One-third of the hexagonal unit cell shown in Figure 2.2 can be considered as the primitive hexagonal unit cell since all crystal structures with a hexagonal symmetry can be created out of it, such that it leads to the creation of a HCP unit cell as well as the graphite crystal structure. A HCP unit cell (Figure 2.7) is quite different from the cubic ones that we discussed until now. Nonetheless, an ideal HCP crystal structure is as close-packed as the FCC one (packing factor of 0.74). This crystal structure is created by alternate layers of atoms on top of each other (see Section 2.1.3). This crystal structure has two characteristic lengths, a, in the base of the unit cell and the height of the unit cell along the c-axis (Figure 2.7). In fact, there are three separate coplanar axes with lattice vectors of a1, a2, and a3 at 120° apart even though they are all equal in length. The upper, middle, and lower

horizontal planes are called the basal planes. Planes (six vertical faces in a HCP unit cell) perpendicular to the basal planes are known as the prismatic planes. Planes other than the basal and prismatic planes are known as the pyramidal planes. This crystal structure has a coordination number of 12 (note the same as for FCC).

With a close examination of the HCP unit cell, the effective number of atoms can be easily calculated. Two atoms present on the upper and lower bases are shared by 2 HCP unit cells, whereas the corner atoms (12 of them) are shared by 6 unit cells, and 3 atoms reside entirely inside the unit cell. So, the count becomes (2 x 1/2) + (12 x 1/6) + (1 x 3) = 6.

The relation between the lattice constant a and the atom radius (r) is simple, a = 2r. However, in order to get a relation between the other lattice constant c and r, one needs to know little further. The HCP unit cell is inherently anisotropic. This implies that certain fundamental characteristics along the a — and c-axes are different. One measure of this anisotropy is given by the c/a ratio. An ideal HCP unit cell has a c/a ratio of 1.633 (can be shown from the geometry) based on a hard sphere model. But in reality, HCP metals are hardly ideal in dose packing, and thus the packing factors also change from metals to metals although slightly. Table 2.2 lists a number of HCP metals with different c/a ratios. The different c/a ratios are thought to be a result of the specific electronic structures of the atoms. Earlier it was thought that the c/a ratios influence the primary deformation mode, and that is why the corresponding dominant deforma­tion modes on the right-hand side column of Table 2.2 are also included. However, later this conjecture has been proved otherwise noting the glaring exception of Be (low c/a ratio, yet the primary deformation mode is basal). This issue will be further dis­cussed in Chapter 4 when we learn more about the slip deformation mode.

■ Example 2.2

Beryllium (Be) is used in LWRs as reflectors. Be has a HCP crystal structure at temperatures <1250 °C. Be has the lattice constants — a = 0.22858 nm and c = 0.35 842 nm (i. e., c/a ratio of 1.568) during normal conditions. The atomic radius of a Be atom is 0.1143 nm. What is the atomic packing effi­ciency of the unit cell? Discuss the significance of the computed result. Solution

Volume of atoms in the unit cell
Volume of the unit cell

Beryllium has a HCP crystal structure. The volume of a hexagonal prism (unit cell) is given by V HCP = (3/3a2/2)c, where a is the length of the prism base edge and c is the height of the hexagonal prism.

As a = 0.22858 nm and c = 0.35842 nm, VHCP = 0.04 865 nm3 (this is the input value in the denominator of the above expression for the PE).

Now we need to find out how much of the volume of the hexagonal prism is occupied by the atoms. The effective number of atoms in a HCP unit cell is 6. The atomic radius of each Be atom is 0.1143 nm. Therefore, the total volume of the atoms in the unit cell is 6 x (4/3)4(0.1143 nm)f = 0.03751 nm3.

Therefore, PE = 0 03751 nm = 0.77. Hence, the atoms must be spheroidal 0.04865 nm3

rather than spherical if they are assumed to be in contact.

The packing efficiency of a HCP unit cell is 0.74 only when the ideal c/a ratio is maintained. However, in reality, it is not and the packing efficiency of the HCP metals will always be little off from the ideal value. In the case of Be, the packing efficiency is slightly higher than the ideal.

2.1.3