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Mechanical and thermodynamic properties of α-UH3 under pressure
Journal of Alloys and Compounds 604 (2014) 171–174
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Journal of Alloys and Compounds journal homepage: www.elsevier.com/locate/jalcom
Mechanical and thermodynamic properties of a-UH3 under pressure Chao Zhang a,⇑, Hong Jiang a, Hong-Liang Shi b, Guo-Hua Zhong c, Yue-Hua Su a a
Department of Physics, Yantai University, Yantai 264005, China Beijing Computational Science Research Center, Beijing 100084, China c Center for Photovoltaics and Solar Energy, Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences and The Chinese University of Hong Kong, Shenzhen 518055, China b
a r t i c l e
i n f o
Article history: Received 29 October 2013 Received in revised form 15 February 2014 Accepted 12 March 2014 Available online 24 March 2014 Keywords: Actinide alloys and compounds High pressure Mechanical properties Thermodynamic properties First-principles calculations Uranium trihydride
a b s t r a c t The mechanical and thermodynamic properties of a-UH3 under pressure have been examined by firstprinciples pseudopotential plane-wave calculations based on the density functional theory. In order to describe the strong on-site Coulomb repulsion among the localized 5f electrons, the generalized gradient approximation and local density approximation plus a Hubbard parameter (GGA + U and LDA + U) formalisms have been adopted for the exchange correlation term. The pressure evolution of elastic constants of a-UH3 has been revealed, and a-UH3 is found to be mechanically stable at least up to 20 GPa. Under pressure, the low-frequency vibration modes of a-UH3 change slightly, while the high-frequency vibration modes shift upward. Our study is expected to facilitate the understanding of uranium hydrides and their applications. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction Actinide elements and compounds exhibit interesting physical and chemical properties due to the complex nature of 5f electrons, and have been extensively studied experimentally [1–5] and theoretically [6–9]. Uranium is the heaviest element in nature with any practical abundance. Thus, it attracts much attention than the other actinides. Uranium metal can react with hydrogen to form uranium hydrides. In solid state, uranium hydride exists mainly in the form of cubic trihydride with two allotropes, i.e., a-UH3 and b-UH3, and a-UH3 could transforms into b-UH3 above room temperature [1,10]. The uranium hydrides have rich physical properties, and play an important role in nuclear fuels and hydrogen storage. Furthermore, the uranium hydrides have been taken as starting materials to reactive uranium power along with various uranium carbides, nitride, and halide. Therefore, understanding the physical, mechanical, and thermal properties of uranium hydrides is of highly desirable. First-principles band-structure calculations can provide valuable insight into physical, elastic, and thermal properties of materials. However, conventional density function theory (DFT), which applies the local density approximation (LDA) or the generalized gradient approximation (GGA), underestimates the strong on-site ⇑ Corresponding author. Tel.: +86 5356902506. E-mail address: [email protected] (C. Zhang). http://dx.doi.org/10.1016/j.jallcom.2014.03.071 0925-8388/Ó 2014 Elsevier B.V. All rights reserved.
Coulomb repulsion of 5f electrons in U atom and fails to describe the properties of uranium compounds. For example, such approaches predict a fluorite metallic ferromagnetic ground state for UO2, while the observed ground state is an antiferromagnetic insulator with a band gap of about 2.0 eV [11]. To overcome this drawback in calculations of actinide compounds, the effective modification of pure DFT which calls LDA/GGA + U scheme has been widely used. This method captures well the localization effect of f electrons, allowing successful description of actinide compounds, such as uranium oxides [12–14] and plutonium oxides [15,16]. Compared to actinide oxide, only a limited uses of DFT to actinide hydrides in the literature [17–20], especially using LDA/GGA + U methods. The electronic properties of a-UH3 at ambient pressure have been investigated within the LDA and LDA + U formalisms [17]. Including the Hubbard U term, the uranium 5f electronic states are represented by spectral envelope consisting of multiple peaks, which weakens the metallicity of a-UH3 and enhances the ionic character of uranium–hydrogen bonds. In addition, the phonon dispersion curves show different shapes as compared to those obtained by pure DFT calculations. Moreover, the intriguing 5f electronic states in UH3 draw the attention of the scientific community. The earlier nuclear magnetic resonance study suggests the 5f electrons in UH3 to be localized [10,21]. Band structure calculations come to the conclusion that there are two types of f character in these materials: localized and itinerant
