Eur. J. Mineral. 1998,10,621-691

Spontaneous strain as a determinant of thermodynamic properties for phase transitions in minerals

MICHAEL A. CARPENTER, EKHARD K.H. SALJE and ANN GRAEME-BARBER

Department of Earth Sciences, University of Cambridge, Downing St., Cambridge CB2 3EQ, England

Abstract: Lattice parameters are geometrical properties of a crystal, but their variations at phase transitions can be formalised for thermodynamic analysis using the concept of spontaneous strain. As with many other physical properties, spontaneous strains consist of up to six independent components forming a symmetric second-rank tensor, and are subject to the constraints of symmetry. Technical aspects of reference states, principal strains, scalar strains, volume strains, etc., are summarised in this review, and sets of equations defining the individual strain components in terms of lattice parameters for different changes in crystal system are also listed. The relationship between any spontaneous strain, e, and the driving order parameter, ß, for a phase transition is explored by first considering a general free-energy expansion of the form: G(ß,e)=L(ô)+ x wfô"+|Σ<^* i,m,n i,k

L(Q) is a standard Landau expansion, the coefficient λ describes coupling between Q and e, and the last term describes elastic energies. The exponents m and n depend on the symmetry properties of both Q and e, but, in general, only one coupling term is needed to account for each strain component. Characteristic relationships are then e oc Q for symmetry-breaking strains when e and Q have the same symmetry, and e 2 2 oc Q for all other strains. The volume strain, Vs, is generally expected to vary linearly with ß . The principles of strain analysis are illustrated for phase transitions in a selection of minerals and model systems. These include: AS2O5, albite, tridymite, anorthite, leucite, calcite, NaNO3, quartz, Pb3(PO4)2, cristobalite, NaMgF3 perovskite, K2Cd2(SO4)3 langbeinite, BiVO4 and BaCeO3 perovskite. The same overall approach applies whether the transitions occur in response to changing pressure or temperature. It can also be successful when lattice-parameter data for minerals displaying cation order/disorder phenomena are collected at room temperature (and pressure), rather than in situ at high temperatures (or pressures). When atomic ordering does not lead to a symmetry change (non-convergent ordering), the spontaneous strains are expected to vary as e ocVs oc ß. Landau theory provides a convenient theoretical framework for the quantitative thermodynamic analysis of all these materials.

Key-words: phase transitions, spontaneous strain, strain/order parameter coupling, Landau theory, non-convergent ordering.

0935-1221/98/0010-0621 $ 17.75 © 1998 E. Schweizeibart'sche Verlagsbuchhandlung, D-70176 Stuttgart

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Contents 1. Introduction 623 2. Spontaneous strain: the formal basis 626 2.1 Tensor analysis 626 2.2 Symmetry constraints I: nature and orientation of the strain quadric 629 2.3 Symmetry-breaking and non-symmetry-breaking strains 634 2.4 Symmetry constraints II: the relationship between spontaneous strain and the order parameter 635 2.5 Symmetry-adapted strains 638 3. Practical determination of spontaneous strains from experimental data 639 3.1 General equations 640 3.2 Strategies for extrapolating lattice-parameter data 644 3.3 Volume strain 645 3.4 Orientation of the principal strain directions 646 4. Some examples of spontaneous strains in materials undergoing structural phase transitions in response to changing temperature 646

4.1e,ocß, ys«o 647

As2O5 647 Albite 647 Anorthite 650 Tridymite 650 2 4.2 et oc ß , Vs « 0 651 Anorthite 651 4.3^-0,^*0 653 Leucite 653 4.4eI-ocß2> vs±0 655

NaNO3, CaCO3 655 Tridymite 657 Kaliophilite 658

Pb3(PO4)2 658 Cristobalite 662 Neighborite 664 4.5 Materials with higher-order couplings 665

K2Cd2(SO4)3 665 BiVO4 668 Quartz 670 BaCeO3 perovskite 674 5. Structural phase transitions driven by changing pressure 676

BiVO4 676 Cristobalite 678 6. Spontaneous strains determined from room-temperature lattice-parameter data 678 Anorthite 678 Staurolite 681 Feldspar 681

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7. Spontaneous strain and non-convergent ordering processes 682 MAI2O4 spinel 683 Potassium feldspar 683 8. Conclusion 684 Acknowledgements 685 References 685

1. Introduction

One of the most significant physical effects that accompanies a structural phase transition is lattice strain. It arises because crystal structures almost invariably undergo some distortion or lattice relaxation on a macroscopic scale as a consequence of cooperative changes that occur on a microscopic scale. The local processes might be ordering of atoms between crystallographic sites, the development of local magnetic or electric dipoles, or simply small atomic displacements, for example. Lattice distortions can represent the predominant driving force for a transition, as in some proper ferroelastic transitions, or they can be a secondary response to some other driving mechanism. In either case, total strains of between -l%c and -5% are typically observed. Given that strains of as little as l%c can be detected in lattice-parameter data collected using quite routine diffraction techniques, and that the level of precision can be increased to ~1 in 106 using specialised X-ray cameras, strain measurements can provide particularly valuable and detailed insights into the nature and mechanisms of most phase transitions. By analogy with the spontaneous magnetisation which occurs at a ferromagnetic transition and the spontaneous polarisation which occurs at a ferroelectric transition, the lattice distortion associated with a phase transition is usually referred to as a spontaneous strain, £s, (Aizu, 1970; Newnham, 1974). This strain is additional to the background effects of thermal expansion and must be separated from the normal response of a crystal to changing temperature or pressure. The analogy is potentially misleading in the sense that a true ferroelastic transition necessarily involves a change in crystal system (Aizu, 1970; Abrahams, 1971, 1994; Newnham, 1974; Wadhawan, 1982; Bulou et al., 1992; Salje, 1993), whereas spontaneous strains can accompany any phase transition. In so-called co-elastic transitions, both the high- and low- symmetry phases may belong to the same crystal system and yet still display large lattice relaxations (Salje, 1993). Such distinctions are largely a matter of description, however, and, no matter whether a transition is ferroelastic, ferroelectric, ferromagnetic, etc., symmetry plays a determinative role in constraining how a structure may distort. As with other macroscopic physical properties, any spontaneous strain must conform to Neumann's principle (Nye, 1985); that is, it is governed by the point-group symmetry of the crystal. It is formally a second-rank tensor property. In the last few years, the concept of spontaneous strain has been applied to a number of minerals and has led to advances in the quantitative description of the diverse phase transitions that occur within them. Some reviews of this work include Carpenter (1988, 1992a), Putnis (1992), Salje (1991a, 1992a and b, 1993) and Redfern (1995). Absent from the Earth Sciences literature, however, are: (a) a summary of the formal theory which underpins the definition of a spontaneous strain, (b) a general analysis of how symmetry constrains the relationship both

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between individual strain components, or between strain components and the macroscopic order parameter, and (c) a systematic description, with a listing of all the appropriate equations, of how strains are determined from lattice-parameter data. The primary objective of the present review is to correct this deficiency by summarising the necessary theoretical background and then illustrating its application to real systems. Before embarking on technical aspects of tensor and symmetry analysis, it is worth reflecting on the general implications of any lattice distortion accompanying a phase transition.

Firstly, the magnitude of £s represents a measure of the extent of transformation at a given pressure and temperature. For thermodynamic purposes, the extent of transformation is usually specified in terms of an order parameter, Q, which is scaled to vary between zero in the high- symmetry phase and unity in the fully transformed low-symmetry phase at 0 K. In effect, spontaneous strains provide an indirect measure of this order parameter, and the interaction between £* and Q is often described as strain/order parameter coupling. An inevitable consequence of such coupling is that the correlation length of Q will tend to become comparable with the interaction length of £>. Some indication of what this length scale might be is provided by the distance over which a kink in one twin wall of a feldspar influences the morphology of adjacent walls. Figure 7.10 of Salje (1993) shows an interaction distance of at least 0.1 µm, for example. At a microscopic level the structure locally presumably "sees" neighbouring areas via the bending of semi-flexible covalent or partly covalent bonds. The distances involved are much greater than nearest-neighbour interactions. If Q develops an interaction length of this order of magnitude through its coupling with £^, fluctuations in Q will tend to be suppressed. The phase transition is then more likely to conform to the predictions of Landau theory, not only over large temperature intervals, but also at temperatures close to the equilibrium transition temperature. For more formal considerations of this point, and of the role of dimensionality in promoting mean-field behaviour, readers are referred to Cowley (1976), Folk et al. (1976, 1979), Als- Nielsen & Birgeneau (1977), Salje et al. (1987), Carpenter & Salje (1989), Salje (1990) and Sollich et al. (1994). For minerals with framework structures, it is found that transitions with strong coupling between Q and £$ display the expected classical behaviour. By way of contrast, strains at the cubic ^ tetragonal transition in SrTi03 are less than 1 %o (using data of Sato et al., 1985), and there is a temperature interval of -5 K below the transition point over which non- classical (mean-field) variations of the order parameter are found (Riste et al., 1971; Mtiller & Berlinger, 1971). Secondly, elastic lattice distortions may play a dominant role in controlling the microstructures which develop as a transition proceeds. For example, the orientation of twin boundaries and the development of tweed textures can be understood in terms of minimum strain configurations (Sapriel, 1975; Wadhawan, 1982; Salje, 1991a and b, 1993; Putnis & Salje, 1994; Bratkovsky et al, 1994a and b). Conversely, observations of the microstructures themselves should provide important evidence concerning the nature and extent of strain effects under both equilibrium and non-equilibrium conditions. Thirdly, the excess energy due to a phase transition contains a component due to the elastic energy. This elastic energy is given by Hooke's law, and, in its most simplified form, can 2 be represented as ^C°£s , where C° represents some function of the elastic constants. The elastic energy may vary from being a substantial proportion of the total excess energy to being only a small fraction of it. Such a direct contribution implies that variations in the elastic

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constants can themselves provide the driving mechanism for a phase transition. Alternatively, if they are not primarily responsible for the transition, they may change as a secondary consequence of coupling between the spontaneous strain and the driving order parameter. Some of the crucial relationships between strains and elastic properties are dealt with in a companion paper (Carpenter & Salje, 1998, this issue).

Finally, even in systems where both £s and the elastic energy are small, strain/order parameter coupling can have a considerable bearing on the overall transition behaviour. If more than one order parameter operates and each has an associated strain, they can influence each other via those strain components which are common to both. A change in Q\, say, induces a small strain which then induces a change in Q

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2. Spontaneous strain: the formal basis

A single crystal cut into the form of a perfect sphere might show two types of deformation as it cooled slowly from some high temperature. It would deform continuously because of the effects of thermal expansion, and, if there was a phase transition, any spontaneous strain would be apparent as an additional change in shape. Ignoring possible complications due to twinning, the sphere would become an ellipsoid with principal axes which varied in length, and also, perhaps, in orientation, as some complex function of temperature. In order to distinguish between the two contributions, consider a second imaginary sphere drawn in the crystal. This sphere would remain undeformed as long as the crystal was in its high-symmetry state and would deform only in response to the phase transition. It too would become some general ellipsoid below the transition point, but its change in size and shape would depend directly on the extent of the transition, i.e. on the magnitude of ß, and on the nature of the symmetry change. Such distortions are described quantitatively using symmetric second-rank tensors. Not surprisingly, the tensor analysis of spontaneous strains is similar to the treatment of thermal expansion. The principal results in this section are taken from standard texts, and readers are referrred to Nye (1985) or Landau & Lifshitz (1986) for more comprehensive background information.

2.1 Tensor analysis

As a consequence of the phase transition, a point on the surface of the imagined sphere, given by the coordinates Xj (j = 1 - 3) along fixed, orthogonal reference axes, 1, 2 and 3, would be translated to a point on the strain ellipsoid by a vector ut (/ = 1 - 3) where:

m = etjXj (1).

This is illustrated in Fig. 1, where the sphere has unit radius. The strain tensor [βij\ is:

en en *i3 w- (2) • *33.

A Axis 2

Fig. 1. Section through a sphere of unit radius drawn within the high-symmetry form of a crystal which deforms into an ellipsoid due to a phase transition. The point Xj on the surface of Axis 1 the sphere is translated by the vector u t.

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The components β2\ (= ^12), ^31 (= £13), and £32 (= ^23) have been replaced by dots to emphasise that there are only six independent components in a symmetric second-rank tensor. Each component is a dimensionless quantity which is small with respect to unity. Components e\\, ^22 and 233 represent extensions parallel to the three reference axes, while e\2 arises from the change in angle between two lines which were parallel to axes 1 and 2 before the deformation, e^ arises from the change in angle between two lines which were originally parallel to axes 1 and 3, and é?23 fromth e angle between two lines which were parallel to axes 2 and 3. A symmetrical tensor may be referred to its principal axes by diagonalisation to give:

«11 e12 e13 £l 0 0 e e -> 0 0 22 2i e2 (3) • e33_ 0 0 £3

where the eigenvalues £1, £2 and £3 are the tensile strains along the principal axes, X\, X2 and X3, of the strain ellipsoid. These axes always remain perpendicular to each other, though they may rotate with respect to the original reference axes. Orientational relationships between the two sets of axes are given by the eigenvectors of the strain tensor. The original imagined sphere in the high-symmetry form is thus deformed into an ellipsoid whose surface is described by:

,/2 v/2 /2 *i = 1 (4). (i + etf (i + etf (l + e,)'

Here the coordinates x[, x2 and x'^ refer to the X\, X2 andX3 axes. The fractional change in volume, or volume strain, Vs, is given by:

ys = (i+£1)(i+£2)(i + β3)-i (5)

which, for strains of less than a few %, is approximated by:

V. = £1 + £9 + £<* (6).

As will be seen, it is instructive to describe the magnitude and orientation of the strain which caused the sphere to become an ellipsoid in terms of a representation surface defined by the equation:

Σeijxixj=±l (7)

where ij =1-3 and *,-, XJ are coordinates of the original reference system. Nye (1985) refers to this surface as the strain quadric. Taking the principal strain directions as axes, equation 7 becomes:

£\X\ H~ £2^2 "^ ^3^3 — —1 (8).

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The plus or minus sign is added to the right hand side as a simplification to keep the dimensions of the quadric real. An important distinction to be noted is between the strain ellipsoid, which is the shape that the imagined sphere adopts after deformation, and the strain quadric, which describes the relationships between the strain components used to describe the deformation. A simple example of the type of representation surface which equation 8 describes is illustrated in Fig. 2 for the special case of £\ * £2 and £3 = 0, with £\ > 0, £2 > 0. In the X1-X2 plane the surface is an ellipse with principal radii 1/ΛJ£\ and 1/Λ/£^- The linear expansion in a general direction parallel to is given by l/^f£p and the direction of the translation vector wp is given by the radius normal at P (Fig. 2a). In three dimensions the representation surface is an elliptic cylinder (Fig. 2b).

X3 (a) X2| 0>) fwp

Fig. 2. Representation surface (strain quadric) for £x Ψ £2 and £3 = 0. (a) In the Xλ-X2 plane the surface is an ellipse

with principal radii l/^JE^ and l/^£^. The extension parallel to OP is l/ΛJ£^ and the direction of the translation

vector wP is given by the radius normal at P. (b) In three dimensions the surface is an elliptic cylinder.

A second example is shown in Fig. 3 for £1 > 0, £2 < 0 and £3 = 0. In the X1-X2 plane the 2 2 ==+ surface is a hyperbola (Fig. 3a). The solid lines in Fig. 3a show £xx{ + £2^ l> and the 2 2 dashed lines show £ix{ + £2X2 =-1. The magnitude of a linear expansion in a general direction is given by the distance l/^fSp and the direction of the displacement vector wp is normal to the tangent at P. Along the X2 axis there is a contraction, rather than an expansion, as signified by the negative value of £2. The magnitude of the contraction in a general direction is given by 1/Λ/—£R and the direction of WR is inwards from the curve for 2 2 β^j + £2*2 =-1. In three dimensions the surface is a hyperbolic cylinder (Fig. 3b). Other representation surfaces are possible, depending on the signs and magnitudes of S\, £2 and £3. They are illustrated in standard texts such as Nye (1985) and Bronshtein & Semendyayev (1985). A measure of the total strain due to a phase transition is provided by the scalar

spontaneous strain, £jS. Three alternative definitions have been used in the past. Aizu (1970) defined £sS in terms of all the individual strain components as:

£ss= !Σé (9) «»7

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Fig. 3. Strain quadric for £x > 0, &i < 0 and £3 = 0. (a) In the X{-X2 plane the surface is a hyperbola. The magnitude "* / /— "* / / of extension in the direction OP is l/«j£? and the magnitude of contraction in the direction OR is I/^-CR • Solid 2 2 x lines indicate Eλx[ +£2x£ =+1 and dashed lines indicate Exx[ +£2 i' = -l- G>) In three dimensions the 2 2 surface is a hyperbolic cylinder (only Eλx[ + £2x' = +1 shown).

with ij =1-3. Salje (1993) gave:

£ (10) ss=ΛΣ*f

: with i=l- 6, where ef- is in Voigt notation (e\ = e\\, β2 = ^22* £3 = ^33> ^4 = 2^23* ^5 = 2^13, ^ 2^12). In some cases it is convenient to use (from Redfern & Salje, 1987):

£ £ ss=ΛΣ i (11)

with i = 1 — 3, where E\y £2 and £3 are the principal strains. The choice is arbitrary and it is only necessary to make sure that self-consistency is maintained in calculations involving ^s, or in comparisons of values of £jS for different phase transitions.

2.2 Symmetry constraints I: nature and orientation of the strain quadric

The spontaneous strain does not arise as a consequence of any applied external stress. It is sometimes referred to as a "self-strain" and, as with other physical properties, it must show symmetry which is consistent with the point-group symmetry of the crystal (Neumann's principle). Unlike properties such as thermal expansion, however, ^ may arise in association with a change in symmetry. The point groups of both the high- and low-symmetry forms must be considered when assessing which strain components are strictly zero, which can be non-zero, and how many can vary independently. The symmetry arguments can be illustrated qualitatively by taking two examples. Firstly, in a crystal which changes from cubic to tetragonal, the resulting spontaneous-strain ellipsoid will keep its axes parallel to the conventional crystallographic axes (^12 = ^13 = ^23 = 0) and will

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have a circular section perpendicular to the retained tetrad. Changes in the lattice repeat parallel to the crystallographic z-axis can be accompanied by changes in the lattice repeats parallel to the crystallographic x- and y-axes, but any expansion (or contraction) along x must be equal to the

expansion (or contraction) along y. Thus the relationships eu = e22 *e33 ^0 apply in the spontaneous-strain tensor. If it is also the case that the distortion occurs without a change in volume, the expansion (or contraction) parallel to z must balance the combined contraction (or

expansion) parallel to x and y, Le. en+e22+e33=Q and, hence, e33 =-{en + e22). The equation for the strain quadric would be:

enx?+e22X2+e33x$ = l (12)

where x\, x2 and x3 are along axes parallel to the crystallographic x-, y- and z-axes respectively. There is only one independent strain component in this case and rewriting equation 12 in terms of e\\ alone gives:

eγγx\ +eux2- 2^*3 = 1 (13).

The representation surface which this equation describes thus has symmetry that depends on the nature of the symmetry change at the phase transition. For e\\ > 0, the surface would be a hyperboloid of one sheet (Fig. 4a), and, for e\\ < 0, it would be a hyperboloid of two sheets (Fig. 4b).

Fig. 4. Strain quadrics for three sets of conditions: (a) eu = e22 > 0, £33 < 0, a hyperboloid of one sheet; (b) eu = e22

< 0, e33 > 0, a hyperboloid of two sheets; (c) eu = e12 > 0>^33 > 0» an ellipsoid. Note that only the curves for eux\ + eiix\ + ^33*3 = +1 are shown in (a) and (b).

