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Volume 70 
Part 1 
Pages 49-63  
January 2014  

Received 9 July 2013
Accepted 9 October 2013
Online 17 December 2013

Atomic order in the spinel structure - a group-theoretical analysis

aSouth Russia State Technical University (Novocherkassk Polytechnic Institute), 346400, Novocherkassk, Russia, and bSouth Scientific Center, Russian Academy of Sciences, 344006, Rostov-on-Don, Russia
Correspondence e-mail: valtalanov@mail.ru

Group-theoretical methods of the Landau theory of phase transitions are used to investigate the structures of ordered spinels. The possibility of the existence is determined of 305 phases with different types of order in Wyckoff position 8a (including seven binary and seven ternary cation substructures), 537 phases in Wyckoff position 16d (including eight binary and 11 ternary cation substructures), 595 phases in Wyckoff position 32e (including seven binary and four ternary anion substructures) and 549 phases with simultaneous ordering in Wyckoff positions 8a and 16d (including five substructures with binary order in tetrahedral and octahedral sublattices, two substructures with ternary order in both spinel sublattices, and nine substructures with different combined types of binary and ternary order). Theoretical results and experimental data are compared. Calculated structures of the spread types of ordered low-symmetry spinel modifications are given.

1. Introduction

The spinel structure was discovered independently by Bragg (1915[Bragg, W. H. (1915). Philos. Mag. 30, 305-315.]) and Nishikawa (1915[Nishikawa, S. (1915). Proc. Tokyo Math.-Phys. Soc. 8, 199-209.]) almost a century ago. At present, substances with this structure are of great interest for a wide community of chemists, physicists, mineralogists, metallurgists and specialists in material science. Initially, spinel applications were based on their magnetic and electric properties (Krupichka, 1976[Krupichka, S. (1976). Physics of Ferrites and Related Magnetic Oxides, Vol. 1, p. 355. Moscow: Mir.]; Talanov, 1986a[Talanov, V. M. (1986a). Energetic Crystallochemistry of Multisublattice Crystals. Rostov-on-Don: RSU. ]). Later their other unique physical properties were discovered. It was revealed that among spinels there were superconductors (LiTi2O4, CuRh2S4, CuV2S4, CuRh2Se4) (Johnston et al., 1973[Johnston, D. C., Prakash, H., Zachariasen, W. H. & Viswanathan, R. (1973). Mater. Res. Bull. 8, 777-784.]; Robbins et al., 1967[Robbins, M., Willens, R. H. & Miller, R. C. (1967). Solid State Commun. 5, 933-934.]), crystals with superionic conductivity (Li2MCl4, where M = Mg, V, Mn, Fe, Cd, bromide spinels and others) (Kanno et al., 1981[Kanno, R., Takeda, Y. & Yamamoto, O. (1981). Mater. Res. Bull. 16, 999-10058.]; Lutz et al., 1985[Lutz, H. D., Schmidt, W. & Haeuseler, H. (1985). J. Solid State Chem. 56, 21-25.], 1997[Lutz, H. D., Partik, M., Schneider, M. & Wickel, Ch. (1997). Z. Kristallogr. 212, 418-422.]; Schmidt & Lutz, 1984[Schmidt, W. & Lutz, H. D. (1984). Ber. Bunsenges. Phys. Chem. 88, 720-723.]), numerous materials for electrodes in chemical current sources (Thackeray, 1997[Thackeray, M. M. (1997). Prog. Solid State Chem. 25, 1-71.]; Ezikian et al., 1988[Ezikian, V. I., Ereyskaya, G. P., Talanov, V. M. & Hodarev, O. N. (1988). Electrochimiya, 24, 1599-1604.]; Talanov et al., 2007[Talanov, V. M., Shirokov, V. B., Torgashev, V. I., Berger, G. A. & Burtsev, V. A. (2007). Phys. Chem. Glass, 33, 822-834.]) and multiferroics (CoCr2O4) (Ederer & Spaldin, 2004[Ederer, C. & Spaldin, N. A. (2004). Nat. Mater. 3, 849-851.]; Yamasaki et al., 2006[Yamasaki, Y. Miyasaka, S. Kaneko, Y., He, J.-P., Arima, T. & Tokura, Y. (2006). Phys. Rev. Lett. 96, 207204-207208.]; Torgashev et al., 2012[Torgashev, V. I., Prokhorov, A. S., Komandin, G. A., Zhukova, E. S., Anzin, V. B., Talanov, V. M. R., Bush, A. A., Dressel, M. & Gorshunov, B. P. (2012). Phys. Solid State, C54, 330-339.]; Pollert, 1988[Pollert, E. (1988). React. Solids, 5, 279-291.]; Gorjaga et al., 1990[Gorjaga, A. N., Talanov, V. M. & Borlakov, K. S. (1990). Multiferroic Substances, pp. 79-85. Moscow: Nauka.]). Some spinels are used in the production of ceramic materials, fireproofs and thermoresistant dyes (Krupichka, 1976[Krupichka, S. (1976). Physics of Ferrites and Related Magnetic Oxides, Vol. 1, p. 355. Moscow: Mir.]; Pollert, 1988[Pollert, E. (1988). React. Solids, 5, 279-291.]; Gorjaga et al., 1990[Gorjaga, A. N., Talanov, V. M. & Borlakov, K. S. (1990). Multiferroic Substances, pp. 79-85. Moscow: Nauka.]). Most of these spinel properties depend largely on atomic ordering. Let us explain this idea.

The general chemical formula of the spinels is AB2X4. The spinel structure has the space group [Fd\bar3m]. It represents the closest cubic threefold layer packing of X atoms. The elementary spinel cell consists of eight cations which occupy Wyckoff position 8a, 16 cations in Wyckoff position 16d, and 32 anions X in Wyckoff position 32e. The cation distribution on the non­equivalent crystallographic positions 8a and 16d can be described by the formula (A[lambda]B1-[lambda])8a[A1-[lambda]B1+[lambda]]16dX4, where [lambda] is the so-called degree of inversibility of the spinel structure. The parameter [lambda] depends on the chemical nature of the substance and the conditions of spinel preparation, temperature and pressure. In more complicated spinels containing certain types of cations, it is necessary to introduce more parameters of the [lambda] type to describe the structure.

With changing the temperature (T), the concentration of solid solution (x) or the pressure (P), phase transitions can occur that are accompanied by cation redistribution and reorganization of the spinel structure. Four types of spinel structure reorganizations are possible.

(i) Isostructural transitions. The degree of inversibility of the spinel structure [lambda] changes by jumping, but the crystal symmetry and its structural type do not change. In this case, the structural transformation is connected with cation redistribution among the nonequivalent crystallographic positions 8a and 16d. The order parameter of this isosymmetrical and isostructural transition is [lambda] = [lambda](T, x, P). The isostructural transition theory for spinels was first proposed by Talanov & Bezrukov (1985[Talanov, V. M. & Bezrukov, G. V. (1985). Solid State Commun. 56, 905-908.], 1990[Talanov, V. M. & Bezrukov, G. V. (1990). Solid State Commun. 75, 601-604.]). A general theory of isostructural transitions for different classes of crystals was developed by Talanov & Bezrukov (1986a[Talanov, V. M. & Bezrukov, G. V. (1986a). Phys. Status Solidi A, 96, 475-482.],b[Talanov, V. M. & Bezrukov, G. V. (1986b). Phys. Status Solidi A, 97, 111-119.]) and Talanov et al. (1989[Talanov, V. M., Bezrukov, G. V. & Men, A. N. (1989). Phys. Status Solidi A, 116, 603-613.]).

(ii) Reconstructive transitions. At a reconstructive phase transition fundamental structural reorganization takes place (Toledano & Dmitriev, 1996[Toledano, P. & Dmitriev, V. P. (1996). Reconstructive Phase Transitions: in Crystals and Quasicrystals. Singapore: World Scientific.]). Examples of reconstructive phase transitions are spinel-olivine, spinel-phenacite and other transitions.

(iii) Phase transitions of the second order. In this case, continuous change of the spinel structure and its thermodynamic states, caused by atomic displacements and atomic redistribution on Wyckoff positions, takes place (Landau & Lifshitz, 1980[Landau, L. D. & Lifshitz, E. M. (1980). Statistical Physics. Part 1. Oxford: Pergamon.]; Gufan, 1982[Gufan, Yu. M. (1982). Structural Phase Transitions. Moscow: Nauka.]; Toledano & Toledano, 1987[Toledano, J.-C. & Toledano, P. (1987). The Landau Theory of Phase Transitions. Singapore: World Scientific.]). There are limiting group-subgroup relationships for these transitions.

(iv) Phase transitions of the first order close to the second order. At such structural transitions the thermodynamic state changes by jumping (and therefore they are phase transitions of the first order), but fundamental spinel structural transformation does not occur (Gufan, 1982[Gufan, Yu. M. (1982). Structural Phase Transitions. Moscow: Nauka.]). There are limiting group-subgroup relationships for these transitions too.

In this article we will consider possible types of atom ordering (possible superstructures) in spinel structures. By definition, a a superstructure cannot be formed as a result of an isostructural transition. The remaining three mechanisms can lead to superstructure formation. We will limit our analysis to the third and fourth types of spinel structure reorganization.

The problem of finding possible superstructures is of great scientific and practical interest. Superstructure formation is accompanied by the appearance of new physical properties in a substance. For example, cation ordering is accompanied by new sublattice formation. This enables ferrimagnetism (Krupichka, 1976[Krupichka, S. (1976). Physics of Ferrites and Related Magnetic Oxides, Vol. 1, p. 355. Moscow: Mir.]). An example is the spinel Cu+[Ni1/2+2Mn3/2+4]O4 with cation order in the octahedral positions (Krupichka, 1976[Krupichka, S. (1976). Physics of Ferrites and Related Magnetic Oxides, Vol. 1, p. 355. Moscow: Mir.]). A low-temperature phase transition to an ordered magnetite phase is accompanied by an anomaly of the specific thermal capacity, conductivity and also a change of approximately two orders of magnitude of the magnetic crystallographic anisotropy (Gorjaga et al., 1990[Gorjaga, A. N., Talanov, V. M. & Borlakov, K. S. (1990). Multiferroic Substances, pp. 79-85. Moscow: Nauka.]). Abnormally high superionic conductivity is found in chloride spinels near the temperature of cation ordering (Kanno et al., 1981[Kanno, R., Takeda, Y. & Yamamoto, O. (1981). Mater. Res. Bull. 16, 999-10058.], 1986[Kanno, R., Takeda, Y., Yamamoto, O., Cros, C., Gang, W. & Hagenmuller, P. (1986). J. Electrchem. Soc. 133, 1052-1056.], 1987[Kanno, R., Takeda, Y., Takahashi, A., Yamamoto, O., Suyama, R. & Koizumi, M. J. (1987). J. Solid State Chem. 71, 196-204.]; Lutz et al., 1985[Lutz, H. D., Schmidt, W. & Haeuseler, H. (1985). J. Solid State Chem. 56, 21-25.], 1997[Lutz, H. D., Partik, M., Schneider, M. & Wickel, Ch. (1997). Z. Kristallogr. 212, 418-422.]; Schmidt & Lutz, 1984[Schmidt, W. & Lutz, H. D. (1984). Ber. Bunsenges. Phys. Chem. 88, 720-723.]; Wussow et al., 1989[Wussow, K., Haeuseler, H., Kuske, P., Schmidt, W. & Lutz, H. D. (1989). J. Solid State Chem. 78, 117-125.]).

Enumeration of the possible superstructures for simple lattices has been carried out previously in various model approximations (Bragg-Williams; see, for example, Smirnov, 1966[Smirnov, A. A. (1966). Molecular-Kinetic Theory of Metals. Moscow: Nauka.]), a molecular field (see, for example, Vaks, 1973[Vaks, V. G. (1973). Introduction to the Microscopic Theory of Ferroelectricity. Moscow: Nauka.]), static concentration waves (see, for example, Khachaturyan, 1974[Khachaturyan, A. G. (1974). Theory of Phase Transformations and Structure of Solid Solutions. Moscow: Nauka.]), and also by a simple search of options of `colouring' the positions of a crystal structure (Smirnova, 1956[Smirnova, N. L. (1956). Kristallographiya, 1, 165-170.]).

