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Exploring the electronic structure and thermal properties of UAl3 using density functional theory calculations
Journal of Physics and Chemistry of Solids 136 (2020) 109179
Contents lists available at ScienceDirect
Journal of Physics and Chemistry of Solids journal homepage: http://www.elsevier.com/locate/jpcs
Exploring the electronic structure and thermal properties of UAl3 using density functional theory calculations Brindaban Modak a, K. Ghoshal b, K. Srinivasu a, **, Tapan K. Ghanty a, * a b
Theoretical Chemistry Section, Bhabha Atomic Research Centre, Homi Bhabha National Institute, Mumbai, 400085, India Nuclear Projects Safety Division, Atomic Energy regulatory Board, Mumbai, 400094, India
A R T I C L E I N F O
A B S T R A C T
Keywords: Uranium-aluminium intermetallics Electronic structure First principles calculations Quasi-harmonic approximation Thermal properties
Among the intermetallic compounds of uranium, U-Al alloy is found to be promising nuclear fuel for high-power research reactors due to its high thermal conductivity and structural stability. U-Al system can form three different intermetallic compounds namely, UAl2, UAl3 and UAl4. Here, we systematically explored the electronic structure and various thermal properties of the cubic UAl3 system. Pseudo-potential plane wave based DFT calculations are used along with the spin-orbit coupling. The structural parameter calculated using the gener alised gradient approximation is comparable to the experimental results. The phonon dispersion plot indicates the dynamic stability of the structure. Thermophysical properties including free energy, molar specific heat, coefficient of thermal expansion, bulk modulus, etc. have been evaluated within the framework of quasiharmonic approximation. Both the electronic and lattice contribution to thermal conductivity has been evalu ated through the Boltzmann transport theory.
1. Introduction Depleting fossil fuel resources and the adverse environmental pollution resulting from their combustion have been considered as two major challenges in near future due to the rapidly increasing world energy demands [1]. Currently, fossil fuels contribute to nearly 80% of the total global energy supply [2] and combustion of fossil fuels can lead to global warming and extreme climate change due to the greenhouse effect. Therefore, it is essential to find alternate strategies for energy production from environment friendly, sustainable and renewable re sources thereby minimizing the use of fossil fuels. Nuclear energy is considered as one of the promising means of carbon free energy [3–6] which requires an in-depth understanding about different materials and processes for its safe operation. Since the cost of fabrication, transportation and reprocessing of fuel are increasing rapidly, researchers are trying to find alternate method to maintain reactor fuel cost within limits [7,8]. Extending fuel lifetime without compromising reactor safety can reduce the fuel cost to signif icant extent. However, to achieve this, fuel burn-up performance has to be increased, which can be attained by using enriched uranium fuel. Recently, an emergence trend has been shown in developing fuel
materials with higher density of uranium. Although, uranium metal shows highest uranium density, it is not useful for fuel applications because of its poor irradiation stability. Interestingly, irradiation sta bility has been found to be significantly increased when uranium in termetallics are considered. Among different uranium intermetallic compound U-Al alloy is found to be very popular due to its high thermal conductivity and structural stability over a wide range of temperature [9–11]. Dispersion of uranium aluminide and aluminium as a fuel ma terial has been used in several high power test reactors, including Advanced Test Reactor, Experimental Test Reactor and Materials Test Reactor. At present research reactor mainly employs UAl3 fuel material in dispersion form in an inert matrix [12,13]. Depending on fabrication process U-Al alloy may contain a mixture of UAl2, UAl3, and UAl4 [14–16]. The UA12 is found to be highly reactive and converted to UAl3 or UAl4 by reacting with excess aluminium. Therefore, UAl3 and UA14 are found to be the major components of fuel plate. Interestingly, it has been reported that the principal crystalline constituent in the U-Al alloy is UAl3. A large number of studies have been reported in last few years to explore the performance of UAl3 as a fuel in nuclear reactors [17–24]. Kassner et al. [25] studied thermodynamic properties of
* Corresponding author. ** Corresponding author. E-mail addresses: [email protected] (K. Srinivasu), [email protected] (T.K. Ghanty). https://doi.org/10.1016/j.jpcs.2019.109179 Received 4 July 2019; Received in revised form 27 August 2019; Accepted 3 September 2019 Available online 4 September 2019 0022-3697/© 2019 Elsevier Ltd. All rights reserved.
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uranium-aluminium systems. Pascuet et al. [26] developed interatomic Al-U potential to study diffusion properties of defects as well as species in the U-Al intermetallics. Apart from these experimental reports, there are some theoretical studies based on empirical and ab initio calcula tions on electronic structure of UAl3. As for example, Cornelius et al. have studied the electronic properties of UAl3 under high magnetic field [27]. Tan et al. [28] employed DFT with full-potential linear muffin-tin orbitals calculation to investigate the electronic properties of 5f states of UAl3. Castleman et al. [29] investigated the interdiffusion in uranium-aluminium alloy system over a range of temperature and pressure. Nazar�e et al. [30] investigated different properties of UAlx-Al dispersion for thermal high-flux reactor. Pearce et al. [31] studied the behaviour of the thermal expansion of UAl3. Nagarajan et al. [32] investigated enthalpy of formation of UAl3 and UAl4 using high tem perature solution calorimetry. In process of revision of our study, we also found a very recent study by Mei et al. [33] where they have studied different thermal properties of U-Al compounds using first principles calculations. To the best of our knowledge, theoretical studies related to thermal properties of UAl3 are very rare. Therefore, in order to fully exploit UAl3 as a fuel for high power research reactor, it is highly desirable to investigate the electronic structure and thermal properties of UAl3. In the current study, we have systematically investigated the electronic structure and thermophysical properties of UAl3 employing the first principles calculation along with the quasi-harmonic approxi mation. Both the Electronic and lattice contributions to the thermal conductivity have been calculated through the Boltzmann transport theory. Properties calculated in our present study can give valuable in sights in designing UAl3 based nuclear fuels.
