 3 of The Stopping and Ranges of Tons in Matter, edited by J. F. Ziegler (Pergamon Press, New York, 1977). 9. J. Lindhard, M. Scharff, and H. E. Schiott, Mat. Fys. Medd. Danske Vid. Selsk. 33, no. 14 (1963). 10. In the original experiments, no fusion could be detected unless both beam and target contained deuterium. However, more recent experiments show a nonzero (though greatly reduced) yield when the D2O in the beam is replaced by H2O. (L. Friedman, private communication.) 515 11. Note that the calculations reported in this paper are functionally equivalent to predictions of fusion yield from impact of bare deuterons on TiD. Thus, the theoretical fusion yield given in the figure for a (D«O) cluster of energy E = 300 keV is just 2n times the yield predicted for a deuteron of initial energy EdQ = (Md/nMD Q)E ~ E/lOn. Figure Caption Fusion yield, D(d,p)T reactions per 300-keV (D^OL cluster, versus n. The JJ II circles are calculated from Eq. (2); the squares are experimental results from Beuhler et al.1 516 ORNL-DWG 89-17062 I -10 CO c -15 o o CO CD -20 3 Q Q LU -25 6) o -30 1 10 50 NUMBER OF D2O PER CLUSTER Fig. 1. 517 Bransstxahlung Induced by 50 HeV H° Bombardment C. Stein, K.W. Habiger, and B.R. Smith Weapons laboratory Kirtland AFB, Hew Mexico 87117-6008 and Oakley H. Crawford and R.H. Ritchie Health and Safety Research Division Oak Ridge National laboratory Oak Ridge, IN 37831 518 ABSTRACT Three different thicknesses of aluminum targets, 0.0025 cm, 0.025 on, and 0.32 cm were irradiated with 50-MeV neutral hydrogen atoras and the resultant bremsstrahlung radiation measured. The experimental results are compared with calculations that include bremsstrahlung by the projectile's electron and its cascade, bremsstrahlung by the cascade electrons that are generated by the projectile nucleus and quasifree- electron bremsstrahlung. The theoretical and experimental values of the total yield agree within a factor of three for all target thicknesses and all bremsstrahlung energies. Bremgstrahlung Induced by 50 MeV H° Atom Bontoardment I. Introduction The continuum of x rays induced by ion bombardment of solids is important as a background in elemental analysis by particle-induced x-ray emission (FIXE) . The variety of physical processes involved in these emissions has stimulated a great deal of interest in recent years. In the present work, the effect of the thickness of the target on the bremsstrahlung yield was determined by the 50 MeV H° bombardment of aluminum foils of three thicknesses; 0.0025 cm, 0.025 cm, and 0.32 cm. The experimental results are compared with calculations that include bremsstrahlung by the projectile's electron and its cascade, bremsstrahlung by the cap™** electrons that are generated by the projectile nucleus and quasifree-electron bremsstrahlung. Relativistic formulae are used. The slowing-down spectra of both the projectile nucleus and the electrons are utilized, as well as the spectrum of cascade electrons. The electron velocity is considered to be isotropic when calculating the angular distribution of the radiation and the spacial distribution of cascade electrons is accounted for in an approximate fashion, as is the loss of electrons through the surface. II. Experimental Procedures All of the experiments were carried out on beam line "A" at Argonne National laboratory using 50 MeV neutral hydrogen. Immediately prior to, and subsequent to, each individual series of experiments the neutral beam was allowed to traverse the specimen chamber, pass through a metal foil sealing the beam line and after traveling a short distance in air, to strike 519 the Faraday cup at the beam stop without interacting with a target. iMs permitted the fluence of hydrogen atoms striking the target foils to be measured directly. The estimated accuracy of the Faraday cup's determination of the flux in the beam stop is ± 5%. the typical pulse width was approximately 55 microseconds and the current was 20 nanoamp6. . The x-ray data were recorded with a liquid nitrogen cooled silicon detector (SiLi) (lithium drifted) made by Princeton Gamma Tech (PGT). Energy calibration of the spectra was obtained by using known standard elements such as: 99.995% aluminum and 99.9995% copper, as well as the K° and K<*_ lines of tantalum for the lew, middle, and upper energy ranges of interest, respectively. The detector efficiency was calibrated with a Fe source, produced by Isotope Products laboratories, which is traceable to the National Bureau of Standards. The calibration was carried out at atmospheric pressure where the absorption due to the air between the source and the detector reduces the signal by a factor of 0.225. This, together with the reduction of 0.994 due to the 0.00033 inch of beryllium detector window, decreases the source's intensity by a factor of 0.224. The source itself experienced an emission rate reduction from its original value of 2.98x10 to 2.91x10 photons/second due to radioactive decay over the period of 25 days between the date of its certification and the carrying out of these experiments. Since, the geometric factor comprising area of detector and distance between detector and sample is 5.17xlO~ the Fe source flux at the detector is (2.91X106) (5.17xlO~5) (0.224) • 33.7 photons per second, which should have produced a flux at the detector of 57,290 photons. The total counts under the experimentally rtPtgrminpflspectrum' s two iron peaks was 55,448, so that the detector efficiency in the range near 6 kev is calculated to be 96.7%. Comparing this to the efficiency curve supplied by the manufacturer, which is listed at 99.29% at 6 keV, yields an uncertainty in the detector efficiency of about 3%, The bremsstrahlung radiation produced by 50 MeV neutral hydrogen atoms on the aluminum foils was measured between 3 keV and 23 kev for the 0.0025 cm target; between 3 keV and 28 keV for the 0.025 cm target and between 3 keV and 35 keV for the 0.32 target. III. Theory Three mechanisms for continuum x-ray production are significant under 520 the following conditions: (1) The specimen's thickness exceeds the projected range of the incident and secondary electrons; (2) we assume that the velocity of the projectile is large compared with the orbital velocities both of its own electron, and of most of the target's electrons; and, (3) the electrons carried by the ion or atom are stripped from it at a negligible depth in the target. These mechanisms are: electron bremsstrahlung (EB), cascade electron bremsstrahlung (CEB), and quasifree- electron bremsstrahlung (QFEB). The first of these is due to the projectile's electrons, while the other two are caused by its nucleus. These mechanisms are treated separately below. Although a relativistic formulation is employed, it is assumed that the projectile is not highly relativistic. We take the projectile ions or atoms to be incident normal to the surface of a plane-bounded solid, and calculate the differential yield of photons escaping the target in the backward hemisphere, i.e., from the surface through which the projectiles enter. Electron Bremsstrahlung (EB) The doubly differential yield, d2Y/dkdfl, is defined as the number of photons escaping the target with energy k in direction 6, per incident atom, per unit photon energy, per steradian. Then, dn' /dkdft is the portion of the doubly differential yield that is due to penetration by the electron of the incident atom. Since the angle dependence of the bremsstrahlung differential cross section is not great for speeds that are not highly relativistic, we have ignored the angular dependence of the differential production cross section. In the following approximation, d ¥_ /dkdft is given, for targets thicker than the projected range of the incident electrons, by the following expression: - SJ dT where & is the number of atoms striking the target per unit time, ./•f and Tn ara, nqpactivttly, tha initial number and initial kinetic energy of electrons in the incident ion or atom, N is the number density of target 521 atans, i. is the target thickness, and do^dklKT) is the energy-differential cross section for production of bremsstrahlung of energy k by electrons of kinetic energy T (and velocity v) in collision with target atoms, d n/dxdT is the number of electrons in the target at depth x that nave kinetic energy T, per unit x per unit T, n is the absorption coefficient of the solid for photons of energy k and # is the angle of observation, relative to the outward normal of the back surface of the target. We are concerned in this paper with emission into the backward direction, i.e., through the surface that is exposed to the incident beam, where 9< w /2. It is often the case that the projected range of projectile electrcns is less than the extinction length, n~ , of x rays in the target. By assumption, this range is less than the thickness of the target, as well. Then, a precise description of the range distribution of the electrons is not necessary, and one may replace d n/dxdT by i[ysrR^Q{1Q) ] • dn/dT, where it] is the Dirac delta function, and R^CTQ) is the mprjian projected range of electrons with initial energy TQ. Although this is rather a crude approximation, it is useful. The effect upon the yield of the electron range is obtainable, approximately, by comparison with a calculation in which R~Q is replaced by zero. This effect is seen to be negligible for the most part in the present application, as discus sod below. In the continuous-slowing-down approximation (CSDA), the quantity •^"•^vdn/dT is given by Se(T)~T?(T,T0), where Se(T) is the stopping power for electrons (energy loss per unit distance traveled) and R(T,TQ) is the ratio of the flux of all electrons (primary plus cascade) of kinetic energy, T, to that of primary electrons alone . Ccnbining the above approximations gives, finally: die Bremsstrahlung produced by collision of the electron with the target electrons is in principle included in d piectron Brernsstrahlung One mechanism of production of continuum x rays by bombardment by awift 522 protons is generation of a cascade of fast electrons, which produce bremsstrahlung in colliding with atoms of the target. The model used here for CEB is similar to the one discussed above for EB. Isotropic emission is assumed, and secondary electrons (arising from protcn-target-atcn collisions) created at depth x with kinetic energy TQ are assumed to produce bremsstrahlung at depth x + 850 (TQ). The doubly differential yield is given by d2Y z 2 N T CEH p f* r 0>nax d.¥. dk dO B 4n I dx I dT0 T J 0 T R(T.T ) dio l where Z is the atomic number of the projectile and ^Omax is the largest value of kinetic energy that is less than or equal to the kinetic limit Tm and satisfies the condition ^o^Qmax^- A~ x" Tm' tne Maximum energy that can be transferred to a free electron at rest by an ion of velocity v_, is 2 2 *^ approximated by 2mv /{I - fi ), with ft =* v /c, and ID and M are the rest masses of the electron and the projectile nucleus, respectively, d i/fvdxdT- is the number of secondary electrons created per target atom at depth x with initial kinetic energy TQf per unit x par unit TQ, by a projectile nucleus moving in the x direction, and is approximated here by the following extension of a formula given by Fano4: for TQ < Tm, where n^ and Z^ are, respectively, the number of electrons and the ionization potential of the ith shell, in the target atcm and v is the velocity (taking account of slowing down) at depth x of the projectile. The above formulation gives a zero yield for photons of energy greater than T^Q (the Tm value of the incident ions), where the yield is in reality finite, though falling rapidly with k. This is adequate for the present application, where the measurements do not approach or exceed T__. 523 Bremsstrahlunct When an ion's velocity is large compared with that of most of the target electrons, quasifree-electron bremsstrahlung (QFEB) ' is generated. In the frame of the moving ion, the mechanism is bremsstrahlung by the target electrons. The doubly differential yield for this process is given by the following expression: ;Z 2 I dx E (k.fl.v ) exp{- (5) 1 p J p Q dkdO where Zfc is the atomic number of the target atoms, v is the velocity of the projectile nucleus at depth x, taking account of the stopping power and d <7 /dkdJ] is the doubly differential bremsstrahlung production cross section for electron collisions with the projectile nucleus, in a frame in which the electron is initially stationary. This cross section is approximated here fay the product of the plane-wave Bom approximation expression6 and the Elwert factor7, as follows: 1 f Z Z ft ln 3 E *" t P T^k o I I ' ° • T (» • P ) < * Inf f-i-J - i p (3 co«2 0 -1) (6) 2 when* p - 1 - k/Tr,aSV137, aQ is the Bohr radius, and f£ is the Elwert factor , given by z a with y1 « p /^p sand y2* Z_ Calculations Calculated values of the doubly differential bremsstrahlung yield d2Y/dkdn , as well as the separate contributions of the three mechanisms, are shown in Figs 1, 2, and 3 for 50 MeV hydrogen atoms striking aluminum targets at normal incidence, with an angle of observation, e , of 45°. Ihree different target thicknesses are considered. The values of T and ru Tg for 50 MeV protons are 27.2 and 106 keV, respectively. Values of the bremsstrahlung production differential cross section da./ The doubly differential yield due to the incident electron (and the cascade electrons generated by it), d Y™ /'dkdft , is shown as the dashed curve in Figures 1, 2, and 3. The EB yield goes to zero at k = Tfl = 27.2 keV. Note that this contribution to the yield is the same for all three targets. This is because they are all thicker than the median projected range of the projectile electron (treated as a 27.2 keV free electron), 2x10 an. The yields from the C3B and QFEB mechanisms are given by the dotted- chain and dashed-chain curves, respectively. As shown in Figures l, 2, and 3, the theory predicts that yields from CEB and from QFEB are of the same order of magnitude, with CEB being the more important mechanism of the two, throughout most of the photon energy range below TQ, the initial value of 1 14 Tr. This is in agreement with the calculation of Ishii and Horita . The QFES yield goes to zero at k = T^ (i.e., where k equals the T value of the proton at its initial velocity), in the present model. Referring to calculations (done at lower energies) that take account of the electron's motion in the initial state, one expects a finite decrease in the QFEB yield at k w 27 keV, followed by a tail extending to higher k. Since the predicted QFEB yield is consistently less than the CEB yield, it 525 is considered unlikely that a more precise treatment of the QFEB process would alter the combined yield significantly. At the lowest proton energies considered, the yields from the CEB and QFEB mechanisms are seen to be practically independent of target thickness. This is because all three targets are thick conpared with the photon extinction depth J*"1 cos 9, which is 3xlO~4cn at k = 3 keV. However, at higher k (i.e., where n'1 cos 6 is greater than target thickness), both the CEB and the QFEB yields increase with thickness, since these targets are range-thin for the 50 MeV proton. 2 Eqs. (2) and (3) for d^^ /dkdfc and d YaB /dkds? contain J%0(T0) as a sinple way to take account of the «ffacts of electron ranges on the x-ray yields, as discussed above. These effects, which reduce the x-ray yields, are probably exaggerated in those equations. In order to estimate the magnitudes of these effects (and hence the possible error), calculated yields are conpared with a second set of confutations, in which R_o is replaced by zero. In the EB process, when IU. is included, the yield is reduced at the lowest k values (by 24% at k =•= 3 keV and 7% at 5 keV), but is reduced by less than 1% for k > 10 keV. In the CEB mechanism, the factor exp ("MRgQ /cos 6) in Eq. (3) leads to a reduction of the yield at small k (by 54% at k = 3 keV, 30% at 5 keV, 7% at 10 keV, and 1.5% at 20 keV). In addition, there is the dependence3 of T^^ on R^. This has less than a 1% effect on the 0.32 an target and less than 6% on the 0.025 cm target, but beoomes significant for the 0.0025 en target, where the reduction is negligible at k « 3 keV, but rises slowly to 40% at 20 keV, 60% at 50 keV and 70% at 70 keV. As far as the total yield, d2Y/dkdfi , is concerned, the errors introduced in the present calculations by the approximate treatment of electron range are probably well below 10%, for photon energies above 4 keV, except for the yield of k > 20 keV photons from the 0.0025 cm target. IV*. Experimental Results Figure 4 compares the experimentally determined bremsstrahlung yields from the 0.0025 cm, 0.025 cm, and 0.32 cm aluminum foils, with the theoretical predicted total yields for these thicknesses. It is seen that at 20 keV, for example, there is almost an order of magnitude change in the experimentally determined brensstrahlung yield corresponding to the almost order of magnitude change in the foil thicknesses. This is due to the sharp differences in slope of the yield versus photon energy curves as a function 526 of foil thickness. The sharp peaks which appear on each of the experimental curves are due to characteristic K a x-ray lines and toaT line. The characteristic lines are due to the beam scattering from the target foils and striking the walls of the stainless steel test chamber. Chromium, iron, cobalt, and nickel K a lines can be seen at 5.4 keV, 6.4 keV, 6.9 keV, and 7.5 keV, respectively. The 7 line at 14.4 keV is due to the reaction: 56Fe30 + lpo 57^30 + 14>4 tevl6> V. Discussion A comparison of the theoretically predicted bremsstrahlung yields with experimentally determined values is shown in Figure 4. It can be seen that the agreement is relatively good and that the shape of the theoretical curves, their slopes, and their quantitative yields are very close to the experimental curves for all three thicknesses of aluminum targets. The experimental yields are larger than the theoretical values by factors which range between 2 and 3, with the error increasing with foil thickness but decreasing, somewhat, with photon energy. A review of the potential errors introduced by assumptions made in developing the theoretical yields cannot account for differences of factors of 2 or 3. 2 2 The terms used in the equations for d YTO /dkdfi , d Y_n /dkdfi , and C*I3 \ Tip OFES /dkdJ] were obtained from experimental data or empirical tables. This includes values for the bremsstrahlung production differential cross- sections, do/dk, the cascade factor R chamber, sane of which enter the detector. Moreover, the scattered bean will strike the target foils again, producing additional bremsstrahlung radiation. It is noted in Figure 4, that the difference between the experimental curves and the theoretical curves increases with thickness. This would be consistent with the suggestion that thicker targets produce more divergence of the beam and thus more "extraneous" bremsstrahlung radiation from the walls of the chamber. Further support for this suggestion canes from noting that the intensity of the 14 keV gamma peak increases by almost an order of magnitude as the thickness of the target increases from 0.0025 cm to 0.32 cm. While a quantitative analyses has not been carried out to determine the extent of this experimentally produced "extraneous" Bremsstrahlung, a factor of 2 to 3 above the theoretically predicted value does not seem extreme. Acknowledgments This research was supported jointly by the SDIO ISH-4 program and under DNA Task Code X99GPXXD and Work Unit Code 00003, sponsored by the Weapons Laboratory, under Interagency Agreement DOE No. 1418-1418-A1 and the Office of Health and Environmental Research, U.S. Department of Energy, under Contract DE-ACO5-840R214OO with Martin Marietta Energy Systems, Inc. The authors gratefully thank the Argonne National Laboratory's staff who operated the Neutral Particle Beam Test Stand for their assistance in providing us with energetic atoms and D.A. Cushing and S.S. MbCready of the Air Force Weapons laboratory for their computer code assistance. 1. F.Folkmann, C.Gaarde, T.Huus and K.Kenp, Nucl. Instr. Meths. U§, 487 (1974). 2. L.V.Spencer and F.H.Attix, Pad. Res. 3, 239 (1955). 3. Secondary electrons are assumed to escape without producing radiation, if and only if, they are born at a depth greater than I - RP50(T0), where TQ is their initial kinetic energy. 4. U.Fano, in "Studies in Penetration of Charged Particles in Matter", Committee on Nucl. Sci., Nucl. Sci. Series Rept. No. 39, P.287, (Nat. Acad. of Sci.-Nat. Research Council Publication 1133, Washington O.C., 1S64). 5. H.W.Schnopper, J.P.Oelvaille, K.Ralata, A.R.Sohval, H.Abdulwahab, K.W.Jones and H.E. Wagner, Phys. Lett. 47A, 61 (1974). 528 6. A.Yanadara, K.Ishii, K.Sera, M.Sebata and S.Merita, rhys. Rsv. A 22> 24 (1931). 7. G.Elwert, tan. Phys. 34, 178 (1939). 3. P..H.Pratt. H.K.Tseng, C.M.Dse, L.Kissel, C.MacCallun, and M.Riley, At. Cata and Nucl. Data Tables 20, 175 (1977). 9. R.O.Iane and D.J.Zaffarano, Phys. Rev. 94, 960 (1954). 10. Internat. Comission on Radn. Units and Measurements, stopping Powars for Electrons and Positrons. ICHJ Report 37 (ICPU, Bethesda KD) 76, € (1984). 11. J.F.Ziegler, Handbook of Stepping Cross Sections fcr Eneraetic Ions in All Elements. Vol. 5 of Tne Stoppirq and Baryyes of Isns in Matter. Ed. by J.F.Ziegler (Pergamon, Hew Vork, 1980), p. 141. 12. H.J.Kagenaim, W.Gudat and C.Kunz, DESY Report SH-74/7 (1574J. 13. H.A.Bearden and A.F.Burr, Re-zs. ^SDd. ?hys. 22/ 125 (1S67). 14. K.Ishii and S.Morita, tiucl. Instr. Meths. B3, 57 (1934). 15. K.Ishii arjd S.Merita, Phys. ?£r/. A 30, 2273 (1934). 