Cubic Crystals
- Practical Electron Microscopy and Database -
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If all of the atoms in a crystal are the same size, the maximum number of atoms that are coordinated around any individual is 12 (also called 12-fold coordination) in which there are two ways that atoms can be packed. For a single layer of atoms with an equal size, there are two kinds of voids between the atoms as shown in Figure 3547a (a):
i) B voids with triangles pointing up.
ii) C voids with the triangles pointing down.

As shown in Figure 3547a (b), if the second layer of atoms is added and its atoms occupy the space above the B voids and the third layer of atoms is added and its atoms occupy the positions over the C voids in the A layer, this will result in cubic closest packing with a stacking sequence of ABC ABC ABC.... etc.

 (a) (b)

Figure 3547a. Formation of cubic closest packing: (a) Single layer of atoms, and (b) Formed cubic closest packing structure.

According to Radius Ratio Rule, the limiting cation-to-anion radius ratios for cubic ionic lattices can be obtained as shown in Table 3547a.

Table 3547a. Cation-to-anion radius ratios (r+/r-) for coordination numbers (CN) 12 and 8.

CN
Crystal type
Geometric shape

Material examples and their r+/r-

1.0
12 Hexagonal or cubic closest packing   Metals
0.732 - 1.0
8 Cubic (i) Binary AB: CsCl (1.13), KCl (0.91), CsBr (0.862); (ii) Binary AB2: CaF2 (0.727); (iii) Others: NH4Br

Close-packing of equal spheres (e.g. the same atoms in a crystal) can form the trigonal, hexagonal or cubic crystal systems. The structure belongs to the trigonal system if the structure has the minimum symmetry, while it belongs to the hexagonal system if it has a 63 axis of symmetry.

Table 3547b and Figure 3547b show the cubic crystal systems and the schematic illustrations of the cubic lattices, respectively.

Table 3547b. Cubic crystal systems.

Crystal
family
Crystal
system
Required
symmetries
of point group
Point
group
Space
group
Bravais
lattices
Lattice
system
Cubic
4 three-fold axes of rotation 5 36 3 Cubic

Figure 3547b. Schematic illustrations of the Bravais lattices of cubic crystals.

Table 3547c. Relationship between Laue classes and point groups.

System
Essential symmetry
Laue class (Diffraction symmetry)
Point Groups (Hermann–Mauguin notation)
None 1, -1
2/m 2/m 2, m, 2/m
222 or 2mm
mmm
mmm 222, mm2, mmm
4/mmm
4/m
4, -4, 4/m
4/mmm 422, -42m, 4mm, 4/mmm
3 3, -3
-3m1 321, 3m1, -3m1
-31m 312, 31m, -31m
6/mmm
6/m 6, -6, 6/m
6/mmm 622, -62m, 6mm, 6/mmm
Cubic
23 m3m
m-3 23, m-3
m-3m 432, -43m, m-3m

Table 3547d lists the characteristics of the three cubic Bravais lattices.

Table 3547d. The only three cubic Bravais lattices.

Lattice
Number of
lattice points
per unit cell
Number of
atoms per unit cell
Nearest distance
between lattice points
Maximum packing
Maximum packing condition
Density (or fraction of packing, Vatom/Vcell)
1 1 When the adjacent atoms touch each other along the edge
of the cube
52.4%
2 2 When the adjacent atoms touch each other along the body diagonal of the cubic cell 68.0%
4 4 When the adjacent atoms touch each other along the face diagonal of the cubic cell 74.0%
4 8 34 %

Table 3547e. Some symmetrical diffraction patterns of cubic crystals.

Zone axis
[100]
[110]
[111]
Symmetry
Square
Rectangular
Hexagonal
Aspect Ratio
1:1 1: for BCC, SC (almost hexagonal for FCC) Equilateral

Cubic lattices have the highest degree of symmetry of any Bravais lattices.

The lattice spacing of cubic structures may be given by, (You can download the excel file for your own calculations)

------------------------------- [3547a]

where,
a -- The lattice constant.
hkl -- The Miller indices.

The interfacial angles of cubic crystals can be given by,

------------------------ [3547b]

Using Equation 3547b, we can obtain Table 3547f.

Table 3547f.  Interfacial angles of cubic crystals between {h k l} and {h' k' l'} planes.

{h k l}
{h' k' l'} 100 110 111 210 211 221 310
100 0
90.0

110 45.0
90.0
0
60.0
90.0

111 54.7 35.3
90.0
0
70.5
109.5

210 26.5
63.4
90.0
18.4
50.8
71.6
39.2
75.0
0
36.9
53.1

211 35.3
65.9
30.0
54.7
73.2
90.0
19.5
61.9
90.0
24.1
43.1
56.8
0
33.6
48.2

221 48.2
70.5
19.5
45.0
76.4
90.0
15.8
54.7
78.9
26.6
41.8
53.4
17.7
35.3
47.1
0
27.3
38.9

310 18.4
71.6
90.0
26.6
47.9
63.4
77.1
43.1
68.6
8.1
32.0
45.0
25.4
40.2
58.9
32.5
42.5
58.2
0
25.9
36.9
311 25.2
72.5
31.5
64.8
90.0
29.5
58.5
80.0
19.3
47.6
66.1
10.0
42.4
60.5
25.2
45.3
59.8
17.6
40.3
55.1
320 33.7
56.3
90.0
11.3
54.0
66.9
36.8
80.8

321 36.7
57.7
74.5
19.1
40.9
55.5
22.2
51.9
72.0
90.0

331 46.5 13.1 22.0
510 11.4

In crystals, anisotropy of many properties clearly arises because the arrangement of their atoms varies in different directions. In general, cubic crystals is less anisotropic than monoclinic ones because of their greater symmetry, for instance, with respect to electrical conductivity. Therefore, cubic crystals do not exhibit electrical polarization when the temperature is changed, so that their pyroelectric effect is isotropic.

Table 3547g. Other characteristics of cubic structures.

Contents
Page
Angles in unit cells page3555
Volume of unit cells page3033
Bravais lattices page4546
Relationship between three-dimensional crystal families, crystal systems, space group, point group, lattice systems and symmetries page4549

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