1. Perovskite (Pm3 – m)
This primitive cubic crystal structure is a very common structure of crystals of compounds of the type MM′X3. It is named after the mineral perovskite, CaTiO3, which has this crystal structure above 900 °C; at room temperature, CaTiO3 has an orthorhombic crystal structure belonging to the space group Pnma (see Section 2.15).4 In this crystal structure, the calcium (M) and oxygen (X) ions taken together form a c.c.p. arrangement with the titanium (M′) ions in the octahedral voids (Figure 3.22). The coordination number of Ca is 12, that of Ti is 6 and that of O is 2. The atomic coordinates in the primitive unit cell are Ti ( ) 111 ,, , 222 Ca (0, 0, 0) and O ( ) ( ) ( ) 11 11 1 1 22 22 2 2 , ,0 , 0, , , , 0, . The description of the perovskite crystal structure in terms of the packing of titanium octahedra will be deferred to Section 4.4.
There are many slightly distorted forms of the perovskite structure that are important in solid-state devices. At room temperature, barium titanate, BaTiO3 (P4mm), has a crystal structure that is a tetragonally distorted form of the Pm3 – m high-temperature structure. At lower temperatures, further distortions convert this into an orthorhombic crystal structure (Amm2) and a rhombohedral crystal structure (R3m) [10]. Pb(Zr1−xTix)O3, in which the Zr and Ti ions both occupy the M′ position, has two different rhombohedral (R3m and R3c) crystal structures [11] and a tetragonal (P4mm) form at room temperature, dependent on the value of x. The high-temperature superconducting oxide YBa2Cu3O7−d , where d is a small fraction << 1 indicating an oxygen deficiency, has a variety of crystal structures all based on a three-layered distorted perovskite structure [12]. Thus, for example, the superconducting phase for values of x of the order of 0.1 or less has an orthorhombic Pmmm crystal structure. The sheets of Cu and O atoms parallel to (001) in this structure are critical to the superconducting behaviour of this oxide.
2. a-Al2O3 (R3 – c), FeTiO3 (R3 – ) and LiNbO3 (R3c)
Sapphire (a-Al2O3 or corundum) possesses a large unit cell with the oxygen ions in an h.c.p. arrangement. Aluminium ions occupy the octahedral interstices. Since the formula is Al2O3, only two-thirds of these are filled, as shown in Figure 3.23. The structure can be easily described by locating the ‘missing’ aluminium ions. If all the atomic positions are projected on to (0001) of the hexagonal cell and Roman letters denote the oxygen ion positions and Greek letters the positions of missing aluminium ions, the stacking sequence is Ag1 Bg2 Ag3 Bg1 Ag2 Bg3 Ag1 …, where the positions g1, g2, g3 are indicated by their suffixes in Figure 3.23.
The hexagonal unit cell is then six oxygen layers high, and contains six formula units of Al2O3. At room temperature, the lattice parameters of the hexagonal cell are a = 4.759 Å, c = 12.993 Å and so c/a = 2.731. If c/a were to equal 2 2 = 2.828, the oxygen ions would be in an ideal h.c.p. arrangement (the factor of 3 greater than 8/3 arises from the 30° rotation of the x- and y-axes of the rhombohedral unit cell relative to the x- and y-axes defining the ideal h.c.p. arrangement). The lattice translations in (0001) of the triply primitive hexagonal cell are marked in Figure 3.23. The rhombohedral primitive cell contains two formula units of Al2O3 and has a = 5.128 Å and a = 55.27°. If the oxygen ions were in the ideal h.c.p. arrangement, the value of a would be 53.78°.
FeTiO3 (ilmenite, R3 – ) has the same structure as a-Al2O3 except that the Fe and Ti atoms are distributed in alternating octahedral sheets. In LiNbO3 (R3c), the Li and Nb atoms are distributed in the same octahedral sheet so that one species occupies the g2 position while the other occupies the g3 position. The crystal structure of LiNbO3 can also be described as a distorted form of the ideal perovskite structure.
3. Spinel (Fd3 – m), Inverse Spinel and Related Structures
The spinel structure, shown by MgAl2O4 and by other mixed oxides of di- and trivalent metals, has a unit cell containing 32 oxygen ions in almost perfect cubic close packing. Since there would be four oxygen ions per unit cell if only the oxygen ions were present, it is apparent that the edge of the unit cell for spinel is twice that which would be expected for the oxygen ions alone in cubic close packing. Eight of the 64 tetrahedral interstices in the unit cell are filled by the divalent Mg ions and 16 of the 32 octahedral interstices are filled with the trivalent Al ions. The Mg ions considered as a group of ions form a structure of the diamond type (Figure 3.24). One-eighth of the face-centred cubic unit cell is shown in this figure.
Some oxides of composition MM′2O4 show a structure called ‘inverse’ spinel to distinguish it from the ‘normal’ spinel which we have just described. MgFe2O4 is an example of an inverse spinel. In this oxide, the oxygen ions are arranged in cubic closest packing and the same two types of interstice are involved. However, the cations are arranged differently. Ideally, of the 16 iron ions per unit cell, eight occupy the eight tetrahedral interstices. The 16 octahedral interstices are occupied by the remaining eight iron ions and by the eight magnesium ions. The Mg and Fe can occur at random amongst the occupied octahedrally coordinated sites. To emphasize the difference from a normal spinel, the formula of this inverse spinel is sometimes written Fe(MgFe)O4, or generally M′(MM′)O4. In fact, the structure of MgFe2O4 deviates somewhat from this ideal; the number of iron atoms in tetrahedral sites is not exactly equal to the number in octahedral sites.
