Order parameters for phase transitions to structures with one-dimensional incommensurate modulations

Transcription

1 Acta rystallographica Section A Foundations of rystallography ISSN Editor: D. Schwarzenbach Order parameters for phase transitions to structures with one-dimensional incommensurate modulations Harold T. Stokes, ranton J. ampbell and Dorian M. Hatch opyright International Union of rystallography Author(s) of this paper may load this reprint on their own web site provided that this cover page is retained. Republication of this article or its storage in electronic databases or the like is not permitted without prior permission in writing from the IUr. ffl Acta ryst. (27). A63, Harold T. Stokes et al. Order parameters for incommensurate phase transitions

2 Acta rystallographica Section A Foundations of rystallography ISSN Received 2 December 26 Accepted 2 May 27 Order parameters for phase transitions to structures with one-dimensional incommensurate modulations Harold T. Stokes,* ranton J. ampbell and Dorian M. Hatch Department of Physics and Astronomy, righam Young University, Provo, Utah 8462, USA. orrespondence stokesh@byu.edu # 27 International Union of rystallography Printed in Singapore all rights reserved Phase transitions that result in incommensurate structural modulations are widely observed in crystalline solids and are relevant to a broad range of physical phenomena in magnetic, electronic, optical and structural materials. While the (3+)-dimensional superspace-group symmetries associated with onedimensional modulations have been tabulated, the order parameters that produce these modulations have not been explored in detail. Here, using grouptheoretical methods, we present a unique and exhaustive enumeration of the isotropy subgroups (and their corresponding order-parameter directions) belonging to irreducible representations of the (3+)-dimensional superspace extensions of the 23 crystallographic space groups at all incommensurate k points. The vast majority of experimentally observed incommensurately modulated structures have order parameters belonging to one of these subgroups.. Introduction Over the years, the study of structural phase transitions in crystalline solids has greatly benefited from group-theoretical methods (radley & racknell, 972 irman, 978 Toledano & Toledano, 987 Kovalev, 993 Janssen et al., 24 Howard & Stokes, 25). In the largest class of these transitions, we observe the onset of some distortion which lowers the symmetry of a parent phase. In this paper, we use the word distortion in a general sense to represent any physical order parameter that lowers the symmetry of the parent phase. Distortions can involve not only atomic displacements but also site occupation, electron density and magnetic spin, for example. When the symmetries of the parent and distorted phases have a group subgroup relationship, the distortion can be classified as belonging to one or more irreducible representations (IRs) of the space-group symmetry of the parent phase. This classification gives us predictive power. Given one or more IRs of a space group, we can use group-theoretical methods to calculate the possible subgroup symmetries that can arise from distortions belonging to those IRs. The symmetries of these distorted structures are called isotropy subgroups. For a given set of IRs, the complete list of possible isotropy subgroups is finite. Order parameters are vectors in representation space and determine the distortions that arise in a phase transition. The direction of the order parameter determines the symmetry of the distortion and thus determines the isotropy subgroup symmetry. For multidimensional IRs, there is a many-to-one correspondence between directions of order parameters and isotropy subgroup symmetries. For each isotropy subgroup symmetry, there is a continuous range of directions of the order parameter that produce distortions with that symmetry. About 2 years ago, Stokes & Hatch (988) implemented an algorithm for generating isotropy subgroups and their corresponding order-parameter directions and published a complete list of isotropy subgroups for all IRs associated with special k vectors (k points of symmetry) for each of the 23 crystallographic space groups. Subsequently, the ISOTROPY computer program was created (available over the internet, see Stokes & Hatch, 998). ISOTROPY can find the isotropy subgroups for IRs associated with non-special k vectors having rational components (i.e. general commensurate k vectors) and for coupled IRs that appear simultaneously in a transition. All of these isotropy subgroups describe transitions to structures that are commensurate with the parent phase. Perez-Mato et al. (984a,b) and Janssen & Janner (984) demonstrated how to apply group-theoretical methods to phase transitions which produce modulations that are incommensurate with the periodicity of the parent phase. Incommensurate modulations belong to IRs associated with k vectors having irrational components. We have extended the method of Perez-Mato et al. and have developed a general algorithm for generating the isotropy subgroups, and their corresponding order-parameter directions, for such IRs. The method described here is limited to structures with onedimensional incommensurate modulations but may be easily extended to the higher-dimensional cases. We have implemented this algorithm on computer and have tabulated 7799 isotropy subgroups and their corresponding order-parameter directions for the 558 IRs of the (3+)- dimensional superspace extensions of the 23 crystallographic space groups associated with irrational k vectors. This table is Acta ryst. (27). A63, doi:.7/s

