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Description
Uranium Carbide (UC) is a nuclear fuel material, whose crystal binding may affect doppler broadening of epithermal neutron absorption resonances and scattering behavior of thermal neutrons. Currently, there is no Thermal Scattering Law (TSL) evaluation for UC in the ENDF/B-VIII.0 database. Ab-initio lattice dynamics (AILD) techniques are used to calculate the phonon spectrum using spin-orbit-coupling density functional theory (DFT). The present evaluation details TSL production for Uranium and Carbon in UC, and generates the thermal neutron scattering cross sections.
The thermal neutron de-Broglie wavelength is on the order of magnitude of inter-atomic spacing and its energy is on the order of magnitude of lattice vibrations. The thermal scattering interactions in a crystalline material are therefore a function of its structure and vibrational state availability. In inelastic scattering, energy can be exchanged between a neutron and the atomic lattice through the absorption and emission of elementary excitations (phonons), and elastic scattering occurs at Bragg edges. The incoherent inelastic cross section is therefore calculated with a phonon expansion in the harmonic approximation and the coherent elastic double differential cross section is calculated as a function of the atomic structure factor in the cubic approximation. Bose-Einstein statistics determine availability of lattice vibrations.
UC has a rock-salt crystal structure with Fm-3m symmetry (space group 225), and paramagnetic structure at room temperature. At low temperatures, both semi-itinerant bonding and AFM-I ordering of 5$f$ electrons exists. Capturing the magnetic orientation of adjacent Uranium atoms reduces unit cell structure to tetragonal P4mmm symmetry (space group 123). Electronic structure minimizations in the AFM ground state were conducted using VASP DFT code. These spin-orbit-coupling (SOC) calculations used PAW pseudo-potentials with the GGA-PBE-sol exchange-correlation functional edited for bulk materials, a plane wave-cutoff of 500 eV and 8x8x8 Monkhorst-Pack k-mesh. 1${}^{st}$ order Methfessel-Paxton smearing with a width 0.2 eV was used, and a moderate effective Hubbard value (U - J) of 1.9 eV controlled the degree of itineracy of the 5$f$ Uranium electrons. These parameters ensured that the ground state energy and volume were converged to below 5 meV/atom and 0.1%, respectively. These 2x2x2 cubic supercell calculations (4x4x4 k-mesh, 64 atom) displaced non-equivalent U and C atom sites 0.02 angstrom in $\mathrm{\pm}$ X-direction. The isotropic PM phase is approximated by averaging Hellman-Feynman forces for AFM magnetic orientation in planes perpendicular (001) and parallel (100) to the direction of atomic displacement with (2/3) and (1/3) weights, respectively.
Phonon dispersion relations and phonon DOS were generated in the PHONON code, as a function of the generated Hellman-Feynman Forces. The partial DOS for Carbon and Uranium were processed in $FLASSH$ to generate their TSL, incoherent inelastic, and coherent elastic cross section data. The generalized DOS is constructed by neutron-weighting partial DOS. These calculations assume isotropic compositions of U and C. The free atom cross section and mass are those which correspond to current ENDF evaluations. The total scattering cross section is composed of inelastic and elastic scattering from both U and C. The total cross section for UC is compares favorably to experimental transmission data after accounting for the absorption cross section of Carbon, U-235 and U-238.