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

Abstract

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 Gd_2−yPu_yZr₂O₇ and Gd₂Zr_2−yPu_yO₇ 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.

1. 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 Gd₂Zr₂O_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].

Experimentally, Pu is often substituted by nonradioactive cerium (Ce), since they share the same crystallographic structure, and thermo-physical and chemical properties [10,11]. Zhao et al. synthesized (Gd_1−xCe_x)₂Zr₂O_7+x (0$\le \mathrm{x}\le 0.6$) solid solutions, indicating that Ce³⁺ ions can be incorporated into the Gd³⁺ sites. They proposed that the content of ${\mathrm{Pu}}^{3+}$ immobilized at Gd-site was less than 40 mol% [12]. On the other hand, Gd₂(Zr_1−xCe_x)₂O₇ (0$\le \mathrm{x}\le 1.0$) solid solutions have been synthesized by Reid et al. [13] and Patwe et al. [14]. They found that Ce can be immobilized at Zr sites entirely as Ce⁴⁺, which leads to structural transformation from pyrochlore to fluorite phase, and its composition ranges from Gd₂Ce_0.2Zr_1.8O₇ to Gd₂Ce_1.7Zr_0.3O₇. Similar solution behavior and structural properties for Pu incorporation in Gd₂Zr₂O₇ have been obtained by first-principles calculations [15,16,17]. However, different electronic structures can be obtained for Ce and Pu immobilization at the Gd-site of Gd₂Zr₂O₇, i.e., the band gap increases and reduces when Ce and Pu substitutes for Gd site, respectively [15,18]. These differences mainly result from the different <Ce-O> and <Pu-O> interactions at band edges. The different electronic structure may result in varying mechanical properties. For Young’s modulus, which is described by $\mathrm{E}\propto \frac{M_a}{r_0{}^4},$ with $M_a$ the Madelung constant and r₀ the interionic spacing [19], it is very sensitive to r₀. For ionic crystals, the r₀ is affected by bond interactions. According to these analyses, large discrepancies in electronic structures between Ce and Pu incorporation in Gd₂Zr₂O₇ may lead to different Young’s modulus. This indicates that for the mechanical properties of pyrochlores, Ce may not be a good substitute for Pu, despite the two having several similar thermo-physical properties. Thus far, there are no reports on the mechanical properties of Pu immobilization in Gd₂Zr₂O₇. It is necessary to explore how Pu doping influences the mechanical properties of Gd₂Zr₂O₇, because the knowledge of thermo-mechanical characteristics, for example, elastic moduli and Debye temperature is important for safe fuel disposal [20]. It provides new perspectives into the behavior of actinide incorporation in pyrochlores for their applications in harsh environments.

In this work, the structural, mechanical and electronic properties of Pu incorporation in Gd₂Zr₂O₇ are investigated by the density functional theory plus the Hubbard U method (DFT+U). The remaining part of the paper is structured as follows: Section 2 lists computational details; Section 3 contains our results and discussions, involving the structural stability, elastic constants with elastic moduli of Gd_2−yPu_yZr₂O₇ and Gd₂Zr_2−yPu_yO₇, as well as the ductility, elastic anisotropy, Debye temperature and electronic structures of Pu-doped Gd₂Zr₂O₇. In Section 4, we summarize our conclusions.

2. Computational Details

The density functional theory method within the Vienna Ab-initio Simulation Package (VASP) [21,22] are employed in all the computations. The Hubbard U correction [23] is considered to take into account the strong correlation interaction between 5f electrons of Pu and the effective U values are taken to be 4 eV. The projector augment wave (PAW) method [24] is used to describe the interaction between electrons and ions. As for the exchange-correlation functional, a number of generalized gradient approximation (GGA) functionals have been reported in the literature [25,26,27] and the functional parametrized by Perdew, Burke and Ernzerhof [28] is employed in this work. In the calculations, a 2 × 2 × 2 k-point sampling in reciprocal space is employed, with a cutoff energy of 600 eV for the plane wave basis sets. Figure 1 illustrates the schematic view of geometrical structure for the considered compounds, i.e., Gd_2−yPu_yZr₂O₇ and Gd₂Zr_2−yPu_yO₇ (y = 0, 0.5, 1.0, 1.5, 2.0). The special quasi-random structure method is used to build the structural models for Pu immobilization at Gd-site and Zr-site [29,30,31,32].

