Coupled channel distorted wave approximation

Another approach for developing approximations to CC and CS reactive scattering calculations is to use distorted wave theory. In this approach, one considers that reaction is only a small perturbation on the nonreactive collision dynamics. As a result, the reactive scattering matrix can be approximated by the matrix element of a perturbative Hamiltonian operator using reagent and product nonreactive wavefunctions. Variations on this idea can be developed by using different approximations to the nonreactive wavefunctions. At the top of the hierachy of these methods is the coupled channel distorted wave (CCDW) method, followed by coupled states distorted wave (CSDW). 

Several decoupling approximations have been developed to simplify treatments where many rotational channels are coupled to begin with. A number of calculations have used orbital-rotational decoupling in the body-fixed frame, an il-dominant decoupling, and helicity decoupling. Several of these approaches have been recast in terms of effective Hamiltonians. Other decoupling treatments have extended the distorted-wave approximation by means of exponential operators or with optical potentials. 

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.

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... 

Fig. 10.15. Total ionisation cross section for helium. Experimental data, de Heer and Jansen (1977) full curve, coupled channels optical (equivalent local) (McCarthy and Stelbovics, 1983a) broken curve, distorted-wave Bom approximation.

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. 

Here all matrix elements in the two-level equations (section B2.2.8.4) are included, except the back coupling F yterm which provides the influence of the inelastic charmel on the elastic channel and is required to conserve probability. Distortion of the elastic and outgoing inelastic waves by the averaged (static) interactions V.. and respectively is therefore included. The two-state equations can then be decoupled and effectively reduced to one-channel problems. An analogous static-exchange distortion approximation, where exchange between the incident and one of the target particles also follows from the two-level treatment. 

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