Structure–property relations and thermodynamic properties of monoclinic petalite, LiAlSi₄O₁₀

Large single crystals of natural petalite of gem quality were chosen for sample preparation. The lattice parameters a₁ = 11.724(2) Å, a₂ = 5.128(1) Å, a₃ = 7.622(2) Å and α₂ = 113.07(2)° obtained from x-ray diffraction experiments on a four-circle diffractometer equipped with graphite-monochromatized Mo Kα-radiation agree well with corresponding literature values (Effenberger 1980). Electron microprobe analyses revealed a nearly ideal chemical composition with traces of Na, K, Mn, Mg, Ca, Fe, Ni and Cr (in total less than 0.05% in weight). The density ρ_X = 2.412(2) g cm⁻³ calculated from sample composition and lattice parameters is in excellent agreement with ρ_b = 2.411(2) g cm⁻³ determined by the buoyancy method on large single crystals in pure water at 293 K. Single crystal structure determination and investigations using a polarizing microscope did not reveal any hint of twinning.

The physical properties reported here are referred to a Cartesian reference system with basis vectors e_i, which are connected to the crystallographic basis vectors a_i according to ${\mathbf{e}}_{1}\parallel {\mathbf{a}}_{1}\parallel {\mathbf{a}}_{1}^{\mathrm{orth}}$, ${\mathbf{e}}_{2}\parallel {\mathbf{a}}_{2}\parallel {\mathbf{a}}_{2}^{\mathrm{orth}}$ and e₃ = (e₁ × e₂) $\parallel {\mathbf{a}}_{3}^{\ast }$, where ${\mathbf{a}}_{3}^{\ast }$ denotes the third basis vector of the reciprocal system. This non-standard setting was chosen to take advantage of the perfect cleavage plane parallel to (001) and of the orthorhombic pseudo-symmetry. The basis vectors ${\mathbf{a}}_{i}^{\mathrm{orth}}$ of the pseudo-orthorhombic unit cell are related to those of the monoclinic cell according to ${\mathbf{a}}_{i}^{\mathrm{orth}}={u}_{i j}{\mathbf{a}}_{j}$ with the transformation matrix

$$\begin{eqnarray} ({u}_{i j})=\left (\begin{array}{@{}ccc@{}} \displaystyle \frac{1}{2}&\displaystyle 0&\displaystyle 0\\ \displaystyle 0&\displaystyle 1&\displaystyle 0\\ \displaystyle \frac{1}{2}&\displaystyle 0&\displaystyle 2 \end{array}\right ). \end{eqnarray} \tag{ 1 }$$

The transformation leads to ${\alpha }_{2}^{\mathrm{orth}}=9 0.4^{\circ}$ (Tagai et al 1982).

Rectangular parallelepipeds with edges parallel to e_i and edge lengths l_i of about 6 mm were cut from large single crystals using a low-speed diamond saw and were polished on diamond disks (mesh 600 and 1200). The sample orientation was controlled by optical and x-ray methods. The deviation from the ideal orientation was kept below 0.4° and the deviation from plane parallelism of opposing faces was smaller than  ± 2 μm. A measure of the quality of a sample with respect to geometrical errors is provided by the difference between the density, ρ_b, obtained by the buoyancy method in pure water at room temperature and the geometric density, ρ_g = M/(l₁l₂l₃), calculated from sample mass M and dimensions l_i. These densities agree within 0.7% for all samples (table 2). Additionally, oriented plane-parallel plates with thicknesses between about 4 and 9 mm suitable for dilatometry and the plate-resonance technique were manufactured.

The accurate determination of the temperature dependence of the elastic coefficients requires a correction for changes in sample dimensions and density due to thermal expansion effects. In order to determine the four independent coefficients of thermal expansion of monoclinic petalite, the temperature-induced strains were studied between 100 and 740 K on plane-parallel plates with plate normals parallel to e_i and (e₁ ±  e₃) using a commercial inductive gauge dilatometer type DIL402C from Netzsch equipped with a sample holder made of fused silica and thermocouples of type E. Each sample was measured at least three times with heating/cooling rates of 1 K min⁻¹ under a He purge gas atmosphere. The dilatometer was first calibrated with standard samples made of fused silica.

