The Thermophysical Properties of TcO₂

Abstract

Technetium-99 is a highly radioactive isotope with a long half-life that is common in nuclear waste. It volatizes at a low temperature, which poses a significant challenge to the clean-up and containment processes. Due to difficulties in purifying technetium compounds, their thermophysical properties have not been measured or calculated. Here, first principle methods are used along with the quasi quasi-harmonic harmonic approximation to compute the Debye temperature, volumetric thermal expansion coefficient, bulk modulus, and heat capacity of rutile TcO₂ for temperatures ranging from 0 to 1500 K and applied pressures ranging from 0 to 255 GPa. The computed atomic structures agree well with the results from diffraction measurements. The computed thermophysical properties are in the neighborhood of other rutile metal oxides and, in particular, are within approximately 10–13% of rutile ReO₂, which is frequently used as a substitute for TcO₂ in experimental studies.

1. Introduction

Technetium, Tc, is a transition metal element with no stable isotope that is a by-product of uranium fission. Of particular interest is ⁹⁹Tc, a pure β-emitter, which has a half-life of 2.13 × 10⁵ years. Due to its electronic structure, 4d⁵5s², Tc is capable of forming a wide range of complex compounds, for example, as an oxide, it is known to assume multiple neutral polymorphs of Tc₂O₇, TcO₂, and possibly Tc₂O₅ [1,2,3,4,5,6]. The pentavalent ionic species TcO₄⁻ has high mobility in subsurface soil [7], which makes the cleanup and containment of Tc from waste sources particularly important.

The current long-term containment plan for highly radioactive nuclear waste involves its vitrification, yet this radioactive isotope has a low retention during the conversion to glass [8,9,10,11]. This is attributed to the volatilization of Tc at low temperatures: Tc₂O₇, Tc(VII), boils at 311 °C, and TcO₂, Tc(IV), sublimes at 900 °C [12]. In addition, glasses with high concentrations of Tc are known to precipitate crystalline Tc oxide and alkali pertechnetates, which significantly impact the structural stability of the glass [11,13]. The vitrification and retention of Tc depend upon many factors, including the melter feed chemistry, for example, waste composition and redox, as well as the processing parameters, for example, temperature, feed reaction kinetics, and glass melt bubbling. At this time, the local atomic structure of Tc in glass, the kinetics of crystallization, and the volatilization mechanisms are only partially understood [8,14,15].

Studying Tc compounds is challenging in part due to their highly radioactive nature. Isolating, purifying, and characterizing the crystalline phases is difficult, and for many phases, there is no reported information. First-principles methods have been used to examine the ground-state properties of several Tc oxide phases, including the atomic and electronic structure and the phonon density of states [5,16]. However, the thermophysical behavior at finite temperatures has yet to be reported. Here, first-principles methods are applied in conjunction with energy volume, E(V), and equations of state to predict the characteristics of rutile TcO₂.

2. Methods

First-principles density functional theory [17,18] calculations are performed with the Quantum ESPRESSO 5.3.0 software package [19] using the provided projector augmented-wave type pseudopotentials generated from the Atomic code 10 April 2014 [20,21,22]. The exchange-correlation functional is treated in the modified Perdew–Burke–Ernzerhof type of generalized gradient approximation (PBEsol-GGA) [23,24,25], which is anticipated to be the best potential for this compound. The electronic wavefunction is expressed as a planewave summation truncated at an energy cutoff of 1088 eV, and the charge density is represented on a grid with an energy cutoff of 6122 eV. Brillouin zone integration is performed within the Monkhorst–Pack scheme [26] on an 8 × 8 × 8 k-point mesh. This approach results in the calculated forces being converged to better than 5 meV/Å.

The thermophysical properties of TcO₂ are predicted from the first-principles calculations using the Gibbs2 software package, version 1.0, [27,28] to fit an E(V) equation of state. The Debye–Slater [29,30] and Debye–Grüneisen [31] models are used to approximate the phonon density of states. The Poisson’s ratio, ν, is approximated as 0.25. The well-established error due to the exchange-correlation potential being approximated as a local functional of the charge density is corrected using three empirical energy corrections [27,32]: a rigid correction directly proportional to the volume (PSHIFT), a rigid correction inversely proportional to the volume (APBAF), and a correction to the E(V) curvature (BPSCAL). For the third correction, the bulk modulus, B, is required. Since it has yet to be experimentally measured, a first-principles calculated B, 295.8 GPa, is used for the initial calculations. Ultimately, the physical parameters, ν and B, used in the energy correction are determined through an optimization process.

3. Results and Discussion

TcO₂ has a distorted rutile structure, space group P2₁/c, with room temperature lattice parameters measuring a = 5.6891, b = 4.7546, c = 5.5195 Å, and β = 121.453° [5]. The calculated ground-state structure is shown in Figure 1a and has the lattice parameters a = 5.5909, b = 4.7608, c = 5.5057 Å, and β = 121.275°. The computed atomic positions are given in

Table 1. The calculated TcO₂ lattice vectors in units of Å and atomic fractional coordinates. Each atom has a multiplicity of 4, obeying the symmetry of space group P2₁/c.

