ISSN 00201685, Inorganic Materials, 2011, Vol. 47, No. 9, pp. 990–998. © Pleiades Publishing, Ltd., 2011. Original Russian Text © V.V. Ivanov, V.M. Talanov, V.B. Shirokov, M.V. Talanov, 2011, published in Neorganicheskie Materialy, 2011, Vol. 47, No. 9, pp. 1091–1100.
Crystal Chemistry and Formation Mechanism of Tetragonal MgTi2O4 V. V. Ivanova, V. M. Talanova, V. B. Shirokovb, and M. V. Talanovc a
South Russia State Technical University, ul. Prosveshcheniya 132, Novocherkassk, Rostov oblast, 346400 Russia b Southern Scientific Center, Russian Academy of Sciences, ul. Chekhova 41, RostovonDon, 344006 Russia c Southern Federal University, pr. Stachki 194, RostovonDon, 344090 Russia email:
[email protected] Received December 9, 2010; in final form, March 16, 2011
Abstract—Symmetry methods of the theory of secondorder phase transitions are used to clarify the struc tural mechanism of the transition from the cubic (Fd3m) to the tetragonal (P41212) phase of magnesium titan ite. The proposed mechanism is consistent with all experimental structural data reported to date for magne sium titanite. We analyze the crystalchemical features of the structure of tetragonal MgTi2O4 and infer the presence of metallic pico and nanostructures: two types of Ti2 dimers, two types of helices along the twofold and fourfold axes of the tetragonal cell, and two types of infinite onedimensional chains of titanium ions. DOI: 10.1134/S0020168511090093
INTRODUCTION Recent years have seen considerable progress in studies of mixed transitionmetal oxides whose struc ture has socalled geometrically frustrated lattices [1– 3]. The term geometric frustration refers to structures with local order defined by lattice geometry. AB2O4 spinels have tetrahedral and octahedral cat ion sites. The following distinctive crystalchemical fea ture of the spinel structure is of special importance: the octahedral cations (B cations) make up metallic tetra hedra (Fig. 1a). The network of these tetrahedra is sometimes referred to as a pyrochlore lattice, because the B cations in the pyrochlore structure form the same sublattice as in spinels (Fig. 1b). The B tetrahedra form a geometrically frustrated structure. If the sites of this lattice are occupied by transition metal cations, such spinels have unusual magnetic, electrical, and optical properties. There is now intense research interest in titanite, vanadite, chromite, and iridite spinels [4–8], which have been found to have special features of atomic, electronic, spin, and charge order, associated with the geometry of the B sublattice. In particular, considerable scientific interest has been attracted by magnesium titanite, MgTi2O4 [9–17]. In cubic MgTi2O4, the Ti3+ ions reside in the sites of a “pyrochlore lattice” formed by the octahedral cations. At T ⯝ 260 K, the material undergoes a metal–insula tor transition [9], accompanied by changes not only in its electrical properties but also in its magnetic [10, 14], optical [11], and thermodynamic [14] properties. In addition, this transformation is accompanied by an unusual helical tetragonal distortion of the structure. The resultant lowsymmetry phase has space group P41212 [9]. There is conclusive neutron and Xray dif
fraction evidence of spin dimerization in the tetragonal phase: dimers of the shortest Ti–Ti bonds are in a spin singlet state. This unusual structural state of the tetragonal phase prompts further research into the crystal chemistry and structural mechanism of the helical distortion in mag nesium titanite. We use methods of the theory of sec ondorder phase transitions, which were tested in ear lier structural studies of a variety of spinels [6–8]. In this paper, we propose a structural mechanism of the for mation of lowsymmetry MgTi2O4 and identify and analyze new crystalchemical features of the structure of tetragonal magnesium titanite. STRUCTURAL MECHANISM OF THE Fd3m → P41212 PHASE TRANSITION IN SPINEL MgTi2O4 The key points in selecting a critical irreducible rep resentation (IR) and stationary vector (order parame ter) for describing the phase transition of MgTi2O4 are the reliably identified space group of the tetragonal phase (enantiomorphic space groups P41212 and P43212) and the experimental finding that the volume of the primitive cell of the lowsymmetry phase is twice that of the highsymmetry phase [9]. The primitive cubic spinel cell contains two formula units, so the vol ume of the primitive cell of the tetragonal phase is twice that of the cubic spinel structure. The volume of the tet ragonal unit cell is half that of the cubic unit cell and, hence, there are four MgTi2O4 formula units per unit cell. Grouptheoretical analysis of phase transitions in crystals with space group Fd3m [18–21] indicates with certainty that the critical IR inducing the phase transi tion of MgTi2O4 is τ104. The designation is taken from Kovalev [22]: the first number indicates the wave vector, and the second indicates the number of the IR corre sponding to this wave vector. To the phase identified in
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Fig. 1. (a) Octahedral (B site) cations in the unit cell of the spinel structure (the tetrahedral cations and oxygens are omitted); (b) pyrochlore lattice, formed by the B site of an AB2O4 spinel; (c) Laves polyhedron made up of tetrahedra in a cubic spinel; (d) supertetrahedron.
