First Principles Investigation of the Structure and Properties of Superconducting Cubic Protactinium Hydride PaH₃

Cubic protactinium hydrides are very important existing form in superconducting protactinium hydrogen series. In this work, the ground state structure and properties of cubic PaH ₃ have been studied using the DFT + U method. This systematic study for two bulk properties includes the electronic structures, phonon dispersion curves, structural, mechanical and thermodynamic properties under the effective coulomb U and exchange J PBE + U parameters. Structural relaxation results show that the Pa-H and Pa-Pa distances in α-PaH ₃ are significantly higher than that in β-PaH ₃, and the H-H distances in α-PaH ₃ are slightly smaller than that in β-PaH ₃. For the ground state electronic structures of α-PaH ₃ and β-PaH ₃, we found that α-PaH ₃ and β-PaH ₃ are metallic, and the protactinium 5 f electronic states and hydrogen have obvious bonding effect, resulting in weakening of the material’s metallicity. This is consistent with observations for the other actinide hydrides such as ThH ₃ and UH ₃. The phonon spectrum calculations reveal that the PBE and PBE + U methods give quite different frequencies for the optical branches of phonons of α-PaH ₃ and β-PaH ₃. In addition, by including the vibrational entropy and the ZPE using the phonon frequencies obtained from the optimized unit cells we predict that the β-PaH ₃ phase can not transit into α-PaH ₃ phase above room temperature.

• <i>α</i>-PaH₃ and <i>β</i>-PaH₃
• Electronic Properties
• Mechanical Properties
• Thermodynamic Properties
• Density Functional Theory (DFT)

1. Introduction

In the actinide hydrides series, Thorium (Th), Uranium (U) and Plutonium (Pu) hydrides have been widely studied due to their important role in the nuclear industry. However, there are few reports on others actinide hydrides such as protactinium hydride (Pa-H). Protactinium hydride has generated some interest because protactinium is considered as a component of the conventional nuclear waste and an opportunity for studying the physics and chemistry of actinide-based materials [1] [2] . Moreover, protactinium hydride shows a variety of structural phases under different pressures. Xiao et al. systematically study the structure and superconductivity of protactinium hydride (PaH_n) at different pressures using the first principle [3] . Although the cubic PaH₈ (Fm-3m) exhibits high superconducting critical transition temperature under high pressure, the cubic PaH₃ structure seems to be thermodynamically stable within 100 GPa. Thus, in order to understand the rich structural phases and superconducting properties of protactinium hydride, it is very important to study the ground state structure and physical properties of cubic protactinium hydride.

As early as 1984, Ward et al. experimentally reported structures, physical and chemical properties of a solid solution Pa₃H₄-Pa₃H₅ above 500 K and cubic PaH₃ phase for the Pa-H system [2] . Sellers et al. reported that metal Pa with H₂ reacted at 523 K to form a black β-PaH₃ which was isostructural with cubic β-UH₃ [1] . Dod reported that α-PaH₃ formed between 373 K and 473 K had a cubic α-UH₃-type structure [2] . Moreover, these early studies indicated that Pa-H system shows some similarities with U-H system. This also provides signs for further understanding and potential applications of cubic PaH₃ phase. The neighbouring thorium-hydrides (Th-H) and uranium-hydrides (U-H) systems have been thoroughly investigated [4] [5] [6] [7] and each exhibits unique properties, but the ground state structures and properties of the cubic PaH₃ are not known.

John W. Ward [2] reported that the high temperature phase β-PaH₃ transits irreversibly into the low temperature phase α-PaH₃ above room temperature. Based on the irreversibility of the β to α phase transition, α-PaH₃ is always believed to be a metastable state. To further understand the phase transition process, the ground state properties such as structures, phonon spectrum, electronic, mechanical and thermodynamic properties of cubic PaH₃ phase should be supplied. For actinide-based compounds, the density functional theory (DFT) schemes including the local spin density approximation (LSDA) or the generalized gradient approximation (GGA) fail to get the ground state, while the DFT + U formalisms can effectively get the right ground state such as its neighbor thorium and uranium compounds [8] [9] [10] [11] . Considering that 5f electronic states of cubic PaH₃ system show both localized and itinerant characters [2] , here we present a comparative DFT + U study on the ground-state properties of α-PaH₃ and β-PaH₃.

