Volume 70 Received 9 July 2013  Atomic order in the spinel structure  a grouptheoretical analysis^{a}South Russia State Technical University (Novocherkassk Polytechnic Institute), 346400, Novocherkassk, Russia, and ^{b}South Scientific Center, Russian Academy of Sciences, 344006, RostovonDon, Russia Grouptheoretical 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 lowsymmetry spinel modifications are given. 
The spinel structure was discovered independently by Bragg (1915) and Nishikawa (1915) 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; Talanov, 1986a). Later their other unique physical properties were discovered. It was revealed that among spinels there were superconductors (LiTi_{2}O_{4}, CuRh_{2}S_{4}, CuV_{2}S_{4}, CuRh_{2}Se_{4}) (Johnston et al., 1973; Robbins et al., 1967), crystals with superionic conductivity (Li_{2}MCl_{4}, where M = Mg, V, Mn, Fe, Cd, bromide spinels and others) (Kanno et al., 1981; Lutz et al., 1985, 1997; Schmidt & Lutz, 1984), numerous materials for electrodes in chemical current sources (Thackeray, 1997; Ezikian et al., 1988; Talanov et al., 2007) and multiferroics (CoCr_{2}O_{4}) (Ederer & Spaldin, 2004; Yamasaki et al., 2006; Torgashev et al., 2012; Pollert, 1988; Gorjaga et al., 1990). Some spinels are used in the production of ceramic materials, fireproofs and thermoresistant dyes (Krupichka, 1976; Pollert, 1988; Gorjaga et al., 1990). Most of these spinel properties depend largely on atomic ordering. Let us explain this idea.
The general chemical formula of the spinels is AB_{2}X_{4}. The spinel structure has the space group . 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_{}B_{1})^{8a}[A_{1}B_{1+}]^{16d}X_{4}, where is the socalled degree of inversibility of the spinel structure. The parameter 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 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 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 = (T, x, P). The isostructural transition theory for spinels was first proposed by Talanov & Bezrukov (1985, 1990). A general theory of isostructural transitions for different classes of crystals was developed by Talanov & Bezrukov (1986a,b) and Talanov et al. (1989).
(ii) Reconstructive transitions. At a reconstructive phase transition fundamental structural reorganization takes place (Toledano & Dmitriev, 1996). Examples of reconstructive phase transitions are spinelolivine, spinelphenacite 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; Gufan, 1982; Toledano & Toledano, 1987). There are limiting groupsubgroup 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). There are limiting groupsubgroup 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). An example is the spinel Cu^{+}[Ni_{1/2}^{+2}Mn_{3/2}^{+4}]O_{4} with cation order in the octahedral positions (Krupichka, 1976). A lowtemperature 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). Abnormally high superionic conductivity is found in chloride spinels near the temperature of cation ordering (Kanno et al., 1981, 1986, 1987; Lutz et al., 1985, 1997; Schmidt & Lutz, 1984; Wussow et al., 1989).
Enumeration of the possible superstructures for simple lattices has been carried out previously in various model approximations (BraggWilliams; see, for example, Smirnov, 1966), a molecular field (see, for example, Vaks, 1973), static concentration waves (see, for example, Khachaturyan, 1974), and also by a simple search of options of `colouring' the positions of a crystal structure (Smirnova, 1956).
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 grouptheoretical 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 grouptheoretical approach has already been used for the study of some types of superstructures in spinels (Talanov, 1989, 1990a,b, 1996, 2007; Talanov & Chechin, 1990). 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 of (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.
The grouptheoretical method as a means of studying phase transitions of atom ordering in alloys was first proposed by Lifshitz (1941). This method was developed further by Gufan (1971, 1982), Kovalev (1993), Chechin (1989), Sahnenko et al. (1986) and Vinberg et al. (1974).
The structure of the lowsymmetry ordered phase within Landau's theory can be described by some function (r), for example, a function of the changing density of an electric charge:
Here the index n is the number of the full irreducible representations (irreps) and the index i is the number of their basis functions _{i}^{(n)}. The coefficients c_{i}^{(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: c_{i}^{(n)} c_{i}^{(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 (r) completely defines the symmetry of the lowsymmetry (or dissymmetrical G_{D}) phase. However, when researching the G_{D}phase structure far from the temperature of phase transition, the contribution of noncritical representations can become essential (Sahnenko et al., 1986). 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).