experiments corresponds the oneparameter, sixcom ponent order parameter (C 0 0 0 0 0) [18–20]. In the Fd3m cubic phase of the normal spinel MgTi2O4, the atoms occupy sites of two invariant and one univariant lattice complexes with the following characteristics: Mg in position 8a ( 4 3m symmetry), lattice complex D, coordinates (0, 0, 0); Ti in position 16d ( 3 m), lattice complex T, coordinates (1/8, 1/8, 1/8); and O in position 32c (3m), lattice complex D4xxx, coordinates (xC, xC, xC), where xC = 3/8 + δ = 0.3842, δ = δX = δY = δZ = 0.0092 at 275 K [9] ( 3 δ is the displacement of the oxygen atoms in the [111] direction of the cubic cell). The structural formula of the cubic spinel is Mg 8a Ti 16d O432e. According to Sakhnenko et al. [19, 20], the critical IR τ104 enters into the composition of the mechanical INORGANIC MATERIALS
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representation in positions 8а, 16d, and 32е and also enters into the composition of the permutation repre sentation in position 32е. Therefore, the Fd3m → P41212 phase transition of spinel MgTi2O4 is due to dis placements of the magnesium, titanium, and oxygen atoms and oxygen ordering. With allowance for the dis placements, the atomic coordinates in the basis of the parent cubic cell are Mg (a, 0, 0), Ti (1/8 + b, 1/8 + c, 1/8 + d), and O (3/8 + δX + e, 1/8 – δY + f, 1/8 – δZ + g) and (1/8 – δX + i, 3/8 + δY + k, 1/8 – δZ + m). The atomic displacements convert the facecen tered cubic cell to a primitive tetragonal cell with parameters aT = 2 aC , and cT = aC. . The basis vectors 2 of the Bravais cell of the P41212 phase A1', A2' , A3' are
(
)
related to those of the Fd3m phase ( A1, A2, A3 ) by (Fig. 2)
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Fig. 2. Relationship between the cubic and tetragonal unit cells of magnesium titanite.
⎛ '⎞ ⎜ A 1 ⎟ ⎛ 0.5( A1 − ⎜ '⎟ ⎜ ⎜ A 2 ⎟ = ⎜ 0.5( A1 + ⎜ ' ⎟ ⎜⎝ A3 ⎜ A3 ⎟ ⎝ ⎠
A2) ⎞ ⎟ A2) ⎟ . ⎟ ⎠
The atomic coordinates in the tetragonal cell (x', y', z') can be obtained from those in the cubic cell as follows:
⎛ x' ⎞ ⎛ 1 −1 0 ⎞ ⎛ x ⎞ ⎛1 4 ⎞ ⎜ y' ⎟ ⎜ ⎟ ⎜ ⎟ = 1 1 0 y + ⎜ 0 ⎟. ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎝ z ' ⎠ P 41212 ⎝ 0 0 1 ⎠ ⎝ z ⎠ Fd 3m ⎝ 0 ⎠ This matrix equation takes into account the shift of the coordinate system along the X axis by onequarter of the cubic spinel cell parameter in order to match the theoretical atomic coordinates in the tetragonal cell of magnesium titanite with experimental data reported by Schmidt et al. [9]. The Fd3m → P41212 phase transfor mation involves a transformation of the lattice com plexes of the Fd3m phase [23]:
⎛ ⎞ I cv D1xx ⎛ D ⎞ ⎜ ⎟ v ⎜ T ⎟ → 2⎜ I c D1xx2 yz . ⎟ ⎜⎜ ⎟⎟ v v ⎜ ⎟ 4 D xxx ⎝ ⎠ Fd 3m ⎝ I c D1xx2 yz + I c D1xx2 yz ⎠ P 41212