3. Results and discussion 3.1. Electronic structure UAl3 is known to crystallize in AuCu3 type cubic structure with Pm3m space group [44]. We considered the cubic structure of UAl3 as reported by Cornelius et al. [27] and optimized at PBE level of theory along with the spin-orbit coupling and the corresponding structure is reported in Fig. 1(a). The calculated cell parameter (4.260 Å) is found to be very close to the experimentally reported values [45] and also consistent with the computational studies by Nourbakhsh at similar level of theory [23]. The shortest U-U, U-Al and Al-Al distances are found to be 4.259 Å, 3.011 Å, and 3.011 Å respectively. To verify the effect of the localization of 5f electrons of Uranium, we have also studied the electronic structure using the PBE þ U method. The Ueff value has been scanned from 1.0 to 6.0 and the corresponding results are reported in Table 1. It is found that for all the considered values of Ueff, the cell parameters are overestimated as compared to the experimental results. The optimized cell parameter at Ueff ¼ 5.5 is consistent with the reported results by Nourbakhsh [23] with same Ueff value. Though the DFT þ U method is shown to be advantageous in modelling the uranium com pounds like UO2 and UN [46–48], conventional DFT is found to be sufficient for modelling U and its alloys like U-Zr and U-Al [33,49]. Our calculated results also show that in the case of UAl3, the results from conventional DFT are closer to the experimental results as compared to that from DFT þ U. To further understand the nature of electronic structure, we calculated the electronic band structure along with the electronic density of states and the corresponding plots are reported in Fig. 1(b) and (c). The reported PDOS indicate that the states near the Fermi level are majorly contributed by uranium 5f states indicating that they will play major role in the transport properties of UAl3. It is interesting to note that there exists a significant overlap between the U 6d states and Al 3p states below the U 5f states indicating a strong hy bridization between the Al 3p and U 6d states. In order to analyse the detailed bonding nature between U and Al, in UAl3, we carried out crystal orbital Hamilton population (COHP) calculation for the U-Al bonds in UAl3 using the LOBSTER code [50–53]. In the COHP method, different chemical bonds of a solid can be identified through the parti tioning of the band energies into pairs of orbital interactions. The pos itive overlap population (-pCOHP> 0) indicates the bonding contribution, while negative population (-pCOHP< 0) stands for anti-bonding contribution. Qualitative information of the bond strength can be achieved by taking the integral of -COHP up to the Fermi energy. More the negative value of ICOHP, stronger the chemical bond. As can be seen from Fig. 1(d), that UAl3 is characterised by massively bonding U-Al interaction in the highest occupied band near to the Fermi energy level.
2. Computational details UAl3 system has been modelled through the first principles density functional theory (DFT) calculations as implemented in the Vienna ab initio simulation package (VASP) [34]. To treat the exchange-correlation energy density functional, Perdew, Burke and Ernzerhof (PBE) functional under the framework of generalised gradient approximation (GGA) has been chosen [35]. The ion-electron in teractions have been treated with projector augmented wave (PAW) potentials [36]. Throughout the calculations, spin-orbit interactions are considered for treating the relativistic effects. PAW potentials were built by considering the 6s26p66d25f27s2 and 3s23p1, as the valence electronic configuration for U and Al, respectively. A plane-wave energy cutoff of 450 eV has been used for all the calculations. Unit cell of UAl3 has been optimized with a force cutoff of 0.001 eV Å 1. Sampling of Brillouin zone has been carried out using automatically generated 9 � 9 x 9 k-point mesh using the Monkhorst-Pack method [37]. Density functional perturbation theory (DFPT) has been used to calculate the phonon properties. 2 � 2 x 2 super cells of UAl3 consisting of 32 atoms are considered for calculation of phonon properties. To estimate the thermal properties within the framework of quasi-harmonic approximation (QHA), we measured the phonon properties at multiple volumes close to the equilibrium volume that can be used to measure the thermal prop erties as a function of temperature and volume [38,39]. Phonon dispersion plots and other thermal properties were evaluated from the calculated force constants using the Phonopy package [40]. We have used the VESTA package to generate the reported Figure of UAl3 struc tures [41]. Boltzmann transport theory as implemented in the BoltzTraP code has been used to measure the Electronic thermal conductivity [42]. To calculate lattice thermal conductivity, we have employed the line arized phonon Boltzman transport equation under the relaxation time approximation (RTA) using the PHONO3PY package [43].