The elastic properties were studied with the aid of resonant ultrasound spectroscopy (RUS). A detailed description of the RUS technique can be found elsewhere (Leisure and Willis 1997, Migliori and Sarrao 1997).

For data collection at room temperature the samples were gently fixed between two piezoelectric ultrasound transducer plates (figure 2(a)). One transducer acts as an ultrasound generator and the other as an ultrasound detector. The force acting on the opposed corners of the sample was kept below 0.05 N. This ensured that the experimental setup fulfilled the conditions of a nearly freely vibrating body. Resonance spectra were collected at room temperature in the frequency interval of 100–1400 kHz with a resolution of 0.01 kHz using a network analyzer (HP 4394a from Agilent). In order to excite and detect all resonances four resonance spectra with the sample mounted in different orientations were taken for each sample. An example of an experimentally obtained resonance spectrum at room temperature is depicted in figure 2(b).

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The evaluation of the 13 independent elastic coefficients from the measured resonance frequencies was carried out by a non-linear least-squares procedure in which the observed resonance frequencies were compared to those calculated from the dimensions of the sample, the experimental density and the initial set of elastic coefficients which was derived by the plate-resonance technique (see below). The calculated resonance frequencies were obtained by solving a general eigenvalue problem, the rank of which was equal to the number of basis functions used for the development of the components of the displacement vector. In the non-linear least-squares refinement procedure the quantity

$$\begin{eqnarray} &&{\chi }^{2}=\sum _{i=1}^{n}{w}_{i}({\omega }_{i}^{2}(\mathrm{calc})-{\omega }_{i}^{2}(\mathrm{obs}))^{2} \end{eqnarray} \tag{ 2 }$$

calculated for n circular eigenfrequencies with resonance frequencies f_i = ω_i/2π was minimized by varying the elastic coefficients of the sample. The w_i are individual weights calculated by assuming experimental errors of ±0.1 kHz for each observed resonance frequency. In order to minimize errors due to truncation effects, up to 8775 normalized Legendre polynomials were used in the development of the displacement vector.

The convergence of the refinement procedure depends critically on the proper choice of the initial set of elastic coefficients. Therefore, an appropriate starting model was first determined by applying the plate-resonance technique (for details see Haussühl 2001) to plane-parallel plates of petalite with normals parallel to e_i and ( − e₁ + e₃). These measurements were also required to overcome an intrinsic ambiguity in the interpretation of the RUS spectra that can appear if the point symmetry of the crystal is lower than the one of the sample shape. In the case of our rectangular parallelepipeds with edges parallel to e_i two sets of elastic coefficients related by a non-crystallographic mirror plane perpendicular to e₃ yield free resonances with different eigenvectors but identical frequencies.

The thermoelastic behavior of petalite was studied using a high temperature RUS-device built in-house (Haussühl et al 2011). The temperature stability inside the furnace (type 6.219.1-26 from Netzsch) achieved by a cascading temperature controller (EPC 900 from Eurotherm) was better than  ± 0.5 K. The accuracy of the S-type thermocouple at the maximum temperature was better than  ± 5 K. For the high temperature RUS measurements the samples were slightly clamped at opposite corners between the ends of two horizontally arranged corundum ceramic rods which acted simultaneously as ultrasonic wave guides. The ultrasound generator and detector were mounted on the cold ends of the corundum rods outside the furnace. For signal generation and detection a network analyzer (HP4194a from Agilent) in combination with a high-speed signal amplifier (type 4400 from NF-electronics) was employed. Resonance spectra in the frequency range 100 and 1400 kHz were collected between 293 and 573 K in steps of 20 K in air. At each temperature step the sample was first allowed to equilibrate for at least 30 min before the resonance spectrum was recorded. After extraction of the resonance frequencies the elastic coefficients at each temperature were refined using the set of elastic coefficients evaluated at the previous temperature step as initial values. Further, changes of sample dimensions l_i(T) = l_i(T₀)(1 + ε_ii(T)) and density