16. E.Browne, J.M.Dairiki, R.E.Doebler, A.A.Shihab-Eldin, L.J.Jardine, J.K.Tuli and A.B.Buym, Table of Isctspes Seventh Edition. Ed. by C.M.Lederer and V.S.Shirley (John Wiley & Sans, Inc., ;;ev York, 197SJ, p.165. Figure 1. Calcsalated values of double differential BrersstrahliaTg yield fron a 0.0025 ca thick Al target facccarded with 50 MsV K° ators obser/ed at 45°. Figure 2. Calculated values of doubly differential Breasstrahlung yield froc a 0.025 ca thick Al target borfcarded with 20 HeV H° atens oteer-zed at 45°. Figure 3. Calculated values of doubly differential BrersstraJiling yield fron a 0.32 era thick Al target boccarded with £-0 JfeV H° atone obGerved at 45°. Fi-r-ire 4. Oocparison of experinentai arid thecretical results cf doily differential Brecsstrahiung yields, d^/d/af: , froc three different thidenesses of Al becbarded with 50 MeV H2 atons observed at 45°. 529 20 40 60 PHOTON ENERGY (keV) 530 : EB : QFEB CEB TOTAL 20 40 60 PHOTON ENERGY (keV) 531 10 r : QFEB • : CEB : TOTAL 10 10 h E o H 10"7 z o O I a. 10 -9 10 20 40 60 PHOTON ENERGY « <0 u X <=4 0.32 cm Toj t o. T _ O i to 0.0 7.0 H.O 31-0 J5.0 PHOTON (K«VI 533 EXCITED SUBSTATES MIXING OF FAST HEAVY IONS IN SOLIDS J.P.ROZET. A.CHETIOUI, K.WOHRER. C.STEPHAN. F.BEN SALAH, A.TOUATI. M.F.POL1TIS, D.VERNHET Institut Curie. Section de Physique et Chimie and University P.M.Curie 75231 Paris Cedex 05. CNRS URA 1379. Recently, anomalous population of capture nJ subleveis of Kr * ions emerging from thin soJid foils has been reported (li This effect was tentatively interpreted on the basis of a Stark mixing in the wake electric field of the moving ion (* io V/cm for 35 Mev/u Kr * ions in C). Such a field has been quantitatively observed in a plasma source 12). However, in solid medium, only partial evidence of this phenomenon has been given, through experiments on molecular dissociation (3) and energy splitting of excited states of ions in solids (4). In any case a possible modification of collisional processes in solids by a wake electric field has to be investigated. Some outcomes have been identified for testing ; these include, among other effects, a difference of the polarization of the light emitted by ions emerging from gaseous or solid targets. A weli known drawback of experimental studies with solid targets, however, is the multicollision effets. We have shown (I) from theoretical grounds that the single coilision condition is fulfilled for deepest ionic states when working with the fast heavy ions delivered at GANIL. Indeed mean free paths for the most probable process, nl-nl' excitation, are many times larger than the used target thicknesses (1). We have measured the angular distributions of the 2P-»1S transition following electron capture of 35 MeV/u Kr * ions in various gaseous and solid media. Three Si(Li) detectors were piaced at 60*. 120* and 150* from the beam axis, corresponding to 74*, 132* and 157* angles in the projectile frame. In the case of solid targets, target-detector distances were respectively 96. 125 and 100 mm ; diaphragma of 1.5, 2 and !.S mm were used to reduce the counting rate. In the case of gas targets, target-detector distances were 143, 143 and 110 mm. Very thin solid targets were used for ensuring single collision conditions. Target thicknesses were 18 jig/cm2. 4.6 2 534 and 4.5 ng/cm for C, Si and Cu targets respectively. Solid target purity and thickness were controlled by a backscattering analysis at the 2.5 MV Van de Graaf of GPS in Paris. Gaseous targets were produced with a well-known effusive jet (5). The beam-jet overlap viewed by each detector was calculated on the basis of known density profiles ; the correctness of the calculation was further estasblished by checking that a same zero polarization was found for target K-Xrays in gas as well as solid targets, solid angles being very precisely defined in the second case. Polarizations of the Lyman oc-Xrays emitted by projectiles after capture have been extracted from the best fit of their measured angular distributions. They are displayed in figure 1 : in the quoted error bars, main contributions come from statistical errors, and in the case of gas targets, uncertainties on the ratios of the beam-jet overlap viewed by the detectors (* 67.). In figure 1 are also reported theoretical polarizations of Lyman a lines including cascade contribution ; they were calculated from CDW partial nlm cross sections (6), taking into account spin-orbit depolarization and alignment transfer in the cascades. We note a good agreement between experimental and theoretical values in the case of gas targets. On the other hand a definite solid-gas difference is observed. These differences in relative intensities and polarization of Lyman transitions can be qualitatively explained by a wake electric field-induced Stark mixing of excited sublevels. Ne 0.3 <) Ar 0.2 •< >,. 3.1 C t * Cu 14 18 29 Figure 1 : Polarizations of 2P -» iS Kr35 * transitions 535/^ (1) J.P.ROZET, A.CHETIOUI. P.BOU1SSET, D.VERNHET, K.WOHRER, A.TOUATI, C.STEPHAN and J.P.GRANDIN, Phys.Rev.Lett.58 (1987) 337. (2) J.B.ROSENZWEIG, D.B.CLINE, B.COLE, H.FIGUEROA, W.GAI, R.KONECNY. J.NOREM. P.SCHOESSOW, and J.SIMPSON. Phys. Rev. Lett. 61 C1988) 98. (3) D.S.GEMMELL, J.REMILLIEUX, M.J.GAILLARD, R.E.HOLLAND and Z.VAGER, Phys. Rev. Lett 34 (1975) 1420. (4) S.DATZ, C.D.MOAK. O.H.CRAWFORD, H.F.KRAUSE, P.F.DITTNER, J.GOMEZ del CAMPO. J.A.BIGGERSTAFF, P.D.MILLER, P.HVELPLUND and H.KNUDSEN, Phys.Rev.Lett. 40 (1978) 843. (5) K.WOHRER - Thesis - PARIS (1984) (6) Dz.BELKIC, R.GAYET and A.SAL1N, Comp.Phys.Commun. 32, (1984) 385. 537 Radiation Mechanisms for Microscopic Relativistic Electron Beams Michael J. Moran University of California Lawrence Uvermore National Laboratory Livermore. CA 94550 USA ABSTRACT Relativistic electron beams are used to generate intense beams of coherent photons from the microwave to the x-ray spectral regions. The brightest sources of visible and x-ray radiation currently are based on synchrotron radiation from multi-GeV storage-ring electron beams. Continuing progress with techniques for generating microscopic-diameter (d < 1000 A) relativistic (E > 1 MeV) electron beams is making feasible new kinds of photon sources. These sources are based on diffraction radiation and they will have high radiation efficiencies for visible photon energies. With further improvements in electron beam quality, these sources might eventually be extended into the x-ray spectral region. This work was performed under the auspices of tne U.S. Department of Energy by the Lawrence Livermore National Laboratory under Contract number W- 7405-ENG-48. 538 Relativistic electron beams are used for generation of high power or coherent photon beams with wavelengths from the microwave through the x-ray spectral region. The shorter-wavelength sources tend to rely on highly relativistic beam energies. This trend is evident in synchrotron storage ring facilities where magnetic deflection of multi-GeV electrons generates intense beams of synchrotron x rays. These intense x-ray sources have propelled rapid advances in a wide variety of applications and pure research activities in the x-ray spectral region. The fullest use of these new technologies will be enhanced by the development of less expensive and more efficient photon sources. Continuing advances in storage-ring design that improve the beam quality and lifetime, or reduce the size and cost of the facility, represent one important avenue for improvement. However, and alternate path may be becoming available with the development of microscopic-diameter relativistic electron-beam sources. These sources have been developed in the pursuit of high-energy electron-beam microscopy, but they may &e adaptable to use as coherent photon generators. The design of efficient photon generators that use electron beams with only moderately relativistic energies and microscopic dimensions may rely radiation mechanisms that differ radically from the conventional synchrotron photon sources mentioned above. The paragraphs below will review the basic nature of radiation by relativisttc electrons from the perspective of virtual photons. This approach will be illustrated with examples based on radiation mechanisms relevant to the resent topic. A comparison of the different mechanisms will show that some lend themselves naturally to efficient photon generation with microscopic electron beams. Finally, the characteristics of photon sources based on diffraction radiation (OR) will be discussed. Photon generation by relativistic electrons can be described as removal of "virtual photons" from the time-dependent Coulomb field of the electron. This approach derives from the method of virtual quanta developed independently by C. F. Weizsacker and E. J. Williams in 1934'. in this approach, the radiation generated when an energetic electron interacts with a scattering system is determined by calculating the radiation scattered when the electron is replaced by a "packet" of photons that reproduces the time-dependent Coulomb field of the ongtnal electron. This "packet" is sometimes called the 'virtual* , or "pseudo*. photon field associated with the electron. The virtual photons have a spectral density. N^. given approximately by2: 539 where <* is the fine-structure constant (e*=l/137). E is the electron energy, A is a constant of order unity, and CJ is the angular pnotcn frequency. Thus, the spectral density has a }/v> distribution for photon energies up to the particle kinetic energy, where it is artificially terminated. if the N(j in Eq.(l) represents the spectral density of photons available to be radiated in a given interaction, then Nu also represents the maximum number of photons that can be radiated by a single interaction into dco. The leading term of <* explains the relatively low efficiencies that seem to plague basic radiation mechanisms, even when they are 'driven* by extremely high-energy electrons. Increasing the particle energy does not increase the spectral density of virtual photons appreciably - it only pushes the maximum photon energies to higher values. These considerations also are consistent with more formal calculations of limits on the photon density that can be radiated by a point charge.3 Because of these limitations, the relative efficiency of a given radiation mechanism can be conveniently expressed in terms of the ratio of the radiated spectral density to the No of Eq.(i). One additional feature of £q.(1) should be pointed out. The fine- structure constant is given by <*= e^Tta . If the electron :s replaced by a particle of charge Oe. then <* in eq.(l) will be replaced by 02ot. Thus. under some conditions, aggregates of charge can radiate with dramatically increased efficiency. This is the basic mechanism by which the "bunched" electron beams of free-electron lasers generate extremely mgn intensity coherent photon beams. The mechanism for radiation in the virtual photon description is that some interaction disturbs the phase relationship between the electron and its accompanying virtual photon packet. If either the electron motion or the spatial phase of the photon packet is disturbed by interaction with a scattering system, then the electron may follow a path that differs from the virtual photon's, with the virtual photons then appearing as actual radiation. This process can be described by the product of the virtual photon spectral density and the probability for tfte interaction of interest. Table l uses this approach to give simple descriptions of the spectra! density of radiation associated with a number of common radiation mechanisms. 540 Table 1 Spectral Density from Common Radiation Mechanisms Radiation Mechanism Virtual Photon factor Characteristic Term mv 4 3 iT 2 Bremsstrahlung - -rrlz Transition Radiation5-15 Cherenlcov Radiation6-0 Synchrotron Radiation4-7-*1 Wiggler. Undulator Rad. NK2 o( f 1 Diffraction Radiation6-e Notes: a Bremsstrahlung (BS) describes radiation due to scattering by an atom with nuclear charge Z. Also, m is the electron rest mass, e the electronic charge, and r0 the classical radius of the electron. b Transition radiation (TR) is a surface phenomena that scatters basically the entire virtual photon spectrum - as long as there is at least one formation 541 length of scattering material. Here. Wp is the electron plasma frequency of the material of interest at a vacuum boundary. c Cherenkov radiation (CR) is similar to TR. except that it is a bulk phenomena. Here. n(a>) is the refractive index at frequency u>. The virtual photons are radiated once every formation length. Z, where Z= \B(\F{c/ <* Synchrotron radiation (SR) is emitted in response to the transverse acceleration of an electron due to deflection in a magnetic field. For a single deflection Ks Kcm)B(T). where I is the pathlength of the interaction and B(T) is the magnetic field strength tn tesla. The emission has a broadband distribution which rolls 2 off rapidly for a> > uc . where wavelength given by: X(8) e Diffraction radiation (DR) is generated when an electron passes through a system that is capable of diffracting photons. The radiation pattern represents diffraction of the virtual photon distribution. The function F represents the complex form factor that describes the diffracted radiation fields. The emission distribution often will scale with the parameter r— , where "3 is the electron total energy divided by its rest mass, X is the radiated photon wavelength, and a the characteristic transverse dimension of the radiator. Often, when 2TX=2na . then JFJ>1 . Table I includes a number of different radiation mechanisms. BS can be seen as scattering of the virtual photon field as a result of collisions with nuclei of charge Z. TR and Cfl represent interactions between an electron and the dielectric constant of a continuous, or semicontiniious. medium. Here, a key parameter is the 'formation length' 2. When the "dephasing" between the electron and virtual photons occurs smoothly, the distance required for a relative dephasing of 1 radian is called the formation length. For both TR and CR the virtual photons are radiated when the particle travels ono formation length in the material. For TR this can occur once at every medium boundary,- but for CR this process continues indefinitely through a continuous medium until it is 542 terminated by some other condition (elastic scattering, finite thickness, etc.). The characteristics of synchrotron sources have been discussed in detail by a large number of authors. For the present purposes it is important to note that SR has several major advantages over other mechanisms for photon generation. These sources have allowed existing high-energy synchrotron storage rings to serve both for high-energy physics experiments as well as high-intensity photon sources. The parameters of the stored electron beams allow the use of relatively large deflection magnets having precisely known magnetic field strengths of the order of 0.1 to 1 tesla. This arrangement allows precise prediction of the expected radiation emission distributions. Finally. SR does not require physical interception of the e" beam and allows the use of these sources in storage rings having long beam lifetimes. For lower-energy e~ beams and physically smaller systems SR will begin to have poor emission efficiencies, largely because the parameter "K" will assume lower values. In microscopic wigglers or undulators it will be very difficult to precisely establish high magnetic field strengths and the length 1 also will be reduced. The product of these factors will lead to much smaller values of K. and since the SR emission is proportional to K2, the output efficiencies are liable to suffer extreme reductions, compared to larger systems. Of the mechanisms described in Table I. BS. TR. and CR are highly efficient, but they are relatively undesirable because they require direct physical interaction with physical targets in the beam, interactions of this type tend to reduce the overall system definition by broadening the spatial and energy distributions of the e~ beam and by producing undesired spurious radiation. For some applications, these difficulties sometimes can be overcome with extremely thin targets. DR is an alternate mechanism that will become more and more useful for photon generation as techniques are developed to produce microscopic diameter e~ beams. DR here refers to radiation generated when an electron passes near or through a dielectric structure without actually striking it.8-9 Thus. DR includes processes like the Smith- Purcell effect10 and Qherenkov radiation generated by electrons passing through or near dielectric structures.1 * A feeling for some of the important features of DR as a photon source can be developed with the aid of a simple example: that of DR 543 generated when an electron having speed v passes thrcugn a single s;>t of /i/idth a. Figure 1 snows a scnematic diagram of the situation. T^e spatial origin is cfiosen as the point wnere trie electron passes througft tne slit, and tne plane of the slit is assumed to be normal to tne electron velocity. Figure I S:r,gie-Si:t OR Geometry Ter ttifcaelian3 has used a pseudcpnotcn approximation approacn to ;atain a solution :n closed form for the radiated fieids: (la) • and 4TT*C ! '• \\ '-'"» 544 (1c) Eu = f-ik. . where dN is the number of photons with frequency u that are radiated inro bandwidth do and solid angle dQ. and Ex and Ey are the polarized electric field amplitudes in the x- and y- directions, respectively. Also. <* = to 1/137. k = o)/c. k =ksinecos^. k =ksinesinp. i- x a = — , and y» and y. are the distances of the electron from the top and bottom of the slit, respectively (y» • y. =a , see Fig. 1). Generally speaking. Eq. (1) describes an emitted beam with a half angle of 1/7 centered on trie direction of electron motion. The intensity and polarization of the beam depend on the particle and slit parameters. Figure 2a shows comparisons of angular distributions in the y-z plane of 2-eV TR and DR photons generated by 1.5-MeV electrons. For the TR the real parts of the dielectric constant n(u) are 0.5 and 1.5: and with the imaginary part of n(u) at 0.5 (absorption length = 0.2 pm), the 2-^m thick membrane gave TR from the exit surface only. The TR is calculated with Krogers formulation.12 which also includes contributions from Qerenkov radiation. "•DHLS ;.••• -O-9Q1.S 3 **B -A0R1.S 1 »-5 ••-0R1.S 3.»-S •-OR1.S 1.«-4 -v-TNt.S fl.t.3 • •D'1.5 '•• SB « TR/OR 1.5-nav Cofnparuon «r;ll ol Enns.an Innli'j (a) (b) Figure 2 Comparisons of Diffraction Radiation and Transition Radiation 545 The DR is calculated with Eg.CD. The plots show results for electron passing through the center of slits having a variety of widths a. The Basic angular distributions for TR and DR are similar in that they nave on-axis "zeros" that are associated with the transverse symmetry of the systems. The DR formula scales with the parameter vx/a-, and when jrx/a > 2TT, then the DR intensities can be seen to be comparable to those for TR. Note that as a is reduced further (so that ffx/a »?.TL) cnly small additional increases of the radiated intensity are obtained. Figure 2b shows a similar comparison, except a single slit of width 5 pm is used. The electrons pass through the slit at a variety of heights. Again, the DR intensity becomes comparable to that Trcm TR when ffx/d > 2TC, where d is the distance from the electron path to the closest slit edge. Here, the transverse symmetry is broken and significant on-axis emission intensities are evident. The TS distributions shown above represent efficient emission of the available virtual photon density. Thus, the comparison with TR is ere simple way to gauge the efficiency of DR. Two general features can se seen in the figures above: 1) that OR is efficient when 2C\/d > 2TZ: and 2) that further changes in the system that give ?X/d » 2T: give c-'s minor further increases of emission intensity. However, even when £X/d » 2TT the actual photon emission efficiency is still rather small (=10** ?not/elect). One way to increase the emission efficiency is to increase the number of slits in the source. This approach is the same as with synchrotron radiation, where going from a single magnetic "bend" to a series of bends in a wiggler or undulator increases the source efficiency. Here, however, the perturbation on the electron trajectory is always small, so that a periodic series of slits will be analagous to a highly coherent undulator. Furthermore, if 3"X/d » 27t. then the slit array efficiency also will correspond to a wiggler with K = l. Another feature of the series of slits is that the radiation will tend to Se ftighiy linearly polarized in the direction perpendicular to the slit itself. These characteristics are illustrated in the illustrations below. Figure 3 shows a pseudo 3-D plots of the x- and y-polanzed 2-ev emission far-field angular distributions from a 0.7 pm slit illuminated &y a 1 .S-MeV electron passing through its center. In this and all of ifie plots to follow the units of intensity are photons/eiectrcn-steradaan- (fractional bandwidth). The distributions show the basic features that we have mentioned above: that the distribution has an cn-axis "zero-, and 546 <% (*}) Figure 3 X- and Y- Polarized "On-Axis" Diffraction Radiation Distributions that the output is dominated by y-polanzed photons (peak !x = 0.24 peak ly). Also, the x- and y- polarized photons occupy complementary portions of the output distribution. Figure 4 shows results for a similar situation, except that now the electron is passing at one tenth of the slit height from the top edge. Now that the transverse symmetry is broken, significant on-axis intensities are present, although they are not as evident as in Fig. 2b. The overall intensity is larger and the relative x-polari2ed output is somewhat larger (peak Ix = 0.29 TpGj^) "X" Figure 4 x- and Y- Polarized "Grazing" Diffraction Radiation Distributions 547 Since the DR is spatially coherent with respect to the electron, the field radiated by a series of slits Es can be obtained by adding the field from each slit while keeping track of the relativistic phase shift resulting from the separation between the slits. This sum can be written: N-1 Es = (2) where Ei is the field from a stngle slit (Eq.