Space-groups symmetries translation presence

The internal symmetry of the crystal is revealed in the symmetry of the Bragg reflection intensities, as discussed in Chapter 4. The crystal system is derived by examining the symmetry among various classes of reflections. Key patterns in the diffraction intensities indicate the presence of specific symmetry operations and lead to a determination of the space group. The translational component of the symmetry elements, as in glide planes or screw axes, causes selective and predictable destructive interference to occur. These are the systematic absences that characterize these symmetry elements, described in Chapter 4. 

Symmetry axes can only have the multiplicities 1,2,3,4 or 6 when translational symmetry is present in three dimensions. If, for example, fivefold axes were present in one direction, the unit cell would have to be a pentagonal prism space cannot be filled, free of voids, with prisms of this kind. Due to the restriction to certain multiplicities, symmetry operations can only be combined in a finite number of ways in the presence of three-dimensional translational symmetry. The 230 possibilities are called space-group types (often, not quite correctly, called the 230 space groups). 

Systematic absences (or extinctions) in the X-ray diffraction pattern of a single crystal are caused by the presence of lattice centering and translational symmetry elements, namely screw axes and glide planes. Such extinctions are extremely useful in deducing the space group of an unknown crystal. 

Simple translation is the most obvious symmetry element of the space groups. It brings the pattern into congruence with itself over and over again. The shortest displacement through which this translation brings the pattern into coincidence with itself is the elementary translation or elementary period. Sometimes it is also called the identity period. The presence of translation is seen well in the pattern in Figure 8-1. The symmetry analysis of the whole pattern was called by Budden the analytical approach. The reverse procedure is the... 

As shown in section 2.12.3, the presence of translational symmetry causes extinctions of certain types of reflections. This property of infinite symmetry elements finds use in the determination of possible space group(s) symmetry from diffraction data by analyzing Miller indices of the observed Bragg peaks. It is worth noting that only infinite symmetry elements cause systematic absences, and therefore, may be detected from this analysis. Finite symmetry elements, such as simple rotation and inversion axes, mirror plane and center of inversion, produce no systematic absences and therefore, are not distinguishable using this approach. 

The parallelepiped in Figure 2 is the unit cell of the ammonia crystal phase I. Thus, the ammonia crystal can be regarded as the combination of a pattern of four ammonia molecules (16 atoms) in the unit cell with all possible translations in a cubic primitive lattice. Considerations about crystalline symmetry lead to the conclusion that ammonia in phase I crystallizes according to space group P2i3. Letter P in the symbol stands for primitive lattice, and the other symbols denote the main symmetry operations. The last element in the symbol, 3, indicates the presence of a three-fold axis not aligned with the principal rotation axis (if it was, it would follow letter P), which further indicates that the lattice is cubic. A cubic unit cell is completely specified by just one... 

Let us now consider the necessary conditions for the appearance of phonons in impurity-ion electronic spectra. The presence of a substitutional defect in an otherwise perfect crystal removes the translational symmetry of the system and reduces the symmetry group of the system from the crystal space group to the point group of the lattice site. Loudon [26] has provided a table for the reduction of the space group representations of a face-centered cubic lattice into a sum of cubic point-group representations. A portion of that table is shown in Table 1 here. Consider an impurity ion that undergoes a vibronic electric-dipole allowed transition, with T and Tf the irreducible representations of the initial and final electronic states. Since the electric dipole operator transforms as Tj in the cubic point group, Oh, the selection rule for participation of a phonon is that one of its site symmetry irreducible representations is contained in the direct product T x Ti X Tf. 

In the presence of the external fields the vector potential appears in the Hamiltonian at Eq. (1) and the space translation symmetry is, therefore, lost. However, there exists a generalization, i.e. the phase space translation group [3] which provides a symmetry associated with the CM motion of the system in the presence of the external fields. The new conserved quantity which is the corresponding generalization of the total canonical momentum of the field-free case is the so-called pseudomomentum K [3,4]... 

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