Exchange scattering amplitude

It is due to the exchange scattering amplitude Bn0 that the cross section of excitation of a molecule into a triplet excited state is nonzero (see Section IV.B.3). 

If the velocity of the incident electron is comparable with those of molecular electrons, the former can be exchanged for one of the latter. Such processes are described by the exchange scattering amplitude, the form of which in the first Born approximation has been found by Oppenheimer.126 In the Born-Oppenheimer approximation the exchange amplitude (4.11) acquires the form... 

A U(1,2)A with A the product of free positive -energy projection operators, as in (2.9). The operator U(l,2) = u + U must be chosen so that if the external potential is turned off, the one-photon and two-photon exchange scattering amplitudes are reproduced. (I note in passing that if one uses U = Uj-r rather that one must already include U to get the a Ry level shift correctly ) On comparing the eigenvalues of (4.3) with those obtained from,... 

Figure 9 This figure depicts two kinds of semiclassical paths which contribute to the exchange scattering amplitude for a triatomic system ABC. An equipotential energy surface for the ground adiabatic electronic state of H3 is represented. The OXYZ frame is the same as the OXKYZK frame of Fig. 5 with the subscript suppressed. The conical intersection configurations for H, (equilateral triangles) are represented by points on the OY axis. The dashed path partially encircles the conical intersection line, whereas the solid one does not. (From Ref. 36.)...

The scattering amplitude defined in (4.7) characterizes the so-called direct scattering. However, when the scattered electron is slow, there can also occur processes in which the molecule captures the incident electron and emits one of its own. This sort of scattering is described by the exchange amplitude Bn0(k , k0), the formula for which differs from that for the direct amplitude (4.7) in that in the final state of the system the coordinates of the incident electron are transposed with coordinates of molecular electrons, namely,... 

Of the 16 terms in (9.18) for f = jo = 1/2, 10 can be omitted because they violate conservation of total spin angular momentum on the basis of pure exchange scattering, leaving only the above six terms. Recalling the definitions in section 9.1.2 of the direct and exchange amplitudes / and g (equns. (9.4a) and (9.4b) respectively) one has... 

The data show that the differential cross section is dominated by the direct scattering amplitude / the modulus for the spin-flip amplitude g is in general an order of magnitude smaller. Thus the spin—orbit interaction has only a small influence on the cross section, which is mainly influenced by the Coulomb interaction, exchange, and charge-cloud polarisation. 

Here fp is the scattering amplitude for momentum exchange hk between... 

These equations reduce to the ones derived by van t Hof and Schmidt and also Harris when the scattering amplitudes in the ground and excited states are identical. We further note that in case of negligible exchange (r i2 34 r,2) for both transitions also Lorentzian lineshapes are predicted with widths determined by T,y. The temperature-dependent shift will now be induced by the acoustic phonons of the crystal. 

The pair scattering for two quasiparticles with opposite momenta at the Fermi surface is evaluated adopting the ladder approximation displayed in fig. 5. The problem is therefore reduced to finding the four-point vertex which is irreducible with respect to particle-particle scattering. The basic assumption is that the important structure in the scattering amplitudes comes from exchange of collective modes in the two particle-hole channels. The central quantity of these theories is the dynamic magnetic susceptibility x (fl ) which can be determined... 

With t = 0 the present expression reduces to the result obtained in Eq. (3.28). If, e.g., t = 2, then spectral exchange takes place both within the branches of an isotropic scattering spectrum (Fig. 6.1) and between them. The latter type of exchange is conditioned by collisional reorientation of the rotational plane, whose position is determined by angle a. As a result, the intensity of adsorbed or scattered light is redistributed between branches. In other words, exchange between the branches causes amplitude modulation of the individual spectral component, which accompanies the frequency modulation due to change of rotational velocity. 

In the simplest case, when the scattering system is a hydrogen atom, after integration over spin variables the exchange amplitude in the... 

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