Thermodynamic Green's Function Methods in Neutron Scattering by Crystals

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Abstract
Formulas are derived for the transition probabilities per unit time for both inelastic coherent scattering of neutrons by crystals, and resonant emission of photons and neutrons by nuclei bound in crystals, without making the assumption that the crystal is harmonic. In deriving these transition probabilities, the analytic structure of thermodynamic correlation or Green's functions, considered as functions of complex temperatures and times, is developed and used. In particular a spectral form is found for the phonon Green's function. Only one assumption is made about the crystal, namely that the displacement of the nuclei due to the forces exerted by the neutron in scattering are linear functions of these forces. This leads to an evaluation of the transition probabilities in terms of the exact thermodynamic displacement autocorrelation function. This evaluation obeys the detailed balancing condition, and Placzek's sum rule. A consequence of this evaluation is that the widths of the "one-phonon" peaks in the neutron scattering are exactly equal to the widths of the corresponding phonon states of the crystal.