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Lecture 2: Point Groups and Space Groups in Crystalline Materials

Restrictions on rotational symmetry elements when there is translational periodicity

All crystals show translational symmetry. A given crystal may or may not possess other symmetry

elements. Axes of rotational symmetry must be consistent with the translational symmetry of the

lattice. A one-fold rotation axis is obviously consistent. To prove that in addition only diads, triads,

tetrads and hexads can occur in a crystal we consider just a two-dimensional lattice or net.

Let A, ,A,A in the figure below be lattice points of the mesh and let us choose the direction

AAA so that the lattice translation vector t of the mesh in this direction is the shortest lattice

translation vector of the net. Suppose an axis of n-fold rotational symmetry runs normal to the net at

A. Then the point A must be repeated at B by rotation through an angle n/2ABA . Also,

since A is a lattice point exactly similar to A there must also be an n-fold axis of rotational

symmetry passing normal to the paper through A . This repeats A at B , as shown in the figure

below. Now B and B define a lattice row parallel to AA . Therefore the separation of B and B

must be an integral number times t. Call this integer N. From the figure below, the separation of B

and B is ) cos 2( tt .

Rotation axes and translational symmetry

Therefore the possible values of are restricted to those satisfying the equation

Nttt cos2

or

2

1 cos

N

where N is an integer. Since 1 cos1 the only possible solutions are shown in the table

below. These correspond to one-fold, six-fold, four-fold, three-fold and two-fold axes of rotational

symmetry. No other axis of rotational symmetry is consistent with the translational symmetry of a

lattice and hence other axes do not occur in crystals.

Table: Solutions of 2

1 cos

N

N 1 0 1 2 3

cos 1 0.5 0 0.5 1

0 60 90 120 180

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Macroscopic symmetry elements

In developing the 32 crystallographic point groups it is convenient to have all the macroscopic

symmetry elements represented by axes and to do this we define what are called improper

rotations. These produce repetition by a combination of a rotation and an operation of inversion.

We shall use rotoinversion axes. These involve rotation coupled with inversion through a centre.

An n-fold axis repeats an object by successive rotations through an angle of ./2 n This can be

represented on a stereogram for the five axes compatible with translational symmetry: 1, 2, 3, 4 and

6:

The repetition of an object by a mirror plane, symbol m, and by a centre of symmetry (or centre of

inversion) is shown below.

In (a) the mirror plane lies normal to the primitive circle. It is denoted by a strong vertical line

coinciding with the mirror in the stereographic projection. In (b) the mirror coincides with the

primitive; the dot representing the pole in the northern hemisphere has as its mirror image the circle

shown in the southern hemisphere. In (c) the centre of inversion is at the centre of the sphere of

projection.

An n-fold rotoinversion axis n involves rotation through an angle of n/2 coupled with

inversion through a centre. The rotation and inversion are both part of the operation of repetition

and must not be considered as separate operations. For example, with 2 , we have:

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The operation of the various rotoinversion axes, which can occur in crystals, on a single initial pole

is shown in the diagram below, with the pole of the rotoinversion axis at the centre of the primitive

circle.

1 is identical to a centre of symmetry, 2 is identical to a mirror plane normal to the inversion diad,

3 is identical to a triad axis plus a centre of symmetry and 6 is identical to a triad axis normal to a

mirror plane (symbol 3/m, the sign ‘/m indicating a mirror plane normal to an axis of symmetry).

Only 4 is unique. The operation of repetition described by 4 cannot be reproduced by any

combination of a proper rotation axis and a mirror plane or a centre of symmetry.

The various different combinations of 1, 2, 3, 4 and 6 pure rotation axes and 4,3,2,1 and 6

rotoinversion axes constitute the 32 point groups or crystal classes. These are shown on the next

two pages.

The 32 crystallographic point groups

A derivation of the 32 point groups, or classes, follows the lines of noting that the rotation axes

consistent with translational symmetry are 1, 2, 3, 4 and 6. Each of these existing alone gives in

total five crystal classes.

The consistent combinations of these axes gives another six classes viz. 222, 322, 422, 622, 332

and 432, thus totalling eleven. All of these eleven involve only operations of the first kind.

A lattice is inherently centrosymmetric, and so each of the rotation axes could be replaced by the

corresponding rotoinversion axis, thus giving another five classes, viz. 4,3,2,1 and .6

The remaining sixteen can be described as combinations of the proper and improper rotation axes.

Possible combinations of rotational symmetries

As we have just seen earlier this lecture, the axes of n-fold rotational symmetry which a crystal can

possess are limited to values of n of 1, 2, 3, 4 or 6. These axes lie normal to a net. In principle, a

crystal might conceivably be symmetric with respect to many intersecting n-fold axes. However, it

turns out that the possible angular relationships between axes are severely limited to the six

combinations 222, 322, 422, 622, 332 and 432. Thus, for example, we cannot combine a 6 with a 4

in the same point group.