At the other extreme, the spontaneous-strain ellipsoid for a transition betwen two triclinic forms of a crystal would be expected to have three principal axes that are unrelated in length and unconstrained in orientation with respect to the crystallographic axes. All six of the strain tensor components could take different non-zero values. In this case the form and orientation of the strain quadric cannot be predicted on the basis of symmetry arguments alone.

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A more rigorous expression of the symmetry constraints is now developed by treating a specific crystal system in detail. Consider a crystal with point group 422. There are a number of distinct ways in which its lattice may distort when the symmetry is reduced at a phase transition. Each possibility is associated with a particular irreducible representation of the point group and, hence, with some particular basis functions (Table 1). The basis functions themselves refer, by definition, to three orthogonal axes, and, if they are made up of terms in x2, v2, z2, xy, xz and yz, represent second-rank tensor properties. The equation describing the strain quadric may be constructed quite simply by inspection of the relevant basis functions for each of the possible changes in symmetry, as shown here for each representation in turn.

Table 1. Irreducible representations and basis functions for point group 422.

E 2C4 C2(C|) 2C2 2C2 Basis functions [001] [001] [100],[010] [H0],[110] 2 2 2 A1 1 1 1 1 1 x +y , z

A2 1 1 1 -1 -1 z, l 1 1 -1 1 1 -1 2 2 Γ x y B2 1 -1 1 -1 1 xy E 2 0 -2 0 0 x, y, x, v, xz, yz

Ai In the event that the phase transition involves only a change in translational symmetry and the point group of the low-symmetry phase remains 422, any associated strain must transform (in the group theoretical sense) as the identity representation, Ai. From Table 1, the basis functions for Ai are x2 + y2 and z2. For the symmetry of the spontaneous-strain to be consistent with these, the equation describing the strain quadric must be of the form a(x2 + v2) + 2 bz = 1, which is equivalent to equation 12 with the special constraint that eu = e12 * e33. Any tensile strain along X\ (parallel to the crystallographic x-axis in this case) would be equal to any

strain along Z2 (parallel to the crystallographic y-axis), while any tensile strain along X3 (parallel to the z-axis) would be independent of the others. The remaining strain components, en, e\-$ and

e23, would be strictly zero, since xy, xz and vz are not basis functions belonging to Ai. If en, e22 and £33 were positive, equation 12 would describe a representation surface which has the form of

an ellipsoid with one circular section (Fig. 4c). If £33 was negative and e\\ (= e22) positive, the representation surface would again be a hyperboloid of one sheet (Fig. 4a). For e\\ = e22 < 0,233 > 0, it would be a hyperboloid of two sheets (Fig. 4b). The symmetry of these representation surfaces is consistent with all the symmetry elements of the point group 422.

In this case the sum en +e22 +e33 is not constrained to be zero, so that a change in volume, i.e. a volume strain, can occur.

A2 If the diads parallel to [100], [010], [110] and [ 1 TO] (2 C'2 and 2 C{ in Table 1) were lost during the phase transition, the low-symmetry phase would have point group 4, and the strain

quadric for any spontaneous strain must transform as irreducible representation A2. Basis functions for A2 include z and 1 (a polar vector and a chiral vector, respectively, parallel to the z-axis), but there are none representing second-rank tensor properties. A spontaneous strain

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transforming as A2 is therefore not possible. Physically, z and 1 might relate to the development of a dipole, or some chiral effect such as optical activity.

Bi If the tetrad parallel to [001] (2C4 in Table 1) and the diads parallel to [110] and [ 1 TO] (2C2 in Table 1) were lost in the phase transition, the low-symmetry phase would be 222 (orthorhombic). The spontaneous strain must transform as irreducible representation Bi, for 2 2 which the relevant basis function relating to second-rank tensor properties is x - y 9 (Table 1). The form of the strain quadric must be a(x2 - y2) = 1, and equation 12 is therefore constrained to

have en = -e2i, £33 = 0; strain components e\2, eγ$ and£23 are again strictly zero. In this case a contraction (or extension) parallel to the crystallographic jc-axis is matched by an equal extension (or contraction) parallel to the crystallographic j-axis; the lattice dimension parallel to the z-axis is unchanged. The representation surface is a hyperbolic cylinder similar to the surface shown in Fig. 3b. It has symmetry consistent with the diads parallel to [001], [010] and [100] (C2, 2C2 in Table 1), which are retained in the low-symmetry phase, but inconsistent with the tetrad parallel to [001] and the diads parallel to [110] and [110], which are lost during the

transition. The sum en + e22 + £33 is required by symmetry to equal zero. In other words, there is no volume change and the strain is a pure shear.

B2 If the tetrad parallel to [001] (2C4) and the diads parallel to [100] and [010] (2C2) were lost during the phase transition, the low-symmetry phase would also have point group 222. In this case the strain must transform as the B2 irreducible representation, for which the relevant basis function is xy. All the strain components apart from en are strictly zero, and the strain quadric can be described by:

2^2^X2=1 (14).

The representation surface is identical to the surface derived for Bi above, except that it is rotated through 45° in the x-y plane. An extension (or contraction) parallel to [110] is matched by an equal contraction (or expansion) parallel to [ 110] and, once again, there can be no volume strain.

E If the point group of the low-symmetry phase was 2 or 1, with more than half of the symmetry elements being lost during the transition, the spontaneous strain must transform as the E representation. The basis functions for this representation include x, y, x, y,yz and xz (Table 1). As discussed above, the polar vectors are not relevant since they cannot describe a strain. The other two basis functions represent second-rank tensor properties, and may be used on their own, or in combination, to generate the equation for the spontaneous-strain quadric. The equation describing the strain quadric due to basis function xz is 2eγ$x\xi = 1. This corresponds to a hyperbolic cylinder again, except that it is oriented differently from the representation surface arising from either Bi or B2 above. Because of the additional loss of symmetry associated with a two-dimensional representation, a further strain not specified by the basis function for E can occur. In effect, there is one more degree of freedom for lattice distortions. The new monoclinic cell (point group 2) is derived from the tetragonal cell by the angle ß deviating from 90°. It could inherit the tetragonal condition a = b, but this does not

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apply in the monoclinic system so that a further distortion, involving extension along the x-axis and contraction along the y-axis (or vice versa), automatically becomes possible (see Fig. 5a).

Thus the additional allowed strain has en = -e12 & 0.

2e (a) v

*22

^

Fig. 5. Illustrations of the unit-cell relations for a tetragonal -» monoclinic transition in which the active representation is E of point group 422. (a) en = -e22, ei3 Ψ 0. The monoclinic cell is shown by dashed lines, (b) e13 = -e23, en^0. The distorted cell is shown by dashed lines and the new conventional cell is shown by dotted lines.

A second strain quadric may be derived from a linear combination of the two basis

functions of E (xz and yz) with a different special condition, namely that e\3 is equal and opposite to £23- The equation describing the strain quadric is then given by:

Λ: 2el3xλx3 + 2e23*2 3 - * (15)

where el3 = -e23- This is a hyperbolic cylinder again, but with a different orientation from any of the previous examples. The deviation of ß from 90° (due to e\3) and of a from 90° (due to ^23) releases the constraint that /should equal 90°. Hence e\2 * 0 automatically becomes possible, even though this is not indicated explicitly by the basis functions. The new lattice is still monoclinic (point group 2), however, because of the condition that a and ß both deviate

from 90° by the same amount (from e13 = -e23), and a new conventional unit cell must be selected (Fig. 5b). An alternative linear combination of the basis functions xz and yz leads to a triclinic

lattice (point group 1) when e\3 and -^23 are not constrained to be equal. The representation quadric is a hyperbolic cylinder in yet another orientation, and the additional strains not

specified by the basis functions are eΛΛ-e11^^ and el2*0. Once again, all these combinations do not permit a change in volume, and the strain is a pure shear.

Fortunately, it is not necessary to follow this type of procedure in detail for every phase transition of interest. The relevant information has been tabulated in different ways for easy access. Table 2 is taken from Salje (1993), for example, and summarises all the preceding discussion for point group 422. Similar tables for all equitranslational phase transitions are given by Salje (1993). Character tables for the 32 crystallographic point groups, or the relevant basis functions, are reproduced in many places (e.g. Wooster, 1973; Stokes & Hatch, 1988).

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Table 2. Summary of strain properties for subgroups of point group 422 (from Salje, 1993). Monoclinic subgroups are given with orientations corresponding to the second setting (unique axis parallel to [010]). The terms proper and improper refer to the type of ferroelastic transition, as defined by Wadhawan (1982) and mentioned in section 2.4.

1 Subgroup Active rep. Spontaneous strain No. of elastic Basis functions Proper Improper domains

4 A2 1 z, z 2 2 222 Bi e\ = -ei 2 x -y 222 B2 eβ 2 xy 2 E es e\ = -é?2 4 x, x,y, y xz, yz

2 E e5 = -é?4 eβ 4 x, x, y, y xz, yz 1 E ^4, ^5 ^6> e\ = -β2 8 x, x, y, y xz, yz \

A few technical points are worth including here for completeness. With regard to the way in which basis functions are derived and presented, a progression is adopted such that, if there are vector or second-rank tensor functions which transform as a given irreducible representation, these are always specified. If no such functions exist, then any higher-order tensor functions which transform as the irreducible representation may be given. For example, in a phase transition involving a symmetry change 4/mmm —> 422, the relevant representation is

Aiu. This representation does not have an associated basis function containing only vector or second-rank tensor components. A basis function which does transform as Aiu is x(yz) - y(xz), however. This contains components of a third-rank tensor to describe a pieozelectric effect, say. Similarly, for the symmetry change mhm -> 3m, the lowest-order basis function is {Λxyy} + {yyzz} + {zzxx}> which involves components of a fourth-rank tensor and relates to properties such as elastic constants. In both these cases, the lack of any basis function describing second- rank tensor components implies that there cannot be a spontaneous strain that transforms as the respective irreducible representation. Finally, when the relevant representation for a phase transition is two or three dimensional, a spontaneous strain may be observed even if it does not appear to be an overt consequence of the basis functions for that representation. It is usually advisable not only to consult the relevant tables but also to examine rather carefully the lattice geometry of the high- and low-symmetry forms.

2.3 Symmetry-breaking and non-symmetry-breaking strains Arising from the detailed discussion of individual representations is an important distinction between two types of spontaneous strain. The total strain due to a phase transition, |>totL may be expressed as the sum of two tensors:

hot] = [*sb] + hsb] (16)

11 where [eSD] is the "symmetry-breaking" strain, and [>nsb] is "non-symmetry-breaking strain. As defined here, a symmetry-breaking strain is a strain which transforms as any irreducible representation of the point group of the high-symmetry phase, other than the identity representation. In the example of 422 detailed above, for example, these are the strains

associated with representations Bi, B2 and E. Non-symmetry-breaking strains transform as the

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identity representation (Ai for point group 422 in the preceding section) and are thus always permitted by symmetry, no matter what the precise symmetry change might be. If there is no symmetry-breaking strain, the only spontaneous strain will be [insb]* Whether this is large or small is a matter for experimental verification. The symmetry-breaking strain is a shearing deformation which, by symmetry, cannot = an include a change in molar volume, i.e. £x sb + £2,sb + £3^ 0 (where £i)Sb, £2,sb d £3,sb we the principal strains of the matrix |>sbl)- On the other hand, as discussed for the Ai representation of point group 422, strains that transform as the identity representation can give rise to a change in volume. The volume strain, Vs, may therefore be expressed as:

^s = £l,nsb + £2,nsb + £3,nsb (17)

where £i,nsb> etc., are principal strains of the matrix [ens^\. tensb] is necessarily a pure volume strain if the high-symmetry phase is cubic. In systems with lower symmetry, the deformation it describes frequently includes a volume change but need not necessarily do so; a pure shear is theoretically possible. Typical values of Vs in real systems fall in the range of a few parts per thousand to -5%, and are thus comparable in magnitude to the values observed for symmetry- breaking strains.

2.4 Symmetry constraints II: the relationship between spontaneous strain and the order parameter In order to investigate the manner in which a spontaneous strain influences the thermodynamic behaviour of a crystal, all the individual strain components can be included explicitly in the Landau free-energy expansion. For systems in which the driving order parameter is not itself a strain, as in atomic ordering processes, for example, any strain contributes to the excess free energy through two effects. Firstly, there is the Hooke's law elastic energy, ^^C^e,-^, already mentioned in the introduction. The coefficients Cfk are elastic constants of the high-symmetry phase, with the superscript ° signifying that they are so-called "bare" elastic constants (Carpenter & Salje, 1998). Voigt notation is used to reduce the number of subscripts (i, k = 1 - 6; e\ = e\\, ei = ^22> £3 = £33* £4 = 2^23* £5 = 2ei3, £6 = 2^12). Secondly, there are "coupling" energies due to interactions between the order parameter and the strain. The excess free energy due to a transition may then be expressed as:

G(Q,e) = L(ß)+ £ hm^ΓQ" +\ΣC*eiek (18) i,m,n i,k

where λ( are coupling coefficients, m, n are positive integers, and the free-energy contribution from the order parameter alone is given by the normal Landau expansion:

2 3 4 L(ß) = iα(Γ-Γc)ô +^Ö +^cß + ... (19).

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Note that the subscripts i, m, n on λ are used here only as labels and are not intended to imply a tensor form. Only those strain components, e,-, which are allowed by symmetry, as set out in a previous section, are included. Just as with the terms in Q making up the Landau potential, the new terms in βj must conform to symmetry rules and cannot be added arbitrarily. The formal requirement is that each term must be invariant with respect to all the symmetry operations of the space group of the high-symmetry phase. In group-theory jargon, every term should transform as the identity representation, Γidentity* of the space group of the high-symmetry phase. This rule is obeyed if the product of the irreducible representations that are associated with each of the parameters making up a term contains Γidentity Symmetry analysis of the elastic energy is trivial in this context. If any two components βt and βk are both non-symmetry-breaking strain components they individually transform as ^identity- The product e,-^ then necessarily transforms as Γidentity also. In general, the product of any two components which individually transform as the same representation is allowed, because the square of an irreducible representation always contains the identity representation. Bilinear combinations of strain components which individually transform as different representations will not be allowed. The problem of which coupling terms are allowed is simplified by noting that energy contributions from terms containing ef with m > 1 are likely to be small because the strains are n themselves small quantities. Assuming m = 1, therefore, consider the coupling terms in etQ for one strain component, £,-. For the case of a phase transition in which only e4 deviates from zero, the total excess energy becomes:

2 3 G(Q,e4) = L(Q) + ±C%4el + λ4Xle4Q + λ4X2e4Q + λ4X3e4Q + ... (20).

Now e4Q can transform as the identity representation only if e4 and Q individually transform as the same representation. By definition, Q transforms as the active representation, Ractive> and it follows that e4 must also transform as Ractive for the term in λ4X\e4Q to take any value other than zero. This condition is met only in the case of equitranslational transitions for which the active representation has an associated basis function with second-rank tensor properties (so- called proper ferroelastic transitions). In general, a bilinear coupling term is allowed for some (but not all) transitions associated with the Brillouin zone centre. Conversely, it is strictly zero for all zone-boundary transitions and some zone-centre transitions.

For the next higher-order term, consider three separate situations. If e4 transforms as Ractive* ^4,i,2^4Ö2 wiH be allowed when L(Q) for the transition contains a third-order term in Q, 2 but not otherwise. If e4 transforms as Γidentity* a term in X^XI^AQ will always be allowed 2 because Q necessarily contains Γidentity On the other hand, if e4 transforms neither as Γidentity nor as Ractive» it is necessary to complete a full multiplication of the relevant irreducible representations and test for the presence of the identity representation. This is not a straightforward task for anyone unfamiliar with group theory, but some worked examples are provided by Dvorak (1971), Torres (1975) and Hatch (1981). Further higher-order terms are analysed in an exactly analogous manner, and the criteria can be generalised by referring to the symmetry-allowed terms in L(ß), as follows:

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n n+l (a) If ei transforms as Ractive> all terms in etQ with n > 1 are allowed when Q is present in UQ). n n (b) If ei transforms as Γidentity> all terms in etQ with n > 2 are allowed when Q is present in

(c) If et does not transform as either Ractive or as Γidentity> no general rule applies and the full multiplication of representations must be completed to test for the presence of I identity There is, as yet, little experimental evidence to suggest that coupling terms which are higher order than e;ß3 provide a significant energy contribution. In practice the important coupling may be restricted to bilinear, λj,i,ie,-ß, linear-quadratic, λ/j^iß2, and linear-cubic, λj,i,3^ß3, therefore. The ß ;= α transition in quartz and a cubic ^ trigonal transition in BaCeÜ3 perovskite (discussed in section 4.5, below) appear to be rare examples of systems with higher- order coupling to a volume strain which can be described by a term of the form λeQ4. The relationship between e, and ß is established by considering the conditions for equilibrium. Equation 20 for the excess free energy with respect to one order parameter, ß, and

one strain component, e4, has now been restricted to:

2 3 G(ß,e4) = L(ß) + λ4Xle4Q + λ4X2e4Q + λ4X3e4Q + i Cfrel (21).

At equilibrium, the crystal must be stress-free (G is at a minimum with respect to the strain, e4) giving:

2 3 — = λ4XlQ + λ4X2Q + λ4X3Q + C°44e4 = 0 (22) ôe4

and, hence:

e4 = -[^Q + ^+^QA (23). V ^44 c44 c44 )

Of course, this applies to the relatively uncommon case of a transition with e4 as the sole non­ zero strain component, but only algebraic manipulation is required to show that a general solution will have the form:

2 3 e(=AQ + BQ + CQ (24)

where A, B and C are complicated functions of the elastic constants, Cfk, and coupling coefficients, λ/?i>n. Now, if L(ß) describes a transition in which the third-order term, ß3, is allowed,

equation 24 describes the relationship between ei and ß for β{ transforming as Ractive (proper ferroelastic). Alternatively, if L(ß) contains only even-order terms, ß2, ß4, ß6, etc., the

condition that et transforms as Ractive yields:

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3 e(=AQ+CQ (25).

In most real systems the high-order coupling terms appear to be negligibly small, however, and equations 24 and 25 reduce to:

et=ΛQ (26).

If e\ transforms as some representation other than Ractive> coupling terms of the form λi,i,ieiQ anc* Λi,i,3^'ô3 are strictly zero. Equation 24 then reduces to:

2 et=BQ (27).

In general, individual strain components are expected to be related to Q by either e\ <* Q

or βi oc Q2. The scalar spontaneous strain, £jS, will behave in the same way, though it is important not to mix strain components with different symmetry properties in the calculation of

£jS. The elastic energy must always include all the related Cfke(ek terms, of course. This discussion of strains coupled with Q refers to phase transitions in which the driving order parameter is not itself a spontaneous strain. In (relatively uncommon) true proper ferroelastic systems (using the terminology of Wadhawan, 1982, and see Carpenter & Salje, 1998), the transition occurs as a consequence of the softening of an individual elastic constant or some symmetry-related group of elastic constants. The order parameter is then the symmetry- breaking spontaneous strain, e^, directly. Symmetry analysis of strain couplings is quite

straightforward in this case - the lowest-order coupling between es\> and ensb has the form Misb4-

2.5 Symmetry-adapted strains

For phase transitions in which the high-symmetry phase is cubic and the active representation is doubly degenerate (E), it is often convenient to reduce the number of strain components by expressing them in terms of "symmetry-adapted" strains. The degenerate symmetry-breaking strains are:

«o=*n-*22 (28) et=^j(2*33-*ii-e22) (29)

where the subscript o signifies that e0 is an orthorhombic distortion and the subscript t signifies that et is a tetragonal distortion. The factor l/V3 is included in the definition of et to ensure that the two strains are on the same scale. The symmetry-adapted form of the non-symmetry-

breaking strain, ea, is defined as:

ea = e\\ +*22 + *33 (3°)-

When expressed in this way, e0 and et transform as the E representation, while ea transforms as the identity representation. The order parameter is itself doubly degenerate, and

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the two components may be specified as q\ and q2. The lowest-order coupling term for ea has the

form λeÅql + q%), but is more complicated for e0 and et. If e0 and et have the same symmetry

as q\ and q2, bilinear coupling is expressed as λx(c[\e0 + q2et) + λ2(q\et -q2e0) (Dvorak, 1972).

