Distorted-wave Born

Distorted-wave Born Qualitative agreement with quasiclassical ... 

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).

Fig. 8.12. Differential cross section for the 1 S, 2 S and 2 P states of helium and electron impact coherence parameters (8.40) for the 2 P state at Eo=50 eV. Experimental data for differential cross sections are 1 S, Register, Trajmar and Srivastava (1980) 2 S,2 P, Cartwright et al. (1992). Experimental data for electron impact coherence parameters are crosses, McAdams et al. (1980) squares, Beijers et al. (1987) plus signs, Eminyan et al. (1974). Solid curves, coupled channels optical (equivalent local) (McCarthy et al., 1991) broken curve, distorted-wave Born (Cartwright et al, 1992). From McCarthy et al (1991).

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... 

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... 

Following that we present calculations on resonances in a two-mathematical-dimensional (2ND) model for van der Waals systems. We will compare the complex eigenvalues obtained previonslyCl) by the complex coordinate method with those obtained from the distorted wave Born approximation (DWBA). Part of the motivation for making this comparison is to assess the accuracy of the DWBA before applying it to more realistic problems. 

There are many other approaches to obtain resonance energies and widths, many are reviewed in this volnme. One that we consider in the next two sections is the distorted wave Born approximation (DWBA). In the following section the DWBA is tested against accurate complex coordinate calcnlations reported previously for a collinear model van der Waals system(l). The DWBA is then used to obtain the resonance energies and widths for the HCO radical. A scattering path hamiltonian is developed for that system and a 2ND approximation to it is given for the J>0 state. 

Distorted wave Born and complex coordinate resonances for 4 model van der Waals system. 

Distorted wave Born approximation resonance energies and widths were calculated numerically using Equation 17. for the reduced-dimensionality hamiltonian given by Equation 35. and employing Equation 36. for V (r,t) up to second order. The coefficients a(t) and b(t) were determined numerically from the ab imitio potential surface. Zero-order wavefimctions X,(t) Xj(t) were deter-... 

Table II. Distorted wave Born approximation resonance energies and widths for two-mode HCO (in eV).

We have presented a sample of resonance phenomena and calculations in reactive and non-reactive three-body systems. In all cases a two-mathematical dimensional dynamical space was considered> leading to a great simplification in the computational effort. For the H-K 0 system, low-energy coupled-channel calculations are planned in the future to test the reliablity of the approximations used here, i.e., the scattering path hamiltonian as well as the distorted wave Born approximation. Hopefully these approximations will prove useful in larger systems where coupled-channel calculations would be prohibitively difficult to do. Such approximations will be necessary as resonance phenomena will continue to attract the attention of experimentalists and theorists for many years. 

Bandrauk (1969) used a distorted wave Born approximation to calculate the inelastic cross sections of atomic alkali halogen collisions. He found a forwardly peaked differential cross section without oscillations. His total cross section decreases with E 1/2 which is the high energy asymptotic behaviour of the LZ cross section (37). The magnitude of the cross section is much larger than the LZ-result, however. His results were not essentially different if instead of a Coulombic Hl2(R) he used a constant or screened Coulombic interaction. 

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