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Section 3.1 Crystallography

Group theory is really all about symmetries. In this section we will see how group theory can be applied to study objects with symmetries, such as molecules and crystals. We will see that symmetries are highly constraining, and that studying symmetries of physical objects such as crystals gives rise to highly non-trivial, and perhaps unexpected, results.

One should note that there is extensive literature on this subject, in fact there are whole books dedicated to the study of symmetries of crystals. Physicists and chemists have invented their own notation to denote the symmetry groups, and reading through this literature can quickly become confusing and even overwhelming. So what we will do here is just quickly go through some of the basic results that highlight how group theory is useful to understand the physics and chemistry of crystals.

Subsection 3.1.1 Point groups

Subsubsection 3.1.1.1 Definition

The starting point of the discussion involves the definition of point groups. Point groups are groups of geometric symmetries (or isometries) of objects that fix a point (usually set to be the origin of the coordinate system). The dihedral groups \(D_n\) that we have encountered previously are examples of discrete point groups in two dimensions.

One can think of points groups in \(N\) dimensions as being subgroups of the orthogonal group \(O(N)\text{.}\) Roughly speaking, the elements of points groups correspond to the symmetry operations of rotations, reflections, and “improper rotations” (i.e. elements with determinant equal to \(-1\)). There are infinite point groups (as an example, think of the group generated by a rotation in two dimensions by an irrational number of turns), and finite point groups. In the following we will mostly focus on finite point groups, which are the most interesting for crystallography.

Finite point groups can be classified.

  • In one dimension, there are two finite points groups: the trivial group, and the reflection group about the origin, which is isomorphic to \(\mathbb{Z}_2\text{.}\)
  • In two dimensions, there are two infinite families of finite point groups: the cyclic groups of rotations by angle \(2 \pi/n\text{,}\) which are isomorphic to \(\mathbb{Z}_n\text{,}\) and the dihedral groups \(D_n\text{.}\) However, as we will, if we also impose consistency with the translational symmetries of crystals, only a finite (and very small) number of those survive, which is the essence of the crystallographic restriction theorem.
  • In three dimensions, there is of course more possibilities. There are 7 infinite familes of finite point groups, and 7 additional point groups. For more information on those, see for instance the Wikipedia page on point groups and on point groups in 3D specifically. Again, imposing consistency with translational symmetries lead to the 32 crystallographic point groups.

Subsubsection 3.1.1.2 Molecular point groups

Three-dimensional point groups are sometimes called molecular point groups, because they are important in the study of molecules, see for instance the Wikipedia page on molecular symmetry. For instance, the chemical compound of Xenon tetrafluoride \(Xe F_4\), which consists of four atoms of fluoride placed in a square planar configuration around an atome of xenon, has point group the dihedral group \(D_4\) of symmetries of the square.

It turns out that representations and characters are crucial in the study of molecular symmetry. This is because the orbitals of the molecules transform according to irreducible representations of the point group. Thus, to understand the possible states of the system, it is essential to gain information about irreducible representations of the point group, which is encapsulated in its character table. In fact, a lot of the properties of molecules (such as optical activity, spectroscopy, dipole moments, electronic properties, what not - note that I have no idea what these terms really mean :-) ) can be analyzed using irreducible representations of the point group. If you are not convinced, google orbitals irreducible representations, or something like that. You would be surprised how much group and representation theory appears in quantum chemistry! Representations and characters are so important that Wikipedia even has a list of character tables for chemically important 3D point groups...

Subsection 3.1.2 Space groups

In the description of crystals, we need more than just point groups. We need space groups, also called crystallographic groups. Roughly speaking, space groups are symmetry groups of particular configurations of objects in space. We do not impose the restriction that a point (the origin) is fixed anymore. So we allow symmetries such as translational symmetries, which are crucial in the study of crystals.

In general, one can think of space groups as combinations of point groups with translational symmetries. More precisely, given a space group, there is an abelian subgroup of translations. Such a subgroup gives rise to the structure of a lattice of points that are invariant under these translations. For particular such lattices, there are sometimes extra symmetries (such as rotations, reflections, etc.), which, together with translations, form the structure of the space group.

Just as for point groups, space groups can be classified.

  • In one dimension, there are two space groups (also known as “line groups”): the group of translations, which is isomorphic to the group of integers \(\mathbb{Z}\text{,}\) and the infinite dihedral group, which includes reflections.
  • In two dimensions, there are 17 space groups, also known as wallpaper groups (have a look at this page, it's super fun!). Those are obtained by combining the 5 types of two-dimensional lattices (known as “Bravais lattices”) specifying the translational symmetries with the two-dimensional point groups that are consistent with translational symmetries.
  • In three dimensions, there is a total of 230 different space groups. Those are obtained by combining the 14 types of three-dimensional Bravais lattices with the 32 point groups that are consistent with translational symmetries.