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[22]. Recently, it has been suggested that the 5f electronic states in UH3 films show both localized and itinerant character [2]. The properties of solids as a function of pressure are of fundamental interest to a wide range of condensed matter physics. Under applied pressure, the crystal structure established for solids at ambient change dramatically, causing large effects on physical and chemical properties. Unfortunately, the knowledge on UH3 at high pressures, especially its mechanical and the thermal properties, is very limited [23], which hinders its practical application. This is partially due to the fact that high pressure is not easy to reach/access and control in diamond anvil cell. On the contrary, adjusting pressure in theoretical simulations can be accomplished straight forwardly by varying the size of unit cell. Development of reliable theoretical methods which simulate such properties would significantly help, especially in the field of the nuclear materials, for which adequate experimental data are missed. In this work, we have investigated the high-pressure behavior of a-UH3 up to 20 GPa by using the GGA + U and LDA + U schemes, with a special focus on the mechanical and thermal properties. The paper is organized as follows: the computational details are described in Section 2, while the results are presented and discussed in Section 3. After that, summary of the work is given. 2. Computational method The first-principles calculations were performed within the framework of DFT implemented in the Vienna Ab-initio Simulation Package (VASP) [24]. The all-electron projector augmented wave (PAW) [25] pseudopotential for U and H from the VASP pseudopotential library were used. For U 5f, 6s, 6p, 6d and 7s and for H 1s were treated as valence electrons. The electron wave function was expanded in plane wave up to a cutoff energy of 500 eV. The k-point meshes [26] in the full edge of the Brillouin (BZ) are sampled by 12 12 12 and all atoms are fully relaxed until the Hellmann–Feynman forces become less than 0.02 eV/Å. The exchange and correlation energy was assessed by the GGA [27] and the LDA [28]. The strong on-site Coulomb repulsion among the localized U 5f electrons is described by the DFT + U method [29,30]. The DFT + U functional is adapted from the formalism developed by Dudarev et al. [31]. In this scheme, the total GGA (LDA) energy functional is of the form
EGGAðLDAÞþU ¼ EGGAðLDAÞ þ
U JX ½Tr qr Trðqr qr Þ; 2 r
where qr is the density matrix of f states with spin r, while U and J are the spherically averaged matrix elements of screened Coulomb energy and the exchange energy, respectively. Since only the difference between U and J is important, we label them as U for simplicity. In order to gain insight into the phase stability of a-UH3 under pressure, the phonon band structures and projected phonon density of states (DOS) have been calculated employing the Hellmann–Feynman theorem and the direct method [32] as implemented in the PHONOPY program [33,34]. For the phonon band structure calculation, we used the 2 2 1 supercell containing 32 atoms and the 3 3 7 Monkhorst–Pack k-point mesh for the Brillouin zone integration. The phonon calculations of a-UH3 have been performed within the GGA + U scheme with Hubbard U = 2 eV.
Fig. 1. The lattice constants of a-UH3 as a function of Hubbard U at ambient pressure. The horizontal line indicates the experimental lattice constant (a = 4.16 Å) from Ref. [1].
one can see that within the LDA + U approach, although the calculated lattice parameters still underestimate in a wide range of U, it improves better than the pure LDA by steadily increasing the amplitude of U. In fact, even for U = 5 eV, the calculated lattice parameter a = 4.142 Å is still underestimated. As for GGA scheme, the calculated lattice parameter is estimated for the pure GGA and small Hubbard U. At a typical value U = 2 eV, the GGA + U method gives a = 4.178 Å, which agrees well with the experimental data. Overall, lattice parameters using GGA + U method agree well with the corresponding experimental lattice parameter by considering the Hubbard U of 2 eV. In the remaining part, the mechanical and thermal properties have been calculated with GGA + U method at U = 2 eV. Fig. 2 shows volume per formula unit for a-UH3 as a function of pressure. The directions and movement of atoms under hydrostatic pressure greatly affects the compressibility and bulk modulus of the solids. The volume of a-UH3 decreases with pressure by 0.25 Å3/GPa. The pressure variation of bond distance is shown in inset of Fig. 2. It is clearly shown that the U–U, U–H, and H–H bond distances decrease as pressure increases. From 0 to 20 GPa, the U–U bond distance decreases about 0.17 Å, while the H–H distance decreases 0.1 Å. This indicates that the U–U bond is slightly sensitive to external pressure than H–H bond. The mechanical properties of a-UH3 under pressure have been investigated. The elastic constants have been estimated from the
3. Results and discussion The a-UH3 crystallize in a cubic structure with space group of Pm3n (No. 223). Its unit cell is composed of two UH3 formula units with the U atoms occupying on site 2a (0, 0, 0) and H on site 6c (0.25, 0, 0.5), respectively. The U atoms are coordinated by 12 H atoms in cuboctahedral configuration. To investigate the highpressure behavior of a-UH3, the ground-state properties of a-UH3 at ambient pressure have been calculated. The lattice constants of a-UH3 at ambient pressure as a function of Hubbard U using LDA and GGA methods are shown in Fig. 1. For the pure DFT calculation (U = 0), it is clearly shown that both LDA and GGA underestimate the lattice parameter, especially the value estimated by LDA is 3.8% smaller than the experimental data. It is indicated that the pure GGA formalism works better than pure LDA formalism for the lattice parameter of a-UH3. After turning on the Hubbard U term,
Fig. 2. Volume of a-UH3 as a function of pressure along with the effect of pressure on bond-distance in the inset.
