VI. simpozij HDZZ, Stubičke Toplice HR0500032
ELASTIC SCATTERING OF ELECTRONS AND POSITRONS
Ines Krajcar Bronić Ruder Bošković Institute, Bijenička 54, HR-10000 Zagreb, Croatia e-mail: [email protected]
INTRODUCTION Electrons and positrons interact with matter through several competing mechanisms, elastic and inelastic scattering being the most abundant. Elastic scattering is considered as the non-radiative interaction between the projectile electron or positron and a target atom/molecule in which the internal energy of the target is not changed. In measurements of electron scattering by molecules, pure elastic scattering cannot be resolved from low-energy vibrational or rotational excitations; and the term quasi-elastic scattering is used. Elastic scattering has a prominent influence on the transport of fast electrons and positrons in matter. In elastic collisions, these particles may undergo large deflections and, as a result, the space distribution of dose from electrons and positrons depends strongly on the elastic scattering properties of the medium. The demand for accurate cross sections of electron and positron interactions with different atoms and molecular systems has been growing rapidly in the applied science community. In particular detailed information on elastic scattering of these particles by molecules is required for Monte Carlo simulations in microdosimetry, radiation dosimetry, nuclear medicine, radiation therapy, radiation protection, atmospheric and plasma physics, various electron-spectrotroscopic techniques, etc. A similar need arises in modeling the energy deposition associated with the interaction of any form of ionising radiation with matter.
REPORT COMMITTEE International Commission for Radiation Units and Measurements (ICRU), having headquarters in Bethesda, MD, USA, formed in 2000 a Report Committee that should prepare a new ICRU Report on "Elastic Scattering of Electrons and Positrons" (ESEP). The report should constitute a synthesis of the leading scientific thinking on matters of radiation quantities, units and measurements techniques and provide recommendations that represent an international consensus on these matters [1]. The ICRU sponsors for this activity are Dr. Mitio Inokuti (Argonne National Laboratory, USA, and ICRU) and Mr. Steven M. Seltzer (National Institute of Standards and Technology, USA). Dr. Francesc Salvat (Faculty of Physics, University of Barcelona, Spain) has been appointed as the Chairman of the ESEP Report Committee. The members are Dr. Martin J. Berger (Bethesda, MD, USA, U12004), Prof. Dr. Aleksander Jablonski (Institute of Physical Chemistry,
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Polish Academy of Sciences, Warszawa, Poland), Dr. Ines Krajcar Bronić (Rudjer Bošković Institute, Zagreb, Croatia), Dr. James Mitroy (Faculty of Science, Northern Territory University, Darwin, Australia), Dr. Cedric J. Powell (NIST, USA) and Dr. Leon Sanche (Faculty of Medicine, University of Sherbrooke, Canada). The first meeting of the Report Committee was held in ICRU Headquarters in Bethesda, MD, in April 2000. The objectives and scopes of the report have been elaborated and the outline of the report was discussed. Each Report Committee member received specific tasks and duties. On the second meeting in Bethesda in December 2001, the individual contributions were presented and discussed. The next meeting of the Report Committee was held in Barcelona, Spain, in June 2003. The Report has been presented in its final form, and the discussion was directed into making the report more consistent and homogeneous.
OUTLINE OF THE REPORT The ESEP Report consists of the following chapters: 1. Introduction (including nomenclature, application of data to be given in the Report, and scope of the report); 2. Experimental methods (measurement techniques of differential cross
sections (DCS), integral elastic (ae|) and momentum transfer (amt) cross sections; for gases, liquids and solids); 3. Theoretical background (fundamentals of scattering theory, quantum theory, approximation methods, elastic scattering by molecules, scattering in the condensed phase, positron scattering); 4. Calculations for atoms (numerical calculation methods, properties of the phase shifts, DCS for atoms, high-energy factorization) for energies above 100 eV; 5. Experimental data (comparison of theoretical with experimental data, atomic and molecular gases, condensed phases); 6. Multiple-scattering angular deflections. Two appendices contain details on the theory of relativistic kinematics and the Dirac equation. Bibliography contains more than 300 relevant references. The CD ROM with the original software for calculating DCS will be distributed with the Report. For the calculation of DCSs for free atoms the central optical model potential is used [2]. The type of the incident particle and its energy, the target atom, and the choice of the model can be chosen by the user. A program for calculation DCSs for electron or positron scattering by molecules according to the independent atom approximation will be also available [3].