In considering phase transitions of the `orderdisorder' type, it is convenient to use the scalar function defined on the given Wyckoff position and characterizing the `colours' of positions appearing as a result of a phase transition. Identical (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 G_{0} on the Wyckoff positions in the G_{D}phase structure, it is possible to find the type of superstructure and the structural formula of the G_{D} phase (if we consider that on each Wyckoff position there is one type of atom).
For finding all possible lowsymmetry phases and corresponding order parameters c_{i}, the basic functions _{i} of irreps of a group of structure symmetry of the highsymmetry phase, and also the stratification of Wyckoff positions, we used the grouptheoretical method which was described in detail in Gufan (1971, 1982), Chechin (1989), Sahnenko et al. (1986) and Vinberg et al. (1974). We checked also the separate results obtained by our method by means of the ISOTROPY program (Stokes & Hatch, 2007; Stokes et al., 2002).
Calculation of possible ordered crystal phases by grouptheoretical 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 highsymmetry 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 _{j}. The translations of the extended cell (a_{j}) are defined from exp (ik_{L}a_{j}) = 1, where k_{L} 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 grouptheoretical 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 lowsymmetry 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 facecentred cubic lattice. For this lattice, the first Brillouin zone represents a bodycentred lattice and contains four points of high symmetry, namely: k_{11}(), k_{10}(X), k_{19}(L) and k_{8}(W) (Kovalev, 1993). To these points there are stars of the following wavevectors:
The indexing of vectors and irreps is given accordingly (Kovalev, 1993). There are ten irreps of the wavevector _{11} (four onedimensional, two twodimensional and four threedimensional); four sixdimensional irreps of the wavevector _{10}, four four and two eightdimensional irreps of the wavevector _{9}, and two 12dimensional irreps of the wavevector _{8}. Thus allinall there are 22 irreps.
All irreps of the wavevector _{8} and also two irreps of the wavevector _{10} (_{103} and _{104}) do not satisfy the Lifshitz criterion, i.e. they induce incommensurate phases. Nevertheless, we included these irreps in our analysis, as among the lowsymmetry phases induced by these representations there are also commensurate phases. In the full permutation representation only irreps relating to the wavevectors k_{11}(), k_{10}(X), k_{9}(L) and k_{8}(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 abovementioned wavevectors.
The permutation representation on Wyckoff position 8a of the space group generates 305 lowsymmetry phases. The composition of the permutation representation is as follows:^{1}
The representation k_{11}(_{1}(A_{1g})) always enters into the permutation representations. We do not give it in equation (1) 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 because they are unnecessary for defining possible types of ordered phases.
In Table 1 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 is presented. As seen from Table 1, among these phases there are seven binary (phases 16, 12) and seven ternary (phases 711, 14) superstructures (Table 1). 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 improper order parameters are also given. They are marked by `sec'.

Cation ordering in octahedral spinel positions has been investigated before (Billet et al., 1967). 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 facecentred 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). We obtained other results. Let us consider this problem in detail.
The permutation representation on Wyckoff position 16d of the space group induces 537 lowsymmetry phases. Among them there are 19 phases with binary and ternary atom ordering. The composition of the permutation representation is as follows:
These phases and the stratifications of Wyckoff position 16d of the space group are given in Table 2. Among these phases there are eight binary (phases 16) and 11 ternary (phases 715) superstructures (Table 2).

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) and (2) 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.
The permutation representation on Wyckoff position 32e of the space group generates 595 lowsymmetry ordered phases. The composition of the permutation representation is as follows:
There are only 11 binary and ternary ordered phases. These phases and the stratifications of Wyckoff position 32e of the space group are given in Table 3. As can be seen from Table 3, among these phases there are seven binary (phases 17) and four ternary (phases 811) 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 it is seen that in some cases, on atomic ordering in Wyckoff position 32e of the space group the forced stratification of Wyckoff positions 8a and 16d takes place.

It should be emphasized that among binary superstructures there are three pairs of enantiomorphic modifications of ordered spinels. Irreps of wavevector k_{8} do not generate binary and ternary superstructures.