(U) and 2TCX0 (3) where £=v/c. and / is the distance between slits. -So •X" 'Y* Figure 5 x- and Y- Polarized "Grazing" Diffraction Radiation Distributions from Symmetric Slit Array Figure 5 shows distributions calculated in this way of 2-eV photons from a series of ten 0.7-jim wide slits spaced by 20 ;im, and illuminated by 1.5-MeV electrons passing within one tenth of the slit height from the top edge. This corresponds to a series of siats identical to the one used 548 for Fig. 4. With ten slits, the peak intensity in Fig. (6) is about two decades greater than for the single slit (Fig. (5)). Although the angular- spectral intensity increases with the square or the number of slits, the actual photon yield only increases linearly with the number of slits. The high on-axis intensity peak results from constructive interference between the slits and will depend on the choice or parameters for the calculation. 2o "X" *Y* Figure 6 x- and Y- Polarized "Grazing" Diffraction Radiation Distributions from Asymmetric Slit Array Figure 6 show results from a similar calculation, except that the slits are separated by 40 jim and the slits alternate: as the electron follows its trajectory, it passes first near the top (within one tenth of the slit width) of the slit, then it passes near the bottom of the next slit (within one tenth ...) . then near the top .... until it passes through all ten slits. This approach is somewhat analagous to a magnetic undulate, where the deflection of the electron reverses at each successive magnet pole. Here, the radiated field from the first, third, etc.. slits are still Ei. but the field Ea from the second, fourth, etc.. slit is given by: Ea = -E (4) 549 Tnis result was cntained from Eq. (1) above. The sum for the entire ser.es £t now secomes: N/2 Et = (5) This sum further destroys the symmetry of the system, with resulting asymmetries evident in the low-level portion of the emission distributions (especially the y-polartzed part). 'X' •y- Figure 7 Diffraction Radiation Spectra Symmetric and Asymmetric Slit Arrays As a final demonstration of the nature of these systems. Fig. 7 shews calculations of the on-axis spectra fro 0.1 to 10 Mev ccrrespcnding to Figs. 5 and 6. Tfie solid lines correspond exactly to figs 5 and 6. while the dotted lines correspond to slit-to slit spaemgs of 5 and 10 jjm. respectively. These spectra show that the 'fundamental* emission nas an cn-axis wavelength of about / /(272). as expected from retativistic contraction. Shorter-waveiength harmonics are suppressed By the system symmetries and the ZTX/a 2 2TC. The calculations above show that significant emission fluxes from diffraction radiation can &e achieved with currently avaiia&le high-energy 550 electron-beam microscopes having a beam energy of 1.5 MeV and a beam diameter or less than 2000 A. As the quality improves and the beam enerbgy increases, it may become feasible to design similar sources that would reach into the x-ray spectral region. REFERENCES: 1 Several authors have given discussions of this technique. See W. Heitter, The Quantum Theory or Radiation . Dover Publications. N. Y. (1984), Appendix 6, J.O. Jackson, Classical Electrodynamics . John Wiley & Sons. N. Y. (1975). Section 15.4. or M.L. Ter-Mikaehan. High- Energy Electromagnetic Processes in Condensed Media , Wiley- (nterscience. N. Y. (1972). Chapter 1. 2 W.K. Panofsky and M. Phillips. Classical Electricity and Magnetism . Addison-Wesley Co., Cambridge. Mass. (1955). p. 295. 3 P.L. Csonka. Phys. Rev. A3£. 2196 (1S87). 4 J.O. Jackson. Classical Electrodynamics . John Wiley & Sons. N. Y. (1975). 5 M.L. Cherry, G. Hartmann, D. Miiller. and T.A. Prince. Phys. Rev. DUO. 3594 (1974). 6 J.V. Jelley. Qherenkov Radiation and its Applications , Permagon Press. N.Y. (1958). 7 E-E Koch. ed.. Handbook on Synchrotron Radiation . North-Holland. N.Y. (1983). 8 M.L. Ter-Mikaelian. High-Energy Electromagnetic Processes in Condensed Media , Wiley-lnterscience. N. Y. (1972). Section 31. 9 B.M. Bolotovskii and G.V. Voskresenskii, Sov. Phys. Usp.fi 73 (1966). 10 S.J. Smith and E.M. Purcell. Phys. Rev. §2. 1069 (1953). 11 E.P. Garate. S Moustaizis, J.M. Buzzi. C. Rouille. H. lamain. j. Walsh, and B. Johnson. Appl. Phys. Lett. 48, 1326 (1986). 12 E. Kroger, Z. Physik 213.. 403 (1970) 551 INTERACTION OF LOW-ENERGY ELECTRONS AND POSITRONS WITH CONDENSED MATTER: STOPPING POWERS AND INELASTIC MEAN FREE PATHS FROM OPTICAL DATA J. C. Ashley Health and Safety Research Division Oak Ridge National Laboratory Oak Ridge, Tennessee 37831-6123 USA ABSTRACT An "optical-data model" is described for evaluating energy loss per unit pathlength and inelastic mean free path for low-energy electrons and positrons (< 10 keV) from optical data on the medium of interest. Exchange between the incident electron and electrons in the medium is included. Results from the optical—data model are compared with previous theoretical calculations. 552 INTRODUCTION In the last decade there has been a resurgence of interest in the use of positrons as a probe for studying a wide variety of systems. Various experimental studies and results were reviewed recently by Schultz and Lynn [1]. The properties of positrons make them an important supplement to electron spectroscopic techniques [2] and in some cases prove superior to electrons as test probes. Average quantities such as the inelastic mean free path and energy loss per unit pathlength are important in describing the interaction of electrons and positrons with condensed matter. The inelastic mean free path A, or the average distance traveled by an electron between energy—loss events, is useful in surface spectroscopic techniques (e.g. photoemission) as a rough measure of the depth sampled by the emerging electrons [2]. The mean energy loss per unit distance traveled by an electron or positron, or, in the absence of radiative losses, the stopping power of the medium for these particles, S, is important in predicting dose distributions from beams of electrons or positrons. We have recently described an "optical-data model" for predicting inelastic mean free paths and energy loss per unit pathlength for electrons with energies < 10 keV using measured optical data for the material of interest [3]. This model is readily used to predict S and X for positrons by omitting the exchange correction required for the interaction of an incident electron with the electrons in the medium. That is, to lowest order in particle charge (e2) for this energy region, we assume that the only difference between an electron and positron is due to exchange. The response of a medium to a given energy transfer u and momentum transfer q is described by a complex dielectric function €(q,a>). While in general E 553 may be a tensor, here we assume the medium is homogeneous and isotropic so that E is a scalar quantity which depends on the magnitude of q and not its direction. Given e(q,w) for the medium of interest, the probability of an energy loss w per unit distance traveled by a non-relativistic electron or positron of energy E is [4,5] q+ q_ where q = V 2 [ y* E ± y* E—u ]. This expression for q^ assumes that the energy—momentum transfer relation for the electron or positron moving in the medium is the same as that for a free particle in vacuum. Note that we use atomic units (a.u.) where Tt = m = e = 1. Results will be expressed in more conventional units as needed. Integration of the differential inverse mean free path, T(E,J) over the allowed values of u> yields the inelastic mean free path A through A"1 = [ du> r(E,w) . (2) The mean energy loss per unit pathlength, or the stopping power, is given by S = J dwu T(E,W) . (3) The energy range of interest here is from a few tens of eV to 10 * eV. The upper limit is set by the wide availability of stopping power tabulations [6,7] for E > 10 iceV. Stopping power data for electrons are available for reveral materials for E < 10 keV, some of which is reviewed in Refs. 3 and 7. Calculations for S and A 554 for positrons in this energy region have been based on electron gas models to describe the medium [8,9]. We will briefly review the optical—data model and compare the results with other model calculations. OPTICAL DATA MODEL The material property required to evaluate S and A is the dielectric response function €(q,w). Since most of the information on e is for the q « o limit, i.e. optical data, a model of the medium's response is required to connect e(q,w) to the optical data e(o,w). The extension of the energy—loss function to q > o from the optical limit is made through [3] ,w)] = f du'W Im[-l/e(o,w')] 6{u-{u' + q2/2))/w o (4) which assumes a simple quadratic extension ?nto the energy— and momentum—transfer plane. The energy—loss sum rule, or the oscillator-strength sum rule, Q0 I do; u) Im[-l/€(qtw)] = 2A Z, (5) where nQ is the density of atoms or molecules in the medium with Z electrons per atom or molecule, applied to Eq. (4) is obviously satisfied as long as the input data, the optical energy—loss function, obeys Eq. (5). Results for S and A for 555 electrons in several materials were given in Ref. 3 along with comparisons with other theoretical calculations and with experimental results to validate this approach. S AND A FOR POSITRONS AND ELECTRONS Equations (1) and (4) lead to the differential inverse mean free path OD i,w')] F(E,w',u/) , (6) where F(E,w ,w) - 6{(O - -j - w) 0 («' - to + -j-)/w(w - u') (7) and 0{x) = 1 for x > 0 and 0{x) = 0 for x < 0. The region of integration over w and w' are shown in Fig. 1 in terms of y = w/E and y' = w'/E. For positrons the integration covers the whole region under the curve y' = 2 V 1-y (1 - T/ 1—y )• Thus Eqs. (6) and (7) lead for positrons (p) to E//2 VE) = 21FE f dw'w' Im[-l/€(o,o;')] G (»' /E) (8) P 0 where 556 Gp(a) = 2[6i(2/a) - a - 3a2/4] with s = V 1—2a . For the positron inelastic inverse mean free path E//2 (E) = dw V 2^EJ J ' W-l/e(o,W)] Lp(W /E) (9) 0 where Lp(a) = &[(l-a/2+s)/(l-a/2-s)] a Ai(4/a) - 7a/4 - 33a2/32 In the large E limit, Eq. (8) leads to the stopping power for positrons in the non- telativistic limit given by V 2 where ftp = 4m Z and 1 is the mean excitation eneigy defined by 0 For electrons we include exchange between the incident electron and electrons in the medium. In analogy wiih the structure of the non—relativistic 557 Miller cross section [10,11], define an "exchange-corrected" DIMFP, T , by rexcv(E,w) E ^ J du'W Im[-l/6(o,w ,w',E + W- w) - [F(E,fc/',w) F(E,w',E+w'- w)]* } . (10) The first two terms on the right hand side (rhs) of Eq. (10) account for indistinguishability. That is, an energy transfer u by the "primary" electron reducing its energy to E—u (1st term, rhs) gives an electron which cannot be distinguished from the "secondary" electron of energy E—w (2nd term, rhs) produced by a different energy transfer from the primary electron to the "struck" electron. The third term on the rhs accounts for exchange between the "incident" electron and the struck electron. The expression in {} in Eq. (10) reduces to an expression proportional to the non—relativistic Miller cross section when w and E—u> are large compared with the magnitude of the "binding energy" u'. In addition, as shown below, Eq. (10) leads to the non—relativistic Bethe formula for stopping power. In the absence of a first—principles, quantum—mechanical calculation of indistinguishability and exchange appropriate to the present model system, Eq. (10) should provide a reasonable first approximation to incorporate these corrections. Accounting for exchange and indistinguishability through Eq. (10) will be referred to simply as "exchange" in the following discussions. With Eq. (10), the stopping power for electrons is given by 558 E/2 j(E + W) f [ + [(E + w - w) (E - w)] * - [w(w - W) (E + w - w) (E - w)] * } , (11) where 2w TT The region of integration is over the shaded area in Fig. 1. For metals, where states in the conduction band are filled up to the Fermi energy Ep, there is an additional restriction on energy transfers in calculations of both S and A. The maximum value of u should be the smaller of { j E; E — Ep }. This will restrict the validity of the results for metals to E — Ep >. 3Ep. The integrations over w in Eq. (11) can be done analytically [3]. For our purposes the following approximate expression is sufficient: E/2 dw w ~ iE j ' ' M-Ve(ow')J Gc(«'/E) (12) with 559 3a afa4Al3/2_a?fa4 312 From this form, in the large E limit, we obviously recover the non—relativistic Bethe formula 2 lP . 2E \~T Aside from small relativistic corrections, which could be incorporated if desired [12], Eq. (12) provides the smooth connection at 10 keV to existing tabulations for E > 10 keV. A similar calculation for the electron inelastic inverse mean free path leads to the approximate expression [3] E/2 du} iE J ' I*[-l/6(o,«')] Le(«'/E) , (13) where From the above expressions, in the large E limit, we have 560 E/2 A"1 - A"1 a -I—* \ dw'u' Im[-l/e(o,w')] 6i(4/a) P e ^-irvr J Op2 and Thus, as expected, S > Se and A" > A" (or XQ > A ), but also the difference in electron and positron mean free paths decreases more rapidly than the stopping power difference as you go to larger particle energy. CALCULATIONS OF S AND A FROM OPTICAL DATA For Al we used a revised and carefully checked set of optical data [13,14] to calculate S and A for positrons and electrons with the optical-data model. The results are summarized in Tables I and II. The good agreement of S and A with previous electron—gas model calculations, including inner-shell ionization calculated from generalized oscillator strengths, has been discussed in detail in Ref. 3. The results for S and S for Al are shown in Fig. 2 for E < 10 keV. These P e stopping powers agree to within ~1% with tabulated values [7] at 10 keV; for E > 10 keV, tabulated values are shown as dashed curves in Fig. 2. The slight "structure" in the curves in the region of a few hundred eV is dae to the 561 contribution of L—shell electrons which show an onset at ~ 73 eV and a relative maximum at ~ 100 eV in Im[—l/€(o,u/)]. In Fig. 3 the solid curve shows our results for the positron inelastic mean free path in Al. The dashed curve is the RPA calculation of Nieminen and Oliva [9, Fig. 1] which does not contain inner-shell contributions. From our earlier results for electrons [3] inner-shell ionization amounts to ~14% of X~ at 10 eV and ~11% at 10 eV. The "dot-dash" curve is for interaction of an electron with conduction band electrons only; the difference from the positron mean free path should be small at these higher energies. Although the results from Refs. 8 and 9 (dashed curve) agree reasonably well with our calculation at lower energies, it departs from the expected results at higher energies by almost a factor of 2 at 10 eV. A similar discrepancy is found for S from Ref. 9 where the rate of energy loss is about 1/2 the expected value at 10 eV. The neglect of inelastic interactions with inner-shell electrons and the anomalous drop off of X~ and S at the higher energies may influence other results in Refs. 8 and 9 based on these quantities. For example, the mean energy loss per inelastic interaction, S/A~ , is given in Fig. 4; the full curves for electrons and positrons show large differences from predictions based on interaction with conduction-band electrons only. For Au, Ag, and Cu we rely on the optical data in a DESY report [15]. Note that this data has not been subjected to the careful reanalysis as was done for Al [13,14]. The oscillator-strength sum rule, Eq. (5), provides a useful overall check on the data but does not adequately test the details of the distribution. For Al the DESY data gives a sum-rule value -5% too large leading to --8% larger value of S at 10 eV compared with SQ calculated from the data in Ref. 14. 562 Differences from Bethe-theory values [6,7] at 10 eV for S of Au, Ag, and Cu have been discussed earlier [3]. Results for S and A for Au, Ag, and Cu calculated from the DESY data are shown in Tables I and II. The mean free path ratio XJX , or equivalently the cross-section ratio a ja , is shown as a function of energy in Fig. 5. The positron and electron mean free paths are essentially the same for energies above 2 or 3 keV. A casual look at Fig. 9 in Ref. 8 which compares electron and positron interaction probabilities implies roughly equal A's at ~200 eV. However, the "electron" curve does not include exchange between the incident electron and electrons in the electron gas as we have done in this paper. In Fig. 6 we show the fractional difference in stopping powers for positrons and electrons in Al, Cu, and Au. Data for energies above 10 eV are from Ref. 7. This form of presentation accentuates less prominent features in S as seen for example in Fig. 2 for Al. CONCLUSIONS We have described a simple optical-data model for predicting energy loss per unit pathlength and inelastic mean free path for electrons and positrons with energies < 10 keV. The results for electrons were tested earlier [3] against other theoretical models and experimental data. Much less information is available for testing the predictions for positrons. Overall these calculations seem to provide a reasonable way to estimate S and A. The question of higher-order corrections proportional to the third power of the particle charge (Barkas effect) needs further study. 563 ACKNOWLEDGEMENTS This research was sponsored jointly by the Solid State Sciences Directorate, Rome Air Development Center, under Interagency Agreement DOE No. 0226-0226-A1 and the Office of Health and Environmental Research, U.S. Department of Energy, under contract DE-AC05-84OR21400 with Martin Marietta Energy Systems, Inc. 564 REFERENCES 1. P. J. Schultz and K. G. Lynn, "Interaction of Positron Beams with Surfaces, Thin Films, and Interfaces," Rev. Mod. Phys. 60, 701-779 (1988). 2. A description of various electron spectroscopic techniques can be found in the series Electron Spectroscopy: Theory, Techniques and Applications, edited by C. R. Brundle and A. D. Baker, Academic Press, New York, 1981. 3. J. C. Ashley, "Interaction of Low—Energy Electrons with Condensed Matter: Stopping Powers and Inelastic Mean Free Paths from Optical Data," J. Electron Spectrosc. Relat. Phenom. 46, 199-214 (1988). 4. J. Lindhard, "On the Properties of a Gas of Charged Particles," Kgl. Danske Vid. Sets. Mat. Fys. Medd. 28 (No. 8), 1-57 (1954). 5. R. H. Ritchie, "Interaction of Charged Particles with a Degenerate Fermi-Dirac Electron Gas," Phys. Rev. 114, 644-654 (1959). 6. L. Pages, E. Bertel, H. Joffre, and L. Sklavenitis, "Energy Loss, Range, and Bremsstrahlung Yield for 10-keV to 100-MeV Electrons in Various Elements and Chemical Compounds," Atomic Data 4, 1—127 (1972). 7. M. J. Berger and S. M. Seltzer, Stopping Powers and Ranges of Electrons and Positrons, National Eureau of Standards Report NBSIR 82—2550 (August 1982). Also available as ICRU Report 37 (October 1984). 8. J. Oliva, "Inelastic Positron Scattering in an Electron Gas," Phys. Rev. B 21, 4909-4917 (1980). 9. R. M. Nieminen and J. Oliva, "Theory of Positronium Formation and Positron Emission at Metal Surfaces," Phys. Rev. B22, 2226-2247 (1980). 10. H. A. Bethe and Julius Ashkin in Experimental Nuclear Physics, Vol. 1, edited by E. Segre, Wiley, New York, 1953, pp. 166-357. 565 11. A. S. Davydov, Quantum Mechanics, Addison—Wesley, Reading, Massachusetts, 1968, pp. 404-406. 12. J. C. Ashley and V. E. Anderson, "Interaction of Low—Energy Electrons with Silicon Dioxide," /. Electron Spectrosc. Relat. Phenom. 24, 127-148 (1981); "Energy Losses and Mean ~'ree Paths of Electrons in Silicon Dioxide," IEEE Trans. Nucl. Set NS-28. 4132-1136 (1981). 13. E. Shiles, Taizo Sasaki, Mitio Inokuti, and D. Y. Smith, "Self-Consistency and Sum—Rule Tests in the Kramers-Kronig Analysis of Optical Data: Applications to Aluminum," Phys. Rev. 5 22, 1612-1628 (3 9801. 14. D. Y. Smith, E. Shiles, and M. Inokuti, The Optical Properties and Complex Dielectric Function of Metallic Aluminum from 0.04 t° 10* eV, Argonne National Laboratory Report ANL—83—24 (1983). Available from NTIS, U.S. Department of Commerce, Springfield, Virginia 22161. 15. H.—J. Hagemann, W. Gudat, and C. Kunz, Deutsches Elektronen- Synchrotron Report DESY SR-74/7, Hamburg, May 1984 (unpublished); "Optical Constants from the Far Infrared to the X—Ray Region: Mg, Ai, Cu, Ag, Au, Bi, C, and A12O3," /. Opt. Soc. Am. 65, 742-744 (1975). 566 FIGURES Fig. 1. Region of u>-u' integration for electrons, shaded area, and positrons, entire region under y' = 2 V 1—y (1-V 1—y )• Fig. 2. S and S in Aluminum. e P Fig. 3. Positron inelastic mean free path in Al. Fig. 4. Mean energy loss per inelastic interaction as a function of particle energy. Fig. 5. Ratio of positron to electron inelastic cross sections for Al, Cu, and Au. Fig. 6. Positron-electron stopping power difference as a function of energy. 