Encounter approximation, binary

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. 

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 theoretical descriptions of the ejected electron spectra for heavy ion impact are basically the same as that for electron impact discussed above, except that the theory is simplified for heavy ions because exchange forces are not an issue. One can write the equivalent of Eq. (17) for the binary encounter approximation to the single differential ionization cross sections for bare heavy ion impact [36] as... 

Figure 10 Angular distributions of electrons ejected from helium by 2-MeV protons. The experimental data are from Ref 54 theoretical results are the binary encounter approximation of Bonsen and Vriens [52] and the plan wave is the Born calculation of Madison [51].

Calculations by Gryzinski and Kowalski (1993) for inner shell ionization by positrons also confirmed the general trend. Theirs was essentially a classical formulation based upon the binary-encounter approximation and a so-called atomic free-fall model, the latter representing the internal structure of the atom. The model allowed for the change in kinetic energy experienced by the positrons and electrons during their interactions with the screened field of the nucleus. 

It is clear from (11.1,11.10,11.14) that the differential cross section in the weak-coupling binary-encounter approximation is proportional to the spectroscopic factor S/(a), defined by... 

The existence of a common momentum profile for the manifold a confirms the weak-coupling binary-encounter approximation. Within these approximations we must make further approximations to calculate differential cross sections. For the probe amplitude of (11.1) we may make, for example, the distorted-wave impulse approximation (11.3). This enables us to identify a normalised experimental orbital for the manifold. If normalised experimental orbitals are used to calculate the differential cross sections for two different manifolds within experimental error this confirms the whole approximation to this stage. An orbital approximation for the target structure (such as Hartree—Fock or Dirac—Fock) is confirmed if the experimental orbital energy agrees with the calculated orbital energy and if it correctly predicts differential cross sections. 

Fig. 11.4 illustrates the momentum profiles of the other ion states observed in a later experiment with better energy resolution than that of fig. 11.2. All these states have momentum profiles of essentially the same shape. They are thus identified as states of the same orbital manifold, for which the experiment obeys the criterion for the validity of the weak-coupling binary-encounter approximation. Details of electron momentum spectroscopy depend on the approximation adopted for the probe amplitude of (11.1). The 3s Hartree—Fock momentum profiles in the plane-wave impulse approximation identify the 3s manifold. However, the approximation underestimates the high-momentum profile. 

The 5s manifold shows great complexity. For the lowest state S23.4(5s) = 0.37. This value is considerably lower than many structure calculations predict, but the perturbation calculation of Kheifets and Amusia (1992) obtains 0.384. The orbital energy ess (11.18) is 27.6+0.3eV, which is to be compared to the Dirac—Fock value 27.49 eV. The Hartree—Fock value is 25.70 eV. The criterion for the strength of the perturbation, given by the ratio of the standard deviation to the mean of the 5s manifold is 0.18. The ratios S29.i(5s) S23.4(5s) and S23.4(5s) Z/S/(5s) are compared at different momenta in fig. 11.10. The condition for the validity of the weak-coupling binary-encounter approximation is completely satisfied within experimental error. 

We consider swift ( 10 keV) alpha particles impinging on Ne gas (e.g. at STP) as our test case. As the average distance between Ne atoms is large ( 60 a.u.) compared to the radius of interaction between the projectile and target in such a case, we may employ the binary encounter approximation without problem. This study, then, will consider the dynamics of the interaction of He2+ ions incident on Ne. 

Abstract The momentum representation of the electron wave functions is obtained for the nonrelativistic hydrogenic, the Hartree-Fock-Roothaan, the relativistic hy-drogenic, and the relativistic Hartree-Fock-Roothaan models by means of Fourier transformation. All the momentum wave functions are expressed in terms of Gauss-type hypergeometric functions. The electron momentum distributions are calculated by the use of these expressions, and the relativistic effect is demonstrated. The results are applied for calculations of inner-shell ionization cross sections by charged-particle impact in the binary-encounter approximation. The reiativistic effect and the wave-function effect on the ionization cross sections are discussed. 

The momentum wave functions thus obtained are used to calculate inner-shell ionization cross sections by charged-particle impact in the binary-encounter approximation (BEA) [5]. The wave-function effect and the electronic relativistic effect on the inner-shell ionization processes are studied. 

Inner shell ionization of electrons to the continuum in ion-atom collisions can occur by two different processes. For low Z (projectile) particles on high Z2 (target) atoms the only available process is Coulomb excitation which is variously treated by plane wave Born approximation (PWBA), the binary encounter approximation (BEA), and the semiclassical approximation (SCA). When Z becomes comparable to 7/1 and the ion velocity v is lower than the velocity of the bound electron in question, v, the electrons adjust adiabati-cally to the approach of the two nuclei and enter molecular orbitals (MO) which in the limit of fused nuclei approach the atomic orbitals of the united atom Z = Z + Z2. This stacking of electrons can lead to a promotion of an innershell electron to the continuum or to a vacant outer orbital by direct curve crossing, rotational coupling, or radial coupling between molecular levels when such channels are available. 

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