Symmetry operators space inversion

Invariance of Quantum Electrodynamics under Discrete Transformations.—In the present section we consider the invariance of quantum electrodynamics under discrete symmetry operations, such as space-inversion, time-inversion, and charge conjugation. 

The group-subgroup relation of the symmetry reduction from diamond to zinc blende is shown in Fig. 18.3. Some comments concerning the terminology have been included. In both structures the atoms have identical coordinates and site symmetries. The unit cell of diamond contains eight C atoms in symmetry-equivalent positions (Wyckoff position 8a). With the symmetry reduction the atomic positions split to two independent positions (4a and 4c) which are occupied in zinc blende by zinc and sulfur atoms. The space groups are translationengleiche the dimensions of the unit cells correspond to each other. The index of the symmetry reduction is 2 exactly half of all symmetry operations is lost. This includes the inversion centers which in diamond are present in the centers of the C-C bonds. 

The special class of transformation, known as symmetry (or unitary) transformation, preserves the shape of geometrical objects, and in particular the norm (length) of individual vectors. For this class of transformation the symmetry operation becomes equivalent to a transformation of the coordinate system. Rotation, translation, reflection and inversion are obvious examples of such transformations. If the discussion is restricted to real vector space the transformations are called orthogonal. 

The unit cell considered here is a primitive (P) unit cell that is, each unit cell has one lattice point. Nonprimitive cells contain two or more lattice points per unit cell. If the unit cell is centered in the (010) planes, this cell becomes a B unit cell for the (100) planes, an A cell for the (001) planes a C cell. Body-centered unit cells are designated I, and face-centered cells are called F. Regular packing of molecules into a crystal lattice often leads to symmetry relationships between the molecules. Common symmetry operations are two- or three-fold screw (rotation) axes, mirror planes, inversion centers (centers of symmetry), and rotation followed by inversion. There are 230 different ways to combine allowed symmetry operations in a crystal leading to 230 space groups.12 Not all of these are allowed for protein crystals because of amino acid asymmetry (only L-amino acids are found in proteins). Only those space groups without symmetry (triclinic) or with rotation or screw axes are allowed. However, mirror lines and inversion centers may occur in protein structures along an axis. 

But this does nob end the tale of possible arrangements. Hitherto we have considered only those symmetry operations which carry us from one atom in the crystal to another associated with the same lattice point—the symmetry operations (rotation, reflection, or inversion through a point) which by continued repetition always bring us back to the atom from which we started. These are the point-group symmetries which were already familiar to us in crystal shapes. Now in "many space-patterns two additional types of symmetry operations can be discerned--types which involve translation and therefore do not occur in point-groups or crystal shapes. 

Figure 3.4 The labeled cube 1 has as its only symmetry element an inversion center. In step (a) the inversion operation is carried out on 1. In step (b) we rotate by n about the 2 axis and thus orient it in space so it can be directly superimposed on 3. The labeled cube 3 is the mirror image of 1 obtained by reflection, step (c), in a plane perpendicular to z.

This is a most useful result since we often need to calculate the inverse of a 3 x3 MR of a symmetry operator R. Equation (10) shows that when T(R) is real, I R)-1 is just the transpose of T(R). A matrix with this property is an orthogonal matrix. In configuration space the basis and the components of vectors are real, so that proper and improper rotations which leave all lengths and angles invariant are therefore represented by 3x3 real orthogonal matrices. Proper and improper rotations in configuration space may be distinguished by det T(R),... 

One can also transform symmetry operators whenever the coordinate system itself gets changed, as for instance in selecting an alternate setting for a space group. Then one must use a similarity transformation in the old system let the 4x4 symmetry operator be denoted by Q3, and in the new system as Qy let the coordinate system transformation be represented by the 4x4 matrix S, whose inverse matrix is S-1 then the similarity transformation yields ... 

The space group symmetry regulates the mutual arrangement of the structural units, not only by means of operations of inversion, reflection, and rotation, but also by translations and by symmetry operations with a translational component. 

The operators of discrete rotational groups, best described in terms of both proper and improper symmetry axes, have the special property that they leave one point in space unmoved hence the term point group. Proper rotations, like translation, do not affect the internal symmetry of an asymmetric motif on which they operate and are referred to as operators of the first kind. The three-dimensional operators of improper rotation are often subdivided into inversion axes, mirror planes and centres of symmetry. These operators of the second kind have the distinctive property of inverting the handedness of an asymmetric unit. This means that the equivalent units of the resulting composite object, called left and right, cannot be brought into coincidence by symmetry operations of the first kind. This inherent handedness is called chirality. 

Crystals of pure chiral objects, such as proteins or nucleic acids, cannot crystallize in a space group that contains mirror planes, glide planes, or inversion centers. Therefore, if a crystal is grown and its space groups contains one or more of these symmetry operations, the crystal is not chiral. 

Parity, or space inversion, is the symmetry operation that interconverts a chiral molecule into its mirror image. All coordinates (x, v, <) are replaced everywhere by (—X, — y, —z) under space inversion [2]. A chirality specific response in liquids and gases requires that the isotropic component of the susceptibility is odd under parity. Further, since the isotropic part of any tensor is necessarily a scalar, it follows that pseudoscalars - independent of the choice of coordinate axes and of opposite sign for enantiomers - underlie chiral observables in fluids. The isotropic medium may... 

---