Optical potential equivalent-local

As the positron approaches the target system it interacts with and distorts it, so that the total wave function no longer has the separable form of equation (3.3). Nevertheless, an equivalent Schrodinger equation can be derived for the positron, the solution to which is a function of the positron coordinate rq only, with the correct asymptotic form but at the cost of introducing a non-local optical potential. 

Fig. 8.8 shows that the coupled-channels-optical method with the equivalent-local polarisation potential (McCarthy and Shang, 1992) gives a good semiquantitative description of the experimental data of Williams (1976 ) for elastic differential cross sections below the n—2 threshold. At energies just below the n=3 threshold the resonances affect the n=2 excitations. Fig. 8.9 shows the energy dependence of the integrated cross sections for the 2s and 2p channels. Since a resonance is a property of the compound system, not the channel, the resonances observed in... 

The equivalent-local form of the coupled-channels-optical method does not give a satisfactory description of the excitation of triplet states (Brun-ger et al, 1990). Here only the exchange part of the polarisation potential contributes. The equivalent-local approximation to this is not sufficiently accurate. It is necessary to check the overall validity of the treatment of the complete target space by comparing calculated total cross sections with experiment. This is done in table 8.8. The experiments of Nickel et al. (1985) were done by a beam-transmission technique (section 2.1.3). The calculation overestimates total cross sections by about 20%, due to an overestimate of the total ionisation cross section. However, an error of this magnitude in the (second-order) polarisation potential does not invalidate the coupled-channels-optical calculation for low-lying discrete channels. 

In the absence of independent measurements of the total cross section the total ionisation cross section gives an estimate of the validity of the equivalent-local polarisation potential used for the coupled-channels-optical calculation of fig. 8.13. The calculated value at 40 eV is 5.2 nal, compared with 4.66+0.47 nal measured by Karstensen and Schneider (1975). 

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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