 1. Introduction
 2. Parameterization of the cation frameworks and the O4 tetrahedra in αquartz, GeO2 and βquartz structures
 3. The temperature variation of [Si+PC] parameters for quartz and [Ge+PC] parameters for germanium dioxide
 4. Discussion
 A1. Curvefitting coefficients for αquartz
 A2. Curvefitting coefficients for βquartz
 Supporting information
 References
 1. Introduction
 2. Parameterization of the cation frameworks and the O4 tetrahedra in αquartz, GeO2 and βquartz structures
 3. The temperature variation of [Si+PC] parameters for quartz and [Ge+PC] parameters for germanium dioxide
 4. Discussion
 A1. Curvefitting coefficients for αquartz
 A2. Curvefitting coefficients for βquartz
 Supporting information
 References
research papers
Modelling the structural variation of quartz and germanium dioxide with temperature by means of transformed crystallographic data
^{a}Werkstofftechnik Glas and Keramik, Hochschule Koblenz, Rheinstrasse 56, 56203 HoehrGrenzhausen, Germany
^{*}Correspondence email: thomas@hskoblenz.de
The pseudocubic (PC) parameterization of O_{4} tetrahedra [Reifenberg & Thomas (2018). Acta Cryst. B74, 165–181] is applied to quartz (SiO_{2}) and its structural analogue germanium dioxide (GeO_{2}). In αquartz and GeO_{2}, the pseudocubes are defined by three length parameters, a_{PC}, b_{PC} and c_{PC}, together with an angle parameter α_{PC}. In βquartz, α_{PC} has a fixed value of 90°. For quartz, the temperature evolution of parameters for the pseudocubes and the silicon ion network is established by reference to the structural refinements of Antao []. In αquartz, the curvefitting employed to express the nonlinear temperature dependence of pseudocubic length and Si parameters exploits the model of a firstorder Landau utilized by Grimm & Dorner [J. Phys. Chem. Solids (1975), 36, 407–413]. Since values of tetrahedral tilt angles about 〈100〉 axes also result from the pseudocubic transformation, a curve for the observed nonmonotonic variation of α_{PC} with temperature can also be fitted. Reverse transformation of curvederived values of [Si+PC] parameters to crystallographic parameters a, c, x_{Si}, x_{O}, y_{O} and z_{O} at interpolated or extrapolated temperatures is demonstrated for αquartz. A reverse transformation to crystallographic parameters a, c, x_{O} is likewise carried out for βquartz. This capability corresponds to a method of structure prediction. Support for the applicability of the approach to GeO_{2} is provided by analysing the structural refinements of Haines et al. [J. Solid State Chem. (2002), 166, 434–441]. An analysis of trends in tetrahedral distortion and tilt angle in αquartz and GeO_{2} supports the view that GeO_{2} is a good model for quartz at high pressure.
Keywords: quartz; pseudocubic; INA; tilt angle; structure prediction; GeO_{2}.
1. Introduction
Although the α↔β quartz inversion has been an issue of scientific investigation for some 130 years (Dolino, 1990), a strong stimulus to review current modelling methods for its crystal structures has been provided by the work of Antao (2016). By using synchrotron powder Xray diffraction coupled with Rietveld structure refinements, she extended the range of structural data well into the temperature range of stability of βquartz and provided a set of structural data for α and βquartz with a fine temperature mesh. A total of 67 new structural refinements resulted from her work, 42 for αquartz and 25 for βquartz, thereby providing an extensive dataset for structural analysis.
The foundation of several structural modelling studies of quartz and its homeotypes was laid by Grimm & Dorner (1975), who identified the tilt angle of δ of SiO_{4} tetrahedra about 〈100〉 axes in αquartz as the microscopic in a firstorder Landau model of the α↔β Parameter δ_{0} in equation (1) corresponds to the jump in tilt angle at the transition temperature T_{0}, with T_{c} a scaling parameter.
Grimm and Dorner (1975) assumed regular tetrahedra as a starting point for fitting equation (1) to values of δ derived from crystallographic data. This resulted in the values δ_{0} = 7.3°, T_{0} = 846 K and T_{0} − T_{c} = 10 K. They noted that the accuracy of the crystallographic data then available was insufficient to test the validity of equation (1), further that `a direct measurement of the tilt angle analogous to the case of SrTiO_{3} would be desirable.'
This notwithstanding, equation (1) has been widely adopted in subsequent studies of the temperature dependence of the structures of αquartz and its homeotypes. This can be attributed to the greater suitability of δ or δ^{2} compared to direct temperature as an independent variable when describing the temperature dependence of structural parameters such as the Si—O—Si angle by means of loworder polynomials. It is appropriate, therefore, to regard δ, also denoted Q in later studies, as a temperaturederived tilt angle, irrespective of the degree of agreement with a structurally derived tilt angle.
Carpenter et al. (1998) adopted this approach to derive quadratic relationships between spontaneous strains e_{1} and e_{3} with Q^{2}. They also utilized the structural data of Kihara (1990) for αquartz to reveal a linear relationship between the mean Si—O bond length and the square of the tilt angle. By virtue of her extensive structural dataset, Antao (2016) has further shown that strain parameters e_{1}, e_{3}, (c/a) and V_{s} vary linearly with Q^{2} for αquartz She also proposed linear relationships between atomic parameters z_{O} and x_{Si} with Q. Mean Si–Si distances and Si—O—Si angles were also shown to vary systematically with Q. In both cases, tilt angle was calculated according to the method of Grimm & Dorner (1975) assuming regular tetrahedra.
The structural refinements of Antao (2016) refer to P3_{2}21 (No. 154) for αquartz and P6_{2}22 (No. 180) for βquartz. The coordinates for αquartz correspond to the z(+)setting (Donnay & Le Page, 1978)^{1}. When cooling righthanded βquartz (space group P6_{2}22), formation either of an α_{1} or an α_{2} trigonal structure depends on the sense of the tetrahedral tilting.^{2} These are in space groups P3_{2}21 and P3_{1}21, respectively.
