The crystal structure of indium is very similar to the c.c.p. structure. It is body-centred tetragonal with c/a = 1.521. This axial ratio is such that, if referred to a face-centred tetragonal lattice instead of the conventional body-centred one, the axial ratio c/a = 1.076.
Therefore, the structure can be described as face-centred tetragonal (F4/mmm), with one atom at each lattice point and an axial ratio of 1.076.
2. Mercury (R3 – m)
Mercury also has a structure that can be described as distorted c.c.p. The primitive unit cell has a rhombohedral lattice with one atom at each lattice point (Figure 3.11). The axial angle a is 70.74° and a = 2.992 Å at 78 K. The primitive unit cell of a c.c.p. crystal has a = 60° (Figure 3.1b). Thus the mercury structure can be derived from the c.c.p. crystal structure by compressing it along a body diagonal of the cell. The atoms in the (111) planes of mercury are therefore arranged to form a triequiangular net but the spacing of these planes is too small to allow closest packing of spherical atoms. The nearest neighbours of any one atom are in adjacent (111) planes. The closest-packed directions are 〈100〉 of the primitive rhombohedral cell of side a (Figure 3.11). There are six nearest neighbours at a distance a. Mercury can, of course, also be referred to a rhombohedral face-centred cell containing four atoms to emphasize the relationship to the conventional cell of an c.c.p. crystal. This larger cell for Hg has a′ = 4.577 Å and axial angle a′ = 98.36° at 78 K (see Problem 3.4).
3. b-Sn (I41/amd)
b-Sn, also known as white tin, is the form of tin that is stable at room temperature. It has a body-centred tetragonal lattice with two atoms associated with each lattice point: one at the lattice point and the other at ( ) 1 1, 2 4 0, . The value of c/a at room temperature is 0.546. Each atom has four nearest neighbours at the vertices of an irregular tetrahedron and two more at a slightly greater distance (see Problem 3.5). The structure is a distorted form of the diamond structure (Section 3.4.), which is taken by the form of tin stable below room temperature known as a-Sn or grey tin.
4. Diamond (Fd3 – m)
The elements Si, Ge, a-Sn (the low-temperature allotrope of tin) and crystalline carbon stable at high temperature and pressure (diamond) all have the structure shown in Figure 3.12. The Bravais lattice is face-centred cubic and there are two atoms associated with each lattice point. The coordinates of these atoms can be taken to be (0, 0, 0) and ( ) 111 , , 444 , or, if the origin is taken to be at a centre of symmetry, ( ) 111 , , 888 ± . The coordination number is 4, with the nearest neighbours at a distance 3 /4 a arranged at the corners of a regular tetrahedron, outlined in Figure 3.12.
The atom centres lie at the corners of triequiangular nets in {111} planes. If these are projected on to (111) the stacking sequence of successive (111) planes can be described as CA AB BC CA AB BC. Successive planes are not equally separated from one another (Figure 3.13). The structure is very loosely packed. If equal spheres in contact have their centres at the atom centres, the packing fraction is only π 3 /16 0.34 = . The diamond crystal structure can be regarded as a special case of the sphalerite crystal structure (sub-section 3.5.3), in which the atoms at (0, 0, 0) and ( ) 111 , , 444 are of different types. For this reason, Figure 3.13 can also be used to show the stacking sequence of the sphalerite crystal structure. In sphalerite, there are no longer centres of symmetry, so the space group symmetry is F4 – 3m (Section 3.5).
5. Graphite (P63 /mmc)
The thermodynamically stable crystal structure of carbon at room temperature is graphite, shown in Figure 3.14a. The lattice is hexagonal with four atoms per unit cell, with coordinates (0, 0, 0), ( ) 1 2 0, 0, ( ) 1 2, 3 3 , 0 and ( ) 211 332 , , . At room temperature a = 2.46 Å and c = 6.71 Å; thus c/a = 2.72 is large. The atoms in one (0001) plane are arranged at the corners of regular hexagons. The x- and y-axes of the unit cell are shown in Figure 1.1. The structure can be built up by stacking successive hexagonal sheets of atoms one above the other along the z-axis so that the hexagons are in the same orientation, but with half the corners of the hexagons in one plane lying in the centres of the hexagons in adjacent planes (Figure 3.14a). If the atomic centres are all projected on to (0001), the stacking sequence can be described as ABABAB or ACACAC, where it must be remembered that the letters A, B and C refer to sheets of atoms arranged at the corners of hexagons, rather than at the vertices of equilateral triangles, as in the h.c.p. crystal structure, for example.
Each atom has three nearest neighbours at a separation of a/ 3 in the basal plane. Half of the atoms in any one hexagonal layer have atoms directly above and below them at a distance of c/2. The separation of the nearest neighbours in the (0001) planes is only 1.42 Å (compared with 1.54 Å in diamond) and this is much smaller than the separation of the hexagonal sheets – 3.35 Å. Graphite is therefore said to possess a layer structure because the atoms are strongly bonded within a sheet, and the sheets are only weakly bound to one another through van der Waals forces.
If graphite crystals are ground or otherwise severely deformed at low temperature, another form of graphite can be detected in which the atomic hexagons are arranged in the sequence ABCABCABC or alternatively CBACBACBA. This crystal structure has a rhombohedral Bravais lattice (space group R3 – m). The primitive rhombohedral cell has a = 3.635 Å and a = 39.5° at room temperature. There are two atoms per unit cell with coordinates ±(u, u, u) where u = 1 6 (= 0.167).
The structure of hexagonal boron nitride thermodynamically stable at room temperature is closely related to that of graphite (Figure 3.14b). The atoms occur in hexagonal sheets, but the sheets are stacked directly above one another along [0001] so that the stacking sequence is described as AAAAA…, with unlike atoms above one another in consecutive layers (see Figure 3.14b). In any one sheet there are equal numbers of B and N atoms arranged so that B and N alternate around any one atomic hexagon. At room temperature the B–N separation in the sheets is 1.45 Å and the separation of the sheets is 3.33 Å.
7. Arsenic, Antimony and Bismuth (R3 – m)
The structures of As, Sb and Bi are also based on a primitive rhombohedral Bravais lattice, with two atoms associated with each lattice point. The structure is shown in Figure 3.15. The coordinates of the atoms in the rhombohedral unit cell are ±(u, u, u); u is a little less than 1 4 and a somewhat less than 60° (see Table 3.2). The structure is easily visualized as being made up of sheets of atoms lying perpendicular to [111]. Within each sheet the atoms are arranged at the corners of triequiangular nets. Referring to Figure 3.15 and noting the values of u, the stacking sequence of these nets along [111] is BA CB AC BA… If u is 1 4 , the spacing of adjacent nets will be regular. In fact, each atom in any one sheet has three nearest neighbours in the closest adjacent sheet at d1 (see Table 3.2) and three at a slightly greater distance in the sheet on the other side at d2. The possession of three nearest neighbours fits with the positions of As, Sb and Bi in the periodic table, since each atom is expected to form three covalent bonds. If a = 60° and 1 4 u = in this structure, each atom has six closest neighbours and the arrangement of the atoms is at the points of a simple cubic lattice.
Reference: https://crystal-algerien1970.blogspot.com/2020/03/crystallography-and-crystal-defects.html
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