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New zirconium hydrides predicted by structure search method based on first principles calculations

The formation of precipitated zirconium (Zr) hydrides is closely related to the hydrogen embrittlement problem for the cladding materials of pressured water reactors (PWR). In this work, we systematically investigated the crystal structures of zirconium hydride (ZrHx) with different hydrogen concentrations (x = 0~2, atomic ratio) by combining the basin hopping algorithm with first principles calculations. We conclude that the P3m1 {\zeta}-ZrH0.5 is dynamically unstable, while a novel dynamically stable P3m1 ZrH0.5 structure was discovered in the structure search. The stability of bistable P42/nnm ZrH1.5 structures and I4/mmm ZrH2 structures are also revisited. We find that the P42/nnm (c/a>1) ZrH1.5 is dynamically unstable, while the I4/mmm (c/a = 1.57) ZrH2 is dynamically stable.The P42/nnm (c/a<1) ZrH1.5 might be a key intermediate phase for the transition of {\gamma}->{\delta}->{\epsilon} phases. Additionally, by using the thermal dynamic simulations, we find that {\delta}-ZrH1.5 is the most stable structure at high temperature while ZrH2 is the most stable hydride at low temperature. Slow cooling process will promote the formation of {\delta}-ZrH1.5, and fast cooling process will promote the formation of {\gamma}-ZrH. These results may help to understand the phase transitions of zirconium hydrides.


Introduction
Zirconium (Zr) alloys have been used as cladding material in pressure water nuclear reactors due to their low thermal neutron absorption cross section, good mechanical properties, excellent waterside corrosion resistance, etc. [1]. However, when in long term service, complex zirconium hydrides (ZrH x ) will form in the cladding due to the hydrogen absorption. For decades, several zirconium hydride phases have been identified by experiment, including ζ-ZrH 0.5 [2], γ-ZrH, δ-ZrH 1.5 and ε-ZrH 2 [3].
In recent years, based on first principles calculations, thermal dynamical and mechanical properties of zirconium hydrides have been investigated, and new structures of zirconium hydrides have also been predicted [4][5][6][7][8][9][10][11][12][13]. Ackland found that for ε-ZrH 2 , two structures with different c/a exists and were almost degenerated in energy [14]. These bistable structures were later studied by Zhang et al. [15]. They suggested that the structure with c/a > 1 was dynamically unstable. Domain et al. [10] discovered two P4 2 /nnm ZrH 1.5 structures with different c/a, which are more stable than the stability of these new structures at different temperatures was also studied.

Method and computational details
The structure search was performed with STEPMAX module of the Correlated Electron System Simulation Package (CESSP) [21][22][23][24] package, which is based on the basin hopping (BH) algorithm [25][26][27][28][29] for structure evolution. Geometry optimizations and energy evaluations were carried out using the projector-augmented wave (PAW) method [30] within density functional theory framework by using Vienna ab initio simulation package (VASP) [31][32] and CESSP. Exchange and correlation were treated in the generalized gradient approximation (GGA) as parameterized in the Perdew-Burke-Ernzerhof (PBE) [33][34] functional. The plane-wave cutoff energy was set to be 500 eV to guarantee a total energy convergence of less than 1 meV per atom. The Brillouin zone k-point sampling was performed by Monkhorst-Pack scheme [35] with the smallest allowed k-points spacing of 2π*0.02Å -1 . The geometry optimization is performed using the conjugate-gradient (CG) algorithm until all the forces acting on the atoms are smaller than 1.0e-4 eV/Å. The Methfessel-Paxton function [36] with a smearing width of 0.1 eV was used to perform the integration over eigenvalues. The tetrahedron method with Blöchl corrections [37] was used to obtain the accurate energies. The formation enthalpy of Zr 1-x H x can be calculated as ( )
To take into account the temperature effect, the quasi-harmonic approximation theory is used to calculate the Helmholtz free energy, and both contributions of the electron excitation free energy and the phonon are included [5]. The calculations of phonon and the thermal properties were also performed by phonopy [38] package.  Figure 1.