The permutation representation on Wyckoff positions 8a and 16d of the space group gives rise to 549 lowsymmetry phases. The composition of the permutation representation is as follows:
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 are given in Table 4. As seen from Table 4, 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}.

The most widespread types of cation ordering in positions 8a are realized in structures with space groups [order parameter (), k_{11}(_{4}(A_{2u})] and [order parameter (, 0, 0, 0), k_{9}(_{4})].
Ordered spinel structure with space group . Phases with this symmetry are formed as a result of a phase transition of the second order from cubic spinel. For many spinels (MgAl_{2}O_{4}, LiGaCr_{4}O_{8}, CdIn_{2}S_{4} etc.) a phase transition is really found experimentally (Suzuki & Kumazawa, 1980; Yamanaka & Takeuchi, 1983; Joubert & Durif, 1966). But there are also compounds, for example LiXY_{4}O_{8} (X = Ga, Fe, In; Y = Cr, Rh), that exist in the ordered modification up to the temperature of melting (Joubert & Durif, 1966).
Irrep k_{11}(_{4}(A_{2u})) 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; Talanov, 1986b). The formation of a 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 A^{4a}A^{4c}B^{16e}_{4}X^{16e}_{4} (Talanov, 1986b). Features of the phase structure are discussed in Talanov (1986b, 1996, 2004, 2005) and Talanov et al. (2008). In Fig. 1, the calculated structure of the ordered 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). The compressed tetrahedron of B cations can be considered as a metal cluster. Compressed tetrahedra have the linear size (2^{1/2}/4)a and are located at a distance (6^{1/2}/4)a, where a is the parameter of a cubic elementary cell of a spinel structure.
 Figure 1 Calculated structure of an ordered spinel (space group ). Atomic presentation of the structure (a) and projection along (001) (b) of the ordered spinel structure. 
 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 B_{4} (b), expanded cluster X_{4} (c), expanded cluster B_{4} (d) and compressed cluster X_{4} (e). 
Many ordered spinels have a similar structure, for example Li_{0.5}Mn_{2}O_{4} (Julien et al., 2006), Li_{0.5}Cr_{x}Ga_{2.5x}O_{4} (x = 1.75) (Arsen et al., 1979; Arsen & Lenglet, 1980; Szymczak et al., 1975), Li_{0.5}Fe_{0.5}[Cr_{2}]O_{4} (Gorter, 1954), LiGaCr_{4}O_{8} (Yamanaka & Takeuchi, 1983), GaV_{4x}Mo_{x}S_{8} (0 x 4) (Powell et al., 2007), Cu_{0.5}In_{0.5}Cr_{2}S_{4} (Sadykov et al., 2001; Plumier, Sougi & Lecomte, 1977), Cu_{0.5}Fe_{0.5}Cr_{2}S_{4} (Sadykov et al., 2001), Ag_{0.5}In_{0.5}Cr_{2}S_{4} (Plumier, Lecomte et al., 1977), LiXY_{4}O_{8} (X = Ga, Fe, In; Y = Cr, Rh) (Joubert & Durif, 1966), CdIn_{2}S_{4} (Joubert & Durif, 1966; Cravetskiy et al., 1992), MgAl_{2}O_{4} (Suzuki & Kumazawa, 1980), Li_{0.5}Fe_{1.0}Rh_{1.5}O_{4} (Kang & Kim, 2006), FeIn_{2}S_{4} (Hill et al., 1978) and others.
Ordered spinel structure with space group . 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 A^{2c}_{1/2}A'^{2c}_{1/2}B^{1a}_{1/4}B^{1b}_{1/4}B^{6h}_{3/2}X^{2c}_{1/2}X'^{2c}_{1/2}X^{6h}_{3/2}X'^{6h}_{3/2} (for the rhombohedral representation of the structure) or A^{6c}_{1/2}A'^{6c}_{1/2}B^{3a}_{1/4}B^{3b}_{1/4}B^{18h}_{3/2}X^{6c}_{1/2}X'^{6c}_{1/2}X^{18h}_{3/2}X'^{18h}_{3/2} (for the hexagonal representation of the structure). In Fig. 3 the calculated structure of the ordered phase is shown. The unusual feature of this structure is `molecular geptamers' (Matsuno et al., 2001, 2003; Nishiguchi & Onoda, 2002; Horibe et al., 2006). The molecular geptamers consist of B^{3b} and B^{18h} atoms (Fig. 3d). The total number of B atoms in a metal cluster (geptamer) equals seven. Substances such as AlV_{2}O_{4}, AlV_{2x}Cr_{x}O_{4} and Al_{1x}Mg_{x}V_{2}O_{4} (Matsuno et al., 2001, 2003; Nishiguchi & Onoda, 2002; Horibe et al., 2006), CuTi_{2}S_{4} (Soheilnia et al., 2004) and CuZr_{1.86}S_{4} (Dong et al., 2010) have the structure.