3 available on the internet (Stokes et al., 27) and will hereafter be referred to by the name ISO(3+)D. The scope of this table is limited to one-dimensional incommensurate modulations arising from single uncoupled IRs. While the present discussion aims to describe and illustrate the group-theoretical methods used to generate the ISO(3+)D tables, the tables themselves are still useful to those who are not interested in the underlying methods, as illustrated by the examples in x4. 2. ackground We first review some principles of IRs, order parameters, isotropy subgroups and superspace groups essential for understanding our algorithm. 2.. IRs of three-dimensional space groups IRs of a three-dimensional crystallographic space group G are induced from IRs of G k, the little group of k. This is what is meant by the IR being associated with a k vector. G k is a subgroup of G that consists of all symmetry operators of G which contain point operators that take k into itself (modulo a reciprocal-lattice vector K). Following the treatment of radley & racknell (972), we factorize G into left cosets with respect to G k so that G ¼ P i g i G k where g i are coset representatives (reps). We choose g to be the identity operator so that the first coset is simply G k itself. Let DðgÞ denote the N-dimensional matrix onto which an IR maps symmetry operators g of the parent space group G and let D k ðgþ denote the N k -dimensional matrix onto which an IR k maps symmetry operators of G k, the little group of k. Then the matrices DðgÞ are induced from the matrices D k ðgþ: ðþ DðgÞ ij ¼ M ij D k ðg i gg j Þ ð2þ where M ij ¼ ifg i gg j is in G k and zero otherwise. The subscripts i j refer to blocks of N k -dimensional matrices within DðgÞ. Each allowed IR of G k induces an IR of G. Since we are considering physical distortions, we need realvalued matrices. Generally, the matrices obtained from equation (2) are complex. In the case of type- IRs, the matrices DðgÞ are brought to a real form with a similarity transformation, D R ðgþ ¼SDðgÞS. In the case of type-2 and type-3 IRs, a physically irreducible representation is formed from a direct sum of DðgÞ and its complex conjugate which is then brought to real form with a similarity transformation Isotropy subgroups and order parameters An IR induces distortions which can be decomposed into sets f i g which transform like basis functions of that IR, i.e. in representation space, a space-group operator g acting on one element in the set results in a linear combination of elements in the set given by g i ¼ PN j¼ j DðgÞ ji : When we say that the distortion arising in a phase transition belongs to an IR, we mean that the distortion is some linear combination of distortions i that transform like basis functions of that IR: ¼ PN i¼ i i : The coefficients i are components of an N-dimensional vector g called the order-parameter direction. When a space-group operator g acts on the total distortion, we obtain g ¼ PN i¼ i P N j¼ DðgÞ ij j : omparing equations (4) and (5), we see that the operator g will leave the distortion invariant (g ¼ ) if i ¼ PN j¼ DðgÞ ij j : The symmetry group of the distortion will contain all operators g which satisfy this equation. This collection of operators g is the isotropy subgroup. If we diagonalize DðgÞ, we can see from equation (6) that an isotropy subgroup can only contain operators which map onto matrices DðgÞ with at least one eigenvalue equal to. Isotropy subgroups can be generated without reference to order parameters by testing subgroups G of G with the following two rules (irman, 978 Jaric, 98, 982). (I) Subduction rule. The subduction frequency must be nonzero: iðg Þ¼ X ðgþ 6¼ ð7þ jg j g2g where ðgþ is the character of DðgÞ. In practice, the sum is taken over operators g 2 G that are mapped onto distinct matrices DðgÞ, and jg j is the number of those distinct matrices. (II) hain rule. We require that iðg Þ > iðg Þ for all isotropy subgroups G which are supergroups of G. We first find isotropy subgroups using the subduction and chain rules, and then obtain the corresponding order-parameter directions by solving sets of simultaneous equations for the components of g. We obtain N equations like equation (6) for each generator of the isotropy subgroup. The solution to these equations yields relationships between the components of g and thereby restrict the order-parameter direction to a specific invariant subspace of representation space. Any physical distortion from equation (4) with a direction of g thus restricted will possess the symmetry of the corresponding isotropy subgroup. ð3þ ð4þ ð5þ ð6þ 366 Harold T. Stokes et al. Order parameters for incommensurate phase transitions Acta ryst. (27). A63,