In contrast to these methods, which are limited to a number of coordination spheres, the type of interparticle interaction (pair, threefold etc.) and also a number of components of solid solution, we use a group-theoretical method of the thermodynamic theory of phase transitions. This method allows us to obtain the list of possible superstructures without applying any models.

The aim of this research is the full enumeration of all possible ordered structures of spinels. A group-theoretical approach has already been used for the study of some types of superstructures in spinels (Talanov, 1989[Talanov, V. M. (1989). Phys. Status Solidi A, 115, 1-4.], 1990a[Talanov, V. M. (1990a). Phys. Status Solidi B, 162, 339-346.],b[Talanov, V. M. (1990b). Phys. Status Solidi B, 162, 61-73.], 1996[Talanov, V. M. (1996). Crystallogr. Rep. 44, 929-946.], 2007[Talanov, V. M. (2007). Phys. Chem. Glass, 33, 852-870.]; Talanov & Chechin, 1990[Talanov, V. M. & Chechin, G. M. (1990). Kristallographiya, 35, 1008-1011.]). In addition to these results, we considered secondary parameters of order and cases of simultaneous ordering in the two cation sublattices of the spinel structure. A peculiarity of the spinel structure is that cations of one type can be distributed between Wyckoff positions 8a and 16d. This means that an ordering description can demand two parameters of order. Such calculations have not been carried out before. We also investigate possible superstructures in the 8a and 16d cation positions induced by irreducible representations of the wavevector [\boldkappa_8] of [Fd\bar3m] (this question has also not been considered before). In this work we will present a full picture of the possible types of atomic order in crystals with spinel structures.

The results of the calculations appear cumbersome, and therefore only part of the results will be given concerning binary and ternary cation and anion superstructures. These results may be of particular interest for experimental research.

2. Method of possible superstructure calculation

The group-theoretical method as a means of studying phase transitions of atom ordering in alloys was first proposed by Lifshitz (1941[Lifshitz, E. M. (1941). J. Exp. Theor. Phys. 41, 255-268.]). This method was developed further by Gufan (1971[Gufan, Yu. M. (1971). Phys. Solid State, 13, 225-231.], 1982[Gufan, Yu. M. (1982). Structural Phase Transitions. Moscow: Nauka.]), Kovalev (1993[Kovalev, O. V. (1993). Representations of Crystallographic Space Groups. Irreducible Representations, Induced Representations and Co-representations, edited by H. T. Stokes & D. M. Hatch, p. 349. London: Taylor and Francis Ltd.]), Chechin (1989[Chechin, G. M. (1989). Comput. Math. Appl. 17, 255-278.]), Sahnenko et al. (1986[Sahnenko, V. P., Talanov, V. M. & Chechin, G. M. (1986). Fiz. Met. Metalloved. 62, 847-856.]) and Vinberg et al. (1974[Vinberg, E. B., Gufan, Yu. M., Sakhnenko, V. P. & Sirotin, Y. I. (1974). Kristallographiya, 19, 21-26.]).

The structure of the low-symmetry ordered phase within Landau's theory can be described by some function [Delta][rho](r), for example, a function of the changing density of an electric charge:

[\Delta \rho (r) = \rho ({\bf r}) - \rho _0({\bf r}) = \textstyle\sum\limits_{i,n}^{} {c_i^{(n)}\varphi _i^{(n)}}.]

Here the index n is the number of the full irreducible representations (irreps) and the index i is the number of their basis functions [varphi]i(n). The coefficients ci(n) are the components of order parameters; they depend on the thermodynamic conditions, in particular, on such variables as the pressure P and the temperature T: ci(n) ~ ci(n)(P, T).

Based on the concept of one critical irrep, a large number of variables describing a crystal state is reduced to a small number of variables, being parameters of the order of the phase transformation. Therefore, the parameter of order selects those degrees of freedom according to which a crystal loses stability. The contribution from a critical irrep in [rho](r) completely defines the symmetry of the low-symmetry (or dissymmetrical GD) phase. However, when researching the GD-phase structure far from the temperature of phase transition, the contribution of noncritical representations can become essential (Sahnenko et al., 1986[Sahnenko, V. P., Talanov, V. M. & Chechin, G. M. (1986). Fiz. Met. Metalloved. 62, 847-856.]). Then the interpretation of some experimental data cannot be carried out only by means of a critical irrep. A method of finding noncritical atomic displacements and ordering is proposed (Sahnenko et al., 1986[Sahnenko, V. P., Talanov, V. M. & Chechin, G. M. (1986). Fiz. Met. Metalloved. 62, 847-856.]).

In considering phase transitions of the `order-disorder' type, it is convenient to use the scalar function [varphi] defined on the given Wyckoff position and characterizing the `colours' of positions appearing as a result of a phase transition. Identical [Delta][rho](r) values have the same Wyckoff positions. Knowing the type of stratification of the Wyckoff positions of the initial phase structure with a symmetry of G0 on the Wyckoff positions in the GD-phase structure, it is possible to find the type of superstructure and the structural formula of the GD phase (if we consider that on each Wyckoff position there is one type of atom).

For finding all possible low-symmetry phases and corresponding order parameters ci, the basic functions [varphi]i of irreps of a group of structure symmetry of the high-symmetry phase, and also the stratification of Wyckoff positions, we used the group-theoretical method which was described in detail in Gufan (1971[Gufan, Yu. M. (1971). Phys. Solid State, 13, 225-231.], 1982[Gufan, Yu. M. (1982). Structural Phase Transitions. Moscow: Nauka.]), Chechin (1989[Chechin, G. M. (1989). Comput. Math. Appl. 17, 255-278.]), Sahnenko et al. (1986[Sahnenko, V. P., Talanov, V. M. & Chechin, G. M. (1986). Fiz. Met. Metalloved. 62, 847-856.]) and Vinberg et al. (1974[Vinberg, E. B., Gufan, Yu. M., Sakhnenko, V. P. & Sirotin, Y. I. (1974). Kristallographiya, 19, 21-26.]). We checked also the separate results obtained by our method by means of the ISOTROPY program (Stokes & Hatch, 2007[Stokes, H. T. & Hatch, D. M. (2007). ISOTROPY. http://stokes.byu.edu/iso/isotropy.html .]; Stokes et al., 2002[Stokes, H. T., Kisi, E. H., Hatch, D. M. & Howard, C. J. (2002). Acta Cryst. B58, 934-938.]).

Calculation of possible ordered crystal phases by group-theoretical methods of the phase transition theory may be made in the following way. In the first step, when limiting by maximum possible primitive cell multiplication, a permutation representation of the given crystal is built. This permutation representation of the symmetry group of the initial high-symmetry phase can be built using probabilities of filling the Wyckoff positions by atoms. The dimension of this representation equals the position number in the extended cell. The extended cell, in its turn, is defined by minimum translations [alpha]j. The translations of the extended cell (aj) are defined from exp (ikLaj) = 1, where kL are vectors of all beams L of stars {k}, entering into the channel of the phase transition. We can carry out the research in two ways.

The first way consists of expanding the permutation representation into irreps. For each irrep of the symmetry group of the initial phase symmetry, all possible phases based on the Landau one critical representation are found.

In the second way the permutation representation is considered. All possible phases are found by group-theoretical methods. The permutation representation is not expanded on the irreps. This method is very cumbersome, but the solution of the task will be complete. This approach is used in our work.

The permutation representation has, as a rule, a large dimension and generates a very great number of low-symmetry phases. In this work we give the results concerning only binary and ternary atom ordering for each Wyckoff position of the spinel structure.

The spinel structure has a face-centred cubic lattice. For this lattice, the first Brillouin zone represents a body-centred lattice and contains four points of high symmetry, namely: k11([Gamma]), k10(X), k19(L) and k8(W) (Kovalev, 1993[Kovalev, O. V. (1993). Representations of Crystallographic Space Groups. Irreducible Representations, Induced Representations and Co-representations, edited by H. T. Stokes & D. M. Hatch, p. 349. London: Taylor and Francis Ltd.]). To these points there are stars of the following wavevectors:

[\eqalign{&{\bf k}_{11} = 0,\, {\bf k}_{10} = 1/2({\bf b}_{1} + {\bf b}_{2}), \,{\bf k}_{9} = 1/2({\bf b}_{1} + {\bf b}_{2} + {\bf b}_{3}),\cr &{\bf k}_{8} = {\bf b}_{1}/4 - {\bf b}_{2}/4 + {\bf b}_{3}/2.\cr}]

The indexing of vectors and irreps is given accordingly (Kovalev, 1993[Kovalev, O. V. (1993). Representations of Crystallographic Space Groups. Irreducible Representations, Induced Representations and Co-representations, edited by H. T. Stokes & D. M. Hatch, p. 349. London: Taylor and Francis Ltd.]). There are ten irreps of the wavevector [\boldkappa]11 (four one-dimensional, two two-dimensional and four three-dimensional); four six-dimensional irreps of the wavevector [\boldkappa]10, four four- and two eight-dimensional irreps of the wavevector [\boldkappa]9, and two 12-dimensional irreps of the wavevector [\boldkappa]8. Thus all-in-all there are 22 irreps.

All irreps of the wavevector [\boldkappa]8 and also two irreps of the wavevector [\boldkappa]10 ([tau]10-3 and [tau]10-4) do not satisfy the Lifshitz criterion, i.e. they induce incommensurate phases. Nevertheless, we included these irreps in our analysis, as among the low-symmetry phases induced by these representations there are also commensurate phases. In the full permutation representation only irreps relating to the wavevectors k11([Gamma]), k10(X), k9(L) and k8(W) are included. When calculating the number of binary and ternary phases only those phases were taken into account that were generated by irreps concerning the above-mentioned wavevectors.

3. Types of cation order in tetrahedral spinel positions

The permutation representation on Wyckoff position 8a of the space group [Fd\bar 3m] generates 305 low-symmetry phases. The composition of the permutation representation is as follows:1

[{\bf k}_{8}(\tau_1) + {\bf k}_{9}(\tau_1 + \tau_4) + {\bf k}_{10}(\tau_3) + {\bf k}_{11}(\tau_4(A_{2u})). \eqno (1)]

The representation k11([tau]1(A1g)) always enters into the permutation representations. We do not give it in equation (1)[link] and in the other equations because it does not lead to a symmetry change and a new type of atom ordering. We do not give the multiplicity of irreps entering into permutation representations on the Wyckoff positions of the space group [Fd\bar3m] because they are unnecessary for defining possible types of ordered phases.

In Table 1[link] we give the binary and ternary ordered phases. The total number of these phases is 14. In the same table stratification of Wyckoff position 8a of the space group [Fd\bar 3m] is presented. As seen from Table 1[link], among these phases there are seven binary (phases 1-6, 12) and seven ternary (phases 7-11, 14) superstructures (Table 1[link]). The types of binary superstructures that are possible are 1:1, 1:3, and the ternary ones are 1:1:2, 1:3:4. In Table 1[link] improper order parameters are also given. They are marked by `sec'.

Table 1
Binary and ternary cation ordering in Wyckoff position 8a of the spinel structure

Designations for order parameters: k8 - [theta], k9 - [eta], k10 - [varphi], k11 - [xi]. The superscript index after the closing parenthesis is the representation number according to Kovalev (1993[Kovalev, O. V. (1993). Representations of Crystallographic Space Groups. Irreducible Representations, Induced Representations and Co-representations, edited by H. T. Stokes & D. M. Hatch, p. 349. London: Taylor and Francis Ltd.]) and V'/V is the change of primitive cell volume as a result of the structural phase transition. The superscript index in the structural formula means the type of Wyckoff position according to International Tables for Crystallography.

