3 Formal description ofcrystal structures
3.4 Application of the formalism of point groups
3.4.1 Symmetry of the crystal
Point groups represent the symmetry of a point, and therefore do not directly give information about the periodicity of a crystal structure. However, they do give information about the symmetry of the environments of the atoms in the
Table 3.2 The symmetry of the 32 crystallographic point groups, labelled along the principal symmetry directions in each crystal class.
Triclinic Point Schoenflies 1
group symbol
1 C1 No
1 Ci(S2) Yes
Monoclinic Point Schoenflies 1 [010]
group symbol
2 C2 No 2
m Cs(C1h) No m
2/m C2h Yes 2/m
Orthorhombic Point Schoenflies 1 [100] [010] [001]
group symbol
222 D2(V ) No 2 2 2
mm2 C2v(C1h) No m m 2
mmm D2h(Vh) Yes 2/m 2/m 2/m
Tetragonal Point Schoenflies 1 100 110 [001]
group symbol
4 C4(V ) No 1 1 4
4 S4 No 1 1 4
4/m C4h Yes 1 1 4/m
422 D4 No 2 2 4
4mm C4v No m m 4
4m2 D2d(Vd) No 2 m 4
4/mmm D4h Yes 2/m 2/m 4/m
Trigonal/ Point Schoenflies 1 100 110 [001]
Rhombohedral group symbol
3 C3 No 1 1 3
3 C3i(S6) Yes 1 1 3
32 D3 No 2 2 3
3m C3v No 1 m 3
3m D3d Yes 2/m 1 3
Hexagonal Point Schoenflies 1 100 110 [001]
group symbol
6 C6 No 1 1 6
6 C3h No 1 1 6= 3/m
6/m C6h Yes 1 1 6/m
622 D6 No 2 2 6
6mm C6v No m m 6
6m2 D3h No m 2 6
6/mmm D6h Yes 2/m 2/m 6
Cubic Point Schoenflies 1 100 110 111
group symbol
23 T No 2 1 3
m3 Th Yes 2/m 1 3
432 O No 4 2 3
43m Td No 4 m 3
m3m Oh Yes 4/m 2/m 3
crystal. This can be particularly useful in conjunction with experiments that probe the behaviour of individual atoms. For example, the point symmetry will describe the symmetry of an electric field experienced by an atom, or the shape of the electronic wave functions that may be distorted by the local electric fields. These will influence results from NMR or optical absorption resonance experiments.
The point group symmetry will also represent the macroscopic symmetry of a crystal. For example, it can describe the symmetry of a cut gemstone, or the relationship between anisotropic physical properties and the crystal axes. We will find in Chapter 6 that the point group symmetry can be used to represent the symmetry of diffraction patterns.
3.4.2 Symmetry breaking transformations
One of the important applications of the formalism of point group symmetry is to describe and quantify changes in the structure that involve a reduction in the number of symmetry operations. These changes may arise as a result of a phase transition, or dynamically through lattice vibrations, both of which are discussed in the later chapters of this book.
The use of point group symmetry to describe distortions is illustrated by the example of a crystal with orthorhombic point group mmm containing four atoms in the unit cell. We consider displacements along one direction, say the x-direction, as shown in Fig. 3.13. There are four independent combinations of positive and negative displacements of the atoms, each of which will preserve four symmetry elements and break four others, and these are what are shown in Fig. 3.13.
Two of these combinations involve the loss of the centre of symmetry;
they will each also lose some of the other symmetry operations. The point group mmm has eight symmetry operations, namely the operation that changes nothing, called theidentity operation, the three 2-fold axes, the three mirror planes, and the centre of symmetry. There are two point groups that can be
Fig. 3.13 Four sets of displacements of atoms in point group mmm, with symmetry labelled by the irreducible representations described in the text. The atomic motions in the A1g irreducible representation do not change the symmetry, so for the corresponding picture the atoms are shown in their mean positions.
A1g
B1u B2u
B3g
x
y
generated from mmm by the loss of the centre of symmetry, namely 222 and mm2, both of which have four symmetry operations. Thus the loss of the centre of symmetry will be accompanied by the loss of three other symmetry operations. There is clearly not one unique combination, and it is therefore useful to be able to quantify and characterize all possible combinations.
The characterization of the combinations of symmetry operations that can be preserved and lost in a single deformation of a structure takes us into the realm ofgroup theory. We can only touch on some of the important results here, and in particular we will illustrate how the patterns of deformations that result in a lowering of symmetry can be characterized.