 Figure 3 Fragments of the calculated ordered structure. Presentation of the ordered spinel structure as two tetrahedra (white and grey), formed by two types of atoms A^{6c} and A'^{6c} (a); presentation of the ordered spinel structure as three octahedra types (white, grey and dashed), formed by atoms B^{3a}, B^{3b} and B^{18h} (b); presentation of the ordered spinel structure as two types of `molecules'  geptamers (c); and the `molecule'  geptamer B^{3b}[B^{18h}]_{6} (d). 
Among structures with cation order in the octahedral positions of a spinel, there are ones with space groups P4_{1(3)}22 [order parameter (0, 0, 0, 0, 0, ), k_{10}(_{1})], P4_{3(1)}32 [order parameter (0, , 0, , 0, ), k_{10}_{1}] and Imma [order parameter (0, , 0), k_{11}(_{7})].
The feature of structures of the ordered enantiomorphic P4_{1}22 and P4_{3}22 phases, and also the P4_{3}32 and P4_{1}32 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 lefthand forms, being mirror reflections of each other. By physical properties they are not distinguishable (except by optical activity). In LiFe_{5}O_{8} enantiomorphic modifications exist as various domains in one sample (Van der Biest & Thomas, 1975).
Ordered spinel structure with enantiomorphic space groups P4_{1}22 and P4_{3}22. These structures are formed as a result of a phase transition with a critical parameter of order transformed by the sixdimensional irrep k_{10}(_{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 lowsymmetry phase the anions are ordered (the type of order is 1:1).
The structural formula of the lowsymmetry phase should be A^{4c}B^{4a}B^{4b}X^{8d}_{2}X'^{8d}_{2}. In Fig. 4 the structural features of the enantiomorphic phases are shown. The following substances crystallize in P4_{1}22 and P4_{3}22 structures: LiZnNbO_{4} (Marin et al., 1994), Li_{2}TeO_{4} (Daniel et al., 1977), A_{2}TiO_{4} [where A = Zn (Marin et al., 1994; Billiet & Poix, 1963; Billiet et al., 1963; Vincent & Durif, 1966), Mn (Hardy et al., 1964; Millard et al., 1995) or Mg (Delanoye & Michel, 1969)], Zn_{2}GeO_{4} (Millard et al., 1995; Syono et al., 1971), LiMnNbO_{4} (Shukaev et al., 2007) and Zn_{0.8}Co_{1.2}GeO_{4} (Preudhomme & Tarte, 1980).
