Phase transitions and (p–T–X) behaviour of centrosymmetric perovskites: modelling with transformed crystallographic data

Dedicated to the memory of Christopher Jason Thomas (30.12.1988–21.10.2021).

A reversible transformation of the unit-cell parameters and atomic coordinates of centrosymmetric perovskites ABX₃ into a Cartesian space is defined. Analytical expressions for the three vectors for the pseudocubic cell and three vectors for a BX₆ octahedron are derived for space groups Pbmn, Cmcm, Ibmm, P4/mbm, P4/nmc, I4/mcm and R3c. The following structural parameters may be derived from these vectors: up to six pseudocubic parameters defining octahedral geometry; length- and angle-based octahedral distortion parameters λ and σ; inclination angles of tilted octahedra, θ₁, θ₂ and θ₃; angles of tilt of octahedra; AX₁₂:BX₆ polyhedral volume ratio, V_A/V_B; parameters η_A and η_B defining the relative contraction of inner AX₈ polyhedra and expansion of BX₆ octahedra due to octahedral tilting. The application of these parameters is demonstrated by reference to published crystal structures. The variation of η_A and η_B with temperature in the compositional series Sr_xBa_1–xSnO₃ and Sr_xBa_1–xHfO₃, as well as the temperature series of BaPbO₃ and CaTiO₃, is related to the sequence of phases Pbmn → Ibmm→ Pm3m. Stabilization of the Cmcm phase is likewise interpreted in terms of these two parameters for NaTaO₃ and NaNbO₃. The pressure evolution of the structures of MgSiO₃, YAlO₃, (La_1–xNd_x)GaO₃ (0 ≤ x ≤ 1) and YAl_0.25Cr_0.75O₃ is modelled with the appropriate structural parameters, thereby also addressing the characteristics of the Pbmn → R3c transition. Simulation of MgSiO₃ up to 125 GPa and of YAlO₃ up to 52 GPa in space group Pbnm is carried out by using the Birch–Murnaghan equation of state. In both cases, full sets of oxygen coordinates assuming regular octahedra are generated. Octahedral distortion is also modelled in the latter system and predicted to have a key influence on structural evolution and the sequence of phase transitions. The core modelling procedures are made available as a Microsoft Excel file.

1. Introduction: the modelling of octahedral tilting in perovskites

Synthetic perovskite-related compounds continue to attract the attention of many scientists and technologists, irrespective of whether they are working, for example, on the development of lead-free piezoelectric ceramics (Shrout & Zhang, 2007) or on organolead halide ABX₃ nanocrystals for photochemical cells (Jena et al., 2019). In general, the current availability of high-quality structural data evokes the need for all the nuances of structural change under varying conditions of pressure, temperature and composition] (p–T–X) to be precisely modelled. This expectation is particularly poignant in the case of perovskite-related compounds, ABX₃, since, according to Mitchell (2002a), all elements, apart from the noble gases, can be found in a variant of these. In the pioneering work of Megaw (1973), their structures were regarded as comprising three essential features: (a) tilting of the anion octahedra; (b) displacements of the cations; (c) octahedral distortions. Definitive work by Glazer (1972) led to a system of classification based on the sense of tilt of the three tetrad axes of corner-linked regular octahedra about three perpendicular, or nearly perpendicular crystal axes.

A regular octahedron is characterized by three pairs of opposite vertices that are linked by stalks of equal length passing through its mid-point at right angles to one another. Two lines of regular octahedra are shown in Figs. 1(a) and 1(b), as they arise in space group Pbnm.

Figure 1 Tilting of regular octahedra about the xPC and yPC axes, as in space group Pbnm. Pseudocubic axes xPC, yPC and zPC are directed parallel to orthorhombic vectors , [110 ] and [001], respectively. (a) Tilting around the yPC axis (in clinographic projection) with θy the angles of inclination of stalks of length s to the axis; (b) Tilting around the zPC axis (head-on view showing mirror planes); (c) yPC-axis tilting [as in (a)] viewed along the yPC axis towards the origin; (d) zPC-axis tilting [as in (b)] viewed along the zPC axis towards the origin.

Zigzag chains of adjacent octahedral stalks of uniform length s are formed, these lying in planes PQRS and TUVW. When viewed along the y_PC axis-of-tilting, the red and blue octahedra are seen to be rotated in opposite senses [Fig. 1(c)], leading to Glazer notation . By comparison, the red and green octahedra are rotated in the same sense [Fig. 1(d)]. This leads to the notation c⁺ , since the tilt angle is different. As tilting around the x_PC and y_PC axes is equivalent, the three-dimensional tilt system in Pbnm is denoted by (Glazer, 1972).

In this now well established a^#b^#c^# nomenclature, superscripts # can be +, − or 0, denoting the angles of tilt of neighbouring octahedra along the three pseudocubic axes as in-phase (+), anti-phase (−) or zero. Importantly, Glazer (1972) also showed that the 23 tilt systems for regular octahedra correlated with 15 alternative space groups. As well as aiding the correct interpretation of perovskite diffraction patterns (Glazer, 1975), these correlations stimulated group-theoretical analysis of perovskites, their being confirmed significantly later by Howard & Stokes (1998, 2002). The latter work led to minor modifications to the space groups originally assigned by Glazer and established a top-down, hierarchical system of group–subgroup relationships starting from the cubic aristotype for perovskite structures with tilted, regular octahedra (Fig. 2).

Figure 2 Group–subgroup relationships among the 15 perovskite space groups arising from the tilting of regular octahedra. A dashed line signifies that the corresponding phase transition must be first order (Howard & Stokes, 2002). Reproduced with permission of the International Union of Crystallography.

Two independent contributions concerning octahedral tilting in perovskites were made by Woodward (1997a,b) and Thomas (1996). Woodward (1997a) concluded that perfectly regular octahedra could not be linked together in some tilt systems. The contentious space groups were subsequently narrowed down to Cmcm (tilt system ) and P4₂/nmc () by Howard & Stokes (1998), with only the latter requiring irregular octahedra. The work of Thomas (1996), by comparison, was primarily concerned with the dependence of AX₁₂:BX₆ polyhedral volume ratio, V_A/V_B, on octahedral tilting in orthorhombic and tetragonal perovskites. Inclination angles θ_y and θ_z in Figs. 1(a) and 1(b), and by extension θ_x, were used to relate octahedral stalk lengths s and tilting to the lengths of pseudocubic cell axes 1, 2 and 3 and therefore cell volume [equation (1)].

This form results from the coupling of inclination angles θ₁ and θ₂ when pseudocubic axes 1 and 2 are oriented at approximately 45° to the crystal axes. In untilted structures (with θ₁ = θ₂ = θ₃ = 0), V_A/V_B is exactly equal to five (Thomas, 1989).

Tamazyan & van Smaalen (2007) subsequently argued that inclination angles θ₁, θ₂ and θ₃ defined by Thomas (1996) were unnecessarily influenced by octahedral distortion. They therefore proposed an alternative method of calculating tilt angles, although this was at the expense of losing the simple link to polyhedral volume ratio expressed by equation (1). Later work by Wang & Angel (2011) restored the link between octahedral tilting and ratio V_A/V_B, employing a group-theoretical, rather than a geometrical approach to separate octahedral tilting and distortion. These authors expressed the ratio V_A/V_B as a function of the amplitudes of the normal modes in a cubic perovskite, rather than by direct calculation from anion coordinates in experimentally determined crystal structures.

The conflict between geometrical, i.e. crystal-chemical and group-theoretical methods in describing perovskite structures is somewhat contrived, since both have legitimate fields of application and ultimately have similar aims. This was made clear in seminal work by Knight (2009), who utilized group-theoretical methods to develop a general parameterization of centrosymmetric perovskites based on symmetry-adapted basis vectors of the phase. He also pointed to difficulties with a method earlier proposed by this author for a general crystal-chemical parameterization of centrosymmetric perovskites (Thomas, 1998). Two specific problems were identified, (a) that the method was `geometrically complex'; and (b) that it relied `totally on an empirical analysis of known crystal structures'. He also made the important point, (c), that the A site has a coordination number less than 12 due to a geometrically complex coordination polyhedron (Knight, 2009). In the current work, point (a) is addressed by focusing on the distortion of centrosymmetric octahedra as independent geometrical forms. Point (b) is addressed by deriving analytical expressions dependent on space group for the three stalk vectors defining octahedral geometry. Point (c) is addressed by developing a simple parameterization based on AX₈ sub-polyhedra that correlates with the tilt classification system of Glazer (1972). Details of these methodological improvements are given in the following section.

2. Elements of the revised crystal-chemical method

2.1. Pseudocubic representations of octahedra

In order that an object can be described as an octahedron, it requires three recognisable pairs of opposite vertices that may be linked to each another by vectors. If the ends of the stalks of a regular octahedron [Fig. 3(a)] are randomly displaced by limited amounts [Fig. 3(b)], the resulting form is still recognisable as an octahedron [Fig. 3(c)]. The six independent vertices lead to six sets of [x,y,z ] triplets in Cartesian space, i.e. 18 parameters. If only the form of the octahedron is relevant, and not its absolute position or orientation, these 18 parameters are reduced to 12 by subtracting six parameters: three to fix one vertex in space at [0,0,0] and three to define the orientation of the octahedron. In Fig. 3(c), one vertex has been fixed at [0,0,0] and the octahedron so rotated that the opposite vertex is fixed at [0 ,0, z₁]. A further rotation of the octahedron about the z axis has been carried out to fix a vertex at [0, y₂, z₂]. The total of six coordinate-components equal to zero here signifies a reduction in independent parameters from 18 to 12, i.e. (z₁; y₂z₂; x_ny_nz_n; n = 3,5). If, however, the translations at opposite ends of the three stalks are equal and opposite [Fig. 3(d)], a centrosymmetric octahedron results [Fig. 3(e)], in which the three stalks bis­ect one other at the centre of symmetry. Only one end of a stalk needs to be fixed in space in order to define the position of the opposite end, so that the number of independent parameters is halved to six [Fig. 3(e)]: (z₁; y₂z₂; x₃y₃z₃). This situation applies to all perovskites with B ions located at centres of symmetry.

Figure 3 The number of independent parameters of a generalized, isolated centrosymmetric octahedron is equal to six. Views (a) to (e) discussed in §2.1.

The six independent parameters of a centrosymmetric octahedron may be assigned to three stalk lengths a₁, a₂, a₃ and three angles of intersection of the stalks, θ₁₂, θ₂₃ and θ₃₁. In order to visualize the extent of its distortion, a parallelepiped enclosing the octahedron may be constructed by displacing the three stalk vectors to a common origin (Fig. 4).

