Re-evaluation of the TSL for Yttrium Hydride

Yttrium hydride (YHx) is of interest as a high-temperature moderator material in advanced nuclear reactor systems because of its superior ability to retain hydrogen at elevated temperatures. Thermal neutron scattering laws (TSL) for hydrogen bound in yttrium hydride (H-YH2) and yttrium bound in yttrium hydride (Y-YH2) were previously evaluated by Naval Nuclear Laboratory using the ab initio approach and released in ENDF/B-VIII.0. In that work, density functional theory, incorporating the generalized gradient approximation (GGA) for the exchange-correlation energy, was used to simulate the face-centered cubic structure of YH2 and calculate the interatomic Hellmann-Feynman forces for a 2×2×2 supercell containing 96 atoms. Lattice dynamics calculations using PHONON were used to determine the phonon density of states. The calculated phonon density of states for H and Y in YH2 were then used to prepare H-YH2 and Y-YH2 TSL evaluations, in the incoherent approximation, using the LEAPR module of NJOY. In addition elastic scattering was assumed to be incoherent for both H and Y. While the incoherent elastic scattering approximation is appropriate for H-YH2, it introduces an undesirable approximation for Y-YH2. In this work, we re-evaluate the TSL for Y-YH2 using FLASSH (Full Law Analysis Scattering System Hub). Y-YH2 is evaluated using the FLASSH generalized coherent elastic scattering capability in order to capture the Bragg peaks associated with the YH2 crystal structure which were neglected in the prior NJOY-based evaluation due to limitations in LEAPR. An experimental approach to validate the Y-YH2 TSL using neutron transmission measurements is discussed.


INTRODUCTION
Yttrium hydride (YHx) is of interest as a high-temperature moderator material in advanced nuclear reactor systems because of its superior ability to retain hydrogen at elevated temperatures. The microstructure of YHx at x ~ 1.6, the yttrium hydride composition range typically considered to be of practical value in nuclear reactor design, consists of a two-phase solid solution of α-Yttrium and YH2 phases [1]. The crystal binding effects of hydrogen in YH2 produce narrow, well-separated energy peaks in the neutron scattering cross section of YHx at thermal energies. This behavior is characteristic of many metal-hydrides and greatly influences neutron thermalization effects.
In neutron transport calculations, ENDF thermal scattering law sublibraries are used to capture crystal binding effects on low-energy neutron scattering. The TSL, ܵ(ߙ, ߚ), is a temperature-dependent probability distribution defining the possible scattering states for an incident neutron. It has contributions from distinct effects (d) due to scattered wave interference between different nuclear scattering sites and self (s) noninterference effects [2].
ܵ(ߙ, ߚ) = ܵ ୢ (ߙ, ߚ) + ܵ ୱ (ߙ, ߚ). (1) The momentum and energy transfer resulting from scattering are described by the unitless parameters α and β defined as, where E and E′ are the incident and scattered neutron energies, μ is the cosine of the scattering angle, A is the nuclide to neutron mass ratio, kB is the Boltzmann constant, and T is temperature. In crystals, phonon expansion is used to compute the TSL such that [2,3], which is a summation of increasing orders of the convolution of the vibrational (phonon) spectra, with ‫‬ = 0 representing elastic scattering and ‫‬ > 0 representing inelastic scattering. In current ENDF thermal scattering sublibraries, the incoherent approximation is applied such that distinct effects on inelastic scattering are neglected. Within this approximation, the double-differential scattering cross section is related to the TSL using Fermi's Golden rule as [2], where ߪ ୡ୭୦ and ߪ ୧୬ୡ are the bound coherent and incoherent nuclear scattering cross sections of the nuclide.
The ENDF/B-VIII.0 H-YH2 and Y-YH2 TSL evaluations [4] were generated in the incoherent approximation using the LEAPR module of NJOY based on vibrational spectra calculated using the ab initio approach [5]. Elastic scattering was assumed to be incoherent for both hydrogen and yttrium, such that ܵ ୢ (ߙ, ߚ) = 0. This approximation is reasonable for hydrogen since its nuclear scattering cross section is primarily incoherent. However, this approximation is not appropriate for yttrium since its nuclear scattering cross section is primarily coherent and it neglects the effects of crystal structure important for coherent elastic scattering from yttrium bound in YH2. When developing the ENDF/B-VIII.0 Y-YH2 evaluation, use of this approximation was driven by the inability of the LEAPR module of NJOY [6] to treat general crystal structures when calculating coherent elastic scattering cross sections. LEAPR is only able to accurately treat certain hard-coded crystal structures.
In this work, we re-evaluate the Y-YH2 TSL using the FLASSH (Full Law Analysis Scattering System Hub) code [7]. In particular, Y-YH2 is re-evaluated using the FLASSH generalized coherent elastic scattering capability [8] in order to capture the Bragg edges associated with the YH2 crystal structure which were neglected in the prior evaluation due to limitations in LEAPR.