 Figure 4 Calculated lowsymmetry spinel structure with enantiomorphic space groups P4_{1}22 and P4_{3}22. Atom presentation of the ordered enantiomorphic spinel structures (atoms B^{4a} and B^{4b} 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 P4_{1}32 and P4_{3}32. As stated, these structures are also formed as a result of a phase transition with a critical parameter of order transformed by the sixdimensional irrep k_{10}(_{1}). The critical irrep k_{10}_{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 lowsymmetry phase, octahedral cations and anions are ordered by the 1:3 type in both cases. The structural formula of the ordered phase should be A^{8c}_{2}B^{4b}B^{12d}_{3}X^{8c}_{2}X^{24e}_{6}. Lowsymmetry modifications of LiM_{5}O_{8} (M = Al, Ga) (Ahman et al., 1996; Datta & Roy, 1963; Tarte & Collongues, 1964), Zn_{3}Ni_{2}TeO_{8} (Bayer, 1967), Zn_{2}Co_{3}TeO_{8} (Bayer, 1967), CuMg_{0.5}Mn_{1.5}O_{4} (Blasse, 1966; Vandenberghe et al., 1976), Cu_{1.5}Mn_{1.5}O_{4} (Blasse, 1966; Vandenberghe et al., 1976), Zn_{2}Mn_{3}O_{8} (Joubert & Durif, 1964a), ZnMGe_{3}O_{8} (M = Mn, Mg) (Joubert & Durif, 1964a), M_{2}Ge_{3}O_{8} (M = Zn, Co) (Joubert & Durif, 1964a), ZnMTi_{3}O_{8} (M = Mn, Cd) (Joubert & Durif, 1964a), M_{2}Ti_{3}O_{8} (M = Zn, Mn, Co) (Joubert & Durif, 1964a), V_{2}Co_{3}O_{8} (Joubert & Durif, 1964b), Li_{2}Mn_{3}CoO_{8} (Joubert & DurifVarambon, 1963), Li_{0.5+0.5x}Fe_{2.51.5x}Ti_{x}O_{4} (0 x 0.4, 1.2 x 1.57) (Arillo et al., 2003), Na_{4}Ir_{3}O_{8} (Okamoto et al., 2007), LiMnTiO_{4} (Arillo et al., 2008), Na_{4}Sn_{3}O_{8} (Iwasaki et al., 2002), LiNi_{0.5}Mn_{1.5}O_{4} (Wang et al., 2011), LiFe_{5x}Mn_{x}O_{8} (0 x 1) (Darul et al., 2005), LiFe_{5}O_{8} (Braun, 1952; Widatallaha et al., 2005), LiMg_{0.1}Ni_{0.4}Mn_{1.5}O_{4} (Wagemaker et al., 2004), LiNi_{0.5}Mn_{1.5}O_{4}, LiNi_{0.5}Mn_{1.5}O_{4} (Kim et al., 2004), Li_{1.25}Fe_{0.25}Ti_{1.5}O_{4} (Reale et al., 2003), Li_{2}Mn_{3}MO_{8} (M = Mg, Zn) (Strobel et al., 2003), LiMg_{0.5}Mn_{1.5}O_{4}, CuMg_{0.5}Mn_{1.5}O_{4} (Branford et al., 2002), Li_{x}Mg_{12x}Fe_{2+x}O_{4} (x 0.40) (Antic et al., 2002), LiMg_{x}Mn_{2x}O_{4} (x > 1) (Singh et al., 2009), Li_{2}CoTi_{3}O_{8} (Reeves et al., 2007), Li_{2}Zn_{3}O_{8}, Li_{2}Ge_{3}O_{8} (Hirota et al., 1988), LiM_{0.5}Ti_{1.5}O_{4}, LiM_{0.5}Ge_{1.5}O_{4} (M = Mg, Co, Ni, Zn) (Kawai et al., 1998; Hernander et al., 1996), Li_{1.33x}Co_{22x}Ti_{1+0.67x}O_{4} (Kremenovic & Antic, 2004), Li_{10.5x}Fe_{2.5x}Mn_{22x}O_{4} (Kremenovic & Antic, 2004), LiMn_{2y}Ti_{y}O_{4} (y > 1) (Petrov et al., 2005), LiFe_{0.5}Ti_{1.5}O_{4} (Avdeev et al., 2007) and others have the structures of ordered P4_{1}32 and P4_{3}32 phases. The features of the calculated P4_{1}32 and P4_{3}32 structures are shown in Fig. 5.
 Figure 5 Calculated lowsymmetry spinel structure with enantiomorphic space groups P4_{1}32 and P4_{3}32. Polyhedral presentation of the ordered enantiomorphic spinel structures (atoms B^{4b} and B^{12d} 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 k_{11}(_{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 lowsymmetry 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 lowsymmetry phase should be A^{4e}B^{4a}B^{4d}X^{8h}_{2}X^{8i}_{2}. The calculated structure is shown in Fig. 6. This structure contains chains of octahedra (B^{4a}X^{8h}_{2}X^{8i}_{4} and B^{4d}X^{8h}_{4}X^{8i}_{2}) shared by edges, running along the y and x axes, respectively (Fig. 6c). The lowsymmetry modifications of Li_{2}CoCl_{4} (Wussow et al., 1989), Li_{2}MnBr_{4} (Kanno et al., 1986), Li_{2}MgBr_{4} (Smirnov, 1966), LiVCuO_{4} (Joubert et al., 1965), Li_{1x}CuVO_{4} (0 x 0.2) (Kanno et al., 1992), LiSbZnO_{4} (Keramidebs et al., 1975) and Li_{2}FeCl_{4} (Lutz et al., 1989) have the Imma structure.