4 2.3. Superspace groups In a phase transition to an incommensurately modulated structure, three-dimensional translational symmetry is lost. A translation n of the lattice in the parent phase causes a phase shift k n in the modulation. If the phase shift is irrational, the structure has no translational symmetry in the direction of n, so that its symmetry cannot be described by a crystallographic space group. In such a case, the symmetry of the structure is best described (in the case of one-dimensional modulations) by a (3+)-dimensional superspace group (Janssen et al., 24), where translations along the axis in the fourth dimension (called the t axis) correspond to phase shifts of the modulation. ecause a translation k n along the t axis corresponds to a phase shift which undoes the phase shift caused by the translation n, the combined (3+)-dimensional translation ðn k nþ in superspace is a symmetry operator of the modulated structure. Thus, in superspace, modulated structures still have translational symmetry. Following Janssen et al. (24), we denote an operator in superspace by ~g ¼fðR"Þjðv Þg, where the three-dimensional part frjvg is a point operation R followed by a translation v, and the part in the fourth dimension f"jg is a point operation " (which operates on the t coordinate) followed by a translation along the t axis. [Here, g refers to operators in three-dimensional space, and ~g refers to operators in (3+)- dimensional space.] It is convenient to write the full (3+)- dimensional operator as an affine transformation matrix Að~gÞ which operates on a position vector x, R R 2 R 3 v x y R 2 R 22 R 23 v 3 Að~gÞx ¼ R 3 R 32 R 33 v " A@ We define basis vectors of the superspace lattice, a ¼ða k aþ a 2 ¼ðb k bþ a 3 ¼ðc k cþ a 4 ¼ð Þ z t A : and a position in superspace as x a þ x 2 a 2 þ x 3 a 3 þ x 4 a 4.y changing from x y z t to x x 2 x 3 x 4 coordinates, equation (8) becomes R R 2 R 3 v x R 2 R 22 R 23 v 2 x 2 Að~gÞx ¼ ðþ where and R 3 R 32 R 33 v M M 2 M 3 " A@ M i ¼ "k i þ P j ¼ þ P i k i v i : k j R ji x 3 x 4 A ð8þ ð9þ ðþ ð2þ The requirement that point operations on lattice vectors must result in a lattice vector demands that the values of M i in equation () be integers. Thus, the superspace group may only contain point operators that take k into either (i) k itself (modulo a reciprocal-lattice vector K) in which case " ¼, or (ii) k (modulo K) in which case " ¼. Two superspace groups ~G and ~G 2 are equivalent if there exists a one-to-one correspondence ~g 2i $ ~g i between operators in ~G and ~G 2 such that A 2 ð~g 2i Þ¼S A ð~g i ÞS ð3þ where A and A 2 are affine transformation matrices of the form in equation () and S is an affine similarity transformation matrix, S R S R2 S R3 S v S R2 S R22 S R23 S v2 S ¼ S R3 S R32 S R33 S S M S M2 S M3 S " S A ð4þ such that S R, S M and S " contain only integer elements. The basis vectors a 2i of the lattice in G 2 are given in terms of the basis vectors a i of the lattice in G by a 2i ¼ P4 j¼ a j S ji ð5þ and the position of the origin of G 2 with respect to G is given by ¼ P4 j¼ a j S j5 : 3. Incommensurate modulations ð6þ We next consider the ways we deal with IRs and isotropy subgroups when the k vector is irrational and the isotropy subgroup must have superspace-group symmetry. In order for superspace groups to be subgroups of the parent group, we must first extend the parent group to include translations along the t axis in superspace. Since the modulation amplitude is zero in the parent phase, such translations along the t axis are symmetry operators of the parent phase. Therefore, we define ~G ¼ G E ðþ ð7þ where E ðþ is the one-dimensional Euclidean group of all translations along the t axis. [As in the previous section, G refers to a group of operators in three-dimensional space, and ~G refers to a group of operators in (3+)-dimensional superspace.] 3.. IRs and isotropy subgroups For an incommensurate modulation in the direction of k, the resulting superspace symmetry can only contain point operators that take k into k. In order to generate the Acta ryst. (27). A63, Harold T. Stokes et al. Order parameters for incommensurate phase transitions 367

5 isotropy subgroups of ~G that lead to incommensurate modulations, we find it useful to first identify a suitable subset of ~G that contains only these operators. We will first consider the analogous subset of the three-dimensional space group G, and then extend the results to ~G. In equation (), the first coset is G k which contains all of the operators in G that take k into k itself. Suppose that one of the coset reps g i contains a point operator that takes k into k.we denote this coset rep by gk. (When no such coset rep exists, the following discussion is greatly simplified.) The coset gkg k then contains all of the operators in G which take k into k. We construct a group G k k which contains only the operators in these two cosets: G k k ¼ G k þ gkg: ð8þ IR matrices D k kðgþ for operators g in G k k are easily obtained from equation (2) by truncating the matrices DðgÞ to include only the indices i j for the cosets G k and gkg k. The resulting matrices D k kðgþ are 2N k -dimensional. If g is in G k, the matrix