567 TABLE 1 INEIASTIC MEAN FREE PATHS (IN A) FOE ELECTRONS AND POSITRONS Al Au Ae Cu E-E (eV) A F AP e AP 'e AP 'e \ 40 4.94 3.54 18.3 14.3 15.0 11.8 9.49 7.14 60 4.73 3.76 13.3 10.0 11.8 9.05 7.86 6.06 80 4.99 4.19 9.83 7.24 9.58 7.31 7.39 5.86 100 5.37 4.65 8.20 6.08 8.43 6.48 7.29 5.92 150 6.43 5.83 6.84 5.32 7.26 5.73 7.59 6.40 200 7.46 6.88 6.72 5.45 7.07 5.75 8.08 6.96 300 9.39 8.80 7.40 6.36 7.81 6.74 9.23 8.16 400 11.2 1G.6 8.34 7.42 8.87 7.92 10.4 9.41 500 13.0 12.4 9.33 8.47 9.99 9.11 11.7 10.7 600 14.7 14.1 10.3 9.49 11.1 10.3 12-9 11-9 800 18.0 17.4 12.3 11.5 13.3 12.5 15.4 14.4 1000 21.3 20.6 14.1 13.3 15.4 14.7 17.8 16.9 1500 29.0 28.4 18.6 17.8 20.5 19.7 23.7 22.8 2000 36.4 35.8 22.9 22.1 25.3 24.5 29.3 28.4 3000 50.5 49.9 31.0 30.2 34.4 33.7 40.0 39.1 4000 64.0 63.5 38.8 38.0 43.2 42.4 50.1 49.3 6000 89.9 89.4 53.6 52.8 59.9 59.2 69.6 68.7 8000 115 114 67.7 67.0 76.0 75.2 88.2 87.4 10000 139 138 81.4 80.7 91.5 90.8 106 105 568 TABLE II STOPPING POWERS (IN eV/A) FOR ELECTRONS AND POSITRONS .41 Au Ae Cu E-E (eV) S S S f Se SP e SP e SP e SP 40 4.25 6.65 .864 1.32 .902 1.36 1.86 2.90 60 4.53 6.45 1.85 2.92 1.76 2.75 2.87 4.39 SO 4.37 5.92 3.39 5.44 2.89 4.54 3.52 5.26 IOL 4.12 5.42 4.82 7.67 3.97 6.17 3.97 5.81 150 3.55 4.49 7.29 11.1 6.19 9.37 4.61 6.55 200 3.26 4.12 8.30 12.1 7.46 11.0 5.01 7.02 300 2.93 3.82 8.36 11.4 7.60 10.4 5.39 7.41 400 2.80 3.58 7.85 10.4 7.07 9.30 5.43 7.31 500 2.64 3.36 7.35 9.51 6.51 8.36 5.27 7.01 600 2.49 3.14 6.94 8.87 6.03 7.60 5.13 6.7i SOO 2.22 2.77 6.26 7.90 5.29 6.54 4.71 6.03 1000 2.01 2.47 5.79 7.22 4.86 5.99 4.31 5.44 1500 1.62 1.95 4.93 6.11 4.11 5.06 3.5S 4.35 20UO 1.36 1.62 4.36 5.35 3.59 4.38 3.03 3.64 3000 1.04 1.22 3.57 4.32 2.89 3.47 2.43 2.90 4000 .359 .992 3.04 3.64 2.43 2.88 2.06 2.45 6000 .649 .741 2.39 2.83 1.87 2.18 1.60 5.88 8000 .527 .598 2.02 2.37 1.53 1.77 1.32 1.54 10000 .446 .504 1.75 2.05 1.31 1.51 1.13 1.31 569 0.50 - 3 0.25 - 0.50 y =uVE Fig. 1. Region of _— ^' integration for electrons, shaded area, and positions, entire region under y' =2 -J 1-y f 1—,/ l-y ). ALUtHNUM POSiTWOf* 2- to! Fig. 2. S and S in Aluminum. 570 Fig. 3. Positron inelastic mean free path in Al. ALUMINUM POMTMM "1" a- /o* Fig. 4. Mean energy loss per inelutic interaction as a function of particle energy. Fig. 5. Ratio of positron to electron inelastic cross sections for Al, Cu, and Au. 0.1 1 t t Fig. 6. Positron-electron stopping power difference as a function of energy. 573 CROSS SECTIONS FOR K-SHELL IONIZATION BY ELECTRON IMPACT RICARDO MAYOL ANO FRANCESC SALVAT Facultat de Flsica (ECM). Unlversltat de Barcelona and Socletat Catalana de Flsica (IEC) Diagonal 647. E-08028 Barcelona. SPAIN. A semi-empirical method to compute the cross section for K-shell ionization of atoms by electron Impact is proposed. The method Involves a simple schematlzatlon of the Bethe surface, which is obtained from a hydrogenic optical oscillator strength density model. The use of these oscillator strengths determines the effective number of electrons that participate in ionizing distant collisions, i.e. the contribution of the considered shell to the Thomas-Reiche-ICuhn SUM rule. Close collisions are described by the Moller differential cross section, thus incorporating exchange effects. Density effect corrections are not included in the present approach. The relationship of our approximation with the Weizsacker-Wllllama method of virtual quanta is also discussed. Supported in part by the Comisi6n Interministerial de Clencla y Tecnologla (Spain), contract n. PB86-O589. 574 1. INTRODUCTION. Cross section data for Inner-shell lonization by electron Impact are required for quantitative elemental analysis In three types of material characterization: electron-probe microanalysis (EPMA), Auger electron spectroscopy (AES) and electron energy-loss spectroscopy (EELS). Cross sections for production of ionized states are also needed in the description of the interactions of radiation with matter as well as for a quantitative understanding of radiation damage. Powell (1985) gives a comprehensive review of experimental data, theoretical calculations and seal-empirical foraulas for Inner-shell ionizatlon. Such empirical formulas are very useful in algorithms for elemental quantitative analysis; although they have been derived from a limited base of experimental and calculated cross sections and there is a risk of using them beyond the range of conditions for which they where developed. On the other hand, experimental and theoretical information on the energy-loss differential cross section (DCS) is scarce although this is the key quantity for a detailed description of the production of Ionized states which could be obtained, for instance, through Monte Carlo simulation. In the present communication a simple method to obtain approximate DCS for inner-shell lonization Is described. The method involves a simple Bethe surface model constructed from the optical oscillator strength (00S) density which may be derived from available experimental data or approximated by suitable theoretical calculations. We will consider in detail the lonization of K-shells since In this case the COS can be closely approximated by using a hydrogenic model (Egerton 1986). The evaluation of the DCS for other shells may be performed In a similar way, but then one needs to develop a satisfactory algorithm for determining the component of the 00S associated with ionization of the considered shell. From a particular point of view, the present method is similar to the 'statistical' models of Tung et al. (19793, Ashley 575 (1982.1988) and Perm (1987). Inelastic collisions of electrons with kinetic energy E, with a target atom of atomic number Z are conveniently described in terms of the energy loss W and the recoil energy Q. This last quantity Is defined by (Fano 1963) Q(l+Q/2mc2) - q2/2* or Q « F{cqJ2Vc*l - me2 (1) where q is the momentum transfer. The Born DCS for energy loss W and recoil energy Q Is given by (Fano 1963. Fano and Cooper 1968. Mayol and Salvat 1989) d2 The expression (2) contains a purely kinematic factor and the GOS which embodies a complete description of the target concerning Inelastic scattering (within the Born approximation). In the Halt Q->0, the GOS reduces to the optical oscillator strength df/dW. It Is worth recalling that ths 00S and the cross section for absorption of a photon with energy W are related by 576 df me _ . ——— a- (W) (3) dW 2n**\ ph provided the wave length of the photon Is much larger than the 'radius' of the atomic shell where the photoelectric effect takes place (dlpole approximation). The GOS and the complex dielectric function e(Q,W) describing the response of any (lsotroplc) medium to a small electromagnetic disturbance are related by (Fano 1956, Pines i963) -1 p 1 df(Q.W) Im (4) e(Q.W) 2W Z dW where fl is the plasma frequency corresponding to the total electron p density In the material. I.e. a2 • 4wh2NZe2/m (S3 p N being the number cf atoms per unit volume. Unfortunately, the GOS is analytically known for only the simplest atomic target, namely the hydrogen atom (see e.g. Inokuti 1971). The complex dielectric function of the free electron gas, derived from the random phase approximation, has been given In closed analytical form by Llndhard (1954). GOSs for atoms and ions have been computed numerically by a number of authors (see references In Powell. 1985) using independent electron models. Besides the large calculation effort to obtain each value of these numerical GOS, additional interpolation and/or extrapolation difficulties arise when computing integrals of the DCS corresponding to different measurable quantities. The GOS can be represented as a surface over the plane (U.Q) which is known as the Bethe surface (Inokutl 1971}. For large values of H. the GOS vanishes except for Q«W and the Bethe surface reduces to a ridge, the Beth* ridge, which peaks around the line Q-W. Indeed, in the high-W limit, binding effects are small and the GOS may be evaluated by assuming the target electrons are free and at rest, this 577 > Z <5(W-Q). (6) dW Actually the Bethe ridge has a finite width which arises from the momentum distribution of the atomic electrons. For small recoil energies (Q«W), we have df(Q,W) df(W) » (7) dW dW In the limited range of Q values where this relation holds, the DCS (2) decreases rapidly with Q. It follows that the key quantity to determine the DCS for low-Q excitations Is the OOS. On the other hand, It is known that useful average quantities such as the total Inelastic cross section and the stopping cross section for incident electrons of high kinetic energies are completely determined by the OOS distribution (see Inokuti 1971). 2. APPROXIMATE GENERALIZED OSCILLATOR STRENGTHS. Once the two asymptotic limits (6) and (7) have been specified, the remaining task in order to construct a scheaatlzed Bethe surface model Is to specify a suitable interpolation algorithm to generate the GOS for intermediate recoil energies. If the algorithm is physically sound and simple enough, we expect to obtain accurate DCS with only a moderate amount of computer calculations. Studies along these lines are due to Tung et al. (1979), Ashley (1982.1988) and Perm (1987). The algorithms proposed by these authors have been given in the context of the dielectric formalism, we give here an alternative description to them from the atomistic point of view, i.e. using the GOS concept. The starting point of the algorithms to be described is the OOS. Nowadays, a great deal of experimental information on the OOS distribution is available either in the form of optical data (Palik 1985) or as photoabsorptlon cross sections (Hubbell 1971, Veigele 1973). 578 Tung, Ashley and Ritchie (1979) adopted the 00S obtained from the local plasma approximation (LPA) of Lindhard and Scharff (1953) by using Dirac-Hartree-Slater atomic electron densities pir) computed under Wigner-Seitz boundary conditions. The LPA sets the DOS spectrum as (8) where, W (r) is the local plasmon energy, i.e. p W2 = 4nh'>(r)e2/m. p and x is a constant that Tung et al. set at unity. It may be shown that their approach is equivalent to using the following GOS df(Q.W)' F(W;Q,H) dW CIO) dW L JTung Jn k'JIPi where F (W';Q,W) is the GOS per electron corresponding to Lindhard's ' dielectric function c for an electron gas characterized by the plasmon energy W, thus 2W f -1 F (W ;Q.W) - /« (113 L -r.» ,2 Ic. (W;Q.W) It can be seen that the use of F guarantees the correct hlgh-Q behaviour of the GOS and, moreover, the resulting Bethe ridge has a finite width reflecting the aoaentua distribution of the electrons in the target. Ashley (1982,1988) uses a similar model which incorporates experimental OOSs and accounts for exchange effects in an approximate way. At the same time, to facilitate further calculations he Introduced the following one-mode approximation FA(W;Q,W) » «(W-Wp), WpmW+Q (12) for the GOS per electron. As pointed out by Ashley (1982). thla 579 one-mode approximation gives nearly the same dispersion relation as Llndhard's theory in the low-Q limit as well as leading to the proper high-Q limit, W<*Q. Thus, Ashley's model Is equivalent to the GOS F (W;Q.W) dW. (13) L law I A 0 L J«xp Hereafter. [df/dW] stands for the 00S derived from available exp experimental data. The procedure proposed by Perm (1987) combines the advantages of the two previous models: It incorporates experimental 00S and it uses the Lindhard GOS per electron to generate the GOS for Q>0. Thus. Penn's GO? is given by i (14) F (W';Q.W) dW. These models have mainly been used to compute Inelastic mean free paths of low energy electrons in condensed natter; Ashley (1988) also gives stopping powers. For comparison of the results fro* the different models the reader Is referred to the original papers. We merely indicate that mean free paths from Ashley's and Penn's models agree to within 5'/. for electron energies between 100 eV and 10 keV (Ashley 1988). In what follows we consider the evaluation of the DCS for K-shell lonizatlon of neutral atoms. As mentioned before, the experimental OOS (i.e. the photoelectric cross section) is accurately reproduced by a simple hydrogenlc model (Egerton 1986). Following Egerton. we introduce the screening effects through an effective nuclear charge Z <-0.3, incorporating the screening of the second la electron, and a binding energy reduction E which accounts for the screening of the outer electron shells. This last quantity is obtained from the observed ionizatlon threshold energy U as E -U -U, where U *2?am*/{2hZ). The hydrogenlc OOS for K-shell lonization Is given in terms of the 580 dimensionless variable < =W/U -1 by n df (W) zso U3 exp{-(2/ic)arctan(2»e/[l-ic2])> _£ x 1 if K >0 dW 3 W* l-exp(2w/»c) (IS) 3 W (When Z-1.2, we take U-U and E «O). This OOS has a characteristic H K S saw-tooth profile which starts at the lonlzatlon threshold W»U and decreases monotonlcally as the energy loss Increases. The statistical models described above lead to realistic results for low energy electrons, i.e. when the majority of excitations correspond to the outer shells or to the conduction band. However, they are not suited to describe inner-shell lonlzatlon. To clarify this feature, let us consider the case of K-shell ionization. The model of Perm generates a GOS which does not vanish for energy losses less than the ionization threshold U and, therefore, it gives a finite cross section for excitations with WU +Q where Asley's GOS Is nonvanishing unless E>2U . An alternative aodel was used by Llljequist (1983.1985) and by Salvat et al (1985) to provide siaple estimates of the inelastic cross sections uuited for Monte Carlo simulation of electron transport. For practical reasons, the adopted OOS were only roughly approximate; they were extrapolated for Q>0 by using the function (which plays the role of the GOS per electron) FgCW'iQ.W) « a(W-W) fl(M'-Q) + a(W-O) e(Q-W) (16) This model can now be improved by Incorporating more accurate OOSs. I.e. by considering the GOS 581 •df(Q.Wdf(Q. ) fdf ] F (W;Q.W) dW 1 QW L. h(Q) SCW-Q) (17) The corresponding Bethe surface vanishes for Q>W. In the region Q h(Q) ff-1 (18) •xp Jo [dw_ where the last equality follows froa the Beth* SUB rule. Thus the aodel clearly separates the distant (Q 3. THE ENEBGY LOSS DCS AMD LOW EWEBGY The energy loss DCS Is given by do- p T — dQ (19) dQdW where the integral extends over the Interval of klneaatlcally allowed recoil energies. Let us now evaluate the energy loss DCS froa the COS (17). We begin by considering the contribution due to close collisions, i.e. to excitations In the Bethe ridge. As stated above, the target electrons behave as though they were free and at rest; the effect of binding is included through the h(Q) function which gives the effective number of target electrons that participate In collisions Involving an energy loss W«Q. The relatlvlstlc Born DCS for binary collisions with 582 free electrons at rest is given by the Holler formula do- (E, W) cue I j 1 dW which incorporates exchange effects. Here y»l+E/mcZ and the maximum energy loss is W *E/2. Thus, the energy loss DCS for close collisions is obtained as do- do- (E.W) __£ » h(W) —5 (21) dW dU The energy loss DCS for distant collisions takes the following analytical expression 2 doo- 2ire* 1 df { ,'W(Q +2mc ) 1 d ll"" +ln (22) dW mv2 W dW I IQ (W+2mc2)J where Q_ is the ninlnua recoil energy Q_ - jffE{E+2mc2)1 ±f(E-W)(E-W+2mc2)j H +»2c*l -uc2 (23) It Is well known that the Born approximation overestimates the cross sections for relatively small kinetic energies of the incident electron. This is mainly due to the distortion of the incident electron wave function due to the electrostatic field of the target. This field produces an increase in the effective kinetic energy of the incident electron which is expected to be important In close collisions. Although It Is very difficult to introduce this effect accurately, we can proceed in analogy with the classical theory of binary collisions (see Salvat et al. 1985) and assume that the Incident electron gains a kinetic energy 2U before It Interacts with a target electron which Is bound with binding energy U . The maximum energy loss is taken to be W >(E+U )/2, i.e. the allowed energy losses lie In the interval U(C 4. THE WEIZSACKER-WILLIAMS METHOD OF VIRTUAL QUANTA. This method (Jackson 1975) exploits the fact that the electro- magnetic field produced by a fast charged particle at the position of a point target is equivalent to the superposition of two pulses of plane polarized radiation Impinging on the target in directions parallel and perpendicular to the momentum p of the Incident particle. This method has been applied to the evaluation of inner-shell ionization cross sections by electron impact by Kolbenstvedt (1967) and by Seltzer and Serger (1982). Let us consider an electron with kinetic energy E passing a target atom with impact parameter b. Fourier analysis of the perturbing field of the moving particle yields the following frequency spectrum (energy per unit area and unit frequency interval measured at the target position) 2 dKw.b) e 1 X2r 2 2 2 , ["El [K |—I +y " K f—[I (24) dw it c p b * ' ^ * ' I /J where K and K are modified Bessel functions. The flux of virtual o l quanta, dN(W,b) (W-h») is obtained by using the relation dN(W,b) 1 dl(u.b) dW - — du dW hu du I2S1 Each of these virtual photons with energy larger than the ionization threshold can now ionize the atom by photoelectric effect. The ionization DCS associated with impact parameters larger than b is •in then da- dN(W) — (b>b :) • dN(W) T 2JI b db dW Jfe dU sin 2 tf I 7-,tf ( v \ V ( v > _ fl^v^j tf < v 1—mf^ / v ) J 1 f 01 "\ (^XK IX; & IXJ HX K IX) K IX J • K&.1 i ^ 0 with Mb x 9 _iii. [28) fij-v Using (3) the DCS (26) can be finally written as do- 2ne* 1 df / •, V •> I1* _ (b>b ) = J2XK (x) K (x) - fx r(x)-rtx) (29) dW ain mv2 U dU \ ° 1 L1 ° J/ For low energy losses (x«l) it reduces to Z do- 2it2 e* 1 dff r , „, lm J _ (b>b } * ; 2 In " C30) dW aln mv2 W dW >- This procedure only gives the DCS for distant collisions. Close collisions, corresponding here to impact parameters smaller than b •In may be described as binary collisions with free electrons at rest. i.e. through the Moller DCS (20) da- d The •iniauM impact parameter b is usually taken as the Bohr •In radius of the shell (see Kolbenstvedt 1967). We note that expressions (22) and (30) coincide when the arguments of the logarithms are the same. This leads to the following estimate of the minimum impact parameter b, -1.123h/(2mW)1/z C32) Kin Assuming there Is a correspondence between recoil energy and impact 585 parameter (i.e. large impact parameters correspond to smaii recoil energies and vice uenoa), we are forced to replace W in (32} by the minimal energy loss U since otherwise the DCS for distant collisions will contain contributions from excitations already accounted for In the DCS for close collisions -note that (31) contains Z rather than the fraction h(W) appearing in (21). Therefore we set b =1.123h/(2mU )1/2 (33) •In K It Is Interesting to observe that for the K-shell hydrogenic model without screening U *Zame4/(2h2) and the Bohr radius is a«h2/Zae2: In this case we have b =l.l23h/(2mU )WZ = 1.123 a. (34) tain K in good agreement with the usual estimate b =a. •in Improvement of the computed cross section near the threshold is obtained by Introducing the empirical low energy correction described in the previous section. It may already be anticipated that the cross sections obtained from our model, Eqs. (21) and (22), and from the method of virtual quanta, Eqs. (29)-(31), will be practically equivalent -except near the threshold due to the effect of the h(W) function In Eq. (21). 5. COMPARISON WITH EXPERIMENTAL DATA AMD DISCUSSION. The lonlzatlon cross section of hydrogen is given in Fig. 1 as a function of the kinetic energy of the incident electron. It Is seen that our GOS aodel and the Welzsaclcer-Wllllaas method are in close agreement for energies above the aaxlaua of the cross section and give a good average description of the experimental data In this energy range. The non-relativistlc Born cross section, derived from the GOS of the hydrogen atoa, is also Included for comparison; this cross section does not differ very much froa that obtained with our GOS model without the low energy correction. The iaproveaent due to this correction li seen to be noticeable for energies near the lonlzatlon threshold. 586 lonization cross sections for helium are plotted In Fig. 2. Again, the agreement with experimental data Is quite good for energies above the maximum of the cross section. As for hydrogen, our theoretical calculations shift this maximum to lower energies. A similar shift Is obtained with the Born approximation {Rudge 1968). In practice (see e.g. Kolbensvedt 1967), Weizsacker-Williams calculations are performed by using approximation (30) Instead of the rigurous result (29). This replacement slightly changes the ionizatlon cross section as shown in Fig. 3. The effect of the low energy correction on the Welzsacker-Williams cross section is also indicated. The uncorrected Welzsacker-Wllllams cross section shows a slight elbow at W*2UR due to the sudden start of the close collision contribution. Results for aluminum and nickel are shown in Fig. 4; again the agreement between the theoretical and experimental data is fairly good, even near the thresholds. It Is interesting to note that the ionizatlon cross section takes a minimum value near 1 HeV and it increases monotonically for Increasing energies as a consequence of relatlvistlc kinematics. Fig. 5 contains the K-shell ionizatlon cross section for gold. In this case the agreement Is not so certain because of the scattered experimental data. For high atomic numbers the use of the (non- relatlvistlc) hydrogenic model Is questionable since relativistic effects on the target electron wave function are known to be Important; this deflcency could be partially amended by using experimental photoionizatlon data. 587 REFERENCES. Ashley JC (1982) J. Electron Spectrosc. Relat. Phenom. 28. 177. Ashley JC (1938) J. Electron.Spectrosc. Relat. Phenom. 46. 199. Egerton RF (1986) 'Electron Energy Loss Spectroscopy In the Electron Microscope' (Plenua Press, New York). Fano U (1956) Phys. Rev. 103. 1202. Fano U (1963) Ann. Rev. Nucl. Scl. 13. 1. Fano U and JW Cooper (1968) Rev. Mod. Phys. 40, 441. Hubbell JH (1971) At. Data Tables 3, 241. Inokutl M (1971) Rev. Mod. Phys. 43. 297. Jackson JD (1975) 'Classical Electrodynamics' (Wiley. New York). Kieffer G and GH Dunn (1966) Rev. Mod. Phys. 38. 1. Kolbensvedt H (1967) J. Appl. Phys. 18. 