$$\begin{eqnarray} &&\rho (T)=\rho ({T}_{0})\frac{V({T}_{0})}{V(T)}=\rho ({T}_{0})\frac{{l}_{1}({T}_{0}){l}_{2}({T}_{0}){l}_{3}({T}_{0})}{{l}_{1}(T){l}_{2}(T){l}_{3}(T)} \end{eqnarray} \tag{ 3 }$$

due to thermal expansion effects were taken into account. T₀ = 293 K is the reference temperature and ε_ii denotes the strain along e_i as observed by dilatometry.

The heat capacity of petalite has been obtained in the range from 2 to 395 K on a single crystal of about 15 mg in weight using a Quantum Design PPMS system. Data sets were collected at 120 temperatures where the temperature difference between subsequent steps was increased logarithmically from low to high temperatures. At each temperature, 3 responses to a heat pulse were measured and analyzed by the MultiVu software under which the measurements were run.

While the temperature-induced strains ε_ii could directly be studied on the rectangular parallelepipeds, the component ε₁₃ = (ε' − ε'')/2 of the strain tensor was derived from the longitudinal strains ε' and ε'' observed along e₁ + e₃ and e₁ − e₃, respectively. The thermal expansion of petalite is highly non-linear (figure 3); thus, fourth order polynomials of the type

$$\begin{eqnarray} &&{\varepsilon }_{i j}(T)={\alpha }_{i j}\Delta T+{\beta }_{i j}(\Delta T)^{2}+{\gamma }_{i j}(\Delta T)^{3}+{\delta }_{i j}(\Delta T)^{4} \end{eqnarray} \tag{ 4 }$$

with ΔT = T − T₀ are required for a proper approximation of the strains ε_ij over the entire investigated temperature range. T₀ = 293 K denotes the reference temperature and α_ij, β_ij, γ_ij and δ_ij are the corresponding coefficients of linear, quadratic and higher order thermal expansion. With less than 0.1% the uncertainties of the fit parameters are rather small (table 1). In fact, the reproducibility of the linear thermal expansion coefficients α_ij at room temperature is of the order of  ± 3%.

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A total of three different samples were investigated with resonant ultrasound spectroscopy at room temperature. All observed resonances could be unambiguously identified and were used in the non-linear least-squares procedures. The fully converged sets of elastic coefficients are presented in table 2. In all cases an excellent match between calculated and observed resonance frequencies could be achieved as indicated by the low average and maximum values of Δf_i = |f_i(calc) − f_i(obs)|. This reflects the high precision of the RUS technique. The corresponding longitudinal and shear stiffnesses of the different samples agree within 0.3%. Deviations of up to 0.5 GPa are observed for the small coupling coefficients c₁₅ and c₃₅.

Between room temperature and 573 K the temperature evolution of the elastic coefficients was fully reversible in successive heating and cooling runs within experimental error (figure 4). With the exception of c₅₅ all principal elastic coefficients c_ii exhibit negative temperature coefficients ∂c_ii/∂T at room temperature. However, most of the principal stiffnesses display a distinct non-linear softening with increasing temperature, which can be best approximated by second order polynomials (table 3).

