Читать книгу Crystallography and Crystal Defects - Anthony Kelly - Страница 30

The macroscopically measured properties of a crystal, such as electrical resistance, thermal expansion, optical properties, magnetic susceptibility and the elastic constants, show a symmetry which can be defined and understood without reference to the translational symmetry elements defined by the lattice. If the translational symmetry of the crystal is disregarded, the remaining symmetry elements (such as axes of rotational symmetry, mirror planes and any centre of inversion), themselves consistent with the translational symmetry of the lattice, can be arranged into 32 consistent groups. These are the 32 crystallographic point groups. They are so called because they are consistent with translational symmetry and because all of the symmetry elements in a group pass through a single point and the operation of these elements leaves just one point unmoved – the point through which they pass. Other point groups, such as the point group symmetry of the icosahedron, which we shall consider in Chapter 4 when discussing the point group symmetries exhibited by quasicrystals, are not compatible with translational symmetry.

The axes of rotational symmetry, the mirror plane and the centre of inversion are all called macroscopic symmetry elements because their presence or absence in a given crystal can be decided in principle by macroscopic tests, such as etching of the crystal, the arrangement of the external faces or the symmetry of the physical properties, without any reference to the atomic structure of the crystal. The macroscopic symmetry elements are of two kinds. A symmetry operation of the first kind, such as a pure rotation axis, when operating on a right‐handed object (say) produces a right‐handed object from it, and all subsequent repetitions of this object are also right‐handed. A symmetry operation of the second kind repeats an enantiomorphous object from an original object. The left and right hands of the ideal external form of the human body are enantiomorphously related. The operation of reflection illustrated in Figure 1.12 is an example of a symmetry operation of the second kind since a left‐handed object is repeated from an original right‐handed object. Subsequent operation of the same symmetry element would produce a right‐handed object again and then a left‐handed object, and so forth. A symmetry operation of the second kind therefore involves a reversal of sense in the operation of repetition. Inversion through a centre is also an operation of the second kind.

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.¹ A pure rotation axis is said to produce a proper rotation.

The basic operation of repetition by a rotation axis is shown schematically in Figure 2.1. An n‐fold axis repeats an object by successive rotations through an angle of 2π/n. In the example shown in Figure 2.1, the axis is one of fourfold symmetry. The operations of monad, diad, triad, tetrad, and hexad pure rotation axes on a single initial pole are shown in Figure 2.2. These diagrams are stereograms (see Appendix 2) with the pole of the axis at the centre of the primitive circle. Dots are used to represent poles in the northern hemisphere of projection related to one another by the rotation axis. The numbers below the stereograms give the shorthand labels for the axes 1, 2, 3, 4 and 6, indicating a onefold, twofold, threefold, fourfold, and sixfold axis, respectively. The repetition of an object by a mirror plane, symbol m, and by a centre of symmetry (or centre of inversion) is shown in Figure 2.3. In Figure 2.3a 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 Figure 2.3b 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 Figure 2.3c the centre of inversion is at the centre of the sphere of projection.

Figure 2.1 The basic operation of repetition by a rotation axis. In this example, the axis is a four‐fold symmetry axis

Figure 2.2 Stereograms representing the operation of one‐, two‐, three‐, four‐ and sixfold axes on a single initial pole, represented as a dot

Figure 2.3 The repetition of an object by a mirror plane, (a) and (b), and by a centre of symmetry, (c)

Figure 2.4 The operation of the twofold rotoinversion axis,

Figure 2.5 The operation of the various rotoinversion axes that can occur in crystals

The onefold inversion axis is a centre of symmetry. The operations of the other rotoinversion axes are explained in Figures 2.4 and 2.5. The twofold rotation–inversion axis shown in Figure 2.4 repeats an object by rotation through 180° (360°/2) to give the dotted circle, followed by inversion to give the full circle. Similarly, the threefold inversion axis involves rotation through 360°/3 = 120° coupled with an inversion. In general, an n‐fold rotoinversion axis involves rotation through an angle of 2π/n 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. The operation of the various rotoinversion axes that can occur in crystals on a single initial pole is shown in Figure 2.5, with the pole of the rotoinversion axis at the centre of the primitive circle. The following symbols are used: inversion monad, , symbol ○; inversion diad, , symbol ⋄; inversion triad, , symbol ; inversion tetrad, , symbol ; inversion hexad, , symbol . Inspection of Figures 2.2–2.4 shows that is identical to a centre of symmetry, is identical to a mirror plane normal to the inversion diad, is identical to a triad axis plus a centre of symmetry and 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).² Only is unique. The operation of repetition described by 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 , , , and rotoinversion axes constitute the 32 crystallographic point groups or crystal classes. These 32 classes are grouped into systems according to the presence of defining symmetry elements (see Table 1.3). Stereograms of each of the 32 crystallographic point groups or crystal classes are given in Figure 2.6, following the current conventions of the International Tables for Crystallography [3]. With the exception of the two triclinic point groups, each point group is depicted by two stereograms. The first stereogram shows how a single initial pole is repeated by the operations of the point group and the second stereogram shows all of the symmetry elements present. The nomenclature for describing the crystal classes is as follows. X indicates a rotation axis and an inversion axis. X/m is a rotation axis normal to a mirror plane, Xm a rotation axis with a mirror plane parallel to it and X2 a rotation axis with a diad normal to it. X/mm indicates a rotation axis with a mirror plane normal to it and another parallel to it. is an inversion axis with a parallel plane of symmetry. A plane of symmetry is an alternative description of a mirror plane.

Figure 2.6 Stereograms of the poles of equivalent general directions and of the symmetry operations of each of the 32 crystallographic point groups. The z‐axis is normal to the paper. In all the centrosymmetric classes, positions of centres of symmetry (inversion monads) lying within the primitive of the stereogram are shown.

Source: Taken from the International Tables for X‐ray Crystallography, Vol. 1 [4] and adapted to conform to current notation for the two centrosymmetric point groups in the cubic crystal system.

We shall describe each of the classes in Sections 2.2–2.8. A derivation of the 32 classes follows by noting from Sections 1.5 and 1.6 that the rotation axes consistent with translational symmetry are 1, 2, 3, 4 and 6. Individually, these give in total five crystal classes. Their consistent combinations give another six (see Table 1.2): 222, 322, 422, 622, 332 and 432, thus totalling 11. All of these 11 involve only operations of the first kind. A lattice is inherently centrosymmetric (Section 1.4) and so each of the rotation axes could be replaced by the corresponding rotoinversion axis, thus giving another five classes: , , , and . The remaining 16 can be described as combinations of the proper and improper rotation axes. It is convenient to begin first with the three crystal systems where the angles between the axes are all 90°, then to consider the hexagonal and trigonal crystal systems, before finally turning our attention to the monoclinic and triclinic systems.