Ever since the early crystalchemical treatments of quartz, the view has dominated that the SiO_{4} tetrahedra deviate insignificantly from perfect regularity. Megaw (1973a) states this clearly: `We have already recognized the importance of a regular (or nearly regular) tetrahedron as a structurebuilding unit.' In the seminal work of Grimm & Dorner (1975) in relating tetrahedral tilt angle to the Landau in equation (1), the assumption of regular tetrahedra was maintained as an expedient. Taylor (1984) explicitly called this assumption into question, to quote from his abstract: `Tilting models of framework compounds are critically examined and their failure to match the observed structural behaviour is attributed to changes in tetrahedral distortion. For quartz it appears that during compression the change in tetrahedral distortion is virtually all angular (O—Si—O angles), whereas during the change in distortion is in the Si—O distances. Such behaviour may typify the behaviour of many other framework compounds but the structural data needed to establish this are lacking.'
The current availability of highquality structural data for quartz following the work of Antao (2016) now supersedes the final remark of Taylor for this framework compound. Furthermore, a new approach for quantifying the distortions of O_{4} tetrahedra has recently been proposed by Reifenberg & Thomas (2018). In the latter work, the pressure variation of the structure of the coesite polymorph of SiO_{2} was taken as a basis for defining a general procedure known as a pseudocubic transformation. Just as it is possible to generate a regular tetrahedron from a cube by taking two diagonally related corners of each cube face, the reverse procedure also holds: a regular tetrahedron will generate a regular cube, whereas a distorted tetrahedron will generate a distorted cube known as a pseudocube (Fig. 1). Such a pseudocube is, in general, characterized by six parameters, a_{PC}, b_{PC}, c_{PC}, α_{PC}, β_{PC} and γ_{PC} (Fig. 1), as for a triclinic As shown in Fig. 1(a), the shape of a generalized tetrahedron is also defined by six parameters. It follows that all volumes and types of distortion of tetrahedral O_{4} cages can be quantitatively modelled by pseudocubic transformations.
Whereas the distorted O_{4} tetrahedra in coesite result in six independent pseudocubic parameters, the twofold symmetry axes through their centresofcoordinates in αquartz dictate that two of the pseudocubic angles are equal to 90°.
This is shown in Fig. 2(a), in which pseudocubic axes a_{PC} are oriented parallel to the twofold axes. The faceon view of the pseudocube along the xaxis in Fig. 2(b) shows a parallelogram with twofold symmetry and internal angle α_{PC}.
A secondary result of the pseudocubic transformation is that it allows angles ϕ_{v} and ϕ_{h} to be defined as direct indicators of tetrahedral tilt angle ϕ about the [100] axis, and more generally 〈100〉 axes: a tetrahedral rotation by this angle also leads to a rotation ϕ of its pseudocube about the same axis. However, unlike the tetrahedral edge vectors, the edge vectors of the pseudocube are aligned with the crystal axes. Angle ϕ_{v} is defined as the angle between pseudocubic axis c_{PC} and crystal axis z and angle ϕ_{h} as the angle between pseudocubic axis b_{PC} and its projection in the crystal xy plane. Owing to small deviations of pseudocubic angle α_{PC} from 90°, ϕ_{h} and ϕ_{v} are not exactly equal to each other. Nevertheless, a method is now provided for measuring the tetrahedral tilt angle directly, as sought by Grimm & Dorner (1975). The method does not require any approximations or abstract geometrical reference points other than the crystal axes. In the current work, the dependence of ϕ_{h}, ϕ_{v} and mean tilt angle ϕ = (ϕ_{h} + ϕ_{v})/2 on temperaturederived tilt angle δ (or equivalently Q) are examined, thereby revealing the extent to which equation (1) holds for αquartz.
The significance of a direct measurement of tilt angle may be made clear by comparing the completely general pseudocubic method with alternative structural approaches advocated by Megaw (1973b) and Grimm & Dorner (1975) for quartz, as well as the method of Haines et al. (2003) adopted for the quartz homeotype FePO_{4}. Megaw adopted as a basis an idealized tetrahedron of orthorhombic symmetry, as in βquartz, and maintained this form as an approximation in αquartz. This approach is equivalent to allowing a pseudocube with unequal edge lengths but with angle α_{PC} fixed at 90°. The method of Grimm and Dorner is more restrictive, as it amounts to assuming a regular cube as the pseudocubic form. Haines et al., by comparison, examined the deviations in orientation of tetrahedral edges PR and QS from ±45° (Fig. 3).
Fig. 3 shows the alternative senses of tilt in P3_{2}21 for αquartz and in its enantiomorphic P3_{1}21, in which the 16 structures of GeO_{2} between 294 and 1344 K to be examined here were set (Haines et al., 2002).^{3}
In addition to investigating the validity of equation (1) in describing the temperature variation of tilt angle as determined by the pseudocubic method, an important further aim of this work is to exploit the pseudocubic transformation for the purpose of structure prediction at temperatures outside the ranges of experimental investigation of Antao (2016) and Haines et al. (2002). Since the pseudocubes only relate to the oxygen ions, the silicon or germanium ions are treated in a separate cationic network. This is consistent with the general methodology of ionic network analysis (INA) (Thomas, 2017). In Fig. 4, the positions of the silicon ions along the screw axes in αquartz have been collapsed on to the xy plane, in order to form a twodimensional framework defined by parameters L and Δ. Δ is equal to zero in the highersymmetry βstructure.
The crystal structures of αquartz and GeO_{2} are defined by two unitcell and four positional parameters, i.e. a, c, x_{Si}, x_{O}, y_{O} and z_{O}, which are known collectively as six (d.o.f.). In βquartz, by comparison, there are three degrees of freedom^{4}, i.e. a, c and x_{O}. The question arises as to how many independent transformed parameters are required to define the O_{4} pseudocubes and silicon ion networks in the two quartz modifications. For αquartz, six independent parameters are required, i.e. a_{PC}, b_{PC}, c_{PC}, α_{PC}, L and Δ. These match exactly the six d.o.f. of the structure. In βquartz, just three independent parameters are required, although the pseudocubes and silicon ion network deliver four: a_{PC}, b_{PC}, c_{PC} and L. This disparity is resolved by noting that parameters a_{PC} and b_{PC} are interdependent.^{5} It should also be noted that the tetrahedral tilt angle in αquartz, ϕ, is not a transformed parameter in this sense: if the six crystal structural parameters or alternatively the six independent transformed parameters are known, the value of ϕ follows by calculation.