3. Results and Discussion

3.1. Structural Stability of Pu Incorporation into Gd₂Zr₂O₇

As Pu substitutes for Gd³⁺ and Zr⁴⁺ in Gd₂Zr₂O₇, the corresponding valence states for Pu are Pu³⁺ and Pu⁴⁺, 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 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 PuO₂ and 11.18 Å for Pu₂O₃ 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 Pu₂O₃ at U_eff = 4 eV is 1.757 eV, which corresponds to the experimental value of 2 eV [35]. Thus, we use U_eff = 4 eV in our subsequent calculations for Pu immobilization in Gd₂Zr₂O₇. 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 Gd_2−yPu_yZr₂O₇ and Gd₂Zr_2−yPu_yO₇. The calculated lattice constants, oxygen positional parameter $x_{O48f}$ and bond distances for Gd_2−yPu_yZr₂O₇ and Gd₂Zr_2−yPu_yO₇ are listed in

Table 1. Lattice constant a₀ (Å), O48f positional parameter x and bond distances (Å) for Gd_2−yPu_yZr₂O₇. Eg represents the band gap.

	a0	xO48f	Eg (eV)	d<Gd-O48f>	d<Gd-O8b>	d<Pu-O48f>	d<Pu-O8b>	d<Zr-O48f>
y = 0	10.666	0.339	2.86	2.553	2.309	-	-	2.109
Exp.	10.540 [38]	0.345 [41]		2.483 [38]
Cal.	10.66 [18]	0.339 [18]		2.548 [18]	2.307 [18]			2.110 [18]
y = 0.5	10.703	0.337	2.33	2.561	2.302	2.582	2.369	2.111
y = 1.0	10.736	0.337	2.27	2.578	2.284	2.591	2.366	2.114
y = 1.5	10.768	0.336	2.33	2.574	2.286	2.605	2.349	2.116
y = 2.0	10.802	0.335	2.37	-	-	2.615	2.339	2.117
Exp.	10.70 [39]

and

Table 2. Lattice constant a₀ (Å), O_48f positional parameter x and bond distances (Å) for Gd₂Zr_2−yPu_yO₇. E_g represents the band gap.

	a0	xO48f	Eg (eV)	d<Gd-O48f>	d<Gd-O8b>	d<Pu-O48f>	d<Zr-O48f>
y = 0	10.666	0.339	2.86	2.553	2.309	-	2.109
Exp.	10.540 [38]	0.345 [41]		2.483 [38]
Cal.	10.66 [18]	0.339 [18]		2.548 [18]	2.307 [18]		2.110 [18]
y = 0.5	10.750	0.342	2.33	2.513	2.338	2.296	2.110
y = 1.0	10.836	0.344	1.99	2.555	2.347	2.228	2.114
y = 1.5	10.909	0.350	1.68	2.508	2.364	2.273	2.121
y = 2.0	11.003	0.350	1.75	2.552	2.382	2.234	-

, respectively. The changes of lattice constant and $x_{O48f}$ for Gd_2−yPu_yZr₂O₇ and Gd₂Zr_2−yPu_yO₇ 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 Gd_2−yPu_yZr₂O₇ and Gd₂Zr_2−yPu_yO₇, and it changes more significantly for Gd₂Zr_2−yPu_yO₇ than that for Gd_2−yPu_yZr₂O₇. This is caused by the fact that the effective ionic radius of 1.053 Å [40] for Gd³ is in good agreement with the value of ⁺~1.1 Å [40] for Pu³⁺, but the effective ionic radius of 0.72 Å [40] for Zr⁴⁺ is much smaller than the value of 0.96 Å [40] for Pu⁴⁺. With regard to oxygen positional parameter $x_{O48f}$, 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 Gd_2−yPu_yZr₂O₇, the $x_{O48f}$ changes slightly as the Pu content increases, which indicates that the Gd_2−yPu_yZr₂O₇ remains the pyrochlore structure. Wang et al. [18] has observed similar phenomenon for Gd_2−yCe_yZr₂O₇. For Gd₂Zr_2−yPu_yO₇, we find that the $x_{O48f}$ increases a lot, varying from 0.339 to 0.350, suggesting that the Gd₂Zr_2−yPu_yO₇ tends to be a defect fluorite structure as the Pu content increases [15]. Comparing the lattice constant and oxygen positional parameter for Gd_2−yPu_yZr₂O₇ and Gd₂Zr_2−yPu_yO₇, 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-O_48f> 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-O_48f> distance is consistent with the experimental value [38] and other calculated value [18] of 2.110 Å. For Gd_2−yPu_yZr₂O₇, 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 Gd₂Zr_2−yPu_yO₇, 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−yPu_yZr₂O₇.

