2.2 Equivalent Atoms

One metric for characterizing a unit cell is denoting the number of equivalent atoms in the unit cell. That is, equivalent atoms are the total number of atoms located inside the unit cell (i.e. the cube for cubic unit cells). For example, in our picture of the simple cubic unit cell, we see a total of eight spheres. However, only a fraction of each sphere actually resides inside the cube. Equivalent atoms only accounts for the total number of sphere fractions that are actually located inside the cube. Geometry tells us that a sphere that is centered on the corner of a cube will result in exactly 1/8th of that sphere lying inside the cube. Given that there are eight spheres, each located on a corner, we determine the number of equivalent atoms by totaling up the fractions of each sphere represented in the unit cell.

\[\mathrm{eq.~atoms} = \dfrac{1}{8} ~\left ( \mathrm{8~corner~spheres} \right ) = 1\] The fraction changes depending on the position of the sphere (shown below) and are as follows:

  • Corner: 1/8
  • Face: 1/2
  • Edge: 1/4

A body-centered cubic contains 4 equivalent atoms.

\[\begin{align*} \mathrm{eq.~atoms} &= \dfrac{1}{8} ~\left ( \mathrm{8~corner~spheres} \right ) + 1 ~\left ( \mathrm{1~sphere~inside} \right ) \\ &= 2 \end{align*}\]

A face-centered cubic contains 4 equivalent atoms.

\[\begin{align*} \mathrm{eq.~atoms} &= \dfrac{1}{8} ~\left ( \mathrm{8~corner~spheres} \right ) + \dfrac{1}{2} ~\left ( \mathrm{6~faces~spheres} \right ) \\ &= 4 \end{align*}\]

The table below summarizes some characteristics of these cubic unit cells.


Type Equivalent Atoms Structure Coordination Number Packing Efficiency
Simple cubic 1 6 52%
Body-centered cubic 2 8 68%
Face-centered cubic 4 12 74%


Note that the packing efficiency is a measure of the volume of the cube that is occupied by a sphere. For example, the cube of a primitive cubic has 52% of its volume occupied by spheres located at the lattice points, whereas, the body-centered cubic has an increased packing efficiency of 68% due to the additional sphere located at the center of the cube.