3.2. Thermophysical properties To calculate the properties like free energy, molar specific heat, entropy, coefficient of thermal expansion, and bulk modulus as a func tion of temperature, we adopted the quasiharmonic approximation (QHA). Through the QHA, anharmonic effects are treated in an indirect way of calculating the volume dependent phonon modes around the equilibrium volume which has been reported to give reliable results up to certain temperatures of around 1/2 to 2/3 of the crystal melting points [54]. For this study, we optimized the UAl3 unit cell at 15 different volumes around the equilibrium volume using the constant volume optimization technique. Phonon frequencies of all the 15 opti mized structures were calculated using the DFPT method for a 2 � 2 x 2 super cell. The phonon dispersion plot of the equilibrium geometry along the high symmetric path of the Brillouin zone has been shown in Fig. 2(a). Our calculated phonon dispersion matches well with the pre viously reported results and none of the phonon modes is having 2
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Journal of Physics and Chemistry of Solids 136 (2020) 109179
Fig. 1. (a) Optimized unit cell structure and (b) electronic band structure (c) Total and projected density of states and (d) -pCOHP plots of UAl3 calculated using PBE method.
relation
Table 1 Calculated cell parameter of UAl3 using PBE and PBE þ U methods at different values of U. U (eV)
Cell parameter (A)
PBE U ¼ 1.0 U ¼ 1.5 U ¼ 2.0 U ¼ 2.5 U ¼ 3.0 U ¼ 3.5 U ¼ 4.0 U ¼ 4.5 U ¼ 5.0 U ¼ 5.5 U ¼ 6.0
4.260 4.287 4.296 4.319 4.327 4.338 4.339 4.344 4.349 4.354 4.358 4.362
FðTVÞ ¼ EðVÞ þ
� � �� ℏω! kB T X q ν ln 2sinh N! 2kB T q
(1)
With N! and ω! and representing the total number of wave q q
vectors considered in the first Brillouin zone and the frequency of the νth normal mode at the wave vector ! q respectively. The obtained F(T V) and volumes were fitted to the Birch-Murnaghan equation of state as can be seen in Fig. 3. The equilibrium volume at different temperatures is indicated and can be observed from Fig. 3. The calculated bulk modulus of UAl3 is found to be 87.11 GPa which is consistent with the compu tationally reported value of 86 GPa by Kang et al. [21]. Thermal expansion coefficient has been calculated using the relation � � 1 ∂V α¼ (2) V ∂T p
imaginary frequency indicating the dynamic stability of the considered structure. We have also plotted the site projected phonon density of states in Fig. 2(b) to understand the individual atom contribution to the phonon modes. The PDOS plot indicates that the low frequency acoustic modes are contributed by the U sites whereas the high frequency optical modes are mainly contributed by the Al atoms. Using the phonon spectra results calculated at 15 different cell vol umes close to the equilibrium volume, the Helmholtz free energy as a function of volume and temperature have been calculated using the
The specific heat at constant volume has been calculated as ! 1 X d 1 � � CV ðTVÞ ¼ ℏω! ðVÞ q ν VN! dT exp ℏω! kT 1 q ! q ν q;ν
(3)
and the Cp value is estimated as Cp ðTÞ ¼ CV ðTVÞ þ TBT α2T
3
(4)
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Journal of Physics and Chemistry of Solids 136 (2020) 109179
Fig. 3. Free energy -volume plots in the temperature range of 0–900 K.
Fig. 2. (a) Phonon dispersion plot of UAl3 calculated using PBE method with the Brillouin zone in the inset along with (b) site projected phonon density of states.
The calculated linear thermal expansion coefficient has been re ported in Fig. 4(a) and our results are consistent with the existing experimental and computational results [30,33]. The molar specific heat at constant pressure (Cp) calculated from the QHA using eqn. (5) as well as the electronic contribution incorporated results are reported in Fig. 4 (b). From the plots, it can be seen that the specific heat calculated form QHA alone is getting saturated to the Dulong-Petit value. However, electronic contribution to the specific heat can be important at high temperature because of the metallic nature of UAl3 that can be observed from the electronic DOS. Electronic contri bution to specific heat has been incorporated through the following linear relation with temperature Celp ¼ γT;
(5)
where, γ indicates the coefficient of electronic specific heat, can be calculated from the electronic density of states at the Fermi level using the relation below � 2� π γ¼ (6) :k2B :nðεF Þ; 3 where kB and nðεF Þ; represent the Boltzmann’s constant and the total DOS at the Fermi level. The nðεF Þ value from the total DOS calculated using the PBE method is found to be 5.9 states/eV and the corresponding coefficient of electronic specific heat is calculated as 13.91 mJ/mol-K2.