C. Zhang et al. / Journal of Alloys and Compounds 604 (2014) 171–174 Table 1 Elastic constants C11, C12 and C44, bulk modulus B, shear modulus G, Young modulus Y, and Poisson’s ratio t for a-UH3 under pressure using the GGA + U method with Hubbard U = 2 eV. The elastic constant and moduli are in unit of GPa. Pressure (GPa)
C11
C12
C44
B
G
Y
t
0 5 10 15 20
217 240 260 274 297
37 42 45 46 61
65 70 66 60 60
97 108 117 122 140
74 80 80 78 79
177 193 196 193 200
0.196 0.202 0.220 0.236 0.262
strained structures under GGA + U scheme. For cubic structure, there are three independent elastic constants C11, C12, and C44. Under the Voigt approximation [35], the effective bulk modulus BV and shear modulus GV can be expressed for cubic phase by the expressions:
BV ¼ ðC 11 þ 2C 12 Þ=3 and
GV ¼ ðC 11 C 12 þ 3C 44 Þ=5; respectively. Under Reuss approximation [36], the Reuss bulk modulus BR and Reuss shear modulus GR are expressed as:
BR ¼ ðC 11 þ 2C 12 Þ=3 and
GR ¼ 5ðC 11 C 12 ÞC 44 =½4C 44 þ 3ðC 11 C 12 Þ; respectively. Based on Hill approximation [37], the bulk modulus B and shear modulus G are arithmetic averages of Voigt and Reuss elastic modulus, i.e., B = (BR + BV)/2 and G = (GR + GV)/2, from which the Poisson’s ratio t is given by:
t ¼ ð3B 2GÞ=ð6B þ 2GÞ: The calculated results of mechanical properties of a-UH3 under pressure up to 20 GPa using the GGA + U methods are summarized in Table 1. For the stable cubic structure, the elastic constants satisfy the following mechanical stability criteria: C11 > 0, C44 > 0, C11 C12 > 0, (C11 + 2C12) > 0. It is clearly shown from Table 1 that the elastic constants satisfy the above criteria at ambient and high pressures, and thus a-UH3 is mechanically stable at least up to 20 GPa. There are no experimental data of elastic constants for aUH3 to compare both at ambient and high pressure. However, our results of mechanical properties for a-UH3 at ambient pressure are in reasonable agreement with previous theoretical calculations [17]. Under external pressure, both C11 and C12 apparently increase monotonically with increasing pressure. This results in that bulk
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modulus B increases with pressure and reaches 140 GPa at pressure of 20 GPa. While C44, along with shear modulus G and Young modulus Y, do not show strong pressure dependence. Interestingly, the Poisson’s ratio t increases with pressure, from 0.196 at ambient pressure to 0.262 at 20 GPa. The calculated phonon band structures along some high-symmetry directions in the Brillouin zone and projected phonon DOS of a-UH3 under ambient pressure and 10 GPa are displayed in Fig. 3. Phonon calculations established the dynamical stability of a-UH3 under pressure of in view of the absence of imaginary frequencies. Additional phonon calculations establish the stability of a-UH3 in the pressure range from 0 to 20 GPa. Due to the fact that U atom is much heavier than H atom, the vibration frequency of U atom is apparently lower than that of H atom. Therefore, there is a gap between in the phonon band structures for both a-UH3 at 0 and 10 GPa, as shown in Fig. 3. The heavy U atoms dominate the low-frequency vibrations below 5 THz, while the light H atoms contribute significantly to the high-frequency modes above 25 THz. The phonon results in this work coincide with previous theoretical calculations of a-UH3 by using LDA + U formalism [17]. The experimental data of phonon vibrations for bulk UH3 has been very limited. Only the vibrational modes of UH3 molecule have been studied by infrared spectra [38,39]. It is indicated that the antisymmetric stretch vibration mode of UH3 molecule is about 40.36 THz, which is much larger than the maximum frequency (35 THz) of bulk a-UH3 in this work. Such discrepancies could be attributed to the different forms of UH3. It is noted that the vibrational frequency decreases as a result of polymerization [38]. With increasing