One-third of all compounds of the type MX crystallize in the sodium chloride (halite) structure (Fm3 – m), shown in Figure 3.16. In such compounds the metal atom or ion M is usually smaller than the electronegative element, X. The lattice is face-centred cubic with two different atoms associated with each lattice point: one, say X, at (0, 0, 0) and the other, M, at ( ) 1 2 0, 0, . Each of the two types of atom, considered in isolation, lies upon a facecentred cubic lattice, and each lies at the largest interstice of the other’s face-centred cubic lattice. From the diagram in Figure 3.5 and from inspection of Figure 3.16, it is then clear that each atom has a coordination number of 6, the neighbours being at the vertices of a regular octahedron. There are four formula units per conventional unit cell but there is clearly no trace of the formation of a molecule of MX in this structure.
Each {111} lattice plane specifies a double sheet of atoms, each of the sheets consisting entirely of atoms of one kind arranged at the points of a triequiangular net. If we denote sheets of atoms of one kind with a Greek letter and those of the other with a Roman letter, then the stacking sequence along a [111] direction can be described as A g B a C b A g B a C b … The spacing of the sheets of atoms denoted by letters from the same alphabet is a/ 3, where a is the lattice parameter.
All of the alkali halides, with the exception of CsCl, CsBr and CsI, crystallize with this crystal structure, as do many of the sulfides, selenides and tellurides of Mg, Ca, Sr, Ba, Pb and Mn, as well as the oxides of formula MO of Mg, Ca, Sr, Ba, Cd, Ti, Zr, Mn, Fe, Co, Ni and U and some transition metal carbides and nitrides such as TiC, TiN, TaC, ZrC, ZrN, UN and UC (see Section A7.2 in Appendix 7).
When the metallic ion is of variable valence, crystals with this structure often form with some ion positions unoccupied. For example, FeO crystals would be perfect if they contained only divalent iron, Fe2+ . If some Fe3+ ions are present then for every two Fe3+ ions, one of the sites on the iron sublattice must be empty. Such a structure is called a defect structure (see Section 10.1).
2, Caesium Chloride, CsCl (Pm3 – m)
CsCl, CsBr, CsI, as well as many intermetallic compounds such as CuBe, CuZn, AgCd, AgMg and FeAl (see Section A7.3 in Appendix 7), show the caesium chloride structure (Pm3 – m), which has a primitive cubic lattice with one atom of each kind associated with each lattice point, one, say X, at (0, 0, 0) and the other, M, at ( ) 111 , , 222 (Figure 3.17). The coordination number is 8 for each atom, the nearest neighbours being at 3 /2 a , where a is the lattice parameteق,
3, Sphalerite, a-ZnS (F4 – 3m)
The cubic form of ZnS, designated a-ZnS, shows the sphalerite structure2 (F4 – 3m) shown in Figure 3.18. This is also exhibited by the sulphides, selenides and tellurides of Be, Zn, Cd and Hg and the halides of Cu and AgI: see Section A7.4 of Appendix 7 for a tabulation of selected crystals with the sphalerite structure. The lattice is face-centred cubic with one atom of each kind associated with each lattice point, one at (0, 0, 0) and the other at ( ) 111 , , 444 , so the relationship to the structure of diamond (Section 3.4) is very close (Figure 3.13). Each type of atom considered in isolation from the other lies at the lattice points of a face-centred cubic lattice and each lies in the second largest interstice of the close-packed crystal structure of the other. The coordination number is 4, the nearest neighbours being atoms of the other kind at a distance 3 /4 a arranged at the vertices of a regular tetrahedron. The sequence of (111) planes is stacked at the same intervals as in diamond (Figure 3.12), but alternate planes are occupied by atoms of different chemical species. Thus, following the same nomenclature as for NaCl, the stacking sequence is g A a B b C g A … (Figure 3.13).
4, Wurtzite, b-ZnS (P63mc)
The sphalerite structure is derived from the c.c.p. structure by placing atoms of a different kind to those at the lattice points at every other tetrahedrally coordinated interstice. A related structure, also shown by ZnS, is obtained by filling alternate tetrahedrally coordinated interstices in the h.c.p. structure. This is the b-ZnS or wurtzite structure (P63mc). The lattice is hexagonal with atoms of one kind at (0, 0, 0) and ( ) 211 , , 332 and those of the other kind at (0, 0, u) and ( ) 211 , , 332 + u (Figure 3.18b). The value of u is very close to 0.375, i.e. 3 8 ; however, it is not constrained by symmetry to be exactly 0.375. From Figure 3.8b the relationship to the h.c.p. structure is obvious. Each atom is, of course, tetrahedrally coordinated with four of the opposite kind. The stacking of planes of atoms parallel to (0001) using the same nomenclature as for NaCl and a-ZnS is then A a B b A a B… or equivalently A a C g A a C g A … This structure is shown by a number of compounds (Table 3.3). The axial ratios shown in Table 3.3 and Section A7.5 of Appendix 7 are quite close to the ideal, 1.633, for hexagonal close packing of one type of ion with the other in the tetrahedral interstices.