Figure 4 Visualization of the distortion of octahedra by enclosure in a pseudocube: (a), (b): centrosymmetric; (c), (d) non-centrosymmetric.

As the three independent edge lengths of the parallelipiped are approximately equal, it may also be termed a pseudocube. Such a pseudocube is defined by the same six parameters as the octahedron, i.e. a₁, and . It follows that regular octahedra would lead to pseudocubic representations of octahedra (PCRO) of cubic form.

In the case of the non-centrosymmetric octahedron of Fig. 3(c), the six octahedral vertices do not touch their parallelogram faces at the meeting points of the four quadrants [Fig. 4(d)]. The 12 independent parameters may be accommodated by noting the 2D polar coordinates of the three emergent octahedral vertices, giving rise to additional parameters r₁, and . Thus the PCRO construction is of general validity for visualizing distorted octahedra. Its parameterization may prove to be useful for characterizing non-centrosymmetric and polar perovskites in future work. The transformation between octahedron and PCRO is also reversible.

An advantage of the PCRO visualization method is that the aggregate distortion parameters representing normal and shear distortion developed for pseudocubic representations of tetrahedra (PCRT), and , (Reifenberg & Thomas, 2018; Fricke & Thomas, 2021) may be taken over without modification [equations (2), (3)].

In this connection, the following values are obtained for the centrosymmetric and non-centrosymmetric octahedra of Figs. 3(e) and 3(c), respectively: [λ,σ] = [0.0367, 11.90°]; [λ,σ] = [0.0370, 7.10°]. It follows that the centrosymmetric octahedron displays a higher degree of angular distortion here. Furthermore, if the angular distortion parameters are expressed in radians instead of degrees, the relative degree of shear versus normal distortion can be quantified. Since 11.90° correspond to 0.21 and 7.10° to 0.12 radian, it follows that the shear distortion is greater than the normal distortion in both cases.

2.2. Parameterization of PCRO in terms of three vectors

The octahedra in perovskites are not isolated but form a three-dimensional corner-sharing network. Their crystal structural parameters, i.e. space group, unit-cell parameters and atomic coordinates, deliver full information on octahedral distortion, tilting and connectivity. Importantly, space group symmetry ensures connectivity. Once the distortion and tilting of a single octahedron in a structure with only one symmetry-independent octahedron has been defined, the distortions and tilting of all the other octahedra in the unit cell follow. To convey full information on the tilting of this single octahedron in a structure, its PCRO may be described by the three Cartesian stalk vectors, a₁, a₂ and a₃ along its pseudocube edges. Generalized, analytical expressions for these vectors in different space groups may be derived inductively from known crystal structures. This process is described in §2.2.1 for space group Pbnm with B ions located at 4b special positions. The three vectors defining the axes of the pseudocubic unit cell are likewise defined in this Cartesian space, so that inclination angles of the stalks to the three pseudocubic axes, θ₁, θ₂ and θ₃ may straightforwardly be calculated [see Fig. 1, equation (1) and §2.3]. In addition, the method also allows calculation of tilt angles ϕ_a, ϕ_b and ϕ_c of the octahedra around the three pseudocubic axes, even when the octahedra are distorted.

2.2.1. Space group Pbnm with B ions at 4b positions

The octahedral cage coordinating the B ion at 0½0 in the unit cell of orthorhombic space group Pbnm may be taken, this being the cab setting of space group 62 with standard symbol Pnma. 0½0 is one of the 4b special positions, with X1 ions in 4c special positions and X2 ions in 8d general positions (see Table S1 in the supporting information).¹

Derivation of the analytical form of stalk, or equivalently PCRO vectors a₁, a₂ and a₃ starts by taking an example structure, such as CaTiO₃ at 296 K [Yashima & Ali (2009); ICSD code 162908]: a = 5.3709 Å, b = 5.4280 Å, c = 7.6268 Å; x(O1) = 0.0708, y(O1) = 0.4830, x(O2) = 0.7113, y(O2) = 0.2891, z(O2) = 0.0375; z(O1) has the fixed value of ¼.

The essential step is to convert from numerical fractional coordinates of the titanium ion and its six oxygen neighbours (Table 1, columns 2–4) to analytical fractional coordinates (Table 1, columns 8–10). One way to generate the numerical fractional coordinates would be to use a simple computer program to generate the Cartesian coordinates of the titanium and oxygen ions. (Table 1, columns 5–7). The numerical fractional coordinates (Table 1, columns 2–4) would then follow by multiplying these by the inverse orthogonalization matrix, in this trivial case

Letters a to f of the oxygen ions correspond to the atom labels in Fig. 1(a). They are based on the principle that the octahedral stalk from a to b is oriented closest to the x_PC axis in the positive direction, with the stalks from c to d and from e to f oriented closest to the positive y_PC and z_PC directions, respectively.

Table 1
Calculation of analytical fractional coordinates of the six oxygen anions coordinating the titanium ion in CaTiO₃ located at 0,½,0

Atom	Numerical fractional coordinates			Cartesian coordinates			Analytical fractional coordinates
	x	y	z	xC	yC	zC	x	y	z
Ti (4b)	0	0.5	0	0.0000	2.7140	0.0000	0	½	0
O a (8f)	−0.2113	0.7891	0.0375	−1.1349	4.2832	0.2860	−x(O2) + ½	y(O2) + ½	z(O2)
O b (8f)	0.2113	0.2109	−0.0375	1.1349	1.1448	−0.2860	x(O2) − ½	−y(O2) + ½	−z(O2)
O c (8f)	−0.2887	0.2891	0.0375	−1.5506	1.5692	0.2860	x(O2) − 1	y(O2)	z(O2)
O d (8f)	0.2887	0.7109	−0.0375	1.5506	3.8588	−0.2860	−x(O2) + 1	−y(O2) + 1	−z(O2)
O e (4c)	−0.0708	0.5170	−0.25	−0.3803	2.8063	−1.9067	−x(O1)	−y(O1) + 1	−¼
O f (4c)	0.0708	0.4830	0.25	0.3803	2.6217	1.9067	x(O1)	y(O1)	¼

The analytical expressions in columns 8 to 10 of Table 1 are obtained by inspection: the values in columns 2 to 4 are compared with the starting values of x(O1), y(O1), x(O2), y(O2) and z(O2). It is important to note that the results depend on the convention used for expressing these parameters. For example, the alternative of x(O1) = 0.4292 and y(O1) = −0.0170, although symmetrically equivalent, would lead instead to the results x(O e) = x(O1) − ½ and y(O e) = −y(O1) + ½.

The three vectors for the PCRO are now formed in Table 2 by taking the differences in the analytical fractional coordinates for atom pairs (a,b), (c,d) and (e,f) given in Table 1. Cartesian vectors are formed by multiplying these differences by the orthogonalization matrix

The Cartesian axes are parallel to the axes of the ortho­rhombic unit cell in this case.

Table 2
The three vectors defining the PCRO with mid-point 0,½,0 in space group Pbnm

		Cartesian components			Nearest pseudocubic axes
Stalk	PCRO vector	X	Y	Z	Pseudocubic	Orthorhombic
O b ← O a	a1	a(2x(O2) − 1)	− 2by(O2)	− 2cz(O2)	xPC
O d ← O c	a2	−2a(x(O2) − 1)	−b(2y(O2) − 1)	−2cz(O2)	yPC	[110]
O f ← O e	a3	2ax(O1)	b(2y(O1) − 1)	c/2	zPC	[001]

By adopting this analytical representation, vectors a₁, a₂ and a₃ are no longer tied to the example structure of CaTiO₃: they apply to all perovskites in space group Pbnm, provided that the B ions occupy 4b sites and that the correct convention in choosing the values parameters x(O1), y(O1), x(O2), y(O2) and z(O2) has been followed. All eight crystallographic parameters are involved in defining the geometry of the PCRO in space group Pbnm: a, b, c, x(O1), y(O1), x(O2), y(O2), z(O2). For regular octahedra, six of these eight degrees of freedom (d.o.f.) would be used up in forming octahedra of a particular volume, with five defining the regular form and the sixth the volume. The remaining two d.o.f. would be used to define the unit-cell parameters and octahedral tilting. If the octahedra were only approximately regular, as is generally the case, more d.o.f. would be available for optimizing the octahedral tilting and unit-cell volume, in response to different A and B ion radii or to changing temperature or hydro­static pressure. Thus the idea of structural compromise in forming connected octahedral networks in perovskites can be modelled in response to varying (p–T–X) conditions.

2.2.2. Other space groups

The other space groups analysed in this work are those that commonly arise in experimentally determined crystal structures. They span the following space groups: Pbmn (B ions in 4a sites), Cmcm, Ibmm, P4/mbm, P4₂/nmc, I4/mcm, , . Tables similar to Table 2 are generated for them in §S2 of the supporting information.

2.3. Implementation of the crystallographic to structural transformation in the Microsoft Excel Solver environment

Structural analysis requires a one-way transformation from crystallographic to structural parameters. For example, crystallographic parameters a, b, c, x_i, y_i, z_i are transformed to structural parameters such as PCRO parameters and tilt angles. By comparison, structural prediction requires a reversible transformation between the two parameter sets. The modeller will seek to establish systematic variations in the structural parameters with (p–T–X). If successful, interpolations and extrapolations to other (p–T–X) values can be made before reverse-transforming to crystallographic parameters. This technique has been demonstrated for olivines (Thomas, 2017), coesite (Reifenberg & Thomas, 2018) and quartz (Fricke & Thomas, 2021).

The requirements of ease and flexibility of use together with a reversible transformation suggest that the Microsoft Excel Solver environment is appropriate for applying the method. In this connection, an Excel datafile already programmed is provided in the supporting information. This consists of eight worksheets for the different space groups. A screenshot of the worksheet for Pbnm with B ions in 4b positions is given in Fig. 5.

Figure 5 Screenshot of the Pbnm worksheet, which contains the reference data for CaTiO3 at 296 K (Yashima & Ali, 2009). No refinement has taken place, since the entries in the `Refined data' and `Reference data' boxes are identical.

The user enters the crystallographic data in the `Refined data' box in blocks K3:K5 (light-green background), N4:N8 and N10:N11 (mustard background), whereupon all the structural parameters are automatically calculated by Excel. The computational core of the spreadsheet is in blocks R9:T11 (deep-blue background) and U9:W11 (dark-green background). The formulas in these cells correspond to the entries in Table 2 and the pseudocubic axes in Cartesian coordinates, respectively. Values of all the structural parameters relating to octahedral distortion and tilting (light-blue background) can be seen as ways of describing the numerical values in these two blocks in a structurally meaningful way.