D k kðgþ has the form D k kðgþ ¼ D D 22 ð9þ where D and D 22 are N k -dimensional matrices and is the N k -dimensional zero matrix. In particular, if g is a lattice translation, expði2k nþ D k kðfejngþ ¼ ð2þ expð i2k nþ where is the N k -dimensional identity matrix. If g is in gkg k, the matrix D k kðgþ has the form D k kðfrjvgþ ¼ D 2 D 2 : ð2þ We now extend the above treatment to include translations along the t axis so that equation (8) becomes ~G k k ¼ ~G k þ ~g ~ k G k : ð22þ The IR matrices ~D k kð~gþ for ~g in ~G k k are simply obtained from ~D k k½fðr"þjðv ÞgŠ ¼ ~D k k½fðe Þjð ÞgŠD k kðfrjvgþ ð23þ where, from Perez-Mato et al. (984a,b), ~D k k½fðe Þjð ÞgŠ ¼ expði2þ expð i2þ : ð24þ From these equations, we obtain ~D k k½fðe Þjðn k nþgš ¼ ð25þ as expected. Phase transitions to structures with modulation vector k will have the symmetries of isotropy subgroups of ~G k k. Our task is to find the isotropy subgroups of ~G k k. Since ~G k k contains an infinite number of operators, it is not practical to find the isotropy subgroups by applying the subduction and chain rules to every subgroup of ~G k k. We must somehow limit the number of subgroups to try Translations From equation (25), we see that every IR maps the lattice translations ðn k nþ onto identity matrices. Therefore, these same lattice translations appear in every isotropy subgroup. The collection of all such lattice translations forms a translation group which we denote by ~T. We now determine which translations along the t axis can appear in isotropy subgroups. An operator ~g can be a member of an isotropy subgroup only if at least one of the eigenvalues of ~D k kð~gþ is equal to. Let us denote these operators as being eligible. onsider an operator ~g 2 ~G k. ombining equations (9) and (24), we find that matrix ~D k kð~gþ has the form ~D k k½fðr Þjðv ÞgŠ ¼ D expði2þ D 22 expð i2þ ð26þ where the matrices D and D 22 are the diagonal elements of D k kðfrjvgþ. The eligible operators are therefore those for which expð i2þ is equal to one of the eigenvalues of D or expði2þ is equal to one of the eigenvalues of D 22. This limits us to a finite number of possible values for. Next consider an operator ~g 2 ~g ~ k G k. From equations (2) and (24), we find that the matrix ~D k kð~gþ has the form D ~D k k½fðr Þjðv ÞgŠ ¼ 2 expði2þ D 2 expð i2þ which can be written in the form D S 2 D 2 S ð27þ ð28þ where S ¼ expðiþ : ð29þ expð iþ ecause a similarity transformation does not change the eigenvalues of a matrix, if D k kðfrjvgþ contains at least one eigenvalue equal to, then so will ~D k kðfðr Þjðv ÞgÞ for any value of. Let ~T be the translation group containing operators fðe Þjð Þg for all possible values of. Factorizing ~g ~ k G k into left cosets with respect to ~T, we obtain ~g ~ k G k ¼ P ~g ki T ~ ð3þ i where ~g ki are coset reps. If one operator in a coset is eligible, then all operators in that coset are eligible The group ~H~H We next need to restrict the search for isotropy subgroups to a manageably finite number of possibilities. We first form a set containing every eligible operator in ~G k, and at least one operator from every coset in equation (3) having eligible operators, and then use group multiplication to complete a 368 Harold T. Stokes et al. Order parameters for incommensurate phase transitions Acta ryst. (27). A63,

6 group that we call ~H. ecause the product of two eligible operators is not necessarily eligible, ~H may contain operators which are not eligible. Note that the construction of ~H is not unique. We can choose any operator from the eligible cosets in equation (3). However, we strategically choose them so as to minimize the total number of operators in ~H. We also find that it is always possible to choose operators ~g so that none of the matrices ~D k kð~gþ depend on k. Other possible choices for ~H result in equivalent sets of isotropy subgroups which are related by translations of the origin along the t axis. In the ISO(3+)D tables, we list the generators ~g of ~H along with the matrices ~D k kð~gþ for each of the 558 IRs associated with incommensurate k vectors. The group of distinct matrices ~D k kð~gþ onto which the IR maps the operators ~g of ~H is called the image of ~H. Among these IRs, we find that only 5 inequivalent images of ~H arise. Now consider some isotropy subgroup ~G of ~G. The operators in ~G may be divided into two sets: (i) those contained in ~G k and (ii) those contained in ~g k ~ G k (if any). y definition, every operator in ~G must be eligible. Thus, every operator in set (i) must also be contained in ~H, since ~H contains every eligible operator in ~G k. The operators in set (ii), however, may not be contained in ~H. Here we use the fact that a translation of the origin of ~G along the t axis changes the value of for operators in ~g k ~ G k but not for those in ~G k. We can therefore move the origin of ~G so that at least one of the operators in set (ii) is in ~H. If we do this, then, by group multiplication, all of the operators in set (ii) will be in ~H. In summary, there exists a translation of the origin along the t axis such that every operator in ~G is in ~H. This means that every isotropy subgroup of ~G k k is