Point symmetry operators

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. 

In Figure 2.4, t happens to be parallel to x. Translations are not point symmetry operations because every point in configuration space is translated with respect to the fixed axes OX, OY, OZ. 

A symmetry element (which is not to be confused with a group element) is a point, line, or plane with respect to which a point symmetry operation is carried out. The symmetry elements, the notation used for them, the corresponding operation, and the notation used for the symmetry operators are summarized in Table 2.1. It is not necessary to use both n and n since all configurations generated by h can be produced by n. ... 

Table 2.1. Symmetry elements and point symmetry operations.

Figure 2.6. Projection diagrams showing examples of the point symmetry operators listed in Table 2.1. (a) / (b) C2z (c) /C4+ (d) ay (e) S+.

It follows from Exercise 2.1-3(a) and Example 2.1-1 that the only necessary point symmetry operations are proper and improper rotations. Nevertheless, it is usually convenient to make use of reflections as well. However, if one can prove some result for R and IR, it will hold for all point symmetry operators. 

The complete set of point symmetry operators that is generated from the operators Ri R2... that are associated with the symmetry elements (as shown, for example, in Table 2.2) by forming all possible products like R, Ry and including E, satisfies the necessary group properties the set is complete (satisfies closure), it contains E, associativity is satisfied, and each element (symmetry operator) has an inverse. That this is so may be verified in any particular case we shall see an example presently. Such groups of point symmetry operators are called point groups. For example, if a system has an S4 axis and no... 

Consider the set of point symmetry operators associated with a pyramid based on an equilateral triangle. Choose z along the C3 axis. The set of distinct (non-equivalent) symmetry operators is G li O, O, [Pg.32]

Any set with the four properties (a)-(d) forms a group therefore the set G is a group for which the group elements are point symmetry operators. This point group is called C3v or 3m, because the pyramid has these symmetry elements a three-fold principal axis and a vertical mirror plane. (If there is one vertical plane then there must be three, because of the three-fold symmetry axis.)... 

Exercise 9.2-1 Justify the above statement about the invariance of eq. (5) under point symmetry operations. 

R,S,T general symbols for point symmetry operators (point symmetry operators leave at least one point invariant)... 

T and therefore equal to Jx, Jy, J without the common factor of 1/2 translation operator (the distinction between T a translation operator and T when used as a point symmetry operator will always be clear from the context)... 

Fig. 3 M.C. Escher s Study of Regular Division of the Plane with Human Figures (1944). As in Escher s other tessellation diagrams, translational and point symmetry operations are used to completely fill the plane with repeated, ordered objects. Copyright 2011 The M.C. Escher Company - Holland. All rights reserved, www.mcescher.com...

Write the transformation matrix for each type of point symmetry operation. 

This family of operators can be regarded as an extension of the family of point symmetry operators. Symmorphy is a particular extension of the point symmetry group concept of finite point sets, such as a collection of atomic nuclei, to the symmorphy group concept of a complete algebraic shape characterization of continua, such as the three-dimensional electron density cloud of a molecule. In fact, this extension can be generalized for fuzzy sets. 

All the point symmetry operators R so defined are linear operators. 

As a second example, consider the growth of the diamond lattice about a central point. Space group 227 (Fd3m O ) describes the diamond crystal structure and we see from this information that the structure is non-symmorphic and that while the factor group is of order 48, not all of these symmetry operations are common to the point symmetry group Oh. Table 2.5 lists the point symmetry operations of O, in which the non-primitive translation j)... 

Note that in two different molecules a given common point symmetry operator R can be associated with two different permutation operators P and P depending on the nuclear arrangements. Consequently, the framework groups do contain more information on molecular shapes than point symmetry groups. 

Within the syntopy model, the essential algebraic structure of point symmetry groups is retained (in fact, this structure is extended), and the elements of syntopy groups are derived from ordinary point symmetry operators [252,394,395]. There are, however, alternative approaches for the generalization of symmetry, where fundamentally different algebraic structures are used. 

One such approach, the symmorphy group approach [43,108], is based on the extension of the family of point symmetry operators to a much richer family of operations which preserve the general morphology of objects. (Note that the term "symmorphy" is used in a different sense in the crystallography literature, with reference to the symmorphic space groups of crystallography, also called semi-direct... 

With reference to a given 3D object such as a molecular charge density function p(r), one can take a selection from the transformations in family G, based on the same general condition used above in the case of point symmetry operations indistinguishability of the original and transformed objects. 

Many common objects are said to be symmetrical. The most symmetrical object is a sphere, which looks just the same no matter which way it is turned. A cube, although less symmetrical than a sphere, has 24 different orientations in which it looks the same. Many biological organisms have approximate bilateral symmetry, meaning that the left side looks like a mirror image of the right side. Symmetry properties are related to symmetry operators, which can operate on functions like other mathematical operators. We first define symmetry operators in terms of how they act on points in space and will later define how they operate on functions. We will consider only point symmetry operators, a class of symmetry operators that do not move a point if it is located at the origin of coordinates. 

For each symmetry operator, we define a symmetry element, which is a point, line, or plane relative to which the symmetry operation is performed. The symmetry element for a given symmetry operator is sometimes denoted by the same symbol as the operator, but without the caret ( ). For example, the S5mimetry element for the inversion operator is the origin. The symmetry element of any point symmetry operator must include the origin. If a point is located on the symmetry element for a symmetry operator, that symmetry operator will not move that point. 

---