The next higher-order terms are given by Maeda (1979). If e0 and et do not have the same symmetry as the driving order parameter, as in the case of a zone-boundary transition, the

lowest-order terms will be linear in e0, et and quadratic in q\, q2. A full symmetry analysis is required to find their precise form.

For a tetragonal -> orthorhombic transition, (eλ -e2) is an orthorhombic symmetry-

breaking strain and (el +e2) is a non-symmetry-breaking strain expressed in symmetry-adapted form. The first would have the same symmetry properties as the order parameter, Q, for an

equitranslational transition, giving a permitted coupling term λi(ei-e2)Q. The second transforms as the identity representation, and the lowest-order coupling term allowed by 2 symmetry is λ2{e^ +e2)Q . The more formal derivation of these symmetry-adapted strains from elastic-constant matrices is reviewed by Carpenter & Salje (1998).

3. Practical determination of spontaneous strains from experimental data

If a length, /, of some material is changed by a small amount, 8/, the linear strain is defined as 8Z/Z. For the lattice distortion illustrated in Fig. 6a, there would be two linear tensile strains:

(α - a0)ja0 parallel to the jc-axis, and (b - b0)/b0 parallel to the y-axis, where α, b are the lattice parameters after deformation and α0, b0 are the lattice parameters prior to deformation. A positive strain implies elongation, while a negative strain implies contraction. If the distortion is a shear through the angle 0, the shear strain is defined as tanβ. For the example illustrated in Fig. 6b the shear strain can be rewritten in terms of the old and new lattice parameters as

[b/b0)cosγ. In this definition a positive shear implies that the +x and +y directions are brought closer together (y in Fig. 6b is acute), while a negative shear implies that they move apart (y is obtuse). Extending these definitions to three dimensions is purely a matter of geometry. The only complication is that the equations for the six components of the strain tensor become somewhat cumbersome, particularly for lattices with non-orthogonal axes. Once a fixed

A

1 0Q a Γ"" a l 1 // l K' l K 'b 1 / 1 1 )tL k >• X X

Fig. 6. Illustration of the geometry of shear, (a) Tensile strains parallel to the jc-axis and the y-axis are given by (a-ao)/a0 and (b - b0)/b0 , respectively, (b) Shear strain given by tanÖ = (b/b0)cosγ.

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coordinate system has been chosen, however, a single set of equations can be used to determine the spontaneous strain from measured lattice parameters for phase transitions involving any change of symmetry.

3.1 General equations Following Redfern & Salje (1987), the most general case of a phase transition is one in which both the high- and low-symmetry forms are triclinic. An underlying Cartesian coordinate system, with axes X, Y and Z, is first selected such that Y is parallel to the crystallographic y- axis, Z is parallel to the normal to the (001) plane {i.e. parallel to c*), and X is perpendicular to both. The +X direction is chosen to conform to a right-handed coordinate system (Fig. 7a). Strain component en is then a tensile strain parallel to X, #22 is a tensile strain parallel to 7, etc. These strains are typically small relative to unity, which means that the relatively simple, linear Lagrangian form of the full strain tensor can be used without serious loss of accuracy (Schlenker et al., 1978; Nye, 1985; Landau & Lifshitz, 1986). For the coordinate system used here, the general equations of Schlenker et al. (1978) give the components of the spontaneous-strain tensor as:

asm 7 (31) ^2=^22=7 ! (32) flbsiny0

csinasin ß £1 é?3=é?= e

bcosa cos/? 1 a cosy bcosγ ) ccosa n 0 0 (34) 2* ='23 =5 a c0 sin a0 sin ß0 b0 sin a0 sin ß0 sin ß0 sin γ0 \ o

a sin γcos/?* c sin a cos/?. * Λ β5=ei3 = 5fC 5{C (35) 2 2 a sin γ sin/3 c sin a sinjS, 0 0 0 0 0 ° )

Fig. 7. (Facing page) Orientation of crystallographic axes, x0, y0 and20, and reciprocal lattice axes a0, b0 and c0, of the high-symmetry form at a phase transition, with respect to reference axes X, Y and Z used to specify spontaneous strains, (a) Triclinic parent phase. The low-symmetry phase is constrained only to have c*//Z and yl/Y. (b) Monoclinic parent phase. A triclinic product would have c*//Z and yllY\ the triclinic axes (x, y, z) are shown with small angular distortions relative to the monoclinic axes for illustrative purposes, (c) Parent crystals with orthogonal crystallographic axes. All crystallographic axes are parallel to the reference axes. A monoclinic product crystal would have c*//Z and yllY. (d) For a cubic ^ trigonal transition, the axes of a pseudocubic unit cell of the trigonal phase (A:pc, ypc, zpc) rotate towards (or away from) the [111] direction of the cubic phase; αis the lattice angle of the pseudocubic cell and the subscript c is used to identify axes of the cubic phase, (e) A more convenient setting of the crystallographic axes for a cubic ^ trigonal transition. Axes of a hexagonal unit cell for the trigonal phase are indicated by the subscript H. A subscript c identifies axes of the cubic phase in this orientation, (f) Alternative scheme as used by Salje et al. (1985a) for a monoclinic —> triclinic transition. The product crystal has its crystallographic axes, x, y,z, a*, b*, c*, constrained to have c*//Z and xfIX in this scheme. Small angular changes are shown for illustrative purposes.

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Z c*c*

Z,[lll]c,zH Z ç*,c* (e) (f)

"o * •

|b* X+ - o- X« [101]c, xH Y,[li0]c,y„ Yv b

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1 If αcosr bcosγn ~e6 ~en =—\ (36). 2 2^0siny0 èocosy0

In these equations a, b, c9 a9 ß* and γ refer to the low-symmetry form at a given

temperature (and pressure), and a0, b09 c0, α0, ß0 and γ0 are the cell parameters that the high- symmetry form would possess at the same temperature (and pressure) had the transition not

taken place; ß* and ß0 are reciprocal lattice angles. Of course, the same type of unit cell must be used in each phase to avoid multiplication effects due to the development of superlattices. These equations can be reduced when transitions involving systems with higher symmetry are being analysed. Thus, if the high-symmetry phase is monoclinic and the low- symmetry phase is triclinic the equations simplify (since a0 = γ0 = 90°) to:

eu =—sin/-I (37) ^99 = 1 (38) 22 K

csinαsinß c cos a a cosß0 cos γ *33= —* 1 (39) e23=- (40) c0sinß0 a0sinß0 , c0sinß0

( *\ αsinycosft* csinαcosff *13 — (41) α0sinß0 c0sinß0

a en = ; —cosy (42).

If both the high- and low-symmetry forms are monoclinic, a = γ= 90° and ß = 180 — ß, the strain components become:

" i en= 1 (43) «22=T—1 (44)

csinß e33 = — 1 (45) e23=0 (46) c0sinj30

ccosß acosß0 ei3 = (47) *12 =0 (48). 2 ^c0sinß0 a0sinß0

For a transition in which the high-symmetry phase has orthogonal crystallographic axes and the low-symmetry phase is monoclinic:

b . en = 1 (49) «ä2=T—1 (50) an

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e33 =—sin/J-1 (51) *23 = 0 (52)

1 c cos *i3=d— ß (53) el2=0 (54). 2U

For transitions in which both the high- and low-symmetry forms have orthogonal crystallographic axes the equations reduce, as expected, to:

(55) ^ 1 (56) en= 1 e22=-Γ -1 bo

e33= 1 (57) e23 = el3 =eu=0 (58).

Stereograms illustrating the relationship between the chosen Cartesian axes and the crystallographic axes for monoclinic and orthogonal crystal systems are shown in Fig. 7b and 7c, respectively. Some special cases require slight modifications. For a cubic —> trigonal transition, none of the crystallographic x-, v- and z-axes remain parallel to X, Y or Z. They all rotate, either towards [111] or away from [111], by an amount that depends on the extent of transformation (Fig. 7d). The equations of Schlenker et al. (1978) were derived for Y parallel to the crystallographic v-axis and do not strictly apply in this situation. However, for small distortions from a = ß = γ = 90°, such as 90 - 90.5°, the values of since, sin/3 and sin/ remain sufficiently close to 1 that significant errors are not introduced if equations 31 - 36 are used directly. Taking

the pseudocubic unit cell of the trigonal phase as having lattice parameters αpc (= bpc = cpc) and αPc (= ßVc = 7Pc) then gives:

I *pc 1 (59) e23 : (60). *11 =^22=^33 = — *13 = *12 = £ cosα.p c

Alternatively, and more rigorously, a hexagonal cell may be used to describe the trigonal phase. In this case, the axes must be changed such that [111] of the cubic cell, which becomes the jc-axis of the hexagonal cell (zn), is parallel to Z [lio] of the cubic cell becomes the y-axis of the hexagonal cell (JH) and is parallel to Y. The x-axis of the hexagonal cell (XH) is parallel to [lOlj of the cubic cell. These axes do not rotate as the transition proceeds. Their orientation relationships are shown in Fig. 7e. The hexagonal cell parameters, an and CH, are related to the pseudocubic parameters by:

OL, Yi pc 2 «,p c %=2tfpcsin \-£- (61) cH=3a 1-^sin (62). pc 3

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Reference parameters a0 and c0 for strain calculations refer to the hexagonal setting of the cubic phase, i.e. the values of an,

^1-22=^-1 = ^-1 (63) ea-aL-1-a—1 (64) a0 V2αc c0 V3αc

^23 = *13 = ^12 = ° (65)«

The linear strains now contain contributions from both symmetry-breaking and non-symmetry-

breaking strains. The former is a pure shear with identical form to the tetragonal shear, et, given by equation 29, and the latter is a pure volume strain, ea, given by equation 30. When the high-symmetry phase is trigonal or hexagonal, it may be more convenient to express the lattice parameters in terms of an orthogonal unit cell. For example, in an

equitranslational hexagonal —» orthorhombic transition, equations 55 - 57 can be used if b0 for the hexagonal unit cell is expressed as V3α0. This is then the correct reference dimension for b of the orthorhombic cell. Similarly, in certain instances an alternative choice of Cartesian coordinate system may yield simplified expressions for the individual strain components. These can again be derived by appropriate manipulations of the equations of Schlenker et al. (1978). In the case of a monoclinic -> triclinic transition, setting X parallel to the crystallographic jc-axis, Z parallel to c , and Yperpendicular to X and Z (Fig. 7f; Salje et ai, 1985a) simplifies £23 and en to:

1 csinjScosα ] fgiC× Λ _ If b (66) ^12 = x —cosy (67). c0sinß0 2U

Further empirical simplifications can always be made in the light of experimental data. During the monoclinic —> triclinic phase transition in feldspars, for example, the angle ß varies very little so that ^23 of equation 66 becomes:

eri ~ — —cosα (68).

3.2 Strategies for extrapolating lattice-parameter data The spontaneous strain is an excess property which is defined with respect to the lattice parameters the high-symmetry phase would have at the same temperature and pressure as the low-symmetry phase. In practice it is usually possible only to measure the lattice parameters of the high-symmetry phase above the transition point and the lattice parameters of the low- symmetry phase below the transition point. Various extrapolations or assumptions are therefore needed in order to estimate values for the high-symmetry phase under the external conditions to which the measured values of the low-symmetry phase refer. Three alternative strategies may be adopted:

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(a) The lattice parameters of the high-symmetry phase are taken as those measured at the transition point and the effects of thermal expansion (or compressibility if pressure is varied) in the high-symmetry phase are ignored. While this is sometimes acceptable for large strains, significant errors are likely to ensue. (b) For certain symmetry changes it is possible to treat a lattice parameter of the high-symmetry phase under given conditions as being the average value of two or more of the lattice parameters of the low-symmetry phase under the same conditions. Thus, for example, in a cubic ->

tetragonal transition a0 = {la + c)/3 might be assumed. Similarly, a0 = (a + b)/2 might be 3 assumed for a tetragonal -» orthorhombic transition or a0 = (abcy for a cubic —» orthorhombic transition. Errors are introduced if there is a significant volume strain. (c) The lattice parameters of the high-symmetry phase are extrapolated from measured values above the transition point. This is the preferred approach and has two advantages. Firstly, thermal expansion (or compressibility) of the high-symmetry phase is taken into account. Secondly, symmetry-breaking and non-symmetry-breaking strains can be separated correctly without the need to assume that the latter are small. Collection of data must be extended to well beyond the transition point in order to ensure the highest possible accuracy for the extrapolation.

3.3 Volume strain

Any volume strain, Vs, accompanying a phase transition has the symmetry properties of the identity representation. Volume strains are therefore permitted at any phase transition. Vs is the most straighforward strain parameter to determine since it is given simply by:

V-V ^="V^ (69) *o

where V is the unit-cell (or molar) volume of the low-symmetry form of the crystal at given P,T

conditions, and V0 is the volume of the high-symmetry form extrapolated to the same pressure and temperature. As usual, it is necessary to compare the same type of unit cell in both phases to avoid introducing doubling, tripling, etc., of the unit cell due to superlattice formation. Any volume strain is due to the contribution of a non-symmetry-breaking strain to the

total strain, [eiot\. When [etoi] contains both symmetry-breaking and non-symmetry-breaking parts, the latter can also be detected by testing predicted relationships between individual strain

components. For example, in a tetragonal -» orthorhombic transition, the relations en =-e22 and e33 = 0 might be anticipated. Any violation of these conditions merits closer inspection. If an anomaly arises because the total strain is composite, the two contributions can usually be separated by writing individual strain components in the form of equation 16. In the case of the tetragonal -» orthorhombic example, the relevant components are:

e\ l,tot = e\ 1, sb + e\ l,nsb (7°) *22, tot = *22, sb + *22,nsb (71 )•

anci e = _é? are By symmetry the relationships eUnsb = ^22,nsb n,sb 22,sb expected, implying that:

*ll,sb = ^(*ll,tot ~*22,tot) (72) é?ii>nsb = -(^n,tot + *22,tot) (73).

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Note that the incorporation of symmetry constraints has produced the symmetry-adapted strains

described in section 2.5; en,nsb has the symmetry properties of Γidentity* and, if the transition does not involve a change in translational symmetry, £π,Sb has the symmetry properties of Ractive- The third component cannot be symmetry-breaking and, hence:

*33,tot = *33,nsb (74>-

In this example, the generalised form of equation 17 for Vs then becomes:

^s = el l,nsb + «22,nsb + *33,tot (75)-

Because the volume strain transforms as the identity representation, symmetry analysis

of the permitted coupling between Vs and the driving order parameter is trivial. For any 2 symmetry change, the lowest-order coupling is λVsQ . Systems which show higher-order coupling appear to be rare (see section 4.5, below). Thus a simple measurement - the unit-cell volume as a function of temperature through a phase transition - should yield the form of 2 variation of the order parameter in most cases through the relationship Vs <* Q .

3.4 Orientation of the principal strain directions Once values of the individual strain components have been determined, the precise shape and orientation of the spontaneous-strain quadric can be deduced. This is straightforward for transitions involving phases with orthogonal crystallographic axes, since the principal strains are parallel to the crystallographic axes. In the general case, the principal strains are found by referring the spontaneous-strain tensor to its principal axes, as described by Nye (1985) for thermal expansion. The mathematical operation involves diagonalisation of the 3x3 strain matrix, which is most easily achieved using standard computer software. The eigenvalues are the magnitudes of the principal strains and the eigenvectors give the orientation of the principal strains with respect to the reference axes, X, Y and Z.

4. Some examples of spontaneous strains in materials undergoing structural phase transitions in response to changing temperature

Any reader not previously familiar with the formal analysis presented in the preceding sections should by this stage accept three general conclusions. Firstly, spontaneous lattice distortions associated with a phase transition can be described effectively using the spontaneous-strain tensor. Secondly, values for all six components of this strain tensor are obtained simply by inserting lattice-parameter data obtained at different temperatures (or pressures) into standard sets of equations. Finally, symmetry criteria constrain the possible coupling behaviour between individual strain components and the driving order parameter, such that permitted strain/order 2 parameter dependences are of the form e& <× Q or es\y <× Q 9 depending on the exact symmetry 2 2 change involved, and ens\y °c Q , Vs <* ß . It is always advisable to test predicted relationships, both between individual strain components and between these components and Vs. Deviations from anticipated behaviour, or from relationships predicted between strains and other physical

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properties, such as birefringence or excess heat capacity, should then raise questions relating to the role of non-symmetry-breaking strains, the possibility of higher-order couplings, the operation of more than one order parameter, etc. All these issues are best illustrated using real examples, and phase transitions in a selection of natural and synthetic materials are reviewed in this section. They are separated into four empirical categories according to whether or not there are symmetry-breaking strain components which vary linearly with Q, and whether or not there is a significant volume strain. A fifth category involves materials with suspected higher-order couplings. Voigt notation (see section 2.4) for the strain components is used throughout, as a matter of convenience.

4.1^ocß,ys«0 AS2O5. About the most straightforward strain evolution observed in any material is that associated with the P4\2\2 -» P2\2\2\ (proper ferroelastic) transition in AS2O5 (Redfern & Salje, 1988; Redfern, 1989). This is a zone-centre transition with a change in point group from 422 (tetragonal) to 222 (orthorhombic) at an equilibrium transition temperature of -578 K. The basis function given in Table 2 for the active representation, Bi, is x1 -y2, implying

βγ = -e2 * 0 and e3 = 0 if there is no non-symmetry-breaking strain. Variations of the a and b lattice parameters with temperature are shown in Fig. 8a.

Within experimental uncertainty, the c parameter is unaffected by the transition, i.e. c = c0 and e3=0. The a and b parameters of the orthorhombic phase do not quite diverge symmetrically about the extrapolated value of a0 for the tetragonal phase because there is a small volume reduction associated with the transition, as shown in Fig. 8b (all data from Redfern & Salje,

I • • M 11111 »*?•:::.:: (a)1 150x1010"" 6 (c)J 8.60 F- •*sF 1 •••!!••••• 100 H

O 8.55 h • 1 •K 50 « 8.50 C- H

•

8.45 l*'" 0 L I 1 1 1 1 I 1 1 1 1 1 1 1 1 1 I -L 11111 ULUJ-Ü. LI 1 1 1.1 1 1 il1l I 1 1 1 1 I iH 300 400 500 600 700 300 350 400 450 500 550

u I 1 1 1 1 I 1 1 1 1 I 1 1 1 1 I 1 1 1 1 I 1 1 1 i| (K)

Fig. 8. Spontaneous-strain data for the tetragonal ^ orthorhombic transition in As205 (data from Redfern & Salje, 1988; Redfern, 1988). (a) Lattice-parameter variation with temperature, (b) Variation of unit-cell volume with temperature, showing a small volume change due to the transition. The straight lines drawn in would imply Vs <* T. (c) The square of the symmetry- adapted form of the symmetry-breaking strain is linear with Γ, implying Q2 «= T and, hence, second-order behaviour. 300 400 500 600 700 Γ(K)

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1988). This volume strain can be excluded from the determination of the symmetry-breaking strain components by taking the symmetry-adapted form: b a {ei-e2)J^- ^]J -±) (76). V ao bo ) V ao )

This symmetry-breaking strain transforms as Ractive> so that a linear coupling term, λ{el -e2)Q,

will contribute to the excess free energy, and [ex -e2) <× Q is predicted. The square of [ex -e2) is linear with temperature over the entire temperature interval for which data are available (Fig. 2 8c), suggesting second-order character (ß <× |ΓC - r|) for the transition. The extrapolation to

zero strain yields Tc = 589 K, which is somewhat higher than the value extracted from the data in a slightly different manner by Redfern & Salje (1988). Independent observations of the temperature dependence of optical birefringence (Salje et al., 1987), diffracted X-ray intensities (Schmahl & Redfern, 1988) and hard modes in Raman spectra (Bismayer et ai, 1986) also

reveal second-order character, confirming the linear coupling of Q with (eλ -e2). The small excess volume (—0.002 at room temperature) could be linear with temperature (Fig. 8b), 2 consistent with Vs <* Q °c T.

