Introduction

Barium zirconate, BaZrO3, is a perovskite-structured compound [1] that crystallises with the aristotype [2] crystal structure in space group \(Pm\overline{3} m\) in the temperature range 4.2 K [3] to 1700 K [4]. BaZrO3 is a well-studied ceramic oxide due to a number of technologically useful properties: as a high-temperature proton or oxide ion conductor when aliovalent cations are substituted for zirconium [5,6,7], and as a component of lead-free piezoelectrics [8]. In addition, the high-temperature behaviour of BaZrO3 has been studied as an easily synthesised clone phase of bridgmanite (MgSiO3 perovskite) to infer behaviour of the Earth’s lower mantle [9]. The presence of BaZrO3 in solid solution with other perovskite-structured materials in fission product compounds in uranium–plutonium mixed oxide fuels in fast breeder reactors has led to a significant number of investigations of its high-temperature thermophysical properties including calorimetry, thermal conductivity, microhardness and elastic moduli [4, 10,11,12,13,14,15].

Despite the evidence from low-temperature neutron diffraction measurements that show BaZrO3 is cubic below 10 K [3, 6] with no evidence for superlattice reflections or diffuse scattering at the R or M points of the Brillouin zone, ab initio calculations have shown that an instability exists at the R point where the calculated phonons exhibit negative frequencies [3, 16,17,18]. This instability is associated with a triply degenerate phonon with irreducible representation \({\text{R}}_{4}^{ + }\) in which successive ZrO6 octahedra along the cubic crystallographic axes rotate in an antiphase manner [19]. Results of first principles calculations carried out on BaZrO3 are consistent with a single component of this mode condensing out giving rise to a lower symmetry tetragonal phase with space group I4/mcm [3, 16,17,18]. The deduction that the theoretical ground state of BaZrO3 is tetragonal, in conflict with crystallographic experiments, is not universally accepted, and the stabilisation of the cubic crystal structure due to quantum zero-point vibrations and anharmonic effects stabilising the soft modes has recently been proposed [20]. The effect of yttrium doping for zirconium in BaZrO3 has been the subject of a synchrotron powder diffraction, EXAFS and Raman spectroscopic investigation with the conclusion that for 0.06 < x < 0.15 in BaZr1−xYxO3−δ the crystal structure is tetragonal in space group P4/mbm [21]. This would be consistent with a soft-mode transition associated with a phonon that transforms as the irreducible representation \({\text{M}}_{3}^{ + }\) [19]. Consideration of Fig. 3 of reference [21] shows that no additional superlattice reflections are observed at the M point of the pseudocubic Brillouin zone when compared to the diffraction pattern of the undoped compound (reference [21] Fig. 2), although allowing the unit cell to become tetragonal is demonstrated from fitting of the 200 and 310 Bragg reflections. The quoted fractional coordinates of the anion O2 are simply those of the aristotype transformed to the tetragonal unit cell and hence would not give rise to the expected superlattice reflections. Furthermore, most ab initio calculations of the phonons in BaZrO3 do not show that negative frequencies are associated with the M point, unlike the R point [3, 16,17,18], although the whole branch between the R and the M points is soft according to Bilić and Gale [20]. The deduced space group of BaZr1−xYxO3−δ is therefore open to debate. As a small tetragonal distortion has been proven to occur in BaZr1−xYxO3−δ for 0.06 < x < 0.15, a high-resolution neutron powder diffraction investigation would be beneficial in confirming the structural characterisation of these materials as the appearance of the diagnostic superlattice reflections is principally dependent upon anion fractional coordinates in zone-boundary-tilted perovskite crystal structures.

Further experimental evidence for a lower crystallographic symmetry than the aristotype phase has been afforded by Raman spectroscopic investigations of BaZrO3 [21, 22]. First-order Raman scattering is forbidden for the aristotype perovskite crystal structure; however, a strong Raman spectrum has been experimentally observed, suggesting the presence of localised structural distortions. Despite this, the spectra presented by Shi et al. [22] are not in detailed agreement with those of Giannici et al. [21], suggesting scope for further detailed vibrational spectroscopic investigations.