3.2. Elastic Constants and Elastic Moduli of Gd_2−yPu_yZr₂O₇ and Gd₂Zr_2−yPu_yO₇

Elastic constants are response functions to the external forces and are very important in the materials’ properties [42].

Table 3. Elastic constants (C₁₁, C₁₂, C₄₄, in GPa), bulk modulus B (GPa), shear modulus G (GPa), Young’s modulus E (GPa) of Gd_2−yPu_yZr₂O₇ and Gd₂Zr_2−yPu_yO₇ (0≤ y ≤2).

		C11	C12	C44	B	G	E
Gd2Zr2O7		285.1	102.5	82.1	163.4	85.7	218.8
	Cal [43]	289	103	85	165	88	224
	Exp [45,46]				153	80	205
	Exp [20]				174	92	236
Gd1.5Pu0.5Zr2O7		282.6	105.1	82.7	164.3	85.1	217.6
Gd1.0Pu1.0Zr2O7		278.1	106.9	83.1	164.0	84.1	215.4
Gd0.5Pu1.5Zr2O7		274.9	107.2	82.5	163.1	83.0	213.0
Pu2Zr2O7		270.6	107.3	81.2	161.7	81.4	209.1
	Cal [2]	306	131.8	90.2
Gd2Zr1.5Pu0.5O7		251.9	86.3	67.7	141.5	73.4	187.7
Gd2Zr1.0Pu1.0O7		235.2	83.8	49.8	134.3	58.9	154.2
Gd2Zr0.5Pu1.5O7		242.8	92.0	56.5	142.3	63.4	165.7
Gd2Pu2O7		234.8	87.9	57.8	136.9	63.6	165.3

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 Gd_2−yPu_yZr₂O₇ and Gd₂Zr_2−yPu_yO₇. For Gd_2−yPu_yZr₂O₇, 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 Gd₂Zr_2−yPu_yO₇, 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 greatly affects the mechanical properties.

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.,

$$\begin{gathered} B=B^V=B^R=\frac{C_{11}+2C_{12}}{3}, \\ {G}^V=\frac{C_{11}-C_{12}+3C_{44}}{5}, \\ {G}^R=\frac{5\left(C_{11}-C_{12}\right)C_{44}}{4C_{44}+3\left(C_{11}-C_{12}\right)}, \\ G=G^{VRH}=\frac{G^V+G^R}{2}, \\ E=\frac{9BG}{3B+G}. \end{gathered}$$

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 Gd_2−yPu_yZr₂O₇ and Gd₂Zr_2−yPu_yO₇. 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 Gd_2−yPu_yZr₂O₇ and Gd₂Zr_2−yPu_yO₇. For Gd_2−yPu_yZr₂O₇, 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 A³⁺ fills the interstices [19,52]. Therefore, the substitution of Zr⁴⁺ by Pu⁴⁺ 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 $\mathrm{E}\propto \frac{M_a}{r_0{}^4}$ for ionic bonds, in which M_a represents the Madelung constant and r_o represents the interionic distance [19]. The <Zr-O> bonds in Gd₂Zr₂O₇ are affected little by Pu substitution for Gd-site, leading to slight effects on the Young’s modulus. For Gd₂Zr_2−yPu_yO₇, the <Zr-O_48f> bond length of 2.11 $\AA$ is smaller than the value of 2.26 $\AA$ 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 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−yPu_yZr₂O₇ and Gd₂Zr_2−yPu_yO₇, we analyze the Bader charge to explore the origin of the discrepancy in the elastic moduli between Gd_2−yPu_yZr₂O₇ and Gd₂Zr_2−yPu_yO₇. The Bader charge values are listed in

Table 4. Bader charge (|e|) for each ion in Gd_2−yPu_yZr₂O₇ and Gd₂Zr_2−yPu_yO₇ (y = 0, 0.5, 1.0, 1.5, 2.0).

