First-Principles Study of Thermo-Physical Properties of Pu-Containing Gd2Zr2O7

A density functional theory plus Hubbard U method is used to investigate how the incorporation of Pu waste into Gd2Zr2O7 pyrochlore influences its thermo-physical properties. It is found that immobilization of Pu at Gd-site of Gd2Zr2O7 has minor effects on the mechanical and thermal properties, whereas substitution of Pu for Zr-site results in remarkable influences on the structural parameters, elastic moduli, elastic isotropy, Debye temperature and electronic structure. The discrepancy in thermo-physical properties between Gd2−yPuyZr2O7 and Gd2Zr2−yPuyO7 may be a result of their different structural and electronic structures. This study provides a direct insight into the thermo-physical properties of Pu-containing Gd2Zr2O7, which will be important for further investigation of nuclear waste immobilization by pyrochlores.


Introduction
As the nuclear industry develops fast, ways to treat spent fuel and nuclear waste safely, such as plutonium and minor actinides (Np, Am, Cm), has become an important environmental conservation issue [1][2][3]. It is acceptable to store spent fuel and separated waste in stainless steel vessels in the short term, but in the long term it is hoped that this material will be transformed into more secure and manageable solids [1,4,5]. One method proposed for the treatment of plutonium is immobilization in zirconate pyrochlores, particularly Gd 2 Zr 2 O 7, which has high thermal stability, high chemical durability, and high radiation tolerance [6][7][8][9]. Besides, Gd is an effective neutron absorber [6].

Structural Stability of Pu Incorporation into Gd2Zr2O7
As Pu substitutes for Gd 3+ and Zr 4+ in Gd2Zr2O7, the corresponding valence states for Pu are Pu 3+ and Pu 4+ , respectively. Because in both PuO2 and Pu2O3 the Pu 5f electrons are strongly correlated, Hubbard U correction is thus necessary. In the revised manuscript, we present the density of state distribution for both PuO2 and Pu2O3 at Ueff = 0 eV and Ueff = 4 eV in Figure 2. It is shown that without Hubbard U correction, i.e., at Ueff = 0 eV, the Pu 5f electrons are itinerant and delocalized over the Fermi lever, resulting in metallic states. At Ueff = 4 eV, the Pu 5f electrons are localized and the system becomes insulating, which is consistent with the experimental finding [32]. The calculated lattice constant of 5.46 Å for PuO2 and 11.18 Å for Pu2O3 obtained at Ueff = 4 eV are comparable to the experimental values of 5.39 Å [33] and 10.98 Å [34], respectively. The calculated band gap for Pu2O3 at Ueff = 4 eV is 1.757 eV, which corresponds to the experimental value of 2 eV [35]. Thus, we use Ueff = 4 eV in our subsequent calculations for Pu immobilization in Gd2Zr2O7. On the other hand, the 4 eV for Ueff is also consistent with the value of 4-5 eV that are reported in the literature [36,37].

Structural Stability of Pu Incorporation into Gd 2 Zr 2 O 7
As Pu substitutes for Gd 3+ and Zr 4+ in Gd 2 Zr 2 O 7 , the corresponding valence states for Pu are Pu 3+ and Pu 4+ , respectively. Because in both PuO 2 and Pu 2 O 3 the Pu 5f electrons are strongly correlated, Hubbard U correction is thus necessary. In the revised manuscript, we present the density of state distribution for both PuO 2 and Pu 2 O 3 at U eff = 0 eV and U eff = 4 eV in Figure 2. It is shown that without Hubbard U correction, i.e., at U eff = 0 eV, the Pu 5f electrons are itinerant and delocalized over the Fermi lever, resulting in metallic states. At U eff = 4 eV, the Pu 5f electrons are localized and the system becomes insulating, which is consistent with the experimental finding [32]. The calculated lattice constant of 5.46 Å for PuO 2 and 11.18 Å for Pu 2 O 3 obtained at U eff = 4 eV are comparable to the experimental values of 5.39 Å [33] and 10.98 Å [34], respectively. The