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The Electron-Impact Double Ionization of Atoms: an Insight Into the Four-Body Coulomb Scattering Dynamics

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The Electron-Impact Double Ionization of Atoms: an Insight Into the Four-Body Coulomb Scattering Dynamics Physics Reports 374 (2003) 91–164 www.elsevier.com/locate/physrep The electron-impact double ionization of atoms: an insight into the four-body Coulomb scattering dynamics J. Berakdara;∗, A. Lahmam-Bennanib, C. Dal Cappelloc aMax-Planck-Institut fur Mikrostrukturphysik, Weinberg 2, D-06120 Halle, Germany bLaboratoire des Collisions Atomiques et Moleculairesà (UMR 8625), Bat.ˆ 351, Universiteà de Paris-Sud XI, F-91405 Orsay Cedex, France cInstitut de Physique, Universiteà de Metz 1 Bd Arago, Technopoleˆ 2000, 57078 Metz, France Received 1 May 2002 editor: J. Eichler Abstract Over the past two decades impressive progress has beenmade inthe theoretical andthe experimentalstudy of the multiple excitation and of the complete fragmentation of four-body Coulomb systems. The double ionization of atoms by charged particle impact is employed routinely to prepare and to explore the Coulomb four-body excited states (the two ionized electrons and the scattered charged projectile moving in the ÿeld of the residual ion). The spectrum of this four-body system can be determined experimentally by resolving simultaneously the momentum vectors of all particles. Such a multi-coincidence measurement entails however low counting rates which makes the experimental realization a challenging task. This work gives a brief overview on recent achievements in multi-detection techniques and outlines the various methods to carry out the double ionization experiments induced by electron impact. The advantages and the limits of the various experimental approaches are pointed out. On the theoretical side, serious di9culties are encountered which are prototypical for the theoretical treatment of many-body correlated systems: (A) With increasing number of interacting particles (and hence of degrees of freedom) a direct numerical evaluation of the four-body Green’s function, which encompasses the entire spectrum of the system, becomes a challenge. (B) Due to the non-integrable character of interacting many particle systems, an analytical approach can only be approximate. Inthis report we discuss indetails the various methods that have beenput forward to deal with the four-body problem, including: perturbative many-body treatments (ÿrst and second order theories) and non-perturbative methods as well as pure numerical approaches. Due to the complicated structure of the four-particle continuum spectrum we present and discuss simple qualitative arguments to explain the main features (peaks and dips) that are observed in the experiments. The limitations of these simple methods are illustrated by contrasting the ∗ Corresponding author. Tel.: +49-345-5582666; fax: +49-345-5511223. E-mail address: [email protected] (J. Berakdar). 0370-1573/03/$ - see front matter c 2002 Elsevier Science B.V. All rights reserved. PII: S0370-1573(02)00515-X 92 J. Berakdar et al. / Physics Reports 374 (2003) 91–164 predictions with full numerical calculations and with experimental data. Future directions and possible appli- cations are also discussed. c 2002 Elsevier Science B.V. All rights reserved. PACS: 34.80.−i Contents 1. Introduction ........................................................................................ 93 2. Theoretical concepts ................................................................................. 94 2.1. Formal development ............................................................................ 94 2.1.1. Scattering theory for multi-particle excitations ............................................... 95 2.1.2. Transition probabilities and cross sections ................................................... 98 2.2. Multiple scattering expansion .................................................................... 98 2.2.1. Double ionization pathways ............................................................... 100 2.3. Double ionizationintheperturbative regime ....................................................... 104 2.3.1. Scaling properties of the cross sections ..................................................... 105 3. Calculational schemes................................................................................ 106 3.1. The many-body wave function approach to double ionization......................................... 107 3.1.1. Normalizationof the N-body wave functions ................................................ 113 3.2. First-order perturbationtreatments ................................................................ 114 3.2.1. The three-body Coulomb wave approach: The 3C method ..................................... 114 3.2.2. The convergent close coupling approach: The CCC method ................................... 114 3.3. Beyond the perturbative regime .................................................................. 115 3.3.1. The secondBornapproximation(2BA) ..................................................... 115 3.3.2. The four-body Coulomb wave function: The 6C approach ..................................... 116 3.3.3. Dynamical screening and eHective charges: The C4FS method ................................. 116 3.3.4. The many-body Green function theory: The GF method ...................................... 117 4. Experimental techniques .............................................................................. 120 4.1. Notations ...................................................................................... 120 4.2. Overview onmeasured double ionizationcrosssections ............................................. 121 4.3. Genesis of the (e,3e) experiments ................................................................ 122 4.4. Experimental techniques ......................................................................... 124 4.5. General description of experimental arrangements ................................................... 124 4.6. Comparison of merits and performances ........................................................... 127 4.7. Procedure of oH-line data analysis ................................................................ 129 4.8. Background coincidences subtraction and percentage statistical error................................... 129 4.9. EHective coincidence energy resolution ............................................................ 131 4.10. Absolute scale determination ..................................................................... 131 5. Comparative analysis of the experimental and the numerical results: Helium ................................ 132 5.1. Fully resolved cross sections for the electron-impact double ionization ................................ 132 5.1.1. Selectionrules for the (e,3e) inthe perturbative regime ....................................... 132 5.1.2. Numerical and experimental results ........................................................ 133 5.1.3. Beyond the ÿrst-Born approximation ....................................................... 140 6. Double ionization of many-electron atoms .............................................................. 142 6.1. The double ionization of noble gases ............................................................. 142 6.1.1. Kryptonmeasurements ................................................................... 145 6.1.2. Neonmeasurements ...................................................................... 146 6.1.3. Argonmeasurements ..................................................................... 146 J. Berakdar et al. / Physics Reports 374 (2003) 91–164 93 6.2. The double ionization of magnesium .............................................................. 148 7. Integral cross sections ............................................................................... 149 7.1. Energy partitioning ............................................................................. 149 7.2. Angular correlations in integral cross sections ...................................................... 150 7.3. Angularcorrelationstudies onHe ................................................................ 151 7.4. Asymmetry parameters for (e,3-1e) reactions ....................................................... 154 7.5. Double ionization at low energies ................................................................ 159 8. Conclusions and future directions ..................................................................... 160 Acknowledgements ..................................................................................... 160 References ............................................................................................ 161 1. Introduction In recent years signiÿcant advances have been achieved in controlling and investigating multiple, highly excited states of four-body Coulomb systems. These states are generated in most cases upon the double ionization of atoms following the impact of electrons [1–20], photons [21] or other charged-particles [22–33]. The experimental and the theoretical eHorts are focused on the study of the correlated dynamics of the two ionized electrons and the scattered projectile as they propagate in the ÿeld of the residual ion. For this purpose one measures the double ionization rate while resolving simultaneously the vector momenta of all particles in the continuum. Thus, a multi-coincidence detection has to be utilized which implies low counting rates (as compared to the single particle detection) and makes an experimental realization a challenging task. This obstacle has however been tackled by several research groups in Europe and in the US by developing and employing
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