EXPERIMENTAL TECHNIQUES The experimental techniques used to study electron collisions with atoms and molecules can be broadly classified into two groups: beam experiments (where
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single collisions between individual scattering partners are examined), and swarm experiments (where the derived quantities are extracted from observation of the collective motion of a large number of charged particles, electron swarms). The two techniques are complementary, although often viewed as competitive. The swarm experiments yield absolute values of the elastic momentum transfer cross section amt, while beam experiments of direct attenuation give the elastic cross section, aei, (or total cross section, at), and the crossed-beam experiments provide the elastic differential cross section, DCS or da/dQ. The beam method is applicable to a wide range of energies, except very low energies, and the measurements are straightforward to interpret. However, the problems are: preparation of electron beams with very low energy, limited angular range accessible to measurement, the need of measurements with standard gases to derive absolute data. The swarm method is experimentally easier, and is particularly reliable at low energies (<1 eV), but requires a delicate numerical analysis to derive absolute values of the sought cross sections from the measured transport parameters.
DATA EXAMPLES A large part of the available experimental data pertain to atomic gases, particularly to a limited group of atoms such as hydrogen, the rare gases, the alkalis and the earth alkalis, and some metals. Only a small set of cross-section data is available for more than half of the atoms in the period table, and data for molecules are even more limited. Experimental studies of positron collision have been performed by a limited number of groups. Elastic DCS for electron collisions with argon are presented in Figure 1. Differences between the measured data from different sources [4-14] indicate the magnitude of the uncertainty of experimental data. The calculated DCS (by using the software attached to the Report) exhibit nearly the same minimum at nearly the same angles as the measured DCS. However, the static-exchange potential used in calculation (dashed curves) overestimates the DCS at intermediate and large angles (more at lower energies) because the model neglects the absorption, i.e., the loss of electrons in open inelastic channels. At small scattering angles, and especially for low-energy electrons and positrons, this model underestimates DCS, because it neglects atomic polarizability. Thus, the validity of the static-field approximation is limited to energies above about few keV. The elastic, total scattering and momentum transfer cross sections for electrons in xenon are shown in Figure 2., and compared with the total cross section for positrons. Total cross sections are approximately equal to elastic cross sections at energies below about 10 eV, where no inelastic collisions take place. The a, for positrons and electrons have different energy dependences below 100 eV, but at higher energies the two values approach each other. The crmt in xenon, as
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well as in argon and krypton, exhibit a pronounced feature below 1 eV, namely a minimum in amt known as the Ramsauer-Townsend (RT) minimum). Different sets
of crmt experimental data generally agree very well at all energies, except around the RT minimum.
