Factor group analysis

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... 

It is furthermore possible with the correlation method to divide the lattice vibrations into translational types and librationcd (rotational) types. The same results are obtained with this method as with that of Bhagavantam and Venkatara-yudu. Adams and Newton (28, 29) have recently pubhshed tables which make a simple factor group analysis possible [see also (30)]. 

Adams, D. M., Newton, D. C. Tables for factor group analysis. London Beckman— RIIC. Ltd. 1970. 

Muller 90) discusses the factor-group analysis of potassium perrhenate KRe04, using previously-published infra-red and Raman spectra. The space group is C%h, where site symmetry of the Re04 ion is S4 there are only two formula units per spectroscopic unit cell, and all eight infra-red active modes are found. 

According to the factor-group analysis, there are 18 Raman and 9 IR active new vibrational modes in the low-temperature structure of CuGe03, below the temperature of the spin-Peierls transition Tc=14 K. While two of the Raman-active folded modes were clearly observed in the very first optical experiments, no traces of the IR folded modes could be found for a long time. We have observed IR folded modes for the first time, measuring... 

In the first step, the vibrations of the hypothetical isolated species are analyzed with respect to its point group. Secondly, the symmetry of the species in the static lattice (the site symmetry) is deteiTnined. The site is normally of lower symmetry than the isolated species. If the species are highly symmetrical, splitting of the degenerate vibrations is observed. In the last step, the correlations between the different groups in the unit cell are analyzed (factor group analysis, Sec. 2.7.6). Since the unit cell of the scheelite crystal contains two WO4 species, twice as many vibrations are observed as expected from site group analysis. 

A further problem in the interpretation of vibrational spectra of solid state compounds arises from the different phases of the vibrations in neighboring cells, leading to a wave described by the wave vector k. In the absence of a phase difference, k equals zero. This is the basis for the factor group analysis. If the vibrational motions are oriented parallel to the direction of the wave caused by the phase differences, a longitudinal branch results while transverse branches result from orthogonal vibrational motions. Furthermore, it is necessary to differentiate between optical and acoustic modes. In the optical mode of the NaCl lattice, Na+ and Cl ions have opposite displacements, while the acoustic modes are caused by in-phase motions of Na+ and CF. 

Behringer J (1973) Factor Group Analysis Revisited and Unified, Springer Tracts in Modern Physics, vol 68. Springer, Berlin, p 161... 

Factor group analysis [58] indicates that 429 IR active modes and 432 Raman active modes are expected for the monoclinic structure and 323 IR active modes and 432 Raman active modes are expected for the orthorhombic form. Only 16 components in the IR specrum and 11 components in the Raman spectrum are observed because of the superimposition of many of the expected fundamentals. However, a careful observation of the spectra, with the help of analysis of the perturbations arising from isomorphic substitution and with the aid of derivative spectra, showed the presence of a great number of very weak components (shoulders) in the spectra. 

Ab initio calculations gave vibrational wavenumbers for 9 isomers of CNNS.524 A CARS study of the effects of 266 nm. pulsed laser photodissociation of NCNCS showed that the vapour contained both NCNCS and CNCN.525 IR and Raman spectra, with factor group analysis, gave vibrational assignments for crystalline ammonium dicyanamide, NH4[N(CN)2].526 Variable-temperature Raman spectroscopy was used to follow the solid state transformation of NI I4[N(CN)2] into NCN=C(NH2)2.527 Ab initio and/or DFT calculations gave vibrational wavenumbers for CH2=CH-N=C=X (X = O, Se) 528 NN-C(CN)2 529 nitroso-azide, NNN-N=0, and nitro-azide, NNN-N02.530... 

Raman spectroscopy was used to characterise a range of arsenate minerals of the vivianite type.589 IR and Raman spectra were assigned on the basis of factor group analysis for Cd2As207, Table 13.590 Ab initio calculations have given values for the vibrational wavenumbers for As4Ofi.591... 

As shown in Fig. 6(a), for an isolated water molecule there are three types of librational modes, twist(T), wag(W) and rock(R). Using the factor group analysis for the four molecules in a primitive cell, 12 librational modes in XI phase are divided into... 

The normal modes at the centre of the Brillouin zone, ( 4.3.1), (A = 0) can be deduced by a conventional factor group analysis [4,5]. Since there are two chains in the unit cell, equivalent modes on each chain couple to give an in-phase and out-of-phase pair of modes (Davydov splitting). Above 600 cm the modes can be described using standard group frequency correlations CH2 rock 720-1170 cm, C-C stretch... 

To analyze the spectra of crystals, it is necessary to carry out a site group or factor group analysis, as described in the following subsection. 

A more complete analysis including lattice modes can be made by the method of factor group analysis developed by Bhagavantam and Venkatarayudu [140]. In this method, we consider all the normal vibrations for an entire Bravais cell. Figure 1.42 illustrates the Bravais cell of calcite, which consists of the following symmetry elements ... 

TABLE 1.21. Factor Group Analysis of Calcite Crystal ... 

Table 1.26 summarizes the results obtained in Tables 1.23-1.25. The total number of vibrations including the acoustical modes should be 30 since the Bravais cell contains 10 atoms. These results are in complete agreement with that obtained by factor group analysis (Table 1.21). 

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