Triply-differential cross sections for inner-shell ionization in electron-atom collisions

Triply-differential cross sections for K-shell ionization by fast electron impact are calculated within the first-order Coulomb Born approximation for the case of a coplanar symmetric geometry. Comparison is made with experimental data on 300 keV and 500 keV eC+Cu, Agand Au. Far the two lightertargetr, the binarypeakregion is reasonably well described by a theory which uses semirelativistic electronic eigenfunctions to the target field, provided spin-flip is included. For the gold target, the Coulomb Born approximation seriously overestimates the data, pointing to the necessity of a fully relativistic description of the electronic states. Triply-differential cross sections for inner-shell ionization in electron-atom collisions provide a sensitive test for theoretical models since a coincident detection of the two momentum-analysed outgoing electrons allows for a complete determination of the collision kinematics (McCarthy and Weigold 1976, Ehrhardt ef a/ 1986, LahmamBennani 1991). For fast collisions where a first-order treatment of the electron-electron interaction should be appropriate and polarization effects may be neglected, the basic information to be extracted from (e, 2e) cross sections concerns thus the particulars of the electronic wavefunctions. For low-energy electron scattering it has become standard to use a Hartree-Focktype function for the bound target electron, and numerically generated scattering states multiple partial wave expansion required for the evaluation of the cross section becomes, however, prohibitive at high projectile energies, and such elaborate wavefunctions have up to now only been employed for energies of 150 keV and below (F'indzola and Buie 1988). The (e, 2e) experiments have recently been extended to the relativistic regime with impact energies up to 500 keV (Schiile and Nakel 1982, Ruoff and Nakel 1987, Bonfen et a/ 1991, Walters er a/ 1991). Apart from the distorted-wave Born approximation of Pindzola and Buie (1988), the theoretical approaches for relativistic electron impact ionization are restricted to the plane-wave Born approximation (PWBA; Moller 1932), where the primary electron is described by Dirac plane waves, while the target field is accounted for in the states of the secondary electron (Das 1972, Davidovic et a1 ,710,. JUFn a S"F p'Gsc"pL1u" . * U L & D "C11 L U L 1561. ,(1'&GL L l C i l Y J (111" 1u1 a J y r r L r n ~ r r l G energy sharing between the two outgoing electrons (the secondary electron being much slower than the primary one). However, it has been shown that the plane-wave Born approximation overestimates experiments with a symmetric energy sharing, the more SO, the heavier the target nucleus (Bonfert et a/ 1991). Therefore we have developed a theory, termed Coulomb Born approximation, which comprises the advantages of the PWBA and the distorted-wave theory: The target potential is considered in the states 0953-4075/92/061297+09$04.50