† Corresponding author. E-mail:

‡ Corresponding author. E-mail:

Project supported by the Long-Term Subsidy Mechanism from the Ministry of Finance and the Ministry of Education of China.

The high pressure behaviors of Th_{4}H_{15} and ThH_{2} are investigated by using the first-principles calculations based on the density functional theory (DFT). From the energy–volume relations, the bct phase of ThH_{2} is more stable than the fcc phase at ambient conditions. At high pressure, the bct ThH_{2} and bcc Th_{4}H_{15} phases are more brittle than they are at ambient pressure from the calculated elastic constants and the Poisson ratio. The thermodynamic stability of the bct phase ThH_{2} is determined from the calculated phonon dispersion. In the pressure domain of interest, the phonon dispersions of bcc Th_{4}H_{15} and bct ThH_{2} are positive, indicating the dynamical stability of these two phases, while the fcc ThH_{2} is unstable. The thermodynamic properties including the lattice vibration energy, entropy, and specific heat are predicted for these stable phases. The vibrational free energy decreases with the increase of the temperature, and the entropy and the heat capacity are proportional to the temperature and inversely proportional to the pressure. As the pressure increases, the resistance to the external pressure is strengthened for Th_{4}H_{15} and ThH_{2}.

Thorium and its hydrides are promising materials for advanced nuclear reactors with a high level of safety^{[1–8]} and have attracted much attention. This is mainly due to their strength and stability at elevated temperatures while maintaining the desirable properties for neutron moderation.^{[9]} In general, thorium hydrides crystalize in ThH_{2} and Th_{4}H_{15} phases, which are both metallic solids.^{[2]} Under ambient conditions, the stable thorium dihydride crystallizes in a body-centered tetragonal (bct) ionic structure with space group *I*4/*mmm* (No. 139). There also exists a face-centered cubic (fcc) fluorite-type structure with space group *Fm*-3*m* (No. 225), which is considered as a metastable phase of ThH_{2}. As for the Th_{4}H_{15} phase, it crystallizes in a body-centered cubic (bcc) structure with space group *I*-43*d* (No. 220), which has been reported to have superconductivity with a transition temperature of 8 K.^{[10]} In addition, due to the large hydrogen-to-metal ratio, Th_{4}H_{15} has been considered as a promising candidate for hydrogen storage.^{[11]}

The structural and electronic properties and the optical phonon density of states of thorium hydrides at the ground state have been widely investigated in previous studies.^{[12–19]} In order to support the experimental measurements or predict the microscopic physical properties that cannot be directly measured in the experiments, many theoretical studies were also performed for thorium hydrides. The optical phonon density of states of ThH_{2} was measured through inelastic neutron scattering,^{[14]} and the bulk modulus of ThH_{2} was calculated by the linear muffin-tin orbital (LMTO) method.^{[15]} The electronic properties of thorium hydrides were investigated by Shein *et al.*^{[16]} through the full-potential LAPW (FLAPW) method. Moreover, the structural, elastic, and electronic properties of thorium hydrides at the ground state were studied by the first-principles calculations.^{[17]} Despite abundant researches on thorium hydrides, relatively little is known regarding Th_{4}H_{15}. Until now, the elastic properties, phonon spectra, and thermodynamic properties are still unknown for Th_{4}H_{15}. Besides, there is still a lack of data about the elastic and thermal properties of thorium hydrides under high pressures, where a variety of interesting changes could show up. Furthermore, the stability of such competitive phases of thorium hydrides is also of interest for testing their practical applications.

In this paper, we perform a first-principles study on the elastic and thermodynamic properties of thorium hydrides at high pressures and make a systematic comparison with the corresponding behaviors at the ground state. The calculation method for the elastic constants and the computational details of first-principles are briefly introduced in Section 2. The calculation results are presented and discussed in Section 3. Finally, the conclusion is given in Section 4.

The first-principles calculations were carried out by using the density-functional theory (DFT) as implemented in Vienna *ab initio* simulations package (VASP)^{[20]} with the projector-augmented-wave (PAW) potential methods.^{[21]} The electron exchange and correlation potential was evaluated by the generalized gradient approximation (GGA) introduced by Perdew, Burke, and Ernzerhof (PBE).^{[22]} To ensure the calculation accuracy, the cutoff energy was set to 450 eV, and the self-consistent energy convergence of the calculation was set to 10^{−5} eV. The Brillouin zone (BZ) integration was performed by using 12 × 12 × 12*k*-points for ThH_{2} and 9 × 9 × 9 *k*-points for Th_{4}H_{15} with the Monkhorst–Pack scheme.^{[23]} The atomic structures were fully relaxed until the Hellmann–Feynman forces on all atoms were less than 1.0 × 10^{−5} eV/atom.