calculated band gap for Pu 2 O 3 at U eff = 4 eV is 1.757 eV, which corresponds to the experimental value of 2 eV [35]. Thus, we use Nanomaterials 2019, 9,196 4 of 13 U eff = 4 eV in our subsequent calculations for Pu immobilization in Gd 2 Zr 2 O 7 . On the other hand, the 4 eV for U eff is also consistent with the value of 4-5 eV that are reported in the literature [36,37]. A structural optimization is first performed for both Gd2−yPuyZr2O7 and Gd2Zr2−yPuyO7. The calculated lattice constants, oxygen positional parameter and bond distances for Gd2−yPuyZr2O7 and Gd2Zr2−yPuyO7 are listed in Table 1 and Table 2, respectively. The changes of lattice constant and for Gd2−yPuyZr2O7 and Gd2Zr2−yPuyO7 with Pu concentrations are shown in Figure  3. The calculated lattice constant of 10.666 Å for Gd2Zr2O7 is slightly larger than the experimental value of 10.54 Å [38], while consistent with other calculations of 10.66 Å [18]. The calculated a0 of 10.802 Å for Pu2Zr2O7 is comparable with the experimental value of 10.70 Å [39]. As the Pu content increases, the lattice constant gradually increases for both Gd2−yPuyZr2O7 and Gd2Zr2−yPuyO7, and it changes more significantly for Gd2Zr2−yPuyO7 than that for Gd2−yPuyZr2O7. This is caused by the fact that the effective ionic radius of 1.053 Å [40] for Gd 3 is in good agreement with the value of +~1 .1 Å [40] for Pu 3+ , but the effective ionic radius of 0.72 Å [40] for Zr 4+ is much smaller than the value of 0.96 Å [40] for Pu 4+ . With regard to oxygen positional parameter , the calculated value of 0.339 for Gd2Zr2O7 is smaller than the experimental value of 0.345 [41], and is comparable to the calculated value of 0.339 reported by Wang et al. [18]. For Gd2−yPuyZr2O7, the changes slightly as the Pu content increases, which indicates that the Gd2−yPuyZr2O7 remains the pyrochlore structure. Wang et al. [18] has observed similar phenomenon for Gd2−yCeyZr2O7. For Gd2Zr2−yPuyO7, we find that the increases a lot, varying from 0.339 to 0.350, suggesting that the Gd2Zr2−yPuyO7 tends to be a defect fluorite structure as the Pu content increases [15]. Comparing the lattice constant and oxygen positional parameter for Gd2−yPuyZr2O7 and Gd2Zr2−yPuyO7, we find that the formation of Gd2Zr2O7-Pu2Zr2O7 solid solution is more preferable than that of Gd2Zr2O7 and Gd2Pu2O7. As for bond distances, the calculated <Gd-O48f> distance of 2.553 Å in Gd2Zr2O7 is a little larger than the experimental value of 2.483 Å [38], and is comparable to other calculated value of 2.548 Å [18]. Meanwhile, the calculated value of 2.109 Å for <Zr-O48f> distance is consistent with the experimental value [38] and other calculated value [18] of 2.110 Å. For Gd2−yPuyZr2O7, the <Gd-O48f> and <Pu-O48f> distances increase slightly and the <Gd-O8b> and <Pu-O8b> distances decrease slightly as the Pu content increases. A structural optimization is first performed for both Gd 2−y Pu y Zr 2 O 7 and Gd 2 Zr 2−y Pu y O 7 . The calculated lattice constants, oxygen positional parameter x O48 f and bond distances for Gd 2−y Pu y Zr 2 O 7 and Gd 2 Zr 2−y Pu y O 7 are listed in Tables 1 and 2, respectively. The changes of lattice constant and x O48 f for Gd 2−y Pu y Zr 2 O 7 and Gd 2 Zr 2−y Pu y O 7 with Pu concentrations are shown in Figure 3. The calculated lattice constant of 10.666 Å for Gd 2 Zr 2 O 7 is slightly larger than the experimental value of 10.54 Å [38], while consistent with other calculations of 10.66 Å [18]. The calculated a 0 of 10.802 Å for Pu 2 Zr 2 O 7 is comparable with the experimental value of 10.70 Å [39]. As the Pu content increases, the lattice constant gradually increases for both Gd 2−y Pu y Zr 2 O 7 and Gd 2 Zr 2−y Pu y O 7 , and it changes more significantly for Gd 2 Zr 2−y Pu y O 7 than that for Gd 2−y Pu y Zr 2 O 7 . This is caused by the fact that the effective ionic radius of 1.053 