4785. Liljequlst D (1983) J. Phys D: Appl. Phys. 16. 1567. Hljequist D (1985) J. Appl. Phys. 57. 657. Lindhard J (1954) Dan. Mat. Fys. Medd. 28. n8 1 Lindhard J and M Scharff (1953) Dan. Mat. Fys. Medd. 27. nl5 1 Mayol R and F Salvat (1989) To be published. Palllc ED (1985) (Ed.) 'Handbook of Optical Constants of Solids' (Academic Press, New York). Penn DR (1987) Phys. Rev. B35. 482. Pines D (1963) 'Elementary Excitations of Solids' (Benjamin. New York). Powell CJ (1985) in 'Electron Impact Ionizatlon', Ed by TD Mark and GH Dunn (Sprlnger-Verlag, Wien) Rudge MRH (1968) Rev. Mod. Phys. 40. 564. Salvat F. JD Martinez, R Mayol and J Parellada (1985). J. Fhys. D: Appl. Phys. 18. 299. Scofleld JH (1978) Phys. Rev. A18. 963. Seltzer SM and MJ Berger (1982). National Bureau of Standards Report NBSIR 82-2572. Tung CJ. JC Ashley and RH Ritchie (1979) Surf. Science 81. 427. Veigele WJ (1973) At. Data Tables 5. 51. 588 FIGURE CAPTIONS. FIGURE 1. Ionizatlon cross sections for hydrogeu. The continuous and dot-dashed curves are the cross sections obtained from our GOS model and from the WeizsacJcer-Williams method with low energy correction included. The dashed curve is the result of the Born approximation (Rudge 1968). Experimental data are those reported by Kleffer and Dunn (1966) FIGURE 2. Ionizatlon cross sections for helium. The dashed curve has been obtained from our GOS model without low energy correction. Other details are the same as in Fig. 1. FIGURE 3. Ionlzation cross sections For helium as given by the Welzsacker-Wllliams method. The continuous and dashed curves are the cross section derived froa (29) with and without low energy correction respectively. The dot-dashed curve gives the ionizatlon cross section coaputed froa the approximate DCS (30). FIGURE 4. K-shell ionization cross sections for aluminum and nickel. The continuous and dot-dashed curves give the results from our approximate GOS and froa the Weizsacker-Wllliams method respectively (including low energy correction). The dashed curve Is the theoretical Born cross section coaputed by Scofleld (1978) for Nl. Experimental data are those quoted by Seltzer and Berger (1982). FIGURE 5. K-shell lonizatlon cross section for gold. Details are the saae as in Fig. 4. CROSS SECTION (a.u.) oo to 590 3 • O ^^ o u Ui v% sV) FIGURE 2 591 0.0 FIGURE 3 592 CM (siuoq) FIGURE 4 593 MT (sujoq) NOI1O3S SSOdO 0l««1 FIGURE S 595 THE THEORY OF TRACK FORMATION IN INSULATORS DUE TO DENSELY IONIZING PARTICLES» R. H. Ritchie*''', A. Gras-Marti*, and J. C. Ashley* *Health and Safety Research Division, Oak Ridge National Laboratory, Post Office Box 2008, Oak Ridge, TN 37831-6123 USA ^The University of Tennessee, Department of Physics, Knoxville, TN 37996 USA *Universitat d'Alacant, E-03080 Alacant, Spain ABSTRACT- We have devised a self-consistent scheme for modeling the evolution of the electrons and ions created along the track of a swift, densely ionizing particle passing through an insulator. Numerical implementation of the model has been made. JResearch sponsored jointly by the Office of Health and Environmental Research, U.S. Department of Energy, under contract DE-AC05-84OR21400 with Martin Marietta Energy Systems, Inc. and the Solid State Sciences Directorate, Rome Air Development Center under Interagency Agreement DOE No. 0226-0226—Al. Support from NATO travel grant No. 0142/87 is gratefully acknowledged. 596 INTRODUCTION The structure of displaced atoms generated in solids by swift charged particles has been of great interest in device physics, nuclear technology, radiation physics, and radiobiology for many decades. Although a considerable body of empirical data has been accumulated, basic understanding of the phenomenon is lacking. 2 8 The energy lost in a solid by an ion with speed v > v0 = e fh = 2.2xlO cm/s goes primarily into electronic excitation. However, in insulators the final result is a fairly well-defined track of displaced atoms about the path of the ion as well as associated atomic sputtering from proximate surfaces. To account for this phenomenon it has been proposed that Coulomb repulsion among residual ions in the core region may account for atomic displacement and track registration (1). However, a detailed and quantitative explanation has not been given, although some aspects of this mechanism have been discussed qualitatively (2, 3). It has been assumed (1) that electrons ejected from the core region are captured at appreciable distances from it, allowing Coulomb—driven expansion of the positively charged residual ions to take place. But the mechanism for this capture process is not at all clear. We have devised a semi—analytic model that incorporates the essential physics of this phenomenon and represents the first quantitative, microscopic approach to its elucidation. In our model the delta—ray electrons ejected with a wide distribution of energies from a narrow core region in an insulator evolve under the influence of the self-consistent electric field of the ion core and of the ensemble of delta rays (4). Electrons in the delta ray distribution are assumed to scatter elastically in the medium; to lose energy to excitation of electronic transitions in the medium at 597 higher energies and to phonons in the subexcitation region; and to move in the self-consistent electric field of the system. Upon slowing to thermal energy, electrons drift toward the core region. After entering this region they become a part of the partially neutralized core-plasma (C-P). The C-P expands under the influence of the Coulomb forces in its vicinity. As neutral, unexcited matter is engulfed by the expanding C—P, the ions in the C—P lose energy to it in nuclear collisions and undergo diffusive motion in the expansion process. In addition, excited atoms in the engulfed solid contribute to the kinetic energy of the expanding C—P. Coulomb forces in the C—P region are expected to be large compared with the atomic forces existing in the undisturbed solid at times t < 10*u s. Expansion of the C-P is assumed to occur freely until the self—consistent electric fields at the C-P become comparable with the static fields of the solid, whereupon the expansion is assumed to cease. Delta ray transport has been parameterized for Si(>2. Transport and energy cross sections are taken from earlier work by us (5). Coulombic effects are represented in cylindrical symmetry. The first published account of the C—P model (2) gave quantitative estimates of parameters relevant to physical processes occurring in a non-conducting solid excited by a densely ionizing charged particle. Detailed mathematical description of the transport of deposited energy in this region was indicated but not explored numerically. Subsequently (4) we carried out preliminary calculations of the evolution of the central C—P region including energy transport by recoil atoms and by phonons. These calculations were not sufficiently 598 detailed to predict sputtering yields 01 the properties of regions in which appreciable atomic displacements occur. Watson and Tombrello (3) have given a mathematical treatment of ion induced sputtering from solids based essentially on our C-P model, but termed by them a "Modified Lattice Potential Model." They claim to have given a parameter-free treatment of the sputtering process, but make several key assumptions that seem to be questionable. For example, their application of the C—P concept involves the use of conventional linear theory (4) to describe the spatial dependence of energy deposition by delta rays around a swift ion track. This despite the fact that Coulombic effects are expected to be very large in the cases considered by them and that one would therefore expect their calculated deposition radii to be inappropriate for this system. They assume that energy deposited in the early physical stage spreads out rapidly (in ~10"15 s) to occupy uniformly at a density €c a cylindrical, electrically neutral volume with radius rc0. A central assumption of theirs is that €c is equal to the sublimation energy for molecules of the solid divided by the volume per molecule. With this assumption rco is fixed, given the energy deposited per unit length by the incident ion. For 9.5 MeV 19F ions incident on UF*, rco turns out to be ~44 A according to their calculations. A Thomas-Fermi statistical approach is used by them to compute an "electronic pressure" exerted by the core—plasma on the lattice, in a manner that does not seem to us to be consistent with the known properties of a solid. Other details of their approach are similarly schematic. One of our main objections to their admittedly heuristic application of the C—P model concerns the assumption that the C—P evolves from its initial radius to ~44 A in UF< in ~10"15 s. Numerical work by us indicates that in such systems ionic 599 response is most important in determining the time evolution and that the natural scale of times for such motion is ~10*14 - 10 "13 s. Recently Tombrello (7) has given a phenomenological model for the formation of ion damage tracks in dielectric solids, combining ideas of K-shell excitation and the resulting Auger processes with the Coulomb explosion model of Fleischer, Price and Walker (1). At present our approach appears to be the only one extant that accounts for the essential microscopic physics involved in track formation and sputtering in insulating solids. EVOLUTION OF THE DELTA-RAY DISTRIBUTION It is important to estimate the number of delta-ray electrons that have returned to the core region at a given time after the initial ionizing event. Consider a representative electron that has made n collisions and has experienced the average squared displacement from the origin of r2 and has energy E (8). Then if it experiences An collisions the change in the mean square displacement A(r2) will be given by A(r2) = L2An . (1) Here L(E) is the transport mean free path for collisions in the medium. The kinetic energy of this electron will be diminished in inelastic collisions and in moving in the electric field that exists in the medium. If the energy loss per unit path length is -dE/dx = S(E) then 600 (2) where F(r) is the Coulomb force on the electron. Passing to the limit where the increments become differential quantities and dividing both sides of the equation by dr/dn, one finds In this limit Eq. 1 becomes d(r2) = L2dn = Lvdt (4) which, when substituted in Eq. 3 yields dE = [TE I S(E) L(E)F where m* is the effective mass of the delta-ray electron. We represent the magnitude of the electric force, F, as 2A(r,t)/r for a cylindrically symmetric geometry. Here A(r,t) is the charge per unit length lying inside a cylinder with radius equal to the distance of the electron from the center of the track at time t. From Eq. 4 we have dr _ rET L[E} at-J25* 601 In the present model, an electron is assumed to lose energy in exciting the medium and in moving against the Coulomb field until it reaches thermal energy. Thereafter it is assumed to drift toward the C-P region, obeying the equation (7) where, in general, its mobility n at a point depends on the electric field strength there. EVOLUTION OF THE CORE-PLASMA We assume that, after the initial ionizing event, the core ions move outward from the track axis due to Coulomb repulsive forces. As they move outward they lose energy in collisions with atoms of the solid. Delta ray electrons returning to the core tend to neutralize the C—P region and reduce the Coulombic forces acting there. We represent this region schematically as one with uniform charge throughout a cylindrical volume with radius rx(t) centered on the track axis. Immediately after the initial ionizing event we take the Coulombic force averaged over the C—P region to be given by Aim r.{0)2^, where r (0) is the initial radius, n is the density of ions each with charge Z e. For simplicity we assume that the charge density in the C—P region is uniform. After the time t has elapsed, when the charge of the core has been reduced to the fraction, f(t), of its initial value because of the return of delta rays to the core, we take the average force to be reduced by the factor f(t). From the classical equation of motion one finds that the increase in the kinetic energy of the core ions per unit time per core ion may be written 602 Coulomb within this schematic model. Energy loss to atoms engulfed by the expanding core may be approximated in binary collision theory. We take the differential cross section da for energy loss dT in T in such a collision as (9) where r 2 fMl pZjZ^eV'* 0.8853ao A a ~ 1 l/4 12 12 and [(Zi,M1),(Z2,M2)] is the (charge, mass) of the [incident, struck] particle. The first Bohr radius is a and A^ is a constant that we take here to be 1.4 for low-speed collisions. We assume that Zi = Zj = Z and Mi = M2 = M and use compositional average values for both of these quantities. The rate at which an ion with velocity v will lose energy in atomic collisions -T« is then given by -dE/dt = v p(t) J Tda, where TB is the maximum energy loss o and p(t) is the atomic density at the ion. For simplicity we take p(t) to be the number of engulfed atoms, averaged over the C—P region at time t. Carrying out the integration over T we find that a representative C-P ion will lose energy in collisions with atoms engulfed in the C—P expansion at the rate 603 af 3 2 2 2 where K = J Aj/4 a12 / (2e ZA)^ and ZA is the nuclear charge of the engulfed atoms. Finally, we take the time variation of the energy E (t) of a representative C—P ion to be described by the equation E 1/2 E i "' {JM *) i} • (11) Here 0 is an operator intended to describe, in an approximate way, the net energy loss from the C—P region due to collisions. In this model, energy transferred to engulfed atoms will not result in a decrease in the mean energy of the region if the energy is deposited locally. In this case the operator 0 could be taken as multiplying the bracketed term by zero. On the other hand, if the energy transferred tc engulfed atoms were transported immediately to infinity the operator 9 could be taken to be the unit multiplier. A more realistic choice would represent energy transport out of the C—P region by phonons. Here we approximate the transport process by using a Green function for the transient heat conduction equation with a thermal diffusivity equal to that of an ideal gas of atoms with the composition of the solid. It turns out that for the energy range of interest here, a constant diffusivity equal to that of the undisturbed SiO2 may be used. In order to represent the spatial expansion of the C-P region, a drift-diffusion-type model is employed. The variation of r^t), the C-P radius with 604 time is -T£ rr(t) = a s, + 0 D/2rj (12) where a and 0 are constants of the order of unity, the drift speed s j is d ynnj(o (£) and the diffusion constant, D, is tor T • (") Also atT is the transport cross section for binary collisions and v is the mean C-P ion speed. For collisions in which Mj = Mj = M we take „ _0.79256C 2 and p(t) = nQ [1-rj (0)/^* (t)], where n0 is the atomic density in the undisturbed solid. ENERGY DISTRIBUTION OF DELTA RAYS We use the following expression for the initial energy distribution of the delta rays generated by the ionizing event in terms of the function (10) 605 This is an approximation to the number of delta ray electrons emitted with energy per unit energy interval by a non—relativistic ion having effective charge Zi e and speed v. The medium in which the ion travels is assumed to have Ni electrons per unit volume with ionization energy Ij in the ith energy level. For the effective charge of the ion we take Z* = where Zj is its bare charge. Figure 1 shows a plot of Eq. 16 computed for SiO2- EVALUATION OF THE TRACK STRUCTURE EQUATIONS In the present realization of the model a number of simplifying assumptions are necessary. We take the system to be cylindrically symmetric. The initial radius of the C—P region is assumed to be quickly established in times < 10"16 s following the passage of the swift ion; inner shell excitations are taken to decay primarily by Auger processes and finally to yield ions in the C—P region that have the valence of the isolated atoms making up the solid. A schematic model from which the initial radius of the core r (0) and the initial ionic charge per unit length in the core may be determined is as follows. Take all ions within the radius ^(0) to be stripped of all valence electrons which, at this time, are located just outside the radius ^(0). Compute the Coulombic energy 606 per unit length required to produce this configuration, add to it the energy per unit length necessary to separate the electrons from the ions and to impart to them the mean delta ray energy, 1 - for the charge per unit length at t=0. Here q is the valence of the atoms of the solid, E is roughly the gap energy of the solid (~8.9 eV for SiO2). Given the charge per unit length, one may determine the radius r2 (0) from ^ (0) = (A/rao qy)'. The evolution in time and space of the system of delta ray electrons and the C—P is found by solving Eqs. 5,6,7,11, and 12 self—consistently. The initial distribution of electrons is represented by N discrete groups, each of which decreases in energy according to Eq. 5 and which expands from the core as described by Eq. 6 in the self-consistent field of the C-P and that of the other electron groups. When the energy of a given group has decreased to the thermal range, it is allowed to drift back toward the C-P region. When it arrives there, the net C-P charge is decreased by an amount determined by the charge represented by that group. In the meantime, the C—P region expands under its mutual Coulomb repulsion, increasing in energy due to this expansion, but decreasing in charge due to the teturning electrons. THE YIELD OF SPUTTERED ATOMS The number of atoms sputtered from surfaces proximate and normal to the track axis is estimated by assuming that atoms in the C—P have an isotropic, 607 Maxwellian distribution of velocities and that, at these low energies, ions in the C~P will capture electrons with high probability upon leaving the surface to become neutral. The yield, Y(t), in atoms per incident swift ion is then written as /• 2 |2E Y(t) = / dt' aijV) ^(t'] o where nx (t) is the density of ions in the C-P region at time t and F is the sublimation energy of an atom. 608 Figure 2 shows the stopping power S(E) and the inverse mean free path [L(E),]-1 for electrons in SiO2 solid. Figure 3 gives some of the results of calculations with the C—P equations. These data are all given in a.u. 18 MeV F ion traveling in SiO2. The yield curve saturates at 0.78 atoms per ion, which is in reasonable agreement with the experimental result by Que and Tombrello»of~1.0. Our results show a characteristic non-linear dependence of yield on the stopping power of the solid for the ion that is in general agreement with experimental results for solids. 609 1. R. L. Fleischer, P. B. Price and R. M. Walker, Nuclear Tracks in Solids (University of California Press, Berkeley, 1975). 2. R. H. Ritchie and C. Claussen, Nud. Instrum. Methods 198,133 (1982) 3. C. C. Watson and T. A. Tombrello, Radiat. Eff. §9_, 363 (1985) 4. An earlier version of this model is described in R. H. Ritchie, Proc. 8th Symposium on Microdosimetry, EUR 9395, p. 145 (1983) 5. J. C. Ashley, R. H. Ritchie and 0. H. Crawford, "Energy Loss and Scattering of Subexcitation Electrons in SiCV1 Proc. 10th Werner Brandt Workshop on Penetration Phenomena: Dynamic Interactions of Energetic Probes with Condensed Matter 6. J. Fain, M. Monnin and M Mont ret, Radiat. Res. 51,379 (1974) 7. T. A. Tombrello, Nucl. Instrum. Methods B_l, 23 (1984) 8. A. H. Samuel and J. L. Magee, J. Chem. Phys. 21,1080 (1953) 9. J. Lindhard, V. Nielsen, and M. Scharff, Mat-Fys Medd Kong. Dansk Vid. Sels M, No. 10 (1968); J. Lindhard, M. Scharff, and H. E. Schi^tt, Mat-Fys Medd Kong. Dansk Vid. Sels 7Q, No. 14 (1963) 10. M. E. Rudd, C. A. Sauter and C. L. Bailey, Phys. Rev. 151, 20 (1966); E. J. Kobetich and R. Katz, Phys. Rev. 172, 391 (1968) 11. X. Qiu, J. E. Griffith, W. J. Meng, and T. A. Tombrello, Rad. Effects 70, 231 (1983) 610 Fig. 1. The enSiC>2 computed from Eq. 16. Fig. 2. The stopping power and the inverse mean free path for electrons in S1O2 plotted as a function of electron energy. These data were taken from Ref. 5. Fig. 3. Output from a typical numerical solution of the model equations describing core—plasma evolution in space and time. These calculations were done for a 30 MeV 18F ion traveling in SiO2- The different curves shown are described in the text. -1 10 Initial Distribution 20 MeV CA Inns in Si OB D A 10-e R A Y D .-3 R I 0 N 10" I 0 10 I01 Ener^-eV I03 10 10' Inverse Mean Free Path -1 8 I 10.-e -3 10 Stopping Pouer-eV/A I 10 .