This article is structured as follows. In §2, analytical expressions are given for the values of transformed parameters a_{PC}, b_{PC}, c_{PC}, α_{PC}, L and Δ, henceforth denoted [Si+PC] or [Ge+PC], in terms of crystal structural parameters a, c, x_{Si} or x_{Ge}, x_{O}, y_{O} and z_{O}. An expression is also given for tilt angles ϕ_{v} and ϕ_{h} in terms of crystal structural parameters. In §3.1, the transformed parameters calculated for αquartz are summarized by reference to Table S1 in §4 of the supporting information. Sections §3.2 to §3.4 refer to αquartz: the temperature variation of the three tilt angles ϕ_{v} and ϕ_{h} and mean tilt angle ϕ = (ϕ_{h} + ϕ_{v})/2 is compared to the temperaturederived value of tilt angle according to equation (1) in §3.2. This equation is subsequently exploited as a baseline curve for a quantitative description of the variation of the three tilt angles with temperature. In §3.3, curves are derived for the temperature variation of [Si+PC] parameters in αquartz, with their application for the purpose of structure prediction shown in §3.4. §3.5 deals with GeO_{2} as a whole, referring to Table S2 in §4 of the supporting information. In §3.6, βquartz is likewise dealt with as a whole, with reference made to Table S3. In §4.1 a comparison of the temperature and pressuredependent behaviour of αquartz and GeO_{2} is made, with a discussion of the significance of tetrahedral distortions in framework structures carried out in §4.2.
2. Parameterization of the cation frameworks and the O_{4} tetrahedra in αquartz, GeO_{2} and βquartz structures
The analytical treatment here applies to the three space groups relevant to the experimental data of Antao (2016) and Haines et al. (2002), i.e. P3_{2}21, P3_{1}21 and P6_{2}22. Although the notation x_{Si} is used, it is to be understood that this also applies to the xcoordinate for germanium in the GeO_{2} structure. The equations quoted here are derived as follows in the supporting information: §S1: cationic network parameters L and Δ in αquartz and GeO_{2}; §S2: PC parameters and tilt angles in αquartz and GeO_{2}; §S3: PC parameters in βquartz. These derivations are based on the appropriate symmetry, in order to fix the Si or Ge ions in space and to form connected O_{4} tetrahedral cages.
2.1. The cationic network
The transformations from a and x_{Si} to L and Δ for αquartz are as follows:
In the case of βquartz, the value x_{Si} = 0.5 leads to the results L = a/2 and Δ = 0.
Reverse transformation from [Si+PC] to crystal structural parameters proceeds according to equations (4) and (5).
Quadratic equation (5) follows from equation (1):
The smaller of the two roots corresponds to the value of x_{Si}.
2.2. Pseudocubic parameters and tilt angles in αquartz and germanium dioxide
The six parameters of the pseudocubes for the O_{4} tetrahedra may be calculated as follows from unitcell parameters a and c together with the x, y and z parameters of the oxygen ions:
The expression for parameter b_{PC} depends on whether P3_{2}21 or P3_{1}21 applies, as for αquartz and GeO_{2}, respectively.
[for P3_{2}21],
[for P3_{1}21],
The results for parameter α_{PC} are likewise dependent on the that applies.
[for P3_{2}21] ,
[for P3_{1}21]
In Figs. 3(a) and 3(b), tilt angles ϕ_{v} and ϕ_{h} are shown for tetrahedra with cations at x_{Si}, 0, and x_{Ge}, 0, in αquartz and GeO_{2}, respectively. In both cases,
and
Calculation of the mean tiltangle,
is straightforward. From the geometry in Fig. 3, it follows that
Equation (16) represents an alternative to equations (10) and (11) for calculating the pseudocubic angle α_{PC}. It also reveals how deviations of the pseudocubic angle from 90° result from differences in the values of tiltangles ϕ_{v} and ϕ_{h}.
The INA method demands that reverse transformations from pseudocubic to crystal structural parameters can take place. In this connection, equations (4) and (5) relating to the cationic network enable this for cell parameter a and cation parameter x_{Si}. The remaining four parameters, i.e. c, x_{O}, y_{O}, z_{O}, may be calculated as follows from the pseudocubic parameters. Parameter x_{O} is derived from a_{PC} via equation (6). Parameters c, y_{O} and z_{O} are derived from the values of b_{PC}, c_{PC} and α_{PC} by finding selfconsistent solutions of equations (7) to (11) using numerical methods. These reverse transformations are carried out in §3.4 for αquartz.
2.3. Pseudocubic parameters in βquartz
The six parameters of the pseudocubes for the O_{4} tetrahedra in βquartz may similarly be calculated analytically from unitcell parameters a and c together with the x_{O} parameter of the oxygen ions^{6}:
The lengths of pseudocubic axes a_{PC} and b_{PC} are interdependent, since both are determined by parameters a and x_{O}. Parameter L in the cation network is equal to twice the unitcell parameter a, and the pseudocubes yield values for x_{0} and c by reverse transformation. These transformations are carried out in §3.6.
3. The temperature variation of [Si+PC] parameters for quartz and [Ge+PC] parameters for germanium dioxide
3.1. Parameters calculated for αquartz
[Si+PC] parameters calculated for αquartz from the data of Antao (2016) are listed in Table S1. Also listed are the volumes of the (V_{UC}), tetrahedral volumes (V_{tetra}), the ratios of the volume occupied by tetrahedra to the unitcell volume (3V_{tetra}/V_{UC}), the lengthbased tetrahedral distortion parameters (λ_{PC}) [equation (21); Reifenberg & Thomas, 2018], together with tilt angles ϕ_{v} and ϕ_{h}.
with L_{0,PC} = (a_{PC} + b_{PC} + c_{PC})/3.
3.2. Curvefitting for the temperature variation of tilt angles in αquartz
The correlation of values of δ calculated from equation (1) using the parameters of Grimm & Dorner (1975) with values of ϕ_{v}, ϕ_{h} and ϕ calculated directly from the structural refinements of Antao (2016) via equations (13) to (15) is shown in Fig. 5.
It is observed that the correlation between the black curve and the other three curves is only qualitative. This indicates that, although the predominant contribution to the microscopic Landau
is made by tetrahedral tilting, there will also be a small contribution to this from tetrahedral distortion.Fitting of the curves linking experimental points for ϕ_{v}, ϕ_{h} and ϕ_{m} was carried out by expressing these three parameters as a function of δ, using polynomials of order 3. The fitting coefficients are listed in Table 1.