CALCULATION OF PHONON DOS
YH2 has a CaF2-type face-centered cubic (FCC) structure. The unit cell is composed of 12 atoms with hydrogen atoms (rose) located in the tetrahedral holes between the Y atoms (blue) as illustrated in Fig. 1. The VASP code [9,10] was used to perform first-principles quantum mechanics simulations of the YH2 lattice structure using density functional theory (DFT). The YH2 lattice structure was optimized using a total electronic energy threshold of 10 -6 eV, a 0.11 Å -1 k-point spacing (19×19×19 k-point mesh), a 500 eV planewave cutoff energy, and the GGA-PBE [11] exchange-energy correlation functional. The VASP optimized 0 K lattice constant of a = 5.2032 Å is in excellent agreement with the experimental value of 5.2032 ± 0.0003 Å determined by Daou and Vajda [12] at 90 K using X-ray diffraction measurements. Lattice dynamics calculations were performed to determine the phonon density of states (DOS) for YH2 using the PHONON [13] code. Interatomic Hellman-Feynman forces on a 2×2×2 supercell (96 atoms) with ± 0.02 Å asymmetric atom displacements were determined from a series of VASP calculations using the GGA-PBE pseudopotential, a 500 eV planewave cutoff energy, and a 0.086 Å -1 k-point spacing (7×7×7 kpoint mesh) and a 0.11 Å -1 k-point spacing (19×19×19 k-point mesh) for the electronic energy and DOS contribution to thermodynamic function calculations. Fig. 2 provides a comparision of the calculated H and Y partial phonon DOS for YH2 [5] to the YH2 optical mode phonon DOS measured by Udovic [16]. The phonon DOS has two well-separated regions (due to the large mass ratio between Y and H): an acoustical region (0-0.028 eV) which is the preferred region for heavier Y atom vibrations and an optical region (0.112-0.142 eV) which is the preferred region for lighter H atom vibrations. The optical region is centered at ~0.127 eV which is in agreement with the 0.128 ± 0.004 eV estimate by Flotow [14] derived from heat capacity measurements and the 0.127 ± 0.007 eV value recommended by Rush [15] based on inelastic neutron scattering measurements. As shown in Fig. 2, the YH2 optical mode from high-resolution inelastic neutron scattering measurements by Udovic [16] has a similar bimodal structure centered at ~0.127 eV and is in excellent agreement with the theoretically calculated phonon DOS. Thermodynamic properties of YH2 can also be calculated from the phonon density of states. Fig. 3 shows the calculated heat capacity for YH2 is in agreement with the measurements by Flotow [14].

THERMAL NEUTRON SCATTERING LAW
The Y-YH2 TSL (MAT=55) was re-evaluated at 10 temperatures between 293.6 and 1600 K using the FLASSH code [7]. The generalized coherent elastic scattering module [8] was used to calculate the elastic scattering cross sections (MT=2) in the cubic approximation using the FCC crystal structure of YH2, the room temperature YH2 lattice parameter of a = 5.2085 Å measured by Daou and Vajda [12], and a 89 Y freeatom scattering cross section of 7.688246 b to ensure consistency with the ENDF/B-VIII.0 89 Y evaluation.

