Evaluation of Thermal Neutron Scattering Law and Cross Sections for Calcium Hydride

. Presented here are the calculated thermal scattering law (TSL) and thermal neutron scattering cross sections for Calcium Hydride, hereafter referred to by its chemical symbol CaH 2 . The only other such data prior to this evaluation are thermal neutron scattering libraries in the JEFF database, which do not fully capture the scattering physics of the CaH 2 system. The data in this evaluation are calculated from first principles; Density Functional Theory (DFT) is used to calculate the phonon density of states (DOS), which is the primary input required to derive the TSL. The TSL and cross sections have been evaluated for the three non-equivalent atom cites in the CaH 2 : Ca, H 1 , and H 2 . Each evaluation has been submitted to the NNDC for consideration in the next ENDF/B database release.


Introduction
CaH2 is being considered for use as a moderator material in new reactor designs, including a small modular microreactor [1]. Metal hydrides have long been known to be effective moderators based on various properties; ZrH2 has been successfully used as such in TRIGA reactors for decades. Of crucial importance to the moderating behavior of any material, especially for use in thermal reactors, is the thermal neutron scattering mechanism. Unlike higher-energy cross sections which are determined empirically, thermal neutron scattering cross sections can be derived from first principles [2], [3]. This means that these cross sections can be evaluated for any material, with experimental data serving as validation. This paper describes the process of evaluating new thermal scattering data for CaH2.

Thermal Scattering Theory
The double-differential scattering cross section is related to the probability of a neutron scattering from incident energy into outgoing energy d ′ about ′, through solid angle dΩ about Ω. The derived form is shown in Eq. 1, where is the Boltzmann constant; is the absolute temperature (K); ℎ and are the bound coherent and incoherent neutron scattering cross sections (b), respectively; and ( , ) is the TSL.
The TSL is inherently a material property; it is the Fourier transform of the time-dependent pair correlation function in both space and time. It describes the probability distribution of the energy and momentum states of the material, and thus defines how a thermal neutron may interact with the scattering system. It is comprised of two components, as shown in Eq. 2; the distinct component ( ) contains interatomic interference effects while the self-component ( ) does not.
( , ) = ( , ) + ( , ) The incoherent approximation assumes that the contribution of is negligible to the total TSL ( = 0), and thus the TSL is entirely incoherent. For crystalline materials the harmonic approximation allows the TSL to be calculated via a phonon expansion, which requires knowledge of the phonon DOS. This DOS can be calculated via density functional theory (DFT), which is a method of calculating material properties using a quantum-mechanical model of electron density, in combination with lattice dynamics analysis.

Computational Method
CaH2 has an orthorhombic crystal structure with symmetry described by the Pnma space group (#62). Its unit cell contains three non-equivalent (not related by symmetry) atom sites, labelled in this work as Ca, H1, and H2; Figure 3 shows the CaH2 unit cell. The structural information was fed to the Vienna ab initio Simulation Package (VASP) [4], [5] which performs the DFT calculations, and the MedeA-VASP [6] platform was used to perform a structure optimization. The ab initio lattice dynamics (AILD) DFT calculation was initiated using the projector-augmented wave (PAW) method [7] and a GGE-PBE exchange-correlation functional. The structure optimization informed a plane-wave cut-off energy of 675eV and a 9x9x9 Monkhorst-Pack k-point mesh. VASP and PHONON [8], [9] are used to calculate the Hellmann-Feynman forces and subsequently the phonon dispersion curves and DOS using the dynamical matrix method.

Results
The results of the structure optimization are shown below in Table 1. The lattice constants show excellent agreement with experimental data [10]. This structural data was used to calculate the phonon DOS shown in Figure 2. The DOS has three sets of clustered peaks, corresponding to each of the non-equivalent atoms; the low-energy acoustic modes belong to the relatively high-mass Ca atom and the high-energy optical modes to the H atoms (the more tightly-bound H1 modes are the higher-energy of the two optical groups). These modes are compared to each other on a one-to-one basis, and to experimental data [11]. The experimental data is naturally weighted towards the energy states of the H atoms; when comparing to calculated data, it is important to compare both the energies and relative magnitude of each mode. The portion of the DOS which has the most significant impact on thermal neutron scattering is the beginning of each mode, which will always be occupied regardless of temperature. These crucial regions of the DOS match well the experimental curves, as do the peak energies and relative magnitudes.  The existing CaH2 thermal neutron scattering cross sections in the JEFF libraries [12] were limited by the available software at the time of their evaluation, which has led to nonphysical artifacts in the data. Table 2 shows the bound coherent and incoherent neutron scattering cross sections of Ca and H at room temperature, taken from the NIST database. These specific values are strictly for demonstrative purposes, providing insight into the physics of the CaH2 scattering system. It is clear based on these data that Ca is almost entirely a coherent scatterer while H is almost entirely an incoherent scatterer. Due to aforementioned limitations, the JEFF evaluation only includes the incoherent component of the Ca cross section and, for proper normalization of the total cross section, overestimates it by several orders of magnitude. This is highlighted in Figure 3 below, where the horizontal line is the NIST bound incoherent scattering cross section. An additional notable aspect of the JEFF evaluation is that the scattering behavior of the unique H atom sites was averaged, resulting in lost information.  The phonon DOS was passed to the FLASSH code [13], where a 500-order phonon expansion was used to calculate the TSL using the cubic approximation. Also calculated were the incoherent inelastic cross section for all three non-equivalent atoms, the coherent elastic (cubic approximation) for CaH2, and the incoherent elastic for H1 and H2. As the characteristic coherent cross section for H is non-zero, the interference effects from to the H1 and H2 atom sites cannot be ignored. Due to the limitations of ENDF formatting, the coherent elastic component included with the Ca data is that of the entire compound. Figure 4 is a comparison between the total Ca cross sections of this work and of the JEFF libraries (using the compound coherent elastic in the former). Figure 5 shows Figure 6 is an analysis of the calculated H1 cross section compared to Fermi's prediction for the interaction between a neutron and H in a hydrogenous substance [14]. There is shown to be great agreement with the predicted phenomena; as the mass of the binding atom increases, the data approaches Fermi's values for an effectively infinite binding mass. The shift in peak energies relative to Fermi's calculation indicates anharmonicity in the lattice vibrations of both of the compared hydrides. The ZrH2 data was generated for this comparison and closely matches both experimental [15] and recent AILD data [16]. The hydride data are normalized according to Fermi's work, such that the first local minima are aligned with Fermi's data, and the low-energy limit of the cross section is unity.

Conclusions
The thermal scattering law and cross sections of CaH2 were calculated using ab initio methods. The data were generated for each of the three non-equivalent sites in CaH2: Ca, H1, and H2. The cross sections for each are consistent with the physics of the scattering system -the contributions to Ca in CaH2 are incoherent inelastic and coherent elastic, while the contributions for H1 and H2 are incoherent inelastic and incoherent elastic. This is an improvement over the existing data in the JEFF-3.3 database, particularly for the Ca cross section. The TSLs and cross sections generated in this work have been submitted to the NNDC for consideration in the next ENDF database release, ENDF/B-VIII.1.   Energy (w / hn) Fermi [14] H1_CaH2 H_ZrH2   Fig 6. Comparison of the thermal neutron scattering cross section in a hydrogenous material as derived by Fermi [14], and the calculated cross sections of H1 in CaH2 and H in ZrH2 at 296K.