Symmetry-equivalent elements

The equivalent symmetry element in the Schoenflies notation is the improper axis of symmetry, S which is a combination of rotation and reflection. The symmetry element consists of a rotation by n of a revolution about the axis, followed by reflection through a plane at right angles to the axis. Figure 1.14 thus presents an S4 axis, where the Fi rotates to the dotted position and then reflects to F2. The equivalent inversion axes and improper symmetry axes for the two systems are shown in Table 1.1. 

TABLE 1.1 Equivalent symmetry elements in the Schoenflies and Hermann-Mauguin Systems... 

If a symmetry element A is carried into the element B by an operation generated by a third element X, then of course B can be carried back into A by the application of X x. The two elements A and B are said to be equivalent. If A can be carried into still a third element C, then there will also be a way of carrying B into C, and the three.elements, A, B, and C, form an equivalent set. In general, any set of symmetry elements chosen so that any member can be transformed into each and every other member of the set by application of some symmetry operation is said to be a set of equivalent symmetry elements. 

Class 3 is obtained by introducing a twofold axis of rotation, symbolized by below the motif on the line of translation. The important thing to note here is that in addition to the C2 operation explicitly introduced (and all those just like it obtained by unit translation) a second set of C2 operations, with axes halfway between those in the first set is created. In space symmetry (even in ID space) the introduction of one set of (equivalent) symmetry elements commonly creates another set, which are not equivalent to those in the first set. It should also be noted that had we chosen to introduce explicitly the... 

The mirror plane (two-fold inversion axis) reflects a clear pyramid in a plane to yield the shaded pyramid and vice versa, as shown in Figure 1.12 on the right. The equivalent symmetry element, i.e. the two-fold inversion axis, rotates an object by 180" as shown by the dotted image of a pyramid with its apex down in Figure 1.12, right, but the simultaneous inversion through the point from this intermediate position results in the shaded pyramid. The mirror plane is used to describe this operation rather than the two-fold inversion axis because of its simplicity and a better graphical representation of the reflection operation versus the roto-inversion. The mirror plane also results in two symmetrically equivalent objects. 

The mutual orientation of different symmetry elements is expressed by the sequence in which they are listed. The orientation refers to the coordinate system. If the symmetry axis of highest multiplicity is twofold, the sequence is x-y-z, i.e. the symmetry element in the x direction is mentioned first etc. the direction of reference for a reflection plane is nomal to the plane. If there is an axis with a higher multiplicity, it is mentioned first since it coincides by convention with the z axis, the sequence is different, namely z-x-d. The symmetry element oriented in the x direction occurs repeatedly because it is being multiplied by the higher multiplicity of the z axis the bisecting direction between x and its next symmetry-equivalent direction is the direction indicated by d. See the examples in Fig. 3.7. 

Crystal lattices can be depicted not only by the lattice translation defined in Eq. (7.2), but also by the performance of various point symmetry operations. A symmetry operation is defined as an operation that moves the system into a new configuration that is equivalent to and indistinguishable from the original one. A symmetry element is a point, line, or plane with respect to which a symmetry operation is performed. The complete ensemble of symmetry operations that define the spatial properties of a molecule or its crystal are referred to as its group. In addition to the fundamental symmetry operations associated with molecular species that define the point group of the molecule, there are additional symmetry operations necessary to define the space group of its crystal. These will only be briefly outlined here, but additional information on molecular symmetry [10] and solid-state symmetry [11] is available. 

Notice, indeed, that a space group is a group of symmetry elements. If an atom is placed in a quite general position in the unit cell it is multiplied by the symmetry elements and thus other atoms, exactly equivalent to the first, are found at other positions precisely related to those of the first. Each space group has its own characteristic number of equivalent general positions. 

For information about point groups and symmetry elements, see Jaffd, H. H. Orchin, M. Symmetry in Chemistry Wiley New York, 1965 pp. 8-56. The following symmetry elements and their standard symbols will be used in this chapter An object has a twofold or threefold axis of symmetry (C2 or C3) if it can be superposed upon itself by a rotation through 180° or 120° it has a fourfold or sixfold alternating axis (S4 or Sh) if the superposition is achieved by a rotation through 90° or 60° followed by a reflection in a plane that is perpendicular to the axis of the rotation a point (center) of symmetry (i) is present if every line from a point of the object to the center when prolonged for an equal distance reaches an equivalent point the familiar symmetry plane is indicated by the symbol a. 

The coefficient of the symmetry element, in the top line, tells how many different equivalent operations of this type occur. However, not all symmetry elements of the same type are always equivalent. Thus, in the symmetry group of benzene, the thirteen C2 operations fall into three distinct classes the one dyad perpendicular to the molecular plane, the six that pass through opposite atoms in the ring, and the six that pass in between the atoms, through the centers of the bonds. Each of these operations is equivalent only to the others within the same class. 

The procedure in determining the parameters will naturally vary with circumstances, but a few general principles can be given. First, as to the most convenient method of calculation. We have seen (p. 228) that for any crystal plane hkl the contribution of each atom (coordinates xyz) to the expression for the structure amplitude consists of a cosine term/ cos 2 n hx- -ky- -lz) and a sine term/ sin 2ir hx- -1cy- -lz). Equivalent atoms (those rebated go each other by symmetry elements) have... 

To specify which plane or line of the vector cell contains the desired information, consider the equivalent positions of the atoms in relation to the symmetry element. Suppose the crystal in question has a twofold axis parallel to b. If there is an atom at x, y, z, there is an equivalent atom at x, y9 z (Fig. 227 a). The vector between these (Fig. 227 b)... 

We consider four kinds of symmetry elements. For an n fold proper rotation axis of symmetry Cn, rotation by 2n f n radians about the axis is a symmetry operation. For a plane of symmetry a, reflection through the plane is a symmetry operation. For a center of symmetry /, inversion through this center point is a symmetry operation. For an n-fold improper rotation axis Sn, rotation by lir/n radians about the axis followed by reflection in a plane perpendicular to the axis is a symmetry operation. To denote symmetry operations, we add a circumflex to the symbol for the corresponding symmetry element. Thus Cn is a rotation by lit/n radians. Note that since = o, a plane of symmetry is equivalent to an S, axis. It is easy to see that a 180° rotation about an axis followed by reflection in a plane perpendicular to the axis is equivalent to inversion hence S2 = i, and a center of symmetry is equivalent to an S2 axis. 

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