# Solids

A solid is a phase of matter where the atoms are arranged in a static way. Crystals, a type of solid have atoms arranged in a repeatable pattern. The simplest repeatable pattern is known as a unit cell. The simplest repeatable cubic pattern is referred to as a primitive cubic cell (or simple cubic cell) and contains atoms located on the corners of a cube called lattice points (see figure below)1.

The shaded green cube is the unit cell and repeats itself in every directory (right in the figure above) in three dimensional space. We can determine certain features of this unit cell. The coordination number of the primitive cubic is 6 and is determined by counting the number of atoms a particular atom in the lattice is in contact with (see figure below).2

Additionally, we can determine the number of equivalent atoms in the primitive cubic by treating atoms at the lattice points as spheres and adding up the fraction of each sphere. For example, in the figure below, an atom is centered at each of the 8 lattice points in the primitive cubic. Exactly 1/8th of each sphere lies inside the cube.

Since there are 8 total atoms and 1/8th of each atom lies inside the cube, the number of equivalent atoms in the unit cell is equal to 1 such that

$8~\mathrm{atoms} \times \frac{1}{8} = 1~\mathrm{equivalent~atom}$

## Cubic Unit Cells

There are many types of unit cells. The figure below shows three cubic unit cells. The simple cubic (left) is the primitive cubic unit cell we have discussed above.3

The features of these unit cells are tabulated below.

Cubic Cell Name Number of Equivalent Atoms Coordination Number Packing Efficiency
Primitive Cubic 1 6 52%
Body-Centered Cubic 2 8 68%
Face-Centerd Cubic 4 12 74%

The coordination number assumes no vacant sites (tetrahedral holes).

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.

### Face-centered cubic

ZnS adopts a face-centered cubic. The unit cell below comes from its crystal structure and has vacant tetrahedral sites.

• Coordination Number = 8

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