. Calculated Molar Heat Capacity for YH2
Use of the cubic approximation is appropriate since YH2 has a FCC crystal structure. The inelastic scattering cross sections (MT=4) were calculated in the incoherent approximation using the phonon DOS for Y in YH2 from Fig. 2 (this is the same phonon DOS used to prepare the ENDF/B-VIII.0 Y-YH2 evaluation) and a phonon expansion order of 300. Fig. 4 provides the coherent elastic scattering and inelastic scattering cross sections calculated by FLASSH. Bragg peaks characteristic of the FCC crystal structure are clearly evident in the elastic scattering cross sections and they behave as expected with increasing temperature. In contrast, the inelastic scattering cross section increases in magnitude with temperature and demonstrates the typical 1/v behavior at sub-thermal energies.
The ENDF-6 formatted thermal scattering sublibrary for the revised Y-YH2 TSL was then processed by the NDEX nuclear data processing code [17]. The total scattering cross sections for Y-YH2 as a function of

Validation
Validation of the Y-YH2 TSL will be challenging because thermal neutron scattering in YH2 is dominated by hydrogen scattering effects. Table I compares the bound coherent (ߪ ୡ୭୦ ), bound incoherent (ߪ ୧୬ୡ ), total bound scattering (ߪ ୠ ), and 2200 m/s absorption cross sections for 1 H and 89 Y from the NIST neutron scattering database [18]. Hydrogen provides about 96% and yttrium 4% of the contribution to the total bound scattering cross section for YH2.    (2) Prior neutron transmission measurement of YHx compositions for ‫0.2~ݔ‬ have concentrated on measurement of the H scattering since it dominates and have experimentally removed the contribution from Y [19,20]. These measurements also had insufficient accuracy to resolve the Bragg edges in YHx and could not measure neutron energies below the Bragg cutoff.
However, it should be possible to experimentally resolve coherent scattering effects using a series of neutron transmission measurements of YHx specimens with progressively increasing bulk hydrogen content between 0.25 < ‫ݔ‬ ≤ 2.0. Table II provides a comparison of the contributions to the total bound scattering cross section of YHx from Y and H scattering, normalized on a unit Y atom basis, for several YHx compositions. The Y contribution to the total bound scattering cross section appears to be sufficient to permit the detection of coherent scattering effects for compositions with ‫ݔ‬ ≤ 1.0. These experiments will need to be limited to x > 0.25 due to the usually high solubility limit of H in α-Y [21]. Nevertheless, this threshold is expected to result in a H contribution to the cross section that is sufficiently low to distinguish the Bragg effects. For 0.25 < x < 1.86, YH2 forms a two-phase solid solution with α-Y metal [21]. Consequently, a TSL for α-Y, which has a hexagonal close packed (HCP) crystal structure, will need to be A sub-thermal neutron production target is currently under development at the Rensselaer Polytechnic Institute (RPI) for use in low-energy neutron transmission measurements at the Gaerttner electron linear accelerator (LINAC). The sub-thermal neutron production target is being developed to perform neutron transmission measurements for moderator material and is expected to be able to support measurements with sufficient accuracy to detect Bragg peaks and the change in the total cross section below the Bragg cutoff for material with significant coherent scattering effects.

CONCLUSIONS
The TSL for Y-YH2 has been re-evaluated using the FLASSH generalized elastic scattering capability in order to account for yttrium coherent elastic scattering effects which were neglected in the prior ENDF/B-VIII.0 evaluation due to limitations in the LEAPR module of NJOY. The elastic scattering cross sections were calculated in the cubic approximation using the FCC crystal structure and room temperature lattice parameter of YH2. The resulting elastic scattering cross sections contain Bragg peaks consistent with the YH2 crystal structure. The inelastic scattering cross sections were calculated in the incoherent approximation using the same calculated Y phonon DOS used to prepare the ENDF/B-VIII.0 Y-YH2 evaluation and a phonon expansion order of 300. The ENDF-6 File 7 produced by FLASSH was processed by NDEX and the resulting total scattering cross sections for Y-YH2 exhibit the expected Bragg peaks and cutoff for YH2.
Since thermal neutron scattering in YH2 is dominated by scattering from hydrogen, experimental validation of the Y-YH2 TSL will be challenging. An experimental approach to validate the Y-YH2 TSL using neutron transmission measurement of YHx samples is discussed. The approach is enabled by the sub-thermal target currently under development at the RPI LINAC.