3.3. Curve fitting for the temperature variation of [Si+PC]parameters in αquartz
Values of parameters L, Δ, a_{PC}, b_{PC}, c_{PC} and α_{PC} from Table S1 for temperatures between 298 and 844 K are plotted as points with associated error bars in Fig. 6.
The method adopted for fitting the curves was consistent with the work of other authors (Grimm & Dorner, 1975; Carpenter et al., 1998; Antao, 2016), in that the δ generated by equation (1) was adopted as the independent variable. The fitting coefficients listed in Table 2 relate to the reduced δ′ defined in equation (22)
Here δ_{0} is the parameter of Grimm & Dorner (1975), which is equal to 7.3°. This is their tilt angle at the temperature T_{0}, which is equal to 846 K. Parameter δ(273 K) is calculated by equation (1) to be 16.40°. δ(T) is the tilt angle calculated from equation (1) for a temperature lying between 273 and 846 K. Therefore equation (22) delivers a parameter between 0 and 1 for decreasing temperatures between 846 and 273 K, respectively. The fitted curves are shown in Fig. 6. It should be noted that the use of polynomial coefficients allows parameters L, Δ, a_{PC}, b_{PC} and c_{PC} to vary independently of one another, even though a single Landau calculated from temperature according to equation (1) is at the core of the fitting method. As a formal contribution to the method, the Landau function provides a more linear baseline that enables the fitting of loworder polynomials. If the five parameter values were fitted directly to reduced temperature, a higher order would be required in order to accommodate the significant nonlinearity in the parameter–variation in the region of the i.e. at T ≤ T_{c}. However, such a step would also introduce undesirable shortrange artefacts in the fitted curves of questionable physical basis.

The curve fitted for parameter α_{PC} was calculated from equation (17) utilizing values for tilt angles ϕ_{v} and ϕ_{h} calculated from the coefficients in Table 1 and shown in Fig. 5.
3.4. Structural prediction for αquartz via the INA method
The fine temperaturemesh adopted by Antao (2016) means that there is more to be gained by calculating crystal structures outside the range of 298–844 K than by calculating structures at intermediate temperatures. Therefore four of the temperatures chosen for Table 3, 273 K, 283 K, 293 K and 846 K lie outside this range. A large separation in temperatures of 100 K has been chosen for temperatures within the range given by Antao (2016). Table 3 should be read from the top downwards. The first step is to calculate the Grimm and Dorner δ, via equation (1). Thereafter parameters ϕ_{v} and ϕ_{h} are calculated via equation (A1.1) and the fitting coefficients of Table 1. In Table 3, the equations used to calculate [Si+PC]parameters from α_{PC} down to c_{PC} are listed in the righthand column. Thereafter the equations used to calculate the crystallographic parameters by reverse transformation from [Si+PC]parameters are quoted in this column.
Calculated crystal structural parameters at the ten temperatures chosen are quoted below the horizontal rule in Table 3. The final three parameters, y_{O}, z_{O} and c, were calculated via an iterative process using the GRG algorithm within the Microsoft Excel Solver software environment. Selfconsistent solutions to equations (7), (9) and (10) were sought, using trial values for these three parameters. Their values were refined in order to bring values of b_{PC}, c_{PC} and α_{PC} calculated from these equations into agreement with the values calculated from the coefficients relating to equation (A1.2) and quoted in Table 3. An indication of the selfconsistency of the method is provided by the values of r.m.s. deviation quoted in the final line of Table 3. This parameter is defined in equation (23).
The numbers in the smallest brackets in equation (23) are equation numbers.
3.5. Tilt angles and [Ge+PC]parameters for GeO_{2}
Although the 16 structures of GeO_{2} refer to temperatures between 294 and 1344 K (Haines et al., 2002), the αquartztype structure for GeO_{2} is metastable with respect to a rutiletype phase at temperatures up to ∼1273 K. It is the equilibrium phase only at higher temperatures up to the melting point of ∼1390 K (Liu & Bassett, 1986). Landau parameters T_{c} and T_{0} as for αquartz cannot be derived from structural data, as melting takes place on rising temperature before any such α→β phase transition.
Lengthbased parameters L, a_{PC}, b_{PC}, c_{PC} are larger for GeO_{2} than for αquartz. Values of pseudocubic angle α_{PC} are also uniformly larger, lying in the range 91.11° ≤ α_{PC} ≤ 91.60°, compared to 88.53° ≤ α_{PC} ≤ 90.45° for αquartz. This observation signifies a greater degree of angular distortion of the tetrahedra. Larger values of λ_{PC} also point to tetrahedra that are comparatively more distorted, as discussed further in §4.1. Values of the parameter 3V_{tetra}/V_{UC} are higher for GeO_{2}, this implying larger tilt angles: the greater the degree of tetrahedral tilting, the larger the proportion of space occupied by the tetrahedra. Tilt angles ϕ_{v} and ϕ_{h} are indeed consistently larger than for αquartz, although they span narrower ranges: 22.36° ≤ ϕ_{v} ≤ 25.46°; 23.73° ≤ ϕ_{h} ≤ 26.63°. As for αquartz, the smallest values in each range apply to the highest temperature. The implication is that GeO_{2} at 1344 K is still far away from an α→β phase transition.
The ability to measure tilt angles directly in this work was exploited by adopting mean tilt angle as the δ for GeO_{2} instead of an equation of the form of (1). A quadratic function was fitted to the experimental data for this purpose, as summarized in equation (24).
The following coefficients and r.m.s. deviation apply: a_{0} = 2.6242 × 10^{1}; a_{1} = −4.0000 × 10^{−4}; a_{2} = −1.5343 × 10^{−6}; r.m.s.d.: 0.14°. Just as the thermal Landau allowed lowerorder polynomials to be fitted for αquartz, using the mean tilt angle here fulfils a similar purpose for the GeO_{2} fitting.
Values of parameters L, Δ, a_{PC}, b_{PC}, c_{PC} and α_{PC} from Table S2 for temperatures between 294 and 1344 K are plotted as points with associated error bars in Fig. 7.
For the curvefitting in Fig. 7, the δ generated by equation (24) was adopted as the independent variable. The fitting coefficients listed in Table 4 relate to the reduced δ′ defined in equation (25).