 Figure 6 Calculated lowsymmetry 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). 
The typical examples of anion ordering in Wyckoff position 32e are spinel lowsymmetry modifications with space groups [order parameter (), k_{11}(_{4})], [order parameter (, , ), k_{11}(_{7})], and P4_{1}2_{1}2 and P4_{3}2_{1}2 [order parameter (0, 0, , 0, 0, 0), k_{10}(_{2})].
Ordered spinel structure with space group . We have already considered the structure of the 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 A^{4a} and A^{4c} atoms at all, or only some of these atoms are absent, various types of Chevrel phases can be obtained (Talanov, 2005; Besnard et al., 2003). For example, if there are no A^{4a} and A^{4c} atoms, then a formula B^{16e}_{4}X^{16e}_{4}X'^{16e}_{4} is obtained. This formula corresponds to the Re_{4}Te_{4}S_{4} structure (Fedorov et al., 1994, 1996). If only one type of atom A in the general formula is absent, then we have the formula A^{4a}B^{16e}_{4}X^{16e}_{4}X'^{16e}_{4}. Such a formula describes the structures of GaMo_{4}O_{8} (Besnard et al., 2003) and Re_{4}As_{6}S_{3} (Besnard et al., 2003). If the positions of A^{4a} and A^{4c} atoms are partly occupied, then this formula describes the structures of such substances as Ga_{1.33}Cr_{4}S_{8} (Besnard et al., 2003) and Ga_{0.5}V_{2}S_{2}Se_{2} (Haeuseler et al., 2001).
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 AB_{2}X_{4} spinel, B_{4} clusters from metal atoms and anions are formed (Talanov et al., 2008) (Fig. 2). 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 symmetry (Talanov et al., 2008)].
We believe that the formation of these clusters causes the original physical properties of the Chevrel phases. In Figs. 7 and 8, examples of calculated structures of the Chevrel phases are given. In particular, Re_{4} clusters exist in structures of the Chevrel phases such as Re_{4}As_{6}S_{3} (Figs. 7ac) and Re_{4}Te_{4}S_{4} (Fig. 8).
 Figure 7 Calculated lowsymmetry anionordered spinel structure with space group and structural formula A^{4a}B^{16e}_{4}X^{16e}_{4}X'^{16e}_{4}. Atom presentation of the ordered spinel structure (a), tetrahedra in the ordered spinel structure (b), B^{16e} metallic tetrahedra (c). A substance with composition Re_{4}As_{6}S_{3} has a similar structure. In this structure there are Re_{4} clusters (Besnard et al., 2003). 
 Figure 8 Calculated lowsymmetry anionordered spinel structure with space group and structural formula B^{16e}_{4}X^{16e}_{4}X'^{16e}_{4}. A substance with composition Re_{4}Te_{4}S_{4} has a similar structure. In this structure there are Re_{4} clusters (Fedorov et al., 1994, 1996). 
Ordered spinel structure with space group . This structure is generated by irrep k_{11}(_{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 lowsymmetry 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 lowsymmetry phase should be A^{2c}B^{1a}_{1/2}B^{3d}_{3/2}X^{2c}X^{6h}_{3}. In Fig. 9, the features of the structure of the phases are shown. Ga_{3}O_{3}N has this structure (Boyko et al., 2011). This substance possesses unusual electronic properties. Boyko et al. (2011) considered three models of the Ga_{3}O_{3}N structure, described by space groups Ima2, Imm2 and . 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 . It should be noted that, according to our theory, at the formation of the 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 Ga_{3}O_{3}N should be Ga^{1a}_{1/2}Ga^{3d}_{3/2}Ga^{2c}N^{2c}O^{6h}_{3}.