Results and
Firstly, we would like to discuss the stability of the P3m1 ζ-ZrH 0.5 structure reported by Zhao et al. [2]. During the structure searching, we found the reported In this study, we further checked the stability of the P3m1 structure by phonon calculation, elastic constants calculation and more strict geometry optimization. As is shown in Figure 2a, there are several imaginary frequencies exists in the obtained phonon density of states (DOS), indicating that the P3m1 structure is dynamically unstable. Besides, the results of elastic constants ( Table 2) also showed that the P3m1 structure is mechanically unstable. Then, we optimized the P3m1 structure with higher accuracy settings and turned the symmetry constraints off. The P3m1 structure transforms to a Cm structure (a = 5.44 Å, b = 3.44 Å, c = 12.06 Å, β = 118.5 • ). Phonon DOS simulation shows that the Cm structure is dynamically stable.
The energy of the Cm structure is 30 meV per atom lower by than the original P3m1 structure, which further proved that the P3m1 structure is unstable.
According to Zhao et al. [2], ζ-ZrH 0.5 belongs to the trigonal crystal system. The most stable structure of ZrH 0.5 predicted in our structure search is a cubic crystal with quasi-harmonic approximation calculations of Helmholtz free energy. As shown in Figure 3a, with the increase of temperature, the relative stability of the structure decreases, while the Cmmm and C2/m (S1 and S2) structures of ZrH 0.5 become more stable. The Cmmm and the C2/m (S2) structures turn to be more stable than the structure when the temperature is higher than 900K, while C2/m (S1) becomes as stable as the structure at 1100K. The new P3m1 is also getting more stable as the temperature increases, but it is still less stable than the structure even at 2000K. The existence of ζ-ZrH 0.5 is possibly due to the lattice matching between α-Zr which stabilizes the Zr/ζ-ZrH 0.5 interface, the high transformation barrier to other more stable structures, or the existence of alloying elements in the commercial zirconium alloys used in experimental observations.
Through the analysis of the above results, we conclude that the ζ-ZrH 0.5 structure reported by Zhao et al. is actually unstable. We found a new P3m1 structure which is more stable. However, the most stable ZrH 0.5 structure predicted by us is a structure, but it might be less stable at high temperature.

ZrH 1.0
For ZrH, the γ-ZrH with tetragonal cell has been experimentally observed, but its stability is still in a dispute. Though γ-ZrH was previously reported to be of P4 2 [3].
In this work, the most stable ZrH structure predicted at 0 K is also P4 2 /mmc.
The optimized lattice constants are a = 3.237 Å and c = 5.004 Å, which is in a good agreement with previous DFT result [46]. Though a Cccm structure with its lattice constants being very close to the values reported by Kolesnikov [47] was found during the structure search, further geometry optimization simulation with tighter convergence criteria showed that its actual structure is still of P4 2 /mmc symmetry. The second and third most stable structures of ZrH predicted by us are of Ccce and P222 symmetry. The total energies of these structures are 24.1 and 25.4 meV per atom higher than the P4 2 /mmc structure, respectively. Their lattice constants are listed in Table 1.
Weck et al. also reported two P222 structures [46] which are different from the P222 structure found in our calculations. As a comparison, we optimized the Weck's structures and found that their symmetries are actually P4 2  As is shown in Figure 2b, phonon density of states of P4 2 /mmc, Ccce and P222 structures have no imaginary frequencies. This indicates that these structures are dynamically stable. Furthermore, as shown in Table 2, the calculated elastic constants also show that the P4 2 /mmc structure is mechanically stable.
The finite temperature effect on the structural stability of ZrH was investigated and result is displayed in Figure 3b. As temperature increases, the free energies of the Ccce and P222 structure are getting closer to the P4 2 /mmc structure, but the P4 2 /mmc structure remains to be the most stable structure even up to 2000K.
As a summary, the P4 2 /mmc structure is the most stable ZrH structure at low temperature as well as high temperature based on our calculations.