No. Order parameters Symbol of space group V'/V Translations of primitive cell of spinel structure Structural formula
1 (0, 0, 0, [varphi], [varphi], 0)3 P4122 (No. 91) P4322 (No. 95) 4 a1+a2+a3, 2a2, 2a3 [\underline { A_{1/2}^{(a)}A_{1/2}^{(b)}} B_{1/2}^{(c)}B_{1/2}^{(c)}B^{(d)}X^{(d)}X^{(d)}X^{(d)}X^{(d)}]
2 ([xi])4 [F{\bar 4}3m] (No. 216) 1 a1, a2, a3 [\underline { A^{(a)}A^{(d)}}B_{4}^{(e)}X_{4}^{(e)}X_{4}^{(e)}]
3 ([eta], 0, 0, 0)4 [R{\overline 3}2/m] (No. 166) 2 a1+a2, a1+a3, 2a1 [\underline { A_{1/2}^{(c)}A_{1/2}^{(c)}} B_{1/4}^{(a)}B_{1/4}^{(b)}B_{3/2}^{(h)}X_{1/2}^{(c)}X_{1/2}^{(c)}X_{3/2}^{(h)}X_{3/2}^{(h)}]
4 ([eta], 0, 0, 0)1 [R{\bar 3}2/m] (No. 166) 2 a1+a2, a1+a3, 2a1 [\underline {A_{1/2}^{(c)}A_{1/2}^{(c)}} {B}_{1/2}^{(c)}B_{3/4}^{(d)}B_{3/4}^{(e)}X_{1/2}^{(c)}X_{1/2}^{(c)}X_{3/2}^{(h)}X_{3/2}^{(h)}]
5 (0, 0, 0, 0, [varphi], [varphi])3 Pcmm (No. 51) 2 a2+a3, a1, 2a2 [\underline {A_{1/2}^{(e)}A_{1/2}^{(f)}} B_{1/2}^{(b)}B_{1/2}^{(c)}{B^{(k)}}{X^{(k)}}{X^{(k)}}{X^{(k)}}{X^{(k)}}]
6 ([varphi], -[varphi], -[varphi], -[varphi], -[varphi], [varphi])3 [R{\bar 3}2/m] (No. 166) 4 a1+a2+a3, 2a2, 2a3 [\underline {A_{1/4}^{(c)}A_{3/4}^{(h)}} B_{1/8}^{(b)}B_{3/8}^{(d)}B_{3/4}^{(f)}B_{3/4}^{(h)}X_{1/4}^{(c)}X_{3/4}^{(h)}X_{3/4}^{(h)}X_{3/4}^{(h)}X_{3/2}^{(i)}]
7 (0, [varphi], 0, 0, 0, 0)3 ([xi])4sec [P{\overline 4} m2] (No. 115) 2 a1+a2, a3, 2a1 [\underline {A_{1/4}^{(a)}A_{1/4}^{(c)}A_{1/2}^{(g)}} {B^{(j)}}{B^{(k)}}{X^{(j)}}{X^{(j)}}{X^{(k)}}{X^{(k)}}]
8 (0, [varphi], 0, [varphi], 0, -[varphi])3 ([xi])4sec [P{\overline 4}3m] (No. 215) 4 a1+a2+a3, 2a2, 2a1 [\underline {A_{1/8}^{(a)}A_{3/8}^{(c)}A_{1/2}^{(e)}} B_{1/2}^{(e)}B_{3/2}^{(i)}X_{1/2}^{(e)}X_{1/2}^{(e)}X_{3/2}^{(i)}X_{3/2}^{(i)}]
9 ([eta], [eta], 0, 0)4 (0, 0, 0, 0, [varphi], -[varphi])3sec Bbmm (No. 63) 4 a2+a3, 2a1, 2a2 [\underline {A_{1/4}^{(c)}A_{1/4}^{(c)}A_{1/2}^{(g)}} B_{1/2}^{(e)}B_{1/2}^{(f)}B_{1/2}^{(g)}B_{1/2}^{(g)}X_{1/2}^{(f)}X_{1/2}^{(f)}X_{1/2}^{(g)}X_{1/2}^{(g)}X_{1/2}^{(g)}X_{1/2}^{(g)}{X^{(h)}}]
10 (0, 0, [eta], [eta])1 (0, 0, 0, 0, [varphi], [varphi])3sec Ccmm (No. 63) 4 a1, 2a2, 2a3 [\underline {A_{1/4}^{(c)}A_{1/4}^{(c)}A_{1/2}^{(g)}} B_{1/4}^{(a)}B_{1/4}^{(b)}B_{1/2}^{(d)}B_{1/2}^{(g)}B_{1/2}^{(g)}]X1/2(f)X1/2(f)X1/2(g)X1/2(g)X1/2(g)X1/2(g)X(h)
11 ([varphi]1, -[varphi]1, [varphi]2, -[varphi]2, [varphi]2, [varphi]2)3 C2/m (No. 12) 4 a1+a2+a3, 2a2, 2a3 [\underline {A_{1/4}^{(i)}A_{1/4}^{(i)}A_{1/2}^{(j)}} B_{1/8}^{(c)}B_{1/8}^{(d)}B_{1/4}^{(e)}B_{1/4}^{(g)}B_{1/4}^{(i)}B_{1/2}^{(j)}B_{1/2}^{(j)}]X1/4(i)X1/4(i)X1/4(i)X1/4(i)X1/2(j)X1/2(j)X1/2(j)X1/2(j)X1/2(j)
12 (0, 0, [theta], [theta], 0, 0, 0, 0, [theta], [theta], 0, 0)1 C2/c (No. 15) 4 a1+a2-a3, 2a2, a1+a3 [\underline {A_{1/2}^{(f)}A_{1/2}^{(f)}} B_{1/4}^{(d)}B_{1/4}^{(c)}B_{1/4}^{(e)}B_{1/4}^{(e)}B_{1/2}^{(f)}B_{1/2}^{(f)}]X1/2(f)X1/2(f)X1/2(f)X1/2(f)X1/2(f)X1/2(f)X1/2(f)X1/2(f)
13 (0, 0, [theta], [theta], 0, 0, 0, 0, 0, 0, 0, 0)1 ([xi])4sec [I{\bar 4}2d] (No. 122) 4 a1+a2-a3, 2a2, a1+a3 [\underline {A_{1/4}^{(a)}A_{1/4}^{(b)}A_{1/2}^{(d)}} {B^{(e)}}{B^{(e)}}{X^{(e)}}{X^{(e)}}{X^{(e)}}{X^{(e)}}]
14 (0, 0, [theta], 0, 0, 0, 0, 0, 0, [theta], 0, 0)1 (0, 0, [varphi], [varphi], 0, 0)3sec C2/m (No. 12) 4 a1+a2-a3, 2a2, a1+a3 [\underline {A_{1/4}^{(i)}A_{1/4}^{(i)}A_{1/2}^{(j)}} B_{1/8}^{(d)}B_{1/8}^{(c)}B_{1/4}^{(e)}B_{1/4}^{(g)}B_{1/4}^{(i)}B_{1/2}^{(j)}B_{1/2}^{(j)}X_{1/4}^{(i)}X_{1/4}^{(i)}]X1/4(i)X1/4(i)X1/2(j)X1/2(j)X1/2(j)X1/2(j)X1/2(j)X1/2(j)

4. Types of cation order in octahedral spinel positions

Cation ordering in octahedral spinel positions has been investigated before (Billet et al., 1967[Billet, Y., Morgenstern-Badarau, I. & Michel, A. (1967). Bull. Soc. Fr. Mineral. Cristallogr. 90, 8-19.]). Simple geometrical consideration of possible superstructures with 1:1 order type in an octahedral sublattice of a spinel led to 12 870 variants. The authors limited their analysis to structures with face-centred cells and as such 198 structures are possible. Limitations of a physical character were introduced. Finally, the authors came to the conclusion that two superstructures with 1:1 order existed in an octahedral spinel sublattice (Billet et al., 1967[Billet, Y., Morgenstern-Badarau, I. & Michel, A. (1967). Bull. Soc. Fr. Mineral. Cristallogr. 90, 8-19.]). We obtained other results. Let us consider this problem in detail.

The permutation representation on Wyckoff position 16d of the space group [Fd\bar3m] induces 537 low-symmetry phases. Among them there are 19 phases with binary and ternary atom ordering. The composition of the permutation representation is as follows:

[{\bf k}_{8}(\tau_1 + \tau_2) + {\bf k}_{9}(\tau_1 + \tau_4 + \tau_5) + {\bf k}_{10}(\tau_1 + \tau_3) + {\bf k}_{11}(\tau_7(F_{2g})). \eqno (2)]

These phases and the stratifications of Wyckoff position 16d of the space group [Fd\bar3m] are given in Table 2[link]. Among these phases there are eight binary (phases 1-6) and 11 ternary (phases 7-15) superstructures (Table 2[link]).

Table 2
Binary and ternary cation ordering in Wyckoff position 16d of the spinel structure