Å [40] for Gd 3 is in good agreement with the value of +~1 .1 Å [40] for Pu 3+ , but the effective ionic radius of 0.72 Å [40] for Zr 4+ is much smaller than the value of 0.96 Å [40] for Pu 4+ . With regard to oxygen positional parameter x O48 f , the calculated value of 0.339 for Gd 2 Zr 2 O 7 is smaller than the experimental value of 0.345 [41], and is comparable to the calculated value of 0.339 reported by Wang et al. [18]. For Gd 2−y Pu y Zr 2 O 7 , the x O48 f changes slightly as the Pu content increases, which indicates that the Gd 2−y Pu y Zr 2 O 7 remains the pyrochlore structure. Wang et al. [18] has observed similar phenomenon for Gd 2−y Ce y Zr 2 O 7 . For Gd 2 Zr 2−y Pu y O 7 , we find that the x O48 f increases a lot, varying from 0.339 to 0.350, suggesting that the Gd 2 Zr 2−y Pu y O 7 tends to be a defect fluorite structure as the Pu content increases [15]. Comparing the lattice constant and oxygen positional parameter for Gd 2−y Pu y Zr 2 O 7 and Gd 2 Zr 2−y Pu y O 7 , we find that the formation of Gd 2 Zr 2 O 7 -Pu 2 Zr 2 O 7 solid solution is more preferable than that of Gd 2 Zr 2 O 7 and Gd 2 Pu 2 O 7 . As for bond distances, the calculated <Gd-O 48f > distance of 2.553 Å in Gd 2 Zr 2 O 7 is a little larger than the experimental value of 2.483 Å [38], Nanomaterials 2019, 9, 196 5 of 13 and is comparable to other calculated value of 2.548 Å [18]. Meanwhile, the calculated value of 2.109 Å for <Zr-O 48f > distance is consistent with the experimental value [38] and other calculated value [18] of 2.110 Å. For Gd 2−y Pu y Zr 2 O 7 , the <Gd-O 48f > and <Pu-O 48f > distances increase slightly and the <Gd-O 8b > and <Pu-O 8b > distances decrease slightly as the Pu content increases. Comparing the <Gd-O> and <Pu-O> bonds, we find that the <Pu-O 48f > and <Pu-O 8b > distances are slightly larger than <Gd-O 48f > and <Gd-O 8b > distances, respectively, i.e., Pu substitution for Gd-site leads to small increase in the bonding distance. For Gd 2 Zr 2−y Pu y O 7 , the situation is different. The <Pu-O 48f > bond is about 0.12-0.19 Å larger than the <Zr-O 48f > bond. Simultaneously, the <Gd-O 8b > bond increases a little as the Pu content increases. Consequently, there is a remarkable increase in the lattice constant of Gd 2−y Pu y Zr 2 O 7 .  Comparing the <Gd-O> and <Pu-O> bonds, we find that the <Pu-O48f> and <Pu-O8b> distances are slightly larger than <Gd-O48f> and <Gd-O8b> distances, respectively, i.e., Pu substitution for Gd-site leads to small increase in the bonding distance. For Gd2Zr2−yPuyO7, the situation is different. The <Pu-O48f> bond is about 0.12-0.19 Å larger than the <Zr-O48f> bond. Simultaneously, the <Gd-O8b> bond increases a little as the Pu content increases. Consequently, there is a remarkable increase in the lattice constant of Gd2−yPuyZr2O7.    Elastic constants are response functions to the external forces and are very important in the materials' properties [42]. Table 3 lists the calculated elastic constants along with available experimental and theoretical values. For Gd 2 Zr 2 O 7 , the calculated C 11 , C 12 and C 44 are 285.1, 102.5 and 82.1 GPa, respectively, showing good agreement with other calculations [43]. For Pu 2 Zr 2 O 7 , the calculated Nanomaterials 2019, 9,196 6 of 13 C 11 = 270.6 GPa, C 12 = 107.3 GPa and C 44 = 81.2 GPa differ from reference [2], in which different calculational parameters are employed. It is noted that the elastic stability criteria are satisfied for all the investigated systems, i.e., C 11 > |C 12 |, C 44 > 0, and (C 11 + 2C 12 ) > 0 [44], implying that they are all mechanically stable.  