The purpose of the `Reference data' box is threefold. First, it acts as a repository for the source, reference data (cells with orange background). Secondly, the dependent structural parameters can be calculated for this reference data by clicking on the button in cell C15.<sup>2</sup> Thirdly, both reference data and dependent structural parameters can be used as constraints in structural refinements, which apply to the cells in the `Refined data' box.

The different types of structural parameters, all with light-blue background, are summarized as follows.

a) PCRO parameters a₁, a₂, a₃, θ₂₃, θ₃₁, θ₁₂ are calculated as the lengths and intersectional angles of the vectors a₁, a₂, a₃ in block R9:T11. For example, . Aggregate distortion parameters λ and σ [equations (2) and (3)] are calculated in block P13:P16, the latter quoted in both degrees and radians.

b) Pseudocubic angle γ_PC is calculated as the angle between pseudocubic axes a_PC and b_PC in cell P11: .

c) Inclination angles θ₁, θ₂, θ₃ of the octahedral stalks are quoted in block X9:X11. These describe the relationship of the two sets of vectors in blocks R9:T11 and U9:W11 to one another. With respect to vector a₁ in block R9:T9 and its nearest pseudocubic axis vector a_PC in block U9:W9, the following applies:

The subscripts X,Y,Z here denote the Cartesian X,Y and Z components, respectively. It follows that

Substitution of

and

leads to θ₁ = 12.16°. Angles θ₂ and θ₃ are derived from the corresponding pairs (a₂,b_PC) as well as (a₃,c_PC), respectively.

d) The method of calculation of angles of tilt ϕ_a and ϕ_c in the tilt system of space group Pbnm is shown in Fig. 6. These angles constitute an alternative to the inclination angles for relating the two sets of vectors in blocks R9:T11 and U9:W11 to each other. Since tilt angles are to be calculated for distorted octahedra, the method evaluates the projections of oxygen atoms O a, O b, O c and O d in the planes perpendicular to pseudocubic axes a_PC, b_PC and c_PC. These four atoms also define PCRO vectors a₁ and a₂: vector a₃ does not influence the calculated tilt angles.

Figure 6 Tilted octahedra and basal parallelogram planes, e.g. PQRS, of AX12 polyhedra (in yellow) viewed along the orthorhombic z axis in space group Pbnm. The + and – signs in black denote z heights of the octahedral vertices relative to the z = 0 plane of the diagram. These are due to tilting. Green dashed lines show the directions of pseudocubic axes xPC and yPC, the reference directions for the tilting. Red dotted lines represent the perpendicular planes to the green dashed lines, which are projected as lines in the xy plane. Red circles lie above the red dotted lines at the same z height as the octahedral vertices O a to O d. They are denoted by O a′ to O d′ and result from the four projections (O a → O a′), (O b → O b′), (O c → O c′) and (O d → O d′). c^+ tilting causes the octahedra to be rotated about the orthorhombic z axis (or equivalently, the zPC axis). The relevant tilt angles, ϕc,1 and ϕc,2, are shaded orange. (Vectors p and q are defined in §2.6.)

A tilt of type around a_PC is the angle between vector (O c′-O d′) and its projection in the xy plane. Similarly, an tilt around b_PC is the angle between vector (O a′-O b′) and its projection in the xy plane. Calculated values are shown in cells Y9 and Y10 in Fig. 5, with the mean value in cell Y11 (8.53°, 8.51° and 8.52°, respectively). The observed difference of 0.02° in calculated values results from an interplay of the deviation of γ_PC from 90° and the octahedral distortions.

The in-phase c⁺ tilting causes the projections of the a₁ and a₂ octahedral vectors in the xy plane to be rotated away from pseudocubic axes x_PC and y_PC (Fig. 6). The two resulting tilt angles are given in cells Y12 and Y13 (8.82° and 8.87°), with the mean value in Y14 (8.84°).

The algorithms for calculating these tilt angles are described in §S3 of the supporting information, where reference is also made to their implementation in the Excel file in the supporting information.

e) Unit-cell volume V_UC, polyhedral volumes V_B, V_A and volume ratio V_A/V_B are calculated in blocks P3:P4 and P6:P7. V_UC = and

The latter calculation method is facilitated by the centrosymmetry of the octahedron. Volume ratio V_A/V_B follows from these parameters as [(V_UC/Z) − V_B]/V_B and V_A as V_B(V_A/V_B ).

f) Parameters η_A and η_B are likewise extracted from the core computational data (blocks R9:T11 and U9:W11) and defined in §2.6.

g) Parameters L and ϕ (block N18:19 with brown background) relate to the A ion positions and are independent of the anionic network. They are defined in §2.7.

2.4. Example Solver refinements

Initial insight into use of the Solver is gained here by addressing an issue relevant to the historical development of perovskite structural chemistry: the ability of different space groups to accommodate regular octahedra. A comparison is also made between tilt angles calculated as in point d) of §2.3 and the values yielded by commonly used approximations.

The five degrees of freedom required to define the regular octahedral form (see §2.2.1) result in the constraints a) a₂ = a₁; b) a₃ = a₂; c) θ₂₃ = 90°; d) θ₃₁ = θ₂₃; and e) θ₁₂ = θ₃₁. Since there are eight d.o.f. in total, two further constraints may be applied in the refinement. For example, it may be stipulated that the unit cell and the octahedral volumes remain unchanged: f) V_UC = V_UC,reference and g) V_B = V_B,reference. The coding of these constraints in the Solver in shown in Fig. 7(a), whereby constraints f) and g) also refer to cells $H$3 and $H$4 in Fig. 5. The end-point of this tightly defined refinement is shown in Fig. 7(b): a cubic PCRO of side length 3.9014 Å is generated, corresponding to a regular octahedron with perpendicular stalks. The values of the other structural parameters described in §2.3 appear in the cells with a light-blue background.

Figure 7 (a) Solver settings for generating regular octahedra in space group Pbnm with unchanged unit cell and octahedral volumes. (b) Results of the refinement.

The crystal structures obtained by applying the same refinement conditions to reference structures in all the space groups are given in Table 3. The Solver constraints for the respective space groups are pre-programmed in the Excel file in the supporting information.

Table 3
Summary of Solver refinements by space group corresponding to structures with regular octahedra

The octahedral and unit-cell volumes are identical to those in the reference structures.

Space group	Pbnm (B: 4b)	Pbnm (B: 4a)		Cmcm		Ibmm		P4/mbm	P42/nmc	I4/mcm
Reference system	CaTiO3	LaCr0.7Ni0.3O3		NaNbO3 at 848 K		BaPbO3 at 300 K		NaNbO3 at 888 K	CaMnO3	CaTiO3 at 1523 K		La(Cr0.2Ni0.8)O3
Reference ICSD	162908	173471		192404		154038		280100	670342	162919		173475
a1(refined) (Å)	3.9014	3.9535		3.9495		4.3480		3.9500	3.8074	3.9025		3.8952
a (Å)	5.33816	5.46101		7.85139		6.08315		5.56480	7.52285	5.47004		5.42241
b (Å)	5.44997	5.57887		7.85672		6.14899
c (Å)	7.64264	7.73997		7.89355		8.60287		3.94998	7.43089	7.80495		13.49346
x(O1)	0.07274	0.57385		0.27596		0.05216		0	0.03909	0		0.55170
y(O1)	0.48876	0.49521		0		0		0	−x(O1)	0		0
z(O1)	¼	¼		0		¼		½	¼	¼		¼
x(O2)	0.70974	0.23308		0		¼		0.22810	¼	0.21650
y(O2)	0.28863	0.26621		0.22407		¼		0.72810	0.00306	0.71650
z(O2)	0.03637	0.03692		0.00921		−0.02608		0	0.96043	0
x(O3)				0.25921					¼
y(O3)				0.25096					y(O2)
z(O3)				¼					−z(O2) +
Method	§2.3	§2.3	a)	§2.3	b)	§2.3	d)		§2.3	§2.3	c)	§2.3
ϕa (°)	8.52	8.47	11.91			6.37	8.72		7.13			5.90
ϕa,refined (°)	8.29	8.33	11.80			5.92	8.39		8.89			5.90
ϕb (°)				2.11	4.15				6.02
ϕb,refined (°)				2.12	2.11				8.89
ϕc (°)	8.84	4.90	4.90	5.38	5.38			5.01	9.34	7.63	7.63
ϕc,refined (°)	8.96	3.79	3.79	5.92	5.92			5.01	0.70	7.63	7.63
θ3 (°)		10.97				7.87
θ3,refined (°)		11.80				8.39

For space group P4₂/nmc, the obtaining of regular octahedra in tilt system a⁺a⁺c⁻ is at variance with the conclusion of Howard & Stokes (1998). A resolution of this discrepancy is thought likely to depend on the observed interdependence of the coordinates of atoms O2 and O3 in the refined structure (Table 3), which implies a refinement into higher symmetry or pseudo-symmetry than P4₂/nmc.

The values of tilt angles below the dividing line in Table 3 serve to compare the values calculated by the method given in §2.3 and approximations for tilt angles used by other authors. Following an initial analytical treatment of perovskite tilt angles by Megaw (1973), a more developed treatment according to space group was given by Kennedy, Prodjosantoso et al. (1999) as follows. The four parameterizations a) to d) here correspond to headers a) to d) in the row `Method' in Table 3.

a) Pbnm (B ions in 4a): O2:. ; .

b) Cmcm: O1: ; O2: ; O3: .

; .

c) I4/mcm: O: . .

Mountstevens et al. (2003) augmented this set with a result for space group Imma, which upon resetting in Ibmm, is as follows:

d) Ibmm: O2: . .

This expression is equivalent to Pbnm with u = v = 0.

In general, the agreement between values generated in this work and approximations a) to d) above is satisfactory for the in-phase tilts, ϕ_c. Antiphase angles ϕ_a for space groups Pbnm and Ibmm show poorer agreement. In order to investigate whether this is a systematic discrepancy, the structural data of Kennedy, Prodjosantoso et al. (1999) for CaTiO₃ (Pbnm) at 1273 K were fed into the Excel program, yielding the following two sets of tilt values and inclination angles θ₃: ϕ_a = 6.53°; ϕ_a,refined = 6.59°; θ₃ = 8.92°; θ_3,refined = 9.24°. These are to be compared with the tilt values given by the Kennedy approximation: ϕ_a = 9.19°; ϕ_a,refined = 9.24°. These results and those in the Pbnm (B: 4a) and Ibmm columns of Table 3 lead to the following two conclusions: (1) The ϕ_a angles calculated by the method of §2.3 are systematically lower than those generated by the Kennedy approximation; (2) The Kennedy approximation generates exact values of inclination angle for refined structures with regular octahedra, i.e. θ_3,refined, and not tilt angle ϕ_a as defined in this work.