No. Order parameters Symbol of space group V'/V Translations of primitive cell of spinel structure Structural formula
1 ([xi], -[xi], [xi])7 [R{\overline 3}2/m] (No. 166) 1 a1, a2, a3 [A^{(c)}\underline {B_{1/2}^{(b)}B_{3/2}^{(e)}} {X^{(c)}}X_3^{(h)}]
2 ([varphi], 0, [varphi], 0, [varphi], 0)3 [P{\overline 4} 3m] (No. 215) 4 a1+a2+a3, 2a2, 2a3 [A_{1/8}^{(a)}A_{3/8}^{(c)}A_{1/2}^{(e)}\underline {B_{1/2}^{(e)}B_{3/2}^{(i)}}X_{1/2}^{(e)}X_{1/2}^{(e)}X_{3/2}^{(i)}X_{3/2}^{(i)}]
3 (0, [xi], 0)7 Imma (No. 74) 1 a1, a2, a3 [{A^{(e)}}\underline {{B^{(b)}}{B^{(d)}}}X_2^{(h)}X_2^{(i)}]
4 (0, [varphi], 0, [varphi], 0, -[varphi])1 P4332 (No. 212) P4132 (No. 213) 4 a1+a2+a3, 2a2, 2a1 [{A^{(c)}}\underline {B_{1/2}^{(a)}B_{3/2}^{(d)}} {X^{(c)}}X_3^{(e)}]
5 (0, [varphi], 0, 0, 0, 0)3 [P{\overline 4} m2] (No. 115) 2 a1+a3, a2, 2a1 [A_{1/4}^{(a)}A_{1/4}^{(c)}A_{1/2}^{(g)}\underline {{B^{(j)}}{B^{(k)}}} {X^{(j)}}{X^{(j)}}{X^{(k)}}{X^{(k)}}]
6 (0, 0, 0, 0, 0, [varphi])1 P4122 (No. 91) P4322 (No. 95) 2 a2+a3, 2a2, a1 [{A^{(c)}}\underline {{B^{(a)}}{B^{(b)}}} X_2^{(d)}X_2^{(d)}]
7 (0, [varphi], 0, 0, [varphi], 0)1 (0, 0, 0, [varphi], 0, 0)3sec [P{\overline 4}2_1 m] (No. 113) 4 a1+a2+a3, 2a2, 2a1 [A_{1/4}^{(b)}A_{1/4}^{(c)}A_{1/2}^{(e)}\underline {B_{1/2}^{(e)}B_{1/2}^{(e)}B^{(f)}} X_{1/2}^{(e)}X_{1/2}^{(e)}X_{1/2}^{(e)}X_{1/2}^{(e)}{X^{(f)}}{X^{(f)}}]
8 (0, 0, [varphi], [varphi], 0, 0)1 (0, [xi], 0)7sec Pbmn (No. 53) 2 a1+a3, a2, 2a1 [{A^{(h)}}\underline {B_{1/2}^{(b)}B_{1/2}^{(c)}{B^{(g)}}} {X^{(h)}}{X^{(h)}}X_2^{(i)}]
9 (0, 0, [eta], 0)4 ([xi], -[xi], [xi])7sec [R{\overline 3}2/m] (No. 166) 2 a1, a3, 2a2 [A_{1/2}^{(c)}A_{1/2}^{(c)}\underline {B_{1/4}^{(a)}B_{1/4}^{(b)}B_{3/2}^{(h)}} X_{1/2}^{(c)}X_{1/2}^{(c)}X_{3/2}^{(h)}X_{3/2}^{(h)}]
10 (0, 0, [eta], 0)1 ([xi], -[xi], [xi])7sec [R{\overline 3}2/m] (No. 166) 2 a1, a3, 2a2 [A_{1/2}^{(c)}A_{1/2}^{(c)}\underline {B_{1/2}^{(c)}B_{3/4}^{(d)}B_{3/4}^{(e)}} X_{1/2}^{(c)}X_{1/2}^{(c)}X_{3/2}^{(h)}X_{3/2}^{(h)}]
11 ([varphi], 0, 0, 0, 0, -[varphi])3 (0, 0, [varphi], 0, 0, 0)1sec P4122 (No. 91) P4322 (No. 95) 4 a1+a2+a3, 2a2, 2a3 [A_{1/2}^{(a)}A_{1/2}^{(b)}\underline {B_{1/2}^{(c)}B_{1/2}^{(c)}{B^{(d)}}} {X^{(d)}}{X^{(d)}}{X^{(d)}}{X^{(d)}}]
12 (0, [varphi]1, 0, [varphi]2, 0, -[varphi]1)1 P41212 (No. 92) P43212 (No. 96) 4 a1+a2+a3, 2a2, 2a1 [{A^{(b)}}\underline {B_{1/2}^{(a)}B_{1/2}^{(a)}{B^{(b)}}} {X^{(b)}}{X^{(b)}}{X^{(b)}}{X^{(b)}}]
13 ([xi]1, -[xi]1, [xi]2)7 C2/m (No. 12) 1 a1, a2, a3 [{A^{(i)}}\underline {B_{1/2}^{(b)}B_{1/2}^{(d)}{B^{(e)}}} {X^{(i)}}{X^{(i)}}X_2^{(j)}]
14 ([varphi], [varphi], 0, 0, 0, 0)3 (0, 0, [xi])7sec Pcmm (No. 51) 2 a1+a2, a3, 2a1 [A_{1/2}^{(e)}A_{1/2}^{(f)}\underline {B_{1/2}^{(b)}B_{1/2}^{(c)}{B^{(k)}}} {X^{(k)}}{X^{(k)}}{X^{(k)}}{X^{(k)}}]
15 ([varphi]1, 0, [varphi]2, 0, -[varphi]2, 0)3 [P{\overline 4}2m] (No. 111) 4 a1+a2+a3, 2a2, 2a1 [A_{1/8}^{(a)}A_{1/8}^{(d)}A_{1/4}^{(f)}A_{1/2}^{(n)}\underline {B_{1/2}^{(n)}B_{1/2}^{(n)}{B^{(o)}}} X_{1/2}^{(n)}X_{1/2}^{(n)}X_{1/2}^{(n)}X_{1/2}^{(n)}{X^{(o)}}{X^{(o)}}]
16 (0, 0, [theta], [theta], 0, 0, 0, 0, 0, 0, 0, 0)1 [I{\overline 4}2d] (No. 122) 4 a1+a2-a3, 2a2, a1+a3 [A_{1/4}^{(a)}A_{1/4}^{(b)}A_{1/2}^{(d)}\underline {{B^{(e)}}{B^{(e)}}} {X^{(e)}}{X^{(e)}}{X^{(e)}}{X^{(e)}}]
17 (0, 0, 0, 0, 0, 0, [theta], 0, 0, 0, 0, 0)1 ([varphi], 0, 0, 0, 0, 0)3sec [I{\overline 4}2m] (No. 121) 4 a1+a2-a3, 2a2, a1+a3 [A_{1/8}^{(a)}A_{1/8}^{(b)}A_{1/4}^{(d)}A_{1/2}^{(i)}\underline {B_{1/2}^{(i)}B_{1/2}^{(i)}{B^j}} X_{1/2}^{(i)}X_{1/2}^{(i)}X_{1/2}^{(i)}X_{1/2}^{(i)}{X^{(j)}}{X^{(j)}}]
18 (0, [theta], 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)2 (0, [varphi], 0, 0, 0, 0)3sec [I{\overline 4}2m] (No. 121) 4 a1+a2-a3, 2a2, a1+a3 [A_{1/4}^{(c)}A_{1/4}^{(e)}A_{1/2}^{(j)}\underline {B_{1/2}^{(i)}B_{1/2}^{(i)}{B^j}} X_{1/2}^{(i)}X_{1/2}^{(i)}X_{1/2}^{(i)}X_{1/2}^{(i)}{X^{(j)}}{X^{(j)}}]
19 (0, 0, [theta], [theta], 0, 0, 0, 0, 0, 0, 0, 0)2 [I{\overline 4}2d] (No. 122) 4 a1+a2-a3, 2a2, a1+a3 [A_{1/2}^{(d)}A_{1/2}^{(c)}\underline {{B^{(e)}}{B^{(e)}}} {X^{(e)}}{X^{(e)}}{X^{(e)}}{X^{(e)}}]

The following types of binary superstructures are possible: 1:1 and 1:3. Ternary superstructures of types 1:1:6, 2:3:3 and 1:1:2 are possible. Among these superstructures we have four pairs of enantiomorphic ordered spinel modifications.

When comparing equations (1)[link] and (2)[link] it can be stated that there are the same irreps. This means that, on atom ordering on one Wyckoff position (8a or 16d) according to these irreps, forced atom ordering will take place on the other Wyckoff position.

5. Types of anion order in spinels

The permutation representation on Wyckoff position 32e of the space group [Fd\bar3m] generates 595 low-symmetry ordered phases. The composition of the permutation representation is as follows:

[\eqalignno{&{\bf k}_{8}(\tau_1 + \tau_2) + {\bf k}_{9}(\tau_1 + \tau_4 + \tau_5 + \tau_6) + {\bf k}_{10}(\tau_1 + \tau_3 + \tau_2)&\cr &\quad + {\bf k}_{11}(\tau_4(A_{2u}) + \tau_7(F_{2g}) + \tau_{10}(F_{1u})). & (3)\cr}]

There are only 11 binary and ternary ordered phases. These phases and the stratifications of Wyckoff position 32e of the space group [Fd\bar3m] are given in Table 3[link]. As can be seen from Table 3[link], among these phases there are seven binary (phases 1-7) and four ternary (phases 8-11) superstructures. The appearance of each binary and ternary superstructure is connected with one irrep. Possible types of ordering are the following: types of binary superstructures are 1:1 and 1:3, the type of the ternary one is 1:1:2. From Table 3[link] it is seen that in some cases, on atomic ordering in Wyckoff position 32e of the space group [Fd\bar 3m] the forced stratification of Wyckoff positions 8a and 16d takes place.

Table 3
Binary and ternary cation ordering in Wyckoff position 32e of the spinel structure

No. Order parameters Symbol of space group V'/V Translations of primitive cell of spinel structure Structural formula
1 (0, [xi], 0)10 I41md (No. 109) 1 a1, a2, a3 [{A^{(a)}}B_2^{(b)}\underline {X_2^{(b)}X_2^{(b)}}]
2 ([xi])4 [F{\overline 4}3m] (No. 216) 1 a1, a2, a3 [{A^{(a)}}{A^{(d)}}B_4^{(e)}\underline {X_4^{(e)}X_4^{(e)}}]
3 ([xi], -[xi], [xi])7 [R{\overline 3}2/m] (No. 166) 1 a1, a2, a3 [{A^{(c)}}B_{1/2}^{(b)}B_{3/2}^{(e)}\underline {{X^{(c)}}X_3^{(h)}}]
4 (0, [varphi], 0, [varphi], 0, -[varphi])1 P4332 (No. 212) P4132 (No. 213) 4 a1+a2+a3, 2a2, 2a1 [{A^{(c)}}B_{1/2}^{(a)}B_{3/2}^{(d)}\underline {{X^{(c)}}X_3^{(e)}}]
5 (0, 0, [varphi], 0, 0, 0)2 P41212 (No. 92) P43212 (No. 96) 2 a1+a3, a2, 2a1 [{A^{(a)}}B_2^{(b)}\underline {X_2^{(b)}X_2^{(b)}}]
6 (0, 0, [xi])7 Imma (No. 74) 1 a1, a2, a3 [{A^{(e)}}{B^{(b)}}{B^{(d)}}\underline {X_2^{(h)}X_2^{(i)}}]
7 (0, [varphi], 0, 0, 0, 0)1 P4122 (No. 91) P4322 (No. 95) 2 a1+a2, 2a1, a3 [{A^{(c)}}{B^{(a)}}{B^{(b)}}\underline {X_2^{(d)}X_2^{(d)}}]
8 ([xi], -[xi], 0)10 (0, 0, [xi])7sec Ima2 (No. 46) 1 a1, a2, a3 [{A^{(b)}}{B^{(a)}}{B^{(b)}}\underline {{X^{(b)}}{X^{(b)}}X_2^{(c)}}]
9 ([varphi], -[varphi], 0, 0, 0, 0)1 (0, 0, [xi])7sec Pbmn (No. 53) 2 a1+a2, a3, 2a1 [{A^{(h)}}B_{1/2}^{(b)}B_{1/2}^{(c)}{B^{(g)}}\underline {{X^{(h)}}{X^{(h)}}X_2^{(i)}}]
10 ([varphi], [varphi], 0, 0, 0, 0)2 (0, 0, [xi])7sec Pbnm (No. 62) 2 a1+a2, a3, 2a1 [{A^{(c)}}{B^{(c)}}{B^{(b)}}\underline {{X^{(c)}}{X^{(c)}}X_2^{(d)}}]
11 ([xi]1, [xi]2, -[xi]1)7 C2/m (No. 12) 1 a1, a2, a3 [A^{(i)}B_{1/2}^{(b)}B_{1/2}^{(d)}{B^{(e)}}\underline {X^{(i)}X^{(i)}X_2^{(j)}}]

It should be emphasized that among binary superstructures there are three pairs of enantiomorphic modifications of ordered spinels. Irreps of wavevector k8 do not generate binary and ternary superstructures.

6. Atom ordering in two sublattices

The permutation representation on Wyckoff positions 8a and 16d of the space group [Fd\bar3m] gives rise to 549 low-symmetry phases. The composition of the permutation representation is as follows:

[\eqalignno{&{\bf k}_{8}(\tau_1 + \tau_2) + {\bf k}_{9}(\tau_1 + \tau_4 + \tau_5) + {\bf k}_{10}(\tau_1 + \tau_3)&\cr &\quad + {\bf k}_{11}(\tau_4 (A_{2u}) + \tau_7(F_{2g})). & (4)\cr}]

There are only 16 types of ordered phases if we consider only binary and ternary atomic ordering in each Wyckoff position. These phases and the stratifications of Wyckoff positions 8a and 16d of the space group [Fd\bar3m] are given in Table 4[link]. As seen from Table 4[link], among these phases there are five phases with binary order in tetrahedral and octahedral spinel sublattices of two types: (1:1)8a[1:3]16d, (1:1)8a[1:1]16d; six phases with binary and ternary orders of three types: (1:1)8a[1:1:6]16d, (1:1)8a[2:3:3]16d, (1:1)8a[1:1:2]16d; three phases with ternary and binary orders of two types: (1:3:4)8a[1:3]16d, (1:1:2)8a[1:1]16d; and two phases with ternary order in both spinel sublattices of two types: (1:1:2)8a[2:2:1]16d, (1:1:2)8a[1:1:2]16d.

