Ionization approximations, semiclassical

Due to the complexity of a full quantum mechanical treatment of electron impact ionization, or even a partial wave approximation, for all but relatively simple systems, a large number of semiempirical and semiclassical formulae have been developed. These often make basic assumptions which can limit their range of validity to fairly small classes of atomic or molecular systems. The more successful approaches apply to broad classes of systems and can be very useful for generating cross sections in the absence of good experimental results. The success of such calculations to reproduce experimentally determined cross sections can also give insight into the validity of the approximations and assumptions on which the methods are based. 

By careful inspection of the relationships of the semiclassical close collision approximations and Bethe formula, one can obtain simple and accurate information on ionization cross sections. By the method first proposed by Platzman, and used extensively by others, it is instructive to form the ratio of the differential cross sections [measured c(W,T) or calculated da(W,T)ldW)] to the Rutherford cross section. This ratio, called Y, is mathematically defined as... 

Also shown in Fig. 5 are results for the total ionization yield obtained by the semiclassical Franck-Condon approximation introduced in Eqs. (60) and (58). As discussed above, this theory is only valid for sufficiently short probe pulses. While this condition is well satisfied for probe pulse up to 20 fs duration [panels (a) and (b)], the approximation is seen to introduce spurious structures in the case of 32 fs pulses [panel (c)]. Since the Franck-Condon approximation reduces the cost of explicit pmnp>-probe simulations to the cost of a standard time-dependent wave-packet propagation, one obtains an overall computational speed-up of about two orders of magnitude compared to the full nonperturbative calculation. 

Conversely, for slow collisions the combined system of incoming electron and target molecule has to be considered, leading in the exit channel to a full three-body problem. Quantum-mechanical (approximate) calculations are difficult and have been carried out only for a few selected examples. Therefore, other methods have been developed with the goal of obtaining reasonably accurate cross sections using either classical or semiclassical theories and by devising semiempirical formulae. Some of these concepts are based on the Born-Bethe formula [22] and on the observation that the ejection of an atomic electron with quantum numbers (n,J) is approximately proportional to the mean-square radius of the electron shell (n,J). This leads also to proposed correlations of the ionization cross section with polarizability, dia-... 

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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