If the space lattice is FCC, the lattice constant is given by the formula [4 x r / (2)1/2] and if the space lattice is BCC, then the lattice constant is given by the formula a = [4 x r / (3)1/2].

Body-centered cubic (bcc) lattice: Primitive translation vectors of the bcc lattice (in units of lattice parameter a) are a1 = 슬-½; a2 = – 슬 a3 = ½-슬. The primitive cell is the rhombohedron. The packing ratio is 0.68, defined as the maximum volume which can be filled by touching hard spheres in atomic positions.

Therefore, the volume V of the primitive unit cell is given by V = 1/n where n is the density of points.

The lattice parameter is 0.3571 nm for FCC iron and 0.2866 nm for BCC iron.

A metal is found to have BCC structure, a lattice constant of 3.31 Å, and a density of 16.6 g/cm3. Determine the atomic weight of this element. At 100°C copper (Cu) has a lattice constant of 3.655 Å.

In the bcc structure each atom has c1=8 c 1 = 8 nearest neighbours (coordination number) at a distance of dc1=2r=√32a≈0.866a(3) (3) d c 1 = 2 r = 3 2 a ≈ 0.866 a and c2=6 c 2 = 6 next-nearest neighbours at a distance of dc2=a≈2.3r≈1.15dc1.

Coordination number – the number of nearest neighbor atoms or ions surrounding an atom or ion. For FCC and HCP systems, the coordination number is 12. For BCC it’s 8. Page 12.

Solution:

1. Determine mass of two atoms in a bcc cell: 22.99 g/mol / 6.022 x 1023 mol¯1 = 3.81767 x 10¯23 g (this is the average mass of one atom of Na)
2. Determine the volume of the unit cell: 7.63534 x 10¯23 g / 0.971 g/cm3 = 7.863378 x 10¯23 cm3

Face-Centered Cubic Unit Cell: The total number of atoms present in a face-centered cubic unit cell are four. Hence, the density of a face-centered cubic unit cell is 4 x M / A3 x Na.

Density has the units of mass divided by volume such as grams per centimeters cube (g/cm3) or kilograms per liter (kg/l). A block of wood has a mass of 8 g and occupies a volume of 10 cm3.