Relation between the Vibration Frequencies of a Crystal and the Scattering of Slow Neutrons

Detailed calculations of the scattering of slow neutrons by crystals have so far been made on the basis of the Debye approximation for the vibration frequencies of the crystal. In the present work, a somewhat more realistic set of frequencies has been calculated for the case of aluminum. The results have been used to calculate the angular and energy distribution of neutrons coherently scattered in a one-phonon process. Comparison is made with the results given by the Debye approximation.The distribution function of the calculated frequencies has been used to obtain values for the Debye-Waller factor and for the total incoherent cross section for incident neutrons with long wavelengths.