Crystal Parameter	$\widehat{\mathit{i}}$	$\widehat{\mathit{j}}$	$\widehat{\mathit{k}}$
a1	5.591	0.000	0.000
a2	0.000	4.761	0.000
a3	−2.858	0.000	4.706
Tc	0.26535	0.00459	0.98489
O1	0.10706	0.19253	0.19506
O2	0.39102	0.71259	0.26659

. The resulting unit cell volume is only 1.66% smaller than the reported experimental volume, which validates the theoretical approach used here.

The ground-state unit cell is hydrostatically expanded and compressed to sample the energy surface near the ground state. The resulting E(V) is fit to the Birch–Murnaghan isothermal equation of state [33]. The computed data and fitted curve are shown in Figure 1b.

The Debye–Slater and Debye–Grüneisen thermal models without corrections are used to obtain the volume at an applied pressure of 0 GPa. At a temperature of 15 K, the volumes are 126.3 and 126.1 ų for the Debye–Slater and Debye–Grüneisen models, respectively; increasing the temperature to 298.15 K yields volumes of 126.6 and 126.4 ų. The uncorrected volume as a function of temperature is given in Figure 2 and

Table 2. The calculated TcO₂ unit cell volume in units of ų at temperatures of 15 and 298.15 K at an applied pressure of 0 GPa. The values are reported to three decimal places to facilitate comparison of results.

	Debye–Slater				Debye–Grüneisen				Literature [5]
	UNCORR	PSHIFT	APBAF	BPSCAL	UNCORR	PSHIFT	APBAF	BPSCAL
15 K	126.261	126.965	126.975	126.999	126.102	127.025	127.036	127.057	127.124
298.15 K	126.639	127.362	127.362	127.362	126.418	127.362	127.362	127.362	127.362

, labeled UNCORR.

Empirical energy corrections are used to force the computed volume at 298.15 K to match the experimentally reported volume, 127.4 ų, as shown in Table 2. The corrections applied at 298.15 K substantially improve the accuracy of the calculated volume at 15 K, which confirms that the approach used here is reasonable. In particular, the Debye–Grüneisen model with the BPSCAL correction has a predicted volume of 127.1 ų at 15 K, which matches the experimentally determined volume to within the uncertainty of the method [5], suggesting that this approach will produce the most accurate results of the methods examined here.

The empirical energy corrections require experimental values for ν and B, neither of which are known. The volumes in Table 2 are determined by approximating, where ν = 0.25 and B = 295.8 GPa. It is possible to vary ν and B to find the values that yield the most accurate lattice volume.

Table 3. The calculated TcO₂ unit cell volume in units of ų at a temperature of 15 K and applied pressure of 0 GPa using the Debye–Grüneisen model and the BPSCAL empirical energy correction. The volumes are calculated for Poisson’s ratio values ranging between 0.20 and 0.30 and bulk modulus values ranging between 280 and 320 GPa. The values are reported to three decimal places to facilitate comparison of results. The experimentally determined volume is 127.124 ų [5].

	280	285	290	295	300	305	310	315	320
0.20	127.068	127.077	127.085	127.092	127.100	127.107	127.113	127.120	127.126
0.21	127.061	127.070	127.078	127.085	127.093	127.100	127.107	127.114	127.120
0.22	127.054	127.062	127.070	127.078	127.086	127.093	127.100	127.107	127.113
0.23	127.046	127.055	127.063	127.071	127.079	127.086	127.093	127.100	127.107
0.24	127.038	127.047	127.055	127.063	127.071	127.079	127.086	127.093	127.100
0.25	127.030	127.039	127.047	127.056	127.064	127.071	127.079	127.086	127.093
0.26	127.021	127.030	127.039	127.048	127.056	127.063	127.0709	127.078	127.085
0.27	127.012	127.022	127.031	127.039	127.047	127.055	127.063	127.070	127.077
0.28	127.003	127.013	127.022	127.031	127.039	127.047	127.055	127.062	127.069
0.29	126.994	127.003	127.013	127.021	127.030	127.038	127.046	127.054	127.061
0.30	126.984	126.994	127.003	127.012	127.021	127.029	127.037	127.045	127.052

shows the predicted lattice volume using the Debye–Grüneisen model and BPSCAL correction for a range of ν and B values. The parameter choices ν = 0.20 and B = 315 GPa yield a lattice volume of 127.120 ų, ν = 0.20 and B = 320 GPa yields 127.126 ų, and ν = 0.21, and B = 320 GPa yields 127.120 ų, all of which are near the experimentally reported value 127.124 ų [5]. It is reasonable to conclude that the best choice for ν is between 0.20 and 0.21, and B is between 315 and 320 GPa. The Debye–Grüneisen model with the BPSCAL correction, using ν = 0.20 and B = 320 GPa, is used to predict the thermophysical properties of TcO₂ at ambient conditions, as shown in

Table 4. The thermophysical properties of TcO₂ at ambient conditions computed using the Debye–Grüneisen model and the BPSCAL empirical energy correction with ν = 0.20 and B = 320 GPa.

