Mott cross-section

In the expressions for the BED and BEB cross sections for each orbital the first log term represents large impact parameter collisions dominated by the dipole interaction. The remaining terms represent low-impact parameter collisions, described by the augmented Mott cross section. The second log function describes interference between direct and exchange scattering which is included in the Mott cross section. 

In Eq. (4.18), it is implicitly assumed that the ionization is a direct, one-electron process that is, the contribution of superexcited states to ionization is not included. The latter process is indirect and essentially of a two-electron nature. When the energy loss is much larger than the ionization potential, however, ionization is almost a certainty. For high energies of the secondary electron, Eq. (4.18) approaches the Rutherford cross section, or the Mott cross section if the incident particle is an electron. 

These classical formulas still do not account for the motion of the bound electrons in the atom or molecule. To be more appropriate to the interaction of an incident electron with the bound target electron, one must recognize that the velocity vector of the bound electron can be randomly oriented with respect to the incident electron providing a broadening of the energy of the secondary electron as calculated by the modified Mott cross section. If one integrates over the velocity distribution of the bound electron, the more familiar binary encounter approximation is derived that, in its simplest form, is given by Kim and Rudd [39] as... 

The binary-encounter-dipole (BED) model of Kim and Rudd [31] couples the modified form of Mott cross section [32] with the Bom-Bethe theory [27]. BED requires the differential continuum oscillator strength (DOS) which is rather difficult to obtain. The simplest approximate version of BED is the binary-encounter-Bethe (BEB) [31] model, which does not need the knowledge of DOS for calculating the EISICS. 

Elastic scattering of incident electrons results from the attractive Coulomb force of the nucleus screened by the atomic electron cloud with a differential Mott cross section of... 

The Mott cross-section is just the cross-section for the scattering of a spin 5 particle in the Coulomb field of a massive (spinless) target. The extra factors in (15.1.19) arise (i) because the target has spin 5 and there is a contribution due to the magnetic interaction between electron and muon, and (ii) because the target has finite mass and recoils. 

Darwin, 1929 Mott, 1930). The incident particle has momentum HKg before any interaction its momentum after exciting atoms 1 and 2 respectively into the nth and mth states is represented by hKnm. Mott showed that the entire process has negligible cross section unless the angular divergences are comparable to or less than (K a)-1, where a denotes the atomic size. As Darwin (1929) correctly conjectured, the wavefunction of the system before any interaction is the uncoupled product of the wavefunctions of the atom and of the incident particle. After the first interaction, these wavefunctions get inextricably mixed and each subsequent interaction makes it worse. Also, according to the Ehrenfest principle, the wavefunction of the incident particle is localized to atomic dimensions after the first interaction therefore, the subsequent process is adequately described in the particle picture. 

The quantum-mechanical ionization cross section is derived using one of several approximations—for example, the Born, Ochkur, two-state, or semi-classical approximations—and numerical computations (Mott and Massey, 1965). In some cases, a binary encounter approximation proves useful, which means that scattering between the incident particle and individual electrons is considered classically, followed by averaging over the quantum-mechanical velocity distribution of the electrons in the atom (Gryzinski, 1965a-c). However, Born s approximation is the most widely used one. This is discussed in the following paragraphs. 

The anomalous energy dependence of the elastic scattering cross section in the vicinity of EPs may be understood by reference to the properties of the S-matrix (see Mott and Massey, 1965, Chapter 13), in terms of which the elastic scattering and positronium formation cross sections in the Ith partial wave are expressed as... 

From the knowledge of the S matrix all information concerning elastic and inelastic cross sections can be obtained, via expressions for the scattering amplitudes, by applying the following formulae (Mott and Massey, 1965). 

The atomic displacement cross section o that determines the probability of vacancy formation in the CNS lattice can be calculated on the basis of the analytical approximation (McKinley and Freshbach 1948) of the known Mott formula for the Coulomb scattering of relativistic electrons by the atomic nucleus with the charge Z (Mott 1929). Taking into account that 0=(jr - x)/2, this cross section in the laboratory frame of reference is written as follows ... 

To demonstrate the calculation of the collision quenching (or transfer) cross section, let us consider the case of singlet-singlet quenching given by Eq. (70). Making use of the Bethe integral (Mott and Massey, 1965),... 

The elastic scattering of electrons and positrons in the screened Coulomb field of a target atom has been treated by Mott [8.43, 8.44]. He found that the elastic scattering cross section for positrons is smaller than that for electrons, mainly because of the influence of electron exchange correlation in the latter case. If we compare with the situation for equivelodty heavy ions, there is a considerable angular spread and even backscattering for the particles of electron mass - and more for electrons than for positrons. This results in larger penetration depths for positrons than for electrons (see, e.g., Valkealahti and Nieminen [8.45]). 

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