Whilst the true nature of the ground state of BaZrO3 remains uncertain, cubic, tetragonal or tetragonal without long-range order, a high-pressure phase transition at 17.2 GPa to a long-range ordered tetragonal phase in space group I4/mcm has been observed experimentally [23].

In the work to be presented here, we report the first detailed crystallographic and thermoelastic study of BaZrO3 at ninety temperatures between 4.2 and 450 K. We demonstrate that at low to intermediate temperatures, BaZrO3 perovskite can be treated thermodynamically in a simple self-consistent two-term Debye model [24,25,26].

Method

Neutron powder diffraction data were collected in time-of-flight using the long-flightpath, high-resolution, backscattering powder diffractometer (HRPD) of the ISIS Facility, Rutherford Appleton Laboratory. The advantage of this instrument over constant wavelength instrumentation is that to first order, the resolution function is independent of scattering vector (Q = 4πsin(θ)/λ), and hence, Δd/d is approximately constant over the whole diffraction pattern. This almost constant Q resolution reduces Bragg reflection overlap in a powder diffraction pattern and hence permits the detailed characterisation of two compounds as a mixture under the identical thermodynamic conditions providing the unit cell metrics are well separated. The methodology has been exploited in temperature-dependent crystallographic studies of LaGaO3 [24] and SrZrO3 [25], and BaCeO3 [26] and SrCeO3 [Knight and Bonanos, in preparation] mixtures. In the work to be described below, we have studied a mixture of two cubic perovskite-structured phases, BaZrO3 and KTaO3, and described the evolution of thermophysical and crystallographic properties of the zirconate phase; results derived from the analysis of the tantalate compound will be described elsewhere. For this mixture, there was no appreciable Bragg peak overlap in the backscattering data between the two phases at any of the measured temperatures.

5.428 g of commercially synthesised BaZrO3 powder (Sigma-Aldrich) was annealed at 1773 K under air for 24 h and then intimately mixed with 5.261 g of KTaO3 powder (Sigma-Aldrich) using an agate pestle and mortar. The equimolar admixture was located into a directly heated 1.5-cm-thick aluminium sample can of slab geometry equipped with thin vanadium windows. Specifics of this sample geometry, thermometry and thermal stability have recently been described in detail by Fortes [27] and will not be elaborated on further in this publication. The can was cooled to 4.2 K using an AS Scientific ‘Orange’ cryostat and was then equilibrated for 1 h. Neutron time-of-flight data were then collected in the time-of-flight window 30–130 ms at 4.2 K for 6 µAh, approximately 9 min duration, before the sample was heated to the next temperature, 10 K. To ensure thermal equilibration was achieved, the sample was required to reach the set-point temperature before a 5-min equilibration period was commenced. Data were then collected for 6 µAh and the temperature increased by 5 K, with the whole protocol of heating, equilibration and data collection being repeated up to 445 K. Finally, the sample was heated to 450 K and equilibrated, and data were collected for 40 µAh, approximately 1 h of duration; this procedure was repeated in 50 K steps on cooling to 150 K, then to 75 K and finally at 4.2 K. For all measurements, the temperature variation was better than ± 0.2 K of the set-point temperature.

The neutron time-of-flight data were focused, normalised to the incident flux distribution and corrected for solid angle variation. These data were then further corrected for self-shielding for a sample of nominal composition (BaK)0.5(ZrTa)0.5O3, number density of 4.63 × 1021 cm−3, scattering cross section of 21.6 b [28] and wavelength-dependent absorption for an absorption cross section of 12.0 b at a wavelength of 1.798 Å [28]. Data sets were generated in GSAS [29] format in the time-of-flight range 32–120 ms that corresponds to a d-spacing range of 0.66–2.48 Å for the high-resolution backscattering bank and 0.92–3.44 Å in the high-count-rate 90° detector bank.