Table 4
Simultaneous binary and ternary ordering in Wyckoff positions 8a and 16d of the spinel structure

No. Order parameters Symbol of space group V'/V Translations of primitive cell of spinel structure Structural formula
1 (0, [varphi], 0, [varphi], 0, -[varphi])3,A,B ([xi])4,Asec [P{\overline 4} 3m] (No. 215) 4 a1+a2+a3, 2a2, 2a1 [\underline {A_{1/8}^{(a)}A_{3/8}^{(c)}A_{1/2}^{(e)}} \underline{\underline {B_{1/2}^{(e)}B_{3/2}^{(i)}}} X_{1/2}^{(e)}X_{1/2}^{(e)}X_{3/2}^{(i)}X_{3/2}^{(i)}]
2 (0, 0, [eta], 0)4,A ([xi], -[xi], [xi])7,B [R{\overline 3}2/m] (No. 166) 2 a1, a3, 2a2 [\underline {A_{1/2}^{(c)}A_{1/2}^{(c)}} \underline{\underline {B_{1/4}^{(a)}B_{1/4}^{(b)}B_{3/2}^{(h)}}} X_{1/2}^{(c)}X_{1/2}^{(c)}X_{3/2}^{(h)}X_{3/2}^{(h)}]
3 (0, 0, [eta], 0)1,A,B ([xi], -[xi], [xi])7,Bsec [R{\overline 3}2/m] (No. 166) 2 a1, a3, 2a2 [\underline {A_{1/2}^{(c)}A_{1/2}^{(c)}} \underline{\underline {B_{1/2}^{(c)}B_{3/4}^{(d)}B_{3/4}^{(e)}}} X_{1/2}^{(c)}X_{1/2}^{(c)}X_{3/2}^{(h)}X_{3/2}^{(h)}]
4 (0, [varphi], 0, 0, 0, 0)3,A,B ([xi])4,Asec [P{\overline 4} m2] (No. 115) 2 a1+a2, a3, 2a1 [\underline {A_{1/4}^{(a)}A_{1/4}^{(c)}A_{1/2}^{(g)}} \underline{\underline {{B^{(j)}}{B^{(k)}}}} {X^{(j)}}{X^{(j)}}{X^{(k)}}{X^{(k)}}]
5 ([varphi], 0, 0, 0, 0, -[varphi])3,A,B (0, 0, [varphi], 0, 0, 0)1,Bsec P4122 (No. 91) P4322 (No. 95) 4 a1+a2+a3, 2a2, 2a3 [\underline {A_{1/2}^{(a)}A_{1/2}^{(b)} } \underline{\underline {B_{1/2}^{(c)}B_{1/2}^{(c)}{B^{(d)}}}} {X^{(d)}}{X^{(d)}}{X^{(d)}}{X^{(d)}}]
6 ([varphi], [varphi], 0, 0, 0, 0)3A,B (0, 0, [xi])7,Bsec Pcmm (No. 51) 2 a2+a3, a1, 2a2 [\underline {A_{1/2}^{(e)}A_{1/2}^{(f)}} \underline{\underline {B_{1/2}^{(b)}B_{1/2}^{(c)}{B^{(k)}}}} {X^{(k)}}{X^{(k)}}{X^{(k)}}{X^{(k)}}]
7 (0, [varphi], 0, 0, [varphi], 0)1,B ([xi])4,A (0, 0, 0, [varphi], 0, 0)3,A,Bsec [P{\overline 4}2_1 m] (No. 113) 4 a1+a2+a3, 2a2, 2a1 [\underline {A_{1/4}^{(b)}A_{1/4}^{(c)}A_{1/2}^{(e)}} \underline{\underline {B_{1/2}^{(e)}B_{1/2}^{(e)}{B^{(f)}} }} X_{1/2}^{(e)}X_{1/2}^{(e)}X_{1/2}^{(e)}X_{1/2}^{(e)}{X^{(f)}}{X^{(f)}}]
8 ([xi])4,A ([xi], -[xi], [xi])7,B R3m (No. 160) 1 a1, a2, a3 [\underline {A_{1/2}^{(a)}A_{1/2}^{(a)}} \underline{\underline {B_{1/2}^{(a)}B_{3/2}^{(b)}}} X_{1/2}^{(a)}X_{1/2}^{(a)}X_{3/2}^{(b)}X_{3/2}^{(b)}]
9 ([xi])4,A (0, [xi], 0)7,B Imm2 (No. 44) 1 a1, a2, a3 [\underline {A_{1/2}^{(a)}A_{1/2}^{(b)}} \underline{\underline {{B^{(d)}}{B^{(c)}}}} {X^{(d)}}{X^{(d)}}{X^{(c)}}{X^{(c)}}]
10 ([xi])4,A (0, [varphi], 0, [varphi], 0, -[varphi])1,B P213 (No. 198) 4 a1+a2+a3, 2a2, 2a1 [\underline {A_{1/2}^{(a)}A_{1/2}^{(a)}} \underline{\underline {B_{1/2}^{(a)}B_{3/2}^{(b)}}} X_{1/2}^{(a)}X_{1/2}^{(a)}X_{3/2}^{(b)}X_{3/2}^{(b)}]
11 ([xi])4,A (0, 0, 0, 0, 0, [varphi])1,B C2221 (No. 20) 2 a2+a3, 2a2, a1 [\underline {A_{1/2}^{(a)}A_{1/2}^{(b)}} \underline{\underline {{B^{(c)}}{B^{(c)}}}} {X^{(c)}}{X^{(c)}}{X^{(c)}}{X^{(c)}}]
12 ([xi])4,A (0, 0, [varphi], [varphi], 0, 0)1,B (0, [xi], 0)7,B sec Pmn21 (No. 31) 2 a1+a3, a2, 2a1 [\underline {A_{1/2}^{(a)}A_{1/2}^{(a)}} \underline{\underline {B_{1/2}^{(a)}B_{1/2}^{(a)}{B^{(b)}}}} X_{1/2}^{(a)}X_{1/2}^{(a)}X_{1/2}^{(a)}X_{1/2}^{(a)}{X^{(b)}}{X^{(b)}}]
13 ([xi])4,A ([xi]1, -[xi]1, [xi]2)7,B Cm (No. 8) 1 a1, a2, a3 [\underline {A_{1/2}^{(a)}A_{1/2}^{(a)}} \underline{\underline {B_{1/2}^{(a)}B_{1/2}^{(a)}{B^{(b)}}}} X_{1/2}^{(a)}X_{1/2}^{(a)}X_{1/2}^{(a)}X_{1/2}^{(a)}{X^{(b)}}{X^{(b)}}]
14 (0, 0, [theta], [theta], 0, 0, 0, 0, 0, 0, 0, 0)1,A,B ([xi])4,A sec [I{\overline 4}2d] (No. 122) 4 a1+a2-a3, 2a2, a1+a3 [\underline {A_{1/4}^{(a)}A_{1/4}^{(b)}A_{1/2}^{(d)}} \underline{\underline {{B^{(e)}}{B^{(e)}}}} {X^{(e)}}{X^{(e)}}{X^{(e)}}{X^{(e)}}]
15 (0, [theta], 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)2,B ([xi])4,A (0, [varphi], 0, 0, 0, 0)3,A,Bsec [I{\overline 4}2m] (No. 121) 4 a1+a2-a3, 2a2, a1+a3 [\underline {A_{1/4}^{(c)}A_{1/4}^{(e)}A_{1/2}^{(i)}} \underline{\underline {B_{1/2}^{(i)}B_{1/2}^{(i)}{B^j}}} X_{1/2}^{(i)}X_{1/2}^{(i)}X_{1/2}^{(i)}X_{1/2}^{(i)}{X^{(j)}}{X^{(j)}}]
16 (0, 0, [theta], [theta], 0, 0, 0, 0, 0, 0, 0, 0)2,B ([xi])4,A [I{\overline 4}2d] (No. 122) 4 a1+a2-a3, 2a2, a1+a3 [\underline {A_{1/2}^{(d)}A_{1/2}^{(c)}} \underline{\underline {{B^{(e)}}{B^{(e)}}}} {X^{(e)}}{X^{(e)}}{X^{(e)}}{X^{(e)}}]

7. Discussion

7.1. Cation order in tetrahedral positions of the spinel structure

The most widespread types of cation ordering in positions 8a are realized in structures with space groups [F{\overline 4}3m] [order parameter ([xi]), k11([tau]4(A2u)] and [R{\overline 3}2/m] [order parameter ([eta], 0, 0, 0), k9([tau]4)].

Ordered spinel structure with space group [F{\overline 4}3m]. Phases with this symmetry are formed as a result of a phase transition of the second order from cubic spinel. For many spinels (MgAl2O4, LiGaCr4O8, CdIn2S4 etc.) a phase transition is really found experimentally (Suzuki & Kumazawa, 1980[Suzuki, I. & Kumazawa, M. (1980). Phys. Chem. Miner. 5, 279-284.]; Yamanaka & Takeuchi, 1983[Yamanaka, T. & Takeuchi, Y. (1983). Z. Kristallogr. 165, 65-78.]; Joubert & Durif, 1966[Joubert, J.-C. & Durif, A. (1966). Bull. Soc. Fr. Mineral. Cristallogr. 89, 26-28.]). But there are also compounds, for example LiXY4O8 (X = Ga, Fe, In; Y = Cr, Rh), that exist in the ordered modification up to the temperature of melting (Joubert & Durif, 1966[Joubert, J.-C. & Durif, A. (1966). Bull. Soc. Fr. Mineral. Cristallogr. 89, 26-28.]).

Irrep k11([tau]4(A2u)) enters into the permutation representation of a spinel on positions 8a and 32e, and enters into the mechanical representation of a spinel on positions 16d and 32e (Gufan, 1971[Gufan, Yu. M. (1971). Phys. Solid State, 13, 225-231.]; Talanov, 1986b[Talanov, V. M. (1986b). J. Struct. Chem. 31, 172-176.]). The formation of a [F{\overline 4}3m] phase is accompanied by ordering of tetrahedral cations (1:1 order type) and anions (1:1 order type), and also by displacements of octahedral cations and anions. It was stated by calculation that the general structural formula of the ordered spinel is A4aA4cB16e4X16e4[X'^{16e}_4] (Talanov, 1986b[Talanov, V. M. (1986b). J. Struct. Chem. 31, 172-176.]). Features of the [F{\overline 4}3m]-phase structure are discussed in Talanov (1986b[Talanov, V. M. (1986b). J. Struct. Chem. 31, 172-176.], 1996[Talanov, V. M. (1996). Crystallogr. Rep. 44, 929-946.], 2004[Talanov, V. M. (2004). Nanoparticles, Nanostructures, Nanocomposites. Topical meeting of the European Ceramics Society, pp. 37-38. St Petersburg: VVM. Co. Ltd.], 2005[Talanov, V. M. (2005). Phys. Chem. Glass, 31, 431-434.]) and Talanov et al. (2008[Talanov, V. M., Ereyskaya, G. P. & Yuzyuk, Y. I. (2008). Introduction to Chemistry and Physics of Nanostructures and Nanostructured Materials, edited by V. M. Talanov. Moscow: Academy of Natural Science.]). In Fig. 1[link], the calculated structure of the ordered [F{\overline 4}3m] phase is shown. Metal clusters are the most interesting feature of this structure. Two neighbouring groups of octahedral cations and anions form the expanded and compressed tetrahedra (Fig. 2[link]). The compressed tetrahedron of B cations can be considered as a metal cluster. Compressed tetrahedra have the linear size (21/2/4)a and are located at a distance (61/2/4)a, where a is the parameter of a cubic elementary cell of a spinel structure.

[Figure 1]
Figure 1
Calculated structure of an ordered spinel (space group [F{\overline 4}3m]). Atomic presentation of the structure (a) and projection along (001) (b) of the ordered spinel structure.
[Figure 2]
Figure 2
Structural mechanism of atomic cluster formation. Displacements of octahedral cations and anions in the adjacent octants of the spinel structure (a); compressed cluster B4 (b), expanded cluster X4 (c), expanded cluster B4 (d) and compressed cluster X4 (e).