Figure 4 presents the changes of elastic constants with Pu content for both Gd 2−y Pu y Zr 2 O 7 and Gd 2 Zr 2−y Pu y O 7 . For Gd 2−y Pu y Zr 2 O 7 , as the Pu content increases, the elastic constants are affected minorly. As the Pu concentration increases, the C 11 and C 12 decreases and increases slightly, respectively, and the change in C 44 , is nearly negligible. As for Gd 2 Zr 2−y Pu y O 7 , the variation of elastic constants with Pu content is more considerable. As the y value changes, the C 11 and C 12 first decreases, then increases, and finally decreases again. As for C 44 , it first decreases to y = 1.5, and then increases. Generally speaking, as the Pu content increases, there are more significant changes on Zr-site than Gd-site, meaning that Pu immobilization at Zr-site leads to remarkable variations in the mechanical properties of Gd 2 Zr 2 O 7 . Zhao et al. [47] also reported that Nd substitution of Zr-site of Gd 2 Zr 2 O 7 greatly affects the mechanical properties. Elastic constants are response functions to the external forces and are very important in the materials' properties [42]. Table 3 lists the calculated elastic constants along with available experimental and theoretical values. For Gd2Zr2O7, the calculated C11, C12 and C44 are 285.1, 102.5 and 82.1 GPa, respectively, showing good agreement with other calculations [43]. For Pu2Zr2O7, the calculated C11 = 270.6 GPa, C12 = 107.3 GPa and C44 = 81.2 GPa differ from reference 2, in which different calculational parameters are employed. It is noted that the elastic stability criteria are satisfied for all the investigated systems, i.e., C11 > |C12|, C44 > 0, and (C11 + 2C12) > 0 [44], implying that they are all mechanically stable.   Figure 4 presents the changes of elastic constants with Pu content for both Gd2−yPuyZr2O7 and Gd2Zr2−yPuyO7. For Gd2−yPuyZr2O7, as the Pu content increases, the elastic constants are affected minorly. As the Pu concentration increases, the C11 and C12 decreases and increases slightly, respectively, and the change in C44, is nearly negligible. As for Gd2Zr2−yPuyO7, the variation of elastic constants with Pu content is more considerable. As the y value changes, the C11 and C12 first decreases, then increases, and finally decreases again. As for C44, it first decreases to y = 1.5, and then increases. Generally speaking, as the Pu content increases, there are more significant changes on Zr-site than Gd-site, meaning that Pu immobilization at Zr-site leads to remarkable variations in the mechanical properties of Gd2Zr2O7. Zhao et al. [47] also reported that Nd substitution of Zr-site of Gd2Zr2O7 From the calculated elastic constants, the elastic moduli, including the bulk modulus (B), shear modulus (G) and Young's modulus (E), can be deduced [48][49][50][51], i.e., 12 3 ,

Elastic Constants and Elastic Moduli of Gd2−yPuyZr2O7 and Gd2Zr2−yPuyO7
Here, the Voigt and Reuss evaluations for B and G are represented by V and R, respectively.  [20,45,46] and other calculated [43] results. Figure 5 shows the variation of elastic moduli for both Gd 2−y Pu y Zr 2 O 7 and Gd 2 Zr 2−y Pu y O 7 . For Gd 2−y Pu y Zr 2 O 7 , the elastic moduli change very slightly as the Pu content increases. When Pu is immobilized at Zr-site, there are remarkable changes. The bulk modulus first decreases, reaching a minimum of 134.3 GPa at y = 1.0, then rises up to 142.3 GPa at y = 1.5 and finally decreases to 136.9 GPa at y = 2.0. The shear modulus decreases to 58.9 GPa at y = 1.0 and increases slightly to 63.6 GPa at y = 2.0. The Young's modulus decreases sharply from 218.8 GPa to 154.2 GPa as the y varies from 0 to 1.0, but increases again to 165.7 GPa at y = 1.5, finally changing slightly. The <Zr-O> bonds determine the total stiffness of A 2 Zr 2 O 7 pyrochlore, because the corner-sharing ZrO 6 octahedra constitutes its backbone, and the A 3+ fills the interstices [19,52]. Therefore, the substitution of Zr 4+ by Pu 4+ causes the change of <Zr-O> bonds to <Pu-O> bonds and influences the Young's modulus, especially for these ionic bonds [19,52]. The Young's modulus E is described