Data analysis

Multibank, multiphase crystal structure refinement was performed using the GSAS software suite [29] for a crystallographic model with isotropic atomic displacement parameters for the cations and anisotropic thermal parameters for the anion. Data analysis commenced with the 4.2 K run, which, once converged, formed the starting model for the 10 K run; the refinement process was iterated up to the 450 K measurement. As the crystallographic refinement was facile due to both phases containing atoms in special (fixed) positions in space group \(Pm\overline{3} m\) (see Table 1), convergence was rapid with typical agreement factors of Rp = 0.065, Rwp = 0.078, χ2 = 2.4 for a model with 35 parameters (1 phase fraction, 2 scale factors, 2 lattice parameters, 4 cation isotropic atomic displacement parameters, 4 anion anisotropic atomic displacement parameters, 12 peak shape parameters and 10 background terms). The phase fraction refined to 0.504(2) in excellent agreement with that expected from the individual component masses (0.5). Refinement of the site occupancy factors of all atoms was introduced into the 4.2 K refinement and showed the expected stoichiometry within one estimated standard deviation. All subsequent refinements were hence carried out assuming an ideal composition of BaZrO3. Average estimated standard deviations for the structural parameters of BaZrO3 in the short data collection times were 1.5 × 10–5 Å for the lattice parameter, 4 × 10–4 Å2 and 3 × 10–4 Å2 for the barium and zirconium isotropic temperature factors, respectively, and 7 × 10–4 Å2 and 4 × 10–4 Å2 for the oxygen anisotropic atomic displacement terms u11 and u33 (= u22). The quality of fit to these data is illustrated in Fig. 1 for the 4.2 K measurement, and the refined structural parameters for the nine long runs are listed in Table 1.

Table 1 Crystallographic parameters for BaZrO3 at selected temperatures
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Figure 1
figure 1

Profile refinement fit to the 4.2 K data collected on BaZrO3 using the high-resolution time-of-flight powder diffractometer (HRPD). a Profile refinement fit to the high-resolution backscattering bank; b the simultaneous fit to the high-count-rate, medium-resolution 90° bank. Observed data are shown as points, the calculated pattern as the full line, and reflection markers are for BaZrO3 (upper) and KTaO3 (lower). The difference (observed–calculated) is shown as the full line below the reflection markers for KTaO3

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No evidence for diffuse scattering was observed at the R point positions of the cubic Brillouin zone in the high-count-rate 90° 2θ data sets, the presence of which would support the conclusions of some of the DFT calculations that the ground state of BaZrO3 contains nanoclusters of lower symmetry. A single crystal neutron diffuse scattering of BaZrO3 may shed further light on this conjecture.

Results

Lattice parameter, unit cell volume and characteristic temperatures

Figure 2 shows the temperature dependence of the lattice parameter of BaZrO3 between 4.2 and 450 K. The temperature variation is as expected for a non-magnetic, dielectric material without structural phase transitions: saturation at low temperature, in this case for temperatures below ~ 40 K, and a linear temperature dependence at higher temperatures, above ~ 350 K. The smooth, monotonic variation, however, masks a subtle temperature variation. For many materials which do not undergo structural phase transitions, the temperature variation of the lattice constant(s) from liquid helium temperatures may be empirically fitted using an expression related to an Einstein internal energy function [30], \(a\left( T \right) = a_{0} + {{k_{i} } \mathord{\left/ {\vphantom {{k_{i} } {\left( {\exp \left( {{{E_{i} } \mathord{\left/ {\vphantom {{E_{i} } T}} \right. \kern-\nulldelimiterspace} T}} \right) - 1} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\exp \left( {{{E_{i} } \mathord{\left/ {\vphantom {{E_{i} } T}} \right. \kern-\nulldelimiterspace} T}} \right) - 1} \right)}}\). However, in this case, although the lattice parameter data of BaZrO3 superficially seem well fitted, the difference/e.s.d. (Δ/σ) showed a large systematic oscillatory discrepancy for temperatures below 200 K as illustrated in Fig. 2 by the dot–dash line (r2 = 0.99909). Permitting a second temperature-dependent term to the fitting procedure resulted in the full line shown passing through the experimental lattice parameter data and the full line in the Δ/σ plot (r2 = 0.99983). In this case, there is no systematic behaviour at low temperature and |Δ/σ| is less than 3 for all data below 350 K. The fitting parameters for the lattice parameter are listed in Table 2.