Many ordered spinels have a similar structure, for example [lambda]-Li0.5Mn2O4 (Julien et al., 2006[Julien, C. M., Gendron, F., Amdounib, A. & Massot, M. (2006). Mater. Sci. Eng. B, 130, 41-48.]), Li0.5CrxGa2.5-xO4 (x = 1.75) (Arsen et al., 1979[Arsen, J., Lopitaux, J., Drifford, M. & Lenglet, M. (1979). Phys. Status Solidi, A52, K111-K114.]; Arsen & Lenglet, 1980[Arsen, J. & Lenglet, M. (1980). Mater. Res. Bull. 15, 1681-1689.]; Szymczak et al., 1975[Szymczak, H., Wardzynska, M. & Mylnikova, I. E. (1975). J. Phys. C Solid State Phys. 8, 3937-3943.]), Li0.5Fe0.5[Cr2]O4 (Gorter, 1954[Gorter, E. W. (1954). Philips Res. Rep. 9, 403-443.]), LiGaCr4O8 (Yamanaka & Takeuchi, 1983[Yamanaka, T. & Takeuchi, Y. (1983). Z. Kristallogr. 165, 65-78.]), GaV4-xMoxS8 (0 [less-than or equal to] x [less-than or equal to] 4) (Powell et al., 2007[Powell, A. V., McDowall, A., Szkoda, I., Knight, K. S., Kennedy, B. J. & Vogt, T. (2007). Chem. Mater. 19, 5035-5044.]), Cu0.5In0.5Cr2S4 (Sadykov et al., 2001[Sadykov, R. A., Zaritskiy, V. N., Mezot, J. & Fous, F. (2001). Kristallographiya, 46, 28-32.]; Plumier, Sougi & Lecomte, 1977[Plumier, R., Sougi, M. & Lecomte, M. (1977). Phys. Rev. Lett. A, 60, 341-344.]), Cu0.5Fe0.5Cr2S4 (Sadykov et al., 2001[Sadykov, R. A., Zaritskiy, V. N., Mezot, J. & Fous, F. (2001). Kristallographiya, 46, 28-32.]), Ag0.5In0.5Cr2S4 (Plumier, Lecomte et al., 1977[Plumier, R., Lecomte, M., Miedan-Gros, A. & Sougi, M. (1977). Physica B, 86-88, 1360-1362.]), LiXY4O8 (X = Ga, Fe, In; Y = Cr, Rh) (Joubert & Durif, 1966[Joubert, J.-C. & Durif, A. (1966). Bull. Soc. Fr. Mineral. Cristallogr. 89, 26-28.]), CdIn2S4 (Joubert & Durif, 1966[Joubert, J.-C. & Durif, A. (1966). Bull. Soc. Fr. Mineral. Cristallogr. 89, 26-28.]; Cravetskiy et al., 1992[Cravetskiy, I. V., Culuyk, L. L., Strumban, A. E., Talanov, V. M., Tazlevan, V. E. & Shutov, D. A. (1992). Phys. Solid State, 34, 2927-2930.]), MgAl2O4 (Suzuki & Kumazawa, 1980[Suzuki, I. & Kumazawa, M. (1980). Phys. Chem. Miner. 5, 279-284.]), Li0.5Fe1.0Rh1.5O4 (Kang & Kim, 2006[Kang, K. U. & Kim, C. S. (2006). Hyperfine Interact. 168, 1181-1184.]), FeIn2S4 (Hill et al., 1978[Hill, R. J., Craig, J. R. & Gibbs, G. V. (1978). J. Phys. Chem. Solids, 39, 1105-1111.]) and others.

Ordered spinel structure with space group [R{\overline 3}2/m]. This structure is formed as a result of ordering and displacements of tetrahedral and octahedral cations and anions in the spinel structure. The tetrahedral cations are ordered by the 1:1 type, the octahedral cations by the 1:1:6 type and the anions by the 1:1:3:3 type. The general structural formula of the ordered spinel should be A2c1/2A'2c1/2B1a1/4B1b1/4B6h3/2X2c1/2X'2c1/2X6h3/2X'6h3/2 (for the rhombohedral representation of the structure) or A6c1/2A'6c1/2B3a1/4B3b1/4B18h3/2X6c1/2X'6c1/2X18h3/2X'18h3/2 (for the hexagonal representation of the structure). In Fig. 3[link] the calculated structure of the ordered [R{\overline 3}2/m] phase is shown. The unusual feature of this structure is `molecular geptamers' (Matsuno et al., 2001[Matsuno, K., Katsufuji, T., Mori, S., Moritomo, Y., Machida, A., Nishibori, E., Takata, M., Sakata, M., Yamamoto, N. & Takagi, H. (2001). J. Phys. Soc. Jpn, 70, 1456-1459.], 2003[Matsuno, K., Katsufuji, T., Mori, S., Nohara, M., Machida, A., Moritomo, Y., Kato, K., Nishibori, E., Takata, M., Sakata, M., Kitazawa, K. & Takagi, H. (2003). Phys. Rev. Lett. 90, 096404.]; Nishiguchi & Onoda, 2002[Nishiguchi, N. & Onoda, M. (2002). J. Phys. Condens. Matter, 14, L551.]; Horibe et al., 2006[Horibe, Y., Shingu, M., Kurushima, K., Ishibashi, H., Ikeda, N., Kato, K., Motome, Y., Furukawa, N., Mori, S. & Katsufuji, T. (2006). Phys. Rev. Lett. 96, 086406.]). The molecular geptamers consist of B3b and B18h atoms (Fig. 3[link]d). The total number of B atoms in a metal cluster (geptamer) equals seven. Substances such as AlV2O4, AlV2-xCrxO4 and Al1-xMgxV2O4 (Matsuno et al., 2001[Matsuno, K., Katsufuji, T., Mori, S., Moritomo, Y., Machida, A., Nishibori, E., Takata, M., Sakata, M., Yamamoto, N. & Takagi, H. (2001). J. Phys. Soc. Jpn, 70, 1456-1459.], 2003[Matsuno, K., Katsufuji, T., Mori, S., Nohara, M., Machida, A., Moritomo, Y., Kato, K., Nishibori, E., Takata, M., Sakata, M., Kitazawa, K. & Takagi, H. (2003). Phys. Rev. Lett. 90, 096404.]; Nishiguchi & Onoda, 2002[Nishiguchi, N. & Onoda, M. (2002). J. Phys. Condens. Matter, 14, L551.]; Horibe et al., 2006[Horibe, Y., Shingu, M., Kurushima, K., Ishibashi, H., Ikeda, N., Kato, K., Motome, Y., Furukawa, N., Mori, S. & Katsufuji, T. (2006). Phys. Rev. Lett. 96, 086406.]), CuTi2S4 (Soheilnia et al., 2004[Soheilnia, N., Kleinke, K. M., Dashjav, E., Cuthbert, H. L., Greedan, J. E. & Kleinke, H. (2004). Inorg. Chem. 43, 6473-6478.]) and CuZr1.86S4 (Dong et al., 2010[Dong, Y., McGuire, M. A., Yun, H. & DiSalvo, F. J. (2010). J. Solid State Chem. 183, 606-612.]) have the [R{\overline 3}2/m] structure.

[Figure 3]
Figure 3
Fragments of the calculated ordered [R{\overline 3}2/m] structure. Presentation of the ordered spinel structure as two tetrahedra (white and grey), formed by two types of atoms A6c and A'6c (a); presentation of the ordered spinel structure as three octahedra types (white, grey and dashed), formed by atoms B3a, B3b and B18h (b); presentation of the ordered spinel structure as two types of `molecules' - geptamers (c); and the `molecule' - geptamer B3b[B18h]6 (d).

7.2. Cation order in octahedral positions of the spinel structure

Among structures with cation order in the octahedral positions of a spinel, there are ones with space groups P41(3)22 [order parameter (0, 0, 0, 0, 0, [varphi]), k10([tau]1)], P43(1)32 [order parameter (0, [varphi], 0, [varphi], 0, -[varphi]), k10[tau]1] and Imma [order parameter (0, [xi], 0), k11([tau]7)].

The feature of structures of the ordered enantiomorphic P4122 and P4322 phases, and also the P4332 and P4132 phases, is that among the elements of symmetry of the space group of their structures there is no inversion and no symmetry planes, but only symmetry axes. Such crystals can exist in right- and left-hand forms, being mirror reflections of each other. By physical properties they are not distinguishable (except by optical activity). In LiFe5O8 enantiomorphic modifications exist as various domains in one sample (Van der Biest & Thomas, 1975[Van der Biest, O. & Thomas, G. (1975). Acta Cryst. A31, 70-76.]).

Ordered spinel structure with enantiomorphic space groups P4122 and P4322. These structures are formed as a result of a phase transition with a critical parameter of order transformed by the six-dimensional irrep k10([tau]1). Octahedral cation ordering is of the 1:1 type. It is accompanied by displacements of tetrahedral and octahedral cations and also anions. In the low-symmetry phase the anions are ordered (the type of order is 1:1).

The structural formula of the low-symmetry phase should be A4cB4aB4bX8d2X'8d2. In Fig. 4[link] the structural features of the enantiomorphic phases are shown. The following substances crystallize in P4122 and P4322 structures: LiZnNbO4 (Marin et al., 1994[Marin, S. J., O'Keeffe, M. & Partin, D. E. (1994). J. Solid State Chem. 113, 413-419.]), Li2TeO4 (Daniel et al., 1977[Daniel, F., Motet, J., Philippot, E. & Maurin, M. (1977). J. Solid State Chem. 22, 113-119.]), A2TiO4 [where A = Zn (Marin et al., 1994[Marin, S. J., O'Keeffe, M. & Partin, D. E. (1994). J. Solid State Chem. 113, 413-419.]; Billiet & Poix, 1963[Billiet, Y. & Poix, P. (1963). Bull. Soc. Chim. Fr. pp. 477-479.]; Billiet et al., 1963[Billiet, Y., Poix, P. & Michel, A. (1963). C. R. Acad. Sci. 256, 4217-4218.]; Vincent & Durif, 1966[Vincent, J. C. H. & Durif, A. (1966). Bull. Soc. Chim. Fr. pp. 246-250.]), Mn (Hardy et al., 1964[Hardy, A. Lecerf, A. Ranet, M. & Villers, G. (1964). C. R. Acad. Sci. 259, 3462-3463.]; Millard et al., 1995[Millard, R. L., Peterson, R. C. & Hunter, B. K. (1995). Am. Mineral. 80, 885-896.]) or Mg (Delanoye & Michel, 1969[Delanoye, P. & Michel, A. (1969). C. R. Acad. Sci. 269, 837-838.])], Zn2GeO4 (Millard et al., 1995[Millard, R. L., Peterson, R. C. & Hunter, B. K. (1995). Am. Mineral. 80, 885-896.]; Syono et al., 1971[Syono, Y., Akimoto, S.-I. & Matsui, Y. (1971). J. Solid State Chem. 3, 369-380.]), LiMnNbO4 (Shukaev et al., 2007[Shukaev, I. L., Pospelov, A. A. & Gannochenko, A. A. (2007). J. Solid State Chem. 180, 2189-2193.]) and Zn0.8Co1.2GeO4 (Preudhomme & Tarte, 1980[Preudhomme, J. & Tarte, P. (1980). J. Solid State Chem. 35, 272-277.]).

[Figure 4]
Figure 4
Calculated low-symmetry spinel structure with enantiomorphic space groups P4122 and P4322. Atom presentation of the ordered enantiomorphic spinel structures (atoms B4a and B4b are in the centres of white and dark octahedra) (a); polyhedral presentation of the ordered enantiomorphic spinel structures (b). The line between the structures indicates a mirror.