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Table 3.  Temperature coefficients of elastic moduli of petalite as derived by polynomial regressions of type ${c}_{i j}(T)={c}_{i j}^{0}+{c}_{i j}^{1}(T-{T}_{0})+{c}_{i j}^{2}(T-{T}_{0})^{2}$ with reference temperature T₀ = 295 K from experimental data obtained on sample 2 (cf table 2). The numbers in parentheses are the uncertainties of the last digit as derived from the standard deviations of the corresponding fit parameter. The thermoelastic coefficients T_ij = ∂logc_ij/∂T are for 295 K.

ij	${c}_{i j}^{0}$(GPa)	${c}_{i j}^{1}$(MPa K−1)	${c}_{i j}^{2}$(kPa K−2)	${T}_{i j}={c}_{i j}^{0}/{c}_{i j}^{1}$(10−4 K−1 )
11	91.30(2)	−19.9(3)	+11.7(9)	−2.2
22	93.03(2)	−12.2(4)	+11.2(13)	−1.3
33	152.85(4)	−35.7(6)	+26(2)	−2.3
44	29.31(1)	−2.5(1)	−6.2(2)	−0.85
55	69.73(1)	+0.6(1)	−3.7(5)	+0.09
66	52.60(1)	−0.6(1)	−1.2(3)	−0.11
12	−0.03(2)	−3.8(4)	+7.9(14)	+1267
13	69.07(2)	−23.7(4)	+16.0(15)	−3.4
23	6.47(3)	−6.9(5)	0(2)	−11
15	−0.44(4)	+2.3(7)	+12(2)	−52
25	−1.32(3)	+5.8(5)	+10(2)	−44
35	−1.16(4)	+1.1(7)	+6(3)	−9.5
46	2.41(1)	−0.6(1)	−1.3(4)	−2.5

The experimentally obtained heat capacity is shown in figure 5, where it is compared to data obtained by Hemingway et al (1984) There is no anomaly in the temperature range investigated, i.e. there is no indication of a phase transition. The agreement between the data obtained here on a very small sample of  ≈ 20 mg are, generally, in excellent agreement with the data obtained on a sample weighing 306 g by Hemingway et al (1984). In effect, the data sets are indistinguishable at temperatures T > 10 K. At lower temperatures, there are small discrepancies. The data obtained at the lowest temperatures accessible to Hemingway et al (1984) show an incorrect temperature dependence, in that the heat capacity decreases with increasing temperature. In our data, there is no such anomaly and we are confident that our low temperature data are more reliable than those presented by Hemingway et al (1984). At low temperatures (T < 3 K), the heat capacity depends linearly on T³ (figure 5). Using the low temperature approximation to the Debye model,

$$\begin{eqnarray} &&{c}_{v}=\frac{1 2{\pi }^{4}}{5}n R\left (\frac{T}{{\Theta }_{\mathrm{calorim}}}\right )^{3} \end{eqnarray} \tag{ 5 }$$

where n = 16 is the number of atoms per formula unit, R = 8.314 462 J (mol K)⁻¹, and neglecting the difference between c_v and c_p, the linear dependence gives a Debye temperature Θ_calorim of 551 K. Petalite differs from other compounds we have investigated in that significant deviations from the T³-dependence already begin at 3 K, while for other silicates and related compounds the linear dependence is observed up to 5 or 6 K. As the analysis relies on a T³-dependence, this is a significant difference. Using the heat capacity data, the molar enthalpy of formation (△_fH° = 38 378 ± 15 J mol⁻¹) and the standard molar entropy (S° = 233.9 ± 0.2 J K⁻¹ mol⁻¹) at 298.15 K were calculated. The values agree very well with the previous result (S° = 233.2 ± 0.6 J K⁻¹ mol⁻¹; Hemingway et al 1984).