Figure 2
figure 2

The temperature variation of the lattice parameter of BaZrO3 between 4.2 and 500 K. The lower plot illustrates the difference/estimated standard deviation (Δ/σ) for the two models discussed in the text. The dot–dash line shows Δ/σ for the single-term model indicating a systematic misfit at low temperatures. The full line shows Δ/σ for the two-term model that exhibits no systematic misfit. The dashed line in the upper figure illustrates the linear thermal expansion coefficient calculated from the two-term fitting parameters

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Table 2 Refined fitting parameters for the temperature dependence of the lattice parameter of BaZrO3
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The linear thermal expansion coefficient (αL) for BaZrO3 was calculated using the fitting parameters and is illustrated in Fig. 2 as the dashed line. αL varies from ~ 0 at 4.2 K, increasing monotonically to ~ 3.0 × 10–6 K−1 by 100 K before showing an approximately linear temperature dependence to 250 K (5.4 × 10–6 K−1). Above this temperature, the rate of change of αL with temperature reduces, reaching a value of 6.6 × 10–6 K−1 at 500 K. Based on the fitting coefficients, a high temperature value for αL is estimated as 7.2 × 10–6 K−1. The sub-300 K results are in fair agreement with the average value of αL derived from two points measured by neutron powder diffraction at 100 and 300 K (4.7 × 10–6 K−1) [3]. The estimated high-temperature limit is consistent with high-temperature experimental results derived from X-ray diffraction (6.87 × 10–6 K−1 [9], 7.5 × 10–6 K−1 [31], 8.02 × 10–6 K−1 [32], 7.89 × 10–6 K−1 [33]) and dilatometry (7.13 × 10–6 K−1 [12]).

In contradistinction to the fitting of the lattice parameter to an empirical function of temperature, the temperature dependence of the unit cell volume can be fitted to a variety of statistical mechanical models on the assumption of a Grüneisen approximation to the zero-pressure equation of state. To the first order, the volume will vary as the internal energy (U(T)); \(V\left( T \right) = V_{0} + \left( {{\gamma \mathord{\left/ {\vphantom {\gamma {K_{0} }}} \right. \kern-\nulldelimiterspace} {K_{0} }}} \right)U\left( T \right)\) [34] where γ is a Grüneisen parameter and K0 is the isothermal bulk modulus. Furthermore, as the isochoric heat capacity (cV) is the temperature derivative of the internal energy function at constant volume, a self-consistent solution may be found by fitting the unit cell volume and the isochoric heat capacity simultaneously to the same internal energy function. An Einstein model for the internal energy function with a delta function in the phonon density of states is clearly an over-simplification, and hence, we have used a more realistic Debye model. For this model of the internal energy function, the volume and the isochoric heat capacity as a function of temperature are given by the expressions [30]

$$ V(T) = V_{0} + \frac{{9\gamma Nk_{{\text{B}}} T}}{{K_{0} }}\left( {\frac{T}{{\theta_{{\text{D}}} }}} \right)^{3} \int_{0}^{{\theta_{{\text{D}}} /T}} {\frac{{x^{3} {\text{d}}x}}{{{\text{e}}^{x} - 1}}} $$
$$ c_{V} (T) = 9Nk_{{\text{B}}} T\left( {\frac{T}{{\theta_{{\text{D}}} }}} \right)^{3} \int_{0}^{{\theta_{{\text{D}}} /T}} {\frac{{x^{4} e^{x} {\text{d}}x}}{{({\text{e}}^{x} - 1)^{2} }}} $$

where N is the number of atoms, kB is Boltzmann’s constant and θD is the Debye temperature.