2. Computational Details

In our works, all DFT calculations are performed using the Vienna Ab initio Simulation Package (VASP, version 5.4) [12] [13] . The exchange-correlation functional in the local density approximation (LDA) and the generalized gradient approximation (GGA) as parametrized by Perdew-Burke-Ernzerhof (PBE) [14] [15] were used in this study. A Hubbard-like term was added in order to take into account the strong 5f electron correlations in actinide protactinium cation, according to the DFT + U scheme. The brillouin zones were divided using Monchhorst Pack k-mesh with 9 × 9 × 9 grid for the DFT + U calculations. The kinetic energy cut off of 550 eV was used for plane waves. The energy convergence criterion was 10⁻⁶ eV. The Hellmann-Feynman forces on each atom for ionic relaxation were less than 0.01 eV/Å. The DFT + U approach [16] [17] [18] [19] [20] was carried out to enforce localization of Pa 5f electrons. The total energy functional for this scheme is of the form

E D F T + U = E D F T + U − J 2 ∑ σ [ T r ρ σ − T r ( ρ σ ρ σ ) ] (1)

where ρ σ is the density matrix of Pa-f states, U and J are the spherically averaged screened coulomb energy and exchange energy, respectively.

3. Results and Discussion

3.1. Structural and Mechanical Properties

The crystal structures of α-PaH₃ and β-PaH₃ have been shown in Figure 1(a) and Figure 1(b) respectively. The different magnetic orders (i.e. FM and AFM) are adopted in our DFT + U calculations. In the meanwhile, for α-PaH₃ and β-PaH₃, the relationship between the optimized lattice constant and a series of U values is shown in Figure 2(a) and Figure 2(b) respectively. Our LDA/PBE + U calculated results show that there is no obvious difference between the effect of ferromagnetic and antiferromagnetic order on the lattice parameters of cubic α-PaH₃ and β-PaH₃. In Figure 2(a), when the U is around 2.0 eV, our calculated lattice constant of α-PaH₃ using the PBE + U method is good calculation result compared to experimental data. However, as shown in Figure 2(b), our results for FM phase of β-PaH₃ demonstrates that the lattice constant is close to the experimental value [1] [2] when U is around 4.0 eV. Thus, for cubic α-PaH₃ and β-PaH₃, the lattice constants obtained from the PBE + U computations are in good agreement with the experimental data.

In Table 1, the lattice constant and the interatomic distance of α-PaH₃ and β-PaH₃ using the PBE and PBE + U methods were listed. Our structural relaxation results show that the atomic spacing has obvious difference in α-PaH₃ and β-PaH₃ structures. In α-PaH₃, the Pa–H distance is 2.305 Å, significantly higher than the Pa–H distance in β-PaH₃. Moreover, the Pa–H distance in β-PaH₃ is also higher than that in α-PaH₃ structure. Charge density decomposition using the Bader method suggests that the Pa atoms forming linear chains in β-PaH₃ atoms are bonded: the bond-lengths of 2.555 Å are only slightly larger than the largest nearest-neighbor distance in the metallic Pa phase (0.326 nm). Therefore, the calculated results show that the Pa–H and Pa–Pa distances in α-PaH₃ is significantly higher than that in β-PaH₃, and the H–H distances in α-PaH₃ is slightly smaller than that in α-PaH₃.

