SIMMETRY AND SPACE GROUPS.

SIMMETRY AND SPACE GROUPS

A certain degree of symmetry is apparent in much of the natural world as well as in many of man's creation in art, architecture, and technology. While is perhaps the most fundamental property of the crystalline state. This chapter introduces some of the fundamental concepts of symmetry-symmetry operations, symmetry elements, and the infinite objects (space symmetry) – as well as the way these concepts are applied in the study of crystals.

An object is said to be symmetrical if after some movement, real or imagined, it is or would be indistinguishable (in appearance and other discernible properties) from the way it was initially. The movement, which might be, for example, a rotation about some fixed axis or a mirror like reflection through some plane or a translation of the entire object in a mirror like reflection though a symmetry operation. The geometrical entity with respect to which the symmetry operation is performed – an axis or plane in the examples cited – is called a symmetry element. Various possible symmetry operation and symmetry elements of crystals are considered below.

It is possible not only to determine the crystal system characteristic of a give crystalline specimen by analysis of the intensities of the reflections in the diffraction pattern of the crystal, but also to learn much more about its symmetry, including the Bravais lattice and the probable space group. As indicated in chapter 2, the 230 spaces groups represent the distinct ways of arranging identical objects on one of the 14 Bravais lattices by the use of certain symmetry operations to be described below. The determination of the space group of a crystal is important because it way reveal some symmetry within the contents of the unit cell being considered, and this symmetry may imply that some particular molecule has several parts are equivalent to each other by symmetry. Space group determination also vastly simplifies the analysis of the be diffraction pattern because different regions of the pattern may then be known to be identical. It also greatly simplifies the required calculation because only the contents of the asymmetric portion of the unit cell (the asymmetric unit) need to be considered in detail.

Scrutiny of photographs of crystal diffraction patterns reveals that there are often systematically related position where diffraction Maxima might occur but where, in fact, the observed intensity is zero. For example, if molecules pack in a crystal so that the is a two – fold screw axis parallel to the a – axis (which means that the molecules is moved a distance a/2 and then rotated 180° about the screw axis, so that for every atom at position x there is another at ½ + x), then, as far as h 0 0 reflection are concerned, the unit cell size has been halved and and the reciprocal lattice spacing has doubled. Reflections will the only be observed for even values of h. This situation is made evident by summing, in Eqs. (5.21) and (5.22) for atoms at x and ½ + x. When k and l are zero, A and B are zero if h is odd.

Most, but not all, combinations of symmetry elements give rise to systematic relationships among the indices of some of the “absent reflection.” For example, the only h k 0 reflections with appreciable intensity may be the for which (h + k) is even. Such systematic relationships imply certain symmetry relations in the parking in the structure. Before continuing with an account of methods of deriving trial structures, we present a short account of symmetry and particularly it’s relation to the possible way of packing molecules or ions in a crystal.

SYMMETRY GROUPS

Any isolated object, such as a molecules or a real crystal, can possess point symmetry; that is, any symmetry operation for this object, such as a rotation of, say, 180°, must leave at least one point within the object fixed. On the other hand, an infinite array of points, such as a lattice (or an ideal unbounded crystal structure), has translational symmetry as well, since translation (motion in a straight line, without rotation) along any integral number of lattice vectors move it intro self – coincidence and thus is a symmetry operation. A translation operation leaves no point unchanged since it moves all points equal distance in parallel directions; it is an examples of a space – symmetry operation. Because most macroscopic crystals consist of 10^12 or more unit cells, it is a fair approximation to regard the arrangement of atoms throughout most of a real crystal as possessing translational symmetry. Edge effects normally are completely negligible in structure analysis.

POINT SYMMETRY AND POINT GROUPS

The operations of rotation, mirror reflection, and inversion through a point (see below) are point – symmetry operations, since each leaves at least one point of the object in a fixed position. The geometrical requirements of lattices restrict the number of possible rotational – symmetry elements that a crystal can have. These possible rotational symmetry will now be considered:

1. n – fold rotation axes. A rotation of (360/n)° leaves the object or structure apparently unchanged (self – coincidence). The order of the axis IR said to be n. When n = 1 – that is, a rotation of 360° - the operation is equivalent to no rotation at all (0°), and IR said to be the “identity operation.” A four – fold rotation axis IR shown in figure 7.1a, and is denoted 4. It may have proved that only axes of order 1, 2, 3, 4, and 6 are compatible with structures built on three – dimensional (or even two – dimensional) lattices. Isolated molecules sometimes have symmetry axes of other orders, but when crystal are formed from a molecule with, for example, a five – fold axis, the five – fold axis cannot be a symmetry axis of the crystal. The molecule may still retain its five – fold symmetry in the crystal, but it can never occur at a position such that this symmetry is a necessary consequence of a five – fold symmetry in the crystalline environment. In the words, the five – fold symmetry is local and not crystallographc – that is, not required by the space group.
2. n – fold rotatory – inversion axes. The inversion operation, whit the origin of coordinates as the “center of inversion,” implies that every point x, y, z becomes – x, -y, -z. An n – fold rotatory – inversion axis implies that a rotation of (360/n)° (where n is 1, 2, 3, 4, or 6) followed by inversion though some point on the axis produces no apparent change in the object or structure. The one – fold case, 1, is the inversion operation itself and is often merely is shown in figure 7.1b. In general these axes are symbolized as n.

The rotatory – inversion operations differ from the pure rotation in an important respect: they convert an object into a left hand; on the other hand, a rotatory – inversion axis will, on successive operations, convert a left hand into a right hand, then that back into a left hand, and so on. Objects that cannot be superimposed on their mirror images cannot possess any element of rotatory – inversion symmetry.

1. 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).

1. 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.
2. 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.
3. 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.
4. 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.
5. 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.

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