by E ∝ M a r 0 4 for ionic bonds, in which M a represents the Madelung constant and r o represents the interionic distance [19]. The <Zr-O> bonds in Gd 2 Zr 2 O 7 are affected little by Pu substitution for Gd-site, leading to slight effects on the Young's modulus. For Gd 2 Zr 2−y Pu y O 7 , the <Zr-O 48f > bond length of 2.11 A is smaller than the value of 2.26 A for <Pu-O 48f >, resulting in remarkable effects on the Young's modulus. The bulk modulus, shear modulus and Young's modulus for Nd doping of Gd 2 Zr 2 O 7 have been calculated by Zhao et al. [47], who also reported that Nd immobilization at Zr-site has more remarkable influences on the elastic moduli than that at Gd-site. From the calculated elastic constants, the elastic moduli, including the bulk modulus (B), shear modulus (G) and Young's modulus (E), can be deduced [48][49][50][51], i.e., Here, the Voigt and Reuss evaluations for B and G are represented by V and R, respectively. Table 3 lists the calculated B, G, E, and others' theoretical and experimental values for both Gd2−yPuyZr2O7 and Gd2Zr2−yPuyO7. For Gd2Zr2O7, the calculated B = 163.4 GPa, G = 85.7 GPa, E = 218.8 GPa are comparable with experimental [20,45,46] and other calculated [43] results. Figure 5 shows the variation of elastic moduli for both Gd2−yPuyZr2O7 and Gd2Zr2−yPuyO7. For Gd2−yPuyZr2O7, the elastic moduli change very slightly as the Pu content increases. When Pu is immobilized at Zr-site, there are remarkable changes. The bulk modulus first decreases, reaching a minimum of 134.3 GPa at y = 1.0, then rises up to 142.3 GPa at y = 1.5 and finally decreases to 136.9 GPa at y = 2.0. The shear modulus decreases to 58.9 GPa at y = 1.0 and increases slightly to 63.6 GPa at y = 2.0. The Young's modulus decreases sharply from 218.8 GPa to 154.2 GPa as the y varies from 0 to 1.0, but increases again to 165.7 GPa at y = 1.5, finally changing slightly. The <Zr-O> bonds determine the total stiffness of A2Zr2O7 pyrochlore, because the corner-sharing ZrO6 octahedra constitutes its backbone, and the A 3+ fills the interstices [19,52]. Therefore, the substitution of Zr 4+ by Pu 4+ causes the change of <Zr-O> bonds to <Pu-O> bonds and influences the Young's modulus, especially for these ionic bonds [19,52]. The Young's modulus E is described by E ∝ for ionic bonds, in which Ma represents the Madelung constant and ro represents the interionic distance [19]. The <Zr-O> bonds in Gd2Zr2O7 are affected little by Pu substitution for Gd-site, leading to slight effects on the Young's modulus. For Gd2Zr2−yPuyO7, the <Zr-O48f> bond length of 2.11 Å is smaller than the value of 2.26 Å for <Pu-O48f>, resulting in remarkable effects on the Young's modulus. The bulk modulus, shear modulus and Young's modulus for Nd doping of Gd2Zr2O7 have been calculated by Zhao et al. [47], who also reported that Nd immobilization at Zr-site has more remarkable influences on the elastic moduli than that at Gd-site.  For each ion in Gd 2−y Pu y Zr 2 O 7 and Gd 2 Zr 2−y Pu y O 7 , we analyze the Bader charge to explore the origin of the discrepancy in the elastic moduli between Gd 2−y Pu y Zr 2 O 7 and Gd 2 Zr 2−y Pu y O 7 . The Bader charge values are listed in Table 4. As Pu is immobilized at Gd-site and Zr-site, the average Bader charge for Pu are 2.10 |e| and 2.33 |e|, respectively, corresponding to the nominal +3 |e| and +4 |e| in oversimplified classical model [2]. For Gd 2−y Pu y Zr 2 O 7 , the Bader charge for Gd and Pu ions are similar to each other, i.e., 2.15 and 2.10|e|, respectively. Wang et al. [18] calculated the Bader charge of Gd 2−y Ce y Zr 2 O 7 and reported very similar results. Additionally, the bonding distance for <Gd-O 48f > and <Gd-O 8b > are determined to be 2.57 Å and 2.30 Å, which are comparable to the values of 2.60 Å for <Pu-O 48f > and 2.36 Å for <Pu-O 8b >, respectively. Obviously, the <Gd-O> and <Pu-O> bonding interaction are very similar to each other, which explains why the mechanical properties of Gd 2 Zr 2 O 7 are affected slightly by Pu immobilization at Gd-site. For Gd 2 Zr 2−y Pu y O 7 , the situation is much different. The average Bader charge are 2.33 |e| for Pu ions and 2.26 |e| for Zr ions. Considering that the <Pu-O 48f > distance of 2.26 A is larger than the <Zr-O 48f > distance of 2.11 A and the ionic radius of 0.96 A for Pu ions is larger than that of 0.72 A for Zr ions [40], it is suggested that the <Zr-O> bonds exhibit weaker ionicity than <Pu-O> bonds in Gd 2 Zr 2−y Pu y O 7 . Because of the brittleness of the ionic bonds, the immobilization of Pu at Zr sites will thus increase the ionicity and decrease the elastic moduli. Pugh's indicator ( B G ) is used to reflect the ductility of materials. If B G > 1.75, the material shows ductility; or, it is brittle [54]. Table 5 presents the calculated Pugh's indicators. For Gd 2 Zr 2 O 7 , our value of 1.907 is comparable with the experimental values of 1.913 [45,46], 1.891 [20] and other calculated value of 2.004 [53]. For both Gd 2−y Pu y Zr 2 O 7 and Gd 2 Zr 2−y Pu y O 7 , our calculations show that the Pugh's indicators are all larger than 1.75, implying that all the considered composites are ductile. Poisson's indicator (ν) can be employed to evaluate the relative ductility of materials. When ν is around 0.1, the material shows brittle covalent properties. When ν is bigger than 0.25, it exhibits ductile ionic properties [55]. Table 5 [45,46], 0.274 [20] and other calculated results of 0.286 [53] and 0.273 [43]. The Poisson's ratios for Gd 2−y Pu y Zr 2 O 7 and Gd 2 Zr 2−y Pu y O 7 are all larger than 0.25, as presented in Table 5, i.e., the Pu-substituted Gd 2 Zr 2 O 7 exhibit good ductility. Elastic anisotropy is an important parameter for phase transformations, dislocation dynamics and geophysical applications [56]. Ranganathan and co-workers [57,58] proposed the universal elastic anisotropic index to indicate the elastic anisotropy of cubic crystals. The index A U = 5 G V G R + B V B R − 6 is investigated for all the considered compositions., where A U = 0 describes an isotropic crystal [57,58]. The calculated A U for both Gd 2−y Pu y Zr 2 O 7 and Gd 2 Zr 2−y Pu y O 7 are shown in Table 5. For Gd 2 Zr 2 O 7 , our calculated value of 0.01355 is comparable with other calculated value of 0.00420 [53]. As Pu is immobilized at Gd-site, we find that the A U values for all the compositions are nearly zero, indicating that the Gd 2−y Pu y Zr 2 O 7 compounds are isotropic elastically. As for the immobilization of Pu at Zr-site, the A U values are 0.21353 for Gd 2 Zr 1.0 Pu 1.0 O 7 and 0.10062 for Gd 2 Zr 0.5 Pu 1.5 O 7 , indicative of elastic anisotropy.
The thermal properties of materials can be analyzed by the Debye temperature [4]. Table 5 lists the calculated and available experimental Debye temperature. The calculated θ D value of 580.2 K for Gd 2 Zr 2 O 7 is larger than the experimental result of 513.3 K [20], but shows better agreement with experimental value than another calculated value of 612.9 K [53]. As the Pu concentration increases, the θ D value decreases for both Gd 2−y Pu y Zr 2 O 7 and Gd 2 Zr 2−y Pu y O 7 . In particular, the θ D values for Gd 2 Zr 2−y Pu y O 7 are smaller than those for Gd 2−y Pu y Zr 2 O 7 . These results imply that the Gd 2 Zr 2−y Pu y O 7 compositions have a lower melting point and weaker interatomic binding force than the Gd 2-y Pu y Zr 2 O 7 compositions. Zhao et al. [59] calculated the Debye temperature for Th immobilization at Gd-site and Zr-site of Gd 2 Zr 2 O 7 and observed similar phenomena, i.e., the Th-substituted Gd 2 Zr 2 O 7 have a lower Debye temperature and especially smaller Debye temperature can be obtained by the Th immobilization at Zr-site than that at Gd-site.
