Distorted wave Born approximation

Fig. 7.2. Distorted-wave approximations to a two-channel (CC) calculation of electron—sodium scattering. UDWB, unitarised distorted-wave Bom DWSB, distorted-wave second Born DWBA, distorted-wave Born (Bray et al., 1989).

A number of improvements to the Bom approximation are possible, including higher order Born approximations (obtained by inserting lower order approximations to i jJ into equation (A3.11.40). then the result into (A3.11.41) and (A3.11.42)), and the distorted wave Bom approximation (obtained by replacing the free particle approximation for the solution to a Sclirodinger equation that includes part of the interaction potential). For chemical physics... 

Dewangan, D.P. and Walters, H.R.J. (1977). The elastic scattering of electrons and positrons by helium and neon the distorted-wave second Born approximation. J. Phys. B At. Mol. Phys. 10 637-661. 

In applying the distorted-wave second Born approximation we have the same difficulty as in calculating the optical potential. We must calculate the spectrum of the Green s function of (6.87). The first iteration of (6.87) is written as... 

The calculation describes the differential cross sections and 2 P electron impact coherence parameters quite well. For the 2 P differential cross section it is contrasted with a variant of the distorted-wave Born approximation, first-order many-body theory, where the distorted waves are both calculated in the initial-state Hartree—Fock potential. 

The T-matrix element in the distorted-wave Born approximation is... 

The factorisation is characteristic also of the plane-wave Born approximation, which is (10.30) with distorted waves replaced by plane waves. Here the two-electron T-matrix element is replaced by the two-electron potential matrix element (3.41). 

Fig. 10.4. Factorisation test of the distorted-wave Born approximation for copla-nar asymmetric ionisation from the 2p orbital of neon at = 400 eV, j = 50 eV (Madison, McCarthy and Zhang, 1989). The fast and slow electrons are respectively indicated by the subscripts / and s on the diagram The Bethe ridge condition is df = 20°. Full curve, unfactorised broken curve, factorised.

The validity of the impulse approximation can be tested by factorising the distorted-wave Born approximation in the same way. The differential cross section in the factorised distorted-wave Born approximation, obtained by replacing the two-electron T-matrix element in (10.42) by the potential matrix element (10.36), is compared with that of the full distorted-wave Born approximation in fig. 10.4 for the 2p orbital of neon in coplanar-asymmetric kinematics for =400 eV, s=50 eV. In this case the Bethe-ridge condition is Of = 20°, and p is less than 2 a.u. for 6s between 0° and 120° with this value of 6f. The impulse approximation is verified in Bethe-ridge kinematics for p less than 2 a.u. 

Fig. 10.6. Relative differential cross section for the ionisation of helium (Pan and Starace, 1991). Ef = Eg = 2 dV, (f> = 0, the polar angle is Of = n+Og. Open circles, Schlemmer et al. (1989) solid triangles, Selles, Huetz and Mazeau (1987) full curve, distorted-wave calculation including the screening effect of the final-state electron-electron interaction broken curve, distorted-wave Born approximation.

Fig. 10.7 compares the calculations of Brauner et a/, with the distorted-wave Born approximation and the approximation to (10.13) of Curran and Walters (1987) for a coplanar-asymmetric experiment on hydrogen at Eo = 150 eV. No calculation yields fully-quantitative agreement with experiment in the peak for small values of p, but all describe the relative shape. The cross section that is observed at much larger p is not well described by the distorted-wave Born approximation, but the other two calculations predict the trends better. 

Fig. 10.8. Coplanar asymmetric ionisation from the 2p orbital of argon (Zhang et al, 1992). Eq = 2549 eV, Ef = 1500 eV, Es = 800 eV, Of = 33.8°. Experimental data, Bickert et al (1991) full curve, distorted-wave Born approximation.

The direct amplitudes involving are analogous to the distorted-wave Born approximation and are calculated by (10.31). The T-matrix element in the second amplitude of (10.51), which has the observed resonances, is calculated by solving the problem of electron scattering on He" ". The solution consists of half-on-shell T-matrix elements at the quadrature points for the scattering integral equations (6.87). The same points are used for the k integration of (10.51). 

The case of helium gives a good test of theoretical methods, since there is only one target orbital in the Hartree—Fock approximation. Information is not further lost by a sum over orbitals. There have been several experiments on helium in different kinematic ranges. The distorted-wave Born approximation (McCarthy and Zhang, 1989) gives a good account of them. 

Fig. 10.12. Primary-electron double differential cross section for electron-helium ionisation. Experimental data are due to Muller-Fiedler et al. (1986) (open circles) and Avaldi et al. (1987a) (full circles). Full curves, distorted-wave Born approximation (McCarthy and Zhang, 1989). Cases illustrated are (a) Eq = 100 eV, Ef = 73.4 eV(A), 71.4 eV(B), 55.4 eV(C) (b) Eq = 300 eV, Ef = 235.4 eV (cross section multiplied by 100) (A), 271.4 eV(B) (c) Eq = 500 eV, Ef = 471.4 eV(A), 435.4 eV(B). From McCarthy and Zhang (1989).

Fig. 10.14. Total ionisation cross section for hydrogen. Experimental data, Shah et al. (1987) full curve, convergent close coupling (Bray and Stelbovics, 1992fc) plus signs, coupled channels optical (Bray et al., 1991c), crosses, pseudostate method (Callaway and Oza, 1979) long-dashed curve, intermediate-energy R-matrix (Scholz et al., 1990) short-dashed curve, distorted-wave Born approximation.

The distorted-wave Born approximation for ionisation considerably overestimates the total ionisation cross section for hydrogen below about 150 eV. This is a good indication of its lower limit of validity. 

It is useful to test approximations for the total ionisation cross section of helium, since it is a common target for the scattering and ionisation reactions treated in chapters 8, 10 and 11. Fig. 10.15 compares the data reported as the experimental average by de Heer and Jansen (1977) with the distorted-wave Born approximation and the coupled-channels-optical calculation using the equivalent-local polarisation potential. Cross sections... 

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