Elastic electron-atom scattering takes place if the final state of an atom after the interaction coincides with the initial one. Differential cross-sections for such a process can be calculated in various approximations. The data presented here were obtained using widely exploited formula proposed by Mott [1].
Mott's formula represents the relativistic differential cross-section as a sum of squared modules of the direct and spin-flip scattering amplitudes. The amplitudes can be calculated through the phase shifts of spherical waves, which are obtained by integration of equations for radial wave functions. The equations, in their turn, can be derived by applying the variation principle to the average energy of the whole system (an electron being scattered and an atom). The total relativistic Hamiltonian of the system is a sum of one-electron Hamiltonians of the bound and scattering electrons and the potentials of the electron-electron interactions.
The use of the variation principle leads to the
appearance of the so-called exchange terms and screened nuclear
potential in the equations for the radial wave functions [2,3].
The exchange terms are important only at low energies of
the incident electron when it can be close enough to the scattering atom
and the
overlapping of its wave function with the wave functions of the
shell-electrons is significant.
At high energies of scattering electrons this effect is negligible,
which simplifies considerably the equations for the wave functions.
The screened nuclear potential is related to the charge
density distribution in an atom.
An exact solution requires cumbersome computations.
Elastic scattering model under consideration includes
screening effects only, while the exchange process and
polarization of an atom are not taken into account.
In these computations the analytical approximation for the atomic electrostatic potential given by the Thomas-Fermi-Dirac model [4] and the numerical algorithm described in [5] are used. In the computation of the direct scattering amplitude, partial waves with phase shifts smaller than 10-4 are neglected. The phase shifts for the spin-flip amplitude are calculated until the difference between them and the direct phase shifts becomes less than 10-5.
The calculated data do not pretend to be on highly accurate over the whole range of electron energies. They are in good agreement with some experimental data for energies of the order of 100 eV and higher and describe qualitatively the behaviour of the measured differential cross sections at lower energies (>~ 10 eV). The data calculated using similar algorithms are in common apply in Monte-Carlo simulation of electron transport in solids. The basic assumption which is used in these simulations is that the condensed matter can be modelled as an ensemble of noninteracting atoms. Generally, this is a good assumption for the electron energies of the order of 10 eV and higher. At lower energies another approach should be exploited.