equivalent, via an origin shift along the t axis, to a subgroup of ~H. As a final step, we factorize ~H into left cosets with respect to the translation group ~T: ~H ¼ P i ~h i ~T: ð3þ The number of cosets is finite. Since every isotropy subgroup will contain every element in ~T, isotropy subgroups will contain only whole cosets from equation (3). Finding the non-equivalent isotropy subgroups of ~G k k is now reduced to finding the non-equivalent sets of cosets from equation (3) which form groups satisfying the subduction and chain rules. In Appendix A, we give a detailed example of generating ~H and its isotropy subgroups Violation of the chain rule This is best illustrated by a simple example. Suppose there are two cosets in equation (3) so that where h is the identity operator and ~H ¼ ~h ~T þ ~h 2 ~T ð32þ ~D k kð~h Þ¼ ~D k kð~h 2 Þ¼ ð33þ ð34þ and every operator in ~T is mapped onto the identity matrix. It would appear that there are two isotropy subgroups, ~G ¼ ~h ~T þ ~h 2 ~T ¼ ~H and ~G 2 ¼ ~h ~T ¼ ~T with subduction frequencies ið ~G Þ¼ and ið ~G 2Þ¼2. oth the subduction and chain rules are satisfied. The order parameter for ~G is g ¼ða Þ, and the order parameter for ~G 2 is g ¼ðb cþ, where a b c are arbitrary constants. Suppose that for ~g ¼fðE Þjð Þg we have ~D k kð~gþ ¼ cos 2 sin 2 : ð35þ sin 2 cos 2 Then ~G ¼ ~g ~H~g is an isotropy subgroup equivalent to ~G with order parameter g ¼ðacos 2 a sin 2Þ. Since can take any value, this order parameter can point in any direction in two-dimensional representation space. Thus, for any order parameter ðb cþ, there exists some value of for which the order parameter of ~G is in the same direction. ecause there does not exist any direction of the order parameter for which the operators in ~G 2 are the only operators that satisfy equation (6), ~G 2 is not really an isotropy subgroup of ~G k k, even though it satisfies the chain rule. In the example above, ~h belongs to ~G k and ~h 2 belongs to ~g ~ k G k. The off-diagonal form of the matrix in equation (2) is diagonal in equation (32) because the similarity transformation that brings ~D k kð~h 2 Þ to real form also diagonalizes it. The assignment of operators to ~G k and ~g ~ k G k can be checked in the following way. The eigenvalues of ~D k kð~gþ ~D k kð~h Þ depend on and thus ~h belongs to ~G k. The eigenvalues of ~D k kð~gþ ~D k kð~h 2 Þ do not depend on and thus ~h 2 belongs to ~g ~ k G k. We add an extension to the chain rule: if ~G contains only operators in ~G k and if there exists an isotropy subgroup ~G which contains all of the operators in ~G plus some operators in ~g ~ k G k, we require that ið ~G Þ > ið ~G Þþ. In the above example, the extended chain rule is not satisfied and ~G 2 is not an isotropy subgroup Identifying superspace groups Once we obtain the operators g in an isotropy subgroup, we need to identify its (3+)-dimensional superspace-group symmetry as being equivalent to one of the 775 such groups that have been tabulated (Janssen et al., 24). Let ~G be the group of operators f~g i g in our isotropy subgroup (in the setting of the parent space group), and let ~G 2 be the group of operators f~g 2i g (in the standard setting of one of the supergroups). [ISO(3+)D contains what we consider to be the standard setting of each superspace group.] Our task is to try to find the matrix S in equation (3) for some one-to-one mapping of operators in ~G onto operators in ~G 2. If successful, we have identified the superspace-group symmetry of the isotropy subgroup. Acta ryst. (27). A63, Harold T. Stokes et al. Order parameters for incommensurate phase transitions 369

7 Table Isotropy subgroups of the (3+)-dimensional superspace extension of space group No. 22 I 43d. For each IR, we give the order-parameter direction g, the superspace-group symmetry of the subgroup, the basis vectors of the lattice of the subgroup in terms of the basis vectors of the lattice of the parent group and the origin of the subgroup with respect to the origin of the parent group. k IR g Subgroup asis vectors Origin ¼ðÞ ða Þ 22. I 42dðÞ ð Þ ð Þ ð Þ ð Þ ð=4 3=8 Þ 2 ða Þ 22. I 42dðÞ ð Þ ð Þ ð 2Þ ð Þ ð=4 3=8 =4Þ 3 4 ða b Þ 43.2 Fdd2ðÞss ð Þ ð Þ ð 2Þ ð Þ ð =4 =4 =2 Þ ða a b bþ 24.2 I2 2 2 ðþs ð Þ ð Þ ð Þ ð Þ ð =4 =4 =4 Þ ða b c dþ 5.3 2ðÞs ð Þ ð Þ ð Þ ð Þ ð =4 =4 =2 Þ ¼ð Þ ða bþ 6. R3cðÞ ð Þ ð Þ ð=2 =2 =2 Þ ð Þ ð Þ 2 2 ða bþ 6. R3cðÞ ð Þ ð Þ ð=2 =2 =2 3Þ ð Þ ð Þ 3 3 ða b Þ 46.2 R3ðÞt ð Þ ð Þ ð=2 =2 =2 Þ ð Þ ð Þ ða b a bþ 9. bðþ ð Þ ð=2 =2 =2 Þ ð Þ ð Þ ð =4 =4 =4 Þ ða b a bþ 9. bðþ ð Þ ð=2 =2 =2 Þ ð Þ ð Þ ð =4 =4 =4 Þ ða b c dþ. PðÞ ð =2 =2 =2 Þ ð=2 =2 =2 Þ ð=2 =2 =2 Þ ð Þ ð Þ ¼ð Þ ða Þ 43.3 F2ddðÞ ð Þ ð Þ ð Þ ð Þ ð =4 Þ 2 ða Þ 43.3 F2ddðÞ ð Þ ð Þ ð 2Þ ð Þ ð =4 Þ D ¼ð=2 =2Þ D D ða b Þ 9. bðþ ð Þ ð=2 =2 =2 Þ ð Þ ð Þ ð =4 =4 =4 Þ ða b a bþ 5.4 2ð 2 Þ ð Þ ð 2 Þ ð Þ ð Þ ð =4 Þ ða b b aþ 9. bðþ ð Þ ð =2 =2 =2 Þ ð Þ ð Þ ð Þ ða b c dþ. PðÞ ð =2 =2 =2 Þ ð=2 =2 =2 Þ ð=2 =2 =2 Þ ð Þ ð Þ G ¼ð Þ G G 2 ða b Þ 9. bðþ ð 2Þ ð =2 =2 =2 Þ ð 2Þ ð Þ ð Þ ða a b bþ 5. 2ðÞ ð Þ ð 2 Þ ð Þ ð Þ ð =4 =4 Þ ða b c dþ. PðÞ ð =2 =2 =2 Þ ð=2 =2 =2 Þ ð=2 =2 =2 Þ ð Þ ð Þ ¼ð Þ ða bþ 9. bðþ ð Þ ð=2 =2 =2 Þ ð Þ ð Þ ð =4 =4 =4 Þ 2 2 ða bþ 9. bðþ ð Þ ð=2 =2 =2 Þ ð Þ ð Þ ð =4 =4 =4 Þ A ¼ð Þ A ða Þ 5. 