In the case of Cmcm, good agreement for ϕ_a between §2.3 and Kennedy, Prodjosantoso et al. (1999) is obtained for regular octahedra in the refined structure, but not for the distorted octahedra in the unrefined structure.

A rationalization for the smaller values of ϕ_a generated by the method of §2.3 is that the inclination angle θ₃ of a regular octahedron will always be larger than an angle within a plane of projection perpendicular to a pseudocubic axis (Fig. 6). The former angle is generated by the approximations of Kennedy, Prodjosantoso et al. (1999) and the latter angle by the method of §2.3.

2.5. PCRO parameters, inclination angles and tilt systems with numbers of degrees-of-freedom by space group

Whether regular octahedra can be formed in a particular space group depends on whether the octahedra have sufficient degrees of freedom (d.o.f.). The number of d.o.f. defining the octahedral anionic network is N(UC) + N(X), of which N(tilt) are used to define independent tilt angles. In P4₂/nmc with irregular octahedra, the tilt system is , i.e. N(tilt) = 3. For regular octahedra, this is reduced to with N(tilt) = 2. The parameter denoting the remaining d.o.f. available to construct the octahedra, N(PCRO), is equal to N(UC) + N(X) − N(tilt) (Table 4). In space groups Pbnm and Cmcm, N(PCRO) values equal to six signify a capacity to form regular octahedra independently of the values of the two tilt angles. Although N(PCRO) is less than 6 in Ibmm, I4/mcm, P4/mbm and , this does not preclude the formation of regular octahedra, since the space group symmetry itself provides partial regularity: in Ibmm: a₁ = a₂ and θ₂₃ = θ₃₁; in I4/mcm and P4/mbm: a₁ = a₂ with θ₂₃ = θ₃₁ = θ₁₂ = 90°; in : a₁ = a₂ = a₃ with θ₂₃ = θ₃₁ = θ₁₂. In space group P4₂/nmc, the N(PCRO) value of less than 6 signifies that the two tilt angles in the refined structure will be interdependent.

Table 4
Summary of inclination angles, PCRO parameters and degrees of freedom by space group

Space group	Pbnm	Cmcm	Ibmm	I4/mcm and P4/mbm	P42/nmc
3-axis in equation (1)	z	–	z	z	z	–	–
Inclination angles
PCRO parameter equalities			a1 = a3	a1 = a2		a1 = a2 = a3	a1 = a2 = a3
Tilt system				and a^0a^0c^+	or		a^0a^0a^0
N(UC)	3	3	3	2	2	2	1
N(X)	5	5	2	1	5	1	0
N(tilt)	2	2	1	1	2	1	0
N(PCRO)	6	6	4	2	5	2	1
N(A)	2	2	1	0	0	0	0
N(B)	0	0	0	0	0	0	0

The final two rows in Table 4 relate to the d.o.f. assigned to A and B cations. The structures associated with the A cations are discussed in §2.7, after the primary structures associated with the parameter N(tilt) have been considered in the following sub-section.

2.6. Additional anionic network parameters related to octahedral tilting

Since octahedral tilting per se is a secondary structural feature relative to the BX₆ octahedra, the primary structures resulting from this tilting necessarily concern the coordination of the A ions. It has long been customary to assume AX₈ polyhedra in GdFeO₃-type perovskites such as CaTiO₃ in space group Pbnm (Liu & Liebermann, 1993). Mitchell & Liferovich (2004), adopting the perspective of coordination chemistry, reported an increase in coordination number from 8 to 9 with x in the solid solution series Ca_1–xNa_xTi_1–xTa_xO₃. Mitchell (2002b) states generally that coordination numbers of 8, 9, 10 and 12 are possible, and calculates as an example the volume of uncoordinated space in SrZrO₃ (I4/mcm) as 8.7%. This is the difference between V_A, the volume assigned to AX₁₂ polyhedra from volume filling with BX₆ octahedra, and the volume of a more appropriate AX₈ polyhedron, V(AX₈ ).

From a functional viewpoint, the AX₈ polyhedra in tilted perovskites provide scaffolding to support the eight octahedral faces of the full AX₁₂ polyhedra. In Pbnm, the inner AX₈ polyhedron is formed by the top and bottom parallelogram AX₁₂ faces and four vertical struts (Fig. 8).

Figure 8 AX8 and AX12 polyhedra in space group Pbnm. The two opposite, yellow parallelogram faces are joined by vertical blue struts to form AX8 polyhedra. The eight green faces are shared with BX6 octahedra.

It is now appropriate to interpret Fig. 6 in terms of the effect of octahedral tilting on A-ion coordination. The c⁺ tilting causes parallelograms PQRS to be formed instead of rectangles, with a concomitant reduction in area A(PQRS). The a⁻ tilting, by comparison, leads to equal and opposite displacements of vertices P, Q, R and S along z (denoted by + and −). The volume contributions made by plane PQRS and the other five faces of the AX₈ polyhedron to V(AX₈) are not affected by this tilting, since changes brought about by the equal and opposite displacements relative to the centre-of-symmetry at [0,0,0] cancel one another out.³ The mirror plane through A, B, C and D in all space groups with c⁺ tilting means that the horizontal positions, in this case the x and y coordinates of the bottom and top parallelograms in Fig. 8, are identical. It follows that the formula V(AX₈) = A(PQRS)(c/2) could be used, since the centres of both parallelograms are separated by c/2 in all space groups with in-phase c-axis tilting apart from P4/mbm, where the separation is c. In space groups P4₂/nmc and I4/mcm, to which anti-phase, c⁻ tilting applies, the same formula is valid, since the mean cross-section is A(PQRS ).

Since the component of the tilting is solely responsible for the reduction in A(PQRS) and V(AX₈), the converse applies that A(PQRS) quantifies the extent of this tilting. Without tilting, A(PQRS) would be equal to one quarter of the unit cell cross-sectional area, i.e. (ab)/4 in crystal systems with perpendicular x and y axes. A dimensionless coefficient to quantify the amount of tilting may therefore be defined as follows.

Although the tilting about the x_PC and y_PC axes in Pbnm, more generally the tilting component when applied to all space groups, does not affect V(AX₈ ), it does affect V_B. For a given perovskite compound with unit-cell cross-section ab, this tilting allows larger V_B values than would otherwise have been the case. The greater the degree of tilting, the larger the V_B value. A dimensionless coefficient to quantify this is given by relating V_B to the volume of an upright, untilted octahedron of equal basal area in xy projection, . The in-plane components of vectors p and q in Fig. 6 define the waist of this octahedron of perpendicular height c/2. These depend in turn on PCRO vectors a₁ and a₂.

and

In the presence of tilting, η_B > 1. V_B is calculated as one-sixth of the volume of the PCRO and V_B,ref as twice the volume of a right pyramid of height c/4 with parallelogram base of area

i.e.

Analytical expressions for and are quoted in Table 5. The form of is similar for all space groups, and can generally be traced back to simple expressions involving fractional coordinates. Particularly significant are the fixed, limiting values of and in space groups Ibmm and I4/mcm, respectively. These govern the sequence of phase transitions in series with increasing V_A volume (see §3.1). The corresponding derivations are in §S4 of the supporting information, along with diagrams of the and tilt patterns determining and , respectively. These parameters are also calculated in the Excel datafile in the supporting information.

Table 5
Analytical expressions for structural parameters η_A and η_B to quantify and tilting

Space group	N(tilt)	ηA	ηB
Pbnm (B in 4b)	2	(1 − [3 − 4x(O2)][4y(O2) − 1])
Pbnm (B in 4a)	2	([1 + [4x(O2) − 1][4y(O2) − 1])	As for Pbnm (B in 4b)
Cmcm	2	16x(O1)y(O2)	As for Pbnm (B in 4b)
Ibmm	1	1	As for Pbnm (B in 4b)
P4/mbm, I4/mcm	1	8x(O2)(1− 2x(O2))	1
P42/nmc	3	2a,2b: 4(− + 2y(O3))(+ 2y(O2))	As for Pbnm (B in 4b)
		4d: 4( − 2y(O3))( − 2y(O2))
	1	4x(O)(1 − x(O))	–

Since tilting reduces V(AX₈ ) and therefore V_A, whereas tilting increases V_B, use of V_A/V_B as an indicator of the overall degree of tilting is vindicated. By using parameters and to quantify the tilting, which are simply calculated, the difficulties of defining and calculating tilt angles unequivocally can be circumvented. As may be inferred from the complexity of Fig. 6 and the comparison of calculated tilt angles in the lower part of Table 3, this could be seen as an advantage.

2.7. A cation positions

In earlier work on the parameterization of centrosymmetric perovskites (Thomas, 1998), difficulties were experienced in fixing the positions of the A ions by reference to their coordinating X ions in the AX₁₂ polyhedra. However, subsequent work by Magyari-Köpe et al. (2001, 2002) demonstrated that their positions could be treated as a function of the V_A/V_B ratio.

The essential objective is to quantify the small displacements from geometrically regular positions [0, 0] and [½, ½] in the relevant plane of projection (Fig. 9). The parameterization applied here is limited to a transformation to polar coordinates L and ϕ in Pbnm [Fig. 9(a)], whereby L is twice the displacement of an individual ion. In space group Cmcm, two independent lengths, L₁ and L₂, apply [Fig. 9(b)]. These displacements are either along one axis or non-existent in the other space groups.

Figure 9 A cation positions in space groups (a) Pbnm and (b) Cmcm in 2D projection. Green circles: height +¼; brown circles: height −¼. A network with dashed lines is shown in (a).

3. Analysis of structures at variable temperature and chemical composition

The structural parameters defined in §2 are used here to characterize the sequences of phase transitions observed with increasing temperature and varying chemical composition in centrosymmetric perovskites. Li et al. (2004) identified the following commonly occurring sequence with increasing temperature or mean A ion radius: Pbnm → Ibmm → I4/mcm → . This applies to Sr_1−xBa_xHfO₃ and Sr_1–xBa_xZrO₃ perovskites (Kennedy et al., 2001), as well as to the systems Sr_1–xBa_xSnO₃ and Ca_1–xSr_xSnO₃ (Mountstevens et al., 2003). Identification of the intermediate Ibmm phase was significant, as it had previously been regarded as rare (Kennedy et al., 2001). The temperature range of stabilization of this phase is highly variable, with values greater than 570 K in BaPbO₃ (Fu et al., 2007), of 140 K in SrRuO₃ (Kennedy et al., 2002), 90 K in SrHfO₃ (Li et al., 2004) and less than 20 K in SrRhO₃ (Kennedy et al., 2004). It is characterized by a fixed value of parameter equal to one (Table 5).