Thus, in the tetragonal spinel (P41212) phase of MgTi2O4, the atoms occupy sites of one univariant and three trivariant lattice complexes with the following characteristics: Mg in position 4a (2), I cv D1xx, coordi nates (x, x, 0), where x = 1/4 + a; Ti in position 8b (1), I cv D1xx2yz, coordinates (x1, y1, z1), where x1 = –1/4 + (b – c), y1 = 1/2 + (b + c), and z1 = 1/8 + d; O(1) in position 8b' (1), I cv D1xx2yz, coordinates (x2, y2, z2), where x2 = 1/2 + (e−f), y2 = 3/4 + (e + f), and z2 = 1/8 + g; and O(2) in position 8b'' (1), I cv D1xx2yz, coordinates (x3, y3, z3), where x3 = (i – k), y3 = 3/4 + (i + k), and z3 = 1/8 + m. The structural formula of the tetragonal spinel is Mg 4aTi 82bO 82b'O 82b''. Using a comparative analysis of calculation results and experimental data for the tetragonal phase of mag nesium titanite at 200 K, Schmidt et al. [9] evaluated the displacement of the atoms from their equilibrium position in the cubic cell for Mg (a = 0.0313 Å), Ti (b = 0.0265 Å, c = 0.0271 Å, d = –0.0696 Å), O(1) (e = 0.0072 Å, f = 0.0121 Å, g = –0.1104 Å), and O(2) (i = 0.0066 Å, k = 0.0506 Å, m = 0.0153 Å). The displacements of the atoms in the cubic cell and their positions in projections of the tetragonal and cubic cells are presented in Figs. 3 and 4. INORGANIC MATERIALS
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Fig. 3. (a, b) Cation and (c, d) anion displacements in the cubic cell of MgTi2O4. Only the symmetry axes along the Z axis are indi cated.
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Fig. 4. (a) Cation and (b) anion (two types of oxygen atoms) sublattices of the tetragonal cell of MgTi2O4. INORGANIC MATERIALS
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Fig. 5. Ti4 tetrahedra in the cell of the tetragonal (P41212) phase of magnesium titanite (MgTi2O4): (a) connectivity of the tetrahedra; (b) double helices made up of cornersharing tetrahedra (shown dark) and isolated MgO4 tetrahedra; (c) Laves polyhedron, com posed of six edges and four faces of ten tetrahedra; (d) supertetrahedron made up of tetrahedra (the heavy solid, heavy dashed, thin solid, and thin dashed lines represent the shortest, two intermediate, and longest Ti–Ti bonds).
CRYSTALCHEMICAL FEATURES OF THE STRUCTURE OF TETRAGONAL MGTI2O4 In the cubic spinel structure, the cornersharing Ti4 tetrahedra and isolated MgO4 tetrahedra form two types of double helices along the X, Y, and Z axes, with four fold rotational symmetry 41 and 43. The system of heli ces that share structural elements and are thus linked to one another form a threedimensional connected net work of Ti4 tetrahedra (Fig. 1). The {333} tetrahedron
and {366} truncated tetrahedron, also referred to as the Laves polyhedron (Fig. 1c), can be considered spatial “cells” of this network. Four faces of the Laves polyhe dron have the form of conformationally distorted hexa gons of tetrahedra, which have identical, 3.008 Å sides. A tetragonally distorted spinel structure retains the system of double helices composed of distorted Ti4 tet rahedra and monoclinically distorted isolated MgO4 tetrahedra (Figs. 5a–5c), but the axial symmetry of the INORGANIC MATERIALS
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Fig. 6. Configuration of all the Ti–Ti bonds (a) in the bulk and (b) in the XY plane of the tetragonal phase (2 × 2 × 2 cell) of magne sium titanite: bond lengths of 2.853 (heavy solid lines), 3.007 (heavy dashed lines), 3.014 (thin solid lines), and 2.853 Å (thin dashed lines).
