Generation of an [formula omitted] Covariance Matrix by Monte Carlo Sampling of the Phonon Frequency Spectrum

Abstract

Formats and procedures are currently established for representing covariances in the ENDF library for many reaction types. However, no standard exists for thermal neutron inelastic scattering cross section covariance data. These cross sections depend on the material's dynamic structure factor, or S(α,β). The structure factor is a function of the phonon density of states (DOS). Published ENDF thermal neutron scattering libraries are commonly produced by modeling codes, such as NJOY/LEAPR, which utilize the DOS as the fundamental input and directly output the S(α,β) matrix. To calculate covariances for the computed S(α,β) data, information about uncertainties in the DOS is required. The DOS may be viewed as a probability distribution function of available atomic vibrational energy states in a solid. In this work, density functional theory and lattice dynamics in the harmonic approximation were used to simulate the structure of silicon dioxide (α-quartz) to produce the DOS. A range for the variation in the partial DOS for silicon in α-quartz was established based on limits of variation in the crystal lattice parameters. Uncertainty in an experimentally derived DOS may also be incorporated with the same methodology. A description of possible variation in the DOS allowed Monte Carlo generation of a set of perturbed DOS spectra which were sampled to produce the S(α,β) covariance matrix for scattering with silicon in α-quartz. With appropriate sensitivity matrices, it is shown that the S(α,β) covariance matrix can be propagated to generate covariance matrices for integrated cross sections, secondary energy distributions, and coupled energy-angle distributions.