A second sequence was identified by Ahtee & Darlington (1980) for NaTaO₃: , with further structural work by Kennedy, Prodjosantoso et al. (1999) reporting transition temperature ranges of 738–753 K, 823–863 K and 883–913 K. More recently, Mitchell et al. (2014) reported the same series of phase symmetries for the mineral lueshite, NaNbO₃.

Both sequences are characterized by an increasing V_A/V_B ratio with T or x, which may be attributed either to a larger volume expansion coefficient of V_A relative to V_B or to a compositionally induced expansion of the A-site volume relative to the B site. The term `A-ion perturbation' may be applied in both cases, since a change in the A-site geometry is the dominant driving force causing the whole structure to respond.

A solid solution series with a rising V_A/V_B ratio brought about by reducing V_B has been established by Yang (2008) for LaCr_1–xNi_xO₃ at room temperature. Upon increasing x from 0 to 0.7, a Pbnm → phase transition is observed. In this case, a B-ion perturbation is being applied.

3.1. x- and T-series with A-ion perturbations

3.1.1. Sr_xBa_1–xSnO₃ and Sr_xBa_1–xHfO₃

Table 6 contains structural parameters calculated from the refinements of Mountstevens et al. (2003) for the solid solution series Sr_xBa_1–xSnO₃ and of Li et al. (2004) for the Sr_xBa_1–xHfO₃ series.

Table 6
Structural parameters for two solid solutions spanning space group Ibmm at room temperature

Composition	ICSD	Space group	ηA	ηB	VA (Å^3)	VB (Å^3)	VA/VB	λ	σ
SrSnO3	190602	Pbnm	0.9847 (1)	1.0313 (3)	54.181 (5)	11.454 (4)	4.730 (2)	0.0023 (4)	0.0104 (8)
Sr0.8Ba0.2SnO3	190611	Pbnm	0.9935 (2)	1.0236 (3)	55.10 (6)	11.42 (1)	4.824 (2)	0.0052 (7)	0.0195 (9)
Sr0.6Ba0.4SnO3	190610	Ibmm	1	1.0147 (2)	56.02 (7)	11.40 (2)	4.913 (1)	0.0004 (3)	0.0091 (8)
Sr0.4Ba0.6SnO3	190609	I4/mcm	0.9960 (1)	1	56.9 (1)	11.43 (2)	4.9758 (6)	0.0000 (4)	0
Sr0.2Ba0.8SnO3	190608		1	1	57.52 (8)	11.50 (2)	5	0	0
BaSnO3	190601		1	1	58.071 (4)	11.6142 (8)	5	0	0
SrHfO3	89383	Pbnm	0.9856 (6)	1.034 (1)	55.64 (2)	11.79 (2)	4.719 (8)	0.003 (2)	0.013 (1)
Sr0.8Ba0.2HfO3	55746	Pbnm	0.996 (2)	1.029 (4)	57.16 (6)	11.89 (6)	4.81 (3)	0.033 (6)	0.058 (9)
Sr0.6Ba0.4HfO3	55747	Ibmm	1	1.026 (1)	57.97 (1)	11.96 (1)	4.848 (7)	0.0058 (5)	0.039 (3)
Sr0.4Ba0.6HfO3	55748	I4/mcm	0.98755	1	58.88 (1)	11.95 (1)	4.926 (5)	0.0013 (2)	0
Sr0.2Ba0.8HfO3	55749		1	1	59.7948 (9)	11.9590 (2)	5	0	0
BaHfO3	–		1	1	60.2909 (4)	12.0582 (1)	5	0	0

On reading Table 6 from top to bottom, values of V_A confirm that the increasing mean A-ion radius is the principal driving force for the structural changes in both systems. V_A rises by 7.2% between x = 0 and x = 1 in the tin-containing and by 8.4% in the hafnium-containing series. Furthermore, the larger ionic radius of Hf⁴⁺ compared to Sn⁴⁺ in sixfold coordination [0.71 cf. 0.69 Å; Shannon (1976)] causes the V_A value at a given Sr:Ba ratio to be systematically larger in the system with hafnium as B ion. By comparison, the volume of the B site increases by 1.4% between x = 0 and x = 1 in the tin-containing series compared system compared to 2.3% in the hafnium series. The following ranges of the V_A/V_B parameter span the Ibmm phase in the two systems: 4.824 < V_A/V_B < 4.976 and 4.81 < V_A/V_B < 4.926.

The variation of parameters η_A and η_B with x reveals the mechanisms behind the Pbnm → Ibmm and Ibmm → I4/mcm phase transitions in these series [Figs. 10(a) and 10(b)].

Figure 10 Characterization of phase transitions in terms of tilt-related parameters ηA and ηB: (a) SrxBa1–xSnO3, (b) SrxBa1–xHfO3, (c) BaPbO3, (d) CaTiO3. Circles denote experimentally determined points, which are joined by straight lines. Arrows signify limited flexibility in the positions of the phase transitions.

Reading these two diagrams from left to right, the Pbnm → Ibmm transition is triggered by η_A reaching the limiting value of 1 whilst η_B is still > 1. The range of stability of the Ibmm phase is determined by the x value at which η_B becomes equal to 1, whereupon a phase transition to I4/mcm takes place. The diagrams suggest that the approach of η_A to 1 takes place continuously at the Pbnm–Ibmm boundary. By comparison, the Ibmm–I4/mcm boundary is characterized by a discontinuous step in η_B down to 1. Also characteristic of the latter boundary is a fall in η_A to a value less than 1. This may be understood by considering that η_A and η_B can only both be equal to one in the aristotype phase. However, at the start of the range of stability of the I4/mcm phase, V_A/V_B has not reached the limiting value of 5. The transition from I4/mcm to takes place as soon as V_A has increased sufficiently for this V_A/V_B limit to be reached.

3.1.2. BaPbO₃

The above rationalization can be transferred without modification to series where temperature is responsible for an increasing V_A volume. For example, the Ibmm phase is stabilized over a wide temperature range in BaPbO₃ perovskites, these being of technological relevance due to superconductivity in the Ba-Pb-Bi-O system. Following earlier structural refinements by Moussa et al. (2001) and Ivanov et al. (2001) in the monoclinic space group C2/m on cooling and at room temperature, respectively, Fu et al. (2005) interpreted this monoclinic distortion as probably due to twinning. They also provided four structural refinements in Ibmm at room temperature and at 4.2 K. Upon heating, Fu et al. (2007) reported phase changes to tetragonal I4/mcm and thereafter to at approximate temperatures of 573 and 673 K respectively. Fig. 10(c) is generated from experimental points calculated from the structural refinements of Fu et al. (2005, 2007). The absence of Pbnm symmetry in BaPbO₃ can be ascribed to the V_A/V_B ratio remaining above 4.8320 (8) over the temperature range from 4.2 K to 773 K, this being higher than the threshold values of 4.81 and 4.824 noted in §3.1.1 for Ibmm symmetry. This interpretation is supported by the stabilization of the strontium compound SrPbO₃ in Pbnm (Fu & Ijdo, 1995). The calculated V_A/V_B value here is significantly lower, at 4.418 (2).

Despite the absence of a Pbnm phase in BaPbO₃, Fig. 10(c) shows that the Ibmm → I4/mcm transition is also governed by discontinuous jumps in η_B to 1 and η_A to less than 1, as in Figs. 10(a) and 10(b). The approach to the aristotype phase follows the same principles as noted earlier.

3.1.3. CaTiO₃

The mineral perovskite, CaTiO₃, has not been observed in space group Ibmm, although it exists in Pbnm at room temperature (Sasaki et al., 1987) and at temperatures up to 1373 K (Liu & Liebermann, 1993). Redfern (1996) paved out the sequence of phase transitions at yet higher temperatures through I4/mcm to and aristotype, reporting transition temperatures of 1373–1423 K and at ∼1523 K. Although Kennedy, Howard & Chakoumakos (1999) subsequently raised the possibility of an orthorhombic Cmcm structure being stabilized at around 1380 K, this was refuted by Ali & Yashima (2005), who later generated the nineteen structural refinements upon which the experimental points of Fig. 10(d) are based (Yashima & Ali, 2009). It is logical that no intermediate phase of symmetry Ibmm is formed with increasing temperature, since η_A remains significantly below one over the whole Pbnm range. This means that the degree of AO₈ expansion induced thermally is smaller than the compositionally induced AO₈-expansion analysed in §3.1.1. Stabilization of the Pbnm phase is curtailed by η_B falling to 1 at the Pbnm–I4/mcm boundary. After the temperature of this phase transition has been reached, the overall expansion of the unit cell allows the volumetric requirements of the TiO₆ octahedron to be satisfied without tilting in space group I4/mcm. The remaining tilting allows further comparatively greater AO₈ expansion with increasing temperature up to the temperature of the phase transition to .

In all four series upon which Fig. 10 is based, the changes in tilt systems taking place at the phase transitions, these being critical events, are largely determined by the systematic changes in tilt angles taking place over wide compositional or temperature ranges. Parameters η_A and η_B are ideally suited for encapsulating these changes.

3.1.4. NaTaO₃ and NaNbO₃

These compounds provide an experimental basis for assembling the factors governing stabilization of the Cmcm phase, and more widely, the progression from Pbnm to via Cmcm and P4/mbm instead of Ibmm and I4/mcm. I4/mcm and P4/mbm structures are very similar, the only difference being anti-phase or in-phase c axis tilting, respectively. Furthermore, structural parameters V_A/V_B, η_A and η_B cannot discriminate between these two structures. It is therefore appropriate to regard Cmcm and Imma, in the first instance, as alternative precursor phases to a generic phase of symmetry I4/mcm or P4/mbm.

An analysis of the phases of Cmcm symmetry in terms of V_A/V_B, η_A and η_B within these two compounds is given in Table 7.

Table 7
Focus on the Cmcm phases of NaTaO₃ and NaNbO₃ with comparative data for neighbouring Pbnm and P4/mbm phases

Compound	ICSD	Temperature (K)	VA/VB	ηA1	ηA2	ηB	Reference
Pbnm
NaTaO3	150430	293	4.74 (2)	0.988 (2)		1.032 (3)	Mitchell & Liferovich (2004)
NaNbO3	192400	293	4.802 (5)	0.9868 (6)		1.0206 (6)	Mitchell et al. (2014)
Cmcm
NaTaO3	280099	803	4.899 (2)	0.987 (3)	0.994 (3)	1.0076 (2)	Darlington & Knight (1999)
NaTaO3	241445	778	4.891 (2)	1.002 (2)	0.977 (2)	1.0076 (2)	Knight & Kennedy (2015)
NaTaO3†	239691	783	4.889 (3)	0.991 (5)	0.989 (5)	1.0087 (3)	Arulnesan et al. (2016)
NaNbO3	192404	848	4.928 (3)	1.051 (3)	0.933 (3)	1.0041 (4)	Mitchell et al. (2014)
NaNbO3	192405	873	4.940 (2)	1.063 (3)	0.924 (3)	1.0033 (3)	Mitchell et al. (2014)
P4/mbm
NaTaO3	88377	843	4.945 (1)	0.9907 (2)			Kennedy, Prodjosantoso et al. (1999)
NaNbO3	192406	898	4.9687 (7)	0.9948 (1)		1	Mitchell et al. (2014)

†Doped with 1 mol% K.