C Ciiipta and Rocs (I 975) o--»Ar(/ IS) v Williams and Willis < I 975b) DuHoisand Rudd (1970) |M-'* /:" 300 eV •; • Janson el ul. (1976) • Vuskovk- and Kurcpa (1976) X Srivaslava ol al. (1981) CJ Wagenaar el :il. (I 986) O C'vcjanovic and Crowe ( H llroml-vri; (1 974) o-lft X Wi 11 iciin.s and Willis f 1 975 i ~- • Jansenet al.( 1976) :
Ai • • 1 • • 1 i i • • 1 • • i I • • c —> Ar(Z IS) •» Ar (Z IS) 1...-" /;••••••- 500 eV 800 cV Bromherg (1 974) + Urombery (1974). 700 cV 1 DuBois and Rudd (1976) A IXiRois and Rudd (1 976) \ I.,- * r • Jansenel al. ( 1976) : n ot al. (1976). 750 eV X Iga otal. (1987) I - : \ It' l7 ^—'"'ATE II-'" 1 i | t • 1 III" 81 VI. simpozij HDZZ, Stubičke Toplice, 2005. Xe total c.s. 100 Xe 100 elections • ref. [16] -*- ref. [18] E • ref. [19] 10 10 . ref. [20] o positrons 10 o ref. [19] : 10 D ref. [15] 0.1 • T • ref. [16] 0.1 x ref. [17] o ref. [18] ref. [21] 0.01 0.01 0.1 1 10 100 1000 10000 1E-3 0.01 0.1 1 10 100 1000 10000 E(eV) E(eV) Figure 2. Elastic, aei, and total scattering cross sections, at, (left), and momentum transfer cross section (right) in xenon. Differential cross sections (selected) for elastic electron scattering from water are presented in Figure 3. Excellent agreement between the experimental [22] and calculated [2,3] values is obtained for energies above 400 eV, while for lower energies and especially for small angles the differences are larger. Elastic cross sections for electrons and positrons in water (Figure 4) are different for energies below 1 keV, and for higher energies approach each other. Figure 4 also shows that the calculated aei agree well with various experimental data at energies above 100 eV, and therefore the program which will be distributed with the ESEP Report could be used for estimation of elastic cross sections for electrons and positrons in various molecules at energies above 100 eV. I 'I • E = 100eV 10,-15 o E = 200 eV A E = 300 eV v E = 400 eV • E = 500 eV < E = 700 eV * E = 1000eV CO O 10*1177 Q _ -Q - -O 10" 10" 0 20 40 60 80 100 120 140 160 180 angle (deg) Figure 3. DCS for electron elastic scattering from water. Symbols: experimental data [22], lines: calculations [3]. 82 VI. simpozij HDZZ, Stubičke Toplice, 2005. H2O 10 - 10 o : electron ^v?^"\; -.-ref. [22] ^-^ -A- ref. [28J ^ 1 - •>. ref. [24] - 1 -*— ref. [26] \*<: - -»- ref. [27] : o ref. [25] %\ — • calculated [3] \s positron calculated [3] I 0.1 r 0.1 10 100 1000 10000 E(eV) Figure 4. Elastic cross section for electrons and positrons in water. REFERENCES [I] International Commission on Radiation Units and Measurements (ICRU). Guidance on the Preparation of ICRU Reports by Report Committees, ICRU/96/38, 1996. [2] Salvat F. Phys Rev. A 2003 ;68: 012708. [3] Salvat F, Jablonski A, Powell C. Computer Phys Comm 2005;165:157-90. [4] Gupta SC, Rees JA. J Phys B: Atom Mol Phys 1975;8:1267-1274. [5] Williams JF, Willis BA.J Phys B: Atom Mol Phys 1975;8:1670-1682. [6] DuBois RD, Rudd ME. J Phys B: Atom Mol Phys 1975;8: 1474-1483. [7] Jansen RHJ, de Heer FJ, Luyken HJ, van Wingerden B, Blaauw HJ. J Phys B: Atom Mol Phys 1976;9: 185-212. [8] Vušković L, Kurepa MV. J Phys B: Atom Mol Phys 1976;9: 837-842. [9] Srivastava SK, Tanaka H, Chutjian A, Trajmar S. Phys Rev 1981;A 23: 2156-2166. [10] WagenaarRW, de Heer FJ.J Phys B: Atom Mol Phys 1980;13:3855-66. [II] Cvejanović D, Crowe A. J Phys B: Atom Mol Opt Phys 1997;30:2873-87. [12] Panajotović R, Filipović D, Marinković B, Pejčev V, Kurepa M, Vušković L. J Phys B: Atom Mol Opt Phys 1997;30: 5877-5894. [13] Bromberg JP. J Chem Phys 1974;61: 963-969. [14] Iga I, Mu-Tao L, Nogueira JC, Barbieri RS. J Phys B: Atom Mol Phys 1987;20: 1095-1104. [15] Gibson JC, Lun DR, Allen LJ, McEachran RP, Parceli LA, Buckman SJ. J Phys B: At Mol Opt Phys 1998;31:3949-3964. [16] Hayashi M. U: IAEA-TECDOC-506: Atomic and Molecular Data for Radiotherapy, IAEA, 1989, str. 193-199. [17] Ester T, Kesler J. J Phys B: Atom Mol Opt Phys 1994;27: 4295-4308. [18] Nishimura H, Matsuda T, Danjo A. J Phys Soc Jpn 1987;56: 70-78. 