Fig. 4. Calculated (a) linear thermal expansion coefficient and (b) molar spe cific heat of UAl3 with and without the electronic contributions along with experimental results. 4
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Journal of Physics and Chemistry of Solids 136 (2020) 109179
Thus, we have calculated the specific heat incorporating both lattice and electronic contribution, and plotted in Fig. 4(b). It has been indicated that the specific heat calculated using both lattice and electronic contribution is much closer to the experimentally reported value [55]. Therefore, it can be concluded that the contribution of the electronic part to the molar specific heat becomes significant at high temperature.
calculate the electronic transport properties like electrical and thermal conductivities as a function of chemical potential and temperature from the band energies obtained. Since these transport properties are calcu lated using the constant relaxation time (τ) and rigid band approxima tions, they are not absolute and are function of the relaxation time, τ. To estimate the τ value from our calculated σ/τ results, resistivity data has been taken from the previously reported results. However, we found three different resistivity data reported for UAl3 and are found to be quite different from each other. We calculate the τ value at 280 K and Fermi energy using the resistivity from all three different data. The relaxation time is measured to be 2.9 � 10 15 s, 6.63 � 10 15 s and 15.7 � 10 15 s from the resistivity data reported by Buschow et al. [58], Aoki eta al [53]. and Cornelius et al. [27], respectively. The calculated thermal conductivities using three different τ values are plotted against the temperature as reported in Fig. 5(a). Using the electrical and thermal conductivities calculated from Boltzmann transport theory, we also measured the Lorentz number, L(T) using the Wiedemann–Franz law, which states that the ratio of elec tronic thermal conductivity (κ) and electrical conductivity (σ) of a metal is directly proportional to temperature (T):
3.3. Thermal conductivity One of the major factors affecting the operating temperature of the fuel is the thermal conductivity of the fuel material. The heat transfer rate has been found to be dependent on the temperature gradient and the material thermal conductivity. Thus, material with low thermal conductivity shows lower rate of heat transfer than that of the materials with high thermal conductivity [56,57]. Therefore, calculation of ther mal conductivity in the present study will provide valuable information related to the heat transfer property of UAl3. Thermal conductivity of the nuclear fuel is a key property for safe operation and it mainly consists of electronic and lattice contributions. UAl3 being metallic in nature, electronic contribution is expected to be dominant rather than the phonon contribution to thermal conductivity. Here, we calculated both the components of thermal conductivity using the Boltzmann transport theory. A non self-consistent electronic structure calculation using the optimized structure of UAl3 at a 9 � 9 x 9 k-point mesh has been carried out using a denser k-point mesh of 25 � 25 x 25 with 2613 irreducible and 15625 total k-points. Boltzmann transport theory has been used to
κ
σ
(7)
¼ LT;
Our calculated Lorentz number is found to be around 2.8 � 10 K which is close to the theoretical value of L (2.45 � 10 8 W Ω K metals measured by treating electrons as classical gas. 2
8 2
WΩ ) for
Fig. 5. (a) Electronic thermal conductivity measured from Boltzmann transport theory and (b) Phonon lifetime (c) Lattice thermal conductivity and (d) Lattice and total thermal conductivity with electronic component computed from the resistivity data from Akoi et al. 5
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Journal of Physics and Chemistry of Solids 136 (2020) 109179
For calculating the lattice thermal conductivity, second and third order force constants were calculated using a finite displacement method with a displacement of 0.03 Å. 4 � 4 x 4 and 2 � 2 x 2 supercell of cubic UAl3 with 256 and 32 atoms were considered for calculating the second and third order force constants respectively. For calculating the third order force constants, we considered 135 supercell structures without any real-space cut-off distance approximation and force con stant calculations were carried out. A k-point mesh of 21 � 21 � 21 has been employed to calculate the lattice thermal conductivity. The calculated phonon lifetime is plotted in Fig. 5 (b). The plot shows that the low frequency acoustic phonons have long lifetime as compared to the high frequency optical phonons indicating that the former with long lifetimes can contribute more to the thermal conductivity. The calcu lated lattice thermal conductivity (KL) is reported in Fig. 5 (c) and it clearly shows that KL decreases as the temperature increases which can be expected due to the better phonon dispersion at high temperature. Fig. 5 (d) shows the lattice and overall thermal conductivities plotted as a function of temperature. This clearly indicates that the lattice contri bution is important at very low temperatures and become negligible at higher temperatures when compared to the electronic contribution.
