A lattice constant describes the spacing between adjacent unit cells in a crystal structure. The unit cells or building blocks of the crystal are three dimensional and have three linear constants that describe the cell dimensions. The dimensions of the unit cell are determined by the number of atoms packed into each cell and by how the atoms are arranged. A hard-sphere model is adopted, which allows you to visualize atoms in the cells as solid spheres. For cubic crystal systems, all three linear parameters are identical, so a single lattice constant is used to describe a cubic unit cell.
1. Identify the Space Lattice
Identify the space lattice of the cubic crystal system based on the arrangement of the atoms in the unit cell. The space lattice may be simple cubic (SC) with atoms only positioned at the corners of the cubic unit cell, face-centered cubic (FCC) with atoms also centered in every unit cell face, or body-centered cubic (BCC) with an atom included in the center of the cubic unit cell. For example, copper crystallizes in an FCC structure, while iron crystallizes in a BCC structure. Polonium is an example of a metal that crystallizes in a SC structure.
2. Find the Atomic Radii
Find the atomic radius (r) of the atoms in the unit cell. A periodic table is an appropriate source for atomic radii. For example, the atomic radius of polonium is 0.167 nm. The atomic radius of copper is 0.128 nm, while that of iron is 0.124 nm.
3. Calculate the Lattice Constant
Calculate the lattice constant, a, of the cubic unit cell. If the space lattice is SC, the lattice constant is given by the formula a = [2 x r]. For example, the lattice constant of the SC-crystallized polonium is [2 x 0.167 nm], or 0.334 nm. 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].