The elastic constants were calculated by means of a Taylor expansion of the total energy *E*(*V*,*δ*) based on the following law:^{[24–26]}

*E*(

*V*

_{0},0) and

*V*

_{0}are the total energy and the volume of the equilibrium cell with no strain, respectively,

*τ*

_{i}is an element in the stress tensor,

*ξ*

_{i}is a Voigt factor,

^{[10]}and

*C*

_{ij}is the elastic constant. For cubic structures, fcc ThH

_{2}and bcc Th

_{4}H

_{15}, there are three independent elastic constants (

*C*

_{11},

*C*

_{12}, and

*C*

_{44}). Thus, the elastic constants can be calculated with three different strains. The bulk modulus

*B*and shear modulus

*G*were obtained by the Voigt–Reuss–Hill (VRH) approximation,

^{[27–29]}i.e.,

_{2}, the six independent elastic constants,

*C*

_{11},

*C*

_{12},

*C*

_{44},

*C*

_{13},

*C*

_{33}, and

*C*

_{66}, can be obtained with six different strains. The Young modulus

*E*and Poisson’s ratio

*ν*were obtained by the following formulas:

*E*= 9

*BG*/(3

*B*+

*G*) and

*ν*= (3

*B*− 2

*G*)/[2(3

*B*+ 2

*G*)]. The supercell approach and the small displacement method were used to calculate the phonon dispersions in the BZ and the corresponding phonon density of states (DOS) with the help of the PHONOPY code.

^{[30]}For the phonon calculations, the 2 × 2 × 2 supercells for bct and fcc phases containing 48 and 96 atoms were constructed, respectively. As for the bcc Th

_{4}H

_{15}, the 2 × 2 × 1 supercell containing 152 atoms was used. The quasi-harmonic approximation (QHA) method

^{[31]}was employed to calculate the thermodynamic properties associated with the phonon frequencies.

The Helmholtz free energy *F* (*T*,*V*) was calculated by

*E*(

*V*) is the ground state total energy at a given unit cell volume

*V*,

*F*

_{vib}(

*T*,

*V*) is the vibrational energy of the lattice ions, and

*F*

_{el}(

*T*,

*V*) is the thermal electronic contribution to the free energy, which can be expressed by electronic entropy

*S*

_{ele}(

*T*,

*V*) and energy due to the electron excitations

*E*

_{ele}(

*T*,

*V*),

*F*

_{vib}(

*T,V*) can be calculated by

*ω*is the phonon frequency and

*g*(

*ω*) is the phonon density of states (DOS). The

*S*

_{ele}(

*T*,

*V*) and

*E*

_{ele}(

*T*,

*V*) can be calculated by

*n*(

*ε*,

*V*) is the electronic density of states,

*f*is the Fermi–Dirac distribution, and

*ε*

_{F}is the Fermi energy.

In the present study, we investigated three phases of thorium hydrides, i.e., bct ThH_{2}, fcc ThH_{2}, and bcc Th_{4}H_{15}(Fig. *I*4/*mmm*. The Th and H atoms are located at the 2a (0, 0, 0) and 4d (0, 0.5, 0.25) sites (in Wyckoff notation), respectively (Fig. *a* and *c* are 4.073 Å and 4.901 Å, respectively, which are in good agreement with the experimental values of 4.10 Å and 5.03 Å.^{[18]} The fcc fluorite-type structure with space group *Fm*-3*m* is recognized as a metastable phase of ThH_{2} (Fig. _{2} is 5.487 Å, being in excellent agreement with the experimental value of 5.489 Å.^{[5]} Besides, the energy–volume relationships of bct and fcc ThH_{2} were calculated to investigate the stability of ThH_{2} in energy, as displayed in Fig. ^{3} and 41.31 Å_{3}, respectively, while the bcc Th_{4}H_{15} phase has space group *I*-43*d*. In this structure, the Th atoms locate at the 16c (0.2087, 0.2087, 0.2087) site and the H atoms have two internal coordinates of 12a (0.375, 0, 0.25) and 48b (0.372, 0.219, 0.404) (Fig. ^{[6]} It is also found that the calculated equilibrium lattice parameter of bcc Th_{4}H_{15} is 9.130 Å, being in agreement with the experimental value of 9.11 Å.^{[19]}

Elastic constants are important mechanical parameters of solid materials, which determine the stiffness of a crystal against the external strain. Thus, we calculate the elastic constants to study the mechanical properties of thorium hydrides under different pressures. The calculated lattice parameters, elastic constants *C*_{ij}, bulk modulus *B*, shear modulus *G*, Young’s modulus *E*, and Poisson’s ratio *ν* for bct ThH_{2}, fcc ThH_{2}, and bcc Th_{4}H_{15} at different pressures are listed in Table *C*_{11} > 0, *C*_{33} > 0 *C*_{44} > 0, *C*_{66} > 0, (*C*_{11} − *C*_{12}) > 0, (*C*_{11} + *C*_{33} − 2*C*_{13}) > 0, and 2(*C*_{11} + *C*_{12}) + *C*_{33} + 4*C*_{13} > 0. Obviously, the bct phase ThH_{2} is mechanically stable since its elastic constants satisfy the above mechanical stability criteria.^{[23]}

As for the cubic structure, the structural stability criteria are *C*_{11} > 0, *C*_{44} > 0, *C*_{11} > |*C*_{12}|, (*C*_{11} + 2*C*_{12}) > 0. From the criteria, it is found that the bcc Th_{4}H_{15} is mechanically stable, while the fcc ThH_{2} is unstable since *C*_{11} − *C*_{12} is negative. Nearly all the elastic constants increase with increasing pressure except *C*_{12} and *C*_{13} for the bct phase and *C*_{11} for the fcc phase (Fig. *B* and Shear modulus *G* changing with pressure can be evaluated from the VRH approximation,^{[27–29]} and the results are also shown in Fig.