The total and projected density of state (DOS) distributions for Gd 2−y Pu y Zr 2 O 7 and Gd 2 Zr 2−y Pu y O 7 are illustrated in Figures 6 and 7

Conclusion
The mechanical and electronic properties of Pu-containing Gd2Zr2O7 are studied by a DFT+U method. For Gd2−yPuyZr2O7 and Gd2Zr2−yPuyO7, the elastic stability criteria are satisfied for all the calculated elastic constants, i.e., all the compounds are mechanically stable. As Pu immobilizes at Gdsite in Gd2Zr2O7, because the bonding distance and covalency of <Gd-O> and <Pu-O> bonds are comparable to each other, the elastic constants, elastic moduli, elastic isotropy and Debye temperature of Gd2Zr2O7 are all affected a little. As for Gd2Zr2−yPuyO7, the elastic constants and elastic moduli change remarkably as compared with Gd2Zr2O7. The substitution of Pu for Zr sites increases the ionicity and decreases the elastic moduli, because the <Zr-O> bonds exhibit weaker ionicity than <Pu-O> bonds. In addition, the Debye temperature is decreased and the band gap is greatly reduced. Our calculations suggest that the Gd2Zr2O7 is a promising material for immobilizing nuclear waste such as Pu, while the thermo-physical of Gd2Zr2O7 may be influenced significantly after nuclear waste incorporation.

Conclusion
The mechanical and electronic properties of Pu-containing Gd2Zr2O7 are studied by a DFT+U method. For Gd2−yPuyZr2O7 and Gd2Zr2−yPuyO7, the elastic stability criteria are satisfied for all the calculated elastic constants, i.e., all the compounds are mechanically stable. As Pu immobilizes at Gdsite in Gd2Zr2O7, because the bonding distance and covalency of <Gd-O> and <Pu-O> bonds are comparable to each other, the elastic constants, elastic moduli, elastic isotropy and Debye temperature of Gd2Zr2O7 are all affected a little. As for Gd2Zr2−yPuyO7, the elastic constants and elastic moduli change remarkably as compared with Gd2Zr2O7. The substitution of Pu for Zr sites increases the ionicity and decreases the elastic moduli, because the <Zr-O> bonds exhibit weaker ionicity than <Pu-O> bonds. In addition, the Debye temperature is decreased and the band gap is greatly reduced. Our calculations suggest that the Gd2Zr2O7 is a promising material for immobilizing nuclear waste such as Pu, while the thermo-physical of Gd2Zr2O7 may be influenced significantly after nuclear waste incorporation.

Conclusions
The mechanical and electronic properties of Pu-containing Gd 2 Zr 2 O 7 are studied by a DFT+U method. For Gd 2−y Pu y Zr 2 O 7 and Gd 2 Zr 2−y Pu y O 7 , the elastic stability criteria are satisfied for all the calculated elastic constants, i.e., all the compounds are mechanically stable. As Pu immobilizes at Gd-site in Gd 2 Zr 2 O 7 , because the bonding distance and covalency of <Gd-O> and <Pu-O> bonds are comparable to each other, the elastic constants, elastic moduli, elastic isotropy and Debye temperature of Gd 2 Zr 2 O 7 are all affected a little. As for Gd 2 Zr 2−y Pu y O 7 , the elastic constants and elastic moduli change remarkably as compared with Gd 2 Zr 2 O 7 . The substitution of Pu for Zr sites increases the ionicity and decreases the elastic moduli, because the <Zr-O> bonds exhibit weaker ionicity than <Pu-O> bonds. In addition, the Debye temperature is decreased and the band gap is greatly reduced. Our calculations suggest that the Gd 2 Zr 2 O 7 is a promising material for immobilizing nuclear waste such as Pu, while the thermo-physical of Gd 2 Zr 2 O 7 may be influenced significantly after nuclear waste incorporation.