Angular differential scattering momentum

In simplifying the derivation of the paramagnetic differential scattering cross section it is common to assume LS coupling [3, 48]. This assumption states that for individual electrons of spin s and orbital angular momentum 1 ... 

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). 

Consider an atom-diatom A + BCiE J ) collision at total energy E with AB(EAg,jAg) + C as one possible product. Here EBC,jgC are the internal energy and rotational angular momentum of the BC diatom and EAB,jAB are the analogous AB variables. The differential cross section for scattering into dEAB djAB (other cross sections have analogous formulations) is given by... 

The differential cross section for elastic scattering of a charged particle by a nucleus is given in terms of the asymptotic phase shifts di of the partial wave of angular momentum 1% by the formula (Schiff )... 

The decrease at larger angles of the differential cross-section for inelastic scattering can be understood qualitatively in terms of a constructive interference from the many high angular momentum components which are present in a collision of a fast nucleon with a nucleus. Even in a nucleus as light as A1 and at an energy as low as 100 Mev kR 10, and 10 phase shifts need to be considered. 

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