Crystallographic point groups

Crystallographic point groups are point groups to which crystals are assigned according to their inherent symmetries. They are used for analysing and predicting physical properties of the assigned crystals.

Although there are an infinite number of three-dimensional point groups, only 32 of them are crystallographic point groups. This is due to the crystallographic restriction theorem, which can be proven with trigonometry or linear algebra. The linear algebra proof is as follows:

In crystallography, a lattice point in a three dimensional vector space is described by the position vector $\boldsymbol{\mathit{r}}_i$ in the form:

$\boldsymbol{\mathit{r}}_i=u\boldsymbol{\mathit{a}}+v\boldsymbol{\mathit{b}}+w\boldsymbol{\mathit{c}}$

where the components $u$ , $v$ , $w$ are integers and $\boldsymbol{\mathit{a}}$ , $\boldsymbol{\mathit{b}}$ , $\boldsymbol{\mathit{c}}$ are primitive translation vectors or basis vectors.

A symmetry operation $A$ , e.g. a rotation by $\frac{2\pi}{n}$ , maps $\boldsymbol{\mathit{r}}_1$ to $\boldsymbol{\mathit{r}}_2$ , where the components of $\boldsymbol{\mathit{r}}_2$ are again integers. Such an operation is always possible only if all the entries of $A$ are integers.

Hence, the trace of $A$ , i.e. $Tr(A)$ , is also an integer. If we perform a similarity transformation on the rotation matrix $A$ , i.e. $A'=P^{-1}AP$ , such that $A'$ is with respect to an orthonormal basis for $\mathbb{R}^3$ , $A'$ is represented by the following matrices:

$A_x'=\begin{pmatrix}1&0&0\\0&cos\theta&-sin\theta\\0&sin\theta&cos\theta\end{pmatrix}\;\;A_y'=\begin{pmatrix}cos\theta&0&sin\theta\\0&1&0\\-sin\theta&0&cos\theta\end{pmatrix}\;\;A_z'=\begin{pmatrix}cos\theta&-sin\theta&0\\sin\theta&cos\theta&0\\0&0&1\end{pmatrix}$

Since $Tr(A)$ is invariant under a similarity transformation, $Tr(A')=2cos\theta+1=2cos\frac{2\pi}{n}+1$ must also be an integer. As we know, $-1\leq cos\frac{2\pi}{n}\leq 1$ , and so, $2cos\frac{2\pi}{n}+1=-1,0,1,2,3$ , which implies that $n=1,2,3,4,6$ . In other words, the rotational symmetry operations of a crystal are restricted to $C_1,C_2,C_3,C_4,C_6$ . This is known as the crystallographic restriction theorem.

To derive the 32 crystallographic point groups, we also need to consider the symmetry operation $S_n=\sigma C_n$ , as it has rotation components. Applying the same logic as above, the entries of the matrix $S_n$ in the primitive translation vector basis must be integers. An example of the transformed matrix $S_n'$ with respect to an orthonormal basis for $\mathbb{R}^3$ is:

$S_n'=\sigma_{xy}C_{n,z}=\begin{pmatrix}1&0&0\\0&1&0\\0&0&-1\end{pmatrix}\begin{pmatrix}cos\theta&-sin\theta&0\\sin\theta&cos\theta&0\\0&0&1\end{pmatrix}=\begin{pmatrix}cos\theta&-sin\theta&0\\sin\theta&cos\theta&0\\0&0&-1\end{pmatrix}$

Once again, $Tr(S_n')=2cos\frac{2\pi}{n}-1$ is an integer. We have, $2cos\frac{2\pi}{n}-1=-3,-2,-1,0,1\implies n=1,2,3,4,6$ , which allows us to conclude that the $S_n$ symmetry operations of a crystal are restricted to $S_1,S_2,S_3,S_4,S_6$ .

If we disregard point groups whose elements include $C_5,C_{n> 6},S_5,S_{n > 6}$ , we are left with the following 32 crystallographic point groups:

1. $C_{1v}=C_{1h}$
2. $D_{1}=C_{2}$
3. $D_{1h}=C_{2v}$ in non-standard orientation
4. $D_{1d}=C_{2h}$ in non-standard orientation
5. $S_{8}$ is an element of $D_{4d}$
6. $S_{12}$ is an element of $D_{6d}$
7. $S_{1}=C_{1h}$
8. $S_{3}=C_{3h}$