The temperature dependence of the unit cell volume is illustrated as the upper graph in Fig. 3. Preliminary fitting of the unit cell volume to the Debye internal energy function found the identical systematic discrepancy in the Δ/σ plot that was observed when fitting the lattice parameter to the single-term energy function. This is not unexpected as the Debye temperature derived from heat capacity data is generally temperature dependent [30]. Within the simple model for BaZrO3 we are proposing, the discrepancy in Δ/σ indicated the necessity of applying a second temperature-dependent term to both functions. The volume and isochoric heat capacity are then given by

$$ V(T) = V_{0} + z\frac{{9\gamma_{1} Nk_{{\text{B}}} T}}{{K_{0} }}\left( {\frac{T}{{\theta_{{{\text{D}}1}} }}} \right)^{3} \int_{0}^{{\theta_{{{\text{D}}1}} /T}} {\frac{{x^{3} {\text{d}}x}}{{{\text{e}}^{x} - 1}} + \left( {1 - z} \right)} \frac{{9\gamma_{2} Nk_{{\text{B}}} T}}{{K_{0} }}\left( {\frac{T}{{\theta_{{{\text{D}}2}} }}} \right)^{3} \int_{0}^{{\theta_{{{\text{D}}2}} /T}} {\frac{{x^{3} {\text{d}}x}}{{{\text{e}}^{x} - 1}}} $$
$$ c_{V} (T) = 9zNk_{{\text{B}}} T\left( {\frac{T}{{\theta_{{{\text{D}}1}} }}} \right)^{3} \int_{0}^{{\theta_{{{\text{D}}1}} /T}} {\frac{{x^{4} e^{x} {\text{d}}x}}{{({\text{e}}^{x} - 1)^{2} }}} + 9\left( {1 - z} \right)Nk_{{\text{B}}} T\left( {\frac{T}{{\theta_{{{\text{D}}2}} }}} \right)^{3} \int_{0}^{{\theta_{{{\text{D}}2}} /T}} {\frac{{x^{4} e^{x} {\text{d}}x}}{{({\text{e}}^{x} - 1)^{2} }}} $$

where z is a mixing parameter: 0 < z < 1.

Figure 3
figure 3

The simultaneous fit to the unit cell volume (upper) and isochoric heat capacity (lower) for BaZrO3 using the two-term Debye model described in the text: circles data, full line fit to these data. The dashed line in the upper figure illustrates the volume expansion coefficient calculated using the fitting coefficients. The dot–dash line shows the contribution of the lower Debye temperature term to the volume expansion coefficient, and the upper shows the contribution from the higher Debye temperature. The dashed line in the lower figure illustrates the inadequacy of the single-term Debye model to fit the isochoric heat capacity

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The molar isobaric heat capacity (cP) for BaZrO3 has been reported in the temperature interval 1.6–301.3 K at 375 temperatures [35], and these data have been converted to the isochoric form using the standard thermodynamic transformation \(c_{V} = c_{P} - T\alpha_{{\text{V}}}^{2} K_{0} V_{{\text{m}}}\)[30], where Vm is the molar volume and αV is the volume expansion coefficient. In the absence of experimental results of the temperature variation of the bulk modulus of BaZrO3, K0 was considered to be a constant, 189 GPa, derived from the ambient temperature equation of state [23]. The volume expansivity and molar volume were calculated from fitting the unit cell volume to a simpler two-term Einstein model which was an adequate approximation for purposes of the transformation. The calculated isochoric heat capacity of BaZrO3 is shown in the lower graph of Fig. 3.