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4. Conclusions We have systematically studied the geometry, electronic structure and various thermal properties of UAl3 through the first principles cal culations along with the quasi harmonic approximation. Present study reveals that the calculated lattice parameter from PBE method is found to be close to the experimentally reported value whereas the PBE þ U method is shown to overestimate. The calculated electronic density of states indicate that UAl3 is metallic in nature and states near the Fermi energy are predominantly coming from U 5f states. This is further sup ported by the COHP calculation. Various thermal properties like, free energy, molar specific heat, bulk modulus, thermal expansion coeffi cient have been computed and compared with the available experi mental results. Present study indicates that the contribution of electronic component to the molar specific heat becomes significant at high tem perature. Both the electronic and phonon components of thermal con ductivity have been measured using the Boltzmann transport theory, which gives important insight in the heat transfer phenomena of UAl3. Thermal conductivity results indicate that the phonon contribution is important at very low temperatures and becomes insignificant as the temperature increases. Acknowledgment We like to thank the BARC computer division for providing the highperformance computational facility. We sincerely thank Dr. S. Kapoor for his valuable support and encouragement. The work has been sup ported by Department of Atomic Energy, India under project XII-N-R&D02.04/Theoretical & Computational Chemistry of Complex Systems. References [1] [2] [3] [4]
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Contents lists available at ScienceDirect
Journal of Physics and Chemistry of Solids journal homepage: http://www.elsevier.com/locate/jpcs
Exploring the electronic structure and thermal properties of UAl3 using density functional theory calculations Brindaban Modak a, K. Ghoshal b, K. Srinivasu a, **, Tapan K. Ghanty a, * a b
Theoretical Chemistry Section, Bhabha Atomic Research Centre, Homi Bhabha National Institute, Mumbai, 400085, India Nuclear Projects Safety Division, Atomic Energy regulatory Board, Mumbai, 400094, India
A R T I C L E I N F O
A B S T R A C T
Keywords: Uranium-aluminium intermetallics Electronic structure First principles calculations Quasi-harmonic approximation Thermal properties
Among the intermetallic compounds of uranium, U-Al alloy is found to be promising nuclear fuel for high-power research reactors due to its high thermal conductivity and structural stability. U-Al system can form three different intermetallic compounds namely, UAl2, UAl3 and UAl4. Here, we systematically explored the electronic structure and various thermal properties of the cubic UAl3 system. Pseudo-potential plane wave based DFT calculations are used along with the spin-orbit coupling. The structural parameter calculated using the gener alised gradient approximation is comparable to the experimental results. The phonon dispersion plot indicates the dynamic stability of the structure. Thermophysical properties including free energy, molar specific heat, coefficient of thermal expansion, bulk modulus, etc. have been evaluated within the framework of quasiharmonic approximation. Both the electronic and lattice contribution to thermal conductivity has been evalu ated through the Boltzmann transport theory.
1. Introduction Depleting fossil fuel resources and the adverse environmental pollution resulting from their combustion have been considered as two major challenges in near future due to the rapidly increasing world energy demands [1]. Currently, fossil fuels contribute to nearly 80% of the total global energy supply [2] and combustion of fossil fuels can lead to global warming and extreme climate change due to the greenhouse effect. Therefore, it is essential to find alternate strategies for energy production from environment friendly, sustainable and renewable re sources thereby minimizing the use of fossil fuels. Nuclear energy is considered as one of the promising means of carbon free energy [3–6] which requires an in-depth understanding about different materials and processes for its safe operation. Since the cost of fabrication, transportation and reprocessing of fuel are increasing rapidly, researchers are trying to find alternate method to maintain reactor fuel cost within limits [7,8]. Extending fuel lifetime without compromising reactor safety can reduce the fuel cost to signif icant extent. However, to achieve this, fuel burn-up performance has to be increased, which can be attained by using enriched uranium fuel. Recently, an emergence trend has been shown in developing fuel
materials with higher density of uranium. Although, uranium metal shows highest uranium density, it is not useful for fuel applications because of its poor irradiation stability. Interestingly, irradiation sta bility has been found to be significantly increased when uranium in termetallics are considered. Among different uranium intermetallic compound U-Al alloy is found to be very popular due to its high thermal conductivity and structural stability over a wide range of temperature [9–11]. Dispersion of uranium aluminide and aluminium as a fuel ma terial has been used in several high power test reactors, including Advanced Test Reactor, Experimental Test Reactor and Materials Test Reactor. At present research reactor mainly employs UAl3 fuel material in dispersion form in an inert matrix [12,13]. Depending on fabrication process U-Al alloy may contain a mixture of UAl2, UAl3, and UAl4 [14–16]. The UA12 is found to be highly reactive and converted to UAl3 or UAl4 by reacting with excess aluminium. Therefore, UAl3 and UA14 are found to be the major components of fuel plate. Interestingly, it has been reported that the principal crystalline constituent in the U-Al alloy is UAl3. A large number of studies have been reported in last few years to explore the performance of UAl3 as a fuel in nuclear reactors [17–24]. Kassner et al. [25] studied thermodynamic properties of
* Corresponding author. ** Corresponding author. E-mail addresses: [email protected] (K. Srinivasu), [email protected] (T.K. Ghanty). https://doi.org/10.1016/j.jpcs.2019.109179 Received 4 July 2019; Received in revised form 27 August 2019; Accepted 3 September 2019 Available online 4 September 2019 0022-3697/© 2019 Elsevier Ltd. All rights reserved.
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uranium-aluminium systems. Pascuet et al. [26] developed interatomic Al-U potential to study diffusion properties of defects as well as species in the U-Al intermetallics. Apart from these experimental reports, there are some theoretical studies based on empirical and ab initio calcula tions on electronic structure of UAl3. As for example, Cornelius et al. have studied the electronic properties of UAl3 under high magnetic field [27]. Tan et al. [28] employed DFT with full-potential linear muffin-tin orbitals calculation to investigate the electronic properties of 5f states of UAl3. Castleman et al. [29] investigated the interdiffusion in uranium-aluminium alloy system over a range of temperature and pressure. Nazar�e et al. [30] investigated different properties of UAlx-Al dispersion for thermal high-flux reactor. Pearce et al. [31] studied the behaviour of the thermal expansion of UAl3. Nagarajan et al. [32] investigated enthalpy of formation of UAl3 and UAl4 using high tem perature solution calorimetry. In process of revision of our study, we also found a very recent study by Mei et al. [33] where they have studied different thermal properties of U-Al compounds using first principles calculations. To the best of our knowledge, theoretical studies related to thermal properties of UAl3 are very rare. Therefore, in order to fully exploit UAl3 as a fuel for high power research reactor, it is highly desirable to investigate the electronic structure and thermal properties of UAl3. In the current study, we have systematically investigated the electronic structure and thermophysical properties of UAl3 employing the first principles calculation along with the quasi-harmonic approxi mation. Both the Electronic and lattice contributions to the thermal conductivity have been calculated through the Boltzmann transport theory. Properties calculated in our present study can give valuable in sights in designing UAl3 based nuclear fuels.
