Semi-classical cross sections

Differential cross section Deflection function. First we describe methods which take advantage of the close relationship between semi-classical cross sections and deflection function as outlined in Section III. A procedure which uses nearly all measurable quantities has been proposed and applied by Buck (1971). In order to unfold the multivalued character of 6(9), the deflection function is separated into monotonic functions g/[b) such that 0(6) = . gj(6)and6 = g (9). The g are represented by the usual functional approximations made in the semiclassical scattering theory ... 

One of the most important examples of this is the semi-classical approximation to quantum mechanics in 1959 Ford and Wheeler [65] described the explicit sequence of approximations by which the rigorous quantum cross section of (2.1)-(2.3) degenerates to the completely classical cross section,... 

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. 

Spectral lineshapes were first expressed in terms of autocorrelation functions by Foley39 and Anderson.40 Van Kranendonk gave an extensive review of this and attempted to compute the dipolar correlation function for vibration-rotation spectra in the semi-classical approximation.2 The general formalism in its present form is due to Kubo.11 Van Hove related the cross section for thermal neutron scattering to a density autocorrelation function.18 Singwi et al.41 have applied this kind of formalism to the shape of Mossbauer lines, and recently Gordon15 has rederived the formula for the infrared bandshapes and has constructed a physical model for rotational diffusion. There also exists an extensive literature in magnetic resonance where time-correlation functions have been used for more than two decades.8... 

Spin-orbit relaxation of Rb(52P) appears to be abnormally slow in all the inert gases except helium (cross-section 10 17 cm2). In fact, according to Pitre et a/.123, the relaxation rate in Kr and Xe is slower than for the equivalent transitions of atomic Cs, which correspond to a three-fold larger change in internal energy. Pitre et al. discuss complications in the rubidium experiments, including the formation of van der Waals complexes with the inert gases, in order to account for the apparently abnormal relaxation rates. Efficient removal of Rb(5 2P) by radiationless processes could upset the derived rate coefficients. The results were discussed in relation to Zener s semi-classical equivalent of equation (14). 

Abstract. Cross sections for electron transfer in collisions of atomic hydrogen with fully stripped carbon ions are studied for impact energies from 0.1 to 500 keV/u. A semi-classical close-coupling approach is used within the impact parameter approximation. To solve the time-dependent Schrodinger equation the electronic wave function is expanded on a two-center atomic state basis set. The projectile states are modified by translational factors to take into account the relative motion of the two centers. For the processes C6++H(1.s) —> C5+ (nlm) + H+, we present shell-selective electron transfer cross sections, based on computations performed with an expansion spanning all states ofC5+( =l-6) shells and the H(ls) state. 

The H3 potential surface which has been most widely used in transition state,2X4 classical dynamical,213 215 and quantum mechanical dynamical216-218 calculations has been the semi-empirical surface of Porter and Karplus.22 Because of the thin potential barrier of the PK surface, one would expect a larger amount of quantum mechanical tunnelling to be predicted at room temperature (this has been found to be the case in calculations performed by Johnston219 on barriers in the very similar LEPS surfaces). However, Karplus et a/.213-216 compared classical and quantum mechanical calculations on the PK surface and found that reaction cross-sections for both are very similar, and therefore that the tunnelling effect in the Ha system is small. 

Within the separable harmonic approximation, the < f i(t) > and < i i(t) > overlaps are dependent on the semi-classical force the molecule experiences along this vibrational normal mode coordinate in the excited electronic state, i.e. the slope of the excited electronic state potential energy surface along this vibrational normal mode coordinate. Thus, the resonance Raman and absorption cross-sections depend directly on the excited-state structural dynamics, but in different ways mathematically. It is this complementarity that allows us to extract the structural dynamics from a quantitative measure of the absorption spectrum and resonance Raman cross-sections. 

As long as the photodissociation reaction is fairly direct, the time-dependent formulation is fruitful and provides insight into both the process itself and the relationship of the final-state distributions to the absorption spectrum features. Moreover, solution of the time-dependent Schrodinger equation is feasible for these short-time evolutions, and total and partial cross sections may be calculated numerically.5 Finally, in those cases where the wavepacket remains well localized during the entire photodissociation process, a semi-classical gaussian wavepacket propagation will yield accurate results for the various physical quantities of interest.6... 

The theoretical description of the total cross section is easier in the semi classical limit since the angular range is indeed restricted to 9 = 0 so that the transition approximation gives results which are in quantitative agreement with those calculated by quantum mechanics. Using the parabola approximation for the phase shifts in the maximum or the straight line approximation for the deflection function at the zero point we have... 

Fig. 15. Polar differential cross section calculated semi-classically for the charge transfer process Na + I - Na+ + I, (a) Calculation with the complete interference structure with omission of the primary rainbow. (b) Approximate semi-classical calculation taking into account only interferences from net repulsive and net attractive scattering, (c) The full bars indicate maxima observed experimentally for net attractive scattering, the dashed bars for net repulsive scattering. H12(RC) = 0-065 eV angular coupling was neglected. (Delvigne and Los, 1973.)...

These facts indicate that the initial velocity of the target electron has a strong influence on the BEA cross sections and it is important to take into consideration the electron velocity (momentum) distribution in the BEA. The inner-shell ionization cross sections in the BEA with realistic velocity distribution are found to be in agreement with those in the PWBA and in the semi-classical approximation (SCA) [23],... 

The calculation of the Coulomb excitation can be treated by a semi-classical approximation. If v is the velocity, and Zg and Z are the charges on the target nucleus and the bombarding particle respectively, the quantity ri = 2Z Z e l%v is greater than unity, and the cross sections can be calculated from the classical trajectories of the bombarding particles. In general the total cross section for Coulomb excitation for multipole order Z is ... 

Raff, LM, Thompson DL. The classical approach to reactive scattering. In Baer M, Editor. Theory of chemical reaction dynamics, Vol. 3. Boca Raton CRC Press 1985. pp. 1-121. Truhlar, DG, Muckerman JT. Reactive scattering cross section. Ill Quasiclassical and semi-classical methods. In Bemtein RB, Editor. Atom-molecule collision theory. New York Plenum Press 1979, pp. 505 6. 

To evaluate a cross section we need to perform calculations of the reactive scattering S-matrix for many J values. In a semi-classical approach J is related to the orbital angular momentum of the relative motion of the collision partners. Approximately,... 

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