Crystal symmetry 1. Point-symmetry operations

3.3.1 Point symmetry

There are many ways that symmetry operates on a structure. The lattice itself has a symmetry: any structure associated with one lattice point will be replicated at all other lattice points. It is this lattice symmetry that allows us to distinguish crystals from glasses. This is an example of translational symmetry. However, there are other types of symmetry within crystals that operate on the environment around a single point, which is called point symmetry. The combination of point symmetry and translational symmetry gives the overall symmetry of the crystal. We turn now to explore the details of symmetry in a formal sense, starting with point symmetry (here and Section 3.4), and then characterizing the different types of translation symmetry (Section 3.5.1), and finally considering the combinations of both types of symmetry (Section 3.5.2).

We have defined point symmetry as giving the symmetry seen from a particular point in the crystal. It is important to appreciate that different points in a crystal will have different point symmetry – indeed, most points have no point symmetry at all, and the existence of point symmetry will be restricted to special points, lines or planes in the unit cell.

There are four main types of symmetry operations that are associated with a single point in space, which we will represent through the effect on atomic coordinates. We will use fractional coordinates x, y, z to represent the position of the initial point, and then produce the new fractional coordinates of the new point generated by symmetry. We will denote a sign change by x rather than−x.

3.3.2 The four point symmetry operations

Rotation axes

Rotational symmetry is present if an object comes into coincidence with itself after rotation by an angle of 360^◦/n, where n is an integer. The value of n is used to label the particular symmetry, for example, when n = 2 we have a 2-fold rotation axis. An example of a 4-fold rotation axis is shown in Fig. 3.9.

Fig. 3.9 Example of the operation of a 4-fold rotation symmetry, seen in the way the molecule is copied by 90^◦rotations.

In principle, rotational symmetry can exist with any values of n. However, the only values of n that are compatible with the existence of a lattice are n= 1, 2, 3, 4 and 6. Objects with other types of rotational symmetry cannot be packed together to completely fill space. The case of n= 5 is interesting because of the discovery of this type of symmetry in diffraction patterns of some metallic alloys – we will discuss this clear ‘breaking of the rules’ briefly at the end of the chapter.

The effects of rotational symmetry on fractional atomic coordinates are as follows. A 2-fold rotation axis parallel to [001] passing through the origin will lead to an atom at x, y, z being replicated at x, y, z. A 4-fold axis in the same place will lead to the atom at x, y, z also being replicated at y, x, z; x, y, z; and y, x, z. The cases of 3-fold and 6-fold axes are treated in Problem 3.8.

If the only symmetry in a crystal is a 2-fold axis, the crystal lattice will be monoclinic, because the action of a 2-fold axis is to replicate atoms within the same plane normal to the 2-fold axis. This can only be accomplished if the 2-fold axis is parallel to one of the lattice vectors and orthogonal to the other two.

By convention, it is taken that the 2-fold axis is parallel to the b lattice vector.

If the crystal has a single 4-fold axis, the lattice is tetragonal. The presence of a single 3-fold axis gives a trigonal or rhombohedral lattice, and the presence of a single 6-fold axis gives a hexagonal lattice. The cubic lattice is defined by the presence of four intersecting 3-fold axes parallel to the111 directions (and not by the presence of three intersecting 4-fold axes parallel to the100 directions as is commonly supposed). These conditions are summarized in Table 3.1.

There are two conventional representations of rotation axes. Standard crystallographic notation is to represent them by the symbols 2, 3, 4, or 6. A common alternative system, prefered by chemists, is theSchoenflies notation, which is described in Appendix C. For rotational symmetry, the Schoenflies

Fig. 3.10 Illustration of mirror symmetry, shown by the reflected image of the molecule.

notation uses the symbols Cn, where n = 2, 3, 4, 6. However, the translation between the two types of notation is not always as straightforward! Both notations are in common use.

Mirror planes

An object that can be reflected through a plane and come into coincidence with itself hasmirror symmetry. An object can have a single mirror plane, or three orthogonal mirror planes (when combined with rotational symmetry axes there can be additional mirror planes at angles of 45^◦or 60^◦). Mirror symmetry is illustrated in Fig. 3.10.

A mirror plane normal to the[001] direction and passing through the origin of a crystal will cause an atom at x, y, z to be replicated at x, y, z. A single mirror plane is another symmetry element that defines monoclinic symmetry, because the only lattice vector that will not lie in the mirror plane must be normal to it.

By convention, in a monoclinic crystal the lattice vector normal to the mirror plane is the b axis. The presence of two or three (necessarily orthogonal) mirror planes will give orthorhombic symmetry. Mirror planes are represented by the symbol m (the corresponding nomenclature in the Schoenflies system depends on the other types of symmetry present, as noted in Appendix C.)

Centre of symmetry

An object has acentre of symmetry if every point at position x, y, z relative to the centre is also found at x, y, z. A crystal with a centre of symmetry is calledcentrosymmetric. Another term for this symmetry is inversion symmetry.

This symmetry operation is illustrated in Fig. 3.11. It is given the label 1, for reasons that will be discussed below.

The centre of symmetry is important for many physical properties, because if there is a centre of symmetry there can be no dipole moment associated with the positions of charged ions in the unit cell.