pressure, the vibration of U atoms changes slightly, while the vibration frequency range of H atoms move upward. As shown in Fig. 3, the frequency range of H atoms between 25 and 35 THz at ambient pressure shifts to the range of 27.5 and 37.5 THz, and consequently, the gap increases from 20 THz at 0 GPa to 22 THz at 10 GPa. The knowledge of the entire phonon band structures and phonon DOS makes possible the evaluation of several critical thermodynamical quantities for a-UH3 under pressure. Temperature dependent thermodynamical function such as Helmholtz free energy, entropy, and heat capacity at constant volume have been calculated using quasiharmonic approximation [33]. The phonon contribution to the Helmholtz free energy F is given by
F¼
X 1X hxq;m þ kB T hxq;m ln½1 exp hxq;m =kB T 2 q;m q;m
where q and m are the wave vector and band index, respectively. xq,m is the phonon frequency at q and m, and T is the temperature. are the Boltzmann constant and the reduced Planck kB and h
Fig. 3. Phonon band structures and projected phonon density of states for a-UH3 at (a) 0 GPa and (b) 10 GPa using the GGA + U method with Hubbard U = 2 eV.
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4. Conclusion The high-pressure behavior of a-UH3 have been investigated with the GGA + U and LDA + U frameworks. The lattice parameters of a-UH3 at ambient pressure obtained by the GGA + U with effective Hubbard parameter U of 2 eV are agreement with experimental data, and thus Hubbard U of 2 eV was chosen for study of the mechanical and thermal properties. The elastic parameters show that a-UH3 is stable up to at least 20 GPa, which is consistent with phonon dispersion calculation. Using quasiharmonic approximation, it is found that the free energy decreases, while entropy and heat capacity increase with pressure increasing. The obtained pressure dependence of mechanical and thermal properties provides significant implications for application of uranium hydrides at extreme conditions. Acknowledgements This work was supported by the Natural Science Foundation of China (Grant Nos. 11247234, 11304269 and 11274335) and the Shenzhen Basic Research Grant (Nos. JC201105190880A and JC201105190912A). References
Fig. 4. The calculated Helmholtz free energy F (a), entropy S (b), and heat capacity at constant volume CV (c) as a function of temperature at 0 GPa and 10 GPa.
constant, respectively. The heat capacity CV and the entropy S at constant volume are given by
CV ¼
X h xq;m 2 expðhxq;m =kB TÞ kB 2 k T B ½expðhxq;m =kB TÞ 1 q;m
and
S ¼ kB
X q;m
ln½1 expðhxq;m =kB TÞ
hxq;m 1X ; T q;m expðhxq;m =kB TÞ 1
respectively. Fig. 4 shows the temperature variation of Helmholtz free energy, entropy, and lattice specific heat at constant volume of a-UH3 at 0 GPa and 10 GPa. The Helmholtz free energy varies linearly with pressure and decreases with temperature in the studied temperature range. The increase of the Helmholtz free energy at higher pressure reveals that the vibration frequency of atoms, especially H atoms, becomes larger. It is noteworthy that the values of the free energy at zero temperature do not vanish due to the zero point motion. In the temperature of 0–3000 K, the entropy of a-UH3 at ambient pressure is larger than that at 10 GPa. The decrease of entropy at higher pressure suggests the lengthening of the U–H and H–H bond distance. The heat capacity at constant volume of a-UH3 increases significantly above 200 K and approaches a constant value at high temperature for both 0 and 10 GPa. This behavior of temperature dependent CV is also found in lanthanum dihydride [40]. At intermediate temperatures, the temperature dependence of CV is governed by the details of vibration of the atoms. In the temperature range from 300 to 1500 K, the value of CV for a-UH3 at 0 GPa is smaller than that at 10 GPa, while at very high temperature, the CV at 0 GPa and 10 GPa tends to the same value, i.e., 200 J/mol/K.