Table 1

The lattice constant and interatomic distance for α-PaH 3 and β-PaH 3

Compound	Method	a0 (Å)	d (Pa–H) (Å)	d (Pa–Pa) (Å)	d (H–H) (Å)
α-PaH3	PBE	4.135	2.288	4.093	2.047
	PBE + U	4.149	2.305	4.123	2.061
	Expt. [1] [2]	4.150
β-PaH3	PBE	6.615	2.250	3.276	2.537
	PBE + U	6.646	2.261	3.294	2.555
	Expt. [2]	6.648

In Table 2, the elastic constants, bulk modulus, shear modulus, Young’s modulus and Poisson’s ratio of α-PaH₃ and β-PaH₃ using the PBE and PBE + U methods were listed. For cubic α-PaH₃ and β-PaH₃ structures, the elastic constants satisfy the following mechanical stability criteria [21] :

C 11 > 0 , C 44 > 0 , C 11 > | C 12 | , ( C 11 + 2 C 12 ) > 0 (2)

These results indicate that the structure of α-PaH₃ and β-PaH₃ is stable. We have calculated the bulk (B) and shear moduli (G), Young’s modulus (E) and Poisson’s ratio (υ) of α-PaH₃ and β-PaH₃ using the Voigt-Reuss-Hill (VRH) approximations [22] [23] . From Table 2, the obtained bulk and shear moduli of

Table 2

The calculated elastic constants C ij , bulk modulus B, shear modulus G, Young’s modulus E and Poisson’s ratio υ for α-PaH 3 and β-PaH 3 . The elastic constants and moduli are in units of GPa

Compound	Methods	C11	C12	C44	B	G	E	υ
α-PaH3	PBE	316.29	51.40	28.87	139.70	49.08	131.80	0.34
	PBE + U	309.66	70.14	28.26	149.98	44.00	120.24	0.37
β-PaH3	PBE	195.48	79.53	51.94	118.18	39.02	105.45	0.35
	PBE + U	227.97	76.72	39.81	127.14	38.62	105.21	0.36

α-PaH₃ and β-PaH₃ with the PBE+ U are larger than that the PBE. These differences might be owing to the overestimation of lattice constants by PBE + U method. Our obtained Poisson’s ratio of 0.34 - 0.37 for α-PaH₃ and β-PaH₃ is reasonable for being in the range of 0.2 - 0.4 for typical metallic materials.

3.2. Electronic Properties

The total density of states (DOS) and projected density of states (PDOS) of α-PaH₃ and β-PaH₃ are shown in Figure 3 by using the PBE and PBE + U methods. Figure 3 shows the protactinium 5f electrons are mainly distributed and there is no obvious localization feature at the Fermi energy for the electronic structure of α-PaH₃ and β-PaH₃. Although the 5f electron of Pa in these two systems is a strongly electronic correlation effects, Pa-f and H-s electrons have obvious hybrid effect near the Fermi energy level, and further proves that the protactinium 5f electrons have itinerant characters in α-PaH₃ and β-PaH₃. In cubic PaH₃ structure, Pa 5f electrons does not show strong localization characteristics near the Fermi level, but has obvious bonding effect with Hand H 1s electrons. This is very similar to the electronic properties of protactinium hydride (PaH_n) (n = 1 - 9) in Fm3m-PaH₈ superconductive structure studied by Xiao et al. [3] . Thus, for cubic α-PaH₃ and β-PaH₃, the electronic properties for Pa-f and H-s are very important for the superconductivity of Pa-H system.

As shown in Figure 4, the difference charge density distribution is further analyzed for α-PaH₃ and β-PaH₃, to study the bonding characters between protactinium and hydrogen atoms. We can see clearly that the ionic character of the protactinium-hydrogen bonds is enhanced with the PBE + U formalism. This result is in consistent with our previous PDOS analysis.

3.3. Phonon Spectrum and Thermodynamic Properties

Phonon frequency and thermodynamic properties calculations are carried out using the Hellmann-Feynman theorem and the supercell method [24] . A 2 × 2 × 2 supercell is adopted to calculate the force constants, and a 3 × 3 × 3 Monkhorst-Pack k-points mesh is used for integrations over the BZ. The atomic displacements for force constants calculations are all set to 0.02 Å.