ZrH 1.5
For ZrH 1.5 , the experimentally observed structure is a face-centered cubic (fcc) phase with lattice constants of 4.768 Å. By using a unit cell with 10 atoms (Zr 4 H 6 ), we find a cubic structure of -3 Pn m symmetry (space group #224, a = 4.775 Å), which agrees with experimental measurement and previous DFT calculations by Zhu et al. [48].
However, the fcc structure is not the most stable one according to our results of structure search. Based on our calculations, the most stable structure of ZrH 1.5 is a P4 2 /nnm tetragonal structure with lattice constants of a = b = 5.020 Å and c = 4.261 Å (c/a < 1). The Zr matrix of this P4 2 /nnm structure is of I4/mmm symmetry with lattice constants of a = 3.550 Å, c = 4.261 Å, c/a = 1.20. It is worth mentioning that this structure is one of the two structures of ZrH 1.5 reported by Domain et al. [10].
We also build the second P4 2 /nnm structure according to the study of Domain et al. by hand. The optimized lattice constants of this structure are a = b = 4.604 Å and c = 5.110 Å (c/a > 1), and its structural energy is 22.6 meV per atom higher than the P4 2 /nnm (c/a < 1) structure. Nevertheless, the P4 2 /nnm (c/a>1) structure is still 8.2 meV/atom more stable than the 3 Pn m structure. Furthermore, as shown in figure   2(c), the calculated phonon density of state has several minor imaginary frequencies, which indicates that the P4 2 /nnm (c/a > 1) structure suggested by Domain et al. is dynamically unstable. A small displacement was applied to the second P4 2 /nnm structure, and after geometry optimization we got the most stable P4 2 /nnm structure as we predicted.
These two P4 2 /nnm structures are not the only structures more stable than the fcc  [46] and Zheng et al. [4]are also summarized in Table 1.
The above discussions indicate that there are many structures that are more stable than the experimentally observed Pn m , we considered the temperature effect on these structures. As shown in Figure 3c, at 0 K, the phase is less stable than P4 2 /nnm, , P2/c and Ibam phases; as the temperature increases, the phase is getting more stable; at 600 K, the phase becomes more stable than 1 P -, P2/c and Ibam phases; when the temperature is higher than Pn m phase becomes more stable than the P4 2 /nnm structure.
In summary, the P4 2 /nnm ( c/a < 1 ) is the most stable ZrH 1.5 structure at 0 K, while the P4 2 /nnm (c/a>1) structure is dynamically unstable. At high temperature (~1000K), the 3 Pn m structure becomes to be the most stable ZrH 1.5 structure.

ZrH 2
For ZrH 2, ε-ZrH 2 with I4/mmm symmetry (space group #139) and fcc-ZrH 2 with 3 Fm m symmetry (space group #225) have been extensively studied. Both structures were also found in our calculations. The I4/mmm structure is predicted to be most stable. Its lattice constants are a = b = 3.538 Å, c = 4.406 Å, c/a = 1.25, which agrees well with the previous calculation by Chattaraj et al. [49]. We also find another I4/mmm structure with lattice constants of a = 3.278 Å and c = 5.150 Å, c/a =1.570. It can be regarded as a distorted structure of I4/mmm (c/a = 1.25). Its formation energy is only 2.6 meV per atom higher than ε-ZrH 2.
The bistable structures of I4/mmm ZrH 2 have also been discussed by Ackland et al. [14] and Zhang et al. [15]. Zhang  Fm m structure of ZrH 2 found by us was with the lattice constant a = 4.807 Å, which is in good agreement with previous calculations by Zhang et al. [15]. The energy of this 3 Fm m structure is 9.2 meV per atom higher than the stable I4/mmm structure. However, its phonon DOS has a small imaginary vibrational frequency (Figure 2(d)), indicating that the 3 Fm m structure is dynamically unstable. This is also consistent with the previous work of Zhang et al. [15]. By applying random small displacements to the 3 Fm m structure and followed by geometry optimization, we got the ZrH 2 of I4/mmm (c/a = 1.25) structure. As shown in

structure. The 3
R m structure is 6.6 meV per atom higher in energy than ε-ZrH 2 , but it is still 2.6 meV per atom more stable than the 3 Fm m structure. It is possible that the 3 R m structure is also a metastable structure which might be observed experimentally under some conditions. Furthermore, we investigated the finite temperature effect on the different ZrH 2 structures. As shown in Figure 2 and Table 3, the 3 Fm m structure is both dynamically and mechanically unstable, but its Helmholtz free energy is lower than the I4/mmm structure as the temperature is higher than 900 K. However, this structure could be transformed into I4/mmm structure under lower temperature, because the 3 Fm m structure is dynamically unstable.
In summary, two I4/mmm structure different c/a (1.25 and 1.57) are the two most stable ZrH 2 structures. Both of them are dynamically stable.