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Contents lists available at ScienceDirect
Journal of Alloys and Compounds journal homepage: www.elsevier.com/locate/jalcom
Mechanical and thermodynamic properties of a-UH3 under pressure Chao Zhang a,⇑, Hong Jiang a, Hong-Liang Shi b, Guo-Hua Zhong c, Yue-Hua Su a a
Department of Physics, Yantai University, Yantai 264005, China Beijing Computational Science Research Center, Beijing 100084, China c Center for Photovoltaics and Solar Energy, Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences and The Chinese University of Hong Kong, Shenzhen 518055, China b
a r t i c l e
i n f o
Article history: Received 29 October 2013 Received in revised form 15 February 2014 Accepted 12 March 2014 Available online 24 March 2014 Keywords: Actinide alloys and compounds High pressure Mechanical properties Thermodynamic properties First-principles calculations Uranium trihydride
a b s t r a c t The mechanical and thermodynamic properties of a-UH3 under pressure have been examined by firstprinciples pseudopotential plane-wave calculations based on the density functional theory. In order to describe the strong on-site Coulomb repulsion among the localized 5f electrons, the generalized gradient approximation and local density approximation plus a Hubbard parameter (GGA + U and LDA + U) formalisms have been adopted for the exchange correlation term. The pressure evolution of elastic constants of a-UH3 has been revealed, and a-UH3 is found to be mechanically stable at least up to 20 GPa. Under pressure, the low-frequency vibration modes of a-UH3 change slightly, while the high-frequency vibration modes shift upward. Our study is expected to facilitate the understanding of uranium hydrides and their applications. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction Actinide elements and compounds exhibit interesting physical and chemical properties due to the complex nature of 5f electrons, and have been extensively studied experimentally [1–5] and theoretically [6–9]. Uranium is the heaviest element in nature with any practical abundance. Thus, it attracts much attention than the other actinides. Uranium metal can react with hydrogen to form uranium hydrides. In solid state, uranium hydride exists mainly in the form of cubic trihydride with two allotropes, i.e., a-UH3 and b-UH3, and a-UH3 could transforms into b-UH3 above room temperature [1,10]. The uranium hydrides have rich physical properties, and play an important role in nuclear fuels and hydrogen storage. Furthermore, the uranium hydrides have been taken as starting materials to reactive uranium power along with various uranium carbides, nitride, and halide. Therefore, understanding the physical, mechanical, and thermal properties of uranium hydrides is of highly desirable. First-principles band-structure calculations can provide valuable insight into physical, elastic, and thermal properties of materials. However, conventional density function theory (DFT), which applies the local density approximation (LDA) or the generalized gradient approximation (GGA), underestimates the strong on-site ⇑ Corresponding author. Tel.: +86 5356902506. E-mail address: [email protected] (C. Zhang). http://dx.doi.org/10.1016/j.jallcom.2014.03.071 0925-8388/Ó 2014 Elsevier B.V. All rights reserved.
Coulomb repulsion of 5f electrons in U atom and fails to describe the properties of uranium compounds. For example, such approaches predict a fluorite metallic ferromagnetic ground state for UO2, while the observed ground state is an antiferromagnetic insulator with a band gap of about 2.0 eV [11]. To overcome this drawback in calculations of actinide compounds, the effective modification of pure DFT which calls LDA/GGA + U scheme has been widely used. This method captures well the localization effect of f electrons, allowing successful description of actinide compounds, such as uranium oxides [12–14] and plutonium oxides [15,16]. Compared to actinide oxide, only a limited uses of DFT to actinide hydrides in the literature [17–20], especially using LDA/GGA + U methods. The electronic properties of a-UH3 at ambient pressure have been investigated within the LDA and LDA + U formalisms [17]. Including the Hubbard U term, the uranium 5f electronic states are represented by spectral envelope consisting of multiple peaks, which weakens the metallicity of a-UH3 and enhances the ionic character of uranium–hydrogen bonds. In addition, the phonon dispersion curves show different shapes as compared to those obtained by pure DFT calculations. Moreover, the intriguing 5f electronic states in UH3 draw the attention of the scientific community. The earlier nuclear magnetic resonance study suggests the 5f electrons in UH3 to be localized [10,21]. Band structure calculations come to the conclusion that there are two types of f character in these materials: localized and itinerant
172
C. Zhang et al. / Journal of Alloys and Compounds 604 (2014) 171–174
