Differential cross section potential scattering

It is generally accepted that the centrifugal sudden (CS) approximation is the most reliable approximate method. Its results are usually very close to those obtained by ab initio close coupling (CC) calculations. The integral and differential cross-sections of Ar inelastic scattering on nitrogen were performed for a few low-frequency rotational transitions and four different interaction potentials [205]. Much better agreement of CC with CS results was found than with IOS calculations performed in... 

Connor J. N. L., Sun H., Hutson J. M. Exact and approximate calculations for the effect of potential anisotropy on integral and differential cross-sections Ar-N2 rotationally inelastic scattering, J. 

The solution to this problem is to transform, or half-transform, the S matrix from the body-fixed to the space-fixed axis system then to use the known analytic properties of the spherical Bessel functions, which are the solutions to the potential-free scattering problem in the space-fixed axes and finally to transform back to the body-fixed axes and then to use Eq. (4.46) to calculate the differential cross section. 

Equation (4.162) displays clearly how the cross-section is determined from the scattering dynamics in the radial coordinate via the time evolution of the initial state and a subsequent projection onto the final state. The angular momentum L = /l(l + l)K lh is according to Eq. (4.30) in the classical description related to the impact parameter, i.e., L = fivob. Thus, the sum can be interpreted as the contribution of all impact parameters. In the classical description only one impact parameter contributed to the differential cross-section. For a hard-sphere potential, it can be shown that da/dQ = d at low energies, which is four times the classical result in Eq. (4.44). 

The main difference between the new potential and that of Buck et al., i.e. the depth of potential wells, cannot be verified by this high energy scattering calculation in order to provide major insights, work is in progress to compute the MO-VB PES at a higher level of theory and to calculate differential cross-sections at the lower collision energy of 275 cm 1 [69], where total inelastic and some state resolved differential cross sections are available [65]. 

The calculation of Tjo for potential scattering has been discussed in great detail in section 4.5. This is a particular case of the present discussion which is useful for illustration. The differential cross section in this case is given by (4.48,4.112b). 

Fig. 8.2. Differential cross section for the elastic scattering of electrons on hydrogen. Circles, Williams (1975) solid curve, coupled-channels-optical calculation long-dashed curve, one channel with discrete polarisation potential only short-dashed curve, one channel without polarisation potential. Adapted from Bray et al. (1991h).

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