The increasing rates of bulk moduli in these phases are slightly lower than those of shear moduli. It is well known that a high ratio of *B*/*G* is responsible for the ductility of polycrystalline materials. The critical value which separates ductile and brittle materials is around 1.75.^{[32]} Our calculated *B*/*G* decreases from 3.29 to 1.37 for the bct phase with the pressure increasing from 0 to 60 GPa (Fig. *B*/*G* ratio of bcc Th_{4}H_{15} increases firstly from 1.76 to 1.82 with pressure increasing from 0 to 20 GPa, and then decreases to 1.60 with pressure further increasing to 60 GPa (Fig. *B*/*G* ratios of bct and bcc phases are both lower than the critical value 1.75, indicating that thorium hydrides are of greater brittleness at high pressures. The Poisson ratio is defined as the absolute value of the ratio of transverse strain to longitudinal strain, which is used to quantify the stability of the crystal against shear. A larger Poisson ratio stands for a stronger plasticity. The Poisson ratios of thorium hydrides at different pressures are also listed in Table _{2} and bcc Th_{4}H_{15} decrease with the increase of the pressure, indicating that the brittleness of thorium hydrides increases with pressure, in accordance with the above result from the *B*/*G* ratio.

In this work, the phonon dispersion was calculated based on the supercell approach and the finite difference method as implemented in the PHONOPY code. The calculated phonon dispersions for bct ThH_{2}, fcc ThH_{2}, and bcc Th_{4}H_{15} are shown in Fig. _{2} and bcc Th_{4}H_{15}, which indicates that bct ThH_{2} and bcc Th_{4}H_{15} can keep their mechanical stability up to at least 50 GPa. However, for fcc ThH_{2}, there exist imaginary frequencies around the *Γ* point, resulting in that the fcc ThH_{2} is thermodynamically instable. The results are well consistent with our previous mechanical stability analysis for bct ThH_{2}, fcc ThH_{2}, and bcc Th_{4}H_{15}. Since thermodynamic properties play an important role in understanding the thermal response of a solid, it is necessary to know the thermal properties of thorium hydrides under high pressures and temperatures. The vibrational energy, entropy, and heat capacity were evaluated by the quasi-harmonic approximation in the temperature range from 0 to 1500 K (Fig. *C*_{V}) increases exponentially with temperature when *T* < 600 K. With the temperature increasing towards 1500 K, the temperature dependence of the heat capacity becomes weaker and *C*_{V} is governed by the atomic vibrations, approaching to the Dulong–Petit limit 3*nR*, where *R* is the gas constant. The trends of the heat capacity at ambient pressure and high pressure are similar. The specific heat at constant pressure *C*_{P} can be obtained by using the relationship between *C*_{P} and *C*_{V}, i.e., *α*_{V} is defined as *V*(*T*) and the bulk modulus *B*(*T*) can be obtained by fitting the data to the Vinet equation of states. Clearly, the bulk modulus *B*(*T*) decreases along with the increase of the temperature, as shown in Fig. ^{[33,34]} It is also found that the bulk modulus *B*(*T*) increases with pressure to a certain degree, and the resistance to the external pressure is strengthened. The calculated heat capacity *C*_{P} of thorium hydrides are displayed in Fig. *C*_{P} declines slightly.

The high pressure behaviors of Th_{4}H_{15} and ThH_{2} have been studied by using the first-principles calculations based on the density functional theory. From the energy–volume relationship, the bct phase of ThH_{2} is more stable than the fcc phase at ambient pressure. At high pressures, the bct ThH_{2} and bcc Th_{4}H_{15} phases are more brittle than they are at ambient pressure in terms of the calculated elastic constants and the Poisson ratio. The thermodynamic stability of these three phases has been discussed with the calculated phonon dispersion curves. In the pressure domain of interest, the phonon dispersions of bcc Th_{4}H_{15} and bct ThH_{2} are positive, indicating that these two phases are stable, while the fcc ThH_{2} is unstable. The thermodynamic properties including the lattice vibration energy, entropy, and specific heat have been predicted based on the vibrational frequencies for the stable phases. Our results show that the vibrational free energy decreases with the increase of the temperature, while the entropy and the heat capacity are proportional to the temperature and inversely proportional to the pressure. It is found that the bulk modulus *B* increases with pressure, indicating the strengthened resistance to the external pressure for Th_{4}H_{15} and ThH_{2}. The calculated results in this study would be helpful for future experimental measurements on thorium hydrides.

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