The main tool is thecharacter table, which sorts and labels the combinations of symmetry operations that can be broken by a set of single deformations. We consider the following table of symmetry operations that lower the symmetry:
mmm E 2x 2y 2z 1 mx my mz
A1g 1 1 1 1 1 1 1 1 mmm
B1g 1 1 −1 −1 1 1 −1 −1 2/m(x)
B2g 1 −1 1 −1 1 −1 1 −1 2/m(y)
B3g 1 −1 −1 1 1 −1 −1 1 2/m(z)
A1u 1 1 1 1 −1 −1 −1 −1 222
B1u 1 1 −1 −1 −1 −1 1 1 2mm(x)
B2u 1 −1 1 −1 −1 1 −1 1 m2m(y)
B3u 1 −1 −1 1 −1 1 1 −1 mm2(z)
This table has one column for each symmetry operation: E stands for the identity operation; 2x, 2y, and 2z represent the three 2-fold axes; 1 is the centre of symmetry; mx, my, and mzrepresent the three mirror planes. The table lists all combinations in which four of the symmetry operations are lost, together with a deformation in which the symmetry is preserved. In this table, the symbol 1 represents the symmetry operations that are preserved, and−1 represents the symmetry operations that are broken. The first column gives the label of the combination of symmetry operations, and the last column (not usually given in character tables) gives the point group symmetry that results from the loss of the particular combination of symmetry operations. The key to the labels is that the subscripts g and u (from the Germangerade and ungerade for even and odd respectively) indicate that the centre of symmetry is either preserved or lost respectively, A represents the preservation of all the rotational symmetry operations, and B represents the loss of some of the rotational symmetry operations. The numbers in the subscripts simply list similar combinations.
Each row in the table is called anirreducible representation.
The action of the irreducible representation B1gis to lose two rotation axes and the mirror planes normal to these axes. This deformation preserves the centre of symmetry, one rotation axis, and the mirror plane normal to this axis.
Thus the symmetry of the deformed state is 2/m. The actions of deformations of symmetry B2gand B3glead to the same point groups, but with the unique axis lying along different directions of the initial structure. The action of the irreducible representation A1uis to lose the centre of symmetry and the three mirror planes, leading to the lowering of the symmetry to point group 222.
The action of irreducible representation B1uis to lose two of the rotation axes and the mirror plane normal to the third axis, leading to a lowering of the
symmetry to point group mm2. The irreducible representations B2u and B3u
lead to the same change in symmetry, but with the unique axes along different directions.
The example of point group mmm is one of the easier groups to study because there are no subtleties. In the case where there are 3-fold, 4-fold and 6-fold rotation axes different distortions can be symmetrically equivalent; for example, the distortion of a cubic unit cell that elongates the unit cell along the x-direction is symmetrically equivalent to the distortions that elongate the unit cell along the y and z directions. Thus this type of distortion is triply degenerate, and is given the symbol T . Similarly, the two distortions that elongate a tetragonal unit cell along the x and y directions are doubly degenerate, and given the symbol E.
The use of character tables has a number of applications in the study of crystalline materials. In the context of this book, two of the important examples are in the study of displacive phase transitions (Chapter 12) and lattice vibrations (Chapter 8). In the study of phase transitions, there is a change in symmetry due to a specific distortion of the crystal structure. If the unit cell does not change size (by which we mean the lower symmetry unit cell does not span two or more unit cells of the higher symmetry structure), the distortion can be described as an irreducible representation of the point group of the higher symmetry phase.
A similar description is possible if there is a change in size of the unit cell, but in this case the full description of the distortion needs also to take account of the change in translational symmetry as discussed below.
The displacements of atoms in a lattice vibration cause an instantaneous distortion of the unit cell. If the vibration has infinite wavelength (that is, if the vibration causes the same atomic displacements in each unit cell), it is possible to characterize the distortions associated with each vibration as a separate irreducible representation of the point group of the crystal. Methods exist to determine how the vibrations of a crystal decompose into the different available irreducible representations. This is useful because experimental probes such as Raman and infrared spectroscopy are sensitive only to vibrations with effectively infinite wavelength, as discussed in Chapter 10.
For vibrations that do not have infinite wavelength, similar descriptions can also apply. However, because there will be changes in the periodicity of the structure the relevant point group will be lower than that of the actual crystal.
For example, a vibration travelling along the[100] direction in a cubic crystal with a finite wavelength will automatically cause a change in the translational symmetry along[100] compared to the translational symmetry along the other two crystal directions. Therefore the relevant point group from which to take the irreducible representations of the vibrations will be one of the tetragonal groups.