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The elastic behavior of petalite exhibits a pronounced anisotropy as expressed by the ratio c'(max)/c'(min) ≈ 2.0 of the maximum and minimum of the longitudinal elastic stiffness c'(u) = u_iu_ju_ku_lc_ijkl, where u_i are direction cosines (cf table 5). The two nearly equivalent local maxima of c'(u) within the (010) plane, c' = 172.5 GPa along 0.5670e₁ + 0.8237e₃ and c'' = 168.8 GPa along  − 0.5680e₁ + 0.8231e₃ enclosing angles of  − 34.6° and  + 34.5° with the e₃-axis, give rise to a nearly orthorhombic pseudo-symmetry of the corresponding representation surface (figures 9(a)–(c)). Figure 8 shows that the directions of these extreme values almost coincide with the orientations of Si–O–Si-bonds. Within the plane perpendicular to e₃ the longitudinal elastic stiffness reaches only about 50% of its maximum value and behaves nearly isotropically.

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Table 5.  Comparison of elastic properties of selected lithium silicates and α-quartz. ρ density in g cm⁻³, M_W molar weight in g mol⁻¹, c'(max)/c'(min) ratio of maximum and minimum longitudinal elastic stiffness, ${c}_{i j}^{\mathrm{iso}}$ aggregate elastic coefficients in GPa (average of Voigt and Reuss model), ${g}_{i i}^{0}$ eigenvalues of deviations from Cauchy relations in GPa, g mean deviation from Cauchy relations, B (Reuss model) and C bulk modulus and mean elastic stiffness in GPa, S_exp and S_calc experimental and calculated elastic S-value in 10⁻²⁰ N m.

Mineral	α-quartz	Petalite	β-eucryptite	Spodumene
Composition	SiO2	LiAlSi4O10	LiAlSiO4	LiAlSi2O6
Reference	Heyliger et al (2003)	This work	Haussühl et al (1984)	Sondergeld et al (2006)
Si:Al:Li	1:0:0	4:1:1	1:1:1	2:1:1
ρ	2.650	2.411	2.352	3.182
MW	60.09	306.28	126.01	186.10
c'(max)/c'(min)	1.75	2.0	1.42	1.43
${c}_{1 1}^{\mathrm{iso}}$	97.1	107.8	164.5	259.5
${c}_{1 2}^{\mathrm{iso}}$	8.0	20.9	73.1	87.7
${c}_{4 4}^{\mathrm{iso}}=({c}_{1 1}^{\mathrm{iso}}-{c}_{1 2}^{\mathrm{iso}})/2$	44.6	43.5	45.7	85.9
Biso	37.7	49.9	103.5	145.0
B	37.4	45.5	102.7	143.9
${g}_{1 1}^{0}$	−46.3	−53.1	+26.0	−12.4
${g}_{2 2}^{0}$	−46.3	−22.4	+26.0	−4.2
${g}_{3 3}^{0}$	−33.7	−0.7	+22.1	+11.4
g = (g11 + g22 + g33)/3	−42.1	−26.6	+24.7	−1.1
C	52.0	62.9	96.1	146.5
Sexp	195	1338	855	1423
Scalc	195	1154	860	1459

A deeper insight into the nature of the bonding interactions in crystals is provided by the deviations from Cauchy relations, a second-rank tensor invariant of the elasticity tensor with components g_mn = c_ijkle_ikme_jln/2, where e_ijk are the components of the Levi-Civitá tensor (Haussühl 1967, 2007, Schreuer and Haussühl 2005). In crystals with strong ionic bonds, and particularly in those containing aspherical or highly polarizable constituents, the transverse interaction coefficients usually dominate considerably over the corresponding shear stiffnesses, resulting in positive deviations from Cauchy relations. Strong covalent or other bonds with preferential orientation give rise to large shear stiffnesses and therefore cause opposite effects. For example, the relatively high covalency of the Si–O bond leads to strongly negative deviations from Cauchy relations in α-quartz (table 5). In contrast, the eigenvalues ${g}_{i i}^{0}$ of the (g_mn) tensor of β-eucryptite, LiAlSiO₄, are all positive because here 1/3 of the tetrahedral positions are occupied by Li, where the Li–O interactions have a predominantly ionic character.