2ðÞ ð Þ ð 2 Þ ð Þ ð Þ ð =4 Þ GP ¼ð Þ GP GP ða bþ. PðÞ ð =2 =2 =2 Þ ð=2 =2 =2 Þ ð=2 =2 =2 Þ ð Þ ð Þ Our method for finding S is similar to the algorithm used by Hatch & Stokes (985) for identifying the three-dimensional space-group symmetry of isotropy subgroups for commensurate k vectors. Writing equation (3) as SA 2 ð~g 2i Þ¼A ð~g i ÞS and setting S " ¼ (to preserve the right-handedness of the coordinate system), we obtain " 2i ¼ " i ð36þ S R R 2i ¼ R i S R ð37þ S M R 2i þ M 2i ¼ M i S R þ " i S M ð38þ S R v 2i þ S v ¼ R i S v þ v i ð39þ S M v 2i þ 2i þ S ¼ M i S v þ " i S þ i : ð4þ Using these equations, we construct an algorithm for finding the transformation S between two equivalent superspace groups. If such a transformation cannot be found, then the two groups are not equivalent. Step. hoose n generating operators f~g 2i g of ~G 2.Finda mapping of these operators onto n operators f~g i g of ~G and test the one-to-one correspondence ~g 2i $ ~g i. From equation (36), we see that we must choose the elements in ~G such that " i ¼ " 2i. From equation (37), we see that we must also choose the elements in G such that det R i ¼ det R 2i and tr R i ¼ tr R 2i. This means that each pair, R i and R 2i, must be the same physical type of point operator, i.e. twofold rotation, reflection etc. At this point, we specify the translational parts v i i of ~g i only to within an integer. Step 2. Use equation (37) to determine possible integer values for the matrix S R. We use the same algorithm in this step as in Hatch & Stokes (985), and require that det S R ¼ to preserve the right-handedness of the coordinate system. Step 3. Use equation (38) to determine possible integer values for the row matrix S M. Step 4. Use equations (39) and (4) to solve for the column matrix S v and the element S. Since we have only specified v i and i to within an integer, we must solve these equations modulo. We do this by computing the Smith normal form of those equations (Grosse-Kunstleve, 999). This allows us to determine whether or not there are any solutions and, if there are, to find the solutions. Using the Smith normal form is an improvement over the algorithm in Hatch & Stokes (985) IR matrices We obtained the matrices D k ðgþ from tables in racknell et al. (979) (DML), which are extensions of the tables of Miller & Love (967). In some cases, the DML tables contain multiple IRs which belong to the same k-vector type. ecause the results in our ISO(3+)D tables are valid for any choice of parameters, even values outside the first rillouin zone, we list only one k vector of each type. As a result, some of the IRs in DML do not appear in our tables. For example, for space group No. 22 I 43d, k vectors along the F and lines are of the same type. If the line is extended beyond the boundary of the first rillouin zone, it becomes translationally equivalent to the F line. Also, the k vectors in the J, and planes are of the same type. Thus, we do not include the F, J or IRs in our tables. Furthermore, for many space groups, DML omits some k vectors that are present for the holosymmetric space group of the same ravais-lattice type. We include IRs for these k vectors in our tables. For example, for the cubic I ravais 37 Harold T. Stokes et al. Order parameters for incommensurate phase transitions Acta ryst. (27). A63,

8 lattice, the A IRs included for the holosymmetric space group No. 229 Im3m are missing in the DML table of IRs for space group No. 22 I43d, since for I43d the A IR is just a special case of the IR at the general point (GP). We include the A IR in our tables. As we see in Table, in this case, the IR at A generates a different isotropy subgroup from that of the IR at the general point Reliability of the tables We carefully implemented on computer each part of the algorithm and compared the results with particular examples that could be hand-calculated. In fact, parts of the algorithm were developed specifically to address problems discovered in the testing procedure (for example, the violation of the chain rule). Our previous tables of isotropy subgroups (Stokes & Hatch, 988) are virtually error free, and we expect the ISO(3+)D tables to be likewise accurate. The previous tables were printed in a book. The ISO(3+)D tables reside on the internet where we have the possibility to correct errors, if necessary. 4. Examples: using the tables 4.. Isotropy subgroups of I 43d In Table, we extract from ISO(3+)D the isotropy subgroups of the (3+)-dimensional superspace extension of space group No. 22 I 43d. omplex IRs of type 2 or 3 which have been brought to real form are denoted by symbols showing two IRs. For example, 3 4 is the direct sum of 3 and 4 which are complex conjugates of each other. is the direct sum of with itself which is equivalent to its own complex conjugate. The first isotropy subgroup in the table is 22. I 42dðÞ, which arises from the IR associated with the k vector ¼ðÞ. The incommensurate modulation is along the y axis ðþ in the setting of the parent group and along the z axis ð Þin the setting of the superspace group. We see that the third basis vector is ð Þ, which means that the z axis in the superspace-group setting is the y axis in