The unit cell volume and the isochoric heat capacity were fitted simultaneously to the two-term Debye models shown above, originally with the mixing parameter z freely refined. The refinement of this parameter did not converge to 1/3 or 2/3, as would be consistent with Barron’s model [36] that allows different cut-off frequencies for the longitudinal and transverse modes, but to ~ 0.43 which is close to 0.4, the cation fraction in the formula unit. Setting this parameter to 0.4 made little difference to the agreement factors for fitting the two data sets, and hence, this value of the mixing parameter was retained, and we consider the two characteristic temperatures to be broadly related to separated cation and anion behaviour. The full lines on the upper and lower graphs of Fig. 3 show the fitted temperature dependences of the unit cell volume and isochoric heat capacity to the two-term Debye model (r2 = 0.99939). The saturated unit cell volume, characteristic temperatures and Grüneisen constants (assuming K0 = 189 GPa [23]) for the two-term Debye model are listed in Table 3. The weighted average of the two Debye temperatures, 526(1) K, shows good agreement with both ab initio calculation (525 K) [37], ultrasonic pulse echo technique (544 K) [13, 14], and calorimetry, for temperatures greater than ~ 150 K, as illustrated in Fig. 3 of Ahrens and Maier [35]. The inadequacy of a single Debye term to simultaneously fit the heat capacity and volume data can be appreciated by the dashed line on the lower graph, for which the Debye temperature was refined as 471 K and the Grüneisen constant as 1.34.

Table 3 Refined fitting parameters for the temperature dependence of the unit cell volume and isochoric heat capacity of BaZrO3
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The temperature dependence of the volume expansion coefficient of BaZrO3 is shown as the dashed line in the upper graph of Fig. 3 with an estimated high-temperature limit of 2.0 × 10–5 K−1; the individual contributions from the two Debye terms are shown as the dot–dash and dot–dot–dash lines. It can be seen that the contribution from the lower Debye temperature term reaches saturation (0.84 × 10–5 K−1) by ~ 200 K and hence contributes a constant expansion coefficient above room temperature. The higher Debye temperature term only reaches saturation (1.15 × 10–5 K−1) by ~ 800 K. The polyhedral volumes are given by a3/6 for the octahedral site (ZrO6) and 5a3/6 for the dodecahedral site (BaO12), and hence, both have the identical polyhedral expansion coefficient to the volume expansivity of the unit cell.

First-order Raman scattering is symmetry-forbidden from the aristotype crystal structure of perovskite, although, as stated earlier, the Raman spectrum has been experimentally measured and characterised [21, 22]. To our knowledge, a high-pressure study of the Raman spectrum of BaZrO3 has not yet been performed, and hence, we lack experimental measurements of the mode Grüneisen parameters of BaZrO3 to make comparisons with the two values derived from the heat capacity/unit cell volume fitting.

Atomic displacement parameters

The temperature dependence of the isotropic atomic displacement parameters for the cations, the anisotropic and the isotropic equivalent atomic displacement parameter for the anion is shown in Fig. 4. All data exhibit the expected variation with temperature, saturation at low temperature and a linear temperature dependence at higher temperatures. For a Debye oscillator, the temperature variation of the mean square displacement is given by [38]

$$ \overline{{u^{2} }} (T) = \frac{{3h^{2} T}}{{4\pi^{2} Mk_{{\text{B}}} \theta_{{\text{D}}}^{2} }}\int_{0}^{{\theta_{{\text{D}}} /T}} {\frac{{x{\text{d}}x}}{{{\text{e}}^{x} - 1}}} + \frac{{3h^{2} }}{{16\pi^{2} Mk_{{\text{B}}} \theta_{{\text{D}}}}} $$

where M is the mass of the atomic species under consideration. In practice, the zero-point term is too constrained as it only depends on the vibrational Debye temperature θD, and to take account of any static disorder that may also be present, an additional constant is added to the fitting procedure. The full lines shown in Fig. 4 show fits to the temperature variation of the atomic displacement parameters based on this modified Debye model [39]; the fitted values of the vibrational Debye temperatures and the static displacement terms are listed in Table 4. Consideration of the Table 4 shows that the transverse vibrational Debye temperature associated with the anion is significantly higher than that of the barium cation in the large dodecahedral cavity site. The vibrational Debye temperature of the zirconium cation in the constrained octahedral site is significantly higher than that of the barium cation, but still less than that associated with the anion in the isotropic equivalent approximation. The average of the cation vibrational Debye temperatures, 325 K, and that associated with the isotropic equivalent atomic displacement of the anion, 743 K, are well separated and are similar to those resulting from the fitting of the unit cell volume and isochoric heat capacity, 220 K, and 730 K.

