Translational symmetry operators

Another distinction we make concerning synnnetry operations involves the active and passive pictures. Below we consider translational and rotational symmetry operations. We describe these operations in a space-fixed axis system (X,Y,Z) with axes parallel to the X, Y, Z) axes, but with the origin fixed in space. In the active picture, which we adopt here, a translational symmetry operation displaces all nuclei and electrons in the molecule along a vector, say. 

We now define the effect of a translational synnnetry operation on a fiinction. Figure Al.4.3 shows how a PHg molecule is displaced a distance A X along the X axis by the translational symmetry operation that changes Xq to X = Xq -1- A X. Together with the molecule, we have drawn a sine wave symbolizing the... 

Since the physical system (crystal) is indistinguishable from what it was before the application of a space-group operator, and a translational symmetry operator only changes the phase of the Bloch function without affecting the corresponding energy E(k),... 

Figure 8.14 Translational symmetry operations in crystals. A screw axis, symbol nm, involves a translation by a fraction of m/n of the unit cell length followed by a rotation of 360In degrees thus a twofold screw axis, 2[, involves a 180° rotation.

In crystallography, we are concerned with translational symmetry as well as point group symmetry, and this means that we must add two additional translational symmetry operators (Figure 8.14) ... 

Combining the five point group symmetry operators with the two translational symmetry operators gives a total of exactly 230 different possible combinations, called the crystallographic space groups. Every crystal... 

Therefore, consider a crystal with two molecules per cell, equivalent under translational symmetry operations, as in the anthracene crystal. The molecular polarizability tensors of the two host molecules, in positions 1 and 2, for a molecular transition are given the form... 

Matter is composed of spherical-like atoms. No two atomic cores—the nuclei plus inner shell electrons—can occupy the same volume of space, and it is impossible for spheres to fill all space completely. Consequently, spherical atoms coalesce into a solid with void spaces called interstices. A mathematical construct known as a space lattice may be envisioned, which is comprised of equidistant lattice points representing the geometric centers of structural motifs. The lattice points are equidistant since a lattice possesses translational invariance. A motif may be a single atom, a collection of atoms, an entire molecule, some fraction of a molecule, or an assembly of molecules. The motif is also referred to as the basis or, sometimes, the asymmetric unit, since it has no symmetry of its own. For example, in rock salt a sodium and chloride ion pair constitutes the asymmetric unit. This ion pair is repeated systematically, using point symmetry and translational symmetry operations, to form the space lattice of the crystal. 

Symmetry is a property we find in objects with at least one dimension (D) (1-D symmetry of beads on a string 2-D symmetry of objects in a plane 3-D symmetry of objects in space). Empty space has the most symmetry. In zero dimensions, any symmetry is allowed. An object that does not have to fill space can have any arbitrary symmetry (e.g., no symmetry, or a sevenfold rotation axis). However, if this object must fill 2-D or 3-D space, it must meet certain local symmetry requirements, which, coupled with translational symmetry operators, allows the space to be completely filled. 

The regular arrangement of molecules or ions in a unit cell can be described in terms of translational symmetry. One set of units of translation in a crystal structure are the unit-cell dimensions, that is, the vectors between points in the crystal lattice (a, b or c). Other translational symmetry operations use fractions of these translations. Screw... 

It should not be assumed that A = 0 is always the lowest slate. It the basis functions are P orbittils (where the translational symmetry operates along the z axis), lAo will have the most nudes and be highest in energy will have no nodes and be lowest in energy. Sec Footnote 26. 

A fundamental characteristic of spatially periodic systems is the existence of a group of translational symmetry operations, by means of which the repeating pattern may be brought into self-coincidence. The translational symmetry of the array, expressing its invariance with respect to parallel displacements in different directions is represented by a lattice. This lattice consists of an array of evenly spaced points (Fig. 3-13), such that the structural elements appear the same and in the same orientation when viewed from each and every one of the lattice points. Another important property of spatially periodic arrays is the existence of two characteristic length scales, corresponding to the average microscopic distance between lattice... 

In the active picture adopted here the (X,Y,Z) axis system remains fixed in space and a translational symmetry operation changes the (X,Y,Z) coordinates of all nuclei and electrons in the molecule by constant amounts, (A X, A Y, A Z) say. 

The coordinates (XpTpZj) are redundant since they ean be determined from the eondition that the X, Y, Z) axis system has origin at the molecular centre of mass. Obviously, the translational symmetry operation diseussed above has the effect of changing the centre of mass coordinates... 

This definition causes the wavefunction to move with the molecule as shown for the X direction in figure Al.4.3. The set of all translation symmetry operations constitutes a group which we call the... 

Figure Al.4.3. A PH molecule and its wavefunction, symbolized by a sine wave, before (top) and after (bottom) a translational symmetry operation.

It is of interest to note that one may change the translation lattice of Fig. 5.3 by replacing the translation lattice vector c with the molecular helix lattice, keeping the translation symmetries a and This would lead to a match of the molecular helix symmetry with the crystal symmetry and even for irrational helices, a crystal stracture symmetry would be recognized. In fact, a whole set of new lattices can be generated replacing all three translation symmetry operations by helix symmetry operations [5]. Since a 1 1/1 hehx has a translational symmetry, this new space lattice description with helices would contain the traditional crystallography as a special case. 

The next step in data analysis is the determination of systematic absences to find the space group symmetry. Systematic absences are Bragg peaks that are missing in the diffraction data because of a translational symmetry operation and/or preferential orientation of polycrystalline samples. 

The full set of operators for a polymer consists of the product of the operators that move atoms within the unit cell and the translational symmetry operators. 

The wave vector 1< = C" - k is significant in that it characterizes the way in wRich She total wave function transforms under translational symmetry operations of the crystal. Therefore one has mixing among only those zero-order functions having the same reduced value of Treciprocal lattice vector. Further, as was seen in the first example by Frenkel, the Hamiltonian matrix elements formed from localized states (such as Wannier functions)... 

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