Question

Compute and compare the linear densities (Function distance occupied by solid atoms) of the [100], and [111] directions in BCC materials.

Answer 1

For the BCC unit cell, the length of cell side is a, where 3^0.5a = 4R, and R is the atomic radius. The 100 plane of the BCC structure is a square, side length a, containing 1 atom (a quarter of an atom from each of the four corner atoms). So the planar density is 1 atom per a² square meters, or 1 atom per ((16/3)R²) square meters. The fractional planar density is pi R^2/((16/3)R²) square meters, or (3 pi / 16), or 0.589.

The 110 plane of the BCC structure is a rectangle, a by 1.414 a, and it contains 2 atoms. So the planar density is 2 atoms per (1.414 * (16/3) * R^2) square meters, or 1 atom per 3.77R^2 square meters. The fractional planar density is pi R^2 / 3.77 R^2 = 0.833.