Albite. Although the data are less precise, analogous ferroelastic behaviour is displayed in albite with no long-range Al/Si order when it transforms from monoclinic (C2/m) to triclinic (Cl) at -1240 K (Kroll et aU 1980; Carpenter, 1988; Hayward & Salje, 1996). The strain transforms as

the active representation, Bg, for which the basis functions are xy and yz, giving e2$ <* e\2 °c Q as the expected strain/order parameter relationship. Relatively simple expressions for e23 and e\2 in terms of latttice parameters (equations 66 and 67) can be derived using the Cartesian coordinate

system shown in Fig. 7f. The angle ß varies very little at the transition and the ratios c/c0 and b/b0 are close to unity. To a good approximation, therefore, equations 66 and 67 reduce to:

e4 = 2e23 ~ - cos a (77) e6 =2e12~ cos γ (78).

From measurements of excess heat capacity in crystals with a slightly different composition from pure albite, Salje et ah (1985a) concluded that the transition is second order in character. The linear dependence of cos2α* on temperature over a range of -1000 K (Fig. 9a) is then consistent with the expected e$ oc Q relationship. On the other hand, the available data suggest that cosy does not vary linearly with cos Of* (Fig. 9b), implying that e^ shows a different dependence on the driving order parameter for the transition. Contributions to e§ from a non- symmetry-breaking strain can be ruled out as the cause of the disparity, because the identity representation for point group 21m has basis functions JC2, y2, z2 and xz (second monoclinic setting). A non-symmetry-breaking strain component of the form xy, i.e. £6,nst» is therefore not allowed. The explanation is more likely to be in the different strength of coupling between Q

and ^6 relative to Q with e4. At T/Tc = 0.5, e4 is -6% and e$ is -0.3% - coupling between Q and e4 is evidently strong, but coupling between Q and eβ is weak. A small higher-order coupling term would have negligible influence on the evolution of e4 but could be significant with respect to a small coupling term in e^Q. Only even-order terms are present in the Landau

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expansion, L(ß), for this transition, and the next higher-order coupling term allowed by 3 symmetry is λe6Q (see equations 20 and 25). Fig. 9b shows the curve:

e6 « cos γ = 0.0251e4 + 7.784 (79)

drawn through the experimental data points (for e4 « -cosα*). Using the extrapolated value of cos2α* at 0 K to calibrate Q in terms of cosα* (ß = 1 at 0 K and cos2α* = 0.00656 at 0 K, from which Q = (cosα*)/0.081)), allows cosy to be expressed in the same form as equation 25, giving:

3 e6«-0.0020ß -0.0041Ö (80).

Alternatively, as discussed by Carpenter & Salje (1998), the driving mechanism for the transition might be softening of a critical combination of elastic constants, with the order parameter as the symmetry-breaking spontaneous strain itself. A simplification is then to take the larger of the two symmetry-breaking strain components, e±, as the driving order parameter and describe e$ as being coupled to it. In this case the effective coupling is described directly by equation 79.

Z c*

400 600 800 1000 1200 1400 Γ(K) Fig. 9. Strain behaviour at the Cllm ^ Cl transition in albite (NaAlSiaOg) with no long-range Al/Si order, (a) A linear variation of cos2α* with temperature is consistent with cos2α* <× Q2 «* Γ. Filled circles: data from Kroll et al. (1980). Open circles: previously unpublished data of A. Graeme-Barber. The straight line fitted to the new data gives a transition temperature of 1241 K. (b) Cos/(« e6) is not linear with cosα* (« -e4). The curve is a least- squares fit to the data implying cos/ = - 0.0020ß - 0.0041 ô3, where Q = (cosα*) /0.081. (c) Orientation of the principal strain axes with respect to the crystallographic axes calculated for room temperature (assuming only α* and γ change significantly due to the • 11111••11111•111n111ii 111 • • 11 • 111 • π 3 transition); e% « 0, e{ +ve, f3 -ve. The deformation is 0 20 40 60xl0' effectively a pure shear in the plane containing b* and c*. cosα

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There are insufficient data available above Tc to give reliable extrapolations for a0, b0, c0, ß0 and VQ below Tc. Any non-symmetry-breaking strain cannot yet be assessed, therefore, but the volume strain appears to be small. Finally, if it is assumed that only α* and γ change significantly as a direct consequence of the phase transition, the orientation of the principal strains can be estimated by diagonalising the strain matrix containing only e^ and e§. This yields

£i ~ -£3 and £2 « 0. The symmetry-breaking strain can be quite well represented as a pure shear in a plane close to the b -c plane of the albite structure (Fig. 9c).

Anorthite. An equivalent monoclinic ^ triclinic transition is observed in body-centered anorthites from the (Ca,Sr)Al2Si20g solid solution series (Bambauer & Nager, 1981; Tribaudino et aL, 1993; McGuinn & Redfern, 1994a and b, 1997; Redfern et aL, 1997; and references therein). Cos2α is again a linear function of temperature (Tribaudino et al, 1993), at least for some compositions (McGuinn & Redfern, 1997), and cosy is a non-linear function of cosα* (McGuinn & Redfern, 1994b, 1997). (See, also, section 6 below).

Tridymite. Tridymite (SiÜ2) displays a phase transition between hexagonal and orthorhombic forms at -626 K which is believed to involve the symmetry change P63H ^ C222\ (de Dombal & Carpenter, 1993; Cellai etal, 1994,1995). The active representation is E2 of point group 622, which has basis functions x2 - y2 and xy. For an orthorhombic product one component of the two-dimensional order parameter with E2 symmetry is zero (ee = 0). In their symmetry-adapted forms, the symmetry-breaking strain is (e\ - ci), and the non-symmetry-breaking strains are (e\ + e>£) and £3. The e^ components can be expressed in terms of orthorhombic lattice parameters alone (Putnis et at., 1987; de Dombal & Carpenter, 1993) as:

'(qΛ/3)-*Λ ^l,sb ~ ~~e2,sb - (81). (fl/V3) + é

Lattice-parameter variations through this transition are shown in Fig. 10 (from de Dombal & Carpenter, 1993). A large volume strain (Fig. 10c) due almost exclusively to the non-

symmetry-breaking strain component e$ (= (c-c0)/c0) is also apparent (Fig. 10b), but is associated with a different transition at a higher temperature and is discussed in a later section. Any 23 strain associated with the 622 ^ 222 transition is smaller than the resolution of the lattice-parameter determinations. The square of ei^ varies linearly with temperature, implying second-order character (Fig. lOd). If the proposed space-group relations are correct, first-order character might have been expected since a third-order term in Q is permitted by symmetry

(Stokes & Hatch, 1988). This third-order term is presumably small. At T/Tc = 0.5 the symmetry-breaking strain is ~5%o, which is less than the typical strain associated with proper ferroelastic materials. The excess energy associated with the transition is also undoubtedly small, however, since differential scanning calorimetry failed to reveal a measurable anomaly in

Cp at 626 K (Fig. lOe; de Dombal & Carpenter, 1993; Cellai et al, 1994).

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8 i i 1 1 11 1 1"'1 ' ] 7 s^ •Sy £-"6 µ (d) 1 L • I Xyβ - [ ^V •1 Q) 2 H

i 1 L 1 1 1i • - ffK J 200 300 400 500 600 700 800 270 280 290 300 310 320 330 340 350 360 Temperature [βC]

6.22 200 300 400 500 600 700 800 300 400 500 600 Temperature [βC] Fig. 10. Lattice-parameter and spontaneous-strain variations (from de Dombal & Carpenter, 1993), and heat- capacity variations (from Cellai et al.9 1994) at high temperatures in tridymite from the Steinbach meteorite. Two transitions are evident. The first, at -353 °C (626 K), involves a symmetry-breaking hexagonal ^ orthorhombic strain revealed by the a parameter (a). The second, at ~465 °C (738 K), involves only a non- symmetry-breaking strain, e3, revealed by the c parameter (b). A volume anomaly (c) arises entirely from e3. (d) A 2 2 2 linear variation of eu (= elsb <* Q ) is consistent with second-order character for a proper ferroelastic hexagonal

200 300 400 500 600 700 800 ^ orthorhombic transition, (e) No obvious specific heat anomaly is observed at -353 °C, but there is a classical Temperature [°C] second-order step at -407 °C (680 K) and a tail decreasing to the values of the high-temperature phase at -465 °C (Cellai etaU 1994).

4Je;ocg2,V,»0 Anorthite. The /T ;= P\ transition at -510 K in anorthite, CaAbSi20g, is associated with a special point on the Brillouin zone boundary (for a comprehensive bibliography see: Smith & Brown, 1988;^alje, 1994; Angel, 1994). The high- and low-symmetry forms are both triclinic (point group 1). Any spontaneous strain is non-symmetry breaking and transforms as Γicientity (co-elastic behaviour). All six strain components are expected to have different non-zero values, and can only be determined by extrapolating lattice-parameter data from above Tc to give values for α0, b0, c0, etc., below Γc(Redfern & Salje, 1987; Redfern et α/., 1988; Redfern, 1992). Redfern & Salje (1987) showed that the individual strain components for a natural sample with a

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high degree of Al/Si order go continuously to zero at T = Tc, and all display a similar temperature dependence. To a good approximation, the scalar strain (£;S in equation 11) varies linearly with the intensity of a superlattice reflection due to the transition, confirming that it 2 2 scales with Q (Fig. 11a). £ss varies linearly with T (Fig. lib), implying that the transition 4 conforms to Landau tricritical behaviour (ß ©c \TC - J|). Similar analysis of spontaneous strains in crystals with a higher degree of Al/Si disorder show a tendency towards second-order character (Fig. lie; Redfern etal, 1988; Redfern, 1992).

r'* 11111111111111 1 1 1 1 1 1 1 1 1 1 1 1 II 5xl(Γ1 P (a) • M 4 s^ H 3 r >r A

ω _J 2 - ^rm ^^ m H 1 Γ m >^m H 0 j X*x :, 1 1 1 1 1 1 1 1 1 1 1_iLi.ixn_iiL.i 1 i 1 l 0 4 8 12 /025 (arbitrary units)

Fig. 11. Scalar spontaneous strain, £ss, for the 71 ^ PI transition in anorthite (CaAl2Si208). (a) In natural crystals with composition ~An10o> £Ss varies linearly with the intensity of a superlattice reflection (the 025 2 reflection), which is consistent with £ss« Q since I025 ec ß2 (after Redfern & Salje, 1987). (b) In the same 300 400 500 600 700 2 4 sample £^s is linear with T implying ß °c T and Γ(K) tricritical behaviour (after Redfern & Salje, 1987). (c) In natural crystals with composition ~An98Ab2 and a 11111 iii 11111 in 11• • • i M i ii 111 i»l*j slightly lower degree of Al/Si order, the transition has 3 FT 2 5xl0" -•-. £sS oc T and, hence, shows second-order character (ß <× • (c) • T). Note that as T -> 0 K, £ss levels off, showing the 4 jTT T saturation in ß (after Redfern et al., 1988). (d) \ Orientation of the principal strains (€\ +ve, €2 ~ 0, €3 «> 3 \ CΛ A -ve) in different anorthite samples, as calculated for W ?. : \ JL room temperature and pressure (Redfern & Salje, 1987; - % Redfern et al, 1988; Redfern, 1992; Angel, 1992). V 1 . •\ and M refer to natural samples from Val Pasmeda and v I"- H •\* I •• H Monte Somma, respectively; Vh refers to a Val 0 ^••••••••'.„j Lu • •Inul IIIIIHIIIIIII 1111 •111 11 ll Pasmeda sample which had been annealed at high 200 400 600 temperature to induce a small degree of disorder; P Γ(K) refers to the strain arising from the transition induced by increasing pressure.

In this system the non-symmetry-breaking strain tensor describes a deformation which gives a volume strain of only -0.1% at T/Tc =0.5 (based on data of Redfern, 1989) and is thus almost a pure shear. The orientations of the axes of the principal strains are not constrained in a triclinic crystal and might vary with temperature during the transition. Redfern & Salje (1987)

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and Redfern et al. (1988) reported, however, that no such temperature dependence is observed in anorthite, and Angel (1992) found no dependence on pressure. This implies that the orientation depends on the topology of the structure and not on the transition mechanism (Redfern & Salje, 1987). As shown in Fig. lid, the principal strains are not obviously related to any simple crystallographic planes. The behaviour of geological materials is not usually examined below room temperature. Spontaneous strains can be useful for characterising saturation effects, that is flattening out of

the order parameter as T —> 0 K, however, as is evident from the evolution of £jS at low temperatures for a relatively disordered anorthite sample (Fig. lie; after Redfern et al., 1988). Such effects have interesting theoretical implications (Salje et al., 1991).

4.3eiocQ9Vs±0 Leucite. One example of a proper ferroelastic transition which is associated with a large volume strain is the cubic ^ tetragonal transition in leucite (KAlSi2C>6), (see recent reviews by: Boysen, 1990; Palmer, 1990; Heaney & Veblen, 1990; the equivalent transition in Rb+,Cs+, and Fe3+

substituted leucites is described by Palmer et al, 1997). At T/Tc =0.5 the volume strain is -4%. There appear to be two separate transitions with equilibrium transition temperatures separated by -20 K, and the symmetry-breaking strain and the volume strain probably arise from separate effects. With falling temperature the space groups (and point groups) of the successive structures are Icßd (m3m) -> I4\lacd (4/mmm) —> I4\/a {Aim). Both phase transitions are associated with the Brillouin zone centre.

For the strain analysis, the relevant active representations and basis functions are Eg for 2 2 2 2 2 2 2 m3m —» 4/mmm (2z - x - y , x - y ) and A2g for 4/mmm —> 41m (g, xy, x - y ). In the case of Eg (m3m —» 4/mmm) the basis function x2 - y2 describes an orthorhombic strain and is not

relevant in the present context. The symmetry-breaking strain components are related by 2e^s\y = -(^l.sb + £2,sb) and £i)Sb = £2,sb- ^^ non-symmetry-breaking strain must be a pure volume strain. In the case of A2g (4/mmm —> 4/m), there is no change in crystal system and only non-symmetry- breaking strain, transforming as Γjdentity of 4/mmm, can occur. This latter strain consists of three

components, with the relationship ei,nsb = ^2,nsb ^ ^3,nsb» and is likely to be a predominantly volume strain. The temperature interval over which the intermediate phase is stable is too small for changes in lattice parameters associated purely with the m3m —» 4/mmm transition to be characterised easily. The individual strain components can therefore only be measured with respect to the cubic structure. In order to try and separate out different contributions to the total observed strain, the simplest initial assumptions are that the volume strain is associated entirely with the 4/mmm —> 4/m transition, and that the deviation from cubic symmetry may be ascribed anc entirely to the m3m —» 4/mmm transition. The expected relationships Vs <*= ÖA2g * ^sb °° ÖEg may be tested to check these assignments. Measured lattice parameters for leucite are shown as a function of temperature in Fig. 12a (data from Palmer et al., 1989). These were converted to strain components using equations

55 - 58 (with a0 = b0 = c0) and expressed in terms of eu the symmetry-adapted form of the symmetry-breaking strain given by equation 29. The variation in unit-cell volume associated

with the transition is shown in Fig. 12b; the volume strain, Vs, was calculated in the usual way (equation 69). Included in the reflections which appear in diffraction patterns at the Ia3d —>

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1 1 11 3 1 1 | 1 1 1 | 1 1 1 | Λ-3 rn ' ! ' 4 3.5xio' (a) J 1.6x10" Hil l (e) 3 1.4 -j 3.0 136 \ s* •••••••••^ 1.2 -j 2.5 : α°-^-—-"*** •-__ -— / *d 1.0 J 2.0 13.4 -3 ^ 0.8 ./ A 1.5 0.6 Γ •• a •3 -I 1.0 13.2 L. 0.4 h ...-•• •• 1 0.2 0.5 '- -•" i"" Id O.OHMIIIIIMI I I I I I I I I I I I I I I I I I I i m L 13.0 "t 1 1 1 i i 1 i i i 1 i i i 1 • 11 -A 300 400 500 600 700 800 900 400 600 800 1000 1200 Γ(K) T(K) I i I | i I I j I I I | I I I [ I I I [ • 300x10° U (Oj 600xl0"6 =i-r- Π-ΠJΓT r-j -r-n-p i i i | "* 2500 2(X)k H 400 * 2480 - ^ f (b)d 100 [- ^200 ~" 2460 _ 3-^ • _J 0 0 / J I I 1 1 I I I 1 I M I I I I I I I I I 2440 .• H 860 880 900 920 940 2420 • J • Γ(K) 2400 • 111111111111111111111111111111 y -] -40xl0"3 2380 _ • • _3 • • 2360 .•• -j -30 2340 t" 1 1 i i i 1 i i .I...I i i . 1 J 400 600 800 1000 1200 ^ -20 Γ(K) -10

(c)H 0

Fig. 12. Strain behaviour at the cubic ^ tetragonal transition in leucite (KAlSi206). All data from Palmer (1990). (a) Lattice parameters as a function of temperature; a0 refers to the cubic phase (including extrapolation below Γc), while a and c refer to the tetragonal phase, (b) Unit-cell volume; V0 refers to the 2000 4000 6000 cubic phase, (c) Linear variation of the square of the symmetry-breaking tetragonal strain with the intensity of a 72oo (arbitrary units) superlattice reflection that appears at the m3m —> 4/mmm transition, consistent with et <× gEg. Some intensity remains at _ [ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 II -40xl0"3 the 200 position in reciprocal space above Γc, hence the linear least-squares fit does not go through the origin, (d) Linear :l yr<®\ relation between the volume strain, VSi and the intensity of a -30 reflection that appears at the 4/mmm —> Aim transition. This is 2 ^ :; yf ] consistent with Vs <× gA2g . In this case the linear least-squares -20 fit was constrained to pass through the origin, (e) Both V^s and H y^ 4 2 4 et go continuously to zero as T -> Γc, but neither Vs (°c gA2g ) % :\ / i 2 nor e} (<× gEg ) are quite linear with T. (f) Detail from (e) -10 2 2 •Å x \ showing that Vs extrapolates to zero at -918 K and et " *• *s j extrapolates to zero at -930 K. (g) V% varies linearly with et, " S" S J 2 0 fe/^ i consistent with gA2g varying linearly with gEg. The "11 ii 11 11 > i 1 ii 11111111111 d extrapolation to Vs = 0 at et * 0 is consistent with the volume 0 1000 2000 3000 4000 strain being associated with a separate phase transition that has /^ (arbitrary units) a lower Tc than the transition at which the symmetry-breaking strain develops.

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IA\lacd transition is 200. The intensity of this reflection, /200> is expected to vary linearly with

ßlg and the linear variation with e\ evident in Fig. 12c confirms the expected relationship et <× ÖEg- The straight line drawn in Fig. 12c does not go through the origin because some residual intensity is found to remain above 938 K (Palmer et α/., 1989). To a reasonable approximation

Vs varies linearly with the intensity of the 901 reflection, which arises at the IA\lacd -> lA\la

transition (Fig. 12d). This is consistent with the predicted relationship Vs °c ÖA2g- The two strains extracted from high-temperature lattice-parameter data in this way provide some insights into the thermodynamic character of the transition and the nature of

coupling between ÖEg and ÖA2g- Both et and Vs go continuously to zero as a function of temperature, though neither conform exactly to a second-order or tricritical pattern. The parameter e\ is not quite linear with Γ, as it would be if the m3ra -» 4/mmm transition was classically second order, while VJ is not quite a linear function of Γ, as it would be if the 4/mmm —> Aim transition was tricritical (Fig. 12e). Both might be consistent with the solutions to a Landau expansion with significant (positive) values for both fourth and sixth-order terms, however. The fact that VJ extrapolates to zero -12 K below the extrapolation to zero of e\ (Fig. 12f) is consistent with the suggestion that the strains arise separately from two discrete transitions with slightly different critical temperatures.