[22]. Recently, it has been suggested that the 5f electronic states in UH3 films show both localized and itinerant character [2]. The properties of solids as a function of pressure are of fundamental interest to a wide range of condensed matter physics. Under applied pressure, the crystal structure established for solids at ambient change dramatically, causing large effects on physical and chemical properties. Unfortunately, the knowledge on UH3 at high pressures, especially its mechanical and the thermal properties, is very limited [23], which hinders its practical application. This is partially due to the fact that high pressure is not easy to reach/access and control in diamond anvil cell. On the contrary, adjusting pressure in theoretical simulations can be accomplished straight forwardly by varying the size of unit cell. Development of reliable theoretical methods which simulate such properties would significantly help, especially in the field of the nuclear materials, for which adequate experimental data are missed. In this work, we have investigated the high-pressure behavior of a-UH3 up to 20 GPa by using the GGA + U and LDA + U schemes, with a special focus on the mechanical and thermal properties. The paper is organized as follows: the computational details are described in Section 2, while the results are presented and discussed in Section 3. After that, summary of the work is given. 2. Computational method The first-principles calculations were performed within the framework of DFT implemented in the Vienna Ab-initio Simulation Package (VASP) [24]. The all-electron projector augmented wave (PAW) [25] pseudopotential for U and H from the VASP pseudopotential library were used. For U 5f, 6s, 6p, 6d and 7s and for H 1s were treated as valence electrons. The electron wave function was expanded in plane wave up to a cutoff energy of 500 eV. The k-point meshes [26] in the full edge of the Brillouin (BZ) are sampled by 12 12 12 and all atoms are fully relaxed until the Hellmann–Feynman forces become less than 0.02 eV/Å. The exchange and correlation energy was assessed by the GGA [27] and the LDA [28]. The strong on-site Coulomb repulsion among the localized U 5f electrons is described by the DFT + U method [29,30]. The DFT + U functional is adapted from the formalism developed by Dudarev et al. [31]. In this scheme, the total GGA (LDA) energy functional is of the form
EGGAðLDAÞþU ¼ EGGAðLDAÞ þ
U JX ½Tr qr Trðqr qr Þ; 2 r
where qr is the density matrix of f states with spin r, while U and J are the spherically averaged matrix elements of screened Coulomb energy and the exchange energy, respectively. Since only the difference between U and J is important, we label them as U for simplicity. In order to gain insight into the phase stability of a-UH3 under pressure, the phonon band structures and projected phonon density of states (DOS) have been calculated employing the Hellmann–Feynman theorem and the direct method [32] as implemented in the PHONOPY program [33,34]. For the phonon band structure calculation, we used the 2 2 1 supercell containing 32 atoms and the 3 3 7 Monkhorst–Pack k-point mesh for the Brillouin zone integration. The phonon calculations of a-UH3 have been performed within the GGA + U scheme with Hubbard U = 2 eV.
Fig. 1. The lattice constants of a-UH3 as a function of Hubbard U at ambient pressure. The horizontal line indicates the experimental lattice constant (a = 4.16 Å) from Ref. [1].
one can see that within the LDA + U approach, although the calculated lattice parameters still underestimate in a wide range of U, it improves better than the pure LDA by steadily increasing the amplitude of U. In fact, even for U = 5 eV, the calculated lattice parameter a = 4.142 Å is still underestimated. As for GGA scheme, the calculated lattice parameter is estimated for the pure GGA and small Hubbard U. At a typical value U = 2 eV, the GGA + U method gives a = 4.178 Å, which agrees well with the experimental data. Overall, lattice parameters using GGA + U method agree well with the corresponding experimental lattice parameter by considering the Hubbard U of 2 eV. In the remaining part, the mechanical and thermal properties have been calculated with GGA + U method at U = 2 eV. Fig. 2 shows volume per formula unit for a-UH3 as a function of pressure. The directions and movement of atoms under hydrostatic pressure greatly affects the compressibility and bulk modulus of the solids. The volume of a-UH3 decreases with pressure by 0.25 Å3/GPa. The pressure variation of bond distance is shown in inset of Fig. 2. It is clearly shown that the U–U, U–H, and H–H bond distances decrease as pressure increases. From 0 to 20 GPa, the U–U bond distance decreases about 0.17 Å, while the H–H distance decreases 0.1 Å. This indicates that the U–U bond is slightly sensitive to external pressure than H–H bond. The mechanical properties of a-UH3 under pressure have been investigated. The elastic constants have been estimated from the
3. Results and discussion The a-UH3 crystallize in a cubic structure with space group of Pm3n (No. 223). Its unit cell is composed of two UH3 formula units with the U atoms occupying on site 2a (0, 0, 0) and H on site 6c (0.25, 0, 0.5), respectively. The U atoms are coordinated by 12 H atoms in cuboctahedral configuration. To investigate the highpressure behavior of a-UH3, the ground-state properties of a-UH3 at ambient pressure have been calculated. The lattice constants of a-UH3 at ambient pressure as a function of Hubbard U using LDA and GGA methods are shown in Fig. 1. For the pure DFT calculation (U = 0), it is clearly shown that both LDA and GGA underestimate the lattice parameter, especially the value estimated by LDA is 3.8% smaller than the experimental data. It is indicated that the pure GGA formalism works better than pure LDA formalism for the lattice parameter of a-UH3. After turning on the Hubbard U term,
Fig. 2. Volume of a-UH3 as a function of pressure along with the effect of pressure on bond-distance in the inset.