-s 10'1 10* 101 Ehm-af r,3. z 10' 10 0 — fractional core charde 10 CO -1 10 1 • -s -3 0 10 10 10 10 615 12 • ^n Werner 3rar.dc Workshop, San Sebastian H-7 Sept 89 Tra£k_ Sj:r u£tu r_e: Zl£.2R*£±1122.' Progress. Problems. Robert Katz, University of Nebraska, Lincoln NE 68588-0111 USA ABSTRACT Heavy ions interact with detectors largely through the agency or" secondary electrons called delta rays, whose influence 13 best described through the radial distri- bution of the dose they deposit. Monte Carlo calcula- tions of the radial dose distribution from protons in liquid water (Hamm), approximated analytically on the basis of a calculation using electron range energy relations in aluminum and the Rutherford formula for delta ray production with normal electron ejection, and extended to heavy ions by a formula for effective charge (3arkas), compare well with recent measurements (letting) of the average dose distribution from "half clad" If ions in gas. Ve extend our calculations to several solids (Nal, L1F, Si, SiO2) by weighting for electron density and adjusting the radial dose formula so that its integral agrees with tabulated stopping power values (Janni) at all proton speeds. We know of no end points in condensed matter where a model based on primary effects alone, or double ioni- zations, or on the statistical distribution of energy deposited in small volumes (as in micro- or nano- dosimetry), has been demonstrated to lead to experi- mental action cross sections. Dosimetrlc problems with ultrasoft x-rays suggest that electron track ends are of minor importance in radiobiology (3aju and oth- ers ). A calculation of the relative pulse height in Nal(Tl) scintillators agrees with data (Salaraon and Ahlen) with near relativistic heavy ions, contrary to the claim of these authors that this model disagrees with their data, and that a much more complex mechanistic model is required, A calculation of the efficiency of TLD-100 for heavy ions agrees well with data (Montret) that has been thought puzzling. Track theory explains the observed branching with Z when the efficiency is plotted as a function of LET as due to "thindown". A parameter free (point target) calculation of the cross section for single and double strand breaks of SV-40 virus in E» buffer taken as 1-hlt detectors is in reasonable agreement with data (Roots). 616 Problems remain in track theory. We do not know well the radial dose close to or remote from an Ion's path, or from partially clad Ions, or in condensed matter. We continue to seek an appropriate analytic expression for the radial dose distribution in all materials and for all ions and ion speeds. Voc all detectors, particularly not all biologi- cal detectors, are in agreement with the perspectives of the model. We do not know why. We have not yet been able to sort out the basic differences between those end points whose response to radiations is in agreement with track theory and those which are not. For biological systems there is some question about the relation between the plateau value of the inac- tivation cross section and the nuclear size. There is a second question about the shape of the dose response curve at lew coses of x-rays and the R3E for high LET radiations. Track theory demands a "shoulder" at low doses if the H3E exceeds 1, but this is not univer- sally observed. Hadiobiologists also question the use of "target theory", a basic component of track theory. The present track model is perhaps the only glo- bal quantitative model in the study of the response of detectors to radiation. We need others that are equally quantitative and falsifiable and which address radiation effects first through the effects of single ions via a cross section, and then to the effects of beams of particles. When heavy ion effects in condensed matter are considered for other end points (dilitaticn, changes in electrical, magnetic and mechanical properties, latent tracks, color center formation) we may be deal- ing with many hit systems in that several -electrons must pass through as yet unidentified sensitive tar- gets in order that there is an accumulation of sub- detectable damage. There is a temptation to use con- cepts already found wanting in other areas of track physics: track core, LET threshold, thermal spike, ion explosion spike, and to ignore the influence of delta rays because one's detector does not see the effect of single electrons. Here as in investigations of other track based phenomena, study of heavy icn tracks in insensitive (many-hit) nuclear emulsions is likely to be a useful guide to data Interpretation. There we can see how the tracks of heavy ions in many-hit emulsions can be misinterpreted as having an LET "threshold". 617 21 We continue to revise our expressions for the radial dose distribution about the path of a proton, and by use of an expres- sion of Barkas for the effective charge (based on the range of heavy ion tracks of particles up to Argon at energies up to 10 MeV/amu), extend chis to partially clothed ions up to Uraniuai. Most recently we take as a basis the Monte Carlo calculations of Hamm for liquid water, and fit analytic expressions to It. Our analytic expressions are developed from Rutherford scattering of free electrons from heavy ions, and experimentally based power law range energy relations for electrons in aluminum. [Zhang et al., Radiation Protection Dosimetry 13, 215 (1985) [Wallgorski et al., Nucl. Tracks and Radlat. Meas., 11, 309 H986) Our formulas agree well with the latest measurements in gases, by Metting et al. for 5.9 MeV/amu U, 13-17.2 MeV/asiu Ge, and 600 MeV/amu Fe. [Metting et al., P.adiat. ?.es., 116, 185 (1988); PNL-SA-16298, 10'th Symposium on Microdosi.iietr7 1 989• The measurements co not penetrate close to the ion's path, nor do they extend to the outermost reaches of delta ray penetration, where we need {•irx.'ner information. To estimate the radial dose distribution in solids we have simplified our analytic expression, and have chosen to normalize that part of the radial dose from delta rays by correcting for electron density differences in different detectors. The part of the expression which is taken to represent energy deposited by the primary ion is then adjusted so that the radial distribution of dose integrates to the proton stopping power (within 10J) in the new material. In this way we now have developed expressions for the radial dose distribution in LiF, in Nal, in Si, and in SiO2. The radial dose for LiF (TLD-100) and Nal are used in sub- sequent calculations discussed here. Some results are shown in Fig. 1. "Radial Distribution of Dose from Energetic Protons in Several Media". 2. SINGLE AND 20U3LE STRAND 3PEAKS IS SV-£0 DNA IN ED 1HII15 In an extended series of experiments at Berkeley and Darmstadt, Roots measured DMA single and double strand breaks in SV-UO DNA in Eo buffer, considered to emphasize the Indirect effect of radiations- The DNA in buffer was irradiated with a series of heavy ions, to U and with x-rays. [Roots et al., in press. Adv. in Space Research (1989) 618 We have compared the reported cross sections to our calculations for 1-hit detectors In the point carget approximation, taking the D-37 dose to have the experimental value for x-rays. The agree- ment Is quite good for single strand breaks. We muse reduce the experimental D-37 dose by about 20% to obtain the best agreement of calculated with experimental cross sections for double strand breaks. There are some residual disagreements with cross sec- tions from helium and uranium bombardments that need to be sorted out, but we must consider the agreement of this model with DNA strand break data to be quite remarkable. One possible interpretation of the difference in experimental x-ray D-37 values, as between 13 Gy for single strand breaks and 320 Gy for double strand breaks, is that the appropriate target volume is the same :'or both SSB and DSB, being the volume encompassing a full double strand of DNA, but that 20-25 times the chemical pro- duct is needed to make a pair of breaks than a single break. [Katz and Wesely, manuscript in preparation Some results are shown in Fig. 2, "Measured Cross Sections for Single Strand Breaks in SV-UO DNA in £e Buffer, compared to cal- culations of the cross sections for heavy ion inactlvation for a 1-hit detector from track theory", and Fig. 3, "Doucie Strand Breaks in Eo 3uffer". Except for the data from He and U bombard- ments, the average discrepancy between theory and experiment is about 505. While this is larger than we would like we must keep in mind that the measured cross sections from C to Ar vary by two orders of magnitude. 3_. J1MLSE HEIGHT IU NaI(J_l) SCIN£ILLATOBS FROM NEAR RELATIVISTIC _I0NS Pulse height in a scintillation counter is taken to be pro- portional to the path integral of the cross section. Our pro- cedure is to assume the counter is a 1-hlt detector, to assign numerical values to both £„ and a0, to calculate a- as a function of ion energy, and to integrate this from the initial energy to rest. We vary parameters until we find proportionality between experimental pulse height and the integrated cross section. In earlier work [Katz and Kobetich, Phys. Hev. 170. 397 (1968) we found that this model agreed reasonably well with measurement for ions heavier than He, at energies up to 200 MeV, from .Yewman and Steigerc. [Newman and Steigert, Phys. Rev. M8, 1578 (1960) In that work there were clear evidences of "thlndown", in a plot of pulse height vs. track segment LET, where "hooks" were displayed, giving clear evidence that this response was due to delta rays. 619 Prom more recent studies it was asserted that track theory failed [Salamon and Ahlen, Phys. Rev B21, 5026 (1981) to account for the response of this scintillator for near rela- tivistic heavy ions {Ne, Ar, Fe, at initial energy of 550 Mev/amu), and that a much more complex mechanistic explanation incorporating terms like "lonizatlon quenching" and "second order annihilation processes" was needed. We find no difficulty with the track theory model for these data which we fit with the same parameters used for the low energy data of Newman and Stelgert, and suggest that Salamon and Ahlen miscalculated. We see no need for their complex mechanistic explanation. Our results are shown In Figs. U, "Relative Pulse Height for Stopping Ions in Nal(TL), Comparing Track Theory Calculations to Data of Newman and Stelgert for Slow Ions, from He to Ne", and rig. 5, "Relative Pul3e Height for Fast Ions in Track Segment Irradiation, for Ne, Ar, and Fe Bombardment of NaKTl), Comparing Theory to Data of Salamon and Ahlen." [Luo Daling and a. Xatz, Nal(Tl) Scintillatora, to be published (1989) -• H£§PONS_E ££ Ii;5"l£2, 12 Mr £11 I2JSI When the efficiency of TLD-100 to stopping He, Ne, and Kr ions is plotted against the LET of the Ion. the response is mul- tiply valued, with the graph appearing as a series of separated hooks at highest LET. This has been regarded as something of a mystery. [Montret-Brugerolle, Thesis, Univ. of Clermont-Ferrand, France (1980) The response of TLD-100 to gamma rays displays supralinearity at high doses. Many explanations have been offered for supralinear- ity. A leading contender is called the "interacting track hypothesis", which is another version of a "2-hit process". We have taken the response to be due to a mix of 1 and 2-hit tar- gets, and have decomposed the suprallnear response into a 3-hit and a 2-hit component. When we examine published response curves, we find that the parameters of the 1-hit response have essentially the same value for much of the published data, while the numerical value of the 2-hit parameters varies considerably from investigator to Investigator. Further it is found that the supralinearity declines when the irradiating x-ray source has lower and lower energy. Since the average delta ray energy is less than 60 eV, and Is limited from above by kinematic con- straints, we have taken the response to delta rays to be i-hlt (without supralinearity) in the stopping end of an ion's path. Using the fitted value of E, for the 1-hit component we have cal- culated the path Integral of the cross section, have divided this 620 by the average stopping power of the ion, to gee the efficiency of TLD production, and have divided this by E9 for gamma rays to get the relative response. In biological terms this is equivalent to the "RBE" which for a 1-hit detector is a-£a/h. Our results agree well with the measured values of relative response. We explain the "hooks" found by Montret-3rugerolle as due to "thindown". The thindown from the track width regime in a parti- cle track is determined by its speed, Thus the response of par- ticles having the same speed should merge as the particle speed approaches zero. When one plots response against LET, the curves are split by a factor of the square of the effective charge. At low ion speeds the response of TLD's to heavy ions can no longer be approximated as a single valued function of LET. The aystery is resolved by a simple, geometric, track physics explanation. Our results are shown in Fig. 6. "Relative Response of TLD-100 to Stopping He, .Ve, and Kr ions, as measured by Montret-Brugerolle, as Compared to Calculations From The Present Work {interrupted lines), and to Calculations by Fain, Montret, and Sahraoui (a). [J. Fain, M. Montret, and L. Sahraoui, Nucl. Instr. Meth. 375. 37 (1980). [Luo Daling and R. Katz, TLD-100, to be published (1989) Electron track end effects have been called into play by many investigators seeking qualitative explanation of low LET radiation effects, especially (but not exclusively) in radiobiol- ogy. Track ends have been likened to high LET radiations, in support of an argument that the R3E of low energy electrons should be greater than 1. To me these qualitative ideas lacked substance. Nevertheless, the supposed importance of track ends, and of energy deposition in nanometer sized volumes (to replicate ONA strands) has been a preoccupation of many investigators, mak- ing Monte Carlo calculations, measuring biological response to ultrasoft Cu-K, Al-K, C-K characteristic x-rays. Initial -experi- mental results CGoodhead, Thacker, Cox, Int. J. Radlat 3iol. 36, 101 (1978) found an RBE for cell killing from C-K x-rays (of energy 280eV, with photoelectrons of range less than 10 nm) of 3 to ^ confirm- ing these ideas. New experiments (Raju and others) however find the RBE to be close to 1, and attribute the discrepancy to the manner of evaluating the absorbed dose in cells whose thickness approximates the attenuation length of these ultrasoft x-rays. They state that "The simplest interpretation of the present find- ings is that most energy depositions caused by track end effects are not necessarily more damaging than the sparsely ionizing com- ponent". Estimating from their finding that the D-37 dose for C-K x-rays is about 1 Gy, we calculate that a cell of diaaeter 10 microns absorbs 1100 C-K photons for killing. Numbers like this. 621 and 3imllar numbers for killing by 15 keV electrons, where about 500 incident electrons per nucleus result in cell killing (Art Cole), make it difficult to understand why it is thought that survival curves should have an initial linear slope. [Cornforth, Schillacl, Coodhead, Carpentert Wilder, Sebrlng, and Raju, in press, Radlat. Res. 1989. It would be very helpful if Monte Carlo based stereograph!c plots of track structures were made giving the spatial distribution of excitations and ionlzatlons from electrons of initial energy 10, 100, and 1000 ev . These events may well be delocalized, partic- ularly If calculated in the Oak Ridge code for liquid water. It delocalized, the logic which supports the claimed Importance of crack end effects will vanish. We may see that the track end is not a tight cluster of lonizations imitating high LET particles. If so another frequently postulated mechanism of radiation action will be derailed. 6. OTHER AREAS 1) Soft Errors in Computers We note with interest that work at the Naval Research Laboratory has shown that the amount of charge collected on test transistors and multilayer structures penetrated by heavy ions is not the same for two ions of the same LET but different energies. They suggest that track structure considerations are relevant here where heretofore only LET variations were thought signifi- cant . [W. J. Stapor and ?. T. McDonald, J. Appl. Phys 64, M30 (1988) 2) Latent Tracks An interesting speculation has been offered by N. Itoh that there may be a correlation between track etching and the grouping of insulators in terms of the electron-lattice interaction. He suggests that insulators in which etchable tracks are not observed are those in which neither excitons nor holes are self- trapped. Where self trapped excitons are formed above some criti- cal density, they interact to form lattice defects. Since the lifetime of these excitons is short {from milli to nano seconds), the critical density is not formed with x-rays, but can readily be formed with heavy ions. According to this classification MgO and CaO should not form latent tracks while SiO2 should. As yet the criteria are qualitative. Yet this proposal is one in which "dose rate" plays an important role in track formation, and which therefore represents a potentially Important extension of track physics concepts. [N. Itoh, Defect Formation in Insulators under Dense Electronic Excitation, unpublished (1989), Nagoya University, Furocho, Nagoya U61-01, Japan. 622 2) Effects in Bulk Matter There are many ongoing investigations of radiation effects in photoresists ana other polymers, and of the alteration In phy- sical properties after irradiation of a number of materials. In general this worn is isolated fron other studies of track struc- ture, both in terminology and in concept. It may be some time before there will be communication between investigators la these several areas. Just as many radlobiologlsts fall to see the relevance of studies of particle tracks in emulsion to raalo&lol- cgy, so also those concerned with the modification of the proper- ties of material typically do not see the relevence of radiobiQl- ogy to their concerns. -} Alanine Measurement of free radicals, especially in alanine, is ::.der extensive investigation for doslmetry. This was originally proposed by Sradshaw et al. in 1962, and lay dormant for many years until actively revived by Bermann. I noted that it could be considered as a "-hit detector of high £„ , and hence would yield a response to neutron dose only moderately different frcs its response ts gamma rays. With 3eraann we compared Its response to energetic protor.s and neutrons to the predictions of track. theory. Because of the stability of radicals in this material, = T.d its near tissue equivalence we proposed tills material for nigh dose aeassurement for both low and high LET radiations, and as a postal comparison dosimeter. These proposals have caught on ir. recent years and we nave seen a flowering of activity in such investigations. Tor references, and as a recent example, see I M. ?. p.. Waligorski, G. Danialy, Kim Sum Loh and ?.. 1- CONFLICTS C£ J_2_ACK THsga? WITH EXPE3IHEMT Track theory 13 based on an assumed interrelationship between the response of a detector to low LET radlatl^fons, an*J its response to high LET radiations. Indeed the latter is taken to be a geometric transform of the former, ma*ing use of the radial distribution of dose around an ion's path. Thus the r*sponse to low LET radiations is taken to constrain the response to high LET radiations. If a system displays ' -r.it response to gamma rays, the observed cross section must be a sublinear func- tion of LET. If iz displays 2-nlt response to gamma rays, the "ion-kill" cross section must vary nearly quadrat!cally with LET. And conversely. The response to high LET radiations at high dos« [where cross sections are normally determined} should infer the response to low LET radiations at lew doses. This implies that systeas displaying '-hit response to gaaaa rays must display an 23E less than " when irradiated with heavy Ions. And that sys- displaying an H2E greater than 1 when Irradiated with heavy 623 ions should have a shouldered survival curve when irradiated with x-rays. [Xatz, ?hys. Med. 3iol. 23. 909 (1978) One finds conflicting results in both physical and biological systems. Thus che production of color centers by x-rays can be expected to be 1-hit, linear at low dose. 'fet this is not ooserved for Llr (Perez et al, 1989). I find the observed rcon linear response to low doses of gamma rays rather incomprehensi- ble. IA. Perez, E. 3alanzat, and J. Dural, to be published, 1939 The demands of the theory also are contradicted for biological cells. In particular yeast cells are said to display i-hit response to gamma rays, and yet have an aBE greater than 1 (Kiefer). Similarly chromosome aberrations (Geard) are said to have a linear-quadratic response to x-rays while the cross sec- tion after heavy ion irradiation varies quadrat!cally with LET. I do not understand these discrepancies. A new assay introduction, called Dynamic Microscope Image Pro- cessing Scanner (DMIPS) has yielded a more distinct shoulder for survival curves for synchronized 7-1 cells in G-1 phase, irradi- ated hypoxically with 225 keV x-rays, than a conventional assay ot^ the same system. [Blakely et al. Low Dose High LET Oxygen Effects, Poster, 3adiat. Society, Seattle, 1989 The issue is not a trivial one, in its Implications for radiation protection, for the manner in which the equivalent dose of high LET radiations is treated, through a committee assigned "Quality Factor", is based on the belief that dose response curves after irradiations with x-rays display a clear initial linear slope. Contradictory data are often ignored. And the conflict of this belief with track theory is taken to be a minus for the theory, overriding its success in describing the response of cells to high LET radiations. Nor is any consideration given to the large number of Incident electrons or absorbed photons needed to tfill a cell, conceivably implying that a hit by 1 electron or 1 photon is not likely to result in cell death. Then, why a linear ini- tial response? These are problems that we hope will be sorted out. But at this moment it is not clear why some physical and biological end points agree with track theory while others do not. Finally we display In Table 1 an array of detectors to whlcn track theory has been applied, and the very large range IT. £,. of seven orders of magnitude, that characterizes these different materials and end points. In part E, reflects target size, but 624 not completely so, for one can sensitize and desensitize nuclear emulsions of the same grain size so as to vary ED by several ord- ers of magnitude, as in the K series ot emulsions shown in the table. Since Eo directly affects the interaction cross sections, and hence what we might call track size, is should be clear that the "size" of a particle track is not a unique quantity, tor it depends both on the detector and the ion. Rather than "track size" it would be more appropriate to speak of the size of the damaged region in a specific detector, with a specific end point, for Indeed one can measure several different end points having different radiosensitivitles, different histedness, and different "crack size" in a single substance. 