Thus δ′ = 0 at 1344 K and δ′ = 1 at 294 K. The curve fitted for parameter α_{PC} was calculated from equation (17), utilizing values for tilt angles ϕ_{v} and ϕ_{h} calculated from the coefficients in Table 5, using equation (A1.1).


Whereas the curves for L, Δ and c_{PC} lie mostly within the bounds of the error bars of the experimental points, this does not apply to parameters a_{PC}, b_{PC} and α_{PC}. It is further observed that successive experimental points for parameters a_{PC} and b_{PC} lie alternately above and below the fitted curves. At a given temperature, a point lying above the a_{PC} trendcurve corresponds to a point lying below the b_{PC} trendcurve, and vice versa. It transpires that points lying above the a_{PC} curve correspond to crystallographic data obtained from a sample measured with the Special Environment Powder Diffractometer at Argonne National Laboratory, whereas points lying below the curve relate to a different sample from the Polaris medium resolution diffractometer at the Rutherford Appleton Laboratory (Haines et al., 2002). In both cases, the was used in conjunction with timeofflight neutron powder diffraction data.
In view of the uncertainties in the values for parameters a_{PC}, b_{PC} and α_{PC}, it was decided not to proceed with calculations of crystallographic parameters at interpolated temperatures, as carried out in Table 3 for αquartz. However, the separation of values for a_{PC}, b_{PC} and c_{PC} into distinctive valueranges is beyond question, this allowing a subsequent treatment of lengthbased tetrahedral distortion in §4. Owing to the systematic variation with temperature of INA parameters L, Δ and c_{PC}, it is reasonable to assume that the INA method is applicable, in principle, to GeO_{2} over the whole temperature range. The observed fluctuations in the other parameters correlate with two different samples and experimental stations.
3.6. Curvefitting and structural prediction for βquartz
The evolution with temperature of several derived parameters for α and βquartz is shown in Fig. 8, based on the structural refinements of Antao (2016). The unitcell volume increases uniformly with temperature in the αphase and continues to rise beyond the to the βphase to a maximum value at 921 K, before falling back gently with increasing temperature (Antao, 2016). The volumes occupied by the SiO_{4} tetrahedra decrease strongly with temperature in the αphase, this being allowed by the decreasing mean tilt angle, and continue to fall gradually in the βphase. The lengthbased tetrahedral distortion, λ_{PC}, decreases with temperature in both phases, with a jump in values observed at the Values ultimately attained at high temperature in the βphase are lower than in the αphase. Parameter 3V_{tetrahedron}/V_{UC}, which represents the fraction of space occupied by the SiO_{4} tetrahedra, decreases much more strongly in the α than in the βphase. In the former case, the decrease is facilitated by the reduction in mean tilt angle. In the latter, the decrease indicates the stronger relative decrease in tetrahedral volume compared to unitcell volume.
Pseudocubic parameters a_{PC} and b_{PC} for βquartz show a stronger temperaturedependence than c_{PC}, with opposite trends observed for a_{PC} and b_{PC}. Curves have been fitted to the variations for a_{PC} and b_{PC}, since equations (17) and (18) yield, by reverse transformation, values of the a cell parameter and the oxygen x_{O} parameter.
The two parameters δ_{1,PC} and δ_{2,PC} are independent indicators of the deviation from regularity of the tetrahedra in βquartz. They are defined as follows, whereby x_{C} is a reference value equal to (see §S3.2 of the supporting information).
A perfectly regular tetrahedron would have both δ_{1,PC} and δ_{2,PC} equal to zero. The contrary motion of their negative values with increasing temperature in the fourth diagram of Fig. 8 indicates that perfect tetrahedral regularity is not attained in βquartz.
The strong monotonic variation of − δ_{2, PC} with temperature allows a curvefitting from which values of unitcell parameter c can be derived. Taken together, parameters a_{PC}, b_{PC} and δ_{2, PC} with associated curves enable prediction of the structures of βquartz at interpolated temperatures. This procedure is shown in Table 6 for temperatures between 900 and 1200 K in 100 K intervals. The calculation procedure, which uses the coefficients listed in Table 7, may be inferred by reading the table from the top downwards.


4. Discussion
4.1. Comparison of the temperature and pressureevolution of quartz and GeO_{2} structures by means of tetrahedral distortion parameters
The length and anglebased tetrahedral distortion parameters, λ_{PC} and σ_{PC}, introduced by Reifenberg & Thomas (2018) to enable a comparative overview of tetrahedral distortions under varying conditions of temperature and pressure, are plotted in Fig. 9 for αquartz and GeO_{2}. The former corresponds to equation (21) and the latter parameter
takes on the form of equation (28) when expressed in radians for αquartz or GeO_{2}. These two parameters correspond to normal and shear distortions, respectively, and are normalized in order to reflect changes in shape and not volume.
Also plotted are calculated values of mean tilt angle, ϕ, in degrees.
It is observed that λ_{PC} has uniformly higher values in GeO_{2} compared to αquartz at a given temperature or pressure, and further, that the application of hydrostatic pressure increases the lengthbased distortion in both crystal structures. The behaviour of σ_{PC} is more complicated. The red points for αquartz touch the xaxis at circa 640 K, when α_{PC} changes from values above 90° to values below 90° on increasing temperature. The blue points representing GeO_{2} are uniformly higher and show a weak dependence on temperature. By comparison, the application of pressures of up to 5.57 GPa to GeO_{2} causes σ_{PC} to fall off, corresponding to a reduction in α_{PC} from 91.0 to 89.9°. Such a falloff is not observed for αquartz, with a small upwards trend in σ_{PC} seen. This results from α_{PC} values that are consistently larger than 90°.
Although angular distortion σ_{PC} falls with increasing pressure in GeO_{2}, this is not associated with the approach to a as tilt angle ϕ takes on successively higher values with increasing pressure. This is the primary structural response of both GeO_{2} and αquartz to increasing pressure.
Taken together, these results support the view expressed by Glinnemann et al. (1992) that unpressurized GeO_{2} is a good model of the highpressure structure of αquartz: the blue points for unpressurized GeO_{2} and the pink points for αquartz at high pressure occupy similar regions along the yaxis in the three diagrams of Fig. 9.