3. Results and discussion 3.1. Electronic structure UAl3 is known to crystallize in AuCu3 type cubic structure with Pm3m space group [44]. We considered the cubic structure of UAl3 as reported by Cornelius et al. [27] and optimized at PBE level of theory along with the spin-orbit coupling and the corresponding structure is reported in Fig. 1(a). The calculated cell parameter (4.260 Å) is found to be very close to the experimentally reported values [45] and also consistent with the computational studies by Nourbakhsh at similar level of theory [23]. The shortest U-U, U-Al and Al-Al distances are found to be 4.259 Å, 3.011 Å, and 3.011 Å respectively. To verify the effect of the localization of 5f electrons of Uranium, we have also studied the electronic structure using the PBE þ U method. The Ueff value has been scanned from 1.0 to 6.0 and the corresponding results are reported in Table 1. It is found that for all the considered values of Ueff, the cell parameters are overestimated as compared to the experimental results. The optimized cell parameter at Ueff ¼ 5.5 is consistent with the reported results by Nourbakhsh [23] with same Ueff value. Though the DFT þ U method is shown to be advantageous in modelling the uranium com pounds like UO2 and UN [46–48], conventional DFT is found to be sufficient for modelling U and its alloys like U-Zr and U-Al [33,49]. Our calculated results also show that in the case of UAl3, the results from conventional DFT are closer to the experimental results as compared to that from DFT þ U. To further understand the nature of electronic structure, we calculated the electronic band structure along with the electronic density of states and the corresponding plots are reported in Fig. 1(b) and (c). The reported PDOS indicate that the states near the Fermi level are majorly contributed by uranium 5f states indicating that they will play major role in the transport properties of UAl3. It is interesting to note that there exists a significant overlap between the U 6d states and Al 3p states below the U 5f states indicating a strong hy bridization between the Al 3p and U 6d states. In order to analyse the detailed bonding nature between U and Al, in UAl3, we carried out crystal orbital Hamilton population (COHP) calculation for the U-Al bonds in UAl3 using the LOBSTER code [50–53]. In the COHP method, different chemical bonds of a solid can be identified through the parti tioning of the band energies into pairs of orbital interactions. The pos itive overlap population (-pCOHP> 0) indicates the bonding contribution, while negative population (-pCOHP< 0) stands for anti-bonding contribution. Qualitative information of the bond strength can be achieved by taking the integral of -COHP up to the Fermi energy. More the negative value of ICOHP, stronger the chemical bond. As can be seen from Fig. 1(d), that UAl3 is characterised by massively bonding U-Al interaction in the highest occupied band near to the Fermi energy level.
2. Computational details UAl3 system has been modelled through the first principles density functional theory (DFT) calculations as implemented in the Vienna ab initio simulation package (VASP) [34]. To treat the exchange-correlation energy density functional, Perdew, Burke and Ernzerhof (PBE) functional under the framework of generalised gradient approximation (GGA) has been chosen [35]. The ion-electron in teractions have been treated with projector augmented wave (PAW) potentials [36]. Throughout the calculations, spin-orbit interactions are considered for treating the relativistic effects. PAW potentials were built by considering the 6s26p66d25f27s2 and 3s23p1, as the valence electronic configuration for U and Al, respectively. A plane-wave energy cutoff of 450 eV has been used for all the calculations. Unit cell of UAl3 has been optimized with a force cutoff of 0.001 eV Å 1. Sampling of Brillouin zone has been carried out using automatically generated 9 � 9 x 9 k-point mesh using the Monkhorst-Pack method [37]. Density functional perturbation theory (DFPT) has been used to calculate the phonon properties. 2 � 2 x 2 super cells of UAl3 consisting of 32 atoms are considered for calculation of phonon properties. To estimate the thermal properties within the framework of quasi-harmonic approximation (QHA), we measured the phonon properties at multiple volumes close to the equilibrium volume that can be used to measure the thermal prop erties as a function of temperature and volume [38,39]. Phonon dispersion plots and other thermal properties were evaluated from the calculated force constants using the Phonopy package [40]. We have used the VESTA package to generate the reported Figure of UAl3 struc tures [41]. Boltzmann transport theory as implemented in the BoltzTraP code has been used to measure the Electronic thermal conductivity [42]. To calculate lattice thermal conductivity, we have employed the line arized phonon Boltzman transport equation under the relaxation time approximation (RTA) using the PHONO3PY package [43].