Ordered spinel structure with enantiomorphic space groups P4132 and P4332. As stated, these structures are also formed as a result of a phase transition with a critical parameter of order transformed by the six-dimensional irrep k10([tau]1). The critical irrep k10[tau]1 enters into the permutation representation of a spinel on positions 16d and 32e and enters into the mechanical representation of a spinel on positions 8a, 16d and 32e. Therefore, lowering of crystal symmetry is caused by displacements of all types of atoms and ordering of octahedral cations and anions. In the low-symmetry phase, octahedral cations and anions are ordered by the 1:3 type in both cases. The structural formula of the ordered phase should be A8c2B4bB12d3X8c2X24e6. Low-symmetry modifications of LiM5O8 (M = Al, Ga) (Ahman et al., 1996[Ahman, J., Svensson, G. & Albertson, J. (1996). Acta Chem. Scand. 50, 391-394.]; Datta & Roy, 1963[Datta, R. K. & Roy, R. (1963). J. Am. Ceram. Soc. 46, 388-390.]; Tarte & Collongues, 1964[Tarte, P. & Collongues, R. (1964). Ann. Chim. 9, 135-141.]), Zn3Ni2TeO8 (Bayer, 1967[Bayer, G. (1967). J. Less-Common Met. 12, 326-328.]), Zn2Co3TeO8 (Bayer, 1967[Bayer, G. (1967). J. Less-Common Met. 12, 326-328.]), CuMg0.5Mn1.5O4 (Blasse, 1966[Blasse, G. (1966). J. Phys. Chem. Solids, 102, 383-389.]; Vandenberghe et al., 1976[Vandenberghe, R. E., Legrand, E., Scheerlinck, D. & Brabers, V. A. M. (1976). Acta Cryst. B32, 2796-2798.]), Cu1.5Mn1.5O4 (Blasse, 1966[Blasse, G. (1966). J. Phys. Chem. Solids, 102, 383-389.]; Vandenberghe et al., 1976[Vandenberghe, R. E., Legrand, E., Scheerlinck, D. & Brabers, V. A. M. (1976). Acta Cryst. B32, 2796-2798.]), Zn2Mn3O8 (Joubert & Durif, 1964a[Joubert, J.-C. & Durif, A. (1964a). Bull. Soc. Fr. Mineral. Cristallogr. 87, 517-519.]), ZnMGe3O8 (M = Mn, Mg) (Joubert & Durif, 1964a[Joubert, J.-C. & Durif, A. (1964a). Bull. Soc. Fr. Mineral. Cristallogr. 87, 517-519.]), M2Ge3O8 (M = Zn, Co) (Joubert & Durif, 1964a[Joubert, J.-C. & Durif, A. (1964a). Bull. Soc. Fr. Mineral. Cristallogr. 87, 517-519.]), ZnMTi3O8 (M = Mn, Cd) (Joubert & Durif, 1964a[Joubert, J.-C. & Durif, A. (1964a). Bull. Soc. Fr. Mineral. Cristallogr. 87, 517-519.]), M2Ti3O8 (M = Zn, Mn, Co) (Joubert & Durif, 1964a[Joubert, J.-C. & Durif, A. (1964a). Bull. Soc. Fr. Mineral. Cristallogr. 87, 517-519.]), V2Co3O8 (Joubert & Durif, 1964b[Joubert, J.-C. & Durif, A. (1964b). Bull. Soc. Fr. Mineral. Cristallogr. 87, 47-49.]), Li2Mn3CoO8 (Joubert & Durif-Varambon, 1963[Joubert, J.-C. & Durif-Varambon, A. (1963). Bull. Soc. Fr. Mineral. Cristallogr. 86, 92-93.]), Li0.5+0.5xFe2.5-1.5xTixO4 (0 [greater-than or equal to] x [greater-than or equal to] 0.4, 1.2 [greater-than or equal to] x [greater-than or equal to] 1.57) (Arillo et al., 2003[Arillo, M. A., López, M. L., Pico, C., Veiga, M. L., Campo, J., Martínez, J. L. & Santrich-Badal, A. (2003). Eur. J. Inorg. Chem. pp. 2397-2405.]), Na4Ir3O8 (Okamoto et al., 2007[Okamoto, Y., Nohara, M., Aruga-Katori, H. & Takagi, H. (2007). Phys. Rev. Lett. 99, 137207-137211.]), LiMnTiO4 (Arillo et al., 2008[Arillo, M. A., López, M. L., Pico, C. & Veiga, M. L. (2008). Solid State Sci. 10, 1612-1619.]), Na4Sn3O8 (Iwasaki et al., 2002[Iwasaki, M., Takizawa, H., Uheda, K. & Endo, T. (2002). J. Mater. Chem. 12, 1068-1070.]), LiNi0.5Mn1.5O4 (Wang et al., 2011[Wang, L., Li, H., Huang, X. & Baudrin, E. (2011). Solid State Ion. 193, 32-38.]), LiFe5-xMnxO8 (0 [less-than or equal to] x [less-than or equal to] 1) (Darul et al., 2005[Darul, J., Nowicki, W., Piszora, P., Baehtz, C. & Wolska, E. (2005). J. Alloys Compd, 401, 60-63.]), LiFe5O8 (Braun, 1952[Braun, P. B. (1952). Nature (London), 170, 1123-1124.]; Widatallaha et al., 2005[Widatallaha, H. M., Johnson, C., Berry, F. J., Jartych, E., Gismelseed, A. M., Pekala, M. & Grabski, J. (2005). Mater. Lett. 59, 1105-1109.]), LiMg0.1Ni0.4Mn1.5O4 (Wagemaker et al., 2004[Wagemaker, M., Ooms, F. G. B., Kelder, E. M., Schoonman, J., Kearley, G. J. & Mulder, F. M. (2004). J. Am. Chem. Soc. 126, 13526-13533.]), LiNi0.5Mn1.5O4-[delta], LiNi0.5Mn1.5O4 (Kim et al., 2004[Kim, J.-H., Myung, S.-T., Yoon, C. S., Kang, S. G. & Sun, Y.-K. (2004). Chem. Mater. 16, 906-914.]), Li1.25Fe0.25Ti1.5O4 (Reale et al., 2003[Reale, P., Panero, S., Ronci, F., Albertini, V. R. & Scrosati, B. (2003). Chem. Mater. 15, 3437-3442.]), Li2Mn3MO8 (M = Mg, Zn) (Strobel et al., 2003[Strobel, P., Ibarra-Palos, A., Anne, M., Poinsignon, C. & Crisci, A. (2003). Solid State Sci. 5, 1009-1018.]), LiMg0.5Mn1.5O4, CuMg0.5Mn1.5O4 (Branford et al., 2002[Branford, W., Green, M. A. & Neumann, D. A. (2002). Chem. Mater. 14, 1649-1656.]), LixMg1-2xFe2+xO4 (x [greater-than or equal to] 0.40) (Antic et al., 2002[Antic, B., Rodic, D., Nikolic, A. S., Kacarevic Popovic, Z. & Karanovic, L. J. (2002). J. Alloys Compd, 336, 286-291.]), LiMgxMn2-xO4 (x > 1) (Singh et al., 2009[Singh, G., Sil, A. & Ghosh, S. (2009). Physica B, 404, 3807-3813.]), Li2CoTi3O8 (Reeves et al., 2007[Reeves, N., Pasero, D. & West, A. R. (2007). J. Solid State Chem. 180, 1894-1901.]), Li2Zn3O8, Li2Ge3O8 (Hirota et al., 1988[Hirota, K., Ohtani, M., Mochida, N. & Ohtsuka, A. (1988). Nippon Seram. 96, 92-96.]), LiM0.5Ti1.5O4, LiM0.5Ge1.5O4 (M = Mg, Co, Ni, Zn) (Kawai et al., 1998[Kawai, H., Tabuchi, M., Nagata, M., Tukamoto, H. & West, A. R. (1998). J. Mater. Chem. 8, 1273-1280.]; Hernander et al., 1996[Hernander, V. S., Martinez, L. M. T., Mather, G. C. & West, A. R. (1996). J. Mater. Chem. 6, 1533-1536.]), Li1.33xCo2-2xTi1+0.67xO4 (Kremenovic & Antic, 2004[Kremenovic, A. & Antic, B. (2004). Phys. Lett. A, 324, 501-506.]), Li1-0.5xFe2.5xMn2-2xO4 (Kremenovic & Antic, 2004[Kremenovic, A. & Antic, B. (2004). Phys. Lett. A, 324, 501-506.]), LiMn2-yTiyO4 (y > 1) (Petrov et al., 2005[Petrov, K., Rojas, R. M., Alonso, P. J., Amarilla, J. M., Lazarraga, M. G. & Rojo, J. M. (2005). Solid State Sci. 7, 277-286.]), LiFe0.5Ti1.5O4 (Avdeev et al., 2007[Avdeev, G., Petrov, K. & Mitov, I. (2007). Solid State Sci. 9, 1135-1139.]) and others have the structures of ordered P4132 and P4332 phases. The features of the calculated P4132 and P4332 structures are shown in Fig. 5[link].

[Figure 5]
Figure 5
Calculated low-symmetry spinel structure with enantiomorphic space groups P4132 and P4332. Polyhedral presentation of the ordered enantiomorphic spinel structures (atoms B4b and B12d are in the centres of white and dark octahedra) (a); double spiral from tetrahedra and hexahedra (b). The line between the structures indicates a mirror.

Ordered spinel structure with space group Imma. This structure is generated by irrep k11([tau]7). This irrep enters into the mechanical representation of the spinel structure on positions 8a and 32e and enters into the permutation representation of a spinel on positions 16d and 32e. Therefore, a low-symmetry modification of the Imma phase is formed as a result of displacements of tetrahedral cations and anions and also ordering of octahedral cations and anions (in both cases the type of order is 1:1). A structural formula of the low-symmetry phase should be A4eB4aB4dX8h2X8i2. The calculated structure is shown in Fig. 6[link]. This structure contains chains of octahedra (B4aX8h2X8i4 and B4dX8h4X8i2) shared by edges, running along the y and x axes, respectively (Fig. 6[link]c). The low-symmetry modifications of Li2CoCl4 (Wussow et al., 1989[Wussow, K., Haeuseler, H., Kuske, P., Schmidt, W. & Lutz, H. D. (1989). J. Solid State Chem. 78, 117-125.]), Li2MnBr4 (Kanno et al., 1986[Kanno, R., Takeda, Y., Yamamoto, O., Cros, C., Gang, W. & Hagenmuller, P. (1986). J. Electrchem. Soc. 133, 1052-1056.]), Li2MgBr4 (Smirnov, 1966[Smirnov, A. A. (1966). Molecular-Kinetic Theory of Metals. Moscow: Nauka.]), LiVCuO4 (Joubert et al., 1965[Joubert, J.-C., Grenier, J. C. & Durif, A. (1965). C. R. Acad. Sci. 260, 2472-2473.]), Li1-xCuVO4 (0 [less-than or equal to] x [less-than or equal to] 0.2) (Kanno et al., 1992[Kanno, R., Kawamoto, Y., Takeda, Y., Haregawa, M., Yamamoto, O. & Kinomura, N. (1992). J. Solid State Chem. 96, 397-407.]), LiSbZnO4 (Keramidebs et al., 1975[Keramidebs, V. G., Deangelis, B. A. & White, W. B. (1975). J. Solid State Chem. 15, 233-245.]) and Li2FeCl4 (Lutz et al., 1989[Lutz, H. D., Pfitzner, A., Schmidt, W., Riedel, E. & Prick, D. (1989). Z. Naturforsch. Teil A, 44, 756-758.]) have the Imma structure.

[Figure 6]
Figure 6
Calculated low-symmetry ordered spinel structure with space group Imma. Atom presentation of the ordered spinel structure (a), tetrahedra in the ordered spinel structure (b), and two types of octahedra (dark and dashed) in the ordered spinel structure (c).

7.3. Anion order in the spinel structure

The typical examples of anion ordering in Wyckoff position 32e are spinel low-symmetry modifications with space groups [F{\overline 4}3m] [order parameter ([xi]), k11([tau]4)], [R{\overline 3}2/m] [order parameter ([xi], -[xi], [xi]), k11([tau]7)], and P41212 and P43212 [order parameter (0, 0, [varphi], 0, 0, 0), k10([tau]2)].

Ordered spinel structure with space group [F{\overline 4}3m]. We have already considered the structure of the [F{\overline 4}3m] phase. A special case of an ordered spinel structure is the structure of the Chevrel phases. The structure of these phases can be presented as a structure of defect spinels with ordering of tetrahedral cations and anions of the 1:1 type. If there are no A4a and A4c atoms at all, or only some of these atoms are absent, various types of Chevrel phases can be obtained (Talanov, 2005[Talanov, V. M. (2005). Phys. Chem. Glass, 31, 431-434.]; Besnard et al., 2003[Besnard, C, Svensson, C., Stahl, K. & Siegrist, T. (2003). J. Solid State Chem. 172, 446-450.]). For example, if there are no A4a and A4c atoms, then a formula B16e4X16e4X'16e4 is obtained. This formula corresponds to the Re4Te4S4 structure (Fedorov et al., 1994[Fedorov, V. E., Mironov, Yu. V., Fedin, V. P. & Mironov, Yu. I. (1994). J. Struct. Chem. 35, 157-159.], 1996[Fedorov, V. E., Mironov, Y. V., Fedin, V. P., Imoto, H. & Saito, T. (1996). Acta Cryst. C52, 1065-1067.]). If only one type of atom A in the general formula is absent, then we have the formula A4aB16e4X16e4X'16e4. Such a formula describes the structures of GaMo4O8 (Besnard et al., 2003[Besnard, C, Svensson, C., Stahl, K. & Siegrist, T. (2003). J. Solid State Chem. 172, 446-450.]) and Re4As6S3 (Besnard et al., 2003[Besnard, C, Svensson, C., Stahl, K. & Siegrist, T. (2003). J. Solid State Chem. 172, 446-450.]). If the positions of A4a and A4c atoms are partly occupied, then this formula describes the structures of such substances as Ga1.33Cr4S8 (Besnard et al., 2003[Besnard, C, Svensson, C., Stahl, K. & Siegrist, T. (2003). J. Solid State Chem. 172, 446-450.]) and Ga0.5V2S2Se2 (Haeuseler et al., 2001[Haeuseler, H., Reil, S. & Elitok, E. (2001). Int. J. Inorg. Mater. 3, 409-412.]).