Figure 4
figure 4

The temperature dependence of the isotropic atomic displacement parameters for the cations and the anisotropic atomic displacement parameters for the oxygen anion. Full lines show the fits to these data based on the modified Debye model described in the text

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Table 4 Vibrational Debye temperatures and mean square static disorder in BaZrO3
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Thermodynamic Grüneisen parameter

The thermodynamic Grüneisen parameter (γth) for BaZrO3 was calculated for the temperature interval 2–500 K using the standard definition γth(T) = (αV(T)K0Vm(T))/cV(T) [34, 40], with the temperature variation of the molar volume, the isochoric heat capacity and the volume expansivity derived from the fitting coefficients; in the absence of experimental data, the bulk modulus was assumed to be constant. The temperature evolution of γth is illustrated in Fig. 5 and shows only a small dynamic range, decreasing from ~ 1.439 at 2 K to ~ 1.398 at 500 K in good agreement with the weighted mean of the individual Grüneisen constants derived from the two-term Debye fitting to the volume and isochoric heat capacity. Assuming Dulong–Petit behaviour for the isochoric heat capacity and the high-temperature volume expansivity deduced from the fitting procedure, at high temperatures γth(T) ~ 1.6 × 10–7K0(T)Vm(T) ~ 1.36.

Figure 5
figure 5

The temperature variation of the thermodynamic Grüneisen parameter for BaZrO3

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Discussion and conclusions

In the temperature interval 4.2–450 K, high-resolution time-of-flight neutron powder diffraction has shown BaZrO3 to be cubic with the aristotype crystal structure. No evidence was found for the tetragonal distortion of the unit cell predicted by ab initio calculations which would be expected to maximise at the lowest temperature assuming critical behaviour of the ZrO6 octahedral rotation angle. Furthermore, no evidence was found for diffuse scattering at the R point superlattice positions of the cubic Brillouin zone. Analysis of the temperature evolution of the unit cell volume and the isochoric heat capacity is consistent with a simple thermodynamic model that treats BaZrO3 perovskite as a two-term Debye system with separated cation and anion behaviour. Characteristic temperatures derived from simultaneous fitting of the heat capacity and volume are found to be in fair to good agreement with the vibrational Debye temperatures derived from fitting the temperature dependence of the atomic displacement parameters. Assessment of the vibrational Debye temperatures of the cations when compared to the lower characteristic temperature showed poorer agreement for the octahedrally coordinated zirconium cation compared to the 12-fold-coordinated barium cation. However, a re-evaluation of the isochoric heat capacity/volume fitting to a three-term Debye model, allowing the two cations to be associated with independent Debye temperatures, did not improve the quality of fit to these data and hence was rejected on statistical grounds.

The simultaneous fitting of the isochoric heat capacity and unit cell volume to the two-term Debye model has been applied successfully to a number of other oxide and fluoride perovskite-structured compounds (SrZrO3 [25], BaCeO3 [26], CaTiO3 [41], LaGaO3 [24], KZnF3 [42], KCaF3 [43]). From these results, a vibrational density of states (VDOS) can be evaluated from the two Debye cut-off frequencies for comparison with either direct measurements of the phonon dispersion curves or sophisticated ab initio calculations. Despite the simplicity of the model, for the case of KZnF3 [42], good agreement was found for the equivalence of the higher characteristic temperature equating to the high-frequency cut-off in the experimental VDOS and the lower characteristic temperature corresponding to the mean frequency of the total VDOS. Unfortunately, we lack either an experimental or calculated VDOS for BaZrO3 to make this comparison.