4.2. The significance of tetrahedral distortion in quartz
The term distortion implies deviation from an ideal. Two fundamental approaches are available for specifying such an ideal, the first referring to symmetry and the second to structure. The former leads naturally to considerations of group theory and the latter to crystal chemistry. In the case of quartz, as examined here, the corresponds to P6_{2}22 for βquartz. On cooling below 846 K, a displacive to its maximal subgroup P3_{2}21 takes place, this corresponding to αquartz. Bärnighausen (1980) has described this transition as latticeequivalent (translationsgleich). The β→α transition involves the loss of the twofold rotation symmetry in the parent along 〈210〉 axes. It is therefore assigned the index 2 and notation t2.
In terms of structure, the dominant feature observed in the lower symmetry, trigonal phase is tetrahedral tilting around the remaining 〈100〉 twofold axes, along which the Si atoms lie. This twofold symmetry restricts the possible distortions of the SiO_{4} tetrahedra, such that the distortion of the O_{4} cages may be represented by pseudocubes in which two of the angles, β_{PC} and γ_{PC}, are equal to 90° (Fig. 2). A corollary is that four independent parameters are required to describe this distortion. The term pseudocube also implies the existence of an ideal of higher symmetry, i.e. the cube, which would be specified completely by one parameter, a_{PC}, since the following three constraints apply: (i) b_{PC} = a_{PC}; (ii) c_{PC} = a_{PC}; (iii) α_{PC} = 90°. Such a cube corresponds to a perfectly regular O_{4} tetrahedron.
Although _{4} tetrahedra to exist in both β and αquartz, this ideal is not observed experimentally. For βquartz, a regular O_{4} tetrahedron would impose restrictions on both oxygen parameter x_{O} and c/a ratio such that δ_{1,PC} = δ_{2,PC} = 0 [see equations (26) and (27) and the fourth diagram of Fig. 8]. For αquartz, the possibility of the existence of perfectly regular tetrahedra has been addressed by Smith (1963), who showed that this would require the c/a ratio to be less than . Equations (6), (7), (9) and (10) of the current work allow an extension of Smith's analysis to examine the consequences of regular tetrahedra for tilt angle. The above three constraints to form a cube may be applied, together with a fourth constraint that the Si ion be located at the centreofcoordinates of its O_{4}cage.
symmetry allows regular SiOSince Smith's c/a criterion is fulfilled only by the nineteen structural refinements of Antao (2016) at temperatures T ≥ 566 K, one way to address this question is to take the values for a and c at these temperatures and to apply the four constraints in a Microsoft Excel spreadsheet supported by the iterative GRG in the Solver. The spreadsheet used for an example structure at 784 K is shown in Fig. 10(a), with the Solver settings for constraints (i)–(iii) above shown in Fig. 10(b).
The values of cells B5–B7 are allowed to vary subject to the constraints that cells C20–C22 contain values less than 0.00001 at the end of the ). This is zero for an α_{PC} angle of 90°. At the end of the cells B9–B12 (with light brown background) contain the parameters of a perfect cube. Further, the underlying equations, based on symmetry, guarantee that a system of interconnected regular SiO_{4} tetrahedra applies. The resulting oxygen x, y, z parameters necessary for this are given in cells B5–B7 (with yellow background). Significant differences are observed relative to the experimental parameters of Antao (2016) (cells C5–C7), to which irregular tetrahedra with pseudocubic parameters in cells C9–C12 apply. The value of x_{Si} in cell B14 is calculated by applying the fourth constraint relating to the location of the silicon ion at the centreofcoordinates of its O_{4} cage. The associated values of L and Δ, which relate to the Siion framework, are quoted in cells B15 and B16 by application of equations (2) and (3). Equation (13) is used to calculate the tilt angle, ϕ_{tilt}, resulting for the structure with regular tetrahedra. α_{PC} = 90° due to the regular tetrahedral geometry, so that ϕ_{v} = ϕ_{h}. This is is quoted in cell B13, whereby the value of 6.68° is obtained for the a and c cell parameters of Antao (2016) at 784 K. This differs significantly from the experimental value of 10.74° (cell C13).
In this connection, cell C22 contains the difference of the two terms in the numerator of the argument to the arccos function in equation (10It is significant that regular tetrahedra give rise to tilt angles that increase from 1.50 to 8.27° over the temperature range from 566 to 844 K, whereas the distorted tetrahedra in the experimental structures of Antao (2016) have tilt angles that decrease from 13.72 to 8.19° over this range [Fig. 11(a)]. That the primary Landau i.e. tilt angle, should increase with increasing temperature is nonsensical. It follows that distorted tetrahedra in the αphase are necessary for Landau theory to be applicable. This situation is at variance with the behaviour of perovskites, i.e. systems of interconnected octahedra. In this context, the grouptheoretical analysis of Howard & Stokes (1998) found that, of the 15 possible subgroups of cubic corresponding to different tilting patterns, only one was necessarily associated with octahedral distortion. They noted that such distortions were possible and expected in the other systems, but not required by geometry. These perovskite distortions have been analysed by other authors [see, for example, Thomas (1998); Tamazyan & van Smaalen (2007)].
Fig. 11(a) also demonstrates the expected correlation between tilt angle and tetrahedral volume for both regular and distorted tetrahedra. The larger tetrahedral volumes of distorted tetrahedra correlate with larger mean Si—O distances^{7},^{8} as well as angles Δ in the silicon ion framework [Fig. 11(b)]. The only case of parallel trends with temperature between regular and distorted tetrahedra relates to parameter L in the silicon ion framework [Fig. 11(c)]. In general, the distorted tetrahedra in the Antao structures permit relatively longer L values, leading to weaker Si⋯Si repulsions.
Violation of the criterion due to Smith (1963) does not allow a network of regular tetrahedra to be formed for the cell parameters obtained at temperatures below 566 K. His limiting c/aratio of corresponds to a tilt angle of zero. However, equations (6) to (10) allow an interconnected network provided that one of the constraints encoded in cells C20 to C22 of Fig. 10(a) is relaxed. The results yielded by the Microsoft Excel Solver for a representative structure at 345 K are given in Table 8.

