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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 3550a (a):
i) B voids with triangles pointing up.
ii) C voids with the triangles pointing down.
Assuming a hard sphere model, atomic packing factor is defined as the ratio of atomic sphere volume to unit cell volume, which is 74% for both FCC and HCP and 68% for BCC. In general, ~90% elemental metals crystallize into three crystal structures which are BCC, FCC, and HCP.
As shown in Figure 3550a (b), if the second layer of atoms is added and its atoms occupy the space above the B voids followed by the third layer above the A atoms, this will result in hexagonal closest packing (HCP) with a stacking sequence of AB AB AB ....etc and with the caxis perpendicular to the AB AB layers.
Figure 3550a. Formation of hexagonal closest packing (HCP): (a) Single layer of atoms, and (b) Formed HCP structure.
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 3550a and Figure 3550b show the hexagonal crystal systems and the schematic illustrations of the hexagonal lattices, respectively. As indicated in Table 3550a, the structures belonging to the trigonal system can have either a rhombohedral or hexagonal lattice. The rhombohedral lattice are stacked in ABCABC...sequence, while the hexagonal lattice are stacked in AAA...sequence. Both the hexagonal (with 6fold symmetry) and the trigonal (with 3fold symmetry) systems require a hexagonal axial system.
Table 3550a. Hexagonal crystal systems.
Crystal
family 
Crystal
system 
Required
symmetries
of point group 
Point
group 
Space
group 
Bravais
lattices 
Lattice
system 
Hexagonal 
Trigonal 
1 threefold axis of rotation 
5 
7 
1 
Rhombohedral 
18 
1 
Hexagonal 
Hexagonal 
1 sixfold axis of rotation 
7 
27 
Figure 3550b. Schematic illustrations of the Bravais lattices of hexagonal crystals.
Table 3550b. 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 
Hexagonal 

6/mmm

6/m 
6, 6, 6/m 
6/mmm 
622, 62m, 6mm, 6/mmm 

23 
m3m

m3 
23, m3 
m3m 
432, 43m, m3m 
For an ideal HCP structure, the c/a ratio is 1.63299. The volume of an unit cell in a hexagonal crystal is given by,
 [3550]
According to Radius Ratio Rule, the limiting cationtoanion radius ratios for HCP ionic lattices can be obtained as shown in Table 3550c.
Table 3550c. Cationtoanion radius ratios (r^{+}/r^{}) for coordination number (CN) 12.

CN 
Crystal type 
Material examples and their r^{+}/r^{} 
1.0 
12 
Hexagonal or cubic
closest packing 
Metals 
For instance, at room temperature and ambient pressure, Ti (titanium) has a hexagonal closepacked structure (called αphase) with the lattice constants listed in Table 1721a. Its unit cell has two atoms at (1/3, 2/3, 1/4) and (2/3, 1/3, 3/4) and the space group number is 194 (P6_{3}/mmc). At room temperature and high pressure, it changes to the ωphase [1,2] with the lattice constants listed in Table 1721b. Its unit cell has three atoms at (0, 0, 0), (1/3, 2/3, 1/2), (2/3, 1/3, 1/2) and the space group is P6/mmm. The α → ω transition in Titanium is a typical example of martensitic transformation.
Table 3550d. Other characteristics of hexagonal structures.
Contents 
Page 
Angles in unit cells 
page3555 
Volume of unit cells 
page3033 
Close packed planes and directions 
page3029 
Atomic packing factor 
page3030 
Number of lattice points (atoms) per unit cell 
page3032 
Coordination number of atoms 
page3031 
Relationship between threedimensional crystal families, crystal systems, space group, point group, lattice systems and symmetries 
page4549 
Lattice point (or called Motif or basis) 
page3076 
Tables of Burgers vectors of dislocations and g·b 
page1995 
Bravais lattices 
page4546 
Dominating slip planes, slip directions and stable Burgers vector for common crystal structures 
page3557 
[1] Jamieson J.C., Science, 1963, 140, 72; doi:10.1126/science.140.3562.72.
[2] Sikka S.K., Vohra Y.K., Chidaraman R., Prog. Mater. Sci., 1982, 27, 245.
