SIMMETRY AND SPACE GROUPS.

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- Mirror planes. These are designated m. They convert a left – handed molecule into a right – handed molecule. As shown in figure 7.1b, a mirror plane is equivalent to a two – fold rotatory – inversion axis, 2, oriented perpendicular to the plane. The symbol m is more common for this symmetry element.

FIGURE 7.1 SOME SYMMETRY OPERATIONS.

To make the distinction of left and right hands clearer a ring and watch have been indicated on the left hand but not the right (even after reflection from the left hand).

- A four – fold rotation axis, parallel to c and through the origin of a tetragonal unit cell (a = b), moves a point at x, y, z to a point at (y, -x, z) by a rotation of 90° about the axis. The sketch on the right shows all four equivalent points resulting from successive rotations, only two of these are illustrated in the left hand sketch.

- The operation 2, a two – fold rotatory – inversion axis parallel to b and through the origin, converts a point at x, y, z to a point at x, -y, z. One way of analyzing this change it to consider it is as the overall result of, first, a two – fold rotation about an axis through the origin and parallel to b (x, y, z to -x, y, -z) and then an inversion about the origin (-x, y, -z to x, -y, z). This is the same as the effect of a mirror plane perpendicular to the b axis. Note that a left hand has been converted to a right hand. The hand illustrated by broken lines is an imaginary intermediate for the symmetry operation 2.

- A two -fold screw axis, 2^1, parallel to b and through the origin, which combines both a two – fold rotation (x, y, z to -x, y, -z) and a translation of b/2 (-x, y, -z to -x, ½ + y, -z). A second screw operation will covert the point -x, ½ + y, -z to x, 1 + y, z, which is the equivalent of x, y, z in the next unit cell along b. Note that the left hand is never converted to a right hand.

- Some crystallographc four – fold screw axes showing two identity periods for each. Note that the effect of 4^1 on a left hand is the mirror image of the effect of 4^3 on a right hand. The right hand has been moved slightly to make this relation obvious.

- A B – glide plane normal to c and through the origin involves a translation of b/2 and a reflection in a plane normal to c. It converts a point at x, y, z to one at x, ½ + y, -z. Note that left hands are converted to right hands, and vice versa.

The point symmetry operations listed above (1, 2, 3, 4, 6, 1,2 or m, 3,4, and 6) can be combined together in just 32 ways in three dimensions to from the 32 three – dimensional crystallographc point group. There are, of course, other point groups, appropriate to isolated molecules and other figures, containing, for example, five – fold axes. The 32 crystallographc point groups or symmetry classes may be applied to the shapes of crystals or other finite objects; the point group of a crystal may sometime be deduced by an examination of any symmetry in the development of faces. For example, a study of crystals of Beryl shows that each has a six – fold axis perpendicular to a plane of symmetry (6/m) whit two more symmetry planes parallel to the six – fold axis and at 30° to each other (mm). The corresponding point group is designated 6/mmm. This external symmetry is a manifestation of the symmetry in the internal structure it the crystal. Frequently, however, the environment of a crystal during growth is sufficiently perturbed that the external the internal symmetry. Diffraction studies the help to establish the point group as well as the space group.

SPACE SYMMETRY

Combination of the point – symmetry operations whit translation give rise to various kinds of space – symmetry operations in addition to the pure translations.

- n – fold screw axes. These result from the combination of translation and pure rotation and are symbolized by n^r. They involve a rotation of (360/n)° and a translation parallel to the axis by the fraction r/n of the identity period along that axis. A two – fold screw axis, 2^1, is shown in figure 7.1c. If p = n – r, then the axes n^r and n^p are enantiomorphous; that is, they are mirror images of one another, like left and right hands. It is important, however, to note that it is only the axes that are enantiomorphous; structures built on them will not be enantiomorphous unless the objects in the structure are themselves enantiomorphous. Thus a left hand operated on by a 4^1 will give an arrangement that is the mirror image of that produced by the operation of a 4^3 on a right hand, but not, of course, the mirror image of that produced by the operation of a 4^3 on another left hand as shown in figure 7.1d.

- Glide planes. These symmetry elements result from the combination of translation with the mirror operations (or its equivalent, 2 normal to the plane), as illustrated in figure 7.1e. The glide must be parallel to some lattice vector, and, because the mirror symmetry operation – a lattice vector – must be reached after two glide translation. Thus these translation may be half of the repeat distance along a unit – cell edge, in which case the plane is referred to as an a – glide, b – glide, or c – glide, depending on the edge parallel to the translation. Alternatively, the glide may be parallel to a face – diagonal. No glide operation involves fractional translational components other than ½ or ¼ and the latter occurs only for glide direction parallel to a face diagonal or a body diagonal in certain nonprimitive groups.

SPACE GROUPS

It is possible to combine the various pure rotation, rotary inversion, screw axes, and glide plane in just 230 ways compatible with the geometrical requirements of three – dimensional lattice. There are thus 230 three – dimensional space groups, ranging from that with no symmetry other than the identity operation (symbolized by P1, the P implying primitive) to those with the highest symmetry, such as Fm3m, a face – centered cubic space group. These 230 space groups represent the 230 distinct ways in which objects (such as molecules) can be packed in three dimensions so that there contents of one unit cell are arranged in the same way as the contests of every other unit cell.

It is interesting to note that these 230 unique three – dimensional combinations of the possible crystallographc symmetry elements were derived independently in the last two decades of the nineteenth century by Fedorov in Russia, Schönflies in Germany, and Barlow in England. It was not until several decades late that anything was know of the actual atomic structure of even the simplest crystalline solid. Since the introduction of diffraction