C. Zhang et al. / Journal of Alloys and Compounds 604 (2014) 171–174 Table 1 Elastic constants C11, C12 and C44, bulk modulus B, shear modulus G, Young modulus Y, and Poisson’s ratio t for a-UH3 under pressure using the GGA + U method with Hubbard U = 2 eV. The elastic constant and moduli are in unit of GPa. Pressure (GPa)
C11
C12
C44
B
G
Y
t
0 5 10 15 20
217 240 260 274 297
37 42 45 46 61
65 70 66 60 60
97 108 117 122 140
74 80 80 78 79
177 193 196 193 200
0.196 0.202 0.220 0.236 0.262
strained structures under GGA + U scheme. For cubic structure, there are three independent elastic constants C11, C12, and C44. Under the Voigt approximation [35], the effective bulk modulus BV and shear modulus GV can be expressed for cubic phase by the expressions:
BV ¼ ðC 11 þ 2C 12 Þ=3 and
GV ¼ ðC 11 C 12 þ 3C 44 Þ=5; respectively. Under Reuss approximation [36], the Reuss bulk modulus BR and Reuss shear modulus GR are expressed as:
BR ¼ ðC 11 þ 2C 12 Þ=3 and
GR ¼ 5ðC 11 C 12 ÞC 44 =½4C 44 þ 3ðC 11 C 12 Þ; respectively. Based on Hill approximation [37], the bulk modulus B and shear modulus G are arithmetic averages of Voigt and Reuss elastic modulus, i.e., B = (BR + BV)/2 and G = (GR + GV)/2, from which the Poisson’s ratio t is given by:
t ¼ ð3B 2GÞ=ð6B þ 2GÞ: The calculated results of mechanical properties of a-UH3 under pressure up to 20 GPa using the GGA + U methods are summarized in Table 1. For the stable cubic structure, the elastic constants satisfy the following mechanical stability criteria: C11 > 0, C44 > 0, C11 C12 > 0, (C11 + 2C12) > 0. It is clearly shown from Table 1 that the elastic constants satisfy the above criteria at ambient and high pressures, and thus a-UH3 is mechanically stable at least up to 20 GPa. There are no experimental data of elastic constants for aUH3 to compare both at ambient and high pressure. However, our results of mechanical properties for a-UH3 at ambient pressure are in reasonable agreement with previous theoretical calculations [17]. Under external pressure, both C11 and C12 apparently increase monotonically with increasing pressure. This results in that bulk
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modulus B increases with pressure and reaches 140 GPa at pressure of 20 GPa. While C44, along with shear modulus G and Young modulus Y, do not show strong pressure dependence. Interestingly, the Poisson’s ratio t increases with pressure, from 0.196 at ambient pressure to 0.262 at 20 GPa. The calculated phonon band structures along some high-symmetry directions in the Brillouin zone and projected phonon DOS of a-UH3 under ambient pressure and 10 GPa are displayed in Fig. 3. Phonon calculations established the dynamical stability of a-UH3 under pressure of in view of the absence of imaginary frequencies. Additional phonon calculations establish the stability of a-UH3 in the pressure range from 0 to 20 GPa. Due to the fact that U atom is much heavier than H atom, the vibration frequency of U atom is apparently lower than that of H atom. Therefore, there is a gap between in the phonon band structures for both a-UH3 at 0 and 10 GPa, as shown in Fig. 3. The heavy U atoms dominate the low-frequency vibrations below 5 THz, while the light H atoms contribute significantly to the high-frequency modes above 25 THz. The phonon results in this work coincide with previous theoretical calculations of a-UH3 by using LDA + U formalism [17]. The experimental data of phonon vibrations for bulk UH3 has been very limited. Only the vibrational modes of UH3 molecule have been studied by infrared spectra [38,39]. It is indicated that the antisymmetric stretch vibration mode of UH3 molecule is about 40.36 THz, which is much larger than the maximum frequency (35 THz) of bulk a-UH3 in this work. Such discrepancies could be attributed to the different forms of UH3. It is noted that the vibrational frequency decreases as a result of polymerization [38]. With increasing pressure, the vibration of