Use of rational approximations for phonon density of states in evaluation of neutron thermal scattering law

Abstract

We propose and explore a new analytical formulation for evaluation of Thermal Scattering Law (TSL) using continuous phonon spectra for neutron inelastic scattering. It involves approximating Phonon Density of States (PDOS) in terms of rational approximations in energy transfer. Debye Waller coefficient calculated with these forms are in reasonable agreement with LEAPR calculated values. TSL can be expanded as a power series, each term of which can be integrated analytically. The method leads to TSL being a series of exponential cosine integrals which again have closed rational forms in energy transfer. Their shapes are analogous to asymmetric resonance line shapes and show maxima for certain values of energy transfer which could correspond to phonon emission or absorption. The series is shown to be convergent and easy to compute as it does not involve any numerical integration. The approach is applied to Beryllium metal and Graphite which have vastly different PDOS shapes, and is shown to be capable of giving most of the trends shown by TSL and cross section results from NJOY nuclear code processing system, even with smaller number of power series terms. Since TSL can be calculated for both negative and positive energy transfer with equal ease, one can easily check for the validity of the approximation obeying Principle of Detailed Balance (PDB). The results overall have shown promise of our formulation in making a case for rapid calculations of TSL and also for using the rational forms in energy transfer for analysis of experimental/calculated PDOS data.