Fig. 12g shows that et is close to being proportional to Vs and, hence, that ÖEg varies linearly with ÖA2g- This might be explained in terms of linear-quadratic coupling, λßEgOA2g between the two order parameters, as predicted on symmetry grounds by Palmer et al. (1990) and Hatch et al. (1990a).

4.4^00 02,^0

2 Examples of phase transitions with spontaneous strains varying as e,- ©c ß , Vs * 0 fall into two categories. The first category is similar to the II ^ PI anorthite transition in that no change in crystal system is involved (co-elastic behaviour). (In this context, hexagonal and trigonal systems may be regarded as being equivalent). The only allowed strain is non-symmetry breaking. Examples are provided by transitions in NaN03, CaCÜ3, tridymite, kaliophilite and quartz, which all show significant volume strains. Quartz shows some evidence of higher-order strain coupling and is discussed in the next section (4.5). In the second category, there is a change of crystal system, but the active representation is associated with a special point on the Brillouin zone boundary; such transitions are termed improper ferroelastic. The symmetry- breaking strain transforms as a representation of the point group of the high-symmetry phase which is neither the active representation nor the identity representation. Examples in this category which also display significant volume strains are the trigonal ^ monoclinic transition in Pb3(P04)2, with the palmierite structure, the cubic ^ tetragonal transition in cristobalite (SiÜ2), and cubic ^ orthorhombic transitions in certain perovskites.

NaNÖ3, CaCOs. The R3m ^ R3c transition in NaNC>3 and CaCÜ3 involves orientational ordering of the NO3 and CO3 groups. Both the high- and low-symmetry phases belong to point group 3 m and the basis functions for Γidentity of this point group, x2 + v2 and z2, imply that the non-symmetry-breaking strain will have e\ = ei & e$ & 0. Experimental difficulties have so far

prevented lattice-parameter measurements from being made above Tc at one atmosphere

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pressure, with the consequence that values of a0 and c0 can only be estimated on the basis of assumptions concerning thermal expansion of the high-symmetry phases. For each material, measured values of e$, corresponding to extension parallel to the crystallographic z-axis, are approximately one order of magnitude greater than the measured values of e\ (extension

perpendicular to the crystallographic z-axis). The volume strain at T/Tc = 0.5 is ~7% for NaNC>3 and -3% for CaC03, with values of Tc = 553 and 1260 K respectively (Reeder et al., 1988; Dove & Powell, 1989).

Figure 13a shows a linear relationship for calcite between Vs and the intensity of a reflection which is absent in diffraction patterns from the R3m structure but present in patterns from the R3c structure (data from Dove & Powell, 1989). Dove & Powell (1989) assumed that VQ does not vary with temperature but, even with this slightly arbitrary assumption, the 2 correlation seen in Fig. 13a is sufficiently close to confirm the relationship Vs °c Q . Assuming that VQ is constant allows c0 and, hence, e^, to be calculated {a and aQ show very little temperature dependence). From Fig. 13b it is apparent that e\ is linear with Γ, showing that the transition in calcite is tricritical, within experimental uncertainty. Analogous relationships for NaN03 are shown in Fig. 14. In calculating e?> for NaNÜ3, Reeder et al. (1988) allowed a small

temperature dependence for c0 (Fig. 14a). They found that e$ is proportional to the excess entropy due to the transition (Fig. 14b), which is normally expected to be proportional to Q2.

However, e\ is a linear function of T only over a range of temperatures away from Tc (Fig. 14c). The strain measurements are thus sufficiently sensitive to indicate deviations from tricritical behaviour that also appear in measurements of birefringence and of superlattice reflection intensities. These deviations have been attributed to coupling with a second order parameter, perhaps via the common strain, above -460 K (Reeder et al., 1988; Poon & Salje, 1988; Schmahl & Salje, 1989; Harris et al, 1990). As with anorthite, the spontaneous strain clearly shows the onset of order-parameter saturation below room temperature (Fig. 14c).

f ! f BI-I |TI t-| r-T 3 ' l"' ' 1 ' 1 ' 1 1.5x10o-^ h (b) J

1.0 - x^ r

0.5 "- ^v H

^Sçj 0.0 "L.J l_l 1 1.1 ,_. i LA i-U J_I U il 500 1000 1500 400 600 800 1000 1200 h 123 (arbitrary units) Γ(K)

Fig. 13. Strain behaviour at the R3m ^ R3c transition in calcite (CaC03), after Dove & Powell (1989). (a) The volume strain, Vs, varies linearly with the intensity of the 1123 reflection which arises at the phase transition. This 2 2 2 is consistent with Vs <× Q . (b) The non-symmetry-breaking component, e3 (©= Q ), varies with T as e$ «= Γ, consistent with tricritical thermodynamic character for the transition.

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0 100 200 300 400 500 600 Γ(K)

Fig. 14. Strain behaviour at the R3m ^ R3c transition in NaN03 (from Reeder et al., 1988). (a) Lattice- parameter variations with temperature; a and c refer to the R3c phase while £0 is the estimated variation of the c parameter for the R3m phase. Note that a shows very little variation due to the transition {eλ is small), (b) e3 is linear with the excess entropy due to the transition, consistent with e$ «= Q2. (c) e^2 (<× ß4) is linear with T over a temperature range between -180 and -430 K, implying tricritical behaviour. Deviation from this relationship between -430 K and Tc has been attributed to coupling with a second order parameter. Below -180 2 K, the levelling off of e3 is indicative of order- 0 0.04 0.08 0.12 0.16 0.20 0.24 parameter saturation. S (J/gK)

Tridymite. As is evident from Fig. 10, tridymite undergoes a transition at -738 K which is marked by a significant linear strain parallel to c and a significant volume strain (de Dombal & Carpenter, 1993). The lattice-parameter data are not shown in terms of strain components, but it

is clear that both e^ and Vs must be linear functions of temperature. The ideal high-temperature structure of tridymite has Pβ^lmmc symmetry (see Heaney, 1994, for a recent review), and the first lower-symmetry phase to develop during cooling is believed to have space group P63H (de Dombal & Carpenter, 1993; Cellai et α/., 1994). The only permissible strains associated with this symmetry change are non-symmetry breaking, (e\ + e>i) and e^. The linear temperature

dependence of e$ and Vs is consistent with second-order character for the transition. All is not quite as it seems, however, because the associated classical second-order step in Cp actually occurs at -680 K rather than -738 K (Fig. lOe). Between -680 K and -738 K there is only a tail in the heat capacity. The transition thus appears to occur in two parts, a view which is reinforced

by single-crystal diffraction data (Cellai et al.9 1994). In the 680-738 K interval, reflections which indicate the symmetry reduction from Pβ^lmmc to Pβ^H are diffuse. Only below -673 K

do they become sharp, implying that the cause of the Cp anomaly at -680 K is a changeover from short-range to long-range correlations of the order parameter. The underlying mechanisms for such behaviour are discussed in more detail by Cellai et al (1994, 1995). For present purposes, the essential point is that short-range or dynamical effects in a crystal may give rise to

a non-symmetry-breaking strain ahead of a discrete phase transition. This is because en<& is not required to be exactly zero in the stability field of the high-symmetry phase.

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Kaliophilite. Natural crystals of kaliophilite (-Nao.sKo.çsAlSiO.*) show a large volume anomaly associated with a symmetry change which is believed to be Pβ^ll ^ P63 (Cellai et al, 1992). In this case the square of individual strain components and of the volume strain go to zero linearly

with temperature and are thus consistent with tricritical character for the transition (Tc~ 1000 K).

Pbs(P04)2- The active representation for the R3m ^ Cllc transition in Pb3(P04)2 is associated with special points on the Brillouin zone boundary. The change in point group is 3m (trigonal)

^ 21m (monoclinic), and the symmetry-breaking spontaneous strain transforms as Eg of 3m. 2 2 From the basis functions of Eg (JC - v and xz), the relevant strain components will be e\t^, £2,sb> é?5>Sb; they are expected to show the relationships e\^ = -^2,sb * ^5,sb * 0. Non-symmetry- breaking strain components transform as Γidentity of 3m giving e\tns\y = ^2,nsb ^^33,nsb * 0. Eg is not the active representation for the transition; individual strain components and the scalar strain should all vary linearly with Q2. The transition provides an example of improper ferroelastic behaviour (for recent reviews see: Bismayer, 1990a and b; Salje et al., 1993a). There is a relatively complex orientational relationship between the rhombohedral and monoclinic unit cells. Strain effects are most easily examined by transforming the axes of a hexagonal cell for the high-temperature form so that they are the same as those of the low-temperature monoclinic form. Variations in the lattice parameters transformed in this way (Joffrin et al, 1977; Guimaraes, 1979a; Bismayer & Salje, 1981) show the effects of the phase transition quite clearly (Fig. 15, after Salje et al, 1993a). Following Tolédano et al (1975), Guimaraes (1979b) and Salje et al (1993a), the symmetry-breaking strain components can be calculated directly from the monoclinic lattice parameters using:

*l,sb=1—^J (82) e = 5>sb 3αsin/J )

Note that e\ is a linear strain in the direction of the crystallographic v-axis in these definitions. Inherent in equations 82 and 83 is the assumption that the reference-state parameters can be taken as average values from the monoclinic parameters, rather than extrapolating them from the stability field of the high-symmetry phase. There is a significant (positive) volume strain accompanying the phase transition, however, and to avoid introducing errors in the calculated strains it is safer to use extrapolated parameters. Maintaining the coordinate system set out in section 3, the relevant strain components are defined by equations 37 - 42, with a-γ- 90°. The

reciprocal lattice angle ß0 is a fixed quantity (« 76.7°) obtained from:

~V3α cos/^, 7J"HH (84) (34+4cèf

where an and CH are the dimensions of a hexagonal cell for the high-temperature phase (Guimaraes, 1979a). The b and c parameters of the artificial monoclinic cell for the high- symmetry form are, in addition, related by c = V3&. A further consequence of transforming the hexagonal cell into a monoclinic form is that the threefold symmetry axis is perpendicular to

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(100) of the new cell, which means that the strain components are related by ^2,Sb = -^3,sb and ^2,nsb = ^3,nsb- (^1 is the exclusively non-symmetry-breaking strain component in this coordinate system). The symmetry-breaking and non-symmetry-breaking strains are easily separated as:

(85) (86) ^2,sb = 2^2"e3) *2,nsb = 2^2+^3)

where ei and e?> are the total strains given by equations 38 and 39. Finally, there is strong

evidence from Cp and spectroscopic measurements (reviewed in Bismayer, 1990a and b) that local structural changes associated with the transition occur up to -530 - 570 K. To test whether these influence the strains, it is necessary to use only data from above this range to extrapolate the reference-state parameters. In the present case, lattice parameters collected between 530 and

673 K were used to fit VÖ, a0 and b0 (with c0 - V3fc0). The exact lower limit is a matter for somewhat arbitrary choice, but does not materially influence the outcome of the analysis.

:i • • • • 1 • » • ' 1 • • ' • 1 • •• H 103.4 FT 1 1 1 1 1 1 1 1 1 1 1 1 1 ß 13.95 ** *^ A 103.2 : a j C 103.0 << o^^**"^ 13.90 ="^^JV H GQ-102.8 - ^^ m **' H 102.6 13.85 -•• _J : ••* H : •• j 102.4 -1 1.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1.1H I 1 1 1 1 I 1 1 1 1 1 1 300 400 500 600 300 400 500 600 5.68 1 1 1 1 1 1 1 1 1 H I 1 1 1 1 I 1 1 1 1 I 1 1 1 1 I 1 1 1 ^ 5.64 Γ \ H 5.60 Γ \ 5.56 •"•*—]

5.52 -1 1 . 1.1 1 1 , . 1 1 1 1 1 1 1 1 1 _J_fi 300 400 500 600 400 500 600 9.65 LI ' 1 1 1 1 1 1 1 1 | 1 1 1 1 1 1 1 1 IΛ Γ(K) 9.60 Co JM H ~~ " + j Fig. 15. Variation of lattice parameters through the 9 55 H < - / R3m ^ C2/c transition in Pb3(P04)2 (data from Salje mm 0 9.50 Γ T et al., 1993a). Straight lines indicate the fits to data y above 530 K for reference-state parameters. 9.45 - • LLi. 1 .!_. Li. 1 1 ..1 1 1 1 11 1 1 1 i H 300 400 500 600 Γ(K) The strain variations shown in Fig. 16 and 17 (calculated from the lattice parameters in Fig. 15) are broadly consistent with other experimental studies of the phase transition (reviewed

in Bismayer, 1990a and b). All the individual strains, apart from Vs, show a small discontinuity at the equilibrium transition temperature, in keeping with the well established first-order character (Fig. 16). Linear relations between the strains are consistent with the expected

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2 relationships e^ °c ^nsb oc Vs <× Q (Fig. 16), and the linear variations of ef and Vf with temperature are consistent with the transition being close to tricritical in character (Fig. 17). Breaking down the overall strain behaviour into its component parts reveals some more subtle aspects of the transition.

. j a I i i | i i I i [ i 20xl0"3 l l l | i l i IJ 20x10" H ΓSqA* 11111 1 15 '- #SS>n V 5 DRx 10 Γ \ J α t-:+ w, t* ++4.Γ W\ • 3 xa i «5 ^ "e2,mb ^^aS | o B^p»-4M#H -5 pr^» " 1 1 l I l i I l l I 1 I i i i 1 i i i π 300 400 500 600 T(K) 20x10° F- 15xlO"J h

20xlO"J

Fig. 16. Variation of spontaneous strains through the phase transition in Pb3(P04)2 (top left). The individual strain components all appear to show a discontinuity at T^ (~ 453 K). Vs is continuous within experimental uncertainty and shows a tail above the transition point. Strain/strain variations are shown in the other diagrams. Note that e2,sb> e an 2,nsb d ^s behave as a colinear set while any of these plotted against e5 show some systematic anomalies at small str strains (i.e. as T —» Γtr from below). Vs and e2^sb ^ linear with each other even for data from above the transition temperature. As in the case of tridymite, the volume strain does not follow the symmetry-breaking

strains exactly. Instead of showing a discontinuity, Vs is apparently continuous through the transition point. A tail extending up to -500 - 550 K (Fig. 16) mirrors the excess heat capacity

above Ttr found by Salje & Wruck (1983), and implies that the local structural variations modelled by Salje & Devarajan (1981) produce a small volume increase (Salje et al., 1993a). The expansion is entirely within the (100) plane (perpendicular to the threefold axis of the

trigonal phase) since β2,nsb follows Vs closely and e\ is essentially zero above ΓtΓ (Fig. 17). Below the transition point, ef.sb πurrors the square of the birefringence in the (100) plane reported by Wood et al. (1980a). A small excess above the linear trend is observed in both cases between -430 and 453 K (compare Fig. 4 of Wood et al, with Fig. 17). All of the straight lines 2 drawn for ef and Vs in Fig. 17 are least-squares fits to the data between 298 and 430 K. On this 2 ut basis, the same anomaly is also seen in Vs and ef,nsb> ^ appears to be absent from e$ and ef. Strains out of the (100) plane are also distinct from those within the plane in that the linear

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and V trends for ^2,nsb, ef.sb ( s) extrapolate to zero at 445, 443 K (and 442 K) respectively, while the trends for e\ and ef go to zero at 453 and 457 K. The same distinction is seen in Fig. an 16, where ^2,sb d ^2,nsb are linear with each other and with Vs, while any plot of e\ or es against the others shows systematic deviations from linearity for data from the -430 - 453 K range (the smallest non-zero strains).

1 120xl06 • k_iii 1 ' 1 1 1 1 1 1 1T-Γ-ΓT-T- g Jill 1 ! T 1 1 11 1 1 11 1 1 1 1 II iOxlO"6 100 • \" H 50 A 80 | 40 -J V 60 H 30 •if 3 -j 40 Γ H Ü 20 H ΓΓΓT T

20 Γ 10 - -J 0 z...... !&••••*•••••*« 0 1 , . . . 1 , , , . 1 1 1 1. i . . . . 300 400 500 600 300 400 500 600

1 ' ' • i 1 i i i i 1 i i i i | i i i ii m J 1 1 • i I • i i i 1 i i i i 1 i i OOxlO'6 6 250 300xl0"

J t.l.l±J L 200 j J cs 200 'tf 150 -J 100 H 100 H 50 H 0 0 JL_J_L_|__L_ , , , i i 1 i i i i 1 i i i i 1 i i i il -iii i i \ i i • •lii J__JJ 300 400 500 600 T (K)

Fig. 17. The square of each strain due to the transition in Pb3(P04)2 is approximately linear with temperature, consistent with near tricritical character (ß4 ©c T). Each straight line is a least-squares fit to the data between 298 and 430 K. Note that there is a small excess above 2 2 the linear fit between -430 and 453 K for e2,Sb » *2,nsb > 2 2 and Vs , but not obviously for e^- and e5 . Moreover, the 2 linear fits for e^- and e5 extrapolate to zero at 457 and e 2 453 K, respectively, whereas the fits for £2>stΛ 2,nsb and Vs2 extrapolate to zero at 443, 445 and 442 K. 400 500 T (K)

In their Potts model of short-lived monoclinic domains within trigonal Pb3(P04)2, Salje

& Devarajan (1981) argued that the first structural change which occurs as T tends to Γtr from above is a displacement of lead atoms off their sites on the threefold symmetry axes. The new lead positions remain within the (100) plane, which presumably accounts for the restriction of the volume strain to this specific plane. At the transition point the crystal structure undergoes a spontaneous shear, but there appear to be significant differences between the processes inducing the shearing within and perpendicular to (100).

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Cristobalite. The active representation for the Fd3m ^ P4\2\2 (or PA^lγl) transition in cristobalite, Si02, is not associated with the Brillouin zone centre. As with lead phosphate, therefore, the symmetry-breaking strain is coupled with the square of the driving order parameter (Hatch & Ghose, 1991; Schmahl et al, 1992; Schmahl, 1993; Hatch et al., 1994; Heaney, 1994; and references therein). Schmahl et al. (1992) have followed lattice-parameter variations through the transition and characterised its strong first-order nature in terms of the spontaneous strains. Their data, reproduced in Fig. 18a, are used here to illustrate the convenience of expressing strain components in symmetry-adapted forms for high-symmetry systems.