The definitive characteristic of the Cmcm phase is the proximity of η_B values to one in Table 7. This parameter has much larger values in the Ibmm phase, with η_B values greater than 1.03 observed [see Table 6 and Figs. 10(a), 10(b) and 10(c)]. This difference is due to there being only one tilt of type in Cmcm, compared to two in Ibmm (see §S4 of the supporting information). Values of η_B close to one confirm Cmcm as a precursor to an I4/mcm or P4/mbm phase, in which η_B is exactly equal to one. The existence of Cmcm symmetry in CaTiO₃ at ∼1380 K, as proposed by Kennedy, Howard & Chakoumakos (1999), is indeed possible, since η_B is close to one at this temperature.

Also to be noted in Table 7 is the splitting of η_A values made possible by there being two symmetry-independent A sites in Cmcm. Geometry simply requires a mean value 〈η_A〉 = (η_A1 + η_A2)/2 of less than one. This splitting is considerably more marked in NaNbO₃. The ability for the lower η_A value to be significantly less than one could go some way towards rationalizing the phase coexistence of Pbnm and Cmcm phases at room temperature observed in NaTaO₃ by Knight & Kennedy (2015). A further contributory factor towards stabilization of Cmcm in 1 mol% K-doped samples of NaTaO₃ at room temperature (Arulnesan et al., 2016) could be preferential occupation of the sites with higher η_A value by the larger potassium ions, i.e. a partial ordering.

It is speculated that the proximity of η_B to one in Cmcm is conducive to a to a higher temperature phase transition to P4/mbm, whereas the sudden fall in η_B within Ibmm or Pbnm phases observed in Figs. 10(a)–10(d) favours a transition to I4/mcm.

3.2. X-series with B-ion perturbations

Calculated structural parameters for the LaCr_1–xNi_xO₃ solid solution series (Yang, 2008) are given in Table 8.

Table 8
Structural data relevant to the Pbnm → transition at x ∼ 0.7 in LaCr_1–xNi_xO₃

Composition	ICSD	Space group	ηA	ηB	〈ϕc〉 (°)	〈ϕa〉 (°)	(〈ϕa〉 + 〈ϕc〉)/2 (°)	VA (Å^3)	VB (Å^3)	VA/VB
x = 0	173469	Pbnm	0.9925 (1)	1.0355 (7)	5.02 (3)	8.5 (2)	6.78 (8)	48.38 (1)	10.182 (7)	4.751 (4)
x = 0.3	173471	Pbnm	0.9836 (4)	1.0278 (9)	8.67 (5)	5.4 (2)	7.05 (8)	48.28 (1)	10.178 (9)	4.744 (5)
x = 0.6	173472	Pbnm	0.9900 (1)	1.0301 (8)	6.13 (4)	6.5 (2)	6.32 (10)	48.388 (9)	10.152 (8)	4.766 (5)
x = 0.7	173473	Pbnm	0.9983 (3)	1.0254 (5)	6.36 (3)	4.64 (9)	5.50 (5)	48.204 (9)	9.956 (6)	4.841 (3)
x = 0.7	173474		0.9891 (1)			5.97 (3)		47.453 (5)	9.867 (4)	4.809 (2)
x = 0.8	173475		0.9893 (2)			5.90 (5)		47.414 (7)	9.850 (5)	4.814 (3)
x = 0.9	173476		0.9897 (1)			5.79 (3)		47.29 (1)	9.810 (4)	4.820 (2)
x = 1	173477		0.9900 (1)			5.72 (3)		46.778 (3)	9.696 (2)	4.825 (1)

Unlike the structures perturbed by A ions, the variation in η_A and η_B parameters is small and also unsystematic for structures in space group Pbnm. There is no tendency of either parameter to approach one. The expected cross-correlations between parameter pairs η_A ↔ ϕ_c and η_B ↔ ϕ_a are observed within the Pbnm phase field: the greater the magnitude of the deviations of η_A and η_B from one, the larger the 〈ϕ_c〉 and 〈ϕ_a〉 angles. The ϕ_a tilt angle falls to 4.64° and the mean tilt angle (〈ϕ_a〉 + 〈ϕ_c〉)/2 to 5.50° in Pbnm just before the phase transition, with ϕ_a rising to 5.97° in the phase at x = 0.7. A gradual reduction in ϕ_a to 5.72° is observed up to x = 1, this still being a stable rhombohedral phase far from a transition to the cubic aristotype. The gradual fall in V_B with x is consistent with the smaller ionic radius of Ni³⁺ compared to Cr³⁺ (Shannon, 1976) and is responsible for the systematic increase in V_A/V_B with x, which peaks at 4.841 in Pbnm before the phase transition to and falls to 4.809 thereafter. The transition is examined further in §4.4.

4. Modelling the structural variation of Pbmn perovskites with increasing pressure

The structures adopted by perovskites under increasing hydro­static pressure rest on the different compressibilities of the AX₁₂ and BX₆ polyhedra (Angel et al., 2005), such that the volume ratio V_A/V_B may either increase or decrease. In general, experimental difficulties limit the pressure range over which full structure refinements can be obtained, a common practice being to report the variation of unit-cell parameters over a wider pressure range by means of the third-order Birch–Murnaghan equation of state [equation (7)].

Here, coefficient K_T is the bulk modulus and is its derivative with respect to pressure. V₀ represents the reference unit-cell volume. The appropriate notation to use for the variation of cell parameters a, b, c with pressure would be K_a0, , a₀ etc., whereby cubes a³,b³,c³ will have been used for the Birch–Murnaghan fitting. It is also common to quote linear compressibilities , , , whereby , etc., these generally holding over a limited pressure range. By setting values equal to 4, the term in curly brackets is equal to one, giving rise to the second-order Birch–Murnaghan equation.

The primary motivation for studying perovskites under pressure has been to simulate MgSiO₃ perovskite in the lower mantle, for which a phase transition to a post-perovskite phase at pressures > 125 GPa and a temperature of 2700 K has been proposed (Murakami et al., 2004). This system is simulated by assuming regular octahedra in the following sub-section.

4.1. Simulation of MgSiO₃ under pressures of up to 125 GPa

It may be assumed that MgSiO₃ remains in space group Pbnm at all pressures up to ∼125 GPa. Accordingly, the 12 sets of a, b, c unit-cell parameters reported by Vanpeteghem et al. (2006) at pressures up to 10 GPa were used to derive the following Birch–Murnaghan constants to second order for the unit-cell volume and a and c unit-cell parameters⁴: K_T = 252.90 GPa; V₀ = 162.52 ų; K_T,a = 232.11 GPa; V_0,a = 109.08 ų; K_T,c = 240.78 GPa; V_0,c = 328.33 ų.

The assumption was made that these constants could extended to 125 GPa, thereby yielding theoretical a, b and c unit-cell parameters over this whole range. This process was validated by comparing the values obtained (`Extrapolated  B-M') with values quoted by Murakami et al. (2004) and Fiquet et al. (2000). Acceptable agreement is found in Table 9.

Table 9
Validation of Birch–Murnaghan (B-M) constants for use at pressures of up to to ∼125 GPa†

Method/calculation type	p (GPa)	a (Å)	b (Å)	c (Å)	Vuc (Å^3)
Murakami XRD	109	4.325	4.579	6.308	124.9
Murakami MD	109	4.403	4.574	6.410	129.1
Extrapolated B-M	109	4.365	4.559	6.315	125.7
Fiquet synchrotron XRD	40.66	–	–	–	143.26
Extrapolated B-M	40.71	4.571	4.743	6.606	143.22

†V_uc: unit-cell volume.

200 equally spaced pressures were taken within the range up to 125 GPa and a, b, c values calculated from the above Birch–Murnaghan constants. Values of x(O1), y(O1), x(O2), y(O2) and z(O2) were allowed to vary within the core Excel Solver functionality for space group Pbnm (Fig. 7), so as to generate regular octahedra. The resulting variation of parameters η_A and η_B with pressure is shown in Fig. 11(a).

Figure 11 Simulations of MgSiO3 and YAlO3 in terms of ηA and ηB. (a) MgSiO3 with regular octahedra up to p = 125 GPa; (b) comparison of YAlO3 with regular and distorted octahedra up to p = 8 GPa; (c) Extrapolation of YAlO3 with regular and distorted octahedra up to p = 52 GPa. Dashed lines indicate pressures for which structural data are quoted in Table 10.

Parameter η_A decreases from 0.9576 at 0 GPa to 0.9529 at 125 GPa, which indicates an increase in c⁺ tilting. By comparison, η_B increases from 1.0649 at 0 GPa to 1.0939 at 125 GPa, indicating an increase in tilting. The effect of pressure is to drive the structure ever further into the Pbnm phase field. It is likely that the imminent phase transition to post-perovskite at 125 GPa is triggered by the strong O⋯O repulsions that will be associated with this degree of octahedral tilting.

4.2. Simulation of YAlO₃ at pressures up to 52 GPa

Ross et al. (2004) elucidated the response of the perovskite compound YAlO₃ at pressures up to 8 GPa by providing eight structural refinements of a synthetic single crystal. A decrease in octahedral tilting with pressure was found, which was ascribed to the AlO₆ octahedral compressibility being greater than that of the YO₁₂ site. It follows that an approach towards cubic symmetry will occur, similar to the T-series with A ion perturbation in §3.1. The Birch–Murnaghan constants to third order quoted by Ross et al. (2004) were used to generate unit-cell parameters at different pressures, as in §4.1. Two alternative simulations were carried out for pressures up to 8 GPa, the first assuming regular octahedra as in §4.1, and the second exploiting the octahedral distortions reported by Ross et al. (2004) from an analysis of their structural refinements. For this purpose, the linear, decreasing trends in the three different Al—O bond lengths with increasing pressure observed by them were translated into linear relationships for PCRO parameters a₁ to a₃. In addition, linear relationships for PCRO angle parameters θ₂₃, θ₃₁ and θ₁₂ were derived [equations (8)].