two types of helices is 41 and 21, respectively. The spatial “cell” is a distorted Laves polyhedron with identical “hexagonal” faces. The 18 edges of the Laves polyhe dron comprise 6 edges of 6 tetrahedra whose geomet ric centers form an octahedron and 12 edges of 4 tet rahedra that complement it to give a supertetrahe dron (Figs. 5d, 5e). The sides of the hexagons differ from one another and from those in the cubic spinel. In addition to similar Ti–Ti bonds (3.007 and 3.014 Å), there are considerably longer (3.157 Å) and shorter (2.853 Å bonds (Fig. 5). The shortest (s) and longest (l) bonds between the titanium atoms form two types of metallic picostruc tures: isolated Ti2 dimers which are linked to one another through the other two types of intermediate Ti–Ti bonds (i1 and i2) to form a threedimensional net work in the interior of the tetragonal cell (Figs. 6, 7). It is worth pointing out that the two types of inter mediate bonds differ markedly in configuration (Fig. 7). The shorter intermediate bonds (3.007 Å in length) form a system of onedimensional metallic nanostruc tures in the [001] direction of the tetragonal cell: helical configurations (Fig. 7c) isolated from each other and INORGANIC MATERIALS
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from the titanium–titanium dimers (bond sequence in the helix: …i1–i1–i1–i1) (Fig. 7a). The longer Ti–Ti intermediate bonds (3.014 Å in length) form two sys tems of isolated lines oriented alternately in the mutu ally perpendicular directions [100] and [010] of the tet ragonal cell (Fig. 7d). The two systems of networks of intermediate bonds (helical and linear) share lattice sites with each other and with the isolated Ti2 dimers (2.853 Å) and have the form of a threedimensional connected network (Fig. 6). They sort of link the struc ture into a single whole. In the [112] direction of the tet ragonal cell, the bond sequence is …–s–i1–l–i1–…. Note that there are also helical nanostructures in the [001] direction of the tetragonal cell, formed by the two types of Ti2 dimers (Fig. 8). The axial symmetry of these helices is 21, and the bond sequence is …–s–l–s–l–…. The existence of two types of helices upon the forma tion of lowsymmetry forms of crystals from their high symmetry forms lends support to the compensation principle proposed by Talanov [24, 25]. Cooling from 200 to 25 K causes insignificant changes in atomic displacements and in the metrics of the tetragonal phase [9], which have little effect on
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Fig. 7. Bonding configuration in the tetragonal (2 × 2 × 2) cell of magnesium titanite: (a) shortest bond, 2.853 Å; (b) long bond, 3.157 Å; (c, d) intermediate bond lengths of 3.007 and 3.014 Å, respectively.
interatomic distances. According to experimental data [9], the interatomic distance in the Ti–Ti dimer decreases from 2.853 to 2.852 Å. At the same time, even slight changes in the shorter titanium–titanium bonds
in the tetragonal phase below 200 K may have a signifi cant effect on spin–orbit exchange interaction and, hence, on the magnetic and optical properties of mag nesium titanite. INORGANIC MATERIALS
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Fig. 8. (a) Two systems of helical onedimensional nanostructures with axial symmetry 41 and 21 in the structure of the tetragonal spinel phase; (b) bond sequences in the helices: …i1–i1–i1–i1… (bond length, 3.007 Å) and …s–l–s–l… (bond lengths, 2.853 and 3.157 Å). The arrows show the motion direction along the helices upward.
CONCLUSIONS The P41212 phase of magnesium titanite results from displacements of magnesium, titanium, and oxygen ions and 1 : 1 oxygen ordering. The proposed structural mechanism of the formation of tetragonal MgTi2O4 enables theoretical interpretation of all experimental data: the distribution of the atoms over the positions of the lowsymmetry form of magnesium titanite; the weak superlattice reflections associated with the oxygen order in the anion sublattice; and the lack of charge ordering of the titanium ions: these only shift away from the center position of the trigonally distorted octahedra in the cubic spinel struc ture (the six nearest neighbor titanium–titanium dis tances become inequivalent, and two of the six dis tances differ markedly from the Ti–Ti bond distances in the cubic phase). The crystalchemical features of the structure of the lowsymmetry phase of magnesium titanite are respon sible for the existence of metallic pico and nanostruc tures: two types of Ti2 dimers with the shortest and long est Ti–Ti bonds, two types of titanium helices along the INORGANIC MATERIALS
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fourfold and twofold axes of the tetragonal cell, and two types of infinite onedimensional chains of titanium ions. REFERENCES 1. Pauling, L., The Structure and Entropy of Ice and of Other Crystals with Some Randomness of Atomic Arrangement, J. Am. Chem. Soc., 1935, vol. 57, pp. 2680– 2684. 2. Verwey, E.J.W., Electronic Conduction of Magnetite (Fe3O4) and Its Transition Point at Low Temperatures, Nature (London), 1939, vol. 144, p. 327. 3. Anderson, P.W., Ordering and Antiferromagnetism in Ferrites, Phys. Rev., 1956, vol. 102, pp. 1008–1013. 4. Kugel’, K.I. and Khomskii, D.I., Jahn–Teller Effect and Magnetism: Transition Metal Compounds, Usp. Fiz. Nauk, 1982, vol. 136, no. 4, pp. 624–664. 5. Tokura, Y. and Nagaosa, N., Orbital Physics in Transi tionMetal Oxides, Science, 2000, vol. 288, no. 5465, pp. 462–468. 6. Van den Brink, J., OrbitalOnly Models: Ordering and Excitations, New J. Phys., 2004, vol. 6, pp. 1–16. 7. Radaeli, P.G., Orbital Ordering in TransitionMetal Spinels, New J. Phys., 2005, vol. 7, pp. 1–22.
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