Covariance Matrix of a Double-Differential Doppler-Broadened Elastic Scattering Cross Section

Legendre moments of a double-differential Doppler-broadened elastic neutron scattering cross section on U are computed near the 6.67 eV resonance at temperature T = 10 K up to angular order 14. A covariance matrix of these Legendre moments is computed as a functional of the covariance matrix of the elastic scattering cross section. A variance of double-differential Doppler-broadened elastic scattering cross section is computed from the covariance of Legendre moments.


Introduction
A general expression for temperature-dependent Legendre moments of a double-differential Doppler-broadened elastic scattering cross section was derived by Ouisloumen and Sanchez in [1].However, its practical applications were limited to computations of the zeroth-order Legendre moment because higher order Legendre moments entail a timeconsuming computation of a triple-nested integral.A recursive algorithm that transforms this triple-nested integral into a single one via iterative application of the integrationby-parts method was designed and implemented in [2].This algorithm enables accurate computation of Legendre moments of an arbitrary order in a way that bypasses the tedious programming of their explicit analytical expressions.We use this algorithm to compute the first fifteen Legendre moments and their covariance matrix.
We comment on convergence of a Legendre expansion of Doppler-broadened double-differential elastic neutron scattering cross section near a low energy resonance of 238 U. A complementary stochastic treatment of thermal effects in [3] and [4] was used to validate a doubledifferential cross section computed via Legendre moment expansion.We also compute a covariance matrix of Legendre moments and use it to compute a variance of a doubledifferential Doppler-broadened elastic scattering cross section.In Section 2 essential formulae are listed, and in Section 3 numerical results are presented.a e-mail: arbanasg@ornl.govNotice: This manuscript has been authored by UT-Battelle, LLC, under contract DE-AC05-00OR22725 with the U.S. Department of Energy.The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes.

Formalism
A double-differential elastic scattering cross section in the laboratory frame at temperature T can be expanded in Legendre polynomials as where µ lab ≡ cos ∠(v, v ′ ), v and v ′ are the initial and the final velocity corresponding to the initial and the final energy E and E ′ in the laboratory frame, respectively, and P n (µ lab ) are the Legendre polynomials.It is noted that the angular distribution in the lab frame, Eq. ( 1), is anisotropic for n ≥ 1 even when it is isotropic in the center-of-mass frame.
Low-energy elastic neutron scattering in the center-ofmass frame can be approximated to a high accuracy by isotropic s-wave scattering.An expression for the n th -order Legendre moment (in the lab frame) of an isotropic angular distribution in the center-of-mass frame taken from [1] is: (2) where E and E ′ are the incident and outgoing neutron energies in the laboratory frame, T is the temperature in degrees Kelvin, k is the Boltzmann constant, β ≡ (A + 1)/A, A is the target mass in units of neutron mass m, E ′′ (t) ≡ βkT t 2 /A is an energy in the lab frame, and σ tab s (E lab ) is a tabulated elastic scattering cross section 1 at zero degrees Kelvin.The ψ n (t) is computed by Eqs.(13-26) of [2], where ψ n (t) is the ψ n0 (t) in a generalized notation introduced in [2].The integration variable t is dimensionless.
Legendre moments in Eq. ( 2) can be written compactly as functionals of σ tab s : 1 Usually given as a function of energy in the lab frame.
EPJ Web of Conferences DOI: 10.1051/ C Owned by the authors, published by EDP Sciences, 2012