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  in the form:

where the components , ,  are integers and , , are primitive translation vectors or basis vectors.

A symmetry operation , e.g. a rotation by , maps  to , where the components of  are again integers. Such an operation is always possible only if all the entries of are integers.

Hence, the trace of , i.e. , is also an integer. If we perform a similarity transformation on the rotation matrix , i.e. , such that  is with respect to an orthonormal basis for , is represented by the following matrices:

Since  is invariant under a similarity transformation,  must also be an integer. As we know, , and so, , which implies that . In other words, the rotational symmetry operations of a crystal are restricted to . This is known as the crystallographic restriction theorem.

To derive the 32 crystallographic point groups, we also need to consider the symmetry operation , as it has rotation components. Applying the same logic as above, the entries of the matrix  in the primitive translation vector basis must be integers. An example of the transformed matrix  with respect to an orthonormal basis for is:

Once again,  is an integer. We have, , which allows us to conclude that the  symmetry operations of a crystal are restricted to .

If we disregard point groups whose elements include , we are left with the following 32 crystallographic point groups:

    1. in non-standard orientation
    2. in non-standard orientation
    3. is an element of
    4. is an element of


Are the symmetry operations  and affected by the crystallographic restriction theorem?


and the trace of a mirror plane is an integer, e.g. .


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