8. ACKUOWLEDOEMENT This work is sponsored by the U. S. Department of Energy 625 Fig 1 in rr i in i i m i i ill i i in i I in i i in i i in i i in i i in i i in i i in i ill I J-J O i III l i lit i t Iff i T In I r In i • (in • Itn t Itit r IMI I~ 626 Fig 2 Single-S+rand Breaks = ' 'I "I Mil I I III I J III T Mil I M* SV-40 DNfl Inflq. Solution 10"6 t- Data from Roots*, 1989 10-7 10-8 £ U 10-9 E cn in 10-11 I Wai!gorsk!+ Model Calculation Z 10-12 P- £Q - 1.3 x IQI gy : 2 10-13 L_Z o: 6,9,10,18,20,26,28,54 _ E A : 92 10-14 I Mil 1 I 111 I Mil I Mil 1 I I it I 1ft 10° ID2 102 103 104 105 10s LET (MeV g"1 cm2) 627 Fig. 3 Double-5+rand Breaks — i 1 111 1 1 i11 i i >i i til] J ii* I SV-4G DNfl In flq. Solution ~ 10"7 r- Data from Roots+, 1989 -= i . 1.. 10-8 E- i ti l | 9 CN 10- £ U 10-1° / — c cn 10-12 IT r WalIgorskl+ Model Calculation "i 2 I Eo =» 2.5 x 10 Gy I 10-13 — (adjusted value) — - x • 9 - 1 Z o : 6,9,10,18,20,26,28,54 _I I A : 92 10-15 i i til i i ]\\ 1 Mil 1 Mil 1 I 111 t 1 II 10° 101 102 103 104 105 106 LET (MeV g"1 cm2) 14 12 h Nal (Tl) Eo = 700Gy 10 h on a 3 niri 8 00 EXPERIMENT THEORY 100 200 Energy(Hcv) 629 fig 5 lfll 1 il -j r I I I I i I i TT I i I n i I i i_ o LInes i Theory TLD- + Montret's Experimental Data g n Montret*s Calculation 10° to O 10-1 u Kr 18-2 J J LjJ i i ill i i|i 1 ID 102 103 104 10s 106 dE/dR (MeV«cm2/g) i-Mir HAHr-HIT DETECTORS t, (Cray) tlalo|lc*ls Doilntur) Enulslon t. i 1 Chenl9trr fhotoreslsis SSNTD's - target Celts Eauliinns ol Ivli.e • I ca - IE7 H0,«, M, r pulyslyrcnc Trypsln C«-3» f* 1aciua1 das* cellulos* nitrate •X-IT1 K-J flal(TI) in T-7 Frlckt «E3 I. IF K- I DouBlt »tr«nd brfika Glass K.O -ii IOIIS: it: III it:;.1 S»-»0 virus In E. Sol Id Or(.inlu chroaaild ekchannes burr.r Sclnll lUiors chillis* hjusirr ICt IEI ctllj survival, chro»OJU« it-i IE-1 633 THE IRRADIATION OF GLYCYLGLYCINE IN AQUEOUS SOLUTION A CASE STUDY OF CALCULATIONS FROM TRACK STRUCTURE TO BIOCHEMICAL CHANGE J. E. Turner^ Wesley E. Bolch,* H. Yoshida,* K. Bruce Jacobson,* 0. H. Crawford, i R. N. Hamm,i and H. A. Wright* health and Safety Research Division and 3Biology Division, Oak Ridge National Laboratory, P. 0. Box 2008, Oak Ridge, TN 37831-6123 USA 2Texas A&M University, College Station, TX 77843 USA We have carried out detailed Monte Carlo calculations of electron tracks produced by X rays in liquid water. The calculations start with the initial physical interactions and proceed in time with radical formation and subsequent chemical changes. We have treated the explicit chemical changes that occur when the dipeptide glycylglycine is present and have calculated the yields of various chemical products. We have also measured the chemical yields under the same experimental conditions as were assumed in the computations. Excellent agreement is obtained between the results of the measurements and the calculations. This work was carried out as a step toward the study of the more complicated DNA. These studies represent the first detailed linking of the basic physics of radiation interaction with the later observed biochemical effects. Research sponsored by the Office of Health and Environmental Research, U.S. Department of Energy, under contract DE-AC05-84OR21400 with Martin Marietta. Energy Systems, Inc. 634 APPROACH Experimental: Measure chrmical yields of radiogenic species produced by irradiation of biological molecules in aqueous solution. Perform detailed track-by-track Monte Carlo sm time of all events under identical conditions of the Radiation transport through water Eariy radical CVwBPtitation of measurable chemical Interactive: Use experimental data to guide development and choice of mechanistic pathways and unknown pan—rtrri in the •odd Tests of Validity of Model: With no further alienations, (eg., photons and ah*a partirW) and 635 MOLECULAR SPECIES TO BE INVESTIGATED: 1. Pure liquid Water 2. Amino Acid Polymers (Small Polypeptides): Glycylglycine - A "proof-of-principle" test case. 3. DNA: pTp Oligonucleotides Double-stranded segments supF tRNA gene fragment (200 nucleotides) 636 CH£MIST*Y OF OH H H 4 DM or H H + H OH + OH 637 | 10 0 n m / * - \ t l M n s S 1 ° " z \ Z | J * o /•' o • •: - \ f •r- «« I \ • '• 7 ° o « *" z Z k • 100 1 10 0 n m z 'T-'y" z \ E / •-. • TOO r / **-**>'• e SI \ 9 « z z 9 8 o JONAH ond MILLER 1977 • JONAH «t ol. 1976 SUMIYOSHI ond KATAYAMA 1982 7 ^^^^ un THIS CALCULATION 1 9 Co 3 - I i • 11 itil i 1 1 1 1 III! < ill t nil iii iiipl | i i iinil iii inn 10-12 10" i0~t0 10"* io- TIME (s) TiMc-dcpendent yields of Oil radicals and hydraced electrons in liquid water. Solid curves were calculated for 1-HtV electron track segments. INITIAL REACTIONS 4 -* AIM, o> OM4 ORNL-DWG 88-17203 4.0 > a> 3.5 - O O (2 3.0 h 2.5 2.0 UJ 1.5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 GLYCYLGLYCINE CONCENTRATION (M) (SW##TH CMMVO *** AtrAttfftfP (HUM*) 641 XHAfL Acid HAA HA* 4... -> Cl.V6Vl.AMll>*, 642 mi LATC ( > IO s) O) t < : oH -6t£a£* I OH -t^C, -i,xCtC* UK a ii *eu*c£ 13 643 0 \.o -6 -4 -2 0 LOG {TIME (s)) 644 M ~ -4 h — O E ^ -8 h Z LU O O O -12 h O O -16 -8 -6 -4 -2 0 LOG {TIME RESULTS 250* ftp A-*AYS, 2BO CALCULATtpA*: M UM CAM. DAS 1/73 £ ,.11 .*. 22X .•it. •If Sue .0* t .Oi •• AsP .31 .Hi 'OS ,2f 2.7t 2.PI * 9.2J AC6 Lift .ar ...i 9.FI 646 FRACTWrfS MAP€ PAffPKCT AOTM I.0M ©AS Sue *sr . MV7 NH, ° ORNL-DWG 89-14927 A 5-MeV PROTON • -••'"'5 ORNL-DWG 89-14928 B 5-MeV ALPHA PARTICLE %* Cv"" **• * • « * • * K". S' • :•" * •• •'.»' •• D/VA AT •CAM C*MPA*t*0*/ WITH 651 COLLECTIVE ASPECTS OF CHARGED PARTICLE TRACK STRUCTURE' G. Basbas Physical Review Letters, Box 1000, Ridge, New York 11961 USA A. Howie Cavendish Laboratory, Madingley Road, Cambridge CB3 OHE UK R. H. Ritchie Oak Ridge National Laboratory, P. O. Box 2008, Oak Ridge, TN 37831-6123 USA and University of Tennessee, Department of Physics, KnoxviUe, TN 37996 USA Abstract A plasmon generated by a swift charged particle constitutes a coherent excitation about the particle track. We discuss the representation of collective modes in impact parameter space when created by a swift ion or a fast electron, and the decay of these modes into localized excitations. Several alternative spatial representations are considered. We show that the high spatial resolution found in secondary electron emission measurements with scanning electron microscopy is consistent with the existence of the plasmon as an intermediary between the fast incident electron and the measured secondary electrons. 'Research sponsored by the Office of Health and Environmental Research, U.S. Department of Energy, under contract DE-AC05-84OR21400 with Martin Marietta Energy Systems, Inc. Partial travel support from the U.S.—Japan Cooperative Science Program of the National Science Foundation Joint Research Project No. 87-1631 l/MPCR-168, and from a NATO travel grant are gratefully acknowledged. 652 I. Introduction In quantal collision theories, momentum and energy are usually taken to be good quantal variables (Bethe, 1930). Classical collision theory, on the other hand, uses position and time to describe interactions between a probe and a target (Bohr, 1913, 1948). In modern physics, one may wish to express quantal theories in terms of space-like variables. For example, experiments are now common in which one measures, by means of a narrowly focused beam of swift electrons, the distribution in energy of losses experienced in a very small region of spacs. Also, in experiments with channeled ions, and in microdosimetry, one is interested in the spatial coherence of unlocalized excitations created by swift ions and electrons, and their ultimate localization through transfer of energy to, e.g., single-particle excitations. II. Theory The problem of visualizing quantal collisions in the space (and perhaps time) variable has been faced by Bohr (1948), Williams (1945), and by Neufeld (1953), among others. Fano (1970) has pointed out that in condensed media, excitations may be coherent over distances comparable with the Bohr cutoff impact parameter, b = v/wL, where v is the speed of a charged particle in a medium characterized by the resonant frequency u . Thus these excitations may involve collective motion of function of space by several alternative methods. The resulting transforms are compared. We also consider the probability of decay of the initially unlocalized coherent excitations into localized ones. i) The Inverse Mean Free Path Quantum dielectric theory may be used to find the expression Z22e2 -1 Im (1) W2/V2) This is the differential inverse mean free path (DIMFP) for energy loss, \ut, and momentum transfer, KK, to condensed matter characterized by the dielectric 2 2 function €* . The magnitude of the total momentum is We = JI(K2 + w /v )*. The goal here is to express the DIMFP in terms of a spatial variable, that we will interpret as impact parameter, rather than the momentum transfer, tut. ii) The "Van Hove Transform" Van Hove (1954) showed that the first Born approximation cross section of matter for incident neutrons may be factored into a product of functions, one of which contains all of the dependence on the properties of the matter. This function, called the pair distribution function, contains the scattering properties of the system, and depends on energy and momentum transfer to it. When Fourier transformed into a function, G(r,t), of space and time, this is interpreted by Van Hove as describing the quantal average density distribution at r, and t, if a particle is located at the origin of coordinates at t = 0. Van Hove notes that G(r,t) is a complex function because it reflects quantum properties of the system. It has been used extensively in analyzing the properties of dense gases and liquids as inferred from neutron scattering experiments. 654 Pines and Nozieres (1966) state that the comparable quantity in a polarizable system characterized by the dielectric function 6^ ^ is to be interpreted as the function describing correlations between density fluctuations at different space—time points. Rather than dealing with the correlation function G as defined by Van Hove we compute the two-dimensional Fourier transform of the DIMFP in order to compare directly with other forms described below. Thus OD 2 2 2 2 2 2 ) = (2Z e /?tv ) / &u> f d n30(bK) Imf-l/e^/^ + o/*/v ) o (2) where we have integrated over u in order to obtain an easily surveyed result. In Fig. 1 the solid line shows Ayrr(b) versus b for an ion with speed v = 20 a.u. (e.g., a 10 MeV proton) moving in an electron gas having the plasmon energy of 15.4 eV. In computing this result, we have assumed that the plasmon has negligible damping and quadratic dispersion with a cutoff at k = w /vp, where Vp is the Fermi speed for this electron gas. That is, we take Im(-l/e) = (TW /2C*) S(u - w.) ${k — k), where u, = (w + Irk y and 0 is the propagation speed of hydrodynamical disturbances. A simple and easily surveyed result is obtained if the plasmon is approximatedmated as a dispersionlesdispersi s resonance at u>— u and if the integral over K it extended to infinity. Then 2*Z2eV TlV 655 where KQ is the modified Bessel function of the second kind and order zero. For large x, KQ(x) ~ exp(-x)/(2ax)*. In Fig. 1 we have also plotted Ay£(b-1, ) versus b calculated from Eq. 3 as the dashed line for the same conditions as for that computed from Eq. 2. In obtaining this result we have assumed that the plasmon has negligible damping and quadratic dispersion with a cutoff at k = wD/vr,. iii) The Chang—Raman Transform An approach to the problem of expressing quantal probabilities in terms of an impact parameter variable has been advocated by Fano (1970), based on a method introduced into high-energy physics by Chang and Raman (1969). To apply this method to the DIMFP of Eq. 1 we first integrate over all u to find 2 2 f da; -1 d Ac£/d * = zk, We now seek to eliminate K in favor of a spatial variable that we will interpret as an impact parameter. Thus ACR = 77(Zir)3 2 2 d b d /cexp(i (S) 656 The integrand of Eq. 5 is now set equal to the DIMFP in impact parameter space, viz., (C (6) One obtains a useful approximation to this equation for the case where the medium supports plasma oscillations codified in a dielectric function that is sharply peaked about a quadratic dispersion line having the form ^ | )* (7) as indicated above for the hydrodynamical model. In this case one may show that This function is plotted in Fig. 2, as the line labeled CR, for the same conditions assumed in plotting Fig. 1, except that tut = 10.2 eV. The Chang-Raman transform does not seem to be useful for finding the DIMFP expressed as a function of both w and impact parameter. It appears that when the DIMFP has a narrow resonance in the k-w plane such as may occur when plasma oscillations exist in the medium, the transformation to the b variable gives an indeterminate result. 657 iv) The Energy Transfer Transform It is possible to factor the integrand of Eq. 4 in such a way that the DIMFP may be written as a function of energy transfer as well as impact parameter. Writing so that dA V u1r fbw (9) interpreting the integrand as before. We term this the "energy transfer transform1* since it agrees precisely with the formula obtained by computing the energy transferred to the medium at fixed impact parameter, using semi-classical dielectric theory, and then dividing the integrand in the w variable by \u. An equation comparable with Eq. 6 is obtained by integrating Eq. 9 over u. The solid line labeled ET of Fig. 2 shows numerical results fo? the conditions specified for Fig. 1. The curve labeled CR was. computed from Eq. 8. One finds that PET(b) has a rather different dependence on b than does P^R(b). Their n asymptotic forms are Pgj ~exp(—2w b/v)/b and p«j» ~exp(—2w b/v)/b. As we 658 show below, pET(b) corresponds more nearly to the spatial variation of the localization probability than does P^R(b). In Fig. 3 the function of Eq. 9 is plotted in the b-w plane. A simplified representation of the dielectric function of the medium was used in obtaining this plot. It shows the DIMFP for a charged particle with a velocity of 20 a.u. proceeding through an electron gas with plasma energy of 15.4 eV and includes contributions from single-electron transitions as well as plasmons. The plasmon contribution appears as a rather broad resonance in us, peaking at near the plasma energy for b>2 a.u. but showing a ridge concentrated at very small b and extending to large values of w. This corresponds to the well—known fact that encounters at small impact parameters tend to involve large energy transfers to the medium. This feature might be called the "Bohr ridge" in analogy with the Bethe ridge of the DIMFP in the k-w plane. It is interesting that although this function is always real, unlike the Van Hove correlation function, it may be negative over small regions of the b—w plane. An example of this may be seen in the vicinity of the point w~.7 a.u. and b~5 a.u. in Fig. 3. This is an indication of the essential quantal character of the interactions under consideration. However, it should be noted that the DIMFP integrated over w, or ever b, is positive definite. v.) The Localization of Initially Unlocalized Excitations The mathematical problem of representing interactions in condensed media in terms of an impact parameter-like variable is separate from, but related to, that of establishing the location and products of plasmon decay. The plasmon is thought to be an unlocalized excitation in the background of the valence electron gas of a piece of condensed matter. The plasmon in an electron gas is regarded as a plane 659 wave extending over the whole volume of the medium. The electric field of a plasmon in an ideal electron gas is in the same direction as its propagation vector. The electric field of a photon in vacuum is perpendicular to its propagation vector, but otherwise these entities are similar. Frolich and Peltzer (1948) pointed out this similarity and Wolff (1953) noted that a long-wavelength plasmon (k< In this approximation the probability of plasmon (photon) localization at a point in space is proportional to the square of the plasmon (photon) electric field strength at that point. It seems clear that this should be true irrespective of the nature of the solid as long as the plasmon wavelength is much less than that of the electron, i.e. kplasmon«lA-i. We may go beyond the long—wavelength approximation in a simple model of the localization process. Assume that an impurity site in a condensed medium is occupied by an electron in an orbital u (r), situated at r. Let a swift ion with speed v traverse the medium at impact parameter b relative to the impurity. If the eigenenergies and wave functions of the impurity site electron are Jtw and u (r) then the probability PQ that the electron is excited through virtual collective states in the medium from its ground state to the nth excited state is illustrated by the Feynman diagram of Fig. 4. Pn may be written 660 y (10) J 2 where w = w — WL, k = «' + wn0 M> and where it is understood that the integration over « is to include only the region of (k,w) space corresponding to collective states of the medium. Equation 10 has been evaluated for illustrative purposes assuming uo(r) and u,(r) are Slater orbitals and have s- and p-wave character, respectively. Figure 5 shows a plot of the dependence of Pj on b for various final state energies. On the same plot is shown the energy transfer transform vs b computed from the integral over u> of Eq. 9. The medium is taken to be described by the Mermin dielectric function, with %u> = 15.4 eV, a damping constant of 3.8 eV, and a cutoff wave number k = w /vp. Ritchie and Brandt (1975) and Brandt and Ritchie (1973) have discussed qualitatively the effect of plasmon propagation and damping on the localization process. They conclude that these are rather unimportant and that, to a good approximation, the impact parameter b « v/w characterizes the decrease of the c p localization probability with increasing impact parameter. Calculations made with Eq. 10 confirm these conclusions. III. Spatial Resolution in Energy Loss Spectroscopy An estimate of the effective distance from the track of a swift electron at which excitation of an electronic transition with energy transfer fcw will occur can be made on the basis of the duration of the electric impulse experienced at a given impact parameter by a struck electron. This yields the "cutoff' impact parameter bc=v/w. For a 15 eV4oss with 100-keV electrons this comes to ~7 nm. 661 Cheng (1987) has pointed out that this figure is considerably larger than the spatial resolution of 0.4 nm that has been achieved in some experiments (Scheinfein et al., 1985) using the 15-eV loss in Al. He identifies the resolution found with the distance traveled by the plasmon before it decays and gets quantitative agreement between his theory and experiment using reasonable estimates of the plasmon group velocity and lifetime. In our view, this explanation is suspect. It is important to realize that t> is an upper estimate of the impact parameter corresponding to zero scattering angle and thus zero momentum transfer perpendicular to the initial velocity. Larger values of momentum transfer are associated with smaller interaction distances. The strength of the excitation at a given energy loss is in general determined by an integral over momentum transfer and ultimately depends on some function of dbfv, such as the function K (2wb/v) in the case of a planar interface (above) or the function K 2(wb/v)+K 2(wb/v), which is appropriate in the dipole limit for a very small sphere. Recent success (Batson et al., 1986) in the high spatial resolution band gap spectroscopy of defects in semiconductors lends some support to the latter expression. The function K (wb/v) varies quite rapidly with cub/v as its argument increases, particularly in the interval wb/v with 1-nm spatial resolution and 1-eV energy resolution and that reflection SE images of oxidized Cu show oxide islands and details of their interaction with surface steps. The generation of secondary electrons by fast incident electrons is quite complex, involving electron cascade processes created by fast secondaries and the slowing-down of the resulting knock-ons as well es the decay of inner-shell vacancies and collective states in the valence band. The relative importance of these different excitation processes has been considered by a number of authors but less attention has been paid to assessing their degree of localization, i.e., the relevant impact parameter or distance from the electron beam where the secondary is generated. Elaborate calculations have been made for Al (Chang and Everhart, 1977; Rosier and Brauer, 1981a, 1981b, 1988) indicating that plasmon excitation followed by decay into electron—hole pairs makes the dominant contribution to the SE signal. Monte Carlo calculations by Luo and Joy (1988) show that the majority of secondaries originate from plasmon decay. Others (Milne and Echenique, 1985) question whether plasmon decay is sufficiently well-localized to explain their measured high spatial resolution. V. Application of the Impact Parameter Representation (IPR.) In the context of SE we use the Energy Transfer Transform of Eq. 9 above to obtain the distribution in impact parameter of energy deposition in the conduction band of aluminum metal. We take the Mermin-RPA dielectric function of the electron gas (Mermin, 1970) to represent the response of the medium, with a damping constant of 0.5 eV. Figure 6 shows the IPR distribution calculated from Eq. 9 for three different electron energies. To obtain these results we have integrated Eq. 9 over K and u> by numerical quadrature. The somewhat surprising 663 result is that each of the curves decreases as b increases, going asymptotically as exp(—2w b/v) when b —•». It turns out that for each of these primary energies the mean value of b, averaged over these distributions is less than 1 nm. For emphasis 4 O O in plotting the calculated DIMFP values have been multiplied by 2ihv /e u . The small fluctuations in these curves correspond to quantal effects in the plasmon field. These results turn out to be quite insensitive to the damping constant assumed in the dielectric model. Note that propagation and decay of the plasmon are described in detail in this treatment. It appears that the narrow spatial resolution of these IPR distributions is due to the presence of relatively large momentum components in the interaction spectrum of the swift electron and the electron gas. Summary and Conclusions We have considered the impact parameter representation of excitations generated in condensed matter by swift charged particles. The problem of spatial resolution in SEM when low-loss valence excitations occur is addressed. It is shown, using a reasonable model of the DIMFP, that the experimental SE data showing high spatial resolution may be accounted for by the presence of large momentum components in the electron—valence band interaction spectrum. 