ΓT i i i i 1 i i i i 1 i i i i 1 1 1 1 II j I i i i i [ i i i i 1 i i i iΠP ~ 1—r-n 0 CDIDQDQI ••$*•* i #H I i (c) j 7.10 (a) A -10 1 | ~;>J&*"0***~\ q >< 7.05 "53 "20 — 1 H -Λ h 1 1 | j I Z i -j -30 — 1 H * 7.00 z 1 : i -40 J 6.95 e 3 : *±J****^ 3 -50xlO" r*÷-*V*^ H i I i i i i I i i i i i i i i iL±J - IIII 300 400 500 6() 0 300 400 500 600 T (K) T (K) • 1 i i i i 1 i i i • 1 i i II 3 50xl0" Fig. 18. Variations of lattice parameters and (b) • -40 Mjjß spontaneous strains at the cubic ^± tetragonal transition 1 in cristobalite. (a) Lattice-parameter data of Schmahl et * -30 H al. (1992). (b) ea and ex derived from the lattice parameters show a linear relationship, consistent with 2 -20 - -j each varying with Q . (c) Curves through the data points for eΛ and et as a function of temperature are -10 - -J given by equation 94 with Γtr = 532 K and Tc = 282 K. The discontinuities in et and ez are et 0 =-0.0128, eΛO = 0 -0.0391. 1 • • i 1 i i II 1 i • i • Li i~il 0 -5 -10 -15xlO~J &t

The three non-zero strains are e\y e2 and e$, and are given by equations 55 - 57. (The equations used by Schmahl et al., 1992, appear to use an incorrect reference state). They contain symmetry-breaking and non-symmetry-breaking parts, which are equivalent to the symmetry-

adapted forms, et (equation 29) and ea (equation 30), respectively; et is a pure shear and ea is a pure volume strain in this case (ea = Vs to a good approximation). One is a linear function of the other (Fig. 18b), as expected if each varies linearly with Q2. At 1 - 2%, the symmetry-breaking strain is fairly characteristic of a ferroelastic transition, but the volume strain, ~4 - 5%, is exceptionally large. Schmahl et al. (1992) showed that the overall thermodynamic behaviour can be accounted for using a single order parameter rather than the multicomponent order parameter that is permitted by symmetry. The full Landau expansion is given by Hatch & Ghose (1991) and Hatch et al. (1994), but the simplest (much reduced) form is:

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2 4 β 2 2 2 G = ±β(Γ- TC)Q +h<2 +\cQ +λaeaß + λtetQ + ±Cfé +^Ct°et (87). 2 4 6 11

Ca and Ct° are the symmetry-adapted forms of the bare elastic constants (i.e. for the cubic phase), defined as:

C°a=Uc?λ+2C?2) (88) <7=|(cf1-Cf2) (89).

2 The relationships between ea, et and ß are given by:

2 (90) *a=Λô 1 0 (91). C? Ct ^

Equation 87 can also be written in its renormalised form as:

2 4 6 G = ±a(T-Tc)Q +h*Q + ±:cQ (92) 2 4 6

where:

, 2Λ 2λ a t (93).

The transition is first order in character and the standard solution for Q as a function of temperature in the case of a Landau expansion of the form of equation 92 with b < 0 is:

f Λ 2 T-Tc ß =fdi+ i-l (94). 4

2 Substituting ea (or £t) for ß , and taking TtT = 532 K then yields the fit parameters Tc = 282 K, £a,o = -0.0391, et,o =-0.0128, where the latter represent the values of ea and et in the tetragonal phase at 532 K (Fig. 18c). For ΓtΓ - Γc = 250 K it follows that ß0 = 0.797 (equation 94). Equations 90 and 91 yield values of Aa and λt if Ca and Ct° are known. Schmahl et al. quote 5 1 5 1 model values for Cft and Cf2 which give Ca° = 9.08 x 10 J.mole- , Ct° = 8.35 x 10 J.mole" (taking the molar volume of cubic cristobalite as the value of V0 at room temperature, 2.71 x 5 3 4 -1 10 m ). Substituting Q0 for Q, e%0 for ea, and eUo foret gives λa = 5.59 x 10 J.mole ,λt = 1.68 x 104 J.mole"1. Following Schmahl et al. (1992), the latent heat of transformation, L, is taken as -1256 -1 -1 -1 J.mole which yields the Landau coefficient a = 7.42 J.mole .K (L = -jaQ%TiT), and, hence, from standard Landau solutions, b* = -11.68 kJ.mole-1, c = 13.79 kJ.mole-1. Substituting all the fit parameters into equation 93 yields (b* - b) = -7.56 kJ.mole-1, and a value of -4.12 kJ.mole-1

for the unrenormalised fourth-order coefficient, b. The values of TiT and Q0 derived from the

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strain data are indistinguishable from the values extracted from atomic-coordinate data by Schmahl et al. (1992), as are the values of the Landau coefficients. The present calculation gives a much greater renormalisation of the fourth-order coefficient than they reported, however. The phase transition in cristobalite appears to conform closely to the pattern of behaviour described by spontaneous strains incorporated formally into the Landau expansion for a first-order transition.

Neighborite. Strain behaviour associated with the Prrßm ^ Pbnm (zone boundary) transition in NaMgF3 perovskite (neighborite) appears to be more complicated (Zhao et al., 1993a and b, 1994). The lattice-parameter variations of Zhao et al. (1993a) are reproduced in Fig. 19a, and

have been used to derive the strain components e\9 e2 and e$ in the usual way (Fig. 19b). In this case there are three symmetry-adapted strains: eaand et, as for cristobalite, plus e0 (equation 28) to represent the orthorhombic distortion. Each of these strains varies approximately linearly with temperature below -780 K (Fig. 19c). This would be consistent with a classical second-order transition in a material in which each strain is coupled to the square of the driving order parameter. Between 780 K and the transition temperature of -1038 K, however, the relationships are non-linear; e\ behaves linearly in this temperature interval and extrapolates to zero at -1034 K (Fig. 19d) which might imply tricritical behaviour. The form of the excess heat capacity is

3.95 • i mi n i II 111111111111 II 1111 in ii [i

E- + * Λ. fi*4 + + + + + -10 EP * + + 4* 1 (c) q «?" -15 .' . » -20

-25xl(r t*i i II In II In i ill II i 400 600 800 1000 1200 400 600 800 1000 Γ(K) T 00

11111111111111 • •• I | I I I I | • I I I | I I I I | I I IT 1 • • • 11TT1 | j II 1 1 | III 1 |l II|lI II 1 |l III |M 1

:% Ö6 * 10 (b)- 500xl010 h (d)J *2 • • • 5P 400 • **.. • • • • • 300 • • • H ++ + -H- + + + + + -fµ + -^ -*+ •'•HH *3 200 • • -5F- ^ 4 • -j <6< 100 o 9) "^^v^ \ 08 0 to*** ' M 11111111111111111II1111111111»111111111 1111111 J-Ü II II 1 II Mil II 11 in iin iiln 1 400 600 800 1000 400 600 800 1000 Γ(K) Γ(K)

Fig. 19. Strain variations associated with the cubic ^ orthorhombic transition in NaMgF3 perovskite (neighborite). (a) Lattice-parameter data from Zhao et al. (1993a). (b) The linear strain components calculated using equations 55 - 57. (c) Symmetry-adapted strains. Note that each strain varies approximately linearly with T below -780 K. (d) The square of the orthorhombic strain, e0, varies linearly with temperature atΓ > 700 K and extrapolates to zero at 1034 K, which might imply tricritical behaviour.

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also consistent, at least qualitatively, with tricritical character for the transition (Zhao et al, 1993b; Topor et al, 1997). Neither e\ nor e\ display such a simple evolution. As discussed by Zhao et al. (1993b) the order parameter for this cubic ^ orthorhombic symmetry change consists of more than one component, and there need not be a simple linear relationship between them. One possibility is that a tetragonal form has a stability field between the stability fields of the cubic and orthorhombic forms. The equivalent transition in CaTi03 perovskite includes this intermediate form existing over a temperature interval of 100 - 150 K, and the lattice parameters show a remarkably similar pattern of evolution (Redfern, 1996). A full thermodynamic description of the transition requires a Landau expansion in a multicomponent order parameter with coupling between the strains and orthogonal combinations of the order parameter components. Zhao et al. (1993b) provided some evidence that coupling terms which are higher order that λeQ2 might also be required. In NaMgFß there is a crossover at -700 - 800 K between temperature intervals in which the order-parameter evolution is significantly different.

4.5 Materials with higher-order couplings For the materials discussed so far, with the exception of albite, simple relationships between the driving order parameter and components of the spontaneous strain arise because higher-order coupling terms in equation 18 turn out to be small relative to the lowest-order coupling term allowed by symmetry. For albite a non-linear relationship between e^ and e\ (Fig. 9b) is only a minor aberration from the otherwise standard strain behaviour. Truncation of coupling terms in the Landau expansion is necessarily an empirical process, however, and it is necessary to test any proposed strain relations. Phase transitions in K2Cd2(S04)3, BΓVO4, quartz and BaCe03 illustrate some alternative forms of behaviour.

K2Cd2(SÖ4)3. K2Cd2(SÜ4)3 has the langbeinite structure, which consists of individual SO4 tetrahedra linked by Cd2+ ions in distorted octahedral coordination with oxygen, and K+ ions in larger irregular polyhedra. At -432 K there is a first-order phase transition from cubic (P2\3) to orthorhombic (P2i2i2i) (Abrahams et al, 1978; Lissalde et al, 1979; Antonenko et al, 1983; Devarajan & Salje, 1984, 1986; Percival et al, 1989; Percival, 1990; Hatch et al, 1990b; Kaminsky, 1996; Guelylah et al, 1996). The change in point group is 23 (cubic) ^ 222 (orthorhombic) and the active representation is E of point group 23. This is a doubly degenerate representation and the order parameter, Q, therefore has two components, q\ and qi, which are not constrained to vary in a linearly dependent manner in the low-symmetry phase. The two basis functions, 2z2 - x2 - y2 and x2 - y2, of the E representation imply two degenerate

symmetry-breaking strains, et and e0. The third symmetry-adapted strain, ea, is associated with the identity representation. Each of these strains can be constructed in the usual way from the individual linear strains, e\, e>i, £3, defined by equations 55 - 57. The Landau expansion for this transition (Dvorak, 1972; Maeda, 1979; Devarajan & Salje, 1984; Hatch et al, 1990b) is:

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+λ, (qiet +q2e0) + λ2(qle0-q2et)+ ... + λaea (q? + q\) + ... l +LC?(el+e?)+ -CZel (95).

DevaΓajan & Salje (1984) showed that the excess entropy varies in a manner consistent with the solution for a first-order transition derived from an expansion with small third-order terms, a

negative fourth-order term and a positive sixth-order term, jciq^ +q2\ . In other words,

(qf + q2) appears to follow the standard solution, from equation 94:

* \ T-T 1 + 1-1 n (96). (<1\ +<è) = |(^o+?2,o } 4 ^tΓ "" ^C /

Here q\# and #2,0 are the values of the order-parameter components in the orthorhombic phase at

the equilibrium transition temperature, and Tc is the value of Tc renormalised by coupling of the symmetry-breaking strains with q\ and q%. + s The relationship ea (« Vs) <* W\ ^at ea and ea>0 may be substituted for (qf + q\\ and f#lo + q2Λ, respectively, in equation 96. Figure 20 shows a previously unpublished set of lattice parameters through the transition, and Fig. 21a shows the best fit of equation 96 to the values of ea extracted from them.

This fit gives T^ - Tc = 9 K (for T^ taken as 432 K), which falls between 5 K from Cp results (Devarajan & Salje, 1984) and 16 K from elastic-constant data (Antonenko et al, 1983).

•4 | 1 1 1 1 | 1 1 1 1 | 1 1 1.1 | 1 1 π 1 1 1 1 1 1 j 1 1 *l 10.30 Γ"3 1090 iT^ 1 1 1 | 1 1 1 1 | 1 1 1 a " • • 4 n -^""""l 10.28 -"•"""^ H -• b Λ^ 1085 ÷×^ H _j O 10.26 - ^.—"•""/! j w 10.24 :-^^^^l ^ 1080 1 0 : .••• J j ^ 10.22 - a. . • • .] H ^^ " ^ 1075 "y×' J J ^ •• y * 10.20 • •j ~ j 10.18 Γ C - - " H 1070 7 " • tr J : 10.16 TIXJLLII 1..1..1 IJ..I r U. i . 11 1 1 i-S t-Ll. 1 1 1 1 1 1 1 1 1 1 1 Γ • 1 • 1111 • 1 i 300 350 400 450 500 300 350 400 450 500 T (K) T (K) Fig. 20. Lattice-parameter variations through the P2j3 ^ F1\2{1\ transition in K^Cc^CSO.^. These previously unpublished data are similar to the results of Lissalde et al. (1979). Values of a and b immediately below the transition temperature are perhaps anomalous, but a similar dip in values as the transition point is approached from below was found by Lissalde et al. Large damping of the acoustic waves has also been found immediately below the transition point (Antonenko et al, 1983). The absolute value of TÜ was not tightly constrained but was taken as the midpoint between the highest temperature at which the diffraction data showed the orthorhombic phase and the lowest temperature at which the powder diffraction data showed cubic symmetry. This point was then taken to be 432 K.

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l i i i i I i i i i l i i i i-i i i i i I i i ii 140xl0~6 iii i i i | i i i i 1 i i i i | i i i i | i ij WÜ * « •• >». "H (a) 1 120 (b) 1 100 J] S~\ £ • 3 V 80 ^^_ H ^L J +^ 60 \ J N o Λ 3 >*, 40 m "J 20 i H 0 {_..... * J I • I • 111 • 1111,11111 ll • I i • • • I • • i • 1 1 1 1 1 1 1 1 1 1 1 1 1 W 500 300 350 400 450 500 T (K) T (K) 2 2 Fig. 21. Fits of ea and (e0 + et ) against temperature for the first-order transition in K^CdΛSO^. The curves are solutions to a Landau potential of the form of equation 92 (with b* < 0). (a) For ea: 7^ - Tc = 9 K, e^0 =-0.0029. 2 2 2 2 2 2 1/2 (b) For (e0 + e?): Γ* - Γc* = 8 K, (e0>0 + et,0 )" = 0.0069 (and (qho + tf2)0 ) = 0.45).

The strains et and e0 couple with q\ and #2> and, by analogy with other bilinearly coupled

systems, the relationships e\ <*= e% <* ea might be anticipated. These proportionalities are not observed, however, as seen in Fig. 22a. Returning to equation 95, the conditions for equilibrium

with respect to the strains et and e0 yield (see, also, Hatch et al, 1990b):

> = 0 = λ1ft+λ2ft + Cj eo (97) 3*n

λ q + λ q λ 2 2 x (98) en=-

and

= 0 = λxqx - λ2q2 + C?et (99) 3et

^lffl "" ^2^2 (100).

From equations 98 and 100 it is easy to show that:

(«ä^)-^(rf*«ä) (101).

In other words, the sum of the squares of the symmetry-breaking strains should be linear

functions of [q\ +q2) and, hence, also of eΛ. This is indeed the case within experimental

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uncertainty (Fig. 22b), confirming that bilinear coupling terms in the free-energy expansion are sufficient to explain the evolution of the symmetry-breaking strains. The variation of (e% + ef) should therefore provide a good representation of the variation of (qf +#f) witn temperature,

and the same fitting procedure as used for ea (in equation 97) yields Tix - Tc = 8 K and

e e 45 { l,o+ lo) = 0.00685 (Fig. 21b), which gives (ft*o+02,o) = °- - According to this interpretation, the apparently anomalous variations of composite strains such as e\ - e>i, e\ - e^ ei - e3, and of birefringence, found by Devarajan & Salje (1984) arise as a consequence of the fact that q\ and #2 do not vary linearly with respect to each other in the orthorhombic phase. Higher-order strain coupling does not appear to be implicated in this case, therefore. This conclusion is also consistent with a recent interpretation of the detailed atomic structure changes (Guelylahefα/., 1996).

I • ' ' I ' • • I 140xl0"6 T- | ' ' • 1 ' ' » 1 ' ' ' 1 ' ' U 80x10"° (a) 2 .V (b) • H 120 -j* • 3 60 100 J§ J o /-v βr H o *Γ 80 • ^» j 40 jf 3 -j o 0 40 20 oV Γ y/m H 20 7 -3 0 j -. 1 1 1 I 1 1 1— -2 -6xlO"J p———--6xlO~J ^

2 Fig. 22. (a) Neither e0 (open circles) nor e} (filled circles) are linear functions of eA for the phase transition in 2 2 K2Cd2(S04)3. (b) (e0 + et ) is a linear function of ea within experimental uncertainty, consistent with both being 2 2 proportional to (qi + q2 ).

B1VO4. B1VO4 undergoes a phase transition involving the symmetry change I4\/a (Aim) ^ 121a

(21m) at -522 K, for which the active representation is Bg of point group Aim. Details of the transition have been described in a series of papers by David and co-workers, including David & Glazer (1979), David (1983), David & Wood (1983), which are reviewed briefly here. The basis 2 2 functions associated with Bg are x - y and xy, while the basis functions associated with the identity representation are x2 + y2 and z2. Following David & Wood (1983), the equation describing the strain quadric for the symmetry-breaking strain is e\\x2 - e\\y2 + 2c\2xy = 1, and the total strain may be written as:

^ll.tot ^12,tot 0 ^ll.nsb 0 0 ^ll.sb ^12,sb 0 e21,tot ^22,tot 0 = 0 ell,nsb 0 + e21,sb -*ll,sb 0 (102). 0 0 %3,tot_ 0 0 e33,nsb 0 0 0

In terms of the unit-cell parameters, the four non-zero strain components are obtained by setting

cc0,ß0, Y0,cc,ß = 90° in equations 31-36, to give:

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JL «i.tot = ^ -1 (103) e2Λot = A. i (104)

105 *3,tot = — -1 ( ) *6,tot = 2e12ftot = — cos 7 (106).

Note that the unit cell of the low-symmetry phase is described in the first monoclinic setting (unique axis parallel to crystallographic z-axis, γ* 90°) to preserve a simple relationship with the tetragonal unit cell. The symmetry-breaking and non-symmetry-breaking strain components can be separated as:

*6,sb = *6,tot (107) *3,nsb = *3,tot (108)

^ot-^tot=αsinyzfc Ub 2 2a0

e g _ htot + 2.tot _ a sin γ + fc - 2α0 ^nsb - j " £- (110).

The symmetry-breaking strains should couple linearly with the driving order parameter, ß, and the non-symmetry-breaking strains should couple with Q2 (David, 1983). The expected relationships between strain components are, therefore:

«*(-

Changes in lattice parameters as a function of temperature (Fig. 23a - d; data from David & Wood, 1983) yield the individual strain components shown in Fig. 23e. The linear variations

of £i,nsb> ^3,nsb and Vs with temperature suggest that the transition is second order in character,

consistent with birefringence measurements (Wood & Glazer, 1980). The volume strain at T/Tc = 0.5 is -0.8% and, as described above for the transition in Pb3(P04.)2, is positive (the low- symmetry phase has a slightly larger volume than the high-symmetry phase at the same temperature). The symmetry-breaking strains do not conform to the predicted relationships, 2 however, since elsb and e6 do not vary linearly with V( (Fig. 23f); ei,sbis also not proportional to £6 (Fig- 23g). A macroscopic explanation of these anomalies is that the symmetry-breaking strain components couple with higher-order terms in Q to a significant extent. The terms allowed by symmetry are of the form λ\e\^Q, λ^e^Q and λieγ^Q}, ληe^Q3 in this case. Using v{2 to

provide a measure of Q, the observed variations of e\iS\> and e^ can therefore be described by:

% é>lsb=0.126Vs^-0.922Vs (112) and e6 =0.104VS^ -3.196VS^ (113).

Note that the deviation from eγ^ <× V{2 is actually quite small.

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BiVO4 seems to be a rare example of a system in which high-order coupling terms might be significant. Adopting a macroscopic approach of course begs the question of what microscopic process is responsible for the non-linearities. Somewhat surprisingly, no influence of the anomalous symmetry-breaking strain is seen in the variation of optical birefringence for sections cut normal to the diad of the low-symmetry phase (Wood et ah, 1980b; David & Wood, 1983). David & Wood (1983) suggested that an electronic instability at the Bi3+ ion may be implicated. An alternative strategy for describing the non-linear strain/order parameter relationships could have been to assume only bilinear coupling, but with the coefficients λ\ and λβ explicitly dependent on temperature. Since these bilinear coupling terms cause a renormalisation of the second-order terms in the Landau expansion, this is the same as saying that the coupling itself contributes to the excess entropy for the transition. The two models might provide equally precise descriptions of the experimental data, but a clearer understanding of the overall transition mechanism would be needed to discriminate between them.