Rotoinversion axes

The symmetry operation of rotoinversion is a two-stage process. First is the operation of a rotation symmetry, and this is followed by thesecond operation of

Fig. 3.11 Illustration of the operation of a centre of symmetry on a pair of molecules, whose distance in the line of sight is empha-sized by the perspective and the lighter shading used to represent further distances.

a reflection through the centre. The rotation axis is then called therotoinversion axis, and given the symbol n in the international crystallographic notation. It should be noted that the Schoenflies notation involves a different line of thinking for this type of symmetry, as discussed in Appendix C.

The simplest example, 1, is the operation of the centre of symmetry. The operation of a 2 rotoinversion axis along the c axis, is to first transform an atom at x, y, z to x, y, z by the operation of a 2-fold rotation, and then to operate with a cente of symmetry to transform to the final position x, y, z.

This is actually equivalent to a mirror plane whose normal is parallel to[001].

Similarly, as a result of a 4 rotoinversion axis along[001], an atom at x, y, z is first transformed to y, x, z by the operation of a 4-fold rotation, and then to y, x, zby the operation of the centre of symmetry. This point is then replicated by the same two processes to give a point at x, y, z, which in turn is then replicated at y, x, z. An example of the 4 rotoinversion symmetry is shown in Fig. 3.12. The examples of 3 and 6 rotoinversion axes are left to Problem 3.8.

It will be demonstrated that the operation of a 6-axis is equivalent to that of a 3-fold rotation axis normal to a mirror plane.

3.3.3 Combination of symmetry operations

Frequently different symmetry operations may be combined. For example, a 2-fold rotation axis can be normal to a mirror plane, and the combined operation is written as 2/m. All combinations will also generate new symmetry operations. This can easily be demonstrated for the 2/m combination. If we have a point at x, y, z, the 2-fold axis along the z-axis will produce a new point at x, y, z. The action of the mirror plane on these two points will be to generate two new points:

x, y, z → x, y, z x, y, z → x, y, z

The last point is generated from x, y, z by the operation of a centre of symmetry, which means that the 2/m also includes this symmetry operation. It is also generated from the point x, y, z by the same 2-fold axis that gave x, y, z → x, y, z, which highlights the self-consistency.

Fig. 3.12 Illustration of 4 rotoinversion sym-metry, showing four molecules whose dis-tance in the line of sight is emphasized by the perspective.

Some combinations of symmetry operations will actually produce the same result as another symmetry operation. We have already met an example when we noted above that a 6-axis is equivalent to a mirror plane normal to a 3-fold rotation axis (what we might call 3/m). The overall result is that there are slightly fewer distinct combinations of symmetry operations than you might have expected. We now discuss the complete set of combinations appropriate for crystalline materials.

3.3.4 The 32 crystallographic point groups

Three-dimensional combinations of symmetry operations

As well as combining different symmetry operations with a single axial direction, we can also combine symmetry operations in three dimensions. These combinations of symmetry operations are calledpoint groups. From the infinite number of three-dimensional point groups, only those where the rotational symmetry has n = 1, 2, 3, 4, 6 will occur in crystals. This restriction defines the set ofcrystallographic point groups, of which there are 32.

We have anticipated the fact that there are fewer independent combinations of point-symmetry operations than might have been expected, because of the fact that combinations of symmetry operations can automatically lead to the presence of additional symmetry elements. We have previously noted this for the example of 2/m. Another example is the combination of two orthogonal 2-fold rotation axes. We start from a point at x, y, z. The first 2-fold axis, say

one that is along[100], generates the second point x, y, z. The second 2-fold rotation axis, say along[010], will operate on both points, to generate the two new points:

x, y, z → x, y, x x, y, z → x, y, z

The last point, namely that generated by the operation of both rotation axes, could also be generated from the original set of coordinates by operation of a third rotation axis along[001], which is the direction orthogonal to the other two rotation axes. Thus we cannot have only two 2-fold axes, since two orthogonal 2-fold rotation axes automatically generate a third 2-fold rotation axis. This combination is called 222.

Another example is when we start with a 2-fold axis along[100] and add a mirror plane whose normal is orthogonal to the 2-fold axis, say along[010]

(this is in contrast to the 2/m condition discussed above). The action of the mirror plane on the pair of coordinates related by the 2-fold axis gives the new set of coordinates

x, y, z → x, y, z x, y, z → x, y, z

The operation of the combination of the 2-fold rotation axis and the orthogonal mirror plane is equivalent to that of a second mirror plane normal to[001].

Thus we have to have a combination of two mirror planes and a 2-fold axis that is orthogonal to the normals of both mirror planes. This combination is called mm2 (where, unlike our example, the 2-fold axis is conventionally placed along the z axis).

Labelling of the point groups

The labels representing the point groups are given three components to denote the symmetry elements along the three directions, unless there is no particular symmetry along that direction. The component is labelled with either its full symmetry or by a convenient abbreviated symbol, when the abbreviation implies additional symmetry. For example, the orthorhombic point group with a 2-fold axis and mirror plane for each axis would be labelled as_m²_m²_m², but this is abbreviated to mmm because the three 2-fold axes are automatically generated by the three perpendicular mirror planes (Problem 3.4). The complete set of 32 crystallographic point groups is described in Table 3.2. Note that the labelling is a little complicated for the five cubic point groups.