For α-PaH₃ and β-PaH₃, the phonon dispersion curves in the BZ and together with the phonon DOS are shown in Figure 5(a) and Figure 5(b). For cubic PaH₃, the acoustic and optical branches of phonon come from Pa and H vibrations, and Pa atoms are much heavier than H atoms, thus a large gap between acoustic and optical branches can be observed. For the three acoustic branches, the dispersion curves are almost the same in PBE and PBE + U results. While for the optical branches from Figure 5(a), the frequencies calculated with PBE + U are all shifted down by about 2.3 THz that in comparison with PBE. This is because the on-site Coulomb repulsion enhances the ionic character of the Pa-H bonds, and thus reduces the bonding strength. Our studies indicate that the lattice dynamics behaviors of both α- and β-PaH₃ strongly depend on the electronic correlation effects.

The thermodynamic properties such as the Helmholtz free energy F, vibrational entropy S, vibrational energy E, the heat capacity C_v (Figure 6) and the zero point energy (ZPE, Table 3) have been calculated by the density functional perturbation theory (DFPT) [25] [26] . All these properties are expressed per α-PaH₃ and β-PaH₃ unit, and a sample of values at 300 and 500 K is presented in Table 3. For α-PaH₃ and β-PaH₃, the vibrational energy E exhibits a similar trend that increases almost linearly with temperature and tends to display k_BT behavior at high temperatures. With the increase of temperature, the differences between α-PaH₃ and β-PaH₃ becomes even more apparent, which may be attributed to the higher frequencies related to lighter H atom vibrations. And the F decreases almost linearly with temperature. The entropy S increases with the rising of temperature since the vibrational contribution to the entropy increases as the temperature rises. However, Figure 6(b) demonstrates the entropy of β-PaH₃ is significantly higher than that of α-PaH₃, the discrepancies are mainly because of the difference in the number of hydrogen atoms. In addition, the heat capacity C_v increases with the applied temperature. Below a temperature of around 200 K, the C_v increases very rapidly with increasing temperature and displays the T³ law behavior. And above 800 K, the C_v increases slowly with temperature and gradually approaches the Dulong-Petit limit [27] . Hence, the β phase of cubic PaH₃ may not transit into α phase above room temperature owing to the thermodynamic criteria.

Table 3

Predicted thermodynamic properties for α-PaH 3 and β-PaH 3 phases at 300 K and 500 K

Phase	T (K)	Fvib J/(K·mol)	Svib J/mol	CV J/(K·mol)	Evib kJ/mol	ZPE kJ/mol
α-PaH3	300	93.10	95.10	80.98	121.63	107.96
	500	68.65	149.11	130.95	143.21
β-PaH3	300	381.69	386.39	321.21	497.61	442.62
	500	282.99	599.28	515.32	582.63

4. Conclusion

In summary, we have systematically studied the structural, electronic, mechanic and thermodynamic properties of α-PaH₃ and β-PaH₃ by employing the PBE and PBE + U methods. Firstly, to obtain reliable ground state structure and properties of α-PaH₃ and β-PaH₃ we found the effective U parameter. Then, the Pa–H and Pa–Pa distances in α-PaH₃ are significantly higher than that in β-PaH₃, and the H–H distances in α-PaH₃ are slightly smaller than that in β-PaH₃. Moreover, we found that α-PaH₃ and β-PaH₃ are metallic, and the protactinium 5f electronic states and hydrogen have obvious bonding effect. These studies on the ground state structure and properties of cubic PaH₃ system are crucial for further understanding and studying the superconducting properties of metal hydride. Especially for the cubic PaH₃, the weak bonding between Pa and H near the Fermi level provides a useful supplement to the understanding of superconductivity. As a result of effecting the Pa-H bonds, inclusion of the on-site Coulomb interaction largely influences the optical branches of vibration modes for α-PaH₃ and β-PaH₃. We thus suggest that measuring the vibrational or thermodynamic properties may give out information on the correlation strengths of uranium 5f electrons in α- and β-PaH₃. Furthermore, the calculations show that the β-PaH₃ phase can not transit into α-PaH₃ phase above room temperature.

Acknowledgements

The authors thank Doc. Shiyin Ma for useful discussion.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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