Phase transitions of the zirconium hydrides
Based on the experimental observations, as H atoms accumulated, the structures of the zirconium hydrides changes from ζ-ZrH 0.5 , γ-ZrH, δ-ZrH 1.5 to ε-ZrH 2 . Their symmetry varies from P3m1 (trigonal), P4 2 /mmc (tetragonal), 3 Pn m (cubic) and 3 Fm m -finally to I4/mmm (tetragonal). By removing the hydrogen atoms from these structures, we analyzed the structural changes of their Zr matrixes. The Zr matrix of ζ-ZrH 0.5 is still of P3m1 symmetry, but it is very close to the α-Zr with P6 3 /mmc symmetry. For γ-ZrH, the Zr matrix is of the I4/mmm symmetry with c/a~1.55. For δ-ZrH 1.5, the Zr matrix is of the 3 Fm m symmetry. For ε-ZrH 2, the Zr matrix is changed again to be of the I4/mmm symmetry, but the c/a~1.24. Therefore, the structure of Zr matrix changes drastically with the increasing of hydrogen concentration.
However, considering the new structures we found in the structure searching, the structure transitions of γ, δ and ε phases could be explained more clearly.
As is shown in Figure 4

Stability of Zr hydrides at different temperature
In our global minima searching process, the formation enthalpies at different temperature are evaluated at 0 K. Whether these structures remain stable at higher temperature is still an open question. Here we investigated the effect of temperature based on the QHA method.
We also investigated the formation free energy of the following bulk reactions via hydrogen diffusion at different temperatures: (I4/mmm), δ-ZrH 1.5 ( 3 Pn m ) and γ-ZrH(P4 2 /mmc) are used, while for ZrH 0.5, the free energy of the new P3m1 structure found by our structure search is used. Results of the above reactions are illustrated in Figure 5. At low temperature (less than 300 K), reactions of R1, R2, R3, R5, R8, R9 and R10 are endothermic, thus these reactions cannot happen spontaneously. However, reactions of R4 and R6 are exothermic and can happen spontaneously. It can be concluded the stability order of these four hydrides are ZrH 2 , ZrH, ZrH 1.5 , and ZrH 0.5 .
As the temperature increases, the free energies of some reactions change. At 1100K, reactions of R4 and R6 turn from exothermic at low temperature to endothermic, which means that ZrH 1.5 is more stable than ZrH. Meanwhile, R1 turns from endothermic to exothermic, indicating that ZrH 1.5 is also more stable than ZrH 2 .
The stability order of these four hydrides becomes ZrH 1.5 , ZrH 2 , ZrH and ZrH 0.5.
From the above analysis, we can see that as temperature increases, δ-ZrH 1.5 becomes the most stable structure. By fast cooling process, Zr hydride will keep the high temperature structure unchanged, which means that δ-ZrH 1.5 should be the structure observed after fast cooling. At low temperature, reaction R4 becomes exothermic, which means that slow cooling will promote the formation of γ-ZrH.
Our conclusions agree with the recent experiment by Wang et al. [50]. The transition of ZrH 1.5 to ZrH 2 and Zr (reverse reaction of R1) is also exothermic. This reaction is also observed by recent experiment by Maimaitiyili et al [51]. As suggested by Wang et al., the transition from δ to γ or ε is diffusion controlled. Since the diffusion of H in δ-ZrH 1.5 is much slower than in α-Zr [52], γ-ZrH is more likely to be formed than ε-ZrH 2 .
For the formation of ZrH 0.5 , the only possible reactions are R7 at low temperature. In this reaction, ZrH 1.5 decomposes to be ZrH 2 and ZrH 0.5. In the experiment of Zhao et al., the specimens were annealed at 550 • C for 1 h and then slowly cooled to room temperature. Our simulation also agrees with this experiment.
Though reactions of R3, R5, R8, R9 and R10 are all exothermic at low temperature, they might be kinetically prohibited by the high hydrogen diffusion barriers in the Zr hydrides.