X-Ray and Neutron Crystallography rational numbers is a group under Crystal Symmetry Groups multiplication, and both it and the integer group already discussed are examples of infinite groups because they each contain an infinite number of elements. ymmetry plays an important role between the integers obey the rules of In the case of a symmetry group, in crystallography. The ways in group theory: an element is the operation needed to which atoms and molecules are ● There must be defined a procedure for produce one object from another. For arrangeds within a unit cell and unit cells example, a mirror operation takes an combining two elements of the group repeat within a crystal are governed by to form a third. For the integers one object in one location and produces symmetry rules. In ordinary life our can choose the addition operation so another of the opposite hand located first perception of symmetry is what that a + b = c is the operation to be such that the mirror doing the operation is known as mirror symmetry. Our performed and u, b, and c are always is equidistant between them (Fig. 1). bodies have, to a good approximation, elements of the group. These manipulations are usually called mirror symmetry in which our right side ● There exists an element of the group, symmetry operations. They are com- is matched by our left as if a mirror called the identity element and de- bined by applying them to an object se- passed along the central axis of our noted f, that combines with any other bodies. Our hands illustrate this most element to give the second one un- vividly; so much so that the image is changed. In the case of the integers, carried over to crystallography when the identity element is zero because one speaks of a molecule as being either any integer plus zero gives that inte- “right”- or “left”- handed. Those of us ger (a + O = a). who live in an old-fashioned duplex ● For every element of the group, there will also recognize that such houses are exists another element that combines built with mirror symmetry so that the with the first to give the identity arrangement of the rooms, hallways, and element; these are known as inverse doors are disposed about an imaginary elements. The negative integers mirror passing through the common constitute the inverses of the positive wall between the two halves of the integers because their pairwise sums house. There are many other examples all equal zero, the identity element of this kind of mirror symmetry in (a + (–a) = 0). ordinary life. We can also see more ● Group operations in sequence obey complex symmetry in the patterns the associative law. For addition of around us. It can be found in wallpaper integers this means that (a + b) + c = THE MIRROR SYMMETRY OPERATION patterns, floor-tile arrays, cloth designs, a+(b+c). Notice that the commutative flowers, and mineral crystals. The basic law, a + b = b + a, is not required even Fig. 1. A pair of left- and right-"footed” boots mathematics of symmetry also applies though it is true for this particular Illustrates the mirror-plane symmetry operation. to music, dance (particularly folk and group. The right boot can be positioned identically square dance), and even the operations You might be tempted to say that the on the left boot by reflection through a mirror needed to solve Rubik’s cube. positive integers, when related by mul- between them and vice versa. The rules that govern symmetry are tiplication (a x b = c), also constitute found in the mathematics of group the- a group with the identity element now quentially. For example, doing a mirror ory. Group theory addresses the way in being one (a x 1 = a). In fact, the pos- operation twice on a right-handed object which a certain collection of mathemat- itive integers do not constitute a group will, with the first operation, move it to ical “objects” are related to each other. under these conditions because, to obey the left-handed position, and with the For example, consider all the positive the group-theory rules, the noninteger second operation, place it back on its and negative integers and zero. They inverses ( 1 /a) as well as all the ratio- original right-handed position. In fact, can constitute a group because under nal fractions (b/a) would have to be applying a mirror operation twice in certain circumstances the relationships included. The expanded set of positive succession is equivalent to the identity 152 Los Alamos Science Summer 1990 X-Ray and Neutron Crystallography operation, so that a mirror operation is be discussed next) gives only five possi- If the symmetry is local with no its own inverse. ble plane lattices and fourteen possible translation component, then the integer The two operations, mirror and iden- space lattices (Fig. 3). n can take on any value from one to tity, obey the four rules of group theory, The second type of crystallographic infinity. An object that has the extreme and thus constitute one of the simplest symmetry is rotation. For it to be a symmetry groups. A mathematical rep- valid symmetry operation, however, the which an infinitesimally small rotation resentation of these operations is leaves looking the same (ignoring any painted design). However, when the m -1 = m and where n is an integer. The rotation- rotation symmetry is part of a plane- or symmetry operations will then all be space-filling symmetry with translation –‘ = mm = 1. mm multiples of this rotation angle. For ex- operators, only five different rotation ample, if n = 6 the rotation angle is angles (n = 1, 2, 3, 4, or 6) can be Further, a “multiplication table” 60 degrees and the operations can be used. Replication of a unit cell with 1 2 between these two operations can be set represented by the unique set C6, C6, a rotation symmetry other than these 3 1 4 5 6 up to show the products that any pair of C 6 (= C z), C 6, C 6, and C 6 (= I ) in cannot fill a plane surface or three- symmetry operations gives in this finite which the subscript gives the fraction of dimensional space without leaving voids group (Fig. 2). or having overlapping regions. The There are three types of symmetry situation is more complicated in the operations in crystallography. The sim- three-dimensional case because a unit plest type is the set of translation oper- cell may also have different rotation ations needed to fill a two-dimensional symmetry in different directions. Many infinite plane or a three-dimensional in- different groups result from the various finite space. These operations form a combinations of these rotations. group by themselves and have essen- An extension to the concept of ro- tially the same characteristics as the tation symmetry is to include in each example group of integers discussed rotation operator a translation compo- above. The difference is that the trans- nent (Fig. 5). The resulting objects are lation group has two or three sets of helical or screwlike; hence, these oper- integers depending on whether a two- ations are called screw rotations. These dimensional plane or a three-dimen- symmetry operations are most prevalent sional space is filled. These translation in crystal lattices in which the unit-cell operations make the concept of a unit repeat requirement means that the trans- cell possible, because once the unit cell lation operations have the same integer for a crystal is specified, it takes only fraction, or some simple multiple, as the the right combination of translation op- A FINITE SYMMETRY GROUP rotation operations. For example, the erations to construct the full crystal lat- screw rotation 61 describes an opera- tice. Fig. 2. This example of a simple, finite group tion in which the rotation of 60 degrees There is also a type of translation obeying all the rules of group symmetry is accompanied by a translation of 1/6 operation that relates objects within consists solely of the Identity element, 1, of the unit cell along the rotation axis. a unit cell so that the same objects and the mirror-plane symmetry operation,m. The 64 screw rotation has the same 60- are found at coordinates that are half The multiplication table shown above for the degree rotation but this time is accom- multiples of unit-cell distances along group gives the products for any pairwise panied by a translation of 4/6 of the unit two or three of the axes. These last application of the two symmetry operations. cell along the axis. A sufficient num- operations are, for example, responsible ber of these is superimposed to give the for the face- and body-centered lattices a full circle for each operation (here required unit-cell translation (Fig. 5), found in three dimensions (Fig. 3). The 1/6) and the superscript gives the mul- and the resulting arrangement is differ- possible combinations of this full set of tiple of 60 degrees used for the rotation ent from that obtained with a 61 screw 6 translations for plane- and space-filling (Fig. 4). Because C6 is the identity rotation. arrays (along with the restrictions on the operation, these six rotation operations The one facet common to the trans- rotation-symmetry operations that will constitute a group, symbolized by C& lation, rotation, and screw operations is Los Alamos Science Summer 1990 153 X-Ray and Neutron Crystallography THE BRAVAIS SPACE LATTICES Fig. 3. The fourteen unit cells depicted above represent the only possible ways that space can be filled without gaps or overlaps between cells, that is, consonant with the restrictions of translation and rotation symmetry.