the parent space-group setting. From the ISO(3+)D table of IRs, we find that the generators of ~H are mapped onto the matrices in equations (33) and (34). We therefore obtain the same result as in the example in x3.4: there is one isotropy subgroup, its order-parameter direction is ða Þ, and the operators in the isotropy subgroup include all of the operators in ~H. From the basis vectors and origin, we obtain the affine transformation matrix =4 3=8 S A ð4þ which takes operators in the parent space-group setting into operators in the superspace-group setting via equation (3). Table lists four isotropy subgroups for the IR 3 3. From ISO(3+)D, we find there are two generators of ~H, each mapped onto a four-dimensional matrix. y group multiplication, we obtain 36 operators in ~H. The image of ~H is identical to image D36b listed in Stokes & Hatch (988). This image appears for 6 different IRs associated with rational k vectors (e.g. IR H 2 H 3 of space group 47 P3). In each of those cases, there are also four isotropy subgroups with the same order-parameter directions as in Table K 2 SeO 4 At 3 K, K 2 SeO 4 undergoes a transition from a commensurate phase with crystallographic space group 62 Pmcn to an incommensurate phase with k vector ð =3 Þ and superspace-group symmetry P Pnam ¼ PmcnðÞss (Perez- ss Mato et al., 984a,b ummins, 99), which is seen to be equivalent to 62.2 PmcnðÞs in the ISO(3+)D table of superspace groups. Perez-Mato et al. found that this transition is driven by a soft-phonon mode belonging to IR 2.This transition appears in ISO(3+)D under the three-dimensional 2 IR of space group 62 Pnma with order-parameter direction ða Þ. At 93 K, the k vector locks in at ð =3Þ so that the 2 IR produces space-group symmetry 33 P2 cn, a result that we also obtain using the ISOTROPY program TTF TNQ At 54 K, TTF TNQ undergoes a transition from a commensurate phase, space group 4 P2 =c, to an incommensurate charge-density-wave phase (known as the M III phase), with k vector ð=2þ, :295, and superspacegroup symmetry P2 =c ¼ 4:3 P2 =bð 2 Þ (ak & Janssen, 978 Wang et al., 23). We find transitions with this symmetry in ISO(3+)D under IRs W and W 2 of space group 4 P2 =c. The IRs W and W 2 produce modulations which are physically different but which have the same superspace-group symmetry. With TTF at the origin in P2 =c, W produces a structure where the modulated displacements of the TNQ molecules are all in phase in a given () plane but the TTF molecules are not. W 2 produces a structure where the modulated displacements of the TTF molecules are all in phase in a given () plane but the TNQ molecules are not. It is not clear from experimental data in the literature which of these two IRs is active in the M III phase of TTF TNQ. At 49 K, there is another transition to a triclinic incommensurate M II phase with superspace-group symmetry P P ¼ 2: PðÞ, =4 < < =2, :295,. This symmetry is found in ISO(3+)D under the IR for the general point (GP) of space group 4 P2 =c NaNO 2 At 436 K, NaNO 2 undergoes a transition from a commensurate phase, space group 7 Immm, to an incommensurate phase with k vector ¼ð Þ, :8, and superspacegroup symmetry P I2mm ss ¼ 44:2 Imm2ðÞss (Ziegler, 93 Kucharczyk & Paciorek, 985 Mconnell, 99). From Acta ryst. (27). A63, Harold T. Stokes et al. Order parameters for incommensurate phase transitions 37

9 ISO(3+)D, we see that there are four IRs, all of which result in isotropy subgroups with superspace ravais class mmmiðþ. In order to obtain Imm2ðÞss, we must couple two different IRs. From ISO(3+)D, we find the generators of Imm2ðÞss to be ðx x 2 x 3 x 4 þ =2Þ and ðx x 2 x 3 x 4 Þ. These are operators in ImmmðÞs, the isotropy subgroup for IR 4, so that Imm2ðÞss is a subgroup of ImmmðÞs. [ImmmðÞs is also the isotropy subgroup for IR 3 but with the a and a 2 axes interchanged so that its symmetry relative to the parent is actually ImmmðÞs.] In order to reduce the symmetry from ImmmðÞs to Imm2ðÞss, we need a ferroelectric distortion along the a 3 axis. From Table 4 in Stokes & Hatch (988), we find that the IR of mmm which is associated with the z component of a polar vector is 2. Therefore, the transition from Immm to Imm2ðÞss is accomplished by coupling 4 and 2. APPENDIX A Example of generating ~H~H As an example of generating the group ~H, consider the IR A of space group 7 P222.TheA point is at k ¼ð =2Þ.The little group of k consists of all space-group elements containing the point operators x y z and x y z. We choose the coset representative gk ¼ðx y z þ =2Þ. Every operator in G takes k either into k or into k, so that, in this case, G k k ¼ G. The IR matrices DðgÞ are two-dimensional: Dðx y zþ ¼ Dðx y zþ ¼ Dðx y z þ =2Þ ¼ i i Dðx y z þ =2Þ ¼ i i Dðx þ y zþ ¼ expði2þ expð i2þ Dðx y þ zþ ¼ Dðx y z þ Þ ¼ : ð42þ We also have from equation (24) ~Dðx y z t þ Þ ¼ expði2þ expð i2þ : ð43þ The eigenvalues of Dðx y zþ are and of Dðx y zþ are, so the eligible operators in ~G k are ðx y z tþ, ðx y z tþ and ðx y z t þ =2Þ. The eigenvalues of Dðx y z þ =2Þ are and of Dðx y z þ =2Þ are i. Therefore, all of the operators ðx y z þ =2 t þ Þ are eligible, and none of the operators ðx y z þ =2 t þ Þ are eligible. We generate ~H from the