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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 12fold 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.
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 cationtoanion radius ratios for cubic ionic lattices can be obtained as shown in Table 3547a.
Table 3547a. Cationtoanion 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 AB_{2}: CaF_{2} (0.727); (iii) Others: NH_{4}Br 
Closepacking 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 6_{3} 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 threefold 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 




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

m3 
23, m3 
m3m 
432, 43m, m3m 
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, V_{atom}/V_{cell}) 

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 threedimensional crystal families, crystal systems, space group, point group, lattice systems and symmetries 
page4549 