5, Nickel Arsenide, NiAs (P63/mmc)
We described the sodium chloride structure as being derived from the c.c.p. structure by placing a second set of atoms in the octahedrally coordinated largest interstices of the structure. It is not surprising therefore that a structure exists in which atoms of a different kind are placed in the octahedral interstices of the h.c.p. structure. This is the nickel arsenide (nickeline)3 structure (P63/mmc) (Figure 3.19). The lattice is hexagonal, with atoms of one kind at (0,0,0) and ( ) 1 2 0, 0, and those of the other kind at ( ) 211 , , 334 and ( ) 1 2 3 , , 334 . For both kinds of atom the coordination number is 6, but while the sites occupied by one kind of atom, those denoted by open circles in Figure 3.19, lie in positions corresponding to a close-packed hexagonal arrangement (As), the others (Ni) lie at the lattice points of a primitive hexagonal lattice (if the As atoms are ignored), with a repeat distance in the c direction half that of the lattice parameter of the NiAs crystal structure.
This difference in stacking of the M and X atoms is obvious from Figure 3.8a and also by stating the stacking sequence of the planes of atoms along [0001] following the procedure for NaCl. The stacking sequence is A b A g A b A g … The atoms denoted by Greek letters lie in the centre of a trigonal prism of atoms denoted by Roman letters, while the atoms denoted by Roman letters are octahedrally coordinated. The axial ratios of some crystals of sulphides with this structure are listed in Table 3.4; the selenides, tellurides and antimonides of the metals listed often have the same structure: see Section A7.6 of Appendix 7. The values of c/a at room temperature depart from the ideal for hexagonal close packing of one type of ion. Usually the metal atom is situated in the type of site occupied by Ni in NiAs, but in PtB the ‘anti-NiAs’ structure is shown.
6, Calcium Fluoride, CaF2 (Fm3 – m)
If all of the tetrahedral interstices in the c.c.p. crystal structure are filled with atoms of a different kind from those in the c.c.p. crystal structure, we obtain the calcium fluoride (CaF2) or fluorite structure (Fm3 – m) (Figure 3.20). The lattice is face-centred cubic with atoms of one kind, for example Ca, at (0, 0, 0) and equivalent positions, and with F atoms at ( ) 111 , , 444 and ( ) 1 1 3, , 444 and equivalent positions. There are clearly four formula units per unit cell. The coordination number of Ca is 8, with the nearest neighbours being atoms of F arranged at the corners of a cube. If the calcium atoms are ignored, it is apparent that the fluorine atoms lie at the lattice points of a primitive cubic lattice. Each F atom has tetrahedral coordination with four Ca atoms.
The centres of atoms of one kind all form triequiangular nets of points parallel to {111} lattice planes. The separation of the atoms in these places is the same for the two types of atom. The stacking sequence of these planes along [111] can therefore be described as … aBg bCa g Ab aBg bCa … The planes of atoms denoted by successive Roman letters are regularly spaced from one another; the same is true of the planes of atoms denoted by Greek letters. This structure is shown by the fluorides of Ca, Sr and Ba and by the oxides of Zr, Th, Hf and U. Oxides and sulfides of alkali metals also show this structure; in these cases it is sometimes called the antifluorite structure. A tabulation of crystals with the flourite structure is given in Section A7.7 of Appendix 7.
7. Rutile, TiO2 (P42/mnm)
A common structure of compounds with the formula AX2, besides that of fluorite, is the rutile (TiO2) structure (P42/mnm) (Figure 3.21a). It is also sometimes called the cassiterite (SnO2) structure. A number of dioxides and fluorides have this crystals structure, such as those tabulated in Section A7.8 of Appendix 7. The lattice is primitive tetragonal with c/a about 0.65. There are titanium atoms at (0, 0, 0) and ( ) 111 , , 222 and oxygen atoms at ± (u, u, 0) and ( ) 11 1 , 22 2 ±+ − u u, , where u is close to 0.30 in all examples of the structure. Thus, there are two formula units per cell. A plan of the structure is shown in Figure 3.21b, projected down [001]. Each Ti atom is surrounded by six O atoms, but the octahedron of O atoms is not a regular octahedron. It is apparent that the crystal structure of rutile can be described in terms of how these Ti octahedra pack together. This leads conceptually to the more general geometry of the packing of polyhedra, which will be discussed in Section 4.4 in the context of the packing of atoms or ions within very complicated crystal structures,
Reference: https://crystal-algerien1970.blogspot.com/2020/03/crystallography-and-crystal-defects.html
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