 Figure 9 Calculated lowsymmetry ordered spinel with space group . Atom presentation of the ordered spinel structure (a); tetrahedra [A^{2c}X^{2c}X^{6h}_{3}] in the ordered spinel structure (b); distorted hexahedra [X^{2c}_{2}X^{6h}_{6}] with the central atom B^{1a} (c); distorted hexahedra [X^{2c}_{2}X^{6h}_{6}] with the central atom B^{3d} and tetrahedra [A^{2c}X^{2c}X^{6h}_{3}] (d); and hexahedral presentation of the ordered spinel structure (e). 
Ordered spinel structure with enantiomorphic space groups P4_{1}2_{1}2 and P4_{3}2_{1}2. The critical irrep inducing P4_{1}2_{1}2 and P4_{3}2_{1}2phase formation is the sixdimensional representation k_{10}(_{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 lowsymmetry phase should be A^{4a}B^{8b}_{2}X^{8b}_{2}X^{8b}_{2}. The features of the P4_{1}2_{1}2 and P4_{3}2_{1}2 structures are given in Fig. 10. An example of this structure is that of MgTi_{2}O_{4} (Isobe & Ueda, 2002; Schmidt et al., 2004). In this substance at a temperature of 260 K a phase transition occurs, accompanied by formation of tetragonal spinel modifications with space groups P4_{1}2_{1}2 and P4_{3}2_{1}2 (Isobe & Ueda, 2002; Schmidt et al., 2004). Highresolution synchrotron and neutron powder diffraction and also Xray investigation shows the existence of weak superstructure reflections (Schmidt et al., 2004). According to our calculations these reflections are caused by ordering of oxygen ions (Fig. 10) (Talanov et al., 2012). 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 lowsymmetry ordered MgTi_{2}O_{4} spinel modification have been studied (Talanov et al., 2012).
 Figure 10 Calculated lowsymmetry ordered structure of the tetragonal spinel modification of MgTi_{2}O_{4} (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). 
One type of superstructure with simultaneous cation ordering in spinel structure positions 8a and 16d is known. It is an ordered phase of LiZn_{0.5}Mn_{1.5}O_{4} with space group P2_{1}3. This phase is formed by two order parameters: ()^{4} and (0, , 0, , 0, )^{1}. These two irreps form a point group of order 192 in the sevendimensional space of the order parameter. The structural mechanism of the lowsymmetry phase formation is defined by representation k_{10}(_{3}) + k_{11}(_{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 P2_{1}3phase formation is much more complex than that proposed previously (Lee et al., 2002). The calculated structural formula of the lowsymmetry P2_{1}3 phase should be (A^{4a}_{1/2}A'^{4a}_{1/2})[B^{4a}_{1/2}B'^{12b}_{3/2}]X^{12b}X'^{12b}X^{4a}X'^{4a}. Experimental data on the structure of LiZn_{0.5}Mn_{1.5}O_{4}, obtained by neutron diffraction and Xray analyses, agree with the proposed structural formula (Lee et al., 2002). The calculated structure of the P2_{1}3 phase is shown in Fig. 11. The structural formula of LiZn_{0.5}Mn_{1.5}O_{4} should be (Li^{4a}_{0.5}Zn^{4a}_{0.5})[Li'^{4a}_{0.5}Mn^{12b}_{1.5}]O^{12b}O'^{12b}O^{4a}O'^{4a}. The symmetry of the order parameter, and the thermodynamics and mechanisms of formation of the atomic structure of the lowsymmetry ordered cubic LiZn_{0.5}Mn_{1.5}O_{4} oxide spinel have been studied by Talanov & Shirokov (2013).
 Figure 11 Fragments of the calculated lowsymmetry ordered structure of LiZn_{0.5}Mn_{1.5}O_{4}. 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 Mn^{12b} are dashed, octahedra with central ions Li'^{4a} are grey) are shown. 
The atomic ordering in the spinel structure was studied within a unified approach based on the Landau phenomenological phasetransition theory. Grouptheoretical 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 orderparameter symmetry and the space groups of the lowsymmetry spinel modifications, and calculated the stratification of Wyckoff positions 8a, 16d and 32e of the initial highsymmetry phase with a spinel structure at the transition to the lowsymmetry 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 lowsymmetry 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 14 will be helpful in research on the structures of new spinels and their lowsymmetry modifications.
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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