It is easy to show that 222 is a self-consistent set from looking at the relevant orthorhombic point

group – this does not require complex mathematics to appreciate the result. Likewise, but with a

little more thought, we can do this for point group 32 where the four triads along <111> axes force

diads along <100> axes.

For a general mathematical method to describe how possible rotations can be combined, see the

book by Kelly and Knowles.

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Point groups of the triclinic, monoclinic, orthorhombic and tetragonal systems

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Point groups of the trigonal, hexagonal and cubic systems

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Stereographic projection geometry: further examples demonstrating point groups:

Hexagonal system, 6/mmm:

Here, for poles on the primitive, the poles are both normals to planes (hkil) and vectors [hkil].

Monoclinic system, 2/m:

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Space groups

Repetition of an object (e.g., an atom or group of atoms) in a crystal is carried out by operations of

rotation, reflection (or inversion) and translation. Point groups are the consistent combinations of

rotations and inversions that can occur in crystals. The possible translations that can occur in

crystals produce the 14 Bravais lattices, or space lattices.

If we combine the operations of rotation (and inversion) with translation, we produce space groups.

There are 230 different crystallographic space groups and each one gives the fullest description of

the symmetry elements present in a crystal possessing that space group.

Space groups are most important in the solution of crystal structures. They are also very useful

when establishing symmetry hierarchies in phase transitions in minerals such as perovskites, such

as BaTiO3. BiFeO3 is another example of a perovskites; this has the space group R3 c.

When an attempt is made to combine the operations of rotation and translation, the possibility arises

naturally of what is called a screw axis. This involves repetition by rotation about an axis, together

with translation parallel to that axis. Similarly, a repetition by reflection in a mirror plane may be

combined with a translational component parallel to that plane to produce a glide plane.

Examples of a diad rotation, a diad screw axis and a glide plane are shown below.

When we combine the operations of rotation with translation and look for the possible consistent

combinations we can start with the point group and associate the symmetry elements of the point

group with each lattice point of the lattices consistent with that point group.

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From the number of Bravais lattices (14) and number of point groups (32), it might be

expected that 66 space groups might be obtained (2 triclinic, 6 monoclinic, 12

orthorhombic, 14 tetragonal, 5 trigonal with a rhombohedral lattice, 5 trigonal with a

hexagonal lattice, 7 hexagonal and 15 cubic). Seven additional ones arise because of cases

in which the glide planes or screw axes automatically arising are different for different

orientations of the point group with respect to the lattice; e.g. in the tetragonal system 24mI

and mI 24 are distinct space groups. The fourteen space groups for which this consideration

is relevant are Cmm2, Amm2 ( mmC2 ), mP 24 , 24mP , mI 24 , 24mI , 26mP , mP 26 ,

P312, P321, P3m1, P31m, 13mP and mP 13 . Therefore, there are 73 such space groups,

referred to as arithmetic crystal classes in Section 8.2.3 of the fifth edition of the

International Tables for Crystallography, Vol. A (2002).

When the possibilities of introducing screw axes to replace pure rotational axes and glide planes to

replace mirror planes are taken into consideration, the result is to produce a total of 230 different

space groups. Of particular note are the space groups in the trigonal crystal system. Of the 25

trigonal space groups, only 7 have a rhombohedral unit cell (R3, R3 , R32, R3m, R3c, R3 m, R3 c);

the majority have a primitive hexagonal unit cell.

[Buerger in his 1963 book Elementary Crystallography argues that ‘to refer one crystal of the

trigonal class to rhombohedral axes, and another to hexagonal axes, because the lattice types are R

and P respectively, leads to confusion’ and discourages the use of rhombohedral axes for this

reason. It is certainly the case in the scientific literature that the use of rhombohedral axes for

crystals where the space group is one of the seven rhombohedral space groups is markedly less

prevalent than the use of hexagonal axes.]

Glide planes are described as axial glide planes if the translation parallel to the mirror is parallel to

a single axis of the unit cell and equal to one-half of the lattice parameter in that direction. Such

glide planes are given the symbols, a, b or c corresponding to the directions of the glide

translations. A diagonal glide plane involves a translation of one-half of a face diagonal or one-half

of a body diagonal (the latter in the tetragonal and cubic systems), given the symbol n, and a

diamond glide plane involves a translation of one-quarter of a face diagonal, given the symbol d.

The diamond glide plane involves a translation of one-quarter of a body diagonal in the tetragonal

and cubic systems. In centred cells, the possibility of a ‘double’ glide plane arises, given the symbol

e. In this symmetry operation, recognised officially in the most recent edition of the International

Tables for Crystallography and now incorporated into the conventional space group symbols of

five space groups, two glide reflections occur through the plane under consideration, with glide

vectors perpendicular to one another.

