Quantum-mechanical treatment the second Born approximation

2 Quantum-Mechanical Treatment The Second Born Approximation 

In the vicinity of the atomic absorption edges, the participation of free and bound excited states in the scattering process can no longer be ignored. The first term in the interaction Hamiltonian of Eq. (1.11) leads, in second-order perturbation theory, to a resonance scattering contribution (in units of classical electron scattering) equal to (Gerward et al. 1979, Blume 1994)4 

The two terms in Eq. (1.34a) include processes in which the initial photon k0 has been annihilated first, and those in which the final photon k has first been created. In the quantum-mechanical description of the first type of process, the photon k0 is absorbed and then, in a very small time interval, the photon k is emitted through a stimulated emission process. This process and three-beam multiple scattering are illustrated by the Feynman diagrams in Fig. 1.6. 

The first term in Eq. (1.34a) represents the resonance scattering. It becomes large when photon (= Hu ) % E — 0. In comparison, the second term is small and is usually neglected. The imaginary term in the denominator contains T, the inverse lifetime (related to linewidth) of the intermediate state t/t . 

For a system of independently scattering electrons, the resonance scattering amplitude, ignoring the second term, becomes 

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