Quartz. As most recently reviewed by Heaney (1994), Dolino & Vallade (1994) and Dorogokupets (1995), the ß ^ α transition in quartz occurs via an intermediate incommensurate (IC) phase which is stable over a temperature interval of ~1 - 2 K. The influence of this intermediate phase is only seen in experimental data collected with exceptional care close to the transition point, however (e.g., Dolino et aL, 1992; Vallade et α/., 1992; and references therein). Variations in the macroscopic properties of quartz over a much wider temperature interval can be accounted for, to reasonable precision, by ignoring the intermediate phase. The transition is

treated as being first order with a small discontinuity in Q at a transition temperature, ΓtΓ, which falls within the observed hystersis limits for the α form going directly to the ß form on heating and the ß form going to the α form via the IC structure during cooling. On this basis, the order parameter appears to adhere closely to equation 94, the solution derived from a Landau potential having the form of equation 92 with b < 0 (Grimm & Dorner, 1975; Bachheimer & Dolino, 1975; Banda etaU 1975; Dolino & Bachheimer, 1982). The change in space group is P6422 ^ P3221 (or P6222 ;= P3i21), and the change in point group is 622 ^ 32. The active representation is Bi of point group 622, but this representation has no associated basis functions corresponding to strains. The only non-zero strains are associated with Γidentity (non-symmetry-breaking), and are related by e\ = ^2 ^ £3- Substantial changes in the lattice parameters of α-quartz occur in association with the ß ^ α transition (summarised by Ackermann & Sorrell, 1974; Bachheimer & Dolino, 1975; Ghiorso et al, 1979; Dolino & Bachheimer, 1982; and illustrated in Fig. 24a - c). Linear extrapolations of

Fig. 23. (Facing page) Strain behaviour at the tetragonal ^ monoclinic transition in BiV04. (a - d) Variation of lattice parameters with temperature (data of David & Wood, 1983, kindly provided by I.G. Wood), (e) Evolution of strain parameters calculated using equations 103 - 110. The non-symmetry-breaking strains eimb, £3)nsband Vs vary linearly with temperature, consistent with second-order behaviour. Data points indicated by crosses and labelled with P in parentheses are the room temperature, 1 bar strains calculated from the high-pressure data of Hazen & Mariathasan (1982). The symmetry-breaking strains are essentially the same whether determined by varying P or Γ, whereas the non-symmetry-breaking strains are larger for the high-P phase transition, (f) Relationship between eλ sb, in e6 and Vs. Neither £lsb (open circles) nor e6 (crosses) vary linearly with Vs for the phase transition as a function of m temperature. The scatter at small values of Vs could be due to experimental factors. Data for the high-P transition (e1>sb filled circles, e6 stars) are more scattered; the straight lines are drawn in as guides to the eye. (g) For the high-Γ transition £lsb and £6are not proportional to each other (dots), whereas for the high-P transition they are linearly related within reasonable experimental uncertainty (crosses).

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-J.| 1111 | 1 1 1 1 | 1 1 1 1 | 1 |1 11 11 1 1 | 1 1 | 1 1 IΛ 312 l) 311 jT J

^ 310 r -j << \ 309

308 | l 1 111111111 1

307 H "l 1 1 1 1 11 1 1 II 1 1 1 1 1 1 i1 1 1i 1• i1 i 1 • i 300 400 500 600

1111.111111 j 11111111111111111111111 ii 90.0 : (d) .

89.9

89.8

89.7

89.6 ' I i i i • I i i i i I i i i i 1 i i i i I i i i i I i i iJ

400 500 300 400 500 600 Γ(K) Γ(K)

i [ i i i i | i i 1 1 | 1 1 1 1 j 1 1 1 1 1 1 1 1 • r^- r^l 1 • ' ' M : ×vs(P) (e) | 12xl0"3

Γ e1>sb(P) 10 Γ * -q

••-. 6l,sb α 8 1^6(P) H l_ "^S,-'i •a ::: *6 -•... j CΛ j Jb 6 = ×^nsb(P)"""-- 4 Γ ×^l,nsb(P) •"•-. "•.".• H e : 3,nsb ""•... Vs •:. π - •••• • •• • ••• —j 2 • a 1 I el,nsb —:::::: •-.::::: H 0 •Jill! ll.J_L_l.-J. L_ 1 1 ...J_J l_J I L_l_ 1—L 1 1 1 11 i i i_l_l 1 1 1 1 1 i i i ii 300 350 400 450 500 550 600 Γ(K) II11111111111111111111111111111111111111 J 10x10" 12xl0" E (g) io F

0.00 0.04 0.08 0.12 (vf2

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TΠTΓΠTJTΓΓ i i i i | i i i i i 5.46 XX X -*,"l 5.45 "(b) Co o 5.44 5.43 - •J 5.42 J fy J 5.41

i i i i i i i i i i i i i

(d)1

-30 E-

-40

-50xl0"J 111 * 400 800 1200 400 800 1200 Γ(K) Γ(K)

-20xlO"J

. -10

Fig. 24. Strain behaviour at the ß ^ α transition in quartz (Si02) (from Carpenter et al.y 1998). (a - c) Variations of lattice parameters with temperature; V0, a0 and c0 are derived by extrapolation of the linear fit to data above the transition temperature. Open symbols indicate data obtained by neutron powder diffraction; crosses indicate data obtained by X-ray powder diffraction, (d) Spontaneous strains calculated from the lattice-parameter variations. Note 2 the large volume strain, which varies up to -5%. (e) Neither ex nor e3 vary linearly with ß (for Ttt = 847 K, Tc = 840 K, Q0 - 0.377). Open symbols are neutron-difraction data. Curves represent relationships of the form of equation 114, with A = -0.01656, B = -0.00501 for ex and A = -0.00903, B = -0.00476 for e3. The fits have not been extended beyond Q2 ~ 0.8 because of the effects of order-parameter saturation below -300 K. Filled circles are data derived from single-crystal X-ray diffraction results of Kihara (1990), for comparison.

α0, cÖ and V0 in the usual manner yield the strain variations shown in Fig. 24d, with values of e\, e3 and Vs at T/Tc * 0.5 of -1.5%, -0.9% and -4% respectively. In order to compare the variations of e\ and e$ with variations of the order parameter for the transition, independently determined values of ΓtΓ = 847 K and Tc = 840 K have been substituted into equation 94 (Q0 = 0.377 from the constraint that Q = 1 at 0 K). These values

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may be justified on the basis of second-harmonic optical generation (SHG) data, which yielded

Γtr - Tc = 7.2 K (Bachheimer & Dolino, 1975), a value of Tc « 841 K from a study of the soft mode in ß-quartz (Tezuka et aL, 1991), and a reanalysis of heat-capacity data (summarised in Carpenter et aL, 1998). From Fig. 24e it is evident that the expected linear relationship between the individual strains and Q2 is not observed. Rather, the strain behaviour seems to be consistent with a dependence of the form:

2 4 et=AQ +BQ (114)

where A and B are coefficients which derive from strain/order-parameter coupling of the form Vô2 + Vô4- Standard models of the phase transition involve rotations of SiÜ4 tetrahedra about the <100> axes (Hochli & Scott, 1971; Megaw, 1973; Grimm & Dorner, 1975; and many subsequent authors). The rotation angle 0 is usually considered to be the microscopic order

parameter; it varies between 0 at T > TtT and -16.3° at room temperature. Rotations of rigid tetrahedra through the angle 0 produce changes in the a and c parameters which depend on cos0 (Taylor, 1972; Megaw, 1973). Expanding cos0 gives:

cos0 = l-^2+^4- ... (115).

For Ψ = 0.28 rad (-16°), the fourth-order term is only -1% of the second-order term, so the linear strains, e\ and e$, should scale closely with , whereas exactly the reverse occurs (Lager et aL, 1982; Carpenter et aL, 1998; Fig. 25a). An effective shear strain A(c/a) could be defined as:

Δ(c/α)=W/ \f h (116)

where (c/a)o is the ratio for ß-quartz extrapolated to T < Γtr. This varies approximately linearly with Q2 between -300 and -800 K (Fig. 25b), but is not a properly defined strain in a normal thermodynamic context. By analogy with other systems, the shearing might be expressed as (e\ - e$), which shows a linear variation with Q2 over the same temperature interval (Fig. 25b). In marked contrast (e\ + e^) is clearly non-linear (Fig. 25b). The two strain contributions (e\ - e?) and (e\ + e-$) do not vary independently, however, because they are not strictly orthogonal eigenvectors of the elastic-constant matrix for crystals with point group 622 (Carpenter et aL, 1998). In other words, they do not correspond to distinct basis functions and do not have the

same distinct properties that the symmetry-adapted forms e0, et and ea have in the case of a cubic —> othorhombic transition, say. They suggest, nevertheless, that the transition mechanism involves an important component of shear which almost exactly matches the SHG data (Fig. 25c; Bachheimer & Dolino, 1975), and that the anomalous strain contributions may be

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associated with some additional volume strain. Certainly, the data do not seem to conform to the relatively simple patterns observed in many other materials. Three strain regimes may be present

in the α-quartz field. For 800 < T < Tti, the strains are smaller than expected. At T < 300 K the effects of order-parameter saturation are clearly visible. In the intermediate interval, there is perhaps also a change in trend at -650 K (Fig. 25).

i i i i I i i i ; i ii \ i i i | i i ijj 12x10"' -30xl(T3 I ](b) "25Γ 20 t, " Γ > t- -lo µ 4 w ^i 1 5 C^ ^-.^^ 2 ~ Γ **eO&******^ "3 0 h ...... ••..••.••.„.••.•...... _J 0 TTΓi'Γ'i 400 800 0.0 0.2 0.4 0.6 0.8 1.0 Γ(K) ô2 Fig. 25. (a) The da ratio for α-quartz should be less than that of ß-quartz if the transition is due to rotations of tetrahedra which remain rigid. Open circles indicate neutron data, crosses -6xl0"3 k indicate X-ray data. Data from Kihara (1990) are also shown for comparison (open diamonds). The observed increase with falling temperature indicates that the tetrahedra become progressively deformed with falling temperature, cla appears to behave in a similar manner to other excess properties and can be converted into an effective shear strain, A(c/a), using equation 116. (b) For 0.3 < Q2 < 0.9, A(c/a) (open triangles) 2 scales linearly with ß (for Γtr = 847 K, Tc = 840 K). ex - e3 (open circles) shows a similar linear variation, but eλ + e3 (filled circles) does not. (c) The fit to a Landau solution for a 400 600 1000 first-order transition (solid line) for eλ - e3 between 300 and Γ(K) 800 K gives Ttr -Tc = 8.4 K (for Γtr = 847 K), and is thus very similar to the variation of the SHG parameter of Bachheimer & Dolino (1975).

BaCeÜ3 perovskite. Another material in which higher-order coupling effects appear to arise is the perovskite BaCeO3. It undergoes a cubic (Pm3m) ^ trigonal (R3c) transition at -1173 K (Knight, 1994; Darlington, 1996). The symmetry change is associated with the R point of the Brillouin zone, and the expected form of the strain/order parameter dependence is ei °c Q2. Lattice-parameter variations, expressed in terms of a pseudocubic unit cell for the trigonal phase, are reproduced from Knight (1994) and Darlington (1996) in Fig. 26a, b. For strain calculations, the pseudocubic unit cell was converted into a hexagonal cell using equations 61 and 62. (The real hexagonal cell is actually larger than this, but the smaller dimensions permit easy comparison with the cubic cell of the reference state). Variations of the shear strain (et) and

volume strain (Vs ~ e^ were then calculated using equations 63, 64, 29, 30. In fitting ac (Fig. 26a), only the four highest temperature data points were used because they fall on a well defined linear trend.

The symmetry-adapted shear strain, et, varies linearly with temperature between -700 and -1090 K (Fig. 26c), with an extrapolation to zero at 1173 K, which is also the transition temperature reported by Knight (1994). The appearance is thus of a classical second-order

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2 transition (ß <* Tc-T), but with some modification to the strain/order parameter evolution as Tc is approached since et actually tends to zero at -1120 K. The non-symmetry-breaking strain ea varies in a non-linear manner and falls to zero at -1190 K (Fig. 26c). In combination, the strain data suggest that a volume strain occurs within the stability field of the cubic phase ahead

of a symmetry-breaking transition point at -1120 K. Below -1090 K, ea scales with et according 2 to the relation ea = 1.068et + 98.89et (Fig. 26d), i.e. following the form of equation 114 if et is proportional to Q2. Order-parameter evolution between 1090 and 1120 K is presumably influenced by the fact that some volume strain has already occurred ahead of the symmetry change. Darlington (1996) pointed out that at least part of the macroscopic strain is due to deformation of the CeC% octahedra. Perhaps higher-order strain coupling effects are more generally associated with shearing of polyhedra, as opposed to simple rotations of rigid units, at phase transitions.

_6xio-3 µ- i ""!' "i1 (c)

o< .:- -3K*

; •• >©•©•»•••••••••- 1 8 a I • • • « • • • I I . . . . ll • • . I • • • • I • • • • I • I i . . . 800 1000 1200 800 1000 1200 Γ(K) Γ(K) I I I I I I I j I I I I I I I I I I I I I I I I I I 11 [i 1111111 ii 111111111 -3.0x10° µ-

90.15 t-(bΓ< I \ <^90.10f Ö 90.05 h

90.00 K 11 i i i i l i 800 1000 1200 Γ(K)

Fig. 26. Spontaneous strains associated with the cubic ^ trigonal transition in BaCe03 perovskite. (a, b) Lattice- parameter data from Knight (1994) and Darlington (1996). ac is a linear fit to the four highest temperature data points for the cubic structure, (c) Variations of the symmetry-adapted strains, calculated using equations 61 - 64, 29, 30. The linear fit to the shear strain, et, between -700 and -1090 K extrapolates to zero at 1173 K. Over the same temperature range, ea (= Vs to a good approximation) varies in a non-linear manner; it remains finite between -1120 and -1190 K, where et = 0. (d) Relationships between ea and et: over the temperature interval -700 - 1120 2 K, ea varies as ea = 1.068et+ 98.89et , which is shown as a curve extending to the origin.

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5. Structural phase transitions driven by changing pressure

In principle, minerals which undergo phase transitions in response to changing pressure display lattice-parameter variations that are subject to exactly the same constraints of symmetry, order- parameter coupling, etc., as all the examples given above for temperature as the external variable. The most immediate practical difference can be that the normal compressibility of silicates is non-linear, while their normal thermal expansion is often linear to a good approximation. This means that extra care must be taken to produce a valid extrapolation from the stability field of the high-symmetry phase into the stability field of the low-symmetry phase when defining the reference parameters. Transitions in BiV04 and cristobalite illustrate some of the additional insights which a spontaneous-strain analysis can provide in studies of structural changes at high pressures.

BWO4, The tetragonal ^ monoclinic transition in B1VO4 has a positive volume strain (Fig. 23b). Increasing pressure at room temperature should therefore favour inversion back to the tetragonal structure - exactly as has been observed (Pinczuk et ah, 1979; Wood et aL, 1980b; Hazen & Mariathasan, 1982; Mariathasan etaL, 1986). Hazen & Mariathasan (1982) measured lattice parameters as a function of pressure and the data they collected, using a diamond cell between 1 bar and 53 kbar, are reproduced in Fig. 27a - d. In calculating the spontaneous strains

(equations 103 - 110), linear extrapolations have been used for a0, c0 and V0. Within experimental uncertainty, the non-symmetry-breaking strains (e\tnsb, ^3,nsb>^s) and the squares of the symmetry-breaking strains (efsb, e\) are all linear with pressure (Fig. 27e, f). This is consistent with the second-order character found in spectroscopic and birefringence studies

(Pinczuk et aL, 1979; Wood et aL, 1980b). Extrapolation of efsb to zero gives a reasonably well constrained transition pressure, Pc, of 12.2 kbar; el shows more scatter but still gives a transition pressure of 11.4 kbar. Vs, ^3,nsb and é?i?nsb yieldPc ~ 13.7 kbar for all data between 1 bar and 11.6 kbar, or 12.7 kbar if the 11.6 kbar data point is excluded from the fit. These values are all lower than the value of 14.5 kbar estimated from the raw lattice parameters by Hazen &

Mariathasan. Pinczuk et aL (1979) obtained Pc = 14.0 ± 0.5 kbar, while linear extrapolation to 298 K of Pc as a function of T from the two data points of Wood et aL (1980b) yields Pc = 12.2 2 ±1.2 kbar. Both elsb and e6 are approximately linear functions of v{ (Fig. 23f), as expected if the symmetry-breaking strains couple bilinearly with Q. Also in contrast with the high-

temperature data, e\iSb appears to vary linearly with eβ (Fig. 23g). The data show considerable scatter, but the overall strain behaviour is at least consistent with a classical second-order transition. There are two clear differences between the pressure-driven and temperature-driven transitions. Firstly, the pressure-dependent strain variations do not show evidence of the higher order coupling found in the temperature-dependent strains. Secondly, while the values of the shear strains at room temperature and pressure are indistinguishable in the two sets of data, an £i,nsb> ^3,nsb d Vs derived from the high-pressure data are a factor of -2 greater than the values derived from the high-temperature data. These two observations may be related, in that the high- pressure tetragonal reference state would be considerably denser than the high-temperature reference state if both could be examined at room temperature and pressure. Perhaps the

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structural effects responsible for the strain anomaly at high T are also responsible for the density difference. Analogous differences also occur at the I\ ^ PT transition in anorthite. In this system, the pressure-dependent strain at room P, T (Angel et al., 1989; Angel, 1992; Hackwell & Angel, 1995; Angel, 1996) tends to be larger than the equivalent temperature-dependent strain (Redfern & Salje, 1987; Redfern et al, 1988), and the structures of the high-symmetry phase just above the high-temperature and high-pressure transition points are not identical (Angel, 1988, 1996). Christy & Angel (1995) suggested that this phenomenon may be more general.

•1 i i i i 1 i i i i 1 i i i i i1 i i i 1 i i i i 1 i ij 3p-ΓT-I-Γ-ΓI ITT i i π | i II i ! i II i | i IJ 5.20 90.0 • • H (a)1 * •' • j 5.18 h -il l (d)j 1 o 89.9 o< 5.16 H •

: ê 1 | rs 5.14 - o -J j 89.8 7 ^β 5.12 1^0° H 1 | J 89.7 5.10 " -4-+ *" ,*>*w m M + k^^^ J :+* +t ^^* 5.08 —^-^ i 89.6 π 5.06 H i i i . 1 i i i i 1 i i i i i1 i i i 1. i til iiH ILl.LilljLXJ-t i • . > • 1 , • . 0 10 20 30 40 50 0 10 20 30 40 50 1 i i i i I i i i i I i i i i I i i i i I i i i i I i .| i i i i 1 i i • i 1 i i • • I • • i i 1 i i 1 1 | 1 U J 11.70 14xlO" *k (b)i (e) J

11.65 — •

X • 1111 1 t 10 h o< 11.60 -j ^S H ^ 11.55 H 11.50 |4 >

11.45 11111 11111 r—•••••••••-••• •*—• •1 » • ' • 1 • • •ill! • • * • J I i i i i I i i i . I i i i i I . i i i I i i i i I i 0 10 20 30 40 50 0 1 0 2(3 30 40 50 -1 i i i i 1 i i i i 1 • • i • 1 i i "I'M 310 • 6 . . , . . (c)1 100x10")-' ωj m 305 80 H - ^\" m ii-ΓTTiir r 60 : v0^v • it . 1 i 300 -1 40 20 J 295 °& J 0 •i. i ii 111 11 1 i i • •In ..In l-J-l Γj "in I. 1 i IJ i 1i i i ill! ,,1X1 1 I, ,i i in 10 20 30 40 50 10 20 30 40 50 P (kbar) P (kbar) Fig. 27. Lattice-parameter and strain variations through the tetragonal ^ monoclinic transition in B1VO4 as a function of pressure, (a - d) Data of Hazen & Mariathasan (1982) collected using a diamond-anvil cell at 298 K and pressures of 1 bar - 53 kbar (a open circles, b crosses. Data kindly provided by R.M. Hazen). Linear extrapolations for α0, c0 and V0 are shown, (e) Non-symmetry-breaking strains. Vs (open circles), £1>nsb (crosses) and e3 nsb (filled 2 circles) (°c Q ) vary approximately linearly with P. The straight line fits give Pc « 13.7 kbar, but if the 11.6 kbar point is excluded from the fit a value of Pc ~ 12.7 kbar is obtained, (f) Symmetry-breaking strains calculated using 2 2 2 equations 103-110. Each of elsb (crosses) and e6 (open circles) (<× Q ) are approximately linear functions of 2 pressure, consistent with second-order character. Pc = 12.2 kbar from a linear least-squares fit to data for elsb between 1 bar and 11.6 kbar.