Chrysoberyl, BeO.Al2O3 (Pnma), is isomorphous with the olivine (Mg1−xFex)2SiO4 group of minerals. In this case the oxygen ions are arranged in a slightly distorted h.c.p. structure [13]. As for the spinels, the metal ions in chrysoberyl are distributed amongst the tetrahedral and the octahedral sites; in the case of chrysoberyl, half of the octahedral sites are occupied by Al ions and one-eighth of the tetrahedral sites by Be.
4. Garnet (Ia3 – d)
The concept of the occupation of the interstices in a structure by different types of ion is useful in describing the garnet structures. While these are perhaps best known as gemstones, garnets are very important in many solid-state physics devices. Various garnets can be ferrimagnetic (e.g. YIG, see below) and can form excellent laser hosts (e.g. YAG).
The garnets occur naturally as the silicates of various di- and trivalent metals. The archetypal garnet mineral has a formula 3MO.M′2O3.3SiO2 or M3M′2Si3O12, where M is a divalent metal ion and M′ a trivalent ion. The technologically important examples of yttrium iron garnet, Y3Fe5O12 (YIG), and yttrium aluminium garnet, Y3Al5O12 (YAG), do not contain silicon. The substitution is possible because in a typical natural garnet such as grossular, 3CaO.Al2O3.3SiO2, it is possible to substitute ‘YAl’ for ‘CaSi’. Replacing all the Ca and Si in this way produces 3YAlO3.Al2O3; that is, Y3Al5O12. Because these substitutions are possible, a general formula {C3}[A2](D3)O12 is often used for garnets, where O denotes the oxygen ion or atom and C, A and D denote cations.
The space group is Ia3 – d and so the crystal structure clearly has a b.c.c. lattice. There are eight formula units per unit cell. There are 96 so-called h sites, which are occupied by oxygen. {C3} denotes an ion in a tetrahedral site; that is, a site surrounded by four oxygen ions. In grossular, 3CaO.Al2O3.3SiO2, these sites would be occupied by silicon. There are 24 of these sites per unit cell. [A2] represents an octahedral site, which is surrounded by six oxygen ions; there are 16 of these per unit cell. These would be occupied by aluminium in grossular. (D3) denotes the so-called dodecahedral site, which is surrounded by eight oxygen ions; this would be occupied by Ca in grossular. This site is variously described as a triangular dodecahedron, hence the name, or as a distorted cube. It is illustrated in Figure 3.25. A triangular dodecahedron is a polyhedron with 12 faces, each of which is a triangle.
The substitution of an enormous number of cations for one another is possible within the garnet structure. For example, trivalent rare earth ions can be introduced. If strict chemical rules of valence applied then {C} sites would be occupied by four valent cations, such as silicon, [A] by trivalent ions, such as Al, rare earth, Y or Fe3+ ions and so on, and (D) by divalent ions, such as Ca, Mg, Fe2+ or others. The ability to substitute two cations to balance the charges of two other cations gives rises to a whole host of possibilities for substitution. Hence, a single simple valence rule cannot be followed. Furthermore, since the substitution of large ions such as the rare earths is possible, the oxygen ions are pushed apart to accommodate the cations, while retaining the cubic I lattice.
5. Calcite, CaCO3 (R3 – c)
Many complicated structures are most easily described as distorted forms of the simpler ones. For instance, the structure of calcite (one form of CaCO3) can be derived from that of sodium chloride by identifying the sodium ions with calcium ones and the carbonate radical CO3 with Cl. If the sodium chloride structure is imagined to be compressed along a [111] body diagonal until the angle between the axes, originally 90°, becomes 101.89°, we produce the doubly primitive cell of calcite containing four formula units. Cleavage occurs parallel to {100} of this cell. The CO3 group is triangular, with the C in the centre of the triangle and the plane of the triangle normal to the direction [111] of this rhombohedral cell. This cell is not a true unit cell – it is a subcell of a true multiply primitive rhombohedral unit cell in which the repeat direction along [111] needs to be doubled.
The primitive unit cell of calcite contains just two formula units of CaCO3. It is shown in Figure 3.26, where it is compared with the cleavage rhombohedral pseudo unit cell. Calcium ions are at (0, 0, 0) and ( ) 111 222 ,, . The centre of the CO3 radical is at ±(u, u, u). Note that u is close to 1 4 in all examples of this structure; in calcite it is 0.259.
The primitive cell of calcite has a = 6.375 Å and α = 46.07°; the cleavage pseudo unit cell has a = 6.424 Å and α = 101.89°, while the true multiply primitive cleavage unit cell has a = 12.828 Å and α = 101.89°. The hexagonal unit cell describing the calcite crystal structure has a = 4.989 Å and c = 17.062 Å.
Reference: https://crystal-algerien1970.blogspot.com/2020/03/crystallography-and-crystal-defects.html
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