	Gd	Pu	Zr	O48f	O8b
Gd2Zr2O7	2.16	-	2.26	−1.25	−1.37
Gd1.5Pu0.5Zr2O7	2.15	2.10	2.27	−1.25	−1.35
Gd1.0Pu1.0Zr2O7	2.13	2.11	2.27	−1.24	−1.35
Gd0.5Pu1.5Zr2O7	2.15	2.09	2.28	−1.24	−1.33
Pu2Zr2O7	-	2.08	2.27	−1.23	−1.32
Gd2Zr1.5Pu0.5O7	2.14	2.36	2.26	−1.25	−1.37
Gd2Zr1.0Pu1.0O7	2.15	2.31	2.27	−1.25	−1.37
Gd2Zr0.5Pu1.5O7	2.15	2.34	2.24	−1.25	−1.36
Gd2Pu2O7	2.16	2.30	-	−1.26	−1.38

. 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−yPu_yZr₂O₇, 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−yCe_yZr₂O₇ 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 Gd₂Zr₂O₇ are affected slightly by Pu immobilization at Gd-site. For Gd₂Zr_2−yPu_yO₇, 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 $\AA$ is larger than the <Zr-O_48f> distance of 2.11 $\AA$ and the ionic radius of 0.96 $\AA$ for Pu ions is larger than that of 0.72 $\AA$ for Zr ions [40], it is suggested that the <Zr-O> bonds exhibit weaker ionicity than <Pu-O> bonds in Gd₂Zr_2−yPu_yO₇. 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.

3.3. Ductility, Elastic Anisotropy, Debye Temperature and Electronic Structures of Pu-Doped Gd₂Zr₂O₇

Pugh’s indicator ($\frac{B}{G}$) is used to reflect the ductility of materials. If $\frac{B}{G}>1.75$, the material shows ductility; or, it is brittle [54].

Table 5. Pugh’s indicator ($B/G$), elastic anisotropy index ($A^U$), sound wave velocity ($v_m$, in m/s), Debye temperature ($\theta$, in K) and Poisson’s ratio ($\nu$) of Gd_2−yPu_yZr₂O₇ and Gd₂Zr_2−yPu_yO₇ (0 ≤ y ≤ 2).

		$\mathit{B}/\mathit{G}$	${\mathit{A}}^{\mathit{U}}$	${\mathit{v}}_{\mathit{m}}$	$\mathit{\theta }$	$\mathit{\nu }$
Gd2Zr2O7		1.907	0.01355	4666.0	580.2	0.277
	Exp. [45,46]	1.913				0.276
	Exp. [20]	1.891			513.3	0.274
	Cal. [53]	2.004	0.00420	4833.5	612.9	0.286
	Cal. [43]					0.273
Gd1.5Pu0.5Zr2O7		1.931	0.00598	4533.7	560.7	0.279
Gd1.0Pu1.0Zr2O7		1.950	0.00105	4367.8	540.2	0.281
Gd0.5Pu1.5Zr2O7		1.964	0.00032	4247.4	522.5	0.282
Pu2Zr2O7		1.987	0.00004	4106.4	503.8	0.285
Gd2Zr1.5Pu0.5O7		1.928	0.04881	4124.6	508.7	0.279
Gd2Zr1.0Pu1.0O7		2.278	0.21353	3591.7	439.5	0.309
Gd2Zr0.5Pu1.5O7		2.243	0.10062	3590.2	435.9	0.306
Gd2Pu2O7		2.151	0.06923	3471.3	418.3	0.299

