Raman tensor

In addition, intensity changes under increasing pressure have been observed. For example, the most intense Raman line at STP conditions is the flg component of v ( 220 cm ), but at about 2 GPa the intensity decreases in favor of the ag component of Vi ( 475 cm ) which on further compression gains more intensity (about a factor of 2 at 5 GPa) [120]. This behavior was explained by the anisotropy of the crystal s compressibihty [139] and differences in the components of the Raman tensor of the two modes [87] with respect to the crystal axes [109]. 

Raman spectroscopy is an inelastic light scattering experiment for which the intensity depends on the amplitude of the polarizability variation associated with the molecular vibration under consideration. The polarizability variation is represented by a second-rank tensor, oiXyZ, the Raman tensor. Information about orientation arises because the intensity of the scattered light depends on the orientation of the Raman tensor with respect to the polarization directions of the electric fields of the incident and scattered light. Like IR spectroscopy, Raman... 

The Raman tensor can be expressed in the coordinate system Oxyz of the molecule (Figure 1) with its principal components aj, a2, and 0(3 such as... 

An approach simpler than the "general" method has been used by Bower and coworkers [52,53] and has proved to be useful to determine the order