Unit cells and the crystal structure

Crystals have an underlying crystal lattice based on the repetition of a parallelpiped throughout space. This parallelpiped is chosen to contain a minimal number of atoms or molecules needed to represent the crystal such that if this single ``cell'' is repeated throughout space, the entire crystal structure can be generated.

The figure below shows a general parallelpiped. 6 numbers are needed

Figure: Six numbers needed to characterize a parallelpiped

to characterize this shape, namely, the lengths of the three unique edges, and the angles made by these edges with the $x$, $y$ and $z$ axes. The lengths are denoted $a$, $b$ and $c$ and the corresponding angles are denoted $\alpha$, $\beta$, and $\gamma$. Once these numbers are known, the volume of the parallelpiped is given by

$$V = abc\sqrt{1-\cos^2\alpha - \cos^2\beta - \cos^2\gamma + 2\cos\alpha\cos\beta\cos\gamma}$$

Alternatively, we can work with an overcomplete set of 9 number, namely, the so-called cell vectors ${\bf a}$, ${\bf b}$ and ${\bf c}$. These are the vectors that lie along the three unique edges. If we know these vectors, then the volume of the parallelpiped is given by

$$V = {\bf a}\cdot({\bf b}\times {\bf c})$$

For example, in a rectangular cell with edges of length $a$, $b$ and $c$, the cell vectors can be taken to lie along the $x$, $y$ and $z$ axes, so we have ${\bf a}= (a,0,0)$, ${\bf b}= (0,b,0)$ and ${\bf c}= (0,0,c)$.

This parallelpiped and the atoms it contains is known as the unit cell of the crystal. Let the unit cell contain $N$ atoms with position vectors ${\bf R}_1,...,{\bf R}_N$. The positions of all other atoms in the crystal can then be generated by

$${\bf R}_1' = {\bf R}_1 + l{\bf a}+ m{\bf b}+ n{\bf c}$$

$$\cdots$$

$${\bf R}_N' = {\bf R}_N + l{\bf a}+ m{\bf b}+ n{\bf c}$$

for any integers $(l,m,n)$. The new cell, with positions ${\bf R}_1',...,{\bf R}_N'$ should look the same as the original unit cell, just translated in space relative to the original cell.

There are 7 basic parallelpiped shapes that fit most crystal structure, and these are shown in the figure below:

Figure: 7 basic unit cell shapes

Example: Consider a hypothetical crystal in which the atoms lie at the centers of cubic cells. the unit cell is a cube of edge length $L$, so that the cell vectors are ${\bf a}= (L,0,0)$, ${\bf b}= (0,L,0)$ and ${\bf c}= (0,0,L)$. The position of the single atom in the unit cell is ${\bf R}= (L/2,L/2,L/2)$. Hence, from the above formula, we can generate the positions of all other atoms in the crystal. For example, setting $l=1, m=0, n=0$, we generate the atom at position ${\bf R}' = {\bf R}+ {\bf a}= (3L/2,L/2,L/2)$. By setting $l=0, m=1, n=0$, we generate the atom at position ${\bf R}'' = {\bf R}+ {\bf b}= (L/2,3L/2,L/2)$, etc.

The most common crystal unit cells that are encountered in nature are known as the simple cubic structure, in which the unit cell contains a single atom at the corner of a cube, the body-centered cubic structure, in which the unit cell contains two atoms, one at the corner of a cube, and the other at the center, the face-centered cubic structure, in which the unit cell contains four atoms, one at the corner and three along the bottom and two side faces, and the side-centered cubic, in which the unit cell contains two atoms, one at the corner and the other at the center of the bottom face (see figure below).

Figure: simple cubic, bcc, fcc and scc lattices