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Cristobalite. Palmer & Finger (1994) have measured lattice parameter variations through a tetragonal ^ monoclinic transition in cristobalite at high-pressure. The transition is associated with a special point on the zone boundary, and all the individual strain components are expected to vary as ei «= ß2. From the unit-cell data it is clear that the transition is first order in character and that the individual parameters do not vary linearly with pressure (Fig. 28a). Note that, in contrast with B1VO4, the low-symmetry phase becomes the stable state on increasing pressure. Also, in calculating the strain components, Palmer & Finger used finite strain formalism for non­ linear extrapolation of the reference states. Their calculated symmetry-breaking and non- symmetry-breaking strains vary in a manner consistent with each being proportional to Q2 (Fig. 28b). The scalar strain (defined in equation 11) then varies in a manner that is entirely consistent with the standard solution for a first-order transition with no odd-order terms permitted in the Landau expansion (Fig. 28c).

Because of their dependence on Q2, volume strains cause a renormalisation of the fourth- order term in Landau expansions. The effect is always to subtract from the value of the unrenormalised coefficient. If it is generally true that volume strains at high-pressure transitions tend to be larger than volume strains at high-temperature transitions, it follows that the former are more likely to show first-order thermodynamic character than the latter. Another example of this might be the IIIc ^ 71 transition in Sr-rich anorthites. As a function of pressure the transition appears to be just first order (McGuinn & Redfern, 1994a), while it is continuous as a function of temperature (Tribaudino et aL, 1993). Within experimental error, the volume strain as a function of temperature is zero (McGuinn & Redfern, 1997).

6. Spontaneous strains determined from room-temperature lattice-parameter data

Strictly speaking, spontaneous strains should be measured under equilibrium conditions. This requires that measurements of lattice parameters be made at a given pressure and temperature from crystals which have been allowed to equilibrate under the same conditions. For transitions which involve only small atomic displacements, the time scale for reequilibration at each new P and T is short. Phase transitions in minerals often involve cation ordering, however, and reequilibration at many different temperatures may not be possible on a sensible experimental time scale. One approach is to anneal a sample for a period of weeks or months in a conventional furnace, quench it, transfer it to a heating X-ray camera and then measure the lattice parameters at the temperature of equilibration. A less laborious procedure is to measure the lattice parameters of the annealed samples at room temperature and use these to calculate the spontaneous strain. Useful empirical calibrations of Q in terms of strain can then be obtained.

Anorthite. Al/Si ordering in anorthite leads to a symmetry change C\ ^ 71. The degree of order produced in crystals annealed at high temperatures tends to be preserved during quenching, even from temperatures close to the melting point (-1557 °C). Changes in lattice parameters observed at room temperature from samples equilibrated at different high temperatures can be formalised in terms of a room-temperature spontaneous strain using the

same arguments and equations as set out earlier. In this case a reference state with parameters α0,

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7.0 | i i i i j i i i i } i i i i j i i i i | i i i i | U| 5C1"1"p-Γrr i TTTTTTTTTp TTTTTTTTΓTTTTJ 6.9 -5 c /2 6.8 m (b) j "2 6.7 -10 o

(a) mi l 6.6 -1Ö V. "j

6.5 -20 I I i I | I 1 1 1 1 1 I 1 I | 1 I I I | II II |] t %lfl

-95 Γi 111 »»••• lllthlllll 111111 ii 11 n in 4.9 K 10 20 30 40 50 60 70 *%.^ a 4.8 β33×l0

4.7 j i i i } i i i i | i i i i 1 i i i l | 1 I 1 l J G 0.10 4.6 0.08 4.5 aJ2 4 0.06 (c) j jU I I | I I II | I I I I | I II I 1 I I I (| 92.5 0.04 H 92.0 .*>*' 0.02 J „ 91.5 A. J O 0.001Hbn i Jj ^ 91.0 ", . , , . , , . . 1 , . , , I.I u_l 1 11 1 1 1 0 12 3 4 5 90.5 ft Pressure [GPa] 90.0 Fig. 28. Zone-boundary tetragonal ^ monoclinic transition in [:[ i u 11111111111111111 u 11 π cristobalite as a function of pressure (from Palmer & Finger, 170 1994). (a) Lattice-parameter variations: individual lattice parameters clearly vary non-linearly with pressure, and extrapolated reference states must therefore also be non-linear. •£ 160 V The monoclinic parameters (subscript m) were scaled to facilitate direct comparison with the tetragonal parameters, (b) i Individual symmetry-breaking strains (such as e ) and non- 75 150 X "•"••-. 13 symmetry-breaking strains (such as e$ vary linearly with each > other, consistent with their both being proportional to Q2. (c) VV4 140 k The total scalar strain, £tot (= £ss as defined according to equation 10) varies in a manner consistent with a classical J-u 2 3 Landau first-order transition. The curve is a fit to an equation Pressure [GPa] of the form of equation 94 with lPtΓ - Pc\ = 0.39 kbar and a discontinuity at PtT from £ss = 0 to £ss,0= 0.0551 (D.C Palmer, pers. comm.).

anc s b0, Cç» (Xo> ß0 * To i provided by crystals with Q « 0 produced from glass of anorthite composition. Both the high- and low-symmetry forms are triclinic and the room-temperature

scalar strain, £sS, has been calculated using equations 31-36 and 11 in the usual way. The 2 expected relationship for a zone-boundary transition, £^s <* ß , has been confirmed by comparing the empirical strains with values of Q derived from determinations of crystallographic site occupancies and with the intensities of superlattice reflections (Fig. 29a, b, c; from Carpenter et al., 1990). Orientations of the three principal strains, 8\, 82 and 83, with

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respect to the crystallographic axes have also been calculated. As shown on the stereogram in Fig. 29d, the strain ellipsoid rotates slightly with changing Q. £\ and £3 are approximately equal in magnitude but opposite in sign, while £2 is close to zero. This is the same as saying that the

volume strain is small, and, indeed, the maximum observed value of Vs, corresponding to a change in Q from ~0 to -0.91, is only -0.4%. The total strain is therefore almost a pure shear.

Fig. 29. Strain behaviour at the Cl ;=± 71 transition in anorthite as determined from measurements of lattice parameters made at room temperature, (a - c) Comparison of Q (Al/Si ordering) from site-occupancy estimates, total scalar spontaneous strain (£ss) and the intensities of all the relevant superlattice reflections (Σ/b) scaled with 2 respect to the sublattice reflections (Σ/a). The linear relations are consistent with Q <* £ss <× Σ4/Σ4. Crosses, data for Val Pasmeda anorthite; circles, data for Monte Somma anorthite (after Carpenter et aL, 1990). (d) Change in orientation of the principal strains with decreasing Q\ £2 is small relative to E\ and £3, and the deformation is virtually a pure shear in the £^£3 plane (after Carpenter, 1992b).

Once an empirical calibration of this type has been obtained, it can be used directly in thermodynamic or kinetic studies. Fig. 30a shows the equilibrium variation of Q for Al/Si ordering in anorthite (from Carpenter, 1992b). The curve represents a first-order Landau solution

with ΓtΓ = 2595 K, TtT - Tc = 312 K and Q0 = 0.65. Comparison of the spontaneous-strain calibration with NMR data then reveals a degree of short-range Al/Si order in excess of that

expected on the basis of the value of Q alone (Phillips et al.9 1992). Alternatively, Fig. 30b shows the time dependence of the scalar strain in ordering experiments using a highly disordered starting material (from Carpenter, 1991). Over a reasonable time interval, the data are consistent with a rate law of the form Q2 <× in /, which is a characteristic solution of the Ginzburg-Landau rate equation for inhomogeneous systems. Salje et aL (1993b) used the same approach to

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calibrate the Ginzburg-Landau coefficients for Al/Si ordering and disordering in anorthite crystals which remained homogeneous with respect to Q.

I Q u1111111111111111 j 111111111111111 ii ii 11 -u o S i^ α8t(a) 0.61- OJ 0.41- Fig. 30. (a) Equilibrium variation of Q (Al/Si ordering) in anorthite derived from spontaneous-strain data extracted from 0.2t- room-temperature lattice-parameter variations (from Carpenter, 1992b). The

0.0 l-i 11 • 111111111 • 111111111111 • I • • •• 11111111 curve is a classical Landau first-order 1200 1400 1600 1800 solution (equation 94). (b) Kinetic evolution of ß using highly disordered Γ(K) starting material (from Carpenter, 1991). Over a significant interval of time the data I 1 l I -T 1— are consistent with a rate law of the form 2 0.008 j- o» - Ö <× ln(time). 0.007

0.006 ?C 0.005 1" i×^ - 0.004 "* 0.003 * 1400 •c x 1300 βc o 1200 βc 0.002 • 1100 •c 0.001

0 i 1 1 *_i_ 1 1 1 L ... i t -4.0 -3.0 -2.0 -1.0 0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 hit (hrs)

Staurolite. Staurolite provides another example of the relationship between cation order and spontaneous strains determined at room temperature (see Hawthorne et al., 1993a, b, c). Associated with the Ccmm ^ Cllm transition are small deviations of the ß-angle from 90°, which Hawthorne et al. (1993b) showed to be linear with an order parameter, ß, derived from cation site occupancies. The symmetry-breaking strain for this transition is e$, which, by symmetry, should couple bilinearly with Q. For small values of (ß - 90), e$ is linear with ß and, hence, with Q. This result confirms an important role for cation ordering in the transition and provides a convenient method for characterising the state of order of natural samples.

Feldspar. Spontaneous strains determined from room-temperature lattice-parameter data also provide a convenient means of characterising the influence of displacive phase transitions across mineral solid solutions. The linear temperature dependence of cos2α* (« e|) for albite with no long-range Al/Si order (Fig. 9a), for example, is mirrored by essentially the same variation when K+ is substituted for Na+ (Carpenter, 1988, 1992a). The Cllm ;= CT transition is classically second order whether T or composition is used as the "external" variable. Similarly, substitution of Ca2+ for Sr2+ in strontian anorthite induces the monoclinic —» triclinic transition discussed

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earlier for temperature and pressure as the external variables (Nager et al, 1970; Tribaudino et al, 1993; McGuinn & Redfern, 1994b, 1997; Phillips et al, 1997; Dove & Redfern, 1997). In this case, cos2α is close to being a linear function of composition in the triclinic stability field (Fig. 31a), mirroring the linear variations with temperature described by Tribaudino et al. (1993). As found for albite (Fig. 9b), cosy is a non-linear function of cosα* (Fig. 31b), suggesting that higher-order coupling of eβ with e$ may be a feature of this type of transition in all feldspars.

• • • • I • • • H • • • n • • • H • • • n • • • •! • • • • i j i 3 1' 1 1 1 ' ' ' 1 ' 1111111111 u 5xl0" P- J • (a)1 -20xl0" (b) . - "" - 4r- -15^1 *8 3 1 8 "1()H 8 2 -5E-I 1 0 0 L i. i • i. i • i • 1 i • i l\ii-i B i B i a 11111 • 11111 • 118111 § 1111 a i 11111 • π : 0 20 40 60 80 100 0 20 60xl0" CaAl2Si2O8 SrAl2Si2O8 mole% Sr-An cosα

Fig. 31. Lattice-parameter variations of the CaAl2Si2O8-SrAl2Si2O8 solid solution at room temperature (data from 2 2 McGuinn & Redfern, 1994b). (a) Cos α* (« e4 <× Q?) is close to being a linear runction_of composition, as would be expected in the case of second-order character for the monoclinic {121c) ^ triclinic (II) transition at 90 mole% Sr-An. (b) Cos7(« e6) is not linear with cosα* (« e4) although both are symmetry-breaking strains which are expected to couple linearly with the driving order parameter.

Composition-dependent strains measured at room temperature are likely to prove illuminating in any solid solution which undergoes a change in symmetry at an intermediate composition. A further example is the cubic ^ tetragonal transition due to Mg/Si ordering in pyrope (Mg3Al2Si30i2) - majorite (Mg4Si4Oi2) garnets. Heinemann et al. (1997) have shown that the symmetry-breaking strain determined by standard X-ray techniques on samples quenched from high pressures and temperatures is consistent with second-order character for the transition.

7. Spontaneous strain and non-convergent ordering processes

In geological materials, so called non-convergent ordering processes are just as important as those associated with discrete phase transitions. The Landau expansion can be modified to treat these systems simply by adding a term linear in Q (Carpenter et al, 1994), which has the effect of ensuring that the fully disordered state (Q = 0) does not have an equilibrium field of stability. Symmetry constraints no longer apply because the ordering does not involve a change in space group. The lowest-order coupling between a generalised strain, e, and the driving order parameter is then invariably of the form λeQ. The form of the full free-energy expansion becomes (from Carpenter et al., 1994):

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2 3 2 G = -hQ + -a(T-Tc)Q +-bQ + ... +λeQ + λ'eQ + ... +^CV (117)

where C° is the bare elastic constant corresponding to the strain of interest. Under equilibrium conditions, the crystal must be at a free-energy minimum with respect to the strain, and, as for almost all the real materials described above, it is likely that only one strain coupling term is needed. Thus the general relationship between the degree of order, Q, and any strain, e, for a non-convergent ordering system is expected to be:

(118).

The geometrical expressions for strain components e\ - e^ still apply, except that the reference state has the same symmetry as the ordered state.

N1AI2O4 spinel. Ordering of Ni and Al between the tetrahedral and octahedral sites of NiAl2Ü4 spinel provides a convenient illustration of the approach. The a lattice parameter of this cubic structure, measured at room temperature, is a linear function of Q from refined site occupancies (Fig. 32; from Carpenter & Salje, 1994a). Fully disordered crystals cannot be prepared, but the

value of a0 is easily obtained by extrapolation to Q = 0. The non-zero strains are simply:

u 1 (119) v.-Z£ (-3C4) (120).

ΓΓΓTTT 11111111 • 1111 M 11 111 • • 1 nil ^1 8.060 Γ H Fig. 32. Variation of the room-temperature cubic lattice parameter, α, with Q for NiAl204 (Carpenter & Salje, 1994a; data from Mocala & Navrotsky, 8.055 H 1989, O'Neill et α/., 1991, and Roelofsen et al, 1992). ß is related to the more commonly used spinel 8.050 7 inversion parameter, x, by Q = 1 - 1.5*. The solid

1111 1 line is a least-squares fit to the data of O'Neill et al.

j (open and filled circles), giving the value of a0 by 8.045 extrapolation to Q = 0. Λ± -0.30 -0.20 -0.10 0.00

Potassium feldspar. Spontaneous strains can be determined in exactly the same way for lower- symmetry systems. Monoclinic potassium feldspar (KAlSisOs), for example, has significant non-convergent ordering of Al and Si between two tetrahedral sites, Tl and T2, that cannot become related by symmetry. The strain components associated with this ordering are given by equations 43 - 48. A 3x3 matrix containing the non-zero components e\, ^2> £3 and e$ can be diagonalised in the usual way to give the principal strains, £\, £2 and £3, and the total scalar strain is still as defined in equation 11. Fig. 33a shows the expected linear relationship between an( e the scalar strain for T1/T2 ordering, £}S,Qt> ^ ^ non-convergent order parameter, Qt, from

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structure refinements. Only data points for triclinic crystals deviate significantly from the linear trend. Two separate ordering processes occur in KAIS13O8. The second gives rise to a symmetry change Cllm ^ C\, and coupling between this and the non-convergent order is sufficiently strong to modify the nature of the transition (Carpenter & Salje, 1994b). Strains associated with the symmetry-breaking order parameter, ß0d, are dominated by the symmetry- breaking components £4 and e& contributions to the non-symmetry-breaking components e\, e>i, £3 and es appear to be smaller than the contributions due to their coupling with ßt- The orientations and magnitudes of principal strains in the triclinic phase associated with each of the two order parameters can therefore be separated with reasonable confidence. They are shown for natural microcline on a stereographic projection in Fig. 33b. The two strain systems are almost exactly orthogonal, with the largest strain due to Qt coinciding with points of almost zero strain due to ßod ordering, and vice versa. There is little, if any, overlap between them, and, therefore, a common strain coupling mechanism for interactions between the two order parameters can be ruled out. An alternative coupling mechanism could be via phonon interactions. This would be strongly temperature dependent, as appears to be required to describe the evolution of K-feldspar in nature (Carpenter & Salje, 1994b).

Fig. 33. Spontaneous strains determined from room-temperature lattice-parameter data for KAlSi308 (from Carpenter & Salje, 1994b). (a) The total scalar strain due to non-convergent ordering, 6ssQt, is a linear function of the non-convergent order parameter, ßt. Only data for triclinic crystals (open circles) diverge from the expected relationship, (b) Orientations of the spontaneous strains in microcline: triangles represent the locations of principal strain axes for ßt ordering and crosses are for ßod ordering. The magnitudes of the principal strains are also specified. Each point of maximum strain forßt ordering occurs at a point of almost zero strain for ß^ ordering, and vice versa.

8. Conclusion

Mineralogists have long made use of the fact that lattice parameters vary systematically with changing degrees of atomic order, or in response to structural phase transitions. The formal framework of theory set out here allows this understanding to be taken one step further. Lattice parameters are essentially only geometrical properties, whereas spontaneous strains derived from them can be related directly to thermodynamic quantities. The step is therefore from standard X-ray data, obtainable in many laboratories, to quantitative thermodynamic analysis.

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Several practical considerations have emerged. Self-evidently, lattice-parameter measurements should be made to the best possible precision and accuracy that is reasonably feasible. The more accurate the measurements are, the more conclusive will be the analysis of fine details in the spontaneous strains calculated from them. Secondly, because of the need to extrapolate reference-state parameters, it is advisable always to continue collecting data beyond a transition point, well into the stability field of the high-symmetry phase. Much of the existing data in the literature are inadequate in this respect. Finally, every calculated strain component should be compared with every other component, on a routine basis, to test for internal consistency and for conformity to expected symmetry relations. Further comparison with an independent measure of the order-parameter behaviour, from spectroscopy, optical birefringence or heat-capacity studies, say, should then provide additional validation (or refutation) of a proposed thermodynamic model. Observations of spontaneous strains are not sufficient by themselves to quantify fully the thermodynamic changes accompanying a phase transition, but they certainly provide a powerful method for characterising structural evolution.

Acknowledgements: Much of this work was undertaken while MAC and EKHS were in receipt of a research grant from the Natural Environment Research Council (NERC) of Great Britain (GR3/8220), and MAC held a Royal Society Leverhulme Trust Senior Research Fellowship. Some of the equipment for analysis of X-ray diffraction data was funded by grants from NERC (GR9/1285) and The Royal Society (14906) to D.C. Palmer. This support is gratefully acknowledged. I. Wood, S.A.T. Redfern and R.M. Hazen are thanked for providing experimental data and D.C. Palmer for permission to reproduce figures published elsewhere. Editorial handling of as long a paper as this is not straightforward; R.J. Angel and C. Chopin are particularly thanked for their patience in this matter.

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Received 13 May 1997 Accepted in revised form 27 February 1998

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