83 VI. simpozij HDZZ, Stubičke Toplice, 2005. [19] Dababneh MS, Hsieh Y-F, Kauppila WE, Pol V, Stein TS. Phys Rev A 1982; 26/3: 1252-1259. [20] Wagenaar RW, de Heer FJ. J Phys B: Atom Mol Phys 1980; 13:3855-66. [21] Elford MT, Buckman SJ. U: Landolt-Bornstein, Numerical Data and Functional Relationships in Science and Technology, New Series, Vol. 17, Springer 2000. str. 2-35 - 2-56. [22] Katase A, Ishibashi K, Matsumoto Y, Sakae T, Maezono S, Kurakami E, Watanabe K, Maki H. J Phys B: At Mol Phys 1986;19:2715-2734. [23] Saglam Z, Aktekin N. J Phys B: At Mol Phys 1990;23:1529-1536. and J. Phys. B: At. Mol. Opt. Phys. 1991;24:3491-3496. [24] Sueoka O, Mori S, Katayama Y. J Phys B: At Mol Phys 1986; 19: L373-8. [25] Cho H, Lee H, Park YS. Radiat Phys Chem 2003;68: 115-120. [26] Johnstone WM, Newell WR. J Phys B: At Mol Opt Phys 1991;24:3633-43. [27] Shyn TW, Cho SY. Phys Rev A 1987;36: 5138-5142. [28] Danjo A, Nishimura H. J Phys Soc Jpn 1985;54: 1224-1227. ABSTRACT Electrons and positrons interact with matter through several competing mechanisms, elastic and inelastic scattering being the most frequent. Elastic scattering is a non-radiative interaction between the projectile electron or positron and a target atom/molecule in which the internal energy of the target is not changed. Elastic scattering greatly affects the transport of fast electrons and positrons in matter. In elastic collisions, these particles may undergo large deflections and, as a result, the space distribution of dose from electrons and positrons depends strongly on the elastic scattering properties of the medium. Knowledge of elastic-scattering cross sections of electrons and positrons is needed in modelling the deposition of energy when beams of these particles or any form of ionising radiation interact with matter. Specific application fields range from radiation dosimetry, radiation therapy, radiation processing, radiation sensors, and radiation protection to atmospheric studies, plasma physics and material analysis (e.g., by electron-probe microanalysis, analytical electron microscopy, Auger- electron spectroscopy, and X-ray photoelectron spectroscopy). In 2000 The International Commission for Radiation Units and Measurements (ICRU) established a Report Committee to prepare the ICRU Report on "Elastic Scattering of Electrons and Positrons". The Report contains theoretical background of elastic scattering, description of measurement methods, and comparison of the experimental and theoretical cross sections (differential, integral elastic and momentum transfer. The CD ROM with the original software for calculating differential cross sections will be distributed with the Report. This paper presents several examples of comparison of the experimental and theoretical cross sections. A large part of available experimental data is obtained for atomic gases, particularly for a limited group of atoms such as hydrogen and rare gases. Experimental data for molecules are very limited, as well as for positron scattering. 84 DOZIMETRIJA ZRAČENJA I INSTRUMENTACIJA RADI A TION DOSIMETRY AND INSTR UMENTA TION