There are disappointingly few detailed low-temperature crystallographic studies of perovskite-structured compounds with either barium in the cavity site or zirconium in the octahedral site. To compare and contrast the thermal behaviour of BaZrO3, we are limited to the two compounds that have been studied in sufficient detail: BaCeO3 [26] and SrZrO3 [25]. These results are summarised in Fig. 6 where the temperature dependences of the normalised unit cell volume, the volume expansivity, the normalised polyhedral volumes and the atomic displacement parameters of barium and zirconium are illustrated. Between 1.2 and 500 K, both BaCeO3 and SrZrO3 are orthorhombic, space group Pbnm with the alkaline earth in eightfold coordination [26, 44]. Both phases exhibit octahedral tilting [19] and undergo a series of structural phase transitions with increasing temperature to reach the untilted cubic aristotype crystal structure exhibited by BaZrO3 (BaCeO3 563 K to Ibnn, 673 K to \({\text{F}}\overline{3} 2/{\text{n}}\) 1173 K to \({\text{Pm}}\overline{3} {\text{m}}\) [45], SrZrO3 973 K to Imma, 1103 K to I4/mcm, 1443 K to \({\text{Pm}}\overline{3} {\text{m}}\)[44]). For perovskite-structured materials that crystallise with the aristotype structure, thermal expansion can only occur through an increase in the dodecahedral–anion and octahedral–anion bond lengths. This contrasts with crystal structures based on zone-boundary tilting of the octahedra where the untilting of the octahedra with increasing temperature makes a significantly greater contribution to the volume expansivity than the bond length increase. For both BaCeO3 and SrZrO3, the transition from the lowest temperature phase to the next highest is tricritical in nature [25, 26], and hence, as the Ce–O bond length is larger than the Zr–O bond length, we expect the volume expansion coefficients to be in the order BaCeO3, SrZrO3, BaZrO3, which is as measured (Fig. 6a).

Figure 6
figure 6

Comparison of the temperature dependences of the crystallographic parameters of BaZrO3 with BaCeO3 (Ba in the cavity site) and SrZrO3 (Zr in the octahedral site). a Normalised molar volumes and calculated volume expansion coefficients derived from two-term fitting coefficients. b Normalised polyhedral volumes. c Isotropic atomic displacement parameter for Ba in the modified Debye model. d Isotropic atomic displacement parameter for Zr in the modified Debye model

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The untilting of the CeO6 octahedra increases the polyhedral volume of the irregular BaO8 site in BaCeO3 due to the large magnitude of the Ce–O bond length, whereas the polyhedral volume of the dodecahedral BaO12 site in BaZrO3 is governed by the temperature variation of the Ba–O bond length. The normalised polyhedral volume change of the barium site in BaCeO3 is therefore significantly larger than that in BaZrO3 (Fig. 6b). Whilst the volume of the cavity site in most perovskite-structured compounds is found to increase with increasing temperature, the volume of the octahedral site in zone-boundary-tilted crystal structures is more subtle. For BaZrO3, the octahedral volume expansivity is identical to that of the unit cell volume, i.e. increases with increasing temperature; however, in SrZrO3, the ZrO6 octahedral expansivity is found to be negative (Fig. 6b) with the magnitude of the normalised volume of the ZrO6 octahedron in SrZrO3 being smaller than that determined for BaZrO3.

The atomic displacement parameters for the barium cation in both BaCeO3 and BaZrO3 are similar in magnitude and temperature dependence with the vibrational Debye temperature of BaZrO3 (224(1)  K) only marginally larger than BaCeO3 (200(2)  K) [26] (Fig. 6c). By contrast, the atomic displacement parameters for the zirconium cation in BaZrO3 and SrZrO3 are significantly different in magnitude but exhibit a similar temperature dependence with the vibrational Debye temperature of SrZrO3 (499(9) K) [25] exceeding that of BaZrO3 (426(7)  K) (Fig. 6d). Estimates of the static disorder for the zirconium site in BaZrO3 are approximately 4.5 times larger than the corresponding site in SrZrO3, possibly suggesting the presence of eightfold systematic disorder as observed in BaTiO3 [46].

The recently reported float-zone growth of large single crystals of BaZrO3 [47] would be ideal for a single crystal neutron diffuse scattering study that would permit investigation of any positional disorder of the zirconium site and also the investigation of the proposed glassy structural state.