3.2. Thermophysical properties To calculate the properties like free energy, molar specific heat, entropy, coefficient of thermal expansion, and bulk modulus as a func tion of temperature, we adopted the quasiharmonic approximation (QHA). Through the QHA, anharmonic effects are treated in an indirect way of calculating the volume dependent phonon modes around the equilibrium volume which has been reported to give reliable results up to certain temperatures of around 1/2 to 2/3 of the crystal melting points [54]. For this study, we optimized the UAl3 unit cell at 15 different volumes around the equilibrium volume using the constant volume optimization technique. Phonon frequencies of all the 15 opti mized structures were calculated using the DFPT method for a 2 � 2 x 2 super cell. The phonon dispersion plot of the equilibrium geometry along the high symmetric path of the Brillouin zone has been shown in Fig. 2(a). Our calculated phonon dispersion matches well with the pre viously reported results and none of the phonon modes is having 2
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Journal of Physics and Chemistry of Solids 136 (2020) 109179
Fig. 1. (a) Optimized unit cell structure and (b) electronic band structure (c) Total and projected density of states and (d) -pCOHP plots of UAl3 calculated using PBE method.
relation
Table 1 Calculated cell parameter of UAl3 using PBE and PBE þ U methods at different values of U. U (eV)
Cell parameter (A)
PBE U ¼ 1.0 U ¼ 1.5 U ¼ 2.0 U ¼ 2.5 U ¼ 3.0 U ¼ 3.5 U ¼ 4.0 U ¼ 4.5 U ¼ 5.0 U ¼ 5.5 U ¼ 6.0
4.260 4.287 4.296 4.319 4.327 4.338 4.339 4.344 4.349 4.354 4.358 4.362
FðTVÞ ¼ EðVÞ þ
� � �� ℏω! kB T X q ν ln 2sinh N! 2kB T q
(1)
With N! and ω! and representing the total number of wave q q
vectors considered in the first Brillouin zone and the frequency of the νth normal mode at the wave vector ! q respectively. The obtained F(T V) and volumes were fitted to the Birch-Murnaghan equation of state as can be seen in Fig. 3. The equilibrium volume at different temperatures is indicated and can be observed from Fig. 3. The calculated bulk modulus of UAl3 is found to be 87.11 GPa which is consistent with the compu tationally reported value of 86 GPa by Kang et al. [21]. Thermal expansion coefficient has been calculated using the relation � � 1 ∂V α¼ (2) V ∂T p
imaginary frequency indicating the dynamic stability of the considered structure. We have also plotted the site projected phonon density of states in Fig. 2(b) to understand the individual atom contribution to the phonon modes. The PDOS plot indicates that the low frequency acoustic modes are contributed by the U sites whereas the high frequency optical modes are mainly contributed by the Al atoms. Using the phonon spectra results calculated at 15 different cell vol umes close to the equilibrium volume, the Helmholtz free energy as a function of volume and temperature have been calculated using the
The specific heat at constant volume has been calculated as ! 1 X d 1 � � CV ðTVÞ ¼ ℏω! ðVÞ q ν VN! dT exp ℏω! kT 1 q ! q ν q;ν
(3)
and the Cp value is estimated as Cp ðTÞ ¼ CV ðTVÞ þ TBT α2T
3
(4)
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Journal of Physics and Chemistry of Solids 136 (2020) 109179
Fig. 3. Free energy -volume plots in the temperature range of 0–900 K.
Fig. 2. (a) Phonon dispersion plot of UAl3 calculated using PBE method with the Brillouin zone in the inset along with (b) site projected phonon density of states.
The calculated linear thermal expansion coefficient has been re ported in Fig. 4(a) and our results are consistent with the existing experimental and computational results [30,33]. The molar specific heat at constant pressure (Cp) calculated from the QHA using eqn. (5) as well as the electronic contribution incorporated results are reported in Fig. 4 (b). From the plots, it can be seen that the specific heat calculated form QHA alone is getting saturated to the Dulong-Petit value. However, electronic contribution to the specific heat can be important at high temperature because of the metallic nature of UAl3 that can be observed from the electronic DOS. Electronic contri bution to specific heat has been incorporated through the following linear relation with temperature Celp ¼ γT;
(5)
where, γ indicates the coefficient of electronic specific heat, can be calculated from the electronic density of states at the Fermi level using the relation below � 2� π γ¼ (6) :k2B :nðεF Þ; 3 where kB and nðεF Þ; represent the Boltzmann’s constant and the total DOS at the Fermi level. The nðεF Þ value from the total DOS calculated using the PBE method is found to be 5.9 states/eV and the corresponding coefficient of electronic specific heat is calculated as 13.91 mJ/mol-K2.