The parameters obtained are strongly dependent on the constraints applied. Tilt angle ϕ is highly variable and is to be compared with the experimental mean tilt angle at this temperature of 15.36°. Given this sensitivity, a further issue is to examine the pseudocubic parameter combinations that apply to all the experimental structures of Antao (2016). To this end, the polynomials in Table 2 for these parameters are plotted as a function of reduced Landau δ′ [as defined in equation (22)] in Fig. 12. Since it is not possible to adopt a regular tetrahedron (or equivalently perfect cube) as a reference over the whole temperature, another method of normalization has been adopted: values of a_{PC}, b_{PC} and c_{PC} have been divided by the mean of the three values at each temperature. In the case of α_{PC}, absolute values have been divided by their median value (89.48°) over the whole temperature range. The corresponding normalized parameters, for which expansion/contraction effects have been factored out, are denoted by a_{PC}′, b_{PC}′, c_{PC}′ and α_{PC}′.
The modes of distortion of the O_{4} tetrahedra vary over the temperature range investigated, with the extent of the variation in parameters increasing in the order a_{PC}′ < b_{PC}′ < c_{PC}′. The values of parameters a_{PC}′ and c_{PC}′ approach each another as δ′ → 1. This behaviour is close to the third pair of constraints in Table 8, for which the maximum tilt angle ϕ is observed. The unique increase in with increasing (→ tilt angle) may be rationalized by noting that increased tilt angles allow progressively larger values of c_{PC} to be accommodated for a given c cell parameter. This analysis also allows an independent assessment of the validity of the rigid unit (phonon) mode (RUM) approximation, according to which displacive phase transitions in framework structures occur without any significant distortion of the MO_{4} tetrahedra (O'Keeffe & Hyde, 1976; Giddy et al., 1993).
The ability to generate structural models for αquartz with alternative modes of tetrahedral distortion within the Microsoft Excel Solver, as shown in Fig. 10 and Table 8, is a useful byproduct of the approach. Since this activity can be conducted independently of experimental diffraction data, it constitutes a simple, but versatile modelbuilding method. It is likely to be useful for the modelling of auxetic (i.e. negative Poisson's ratio) or nonauxetic behaviour of αquartz subject to different constraints. Pioneering modelling work has been carried out here by Alderson & Evans (2009), in which alternative combinations of tetrahedral rotation and dilation were examined.
Since the [Si+PC] parameters can be reversetransformed to crystallographic parameters, the method also allows the prediction of ; Reifenberg & Thomas, 2018) will be of benefit when carrying out structural refinements of lower symmetry structures. The use of alternative, grouptheoretical methods in this context was pioneered by Stokes & Hatch (1988) and resulted in the ISOTROPY suite of programs (https://iso.byu.edu/iso/isotropy.php). In particular, the ISODISTORT webbased tool (Campbell et al., 2006), which acts as a gateway to the ISOTROPY suite, is geared towards analysing structural distortions. This proceeds by identifying the irreducible representations of parent space groups that are associated with distortions in their subgroups. It would therefore be worthwhile to attempt a synthesis of the two approaches towards the quartz both crystal chemical and grouptheoretical.
at interpolated or extrapolated temperatures. This process, along with INA methods in general (Thomas, 2017It is not surprising that length and anglebased parameters vary smoothly with temperature and pressure, since they fundamentally reflect the interactional potential energies and vibrational energies of the ions. This observation underlies the importance of crystallographic experiments carried out under variable (p,T) conditions: they probe structural space. Furthermore, when lengths and angles are calculated, the complementary and atomic positional crystallographic parameters are combined in a Cartesian space that is conducive to establishing smooth trends with (p,T). This is the essential purpose of the transformation from crystallographic to [Si+PC] or, more generally, INA parameters. It therefore constitutes a technique that could become widely used in the of structures examined under variable (p,T)conditions.
It is intended to extend the current method to formulate more detailed structurepieozelectric property relationships for singlecrystal phosphates and arsenates (ABO_{4}; A = B,Al,Ga,Fe; B = P,As) (Baumgartner et al., 1984, 1989; Sowa, 1991, 1994; Nakae et al., 1995; Haines et al., 2004). These are homeotypic with αquartz and GeO_{2}. However, the presence of two different cations leads to two symmetryindependent tetrahedra in the For this reason, their structures have not been analysed here. However, continued application of the tilted regular tetrahedron model to these materials (Krempl, 2005) points to a need to discriminate more clearly between tetrahedral tilt and distortion in these materials.
The additional insight regarding tetrahedral distortions in quartz made possible by the data of Antao (2016) signals how highquality crystallographic data can also contribute to a deeper understanding of phase transitions. This should act as a spur towards the more regular collection of crystallographic data of superior quality.
In seeking a microscopic interpretation of the Landau ), that (purely) `tilting models of framework compounds fail to match the observed structural behaviour', has been addressed.
attention in the literature has been focused until now on the tilt angle of the tetrahedra. This is indeed the dominant contribution. However, the inability of regular tetrahedra to generate appropriate values of tilt angle, as found here, demonstrates the importance of also taking tetrahedral distortion explicitly into consideration. Thus the comment of Taylor (1984In general, the potential of crystal chemistry is far greater than merely offering a descriptive postrationalization of experimentally determined structures. It is also able to offer a predictive framework for detailed dialogue with experiment.
APPENDIX A
A1. Curvefitting coefficients for αquartz
A2. Curvefitting coefficients for βquartz
Polynomials of the form of equation (A1.2) were employed with n = 4 for parameters a_{PC} and b_{PC} and n = 3 for parameter −δ_{2,PC}. The reduced parameter applies here, as defined in equation (A2.1), whereby the temperatures of 860 K and 1235 K correspond to the minimum and maximum temperatures of the structures reported by Antao (2016) for the βphase. Fitting coefficients a_{i} are listed in Table 7.
Supporting information
The supporting information contains a derivation of the equations in Section 2 of the main article from first principles. DOI: https://doi.org//10.1107/S2052520621002717/ra5093sup1.pdf
Footnotes
^{1}In order to generate the coordinates of Donnay & Le Page (1978), it would be necessary to shift the origin by from the origin in International Tables of Xray Crystallography (Hahn, 1995). The latter was adopted by Antao (2016).
^{2}The α_{1} and α_{2} structures are Dauphiné twins related by a 180° rotation about the threefold axis (Antao, 2016).
^{3}GeO_{2} was set in the r(+) setting according to Donnay and Le Page (1978). In order to generate the coordinates of Donnay and Le Page, it would be necessary to shift the origin by from the origin in International Tables of Xray Crystallography (Hahn, 1995). The latter was used by Haines et al. (2002).