Solver-based refinements in Pbnm were carried out at equally spaced pressure values, the spacing determined by covering the pressure range from 0 to 60 GPa with 200 values. Parameters x(O1), y(O1), x(O2), y(O2) and z(O2) were allowed to vary in refinements with end-point determined by the minimum deviation from conditions (8). The deviation was observed to range between 0.001 and 0.050%.

The results of both simulations are shown in Fig. 11(b) and in that part of Fig. 11(c) with the blue-shaded background. The slope of both η_B curves is negative, compared to the positive slope of η_B for MgSiO₃. The slope of η_A for the simulation with regular octahedra is positive, compared to the negative slope for MgSiO₃. The opposite behaviour to MgSiO₃ is shown by YAlO₃, which shows a gradual progression away from Pbnm towards higher symmetry. The curve for η_A for distorted octahedra has a shallow minimum at ∼3.4 GPa, which is more clearly seen in Fig. 11(c). Thereafter η_A rises, as for undistorted octahedra.

In a computational experiment, the pressure was allowed to rise towards 60 GPa, with unit-cell constants calculated from the Birch–Murnaghan constants of Ross et al. (2004) for the pressure range up to 8 GPa. Fig. 11(c) shows that the curvature of both pairs of curves changes sign as the octahedra change from regular to distorted. Since the constraints in equations (8) only apply up to 8 GPa, it became increasingly difficult to reach low deviations at the end-points of the refinements. The maximum deviation amounted to 1.08%, which was obtained at the highest pressure.

The ascending curve for η_A with distorted octahedra in Fig. 11(c) reaches zero at a pressure of ∼41.5 GPa. This would be consistent with a phase transition to Ibmm [cf. Figs. 10(a) and 10(b)]. By comparison, the η_B curve is the first to reach zero in the simulation with regular octahedra. The corresponding pressure is ∼51.6 GPa. This would be consistent with a phase transition to I4/mcm or P4/mbm [cf. Fig. 10(d)]. The asymptotic approach of the η_B curve to 1 for regular octahedra gives rise to a broad pressure range in which Cmcm could be stabilized as well as Pbnm prior to this phase transition. This phenomenon has been observed in NaTaO₃ by Knight & Kennedy (2015) over a broad temperature range, as discussed in §3.1.4. Assuming pattern similarity, a phase change to P4/mbm would be anticipated at 51.6 GPa. More generally, the simulations of YAlO₃, both with and without octahedral distortion, indicate how this distortion is expected to have a direct effect on the sequence of phase transitions and the pressures at which they occur.

In summary, the analysis of (p–T–X)-induced phase transitions in terms of η_A and η_B parameters is a stimulus to further, targeted experimental work on the systems that have been analysed and simulated in §3.1, §4.1 and §4.2.

4.3. Crystal structures generated in the high-pressure simulations

A by-product of the simulations in §3.1 is the generation of full sets of oxygen ion coordinates. Table 10 contains these data corresponding to the conditions denoted by dashed lines in Figs. 11(a) and 11(c). The space group is Pbmn with the B ions in 4b positions. The full set of associated structural parameters is also quoted below the line in the table.

Table 10
Crystallographic and structural parameters generated in the simulations of MgSiO₃ and YAlO₃

Compound	MgSiO3	MgSiO3	YAlO3	YAlO3	YAlO3	YAlO3
Pressure (GPa)	109	125	8	8	31.5	31.5
Octahedra	Regular	Regular	Regular	Distorted	Regular	Distorted
a (Å)	4.3649	4.3278	5.1285	5.1285	5.0308	5.0308
b (Å)	4.5592	4.5263	5.2443	5.2443	5.0751	5.0751
c (Å)	6.3151	6.2628	7.2899	7.2899	7.1349	7.1349
x(O1)	0.1067	0.1083	0.0756	0.0810	0.0470	0.0604
y(O1)	0.4780	0.4775	0.4925	0.4810	0.4965	0.4934
x(O2)	0.6936	0.6933	0.7241	0.7054	0.7309	0.7241
y(O2)	0.3017	0.3019	0.2748	0.2940	0.2687	0.2758
z(O2)	0.0533	0.0542	0.0378	0.0420	0.0235	0.0295
γPC (°)	92.49	92.57	91.28	91.28	90.50	90.50
VA (Å^3)	25.439	24.816	40.386	40.127	37.772	37.651
VB (Å^3)	5.979	5.855	8.630	8.889	7.769	7.890
VA/VB	4.255	4.239	4.680	4.514	4.862	4.772
θx (°)	16.89	17.05	10.27	13.64	6.88	8.90
θy (°)	16.89	17.05	10.27	13.71	6.88	8.91
θz (°)	16.79	17.03	12.06	13.19	7.58	9.72
〈ϕc〉 (°)	12.19	12.24	5.79	10.05	4.32	5.90
〈ϕa〉 (°)	12.05	12.25	8.55	9.48	5.36	6.72
ηA	0.9534	0.9529	0.9898	0.9686	0.9943	0.9893
ηB	1.0910	1.0939	1.0457	1.0549	1.0177	1.0285
a1 (Å)	3.2981	3.2750	3.7272	3.7845	3.5989	3.6193
a2 (Å)	3.2981	3.2750	3.7272	3.7647	3.5989	3.6138
a3 (Å)	3.2981	3.2750	3.7272	3.7437	3.5989	3.6194
θ23 (°)	90	90	90	90.61	90	89.85
θ31 (°)	90	90	90	89.46	90	89.75
θ12 (°)	90	90	90	89.69	90	89.71
λPC	0	0	0	0.0037	0	0.0007
σPC	0	0	0	0.012	0	0.004

In all cases, the effect of increasing pressure is to reduce V_A and V_B. However, the ratio V_A/V_B is reduced in MgSiO₃ and increased in YAlO₃. This signifies movement further into the Pbnm phase field in the former case [Fig. 11(a)] and movement away towards higher symmetry in the latter [Fig. 11(c)]. This is borne out by the changes in inclination angles θ_x, θ_x, θ_x, tilt angles ϕ_a, ϕ_c and tilt-related parameters η_A, η_B.

4.4. Analysis of (La_1–xNd_x)GaO₃ structures at pressures of up to 12 GPa

Apart from the above simulations, the structural parameters used in this work are applied to the structural refinements of Angel et al. (2007) for this 3:3 perovskite solid solution. In carrying out A-ion substitutional perturbations, it was found that a phase transition from Pbnm to takes place under pressure for x values up to 0.20, but not for x = 0.62 or x = 1. The question arises as to whether a particular pattern of structural evolution within the Pbnm phase is associated with a phase transition to . Data for the Pbmn and phases at their lowest and highest investigated pressures for a given x value are given in Table 11.

Table 11
Structural parameters for (La_1–xNd_x)GaO₃ in space groups Pbnm and

x	Space group	p (GPa)	ICSD	ηA	ηB	〈ϕc〉 (°)	〈ϕa〉 (°)	(〈ϕa〉 + 〈ϕc〉)/2 (°)	VA (Å^3)	VA (Å^3)	VA/VB
0	Pbnm	0.0001	160233	0.9926 (2)	1.0406 (7)	4.90 (8)	8.5 (1)	6.69 (7)	48.653 (8)	10.299 (8)	4.724 (4)
	Pbnm	2.038 (10)	160235	0.9939 (2)	1.0382 (7)	4.46 (7)	8.3 (1)	6.39 (7)	48.136 (7)	10.146 (7)	4.744 (4)
		2.356 (7)	160265	0.9868 (3)			6.56 (7)		48.024 (9)	10.066 (8)	4.771 (5)
		8.127 (23)	160270	0.9872 (3)			6.45 (7)		46.745 (8)	9.782 (8)	4.779 (4)
0.06	Pbnm	0.0001	160236	0.9902 (4)	1.044 (2)	5.7 (1)	9.1 (3)	7.4 (1)	48.50 (2)	10.33 (2)	4.693 (9)
	Pbnm	5.30 (5)	160240	0.9947 (4)	1.042 (2)	4.2 (2)	9.1 (3)	6.6 (2)	47.18 (2)	9.98 (2)	4.73 (1)
0.12	Pbnm	0.0001	160241	0.9907 (4)	1.040 (1)	5.5 (1)	8.5 (2)	7.0 (1)	48.55 (1)	10.29 (1)	4.716 (7)
	Pbnm	6.437 (7)	160243	0.9939 (3)	1.039 (1)	4.4 (1)	8.5 (2)	6.5 (1)	46.96 (1)	9.91 (1)	4.741 (6)
		8.020 (2)	160271	0.9875 (5)			6.4 (1)		46.65 (2)	9.75 (1)	4.783 (9)
		9.496 (13)	160272	0.9872 (4)			6.46 (9)		46.37 (1)	9.71 (1)	4.778 (6)
0.20	Pbnm	0.0001	160244	0.9884 (4)	1.0425 (7)	6.2 (1)	8.7 (1)	7.45 (7)	48.404 (9)	10.323 (8)	4.689 (5)
	Pbnm	8.671 (7)	160248	0.9931 (3)	1.042 (1)	4.7 (1)	8.8 (2)	6.79 (9)	46.38 (1)	9.83 (1)	4.720 (6)
0.62	Pbnm	0.0001	160250	0.9807 (5)	1.049 (1)	7.9 (1)	9.1 (1)	8.52 (8)	47.72 (1)	10.35 (1)	4.611 (6)
	Pbnm	9.432	160256	0.9842 (7)	1.046 (1)	7.2 (2)	8.9 (2)	8.0 (1)	45.60 (1)	9.81 (1)	4.648 (7)
1.00	Pbnm	0.0001	160257	0.9749 (5)	1.0537 (9)	9.00 (9)	9.6 (1)	9.28 (8)	47.18 (1)	10.36 (1)	4.555 (6)
	Pbnm	8.292 (9)	160264	0.9762 (4)	1.0514 (8)	8.78 (7)	9.4 (1)	9.11 (7)	45.324 (9)	9.909 (9)	4.574 (5)

The direct influence of the A-ion perturbations is seen in the values of V_A, which decrease downwards in the table with increasing x for the p = 0.0001 GPa values. Since V_B remains approximately constant, a parallel trend of decreasing V_A/V_B ratio with increasing x is generally observed. This is consistent with greater stabilization within the Pbnm phase field. Pressure induces the opposite trend to raising x, since the V_A/V_B ratios for maximum pressures within the Pbnm phase field are uniformly higher than at atmospheric pressure. A parallel increase in η_A and decrease in η_B is observed. At lower x values up to 0.20, which are associated with reduced Pbnm stabilization, the increased pressure induces a phase transition to . Angel et al. (2007) report the following approximate pressures for this phase transition: x = 0: 2.2 GPa; x = 0.06: 5.5 GPa; x = 0.12: 7.8 GPa; x = 0.20: 12 GPa. The expected cross-correlations between parameter pairs η_A ↔ ϕ_c and η_B ↔ ϕ_a are observed within the Pbnm phase field: the greater the magnitude of the deviations of η_A and η_B from one, the larger the 〈ϕ_c〉 and 〈ϕ_a〉 angles. It is proposed that the observed fall in 〈ϕ_c〉 to ∼4.2–4.4° with increasing pressure is the principal driving force for the phase transitions to . After the transitions for x = 0 and x = 0.12, i.e. within the phase field, the ϕ_a tilt angle is larger and approximately equal to the mean tilt angle (〈ϕ_a〉 + 〈ϕ_c〉)/2 in the Pbnm field beforehand. For x = 0.62 and x = 1.00, the critical range of ϕ_c between 4.2 and 4.4° is not reached at the pressures investigated, so that these compounds remain stabilized in space group Pbnm.