eligible operators, ðx y z tþ, ðx y z tþ, ðx y z t þ =2Þ and ðx y z þ =2 tþ. We arbitrarily choose ¼ for the last generator. Through group multiplication, we obtain an additional four operators in ~H: ðx y z t þ =2Þ, ðx y z þ =2 t þ =2Þ, ðx y z þ =2 tþ and ðx y z þ =2 t þ =2Þ. We can generate ~H with two operators: ðx y z t þ =2Þ and ðx y z þ =2 tþ. These two operators are listed in our ISO(3+)D table of IRs in terms of the superspace lattice: ðx x 2 x 3 x 3 þ x 4 þ =2Þ and ðx x 2 x 3 þ =2 x 4 Þ. The IR A is of type 3 because IR A is equivalent to IR A 2. We obtain real matrices from the physically irreducible representation A A 2 which maps operators g onto matrices D A A 2ðgÞ ¼D A ðgþd A 2ðgÞ. Using equation (23), the generators of ~H are thus mapped onto ~D A A 2 ðx y z t þ =2Þ A ð44þ i ~D A A 2 i ðx y z þ =2 tþ i A : i We next bring these matrices to real form with a transformation S ¼ð=2Þ i i A ð45þ i i resulting in S ~D A A 2 ðx y z t þ =2ÞS A S ~D A A 2 ðx y z þ =2 tþs A : ð46þ Note that the choice of S is not unique. We simply require any transformation S which brings the IR matrices to real form. Our computer algorithm found the matrix S shown above. At this point, we determine that the eight matrices in the image of ~H are reducible to block-diagonal form and contain two copies of the image 8a which we found in our previous work (Stokes & Hatch, 988). We find the transformation S 2 ¼ A ð47þ which brings the matrices to block-diagonal form with exact copies of the image 8a as diagonal elements: 372 Harold T. Stokes et al. Order parameters for incommensurate phase transitions Acta ryst. (27). A63,

10 S 2 S ~D A A 2 ðx y z t þ =2ÞS S 2 ¼ S 2 S ~D A A 2 ðx y z þ =2 tþs S 2 A : ð48þ Again, the choice of S 2 is not unique. Our computer algorithm found the matrix S 2 shown above. We label this image 8a8a since it contains two copies of 8a. Among the 558 IRs in our ISO(3+)D tables, we find only 5 inequivalent images of ~H. Each of these images can be brought to a form that contains exact copies of images found in our previous work (Stokes & Hatch, 988). The transformation S 2 is important since it brings uniformity to our tables. IRs with the same images give rise to isotropy subgroups with the same order-parameter directions. Using the two generators above, we can easily generate the eight matrices onto which the IR maps the operators in ~H and use equation (6) to determine the isotropy subgroups of ~H: ½Š ðx y z tþ ðx y z tþ : g ¼ða a b bþ ½2Š ðx y z tþ ðx y z t þ =2Þ : g ¼ða a b bþ ½3Š ðx y z tþ ðx y z þ =2 tþ : g ¼ð a bþ ð49þ ½4Š ðx y z tþ ðx y z þ =2 t þ =2Þ : g ¼ða b Þ ½5Š ðx y z tþ : g ¼ða b c dþ: Subgroups [] and [2] are equivalent, and subgroups [3] and [4] are equivalent (via a similarity transformation). Therefore, there are three non-equivalent isotropy subgroups for the IR A A 2. We gratefully acknowledge helpful discussions with Ivan Orlov and Vaclav Petricek. This research was supported by an award from Research orporation. References ak, P. & Janssen, T. (978). Phys. Rev., 7, irman, J. L. (978). Group Theoretical Methods in Physics, edited by P. Karmer & A. Reickers, pp New York: Springer. radley,. J. & racknell, A. J. (972). The Mathematical Theory of Symmetry in Solids. London: Oxford University Press. racknell, A. P., Davies,. L., Miller, S.. & Love, W. F. (979). Kronecker Product Tables, Vol.. New York: Plenum. ummins, H. Z. (99). Phys. Rep. 85, Grosse-Kunstleve, R. W. (999). Acta ryst. A55, Hatch, D. M. & Stokes, H. T. (985). Phys. Rev., 3, Howard,. J. & Stokes, H. T. (25). Acta ryst. A6, 93. International Tables for rystallography (23). Vol. D, edited by A. Authier. Dordrecht: Kluwer Academic Publishers. Janssen, T. & Janner, A. (984). Physica (Utrecht), 26A, Janssen, T., Janner, A., Looijenga-Vos, A. & de Wolff, P. M. (24). International Tables for rystallography, Vol., edited by E. Prince, pp Dordrecht: Kluwer Academic Publishers. Jaric, M. V. (98). Phys. Rev., 23, Jaric, M. V. (982). Physica (Utrecht), 4A, Kovalev, O. V. (993). Irreducible Representations of the rystallographic Space Groups: Irreducible Representations, Induced Representations and orepresentations. New York: Gorden and reach. Kucharczyk, D. & Paciorek, W. A. (985). Acta ryst. A4, Mconnell, J. D.. (99). Philos. Trans. R. Soc. London. Ser. A, 334, Miller, S.. & Love, W. F. (967). Tables of Irreducible Representations of Space Group and o-representations of Magnetic Space Groups. oulder: Pruett. Perez-Mato, J. M., Madariaga, G. & Tello, M. J. (984a). Ferroelectrics, 53, Perez-Mato, J. M., Madariaga, G. & Tello, M. J. (984b). Phys. Rev., 3, Stokes, H. T., ampbell,. J. & Hatch, D. M. (27). ISO(3+)D, Stokes, H. T. & Hatch, D. M. (988). Isotropy Subgroups of the 23 rystallographic Space Groups. Singapore: World Scientific. Stokes, H. T. & Hatch, D. M. (998). Isotropy Software Suite, stokes.byu.edu/isotropy.html. Toledano, J.-. & Toledano, P. (987). The Landau Theory of Phase Transitions. Singapore: World Scientific. Wang, Z. Z., Girard, J.., Pasquier,., Jerome, D. & echgaard, K. (23). Phys. Rev., 67, 24 Ziegler, G. E. (93). Phys. Rev. 38, Acta ryst. (27). A63, Harold T. Stokes et al. Order parameters for incommensurate phase transitions 373