Subsection 3.1.3 Crystals

We are now ready to discuss symmetry groups of crystals, and the famous crystallographic restriction theorem, which shows how powerful the study of symmetries can be in physics and chemistry.

To start with: what is a crystal? In physics and chemistry, a crystal is a solid material which has a highly ordered microscopic structure: it consists in a lattice of atoms that extends in all directions and is invariant under translations. Examples include snowflakes, diamonds, table salt, etc.

Let us focus on crystals in three dimensions. Mathematically, the lattice of atoms in a crystal is a Bravais lattice: this is an infinite array of discrete points that are generated by a translation:

\begin{equation*} \vec{T} = n_1 \vec{v_1} + n_2 \vec{v_2} + n_3 \vec{v_3}, \end{equation*}

where \(n_1,n_2,n_3 \in \mathbb{Z}\) and \(\vec{v_1}, \vec{v_2}, \vec{v_3}\) are primitive vectors in three dimensions which lie in different directions (not necessarily perpendicular) and span the lattice. Thus a crystal is invariant under translations generated by \(\vec{T}\text{.}\)

Furthermore, a given crystal may be invariant under symmetry operations that leave a point fixed, such as rotations and reflections. The group of such symmetry operations is a point group; in this context, it is known as a crystallographic point group. Together with translations, it forms the space group of the crystal.

We have seen that finite point groups in three dimensions can be classified: there are 7 infinite familes, and 7 additional point groups. But can all of those be point groups of crystals? In other words, can all of these point groups arise as symmetry groups of lattices? The answer to this question is the essence of the crystallographic restriction theorem, and may appear surprising: only a very small, finite, number of point groups are symmetry groups of crystals! For instance, there is no crystal that is invariant under rotations by an angle \(2\pi / 5\text{!}\) Isn't that surprising?

What? Really? Those are the only possibilities? Imposing translational symmetry is that constraining? Let's prove this! There are a number of different proofs of this theorem. We will first present a geometric proof, and then present a representation theoretic, or matrix based, proof, for fun.

First, we forget about the case \(n=1\) since it is trivial, as it involves no rotational symmetry.

Now suppose that a lattice is invariant under rotations by an angle \(2 \pi/n\) about an axis. Consider two lattice points \(A\) and \(B\) in the plane perpendicular to the axis of rotation, and separated by a translation vector \(\vec{v}\) of minimal length (by which I mean that there is no lattice point on the line between \(A\) and \(B\)). Let us rotate point \(A\) about point \(B\) by an angle \(2 \pi/n\) and call the resulting point \(A'\text{.}\) Similarly, let us rotate point \(B\) about \(A\) by the same angle \(2\pi /n\) but in the opposite direction, and call the resulting point \(B'\text{.}\)

Let us call \(\vec{v'}\) the vector joining \(A'\) and \(B'\text{.}\) We notice that \(\vec{v'}\) is parallel to \(\vec{v}\text{.}\) Thus, if rotation by an angle \(2 \pi/n\) is a symmetry of the lattice, it follows that the new translation vector \(\vec{v'}\) must be an integer multiple of \(\vec{v}\text{,}\) that is,

\begin{equation*} \vec{v'} = k \vec{v} \qquad \text{for some } k \in \mathbb{Z}. \end{equation*}

This is because the two points \(A'\) and \(B'\) must be related by a translation generated by \(\vec{v}\text{,}\) and \(\vec{v}\) was chosen to have minimal length.

Now we can relate the length of \(\vec{v'}\) to the length of \(\vec{v}\) and the angle \(2\pi/n\text{.}\) The four points \(A,B,A',B'\) form a trapezium, with angles \(2 \pi/n\) and \(\pi - 2 \pi/n\text{.}\) By basic geometry, we get:

\begin{equation*} |\vec{v'}| = |\vec{v}|+ 2 |\vec{v}|\cos\left(\pi - \frac{2 \pi}{n}\right) = |\vec{v}| \left( 1 - 2 \cos \frac{2\pi}{n} \right). \end{equation*}

Since we must have \(\vec{v'} = k \vec{v}\) for some \(k \in \mathbb{Z}\text{,}\) and hence \(|\vec{v'}| = |k| |vec{v}|\text{,}\) we get the constraint:

\begin{equation*} |k| = 1 - 2 \cos \frac{2\pi}{n}, \end{equation*}

that is, \(1 - 2 \cos \frac{2 \pi}{n}\) must be a non-negative integer. Since \(|\cos \alpha| \geq 1\) for any angle \(\alpha\text{,}\) the only possible values of \(|k|\) are \(|k|=0,1,2,3\text{.}\) Solving for \(n\) for each cases, we get, respectively, \(n = 6,4,3,2\text{.}\) Those are the only possible rotational symmetries of a crystal, which concludes the proof.

