Thermal Neutron Scattering in Graphite

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Title: Thermal Neutron Scattering in Graphite
Author: Al-Qasir, Iyad Ibrahim
Advisors: Prof. Marco B. Nardelli, Committee Member
Prof. Bernard W. Wehring, Committee Member
Prof. K. L. Murty, Committee Member
Prof. Ayman I. Hawari, Committee Chair
Abstract: Generation IV Very High Temperature Reactor (VHTR) concepts, are graphite moderated and gas cooled thermal spectrum reactors. The characteristics of the low energy (E < 1 eV) neutron spectrum in these reactors will be dictated by the process of neutron slowing-down and thermalization in the graphite moderator. The ability to accurately predict this process in these reactors can have significant neutronic and safety implications. In reactor design calculations, the prediction of the thermal neutron environment in the reactor core is possible using scattering cross section libraries. Currently used libraries (ENDF⁄B-VII) are a product of the 1960s and remain based on many physical approximations. In addition, these libraries show noticeable discrepancies with experimental data. In this work, investigation of thermal neutron scattering in graphite as a function of temperature was performed. The fundamental input for the calculation of thermal neutron scattering cross sections, i.e., the phonon frequency distribution and/or the dispersion relations, was generated using a modern approach that is based on quantum mechanical electronic structure (ab initio) simulations combined with a lattice dynamics direct method supercell approach. The calculations were performed using the VASP and PHONON codes. The VASP calculations used the local density approximation, and the projector augmented-wave pseudopotential. A supercell of 144 atoms was used; and the integration over the Brillouin zone was confined to a 3x3x4 k-mesh generated by the Monkhorst-Pack scheme. A plane-wave basis set with an energy cutoff 500 eV was applied. The corresponding dispersion relations, heat capacity, and phonon frequency distribution show excellent agreement with experimental data. Despite the use of the above techniques to produce more accurate input data, the examination of the results indicated persistence of the inconsistencies between calculations and measurements at neutron energies below the Bragg cutoff (˜ 1.8 meV). Consequently, this motivated the examination of the principal assumption in thermal scattering cross section calculations for graphite, i.e., the incoherent approximation. For a strongly coherent scaterer like graphite, the coherent one-phonon scattering law and corresponding cross section were calculated exactly and without approximations. The required input to perform such calculation such as dispersion relations and polarization vectors were taken from the results of the graphite lattice dynamics calculations mentioned above. As a result, significant improvements were achieved especially in the scattering law characteristic behavior at small momentum and energy transfers, and excellent agreement was found between the calculated inelastic scattering cross sections and the experimental data of pyrolytic graphite. Furthermore, a consistent approach for defining the parabolic region in the phonon frequency distribution of graphite for use in calculations using the incoherent approximation was developed. This approach is based on the graphite mean squared displacement and the agreement of the one-phonon cross sections as generated using both the incoherent approximation and the self part of the coherent one-phonon cross section. In this case, the parabolic energy cutoff was found to be 5.60 meV (equivalent to 65 K). Finally, the effect of temperature (anharmoncity) on the phonon frequency distribution was addressed and investigated by estimating the effects of energy shift and broadening of the distribution as a function of temperature. It was found that in graphite at low energies an energy shift is expected towards higher values. This is due to negative Gruniesen parameters. The phonon frequency distribution was broadened using a Lorentzian distribution, where the broadening effect has a linear temperature dependence at high temperatures. Therefore, the broadening and shift operations are two competing processes t low energies, resulting in relative differences in the calculated cross sections of less than 10 % at all temperatures.
Date: 2008-04-27
Degree: PhD
Discipline: Nuclear Engineering

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