4.5. Structural parameters for YAl_0.25Cr_0.75O₃ with locked octahedral tilting

Ardit et al. (2017) subsequently took up the theme of locked octahedral tilting in orthorhombic 3:3 perovskites (A ion +3; B ion +3) by reference to the solid solution YAl_0.25Cr_0.75O₃ in space group Pbnm. Although in general agreement with Zhao et al. (2004) and Angel et al. (2005) in assuming compressibility ratios β(AO₁₂)/β(BO₆) < 1 for a 3:3 perovskite and > 1 for a 2:4 perovskite, their compound showed a compressibility ratio approximately equal to one (Table 12).

Table 12
Structural parameters for eleven structural refinements of Ardit et al. (2017) on YAl_0.25Cr_0.75O₃

p (GPa)	ηA	ηB	〈ϕc〉 (°) (§2.3)	〈ϕa〉 (°) (§2.3)	〈ϕc〉 (°) (Ardit)	〈ϕa〉 (°) (Ardit)	VA (Å^3)	VB (Å^3)	VA/VB
0.05 (5)	0.9557 (9)	1.082 (1)	11.9 (1)	11.7 (2)	12.0 (5)	16.6 (5)	43.52 (2)	10.10 (2)	4.309 (8)
1.44 (9)	0.9604 (4)	1.0868 (7)	11.24 (6)	12.32 (9)	12.7 (5)	16.6 (5)	43.238 (9)	10.030 (8)	4.311 (4)
3.32 (9)	0.9609 (5)	1.0862 (7)	11.18 (7)	12.32 (9)	12.7 (5)	16.5 (5)	42.892 (9)	9.938 (8)	4.316 (4)
4.93 (7)	0.9603 (5)	1.0871 (7)	11.25 (7)	12.36 (9)	12.8 (5)	16.6 (5)	42.561 (9)	9.878 (8)	4.309 (4)
6.36 (8)	0.9630 (5)	1.0890 (7)	10.88 (7)	12.67 (9)	13.1 (5)	16.5 (5)	42.321 (9)	9.812 (8)	4.313 (4)
8.48 (9)	0.9630 (5)	1.0890 (8)	10.88 (7)	12.67 (9)	13.1 (5)	16.6 (5)	41.954 (9)	9.727 (8)	4.313 (5)
10.54 (9)	0.9635 (6)	1.090 (1)	10.81 (9)	12.8 (1)	13.2 (5)	16.6 (5)	41.60 (1)	9.65 (1)	4.312 (6)
12.34 (7)	0.9580 (6)	1.086 (1)	11.57 (8)	12.2 (1)	12.6 (5)	16.5 (5)	41.28 (1)	9.59 (1)	4.303 (6)
14.40 (9)	0.9569 (8)	1.085 (1)	11.7 (1)	12.1 (2)	12.4 (5)	16.6 (5)	40.97 (1)	9.52 (1)	4.302 (7)
16.45 (9)	0.956 (1)	1.084 (2)	11.9 (2)	12.0 (2)	12.4 (5)	16.5 (5)	40.65 (2)	9.45 (2)	4.30 (1)
18.86 (9)	0.959 (2)	1.087 (3)	11.4 (3)	12.6 (3)	13.0 (5)	16.5 (5)	40.30 (3)	9.36 (3)	4.30 (2)

In spite of consistently falling V_A and V_B values with increasing pressure, the V_A/V_B ratio remains approximately constant, as do parameters , , 〈ϕ_c〉 and 〈ϕ_a〉. Discrepancies are observed between the values of the tilt angles calculated according to §2.3 and the values quoted by Ardit et al. (2017) without declaring the method of calculation. This is not unexpected, as discussed in §2.4. Since values of 〈ϕ_c〉 and 〈ϕ_a〉 calculated according to §2.3 show the expected cross-correlations between parameter pairs η_A↔ϕ_c and η_B↔ϕ_a, indirect support is given for the correctness of these calculations.

5. Discussion

The ability of the method to analyse experimental structural data and the sequences of phase transitions between space groups has been demonstrated in §3 and §4. By comparison with the group-theoretical approach, it has not been necessary to relate these explicitly to the cubic aristotype, since deviations of V_A/V_B from the limiting value of 5 in the aristotype provide a direct indication of how far away a given structure is from the aristotype. Parameters V_A, V_B and V_A/V_B continue to yield valuable insight, for example in Table 11. However, the direct calculation of Glazer tilt angles ϕ_c and [ϕ_a, ϕ_b] and associated derived parameters η_A and η_B has allowed a deeper analysis of the structural factors leading to phase transitions than a consideration of these volumes alone. The significance of the generalized algorithms introduced here for the Glazer tilt angles may be assessed by reference to Wang & Angel (2011): `…the decomposition of a perovskite structure including tilted and distorted octahedra by geometric analysis does not result in an unambiguous definition of the Glazer (1972) tilts and the problem is more acute in perovskites with lower space-group symmetries'. These authors noted further that `unambiguous expressions for both the Glazer tilts and their relationship to the V_A/V_B ratio are still to be determined explicitly for each space group, and in a general form'. These observations led Wang & Angel (2011) to resort to group-theoretical methods in order to relate the amplitudes of symmetry-adapted modes to V_A/V_B ratio. The current work, by comparison, has remained strictly based on unit-cell parameters and atomic coordinates. Although it has not provided generalized analytical expressions for these tilt angles, it has led to two generalized algorithms (a) for all centrosymmetric space groups apart from and (b) for space group (and more generally, triclinic space groups). An analytical link between ϕ_c, [ϕ_a, ϕ_b] and V_A/V_B has not been provided, since inclination angles to are more suitable for this purpose [equation (1)]. However, the linked-cell implementation within Excel allows the empirical derivation of numerical relationships between Glazer tilt angles and V_A/V_B.

The methodological innovations of the current work and their benefits are summarized in Table 13.

Table 13
Summary of innovations in the current work

Innovation	Benefits
Quantification of distortion of centro­symmetric octahedra via PCRO.	(a) Direct, concise visualization of octahedral distortion.
	(b) Variation of PCRO parameters rooted in space group symmetry.
	(c) New possibilities for simulating perovskite structures at extrapolated (p–T–X) conditions.
	(d) Many more visualizable structural parameters cf. Thomas (1998).
	(e) Upgradable to non-centrosymmetric octahedra, and so applicable to all perovskites and more widely to all crystal structures containing octahedra.
	(f) Provides summary distortion parameters λ and σ.
Generalized algorithms for calculating tilt angles [ϕa, ϕb] and ϕc.	(a) Extension to space groups Cmcm, P42/nmc and not previously covered by analytical approximations.
	(b) Ability to calculate tilt angles in structures with distorted octahedra without approximation.
Structural parameters ηA and ηB.	(a) ηA emphasizes the importance of AX8 inner polyhedra and focuses on octahedral tilting around the z (3) axis.
	(b) ηB focuses on tilting around the x (1) and y (2) axes.
	(c) Tracking structural evolution via ηA and ηB allows phase transitions between Pbnm and Ibmm, Cmcm, I4/mcm or P4/mbm to be rationalized and anticipated.
	(d) Improved integration with the tilt classification of Glazer (1972), and, by implication, group-theoretical methods.
Implementation in Excel Solver software environment.	(a) Reversibility of transformation between crystallographic and structural parameters.
	(b) The Excel file in the supporting information is a useful resource for calculating tilt angles and other parameters, as well as for refining crystallographic parameters under structural constraints.
	(c) Upgradable to other programming languages (Frontline Systems Inc., 2021).
Simulation of structural development at increasing pressure.	The assumption of regular or idealized distorted octahedra allows the prediction of crystal structure and associated parameters, e.g. ηA, ηB from unit-cell parameters alone. The potential for modelling structures and phase transitions at high pressure is thereby increased.

The question of the relative stability of alternative perovskite phases has been addressed by several workers employing quantum-mechanical methods (Zagorac et al., 2014) as well as the semi-empirical bond-valence method. Woodward (1997b) initiated this discussion by evaluating the ionic and covalent bonding in perovskites and analysing the conditions of stabilization of favoured tilt systems , and a⁰a⁰a⁰. In later work (Lufaso & Woodward, 2001), an algorithm was developed to minimize the so-called global instability index (GII) in alternative tilt systems, this being the r.m.s. deviation between calculated bond valences and ideal cationic valences. The bond-valence method was also the favoured approach of Zhao et al. (2004) in rationalizing the relative compressibilities of the AO₁₂ and BO₆ polyhedra in perovskites. It led to a clear differentiation in behaviour between 2:4 and 3:3 perovskites, which has remained a feature of experimentally led investigations of perovskites under pressure, for example by Ardit et al. (2017).

The potential of direct transformation of crystallographic data into structural parameters for the analysis of sequences of phase transitions has been demonstrated in this work. The set of structural parameters has been extended and a baseline provided for future investigations of non-centrosymmetric perovskites. The intention is to promote more detailed interaction between experimental crystallography and modelling in developing new materials by atomic and molecular design.

6. Related literature

The following references are cited in the supporting information: Hahn (1995), Williams (1971).

Footnotes

¹Since oxides are exclusively considered in this work, the notation `O1,O2…' is used interchangeably with `X1,X2…'.

²This activates a VBA macro, which calculates the values of structural parameters for the cells with a pink background. Although not included in the supporting information, the author is willing to supply an xlsm file containing VBA codes on request.

³Polyhedral volume = . A_i is area of face; h_i is perpendicular distance from face to common apex inside polyhedron.

⁴Values of K_T and V₀ are in good agreement with the values quoted by Vanpeteghem et al. (2006): 253 (1) GPa and 162.51 (2) ų.