Fig. 4. Calculated (a) linear thermal expansion coefficient and (b) molar spe cific heat of UAl3 with and without the electronic contributions along with experimental results. 4
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Journal of Physics and Chemistry of Solids 136 (2020) 109179
Thus, we have calculated the specific heat incorporating both lattice and electronic contribution, and plotted in Fig. 4(b). It has been indicated that the specific heat calculated using both lattice and electronic contribution is much closer to the experimentally reported value [55]. Therefore, it can be concluded that the contribution of the electronic part to the molar specific heat becomes significant at high temperature.
calculate the electronic transport properties like electrical and thermal conductivities as a function of chemical potential and temperature from the band energies obtained. Since these transport properties are calcu lated using the constant relaxation time (τ) and rigid band approxima tions, they are not absolute and are function of the relaxation time, τ. To estimate the τ value from our calculated σ/τ results, resistivity data has been taken from the previously reported results. However, we found three different resistivity data reported for UAl3 and are found to be quite different from each other. We calculate the τ value at 280 K and Fermi energy using the resistivity from all three different data. The relaxation time is measured to be 2.9 � 10 15 s, 6.63 � 10 15 s and 15.7 � 10 15 s from the resistivity data reported by Buschow et al. [58], Aoki eta al [53]. and Cornelius et al. [27], respectively. The calculated thermal conductivities using three different τ values are plotted against the temperature as reported in Fig. 5(a). Using the electrical and thermal conductivities calculated from Boltzmann transport theory, we also measured the Lorentz number, L(T) using the Wiedemann–Franz law, which states that the ratio of elec tronic thermal conductivity (κ) and electrical conductivity (σ) of a metal is directly proportional to temperature (T):
3.3. Thermal conductivity One of the major factors affecting the operating temperature of the fuel is the thermal conductivity of the fuel material. The heat transfer rate has been found to be dependent on the temperature gradient and the material thermal conductivity. Thus, material with low thermal conductivity shows lower rate of heat transfer than that of the materials with high thermal conductivity [56,57]. Therefore, calculation of ther mal conductivity in the present study will provide valuable information related to the heat transfer property of UAl3. Thermal conductivity of the nuclear fuel is a key property for safe operation and it mainly consists of electronic and lattice contributions. UAl3 being metallic in nature, electronic contribution is expected to be dominant rather than the phonon contribution to thermal conductivity. Here, we calculated both the components of thermal conductivity using the Boltzmann transport theory. A non self-consistent electronic structure calculation using the optimized structure of UAl3 at a 9 � 9 x 9 k-point mesh has been carried out using a denser k-point mesh of 25 � 25 x 25 with 2613 irreducible and 15625 total k-points. Boltzmann transport theory has been used to
κ
σ
(7)
¼ LT;
Our calculated Lorentz number is found to be around 2.8 � 10 K which is close to the theoretical value of L (2.45 � 10 8 W Ω K metals measured by treating electrons as classical gas. 2
8 2
WΩ ) for
Fig. 5. (a) Electronic thermal conductivity measured from Boltzmann transport theory and (b) Phonon lifetime (c) Lattice thermal conductivity and (d) Lattice and total thermal conductivity with electronic component computed from the resistivity data from Akoi et al. 5
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For calculating the lattice thermal conductivity, second and third order force constants were calculated using a finite displacement method with a displacement of 0.03 Å. 4 � 4 x 4 and 2 � 2 x 2 supercell of cubic UAl3 with 256 and 32 atoms were considered for calculating the second and third order force constants respectively. For calculating the third order force constants, we considered 135 supercell structures without any real-space cut-off distance approximation and force con stant calculations were carried out. A k-point mesh of 21 � 21 � 21 has been employed to calculate the lattice thermal conductivity. The calculated phonon lifetime is plotted in Fig. 5 (b). The plot shows that the low frequency acoustic phonons have long lifetime as compared to the high frequency optical phonons indicating that the former with long lifetimes can contribute more to the thermal conductivity. The calcu lated lattice thermal conductivity (KL) is reported in Fig. 5 (c) and it clearly shows that KL decreases as the temperature increases which can be expected due to the better phonon dispersion at high temperature. Fig. 5 (d) shows the lattice and overall thermal conductivities plotted as a function of temperature. This clearly indicates that the lattice contri bution is important at very low temperatures and become negligible at higher temperatures when compared to the electronic contribution.
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4. Conclusions We have systematically studied the geometry, electronic structure and various thermal properties of UAl3 through the first principles cal culations along with the quasi harmonic approximation. Present study reveals that the calculated lattice parameter from PBE method is found to be close to the experimentally reported value whereas the PBE þ U method is shown to overestimate. The calculated electronic density of states indicate that UAl3 is metallic in nature and states near the Fermi energy are predominantly coming from U 5f states. This is further sup ported by the COHP calculation. Various thermal properties like, free energy, molar specific heat, bulk modulus, thermal expansion coeffi cient have been computed and compared with the available experi mental results. Present study indicates that the contribution of electronic component to the molar specific heat becomes significant at high tem perature. Both the electronic and phonon components of thermal con ductivity have been measured using the Boltzmann transport theory, which gives important insight in the heat transfer phenomena of UAl3. Thermal conductivity results indicate that the phonon contribution is important at very low temperatures and becomes insignificant as the temperature increases. Acknowledgment We like to thank the BARC computer division for providing the highperformance computational facility. We sincerely thank Dr. S. Kapoor for his valuable support and encouragement. The work has been sup ported by Department of Atomic Energy, India under project XII-N-R&D02.04/Theoretical & Computational Chemistry of Complex Systems. References [1] [2] [3] [4]
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