Factor groups

One of the best available texts describing the principles of Raman scattering from crystals. Includes factor group calculations, polarization measurements, force constant calculations, and many other aspects of crystal physics. 

The permutations of 0[H] have a special effect on the rows of the matrix (1.37) if a permutation moves an element of one row into another row, then the permutation moves all the elements of the one row into the other row. The rows of (1.37) are imprimitive domains of 0[H]. The permutations of 0[H] which leave the r imprimitive domains invariant (the gross permutation of which is the identity) form a subgroup it has order it is the direct product H xHxHx...xH with r factors and is a normal subgroup of C [H], with factor group. 

According to a factor group analysis of R (space group I4cm with Z = 4), vibration modes can be presented in the center of the Brillouin zone (wave... 

Figure 6-3. Top Structure of the T6 single crystal unit cell. The a, b, and c crystallographic axes are indicated. Molecule 1 is arbitrarily chosen, whilst the numbering of the other molecules follows the application of the factor group symmetry operations as discussed in the text. Bottom direction cosines between the molecular axes L, M, N and the orthogonal crystal coordinate system a, b, c. The a axis is orthogonal to the b monoclinic axis.

Tabic 6-2. Correlation diagram of the C2/, point group of the isolated T6 molecule (left column) with the C2i, factor group for solid T(, (right column) via the site symmetry C group (center). L, M, and N indicate the principal molecular transition dipole moments, while a, b, and c arc the crystalline axes. 

The vibrations of the free molecule can be correlated with the vibrations of the crystal by group theoretical methods. Starting with the point group of the molecule Did)> the irreducible representations (the symmetry classes) have to be correlated with those of the site symmetry (C2) in the crystal and, as a second step, the representations of the site have to be correlated with those of the crystal factor group (D2h) [89, 90]. Since the C2 point group is not a direct subgroup of of the molecule and of D211 of the crystal, the correlation has to be carried out in successive steps, for example ... 

Table 2 Correlation of the molecular point group of Sg with the factor group of the orthorhombic crystal (E>4d—>C2— D2h) [88]...

The factor group Dzh of orthorhombic Sg includes an inversion operation therefore, the g-u exclusion principle works resulting in modes of either Raman (gerade, g) or infrared activity (ungerade, u). 

Of the five bending vibrations of the Sg molecule three are Raman active (V2, Vg, Vii) and two are IR active (V4, Vg). Most of the Raman active modes in the crystal could clearly be resolved in spectra at low temperatures and by polarization measurements. For example. Fig. 6 shows the Raman active factor group components of the Vg mode obtained at three different polarizations. In Fig. 7 an analogous IR spectrum is presented. 

The V3 mode observed at about 415 cm is inactive in the free molecule and, therefore, the factor group components are very weak in the Raman... 

Due to the large factor group spHtting of about 25 cm the bands of the torsional vibration are well separated (Fig. 5) and the isotope effect on the frequency was calculated to about 0.6 cm [131]. Figure 14 clearly demonstrates the broadening and splitting of the and bi components of Vg as... 

Observed on the wing of the CS2 bending mode. Occurs in violation of the selection rules of the point group Dsd but is IR active under the Csi factor group of the crystal. Could also be a combination vibration or caused by the CS2 impurity which was present in the sample (see text)... 

The resonance splitting of intramolecular modes in the crystalline state is often called Davydov splitting or factor group splitting . In contrast to the static field effects of the crystal, this splitting is due to the dynamical interaction of the constituents in the primitive cell... 

Six comments are appropriate at this point. Firstly, it is the experience of the reviewer that chemically-similar compounds with very similar infrared and Raman spectra in factor-group split regions are isomorphous. This method is probably at least as reliable as X-ray powder methods when there is a significant change in scattering factors between the two compounds studied (e.g. bromo-derivatives of... 

Adams, D. M., Newton, D. C. Tables for factor group and point group analysis (Beckman-RIIC Limited)... 

A dipole-dipole model reproduces factor group frequencies of i>(CO) in a variety of dinuclear metal carbonyls... 

This is a sub-group of the space group, from which the effects of simple translations have been factored off . Isomorphous means here isomorphous to the factor group . 

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