U atoms changes slightly, while the vibration frequency range of H atoms move upward. As shown in Fig. 3, the frequency range of H atoms between 25 and 35 THz at ambient pressure shifts to the range of 27.5 and 37.5 THz, and consequently, the gap increases from 20 THz at 0 GPa to 22 THz at 10 GPa. The knowledge of the entire phonon band structures and phonon DOS makes possible the evaluation of several critical thermodynamical quantities for a-UH3 under pressure. Temperature dependent thermodynamical function such as Helmholtz free energy, entropy, and heat capacity at constant volume have been calculated using quasiharmonic approximation [33]. The phonon contribution to the Helmholtz free energy F is given by
F¼
X 1X hxq;m þ kB T hxq;m ln½1 exp hxq;m =kB T 2 q;m q;m
where q and m are the wave vector and band index, respectively. xq,m is the phonon frequency at q and m, and T is the temperature. are the Boltzmann constant and the reduced Planck kB and h
Fig. 3. Phonon band structures and projected phonon density of states for a-UH3 at (a) 0 GPa and (b) 10 GPa using the GGA + U method with Hubbard U = 2 eV.
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4. Conclusion The high-pressure behavior of a-UH3 have been investigated with the GGA + U and LDA + U frameworks. The lattice parameters of a-UH3 at ambient pressure obtained by the GGA + U with effective Hubbard parameter U of 2 eV are agreement with experimental data, and thus Hubbard U of 2 eV was chosen for study of the mechanical and thermal properties. The elastic parameters show that a-UH3 is stable up to at least 20 GPa, which is consistent with phonon dispersion calculation. Using quasiharmonic approximation, it is found that the free energy decreases, while entropy and heat capacity increase with pressure increasing. The obtained pressure dependence of mechanical and thermal properties provides significant implications for application of uranium hydrides at extreme conditions. Acknowledgements This work was supported by the Natural Science Foundation of China (Grant Nos. 11247234, 11304269 and 11274335) and the Shenzhen Basic Research Grant (Nos. JC201105190880A and JC201105190912A). References
Fig. 4. The calculated Helmholtz free energy F (a), entropy S (b), and heat capacity at constant volume CV (c) as a function of temperature at 0 GPa and 10 GPa.
constant, respectively. The heat capacity CV and the entropy S at constant volume are given by
CV ¼
X h xq;m 2 expðhxq;m =kB TÞ kB 2 k T B ½expðhxq;m =kB TÞ 1 q;m
and
S ¼ kB
X q;m
ln½1 expðhxq;m =kB TÞ
hxq;m 1X ; T q;m expðhxq;m =kB TÞ 1
respectively. Fig. 4 shows the temperature variation of Helmholtz free energy, entropy, and lattice specific heat at constant volume of a-UH3 at 0 GPa and 10 GPa. The Helmholtz free energy varies linearly with pressure and decreases with temperature in the studied temperature range. The increase of the Helmholtz free energy at higher pressure reveals that the vibration frequency of atoms, especially H atoms, becomes larger. It is noteworthy that the values of the free energy at zero temperature do not vanish due to the zero point motion. In the temperature of 0–3000 K, the entropy of a-UH3 at ambient pressure is larger than that at 10 GPa. The decrease of entropy at higher pressure suggests the lengthening of the U–H and H–H bond distance. The heat capacity at constant volume of a-UH3 increases significantly above 200 K and approaches a constant value at high temperature for both 0 and 10 GPa. This behavior of temperature dependent CV is also found in lanthanum dihydride [40]. At intermediate temperatures, the temperature dependence of CV is governed by the details of vibration of the atoms. In the temperature range from 300 to 1500 K, the value of CV for a-UH3 at 0 GPa is smaller than that at 10 GPa, while at very high temperature, the CV at 0 GPa and 10 GPa tends to the same value, i.e., 200 J/mol/K.
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