Let us now give a representation theoretic, or matrix based, proof. A rotation by an angle \(2 \pi/n\) about an axis can be seen as a linear operator acting on the two-dimensional vector space perpendicular to the axis of rotation. After choosing a basis for this vector space, such a rotation can be written in terms of a \(2 \times 2\) rotation matrix. In the language of group theory, this gives a two-dimensional representation of the cyclic rotation group \(\mathbb{Z}_n\text{.}\) On general grounds, we know how to write such a rotation matrix with respect to an orthonormal basis for \(\mathbb{R}^2\text{:}\)

\begin{equation*} \begin{pmatrix} \cos \frac{2\pi}{n} \amp \sin \frac{2 \pi}{n} \\ - \sin \frac{2 \pi}{n} \amp \cos \frac{2 \pi}{n} \end{pmatrix}. \end{equation*}

This matrix expression for the linear operator is only valid with respect to an orthonormal basis, but its trace (that is, the character of the representation), which is equal to \(2 \cos (2 \pi/n)\text{,}\) is invariant under changes of basis, as we know (that is, the characters are the same for equivalent representations related by similarity transformations). So in a different choice of basis, the \(2 \times 2\) rotation matrix may look different, but its trace will always be equal to \(2 \cos( 2 \pi / n)\text{.}\)

Now, in a lattice, it is appropriate to write down how rotations act with respect to the basis given by the primitive translation vectors \(\vec{v_i}\text{.}\) Those are in general not orthogonal, nor of length one, so the precise matrix representation may look ugly. However, if a rotation by an angle \(2 \pi/n\) is a symmetry of the lattice, then we know that it must map each translation vector into an integer linear combination of the other translation vectors. In other words, with respect to this lattice basis, the rotation matrix corresponding to rotation by an angle \(2 \pi/n\) must have integer entries. In particular, its trace must be integer. But since we know that the trace is invariant under changes of basis, its trace must also be equal to \(2 \cos(2 \pi / n)\text{.}\) Therefore we conclude that \(2 \cos(2 \pi /n ) = k\) for some \(k \in \mathbb{Z}\text{.}\) The only possibilities are \(k = -2,-1,0,1,2\text{,}\) which correspond to, respectively, \(n=2,3,4,6,1\text{,}\) with the last choice corresponding to the identity matrix (i.e. no rotational symmetry). These are the only possible rotational symmetries that are consistent with translational symmetries of a crystal.

This is a highly non-trivial result, and is perhaps unexpected. Why can't we build a lattice that is invariant under five-fold rotations? Or seven-fold rotations? This is related to the fact that we cannot combine objects with a five-fold or seven-fold apparent symmetry in such a way that they completely fill the plane. See for instance this webpage for pretty pictures.

Using the crystallographic restriction theorem, we can enumerate the possible crystallographic point groups in low dimensions.

  • In two dimensions, we had two infinite families of point groups: the cyclic rotations groups (\(\mathbb{Z}_n\)) and the dihedral groups (\(D_n\)). Since crystals can only be invariant under rotations by an angle \(2 \pi/n\) with \(n \in \{1,2,3,4,6\}\text{,}\) we see that the only crystallographic point groups are the five cyclic rotation groups \(\mathbb{Z}_n\) and five dihedral groups \(D_n\) with \(n \in \{1,2,3,4,6\}\text{,}\) for a total of 10 crystallographic point groups in two dimensions.
  • In three dimensions, we started with 7 infinite families and 7 additional point groups. Looking at the rotations that are elements of these point groups, one can conclude that there are only 32 crystallographic point groups in three dimensions.

To summarize, the possible symmetries of crystals are highly constrained! This is neat, and is a nice, direct, application of group theory to physics and chemistry. Crystallography is all about group and representation theory!

To end this section, note that the crystallographic restriction theorem is not the end of the story. The story continues with aperiodic tilings and quasicrystals. It is truly fascinating (and has led to a Nobel prize). Have a look at these Wikipedia pages and further resources if you are interested!

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