In the BCC unit cell, an atom is located at the body center of a cubic unit cell in addition to the eight atoms on the cube corners. The coordination number of this type of crystal structure is 8, that is, each atom is surrounded by eight equidistant neighboring atoms. Figure 2.5a shows a schematic of a BCC unit cell. The effective number of atoms in such a unit cell is (8 x 1/8) +1 = 2, because each corner atom is shared by eight unit cells and the body center atom is fully inside the unit cell to be counted as one. We note that each atom is surrounded by eight nearest neigh­bors and this factor is known as coordination number defined as the number of equivalent nearest neighbors. The packing efficiency or atomic packing factor is

defined as the volume of the space occupied by atoms in a unit cell. Below we pres­ent a detailed procedure for calculating the atomic packing factor of a BCC unit cell. Note that the same method can be applied to calculate the packing factors of virtu­ally any kind of known unit cells.

The packing efficiency PE of atoms in the BCC unit cell is derived as follows:

PE Volume of atoms in the unit cell 2 x (4/3)pr3 Volume of the unit cell a3

^2 x (4/3)p((/3/4)a)3 ^ 0 68 a3 ‘ ‘

Many metals such as a-Fe, b-Zr, Mo, Ta, Na, K, Cr, and so on have a BCC crystal structure. This structure is relatively loosely packed (only 68% of the crystal volume is occupied by atoms) compared to the best packing that can be achieved using hard incompressible solids spheres. Two corner atoms touch the body center atom and create the body diagonal that is closest-packed direction in the BCC unit cell. This also gives us an opportunity to calculate the relation between the atom radius (r) and the lattice constant (a) using the simple Pythagorean rule. The body diagonal length is given by (r + 2r + r) = 4r. The body diagonal makes the hypotenuse of the triangle that a base of a face diagonal (/2a) and a cube edge (a), as shown in Figure 2.5b.

Hence, (4r)2 = (p2a)2 + (a)2 or (4r)2 = 3a2 = (p3a)2.

Or, 4r = P3a. (2.1)

& Example 2.1

Calculate the theoretical density of a-Fe at room temperature.

Solution

The density of any crystal can be calculated from the first principles.

Mass of atoms in the unit cell Г Volume of the unit cell

First, we need to know the crystal structure (BCC) and lattice parameter (a = 0.287 nm) of a-Fe.

Mass of an Fe atom = 55.85 atomic mass unit (or amu) from the defini­tion of atomic weight.

1 amu = 1.66 x 10-27 kg.

Hence,