Petalite presents a case intermediate between β-eucryptite and quartz. As has been discussed before, the bond populations obtained from the DFT calculations clearly show the covalent character of the Si–O and Al–O bonds, while there is no charge accumulation along Li–O contacts, so the interactions within the Li–O tetrahedra are purely electrostatic. The ratio of predominantly covalently bonded (Si,Al)O₄-tetrahedra to ionic LiO₄-tetrahedra is therefore 5:1 in petalite, while it is 1:0 in quartz and 2:1 in β-eucryptite, respectively. One can therefore expect a value of the mean deviation form Cauchy relations g for petalite which is intermediate between those of quartz and β-eucryptite. This is indeed the case (table 5).

The bulk modulus derived here from experiment and theory is about 38% larger than B = 33 GPa derived by Fasshauer et al (1998) from compressibility data, to which a Murnaghan equation of state was fitted. The latter data set was obtained in a very small pressure interval from 1.2–5.7 GPa only and no detailed information on the data analysis is presented. Due to the limited pressure range, the uncommonly small value derived for the pressure derivate, B' = 0.1, by Fasshauer et al (1998) will certainly have a very large error associated with it. We therefore are confident that the data presented here (B = 45 GPa, B' ≈ 3) are superior to the value used by Fasshauer et al (1998). We note that other values employed in the thermodynamic analysis of Li-bearing systems by Fasshauer et al (1998) have also been superseded by new results, which leads us to conclude that this internally consistent database may be of limited accuracy.

It is well established that the bulk modulus of quartz is uncommonly small for silicates, and that this is due to the presence of rigid unit modes, which permit a compression of the structure by a cooperative rotation of quasi-rigid tetrahedra. The bulk modulus of petalite is similar to that of α-quartz, and hence it is likely that in this structure deformations by small changes of O–T–O bond angles are also possible. In β-eucryptite and spodumene the larger number of cation–anion contacts and, respectively, the much higher packing density lead to considerably stiffer crystal structures.

A detailed comparison of the elastic properties of crystal species belonging to different point symmetry groups can be achieved by comparing further scalar invariants of the elasticity tensor, such as the mean elastic stiffness C = (c₁₁ + c₂₂ + c₃₃ + c₁₂ + c₁₃ + c₂₃ + c₄₄ + c₅₅ + c₆₆)/9. In order to take into account the effect of packing, Haussühl (1993) introduced the elastic S-value as the product S = CV_m of the mean elastic stiffness and the molecular volume V_m = M_w/(N_Aρ), where M_w is molar weight, N_A is Avogadro's number and ρ is the density. Under the assumption that the interactions between cations and surrounding anions are similar in a compound X and in its stable constituents X_i the S-value of X can be decomposed into additive contributions S(X_i) according to S(X) = ∑S(X_i). This quasi-additivity rule holds for many ionic crystals and even for many silicates to within about 10% (Haussühl 1993, Schreuer and Haussühl 2005). For petalite the quasi-additivity rule leads to

$$\begin{eqnarray*} &&S({\mathrm{LiAlSi}}_{4}{\mathrm{O}}_{1 0})=1/2 S({\mathrm{Li}}_{2}\mathrm{O})+1/2 S({\mathrm{Al}}_{2}^{\mathrm{IV }}{\mathrm{O}}_{3})+4 S({\mathrm{SiO}}_{2}) \end{eqnarray*}$$

where S(Li₂O) = 228 × 10⁻²⁰ N m, $S({\mathrm{Al}}_{2}^{\mathrm{IV }}{\mathrm{O}}_{3})=5 2 0(5 0)\times 1{0}^{-2 0}$ N m and has been derived for a hypothetical compound with Al in tetrahedral coordination (Schreuer and Haussühl 2005). SiO₂ crystallizes in a number of modifications whose S-values differ by more than a factor of 5 (Schreuer and Haussühl 2005). In the case of β-eucryptite, the use of S(quartz) = 195 × 10⁻²⁰ N m is obvious, since its crystal structure can be considered a derivative of the β-quartz structure. A reasonable agreement between the experimental and calculated S-value of petalite can be achieved if the S-value of α-quartz is used for S(SiO₂). In contrast, the almost closed-packed arrangement of oxygen atoms in spodumene requires the use of S(coesite) (table 5).