664 References P. E. Batson, K. L. Kavanagh, J. M. Woodall, and J. W. Mayer, Electron- Energy-Loss Scattering Near a Single Misfit Location in a Dielectric Medium of Randomly Distributed Metal Particles, Phys. Rev. Lett. 57,2729-2732 (1986) H. A. Bethe, Zur Theorie des Durchgangs schneller Korpuskularstrahlen durch Materie, Ann. Phys. 5, 325-400 (1930) A. L. Bleloch, A. Howie, and R. H. Milne, High Resolution Secondary Electron Imaging and Spectroscopy, Ultramicroscopy 21, 99-110 (1969) N. Bohr, On the Decrease of Velocity of Swiftly Moving Electrified Particles in Passing Through Matter, Phil Mag. 25, 20-13 (1913) N. Bohr, The Penetration of Atomic Particles Through Matter, K. Danske Vidensk. Selsk. Mat-Fys. Mcdd. lg, 1-144 (1948) W. Brandt and R. H. Ritchie, Primary Processes in the Physical Stage, In Proc. Conf. Phys. Mech. Radial Biol pp. 20-46, Natl. Tech. Info. Service, U.S. Dept. of Commerce, Springfield, Virginia 22151, U.S.A. CONF-721001 (1973) N. P. Chang and K. Raman, Impact Parameter Representation and the Coordinate Space Description of a Scattering Amplitude, Phys. Rev. JLgl, 2048-2055 (1969) S. C. Cheng, Localization Distance of Plasmons Excited by High—Energy Electrons, UUramicrosc. 21. 291-292 (1987) M. S. Chung and T. E. Ever hart, Role of Plasmon Decay in Secondary Electron Emission in the Nearly-Free-Electron Metals. Application to Aluminum. Phys. Rev. B15T 4699-4714 (1977) U. Fano, The Formulation of Track Structure Theory, In Charged Particle Tracks in Solids and Liquids, pp. 1-7. The Institute of Physics and the Physical Society, Conf. Series No. 8, London (1970) H. Frolich and H. Peltzer, Plasma Oscillations and the Energy less of Charged Particles in Solids, Proc. Phys. Soc. Ajjg, 525-530 (1955) S. Luo and D. C. Joy, Monte Carlo Calculations of Secondary Electron Emission, Scanning Microscopy 2, 1901 (1988) N. D. Mermin, Lindhard Dielectric Function in the Relaxation-Time Approximation, Phys. Rev. gl, 2362-2363 (1970) R. H. Milne and P. M. Echenique, The Probability of MgO Surface Excitations with Fast Electrons, Solid State Comm. 55, 909-910 (1985) 665 J. Neufeld, Energy—Losses of Charged Particles of Intermediate Energy, Proc. Phys. Soc. Agfi, 489-596 (1953) D. Pines and P. Nozieres, The Theory of Quantum Liquids, W. A. Benjamin, New York (1966) R. H. Ritchie and W. Brandt, Primary Processes and Track Effects in Irradiated Media, In Radiation Research; Biomedical, Physical, and Chemical Perspectives, pp. 315-324, Academic Press, New York (1975) M. Rosier and W. Brauer, Theory of Electron Emission from Solids by Proton and Electron Bombardment, Phys. Stat Sol. b!48. 213-226 (1988) M. Rosier and W. Brauer, Theory of Secondary Electron Emission. II. Application to Aluminum, Phys. Stat. Sol. b!04. 575-587 (1981) M. Rosier and W. Brauer, Theory of Secondary Electron Emission. I. General Theory for Nearly-Free-Electron Metals, Phys. Stat. SoL b!04.161-175 (1981) A. Scheinfein, A. Muray, and M. Isaacson, Electron Energy Loss Spectroscopy Across a Metal-Insulator Interface at Sub-Nanometer Spatial Resolution, Ultramicrosc. 16, 233-240 (1985) L. Van Hove, Correlations in Space and Time and Born Approximation Scattering in Systems of Interacting Particles, Phys. Rev. 9j>,249—262 E. J. Williams, Space-Time Concepts in Collision Problems, Rev. Mod. Phys. 17, 217-245 (1945) P. A. Wolff, Theory of Plasma Waves in Metals, Phys. Rev. §2,18-23 (1953) 666 Fig. 1. A plot of the "Van Hove" inverse mean free path as a function of impact parameter for an ion with speed v = 20 a.u. moving in an electron gas with a plasma energy of 15.4 eV. The dashed line shows a plot of the distribution computed from the approximate form [Equation (3)] while the solid curve was calculated from Equation (2). Fig. 2. A plot of the scaled DIMFP vs. impact parameter b computed from the Chang-Raman transform equation (8) (labeled CR), and from the energy transfer transform equation (9) after integration over w (labeled ET). The plasma energy is taken to be 10.2 eV. Both sets of data have been multiplied by b to emphasize the differences for large b. Fig. 3. The DIMFP for an electron gas, plotted as a function of impact parameter b and energy transfer tuii. The DIMFP is normalized arbitrarily, and b and %u) are expressed in e.u. An approximate expression for 6^ has been used in this calculation; the particle is assumed to have a speed of 20 a.u. and Ku_ = 15.4 eV. This is equivalent to a proton with an energy of 10 MeV in Al metal. Fig. 4. A Feynman diagram representing the process of virtual plasmon creation by a swift charged particle in condensed matter followed by excitation of a real, single electron-hole pair in the medium. Fig. 5. The excitation probability, Pj, of an electron bound to an impurity site, plotted as a function of the impact parameter of the site relative to the trajectory of a charged particle with speed 20 a.u. For comparison, the curve labeled nETn is computed from Equation (9) after integration over a) and represents the DIMFP d2Ag|,/d b for the same particle. Arbitrary normalization of both curves has been used. Both sets of data have been multiplied by b for emphasis. Fig. 6. A plot of dA /db, the inverse mean free path differentia! in impact parameter b, versus b, for three different electron energies, calculated from Eq. 10. For convenience in plotting, the curves have been scaled by multiplying the results from Eq. 10 by the factor (2/wv), where v is the electron speed. 667 \ 7 > "•M 9 Pfl 'Ifl ca 100 impact parameter-b (a.u.) Fig. 1 668 b-Inract P*rwmi*ri.m. Fig. 2 669 Fig. 3 670 Fig. 671 b-Iaract Fig. 5 672 Fi». 6 673 CALCULATIONS OF AUGER-OSCADE-INDUCED REACTIONS WITH HIA IN AQUEOUS SOLUTION* R. N. Harnn. H. A. Wright . and J. E. Turner Health and Safety Research Division Oak Ridge National Laboratory Oak Ridge. Tennessee 37831-6123 R. W. Howe11 and D. V. Rao University of Medicine and Dentistry of New Jersey. Newark, New Jersey 07103 K.S.R. Sastry University of Massachusetts, Amherst. Massachusetts 01003 INTRODUCTION The biological effects of radionuclides incorporated into nannalian cells are of considerable interest for radiation biology and radiation protection. When radionuclides decay by electron capture or internal conversion an inner-atomic-shell vacancy is created. The subsequent Auger transitions produce numerous low-energy electrons whose ranges are of subcellular dimensions In tissue-equivalent matter (1,2). When such Auger emitters are localized on or near the DNA of cell nuclei, severe biological effects are observed, both In vitro and in vivo (2-4), reminiscent of alpha particles of high linear energy transfer (LET). Indeed, recent experiments with the 5.3-MeV alpha emitter Po-210, intracellularly localized, clearly showed that its lethal effects were essentially the same as those of DNA-incorporated 1-125, a prolific Auger emitter (5,6). The relatively high radiotoxicity of Auger emitters has been attributed to the highly localized energy density in the immediate vicinity of their decay sites (7). A more detailed understanding of the equivalent lethality of high-LET alpha particles and DNA-bound Auger emitters may come from track-structure calculations which provide information on the number of direct physical interactions and subsequent indirect chemical interactions with the DNA molecule (7.8). Accordingly, track-structure calculations for a variety of medical and environmental radionuclides (Fe-55. In-Ill, 1-125. Pt-193m. Pt-195m) which emit different numbers of Auger electrons are considered in this work and compared with those for alpha, particle tracks. "Research sponsored in part by the Office of Health and Environmental Research. USDOE. under contract DE-AC05-84OR21400 with Martin Marietta Energy Systems. Inc.. and in part by the USPHS under grant number CA-32877. Present Address: Consultec Scientific. Inc.. 725 Pellissippi Parkway. Khoxville, Tennessee 37932-3300 674 MONTE CARLO SIMULATION OF AUGER CASCADES Simulation of the nuclear and atomic events associated with the decay of several Auger-electron-emitting radionuclides was accomplished using Monte Carlo calculational techniques described in detail elsewhere (9,10). Briefly, at any stage of the radionuclide decay, the simulated decay process proceeds randomly by selecting a transition from the set of all allowed transitions for the nuclear as well as atomic configuration at that stage. As the random selections are made, data concerning the emitted radiations are recorded. This random selection process continues until the ground state of the isotope is reached. Calculations of the energies of Auger electrons produced from a number of decays have been performed for the radionuclides Pt-195m, Pt-193m. 1-125, In-Ill, and Fe-55. The Monte Carlo radiation transport code OREC (8,11-13) has been used to transport the electrons produced during the Auger cascades through liquid water surrounding the decay site and to calculate the physical and chemical interactions produced. In this code, a charged particle and all of its secondary electrons are followed, and the location and type of each inelastic energy loss event is recorded. These physical events occur within 10"1 s and produce excited or ionized water molecules and subexcitation electrons. The program then calculates the location and identity of each reactive chemical species that is produced by 10~12s. These chemical species are then followed as they undergo diffusive motion and react with each other or with, solute molecules that may be present. ' An example of such a calculation is shown in Figure 1. In order to estimate the interactions that might be produced with a DNA molecule, a very simple model has been assumed. A segment of double-stranded DNA is represented as a right circular cylinder of radius 1 nm with "sugar" and "base" reactive sites alternating along two helical strands on the surface. Each strand makes one turn in 3.4 nm and has 20 reactive sites per turn. A description of this simple model has been published (8). For the purposes of this paper two types of interactions with the DNA are considered. During the charged-particle transport the DNA cylinder is treated as though it were water, and if an inelastic energy loss event occurs within the cylinder it is considered to represent a "direct" physical event. No attempt is made to further characterize the event or its consequences. In our calculations there is no limit to the number of these direct events. An "indirect" chemical event is considered to result when a reactive chemical species interacts with a "sugar" or "base" site on the DNA. Only one indirect event can occur with a "sugar" or "base," and hence the maximum number of indirect events is the number of sugars and bases. Although no attempt is made to identify the consequences of these direct or indirect events, it is interesting to compare the relative numbers of such events for various types of radiation. 675 RESULTS Calculations have been performed of the transport of the electrons produced during Auger cascades from decay of the radionuclides Fe-55. In-111, 1-125, Pt-193m, and Pt-195m. In order to get a feeling for the relative density of reactive chemical species produced during such cascades, the positions of chemical species from representative decays are plotted in Figure 2. In the four quadrants of the figure the dots represent the initial positions of the reactive chemical species OH. + 12 H30 , H, and the hydrated electron e" at 10~ s before appreciable aq diffusion has occurred. In each case the positions of the sugar and base sites on the two strands of the DNA are also shown. The radionuclide decay site is taken to be on the surface of the cylinder, i.e. 1 nm from the axis. In the lower right quadrant the positions of chemical species produced by a 5.3-MeV alpha particle passing perpendicular to the DNA cylinder are shown. It should be noted that there is significant statistical variation in the Auger cascades. In each case we have taken examples of decays that have produced average numbers of chemical species. Figure 3 shows the initial distribution of chemical species around the site of the decay of 1-125. It shows the number of each of the 4 reactive species in concentric spherical shells of thickness 1 nm centered at the decay site. The hydrated electrons peak somewhat further from the decay site than the other species. We consider now the number of events that would be produced on DNA as a result of the decay of an Auger emitter. Table I shows the number of direct physical and indirect chemical events that would be produced with a segment of DNA by the decay of various Auger emitters located on the surface at the middle of the cylinder, 1 nm from the cylinder axis. For comparison, similar information is shown for 5.3-MeV alpha particles passing perpendicular to the DNA and 1 nm from the axis of the cylinder and also for 5.3-MeV alpha particles emitted isotropically from a point on the surface at the middle of the cylinder. The radionuclide Po-210 emits a 5.3-MeV alpha particle when it decays, and other studies (5,6) have shown that biological effects similar to those from 1-125 decay are produced (3}. Table I: Numbers of direct physical and indirect chemical interactions produced on DNA by Auger emitters located on the surface of the 1-nm radius DNA cylinder compared with those from 5.3-MeV alpha particles passing 1 nm from the center of the DNA cylinder perpendicular to its axis and with 5.3-MeV alpha particles emitted isotropically at 1 nm from the center of the DNA. Direct Indirect Pt-195m 36 56 Pt-193m 35 52 1-125 21 44 In-Ill 11 30 Fe-55 5 18 5.3-MeV alpha perpendicular 4 29 5.3-MeV alpha isotropic 4 18 676 Figure 4 shows the total number (direct plus indirect) of DNA interactions produced by the decay of various radionuclides as a function of the distance of the decay site from the center of the DNA. Also shown are the number of DNA interactions from 5.3-MeV alpha particles passing perpendicular to the DNA cylinder at various distances from the axis and from 5.3-MeV alpha particles emitted isotropically from points the same distances from the center of the DNA. CONCLUSIONS The essentially simultaneous emission of electrons from an Auger cascade produces a very high energy density within a few nanometers of the decay site. If the Auger emitter is attached to the DNA it can produce damage to the DNA similar to that of a high-LET charged particle. Although the model for DNA used in this paper is very simplistic, it is interesting to compare the relative numbers of interactions produced by various radiations. Experimental results discussed elsewhere (5,6) indicate that the effects of 1-125 decay and Po-210 decay producing 5.3-MeV alpha particles have similar biological effects. However, the calculations in this paper indicate that more DNA interactions are produced per decay by the 1-125 than the Po-210. Thus one might expect the 1-125 to produce more biological effects. However, the alpha particles have a range of ~50 micrometers in unit density matter and will pass entirely through a cell and may interact with the DNA in several places, whereas the high energy density regions from the 1-125 decay are confined to only a few nanometers. REFERENCES 1. Sastry, K.S.R., and Rao, D. V., Dostmetry of Ixw Energy Electrons. In Physics of Nuclear Medicine: Recent Advances, D. V. Rao. R. Chandra, and M. Graham. Eds.. Medical Physics Monograph No. 10. American Institute of Physics, pp. 169-208 (1984). 2. Rao. D. V.. Mylavarapu, V. B.. Govelitz. G. F.. Lanka, V. K.. Sastry, K.S.R.. and Howell, R. W., Biological and Biophysical Dosimetry of Auger-emitters in vivo: A Review. In Selected Topics in Physics of Radiotherapy and Imaging. U. Madhvanath. K. S. Parthasarathy. T. V. Venkateswaran. Eds. Tata McGraw-Hill. New Delhi, pp. 232-258 (1988). 3. Kassis. A. I., Howell. R. W.. Sastry. K.S.R.. and Adelstein. S. J.. Positional Effects of Auger Decays in Mammalian Cells tn Culture. In DNA Damage by Auger Emitters. K. F. Baverstock. D. E. Char 1 ton. Eds.. Taylor & Francis, London, pp. 1-14 (1988). 4. Rao. D. V.. Mylavarapu, V. B.. Sastry. K.S.R.. and Howell. R. W., Internal Auger Emitters: Effects on Spermatogenesis and Oogenesis tn Mice. In DNA Damage by Auger Emitters. K. F. Baverstock and D. E. Charlton. Eds.. Taylor & Francis, London, pp. 15-26 (1968). 677 5. Koweli, R. W.. Narra, V. R., Rao. D. V., and Sastry. K.S.R.. Radiobiological Effects of Intracellular Polonium-210 Alpha Emissions: A Comparison with Auger-emitters. Radiat. Prot. Dosim. (in press). 6. Rao, D. V.. Narra, V. R.. Govelitz, G. F., Lanka. V. K.. Howell. R. W.. Sastry, K.S.R.. In vivo Effects of 5.3-JieV Alpha Particles from Po-210 in Mouse Testts: Comparison with Internal Auger-emitters. Radiat. Prot. Dosim. (in press). 7. Sastry. K.S.R., Howell, R. W.. Rao, D. V.. Mylavarapu. V. B.. Kassis. A. I., Adelstein, S. J.. Wright, H. A.. Kami, R. N.. and Turner, J. E., Dosimetry of Auger-emitters- Physical and Phenomenological Approaches. In DNA Damage by Auger Emitters. K. F. Baverstock, D. E. Char1 ton. Eds., Taylor & Francis, London, pp. 27-38 (1988). 8. Wright. H. A., Nagee, J. L.. Ham. R. N.. Chatterjee. A.. Turner. J. E.. and Klots. C. E.. Calculations of Physical and Chemical Reactions Produced in Irradiated Water Containing OKA. Rad. Prot. Dosim. 13. 133-136 (1985). 9. Howell, R. W.. Sastry. K.S.R.. Hill. H. Z.. and Rao. D. V.. Cis-Platinum-193m: Its Jficrodoslmetry and Potential for Chemo-Auger Combination Therapy of Cancer. In Proceedings of 4th International Radiopharmaceutical Dosimetry Symposium. E. E. Watson and A. T. Schlafke-Stelson, Eds., Oak Ridge. TN. OONF-8511113. pp. 493-513 (1986). 10. Char1ton. D. E., and Booz, J.. A Monte Carlo Treatment of the Decay of 1-225. Radiat. Res. 87, 10-23 (1981). 11. Turner, J. E., Magee. J. L.. Wright. H. A., Chatterjee. A., Hamm, R. N.. and Ritchie, R. H.. Physical and Chemical Development of Electron Tracks in Liquid Water. Radiat. Res. 96. 437-449 (1983). 12. Hamm. R. N.. Turner, J. E.. Ritchie. R. H.. and Wright. H. A.. Calculation of Heavy Ion Tracks in Liquid Water, Radiat. Res. 104. S-20 - S-26 (1985). 13. Turner, J. E., Hamm, R. N. Wright. H. A.. Ritchie. R. H.. Magee, J. L., Chatterjee. A., and Bolch. Wesley E.. Studies to Link the Basic Radiation Physics and Chemistry in Liquid Water, Radiat. Phys. Chem. 32, 503-510 (1988). 678 ( \ t. WH« w-'°i N-924 N-924 N-786 N-785 -f- -H •+• •+• 100 ran WO nm WOrnn WO nm ' • 1.' K 1O'^s 10" s «*•• >**s • t*.; N -496 N- 496 N- 403. N-403 •'*. *.* • **" i WOnm tOO ran 100 nm TOOom Fig. 1. Example of the chemical evolution in the track of a 4-keV election in pure crater from 1012 s to 10' s, when intrattack radical reactions are essentially complete. In these stereo views the electron suns moving upward at the intersection of the axes and stops above, where the familiar "darter" end of the track is seen. Each dot represents an individual reactam produced by ibe electron or one of iu secondary electrons. Intratrack reactions reduce the original number of reactants at 1012 s from N » 924 to N * 403 at 10"7 s. 679 10 nm 125. 5.3-MeV ALPHA Fig. 2. Initial positions of reactive chemical species produced by representative examples of Auger cascades from the decay of the radkmudides Fc-55, In-Ill, and 1-125. Sugar and base sites on two helical strands of a segment of DNA are also shown. The decay site is on the sur&ce at the middle of the cylindrical segment of DNA. The tower right quadrant shows the initial positions of chemical species produced by the passage of a 5.3-McV alpha particle perpendicular to and at the middle of the DNA segment 680 25 1251 AUGER CASCADE c 20 to LJ U UJ 15 Q. CO 10 m 5 & —*- 5 10 15 DISTANCE (nm) Fig. 3. Average number of chemical species initially produced in spherical shells of thickness 1 nm centered at the decay site of 1-125 Auger emitter. l > 1 D Pt-195 A A Pt-193 A 1-125 • • In—111 o o Fe-55 • 5.3-MeV a, PERPENDICULAR 5.3-MeV a, ISOTROPIC 0 2 4 6 8 10 DISTANCE FROM CENTER OF DNA (nm) Fig. 4. Average number of DNA interactions (direct plus indirect) produced by decay of various radionuclides at different distances from the center of a cylindrical segment of DNA and also produced by 5.3-MeV alpha particles passing perpendicular to the DNA cylinder or emitted isotropically from points at various distances from the center of the DNA cylinder. .?»•.