The conventional space group symbol shows that the space groups have been built up by placing a

point group at each of the lattice points of the appropriate Bravais lattice. Thus, for example,

mFm3 (in full, F 4/m 3 2/m) means the cubic face-centred lattice with the point group mm3

associated with each lattice point and mmcP /63 (in full, P m/63 2/m 2/c) is a hexagonal primitive

lattice derived from P6/mmm by replacing the six-fold rotation axis by 36 and one of the mirror

planes by a c axis glide plane. mmcP /63 and P6/mmm share the same point group symmetry. The

point group symmetry of any crystal is derived immediately from the space group symbol by

replacing screw axes by the appropriate rotational axes and glide planes by mirror planes in the

space group symbol.

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Groups, sub-groups and supergroups

It is evident from the discussion on point groups and space groups that each point group and space

group consists of a self-consistent finite set of symmetry operations. Therefore, point groups and

space groups can be described in the formalism of group theory. A group G is a set of elements g1,

g2, g3 ..... gn which conform to the following four conditions:

(i) The product gigj of any two group elements must be another element within the group;

(ii) Group multiplication is associative: (gigj)gk = gi(gjgk);

(iii) There is a unique group element I called the identity element belonging to G so that its

operation on a member of the group gi is such that Igi = giI = gi;

(iv) Each element within the group has a unique inverse, i.e. for each gi there is a unique element

gi1 so that gigi

1 = gi1 gi = I.

Thus, for example, following the example of the point group mmm, the eight group elements within

this point group are: the identity, I, three 180° rotations about the x-, y- and z-axes, three reflections

in the (100), (010) and (001) planes and a centre of symmetry.

A sub-group of these symmetry elements is found in 2mm: the identity, a 180° rotation about the x-

axis and two reflections in the (010) and (001) planes. Other sub-groups are the point groups 222,

2/m, m, 2, 1 and 1. Conversely, the point group mmm is a clearly a supergroup for 2mm and 222.

Similar principles apply to sub-groups and supergroups of space groups. Thus, for example, the

space group P 6 2m is one member of the group whose minimal non-isomorphic supergroup is

P63/mcm; it has as maximal non-isomorphous subgroups P 6 , P321, P31m and Pm2m. (Groups

which are isomorphic are termed such if they have an equivalent group multiplication structure,

while differing in the nature of the elements constituting the groups. Thus, for example, the point

groups 2/m, 222 and 2mm are isomorphic: all three have four elements within the group and the

same multiplication structure within the group.)

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An example of a three-dimensional space group

To illustrate the principles above, an example of an entry in the most recent edition of the

International Tables for Crystallography is shown below. It is apparent from the

HermannMauguin notation of this space group, Pnma, No. 62, that crystals with this space group

symmetry belong to the orthorhombic crystal system, point group mmm. The Schoenflies notation 162hD indicates that the space group is dihedral (or two-sided) from the ‘D’, that it has a 2-fold

rotation axis (from the ‘2’), that there is in addition a mirror plane perpendicular to this 2-fold axis

(from the h subscript) and that it is the 16th of 28 such space groups 12hD ....... 28

2hD (space groups

4774 inclusive), all of which are space groups derived from the mmm point group.

The full HermannMauguin notation for this space group is P 21/n 21/m 21/a, showing that there are

three mutually perpendicular sets of screw diads in addition to the mirror, m, and the diagonal, n,

and axial, a, glide planes. The diagrams of the symmetry elements of the group follow the

principles for planar space groups with the use of standard graphical symbols for n (parallel to

(100)) and a (parallel to (001)). The projections in the top left-hand corner and the bottom right-

hand corner are both down [001], the one in the top right-hand corner down [010] and the one in the

bottom left-hand corner down [100].

The asymmetric unit is ‘a (simply) connected smallest part of space from which, by application of

all symmetry operations of the space group, the whole of space is filled exactly’ (International

Tables for Crystallography, Vol. A). The listing under the heading ‘Symmetry operations’ is a

summary of the various geometric descriptions of each of the eight symmetry operations in the

space group, numbered (1) – (8). Thus, for example, here, the second of these, 2 (0,0,2

1) ,

4

1,0,z

defines the screw diad parallel to [001] intersecting the x-y plane at (4

1,0). The ‘Generators

selected’ define the order in which the coordinates of the symmetrically equivalent positions are

produced, e.g., the coordinates related by symmetry to the general position x, y, z in the space

group. (1) is the identity operation showing that in this case integral lattice translations produce the

equivalent point in adjacent unit cells. After the ‘Positions’, and the ‘Symmetry of special

projections’, both of which are self-explanatory, a summary is given of subgroups and supergroups

related to the space group under consideration. Finally, the reflection conditions listed under

‘Positions’ specify the systematic absences which occur under kinematical diffraction conditions,

such as X-ray diffraction.

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