Angular momentum projections cross-section

The differential cross section is defined for experiments that do not resolve angular-momentum projections or observe polarisations. States with different values of these observables are degenerate. We average over initial-state degeneracies and sum over final-state degeneracies. In the absence of details of the states this is denoted by Lav- The final form of the differential cross section is... 

The primary reason it is difficult to treat angular momentum rigorously is due to the angular momentum catastrophe [58]. As noted in Section III, cross sections and other experimental observables are sums over all relevant total angular momentum quantum numbers, J. Each J represents a quantum dynamics problem to be solved, and the size of the problem increases dramatically with J. For each J, there are Nk projections of K, where Nts = fmax — min + 1- For a fliree-atom system, the minimum value of K, is a function of both J and p, such that = 0 when J and p are... 

It is not difficult to surmise that the final expression for the fully resolved differential photodissociation cross section is extremely complicated (Balint-Kurti and Shapiro 1981). It contains vast quantities of sums and angular momentum coupling elements. Note that the cross section depends explicitly on the magnetic quantum numbers Mi and mj. Somewhat simpler cross section expressions can be derived by averaging over the initial projection quantum number Mi and summing over the final projection quantum number of the rotor, mj. As shown by Balint-Kurti and Shapiro, the angle-resolved cross section then takes on the general form... 

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). 

For collision processes that change the projection of the orbital angular momentum (5O sm), the cross-section can be written as [52]... 

Figure 5.6 The dependence of the partial cross sections cr ij=o( totai) on total angular momentum J, for various total energies. The linear increase for small J results from the 2J + 1 SF projection degeneracy. The exponential decrease for larger J results from the centrifugal force keeping the reactants well separated. The exponential decrease occurs at larger J values for larger initial translational energies, as demonstrated by the two marked total energies Etotai — 0.91 eV and 1.35 eV,...

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