The structures of the Chevrel phases, their properties and features of the structures are genetically connected with the structure of ordered spinels. In particular, on cation ordering of the 1:1 type in tetrahedral positions of AB2X4 spinel, B4 clusters from metal atoms and anions are formed (Talanov et al., 2008[Talanov, V. M., Ereyskaya, G. P. & Yuzyuk, Y. I. (2008). Introduction to Chemistry and Physics of Nanostructures and Nanostructured Materials, edited by V. M. Talanov. Moscow: Academy of Natural Science.]) (Fig. 2[link]). The structure of the Chevrel phases `inherits' four types of clusters from the ordered spinel structure [this fact was established as a result of theoretical research into the structural mechanism of ordered spinel formation with [F{\overline 4}3m] symmetry (Talanov et al., 2008[Talanov, V. M., Ereyskaya, G. P. & Yuzyuk, Y. I. (2008). Introduction to Chemistry and Physics of Nanostructures and Nanostructured Materials, edited by V. M. Talanov. Moscow: Academy of Natural Science.])].

We believe that the formation of these clusters causes the original physical properties of the Chevrel phases. In Figs. 7[link] and 8[link], examples of calculated structures of the Chevrel phases are given. In particular, Re4 clusters exist in structures of the Chevrel phases such as Re4As6S3 (Figs. 7[link]a-c) and Re4Te4S4 (Fig. 8[link]).

[Figure 7]
Figure 7
Calculated low-symmetry anion-ordered spinel structure with space group [F{\overline 4}3m] and structural formula A4aB16e4X16e4X'16e4. Atom presentation of the ordered spinel structure (a), tetrahedra in the ordered spinel structure (b), B16e metallic tetrahedra (c). A substance with composition Re4As6S3 has a similar structure. In this structure there are Re4 clusters (Besnard et al., 2003[Arillo, M. A., López, M. L., Pico, C. & Veiga, M. L. (2008). Solid State Sci. 10, 1612-1619.]).
[Figure 8]
Figure 8
Calculated low-symmetry anion-ordered spinel structure with space group [F{\overline 4}3m] and structural formula B16e4X16e4X'16e4. A substance with composition Re4Te4S4 has a similar structure. In this structure there are Re4 clusters (Fedorov et al., 1994[Fedorov, V. E., Mironov, Yu. V., Fedin, V. P. & Mironov, Yu. I. (1994). J. Struct. Chem. 35, 157-159.], 1996[Fedorov, V. E., Mironov, Y. V., Fedin, V. P., Imoto, H. & Saito, T. (1996). Acta Cryst. C52, 1065-1067.]).

Ordered spinel structure with space group [R{\overline 3}2/m]. This structure is generated by irrep k11([tau]7). This irrep enters into the mechanical representation of the spinel structure on positions 8a and 32e and enters into the permutation representation of the spinel on positions 16d and 32e. This means that the low-symmetry phase formation is connected with displacements of tetrahedral cations and anions and also with ordering of octahedral cations and anions (in both cases the types of order are the same, i.e. 1:3). The structural formula of a low-symmetry phase should be A2cB1a1/2B3d3/2X2cX6h3. In Fig. 9[link], the features of the structure of the [R{\overline 3}2/m] phases are shown. Ga3O3N has this structure (Boyko et al., 2011[Boyko, T. D., Zvoriste, C. E., Kinski, I., Riedel, R., Hering, S., Huppertz, H. & Moewes, A. (2011). Phys. Rev. B, 84, 085203.]). This substance possesses unusual electronic properties. Boyko et al. (2011[Boyko, T. D., Zvoriste, C. E., Kinski, I., Riedel, R., Hering, S., Huppertz, H. & Moewes, A. (2011). Phys. Rev. B, 84, 085203.]) considered three models of the Ga3O3N structure, described by space groups Ima2, Imm2 and [R{\overline 3}2/m]. Study of the local electronic structure of this substance allowed the authors to establish the ordered arrangement of anions and to choose the model of the structure with space group [R{\overline 3}2/m]. It should be noted that, according to our theory, at the formation of the [R{\overline 3}2/m]-phase structure, not only oxygen and nitrogen ordering occurs, but also ordering of two of the three atoms of gallium, located in positions 1a and 3d, takes place. The structural formula of Ga3O3N should be Ga1a1/2Ga3d3/2Ga2cN2cO6h3.

[Figure 9]
Figure 9
Calculated low-symmetry ordered spinel with space group [R{\overline 3}2/m]. Atom presentation of the ordered spinel structure (a); tetrahedra [A2cX2cX6h3] in the ordered spinel structure (b); distorted hexahedra [X2c2X6h6] with the central atom B1a (c); distorted hexahedra [X2c2X6h6] with the central atom B3d and tetrahedra [A2cX2cX6h3] (d); and hexahedral presentation of the ordered spinel structure (e).

Ordered spinel structure with enantiomorphic space groups P41212 and P43212. The critical irrep inducing P41212- and P43212-phase formation is the six-dimensional representation k10([tau]2). It can be shown that the structures of these phases are formed as a result of displacements of all atom types and also by ordering of atoms located in positions 32e of the spinel structure (the type of order is 1:1). The structural formula of the low-symmetry phase should be A4aB8b2X8b2X8b2. The features of the P41212 and P43212 structures are given in Fig. 10[link]. An example of this structure is that of MgTi2O4 (Isobe & Ueda, 2002[Isobe, M. & Ueda, Y. (2002). J. Phys. Soc. Jpn, 71, 1848-1851.]; Schmidt et al., 2004[Schmidt, M., Ratcliff, W., Radaelli, P. G., Refson, K., Harrison, N. M. & Cheong, S. W. (2004). Phys. Rev. Lett. 92, 056402.]). In this substance at a temperature of ~260 K a phase transition occurs, accompanied by formation of tetragonal spinel modifications with space groups P41212 and P43212 (Isobe & Ueda, 2002[Isobe, M. & Ueda, Y. (2002). J. Phys. Soc. Jpn, 71, 1848-1851.]; Schmidt et al., 2004[Schmidt, M., Ratcliff, W., Radaelli, P. G., Refson, K., Harrison, N. M. & Cheong, S. W. (2004). Phys. Rev. Lett. 92, 056402.]). High-resolution synchrotron and neutron powder diffraction and also X-ray investigation shows the existence of weak superstructure reflections (Schmidt et al., 2004[Schmidt, M., Ratcliff, W., Radaelli, P. G., Refson, K., Harrison, N. M. & Cheong, S. W. (2004). Phys. Rev. Lett. 92, 056402.]). According to our calculations these reflections are caused by ordering of oxygen ions (Fig. 10[link]) (Talanov et al., 2012[Talanov, V. M., Shirokov, V. B., Ivanov, V. B. & Talanov, M. V. (2012). Kristallographiya, 58, 80-91.]). The theory predicts that ordering of oxygen ions should be of the 1:1 type. The symmetry of the order parameter, and the thermodynamics and mechanisms of formation of the atomic and orbital structure of the low-symmetry ordered MgTi2O4 spinel modification have been studied (Talanov et al., 2012[Talanov, V. M., Shirokov, V. B., Ivanov, V. B. & Talanov, M. V. (2012). Kristallographiya, 58, 80-91.]).

[Figure 10]
Figure 10
Calculated low-symmetry ordered structure of the tetragonal spinel modification of MgTi2O4 (on the basis of the cubic initial spinel structure). Atom presentation of the ordered spinel structure (a), tetrahedra arrangements in the ordered tetragonal spinel structure (b), and octahedra arrangements in the ordered tetragonal spinel structure (c).

7.4. Atom ordering in two sublattices

One type of superstructure with simultaneous cation ordering in spinel structure positions 8a and 16d is known. It is an ordered phase of LiZn0.5Mn1.5O4 with space group P213. This phase is formed by two order parameters: ([xi])4 and (0, [varphi], 0, [varphi], 0, -[varphi])1. These two irreps form a point group of order 192 in the seven-dimensional space of the order parameter. The structural mechanism of the low-symmetry phase formation is defined by representation k10([tau]3) + k11([tau]4). It appears to be complex and includes: (i) binary cation ordering of the 1:1 type in tetrahedral spinel positions 8a and the 1:3 type in octahedral spinel positions 16d; (ii) ordering of four anions of the 1:1:3:3 type in the initial spinel phase structure; and (iii) all types of atom displacements.

This structural mechanism of P213-phase formation is much more complex than that proposed previously (Lee et al., 2002[Lee, Y. J., Park, S. H., Eng, C., Parise, J. B. & Grey, C. P. (2002). Chem. Mater. 14, 194-205.]). The calculated structural formula of the low-symmetry P213 phase should be (A4a1/2A'4a1/2)[B4a1/2B'12b3/2]X12bX'12bX4aX'4a. Experimental data on the structure of LiZn0.5Mn1.5O4, obtained by neutron diffraction and X-ray analyses, agree with the proposed structural formula (Lee et al., 2002[Lee, Y. J., Park, S. H., Eng, C., Parise, J. B. & Grey, C. P. (2002). Chem. Mater. 14, 194-205.]). The calculated structure of the P213 phase is shown in Fig. 11[link]. The structural formula of LiZn0.5Mn1.5O4 should be (Li4a0.5Zn4a0.5)[Li'4a0.5Mn12b1.5]O12bO'12bO4aO'4a. The symmetry of the order parameter, and the thermodynamics and mechanisms of formation of the atomic structure of the low-symmetry ordered cubic LiZn0.5Mn1.5O4 oxide spinel have been studied by Talanov & Shirokov (2013[Talanov, V. M. & Shirokov, V. B. (2013). Kristallographiya, 58, 296-301.]).

[Figure 11]
Figure 11
Fragments of the calculated low-symmetry ordered structure of LiZn0.5Mn1.5O4. Atom presentation of the ordered spinel structure (zinc ions are not shown) (a) and tetrahedra and octahedra in the ordered spinel structure (b). In (b) two types of octahedra (octahedra with central ions Mn12b are dashed, octahedra with central ions Li'4a are grey) are shown.

8. Conclusions

The atomic ordering in the spinel structure was studied within a unified approach based on the Landau phenomenological phase-transition theory. Group-theoretical methods of the Landau theory were used to investigate the structures of ordered spinels. Based on the hypothesis of one critical representation, we established the order-parameter symmetry and the space groups of the low-symmetry spinel modifications, and calculated the stratification of Wyckoff positions 8a, 16d and 32e of the initial high-symmetry phase with a spinel structure at the transition to the low-symmetry phases. The possibility of the existence of 305 phases with different types of order in position 8a (including seven binary and seven ternary cation substructures), 537 phases in position 16d (including eight binary and 11 ternary cation substructures), 595 phases in position 32e (including seven binary and four ternary anion substructures) and 549 phases with simultaneous ordering in positions 8a and 16d (including five substructures with binary order in tetrahedral and octahedral sublattices, two substructures with ternary order in both spinel sublattices, and nine substructures with different combined types of binary and ternary order) was determined. The structural mechanisms of forming ordered spinel phases were considered. Calculated structures of the spread types of ordered low-symmetry spinel modifications were given.

Comparison of theoretical results and experimental data was made. The absolute part of the experimentally studied superstructures with binary atom ordering agrees with the theoretical results of this work.

Thus, in this work all possible types of binary and ternary ordering of cations and anions in the spinel structure have been determined. We hope that the data of Tables 1[link][link][link]-4[link] will be helpful in research on the structures of new spinels and their low-symmetry modifications.

Acknowledgements

The authors wish to express their thanks to the anonymous referees for valuable remarks which helped to improve the manuscript. We are also very grateful to Professor Dr W. F. Kuhs for his kind interest and encouragement. The results of this work have been obtained with the support of the Ministry of Education and Science of the Russian Federation in the framework of the State task (project N6.8604.2013).

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Acta Cryst (2014). A70, 49-63   [ doi:10.1107/S2053273313027605 ]