Thermophysical Properties, Symbol, Unit	Value
Gibbs free energy, G, kJ/mol	−6.225 × 105
Debye temperature, ΘD, K	891.91
Volumetric thermal expansion coefficient, αvol, 10−5/K	1.484
Isothermal bulk modulus, BT, GPa	320
Adiabatic bulk modulus, BS, GPa	322.59
Grüneisen parameter, γ	1.832
Constant pressure heat capacity, Cp, J/mol·K	50.103
Constant volume heat capacity, Cv, J/mol·K	49.700
Thermal pressure, pth, GPa	2.991
Vibrational contribution to the Helmholtz free energy, Fvib, KJ/mol	21.762
Entropy, S, J/mol·K	32.256

.

The predicted Debye temperature of TcO₂ at ambient conditions is 891.91 K. The Debye temperature is closely correlated to many physical properties, including specific heat, elastic constant, and melting point. The TcO₂ Debye temperature is greater than that reported for the stable phase β-ReO₂, 850 K, another group VII dioxide [34]. For the rutile structure ReO₂, the calculated Debye temperature is 785 K [34]. Compared to other common oxides, the Debye temperature of TcO₂ falls somewhere in between TiO₂ (790 K) [35,36] and Al₂O₃ (950 K) [37].

Figure 3 shows the predicted isothermal bulk modulus as a function of temperature and pressure. At ambient conditions, the calculated modulus, 320 GPa, is approximately that of β-ReO₂, 322 GPa, but lower than the calculated rutile-phase ReO₂, 353 GPa [34]. The high modulus reflects the relatively high bonding strength and density that are typical for most ceramic oxide materials. For comparison, the other rutile-type oxides, such as GeO₂, MnO₂, and SiO₂, have bulk moduli of 270, 280, and 320 GPa, respectively [38,39].

The predicted volumetric thermal expansion coefficient of TcO₂ at ambient conditions is 1.484 × 10⁻⁵ K⁻¹. The room temperature volumetric thermal expansion coefficients of other rutile-type oxides GeO₂, SnO₂ (cassiterite), SiO₂ (stishovite), and MnO₂ (pyrolusite) are 1.41 × 10⁻⁵ K⁻¹, 1.42 × 10⁻⁵ K⁻¹, 1.73 × 10⁻⁵ K⁻¹, and 2.03 × 10⁻⁵ K⁻¹, respectively [40,41,42]. Figure 4 shows the predicted pressure and temperature dependence of the volumetric thermal expansion. The temperature dependence has an S-shaped curve, which is common for many oxides.

Figure 5 shows the computed heat capacities of TcO₂ with respect to constant pressure (C_p) and volume (C_v) as a function of pressure or temperature. The values range from 0 J/mol·K at 0 K to 73.64 J/mol·K for C_v and 79.33 J/mol·K for C_p at 1500 K. C_v reaches the Dulong–Petit limit near 890 K, which coincides with the Debye temperature. Above this, C_v plateaus, but C_p continues to increase due to the volume expansion. The room temperature constant pressure heat capacity of 50.10 J/mol·K is approximately that of other oxides. It is slightly greater than CaO, 42.42 J/mol·K; SiO₂, 44.42 J/mol·K; and BaO, 47.06 J/mol·K, and lower than GeO₂, 51.95 J/mol·K; MnO₂, 54.42 J/mol·K; TiO₂, 55.10 J/mol·K; and ReO₂, 56.60 J/mol·K [43,44].

4. Conclusions

First-principles data are used to predict the thermophysical properties of TcO₂ using the Gibbs2 software version 1.0. The results indicate that TcO₂ is a relatively stiff material with a bulk modulus of 320 GPa. The Poisson’s ratio is between 0.20 and 0.21. The Debye temperature of TcO₂, 891.91 K, falls somewhere in between TiO₂ (790 K) and Al₂O₃ (950 K). The volumetric thermal expansion coefficients of TcO₂ at ambient conditions are 1.484 × 10⁻⁵/K, which is close to those of other rutile-type oxides, such as GeO₂, SnO₂, and SiO₂. The room temperature constant pressure heat capacity of TcO₂, 50.10 J/mol·K, is slightly higher than those of CaO, SiO₂, and BaO but lower than those of GeO₂, MnO₂, TiO₂, and ReO₂. ReO₂, which is sometimes used as a substitute for TcO₂ in studies, has physical properties that are within 10 to 13% of those computed here. This difference is within the uncertainty arising from the approximations used in the calculations, suggesting that in many cases, using ReO₂ may be a reasonable approximation for TcO₂.