^{4}Use of the concept of structural has been made freely in earlier work (Thomas, 2017; Reifenberg & Thomas, 2018).
^{5}This is discussed in §S3 of the supporting information.
^{6}Values of x_{0} in equations (17) and (18) are in accordance with the notation in International Tables for Crystallography (Hahn, 1995) and equal to one half of the values of O_{x} quoted by Antao (2016).
^{7}In a regular quartz, all four Si—O distances are equal. In αquartz with distorted tetrahedra (Antao, 2016), there are two sets of two equal distances.
^{8}The observed decrease in Si—O distances with temperature in the experimental structures may an artefact arising from correlated thermal librations. Antao (2016) advocates a possible correction due to Downs et al. (1992).
Acknowledgements
Open access funding enabled and organized by Projekt DEAL.
References
Alderson, A. & Evans, K. E. (2009). J. Phys. Condens. Matter, 21, 025401. CrossRef PubMed Google Scholar
Antao, S. M. (2016). Acta Cryst. B72, 249–262. CrossRef IUCr Journals Google Scholar
Bärnighausen, H. (1980). MATCH, 9, 139–175. Google Scholar
Baumgartner, O., Behmer, M. & Preisinger, A. (1989). Z. Kristallogr. 187, 125–131. CrossRef ICSD CAS Web of Science Google Scholar
Baumgartner, O., Preisinger, A., Krempl, P. W. & Mang, H. (1984). Z. Kristallogr. 168, 83–91. CrossRef ICSD CAS Web of Science Google Scholar
Campbell, B. J., Stokes, H. T., Tanner, D. E. & Hatch, D. M. (2006). J. Appl. Cryst. 39, 607–614. Web of Science CrossRef CAS IUCr Journals Google Scholar
Carpenter, M. A., Salje, E. K. H., GraemeBarber, A., Wruck, B., Dove, M. T. & Knight, K. S. (1998). Am. Mineral. 83, 2–22. CrossRef Google Scholar
Dolino, G. (1990). Phase Transit. 21, 59–72. CrossRef CAS Web of Science Google Scholar
Donnay, J. D. H. & Le Page, Y. (1978). Acta Cryst. A34, 584–594. CrossRef CAS IUCr Journals Web of Science Google Scholar
Downs, R. T., Gibbs, C. V., Bartelmehs, K. L. & Boisen, M. B. Jr (1992). Am. Mineral. 77, 751–757. CAS Google Scholar
Giddy, A. P., Dove, M. T., Pawley, G. S. & Heine, V. (1993). Acta Cryst. A49, 697–703. CrossRef CAS Web of Science IUCr Journals Google Scholar
Glinnemann, J., King, H. E., Schulz, H., Hahn, T., Placa, S. J., La, & Dacol, F. (1992). Z. Kristallogr. 198, 177–212. Google Scholar
Grimm, H. & Dorner, B. (1975). J. Phys. Chem. Solids, 36, 407–413. CrossRef CAS Web of Science Google Scholar
Hahn, T. (1995). Editor. International Tables for Crystallography, Vol. A, SpaceGroup Symmetry. Dordrecht: Kluwer. Google Scholar
Haines, J., Cambon, O., Astier, R., Fertey, P. & Chateau, C. (2004). Z. Kristallogr. 219, 32–37. Web of Science CrossRef ICSD CAS Google Scholar
Haines, J., Cambon, O. & Hull, S. (2003). Z. Kristallogr. 218, 193–200. Web of Science CrossRef ICSD CAS Google Scholar
Haines, J., Cambon, O., Philippot, E., Chapon, L. & Hull, S. (2002). J. Solid State Chem. 166, 434–441. Web of Science CrossRef ICSD CAS Google Scholar
Howard, C. J. & Stokes, H. T. (1998). Acta Cryst. B54, 782–789. Web of Science CrossRef CAS IUCr Journals Google Scholar
Kihara, K. (1990). Eur. J. Mineral. 2, 63–78. CrossRef ICSD CAS Google Scholar
KimZajonz, J., Werner, S. & Schulz, H. (1999). Z. Kristallogr. 214, 324–330. CAS Google Scholar
Krempl, P. W. (2005). J. Phys. IV Fr. 126, 95–100. Web of Science CrossRef CAS Google Scholar
Liu, L. & Bassett, W. A. (1986). Elements, Oxides, Silicates. HighPressure Phases with Implications for the Earth's Interior, p. 112. New York: Oxford University Press. Google Scholar
Megaw, H. D. (1973a). Crystal structures – a working approach, p. 268. London: Saunders. Google Scholar
Megaw, H. D. (1973b). Crystal structures – a working approach, pp. 453–456. London: Saunders. Google Scholar
Nakae, H., Kihara, K., Okuno, M. & Hirano, S. (1995). Z. Kristallogr. 210, 746–753. CAS Google Scholar
O'Keeffe, M. & Hyde, B. G. (1976). Acta Cryst. B32, 2923–2936. CrossRef CAS IUCr Journals Web of Science Google Scholar
Reifenberg, M. & Thomas, N. W. (2018). Acta Cryst. B74, 165–181. CrossRef IUCr Journals Google Scholar
Smith, G. S. (1963). Acta Cryst. 16, 542–545. CrossRef CAS IUCr Journals Web of Science Google Scholar
Sowa, H. (1991). Z. Kristallogr. 194, 291–304. CrossRef ICSD CAS Web of Science Google Scholar
Sowa, H. (1994). Z. Kristallogr. 209, 954–960. CrossRef ICSD CAS Web of Science Google Scholar
Stokes, H. T. & Hatch, D. M. (1988). Isotropy Subgroups of the 230 Crystallographic Space Groups. Singapore: World Scientific. Google Scholar
Tamazyan, R. & van Smaalen, S. (2007). Acta Cryst. B63, 190–200. Web of Science CrossRef IUCr Journals Google Scholar
Taylor, D. (1984). Mineral. Mag. 48, 65–79. CrossRef CAS Google Scholar
Thomas, N. W. (1998). Acta Cryst. B54, 585–599. Web of Science CrossRef CAS IUCr Journals Google Scholar
Thomas, N. W. (2017). Acta Cryst. B73, 74–86. Web of Science CrossRef IUCr Journals Google Scholar
This is an openaccess article distributed under the terms of the Creative Commons Attribution (CCBY) Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original authors and source are cited.