presents the calculated Pugh’s indicators. For Gd₂Zr₂O₇, 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−yPu_yZr₂O₇ and Gd₂Zr_2−yPu_yO₇, 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 ($\mathsf{\nu })$ can be employed to evaluate the relative ductility of materials. When $\mathsf{\nu }$ is around 0.1, the material shows brittle covalent properties. When $\mathsf{\nu }$ is bigger than 0.25, it exhibits ductile ionic properties [55]. Table 5 lists the calculated, experimental and others’ calculated Poisson’s ratio for both Gd_2−yPu_yZr₂O₇ and Gd₂Zr_2−yPu_yO₇. For Gd₂Zr₂O₇, the calculated Poisson’s ratio of 0.277 could be comparable to the experimental values of 0.276 [45,46], 0.274 [20] and other calculated results of 0.286 [53] and 0.273 [43]. The Poisson’s ratios for Gd_2−yPu_yZr₂O₇ and Gd₂Zr_2−yPu_yO₇ are all larger than 0.25, as presented in Table 5, i.e., the Pu-substituted Gd₂Zr₂O₇ 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 $A^U=5\frac{G^V}{G^R}+\frac{B^V}{B^R}-6$ 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−yPu_yZr₂O₇ and Gd₂Zr_2−yPu_yO₇ are shown in Table 5. For Gd₂Zr₂O₇, 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−yPu_yZr₂O₇ compounds are isotropic elastically. As for the immobilization of Pu at Zr-site, the $A^U$ values are 0.21353 for Gd₂Zr_1.0Pu_1.0O₇ and 0.10062 for Gd₂Zr_0.5Pu_1.5O₇, 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 ${\theta }_D$ value of 580.2 K for Gd₂Zr₂O₇ 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 ${\theta }_D$ value decreases for both Gd_2−yPu_yZr₂O₇ and Gd₂Zr_2−yPu_yO₇. In particular, the ${\theta }_D$ values for Gd₂Zr_2−yPu_yO₇ are smaller than those for Gd_2−yPu_yZr₂O₇. These results imply that the Gd₂Zr_2−yPu_yO₇ compositions have a lower melting point and weaker interatomic binding force than the Gd_2-yPu_yZr₂O₇ compositions. Zhao et al. [59] calculated the Debye temperature for Th immobilization at Gd-site and Zr-site of Gd₂Zr₂O₇ and observed similar phenomena, i.e., the Th-substituted Gd₂Zr₂O₇ 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−yPu_yZr₂O₇ and Gd₂Zr_2−yPu_yO₇ are illustrated in Figure 6 and Figure 7, respectively. Table 1 and Table 2 list the band gap values. It is shown that all the compositions have large band gap values. For Gd₂Zr₂O₇, the calculated band gap is 2.86 eV. For Pu incorporation into Gd-site, the band gap values of 2.27–2.37 eV are smaller than that of Gd₂Zr₂O₇ and nearly independent of the Pu content. For Gd₂Zr_2−yPu_yO₇, the changes in the band gap is more significant, ranging from 2.33 to 1.68 eV. It is indicated that Pu immobilization at Gd-site and Zr-site has different influences on the electronic structure of Gd₂Zr₂O₇. The same conclusion can be drawn from the density of state distribution. For Gd_2−yPu_yZr₂O₇, O 2p orbital dominates and hybridizes with very few Gd 5d, Pu 5f and Zr 4d orbitals in the energy range of −5–1 eV, and the Pu 5f orbitals hybridize with the O 2p orbitals at the bottom of valence band. For Gd₂Zr_2−yPu_yO₇, the O 2p orbital dominates and hybridizes with Pu 5f orbitals and very few Zr 4d and Gd 5d orbitals at the valence band. Obviously, in Gd₂Zr₂O₇, different electronic structures can be obtained from the immobilization of Pu at Gd and Zr sites. These different electronic structures may result in discrepancies in the thermo-physical properties of Gd_2−yPu_yZr₂O₇ and Gd₂Zr_2−yPu_yO₇.

4. Conclusions

The mechanical and electronic properties of Pu-containing Gd₂Zr₂O₇ are studied by a DFT+U method. For Gd_2−yPu_yZr₂O₇ and Gd₂Zr_2−yPu_yO₇, 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 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 Gd₂Zr_2−yPu_yO₇, 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.