Thermal inelastic scattering of cold neutrons in polycrystalline solids

A general theory of the influence of thermal motion on the scattering of slow neutrons in polycrystals is discussed. Unlike Weinstock's earlier treatment of the same problem, we express temperature displacements of the lattice points as a sum of emission and absorption operators. This alternative approach simplifies the calculation of the transition probability to a considerable extent and provides a simple proof of the Debye-Waller factor. The theory is also generalized to the case of multi-phonon processes. General expressions for both the incoherent and coherent cross-sections, corresponding to an 'l' phonon process, are derived. The latter, hitherto not treated rigorously, is examined in detail. It is shown that it can be expressed as a sum of two terms, of which the main term, apart from a constant, is identical with the expression for the incoherent part and the other is a correction term. Both terms are put in 'Placzek' form, and for cold neutrons explicit expressions are obtained for the cases: (i) M $\gg $ 1 and T/$\Theta \geq $ 1, and (ii) M $\sim $ 10 and T/$\Theta \geq $ 0$\cdot $5. Numerical results for magnesium, aluminium, iron, lead and beryllium are discussed and compared with experiment. The agreement is found to be satisfactory.