Among the thermo-mechanical properties investigated here only the fourth-rank tensor (∂c_ij/∂T) of the temperature coefficients of the elastic coefficients shows significant deviations from the orthorhombic pseudo-symmetry (figures 9(g)–(i)). In particular, the two local maximum values, c' and c'', within the (010) plane have different temperature dependences. In conjunction with the negative thermal expansion along e₂ (see below) this probably is indicative of a cooperative rotation of coordination polyhedra.

Remarkable features are the pronounced anisotropy and the non-linearity of the temperature dependence of the elastic stiffnesses (figure 4). For example, between room temperature and 570 K, c₄₄ softens by about 4% whereas the shear stiffnesses c₅₅ and c₆₆ are nearly independent of temperature. Regarding the sign of their second derivatives with respect to temperature the principal elastic coefficients of petalite can be divided into two groups (table 3). With ∂²c_ij/∂T² < 0 the shear stiffnesses develop normally with increasing temperature. However, the longitudinal stiffnesses and the transverse interaction coefficients all show ∂²c_ij/∂T² > 0, i.e. the softening of these elastic coefficients reduces with increasing temperature. An understanding of this observation would require ultrasound and diffraction measurements at higher temperatures, which are, however, difficult due to an often observed spontaneous cleaving of the petalite samples at elevated temperatures.

Petalite shows no observable dissipation phenomena in the investigated temperature and frequency range. This implies that there is no significant diffusion, e.g. of lithium or oxygen vacancies.

The thermal expansion of petalite shows an orthorhombic pseudo-symmetry of the type α₃₃ < 0 ≈ α₁₁ < α₂₂ (cf figures 9(j)–(l)). Components of tensors of different ranks cannot be compared directly, but as usually observed, the direction with maximum thermal expansion almost coincides with that of minimal longitudinal elastic stiffness and vice versa. Negative thermal expansion is observed along e₃. In framework structures, negative thermal expansion is often due to a cooperative rotation of coordination polyhedra, and this is very likely the case in petalite as well. However, a confirmation of this hypothesis would require high temperature diffraction experiments. The thermal expansion provides no indication of a phase transition in the temperature range studied, but it is known that at temperatures T > 1200 K high temperature polymorphs are formed (Tagai et al 1982).

The significant discrepancy between the low temperature heat capacities obtained experimentally and those from model calculations were not to be expected as the theoretical and experimental elastic stiffness coefficients were in good agreement. The low frequency phonon density of states is usually dominated by the dispersion of the acoustic phonons, and once the slopes of the acoustic phonons, i.e. the elastic stiffness coefficients, are well described by a model, the low temperature heat capacity should be also predicted with a reasonable accuracy. A low frequency motion due to ions rattling in a coordination polyhedron slightly too large for them is unlikely, as all cation–oxygen bonds have values typical for tetrahedral coordination and the anisotropic displacement parameters of the cations are not excessively large at ambient temperatures (Effenberger 1980, Tagai et al 1982). However, petalite does have an unusual 180° Si–O–Si bond (Effenberger 1980). In analogy to the well-studied situation in β-cristobalite (Gambhir et al 1999), this points towards the possibility that, in fact, the time and space-averaged structure obtained from diffraction experiments may hide a small tilt of statically or dynamically disordered tetrahedra. This hypothesis is supported by the observation of rather anisotropic, plate-like atomic displacement parameters of those oxygen atoms, which participate in Si–O–Si bonds (Effenberger 1980). Low frequency contributions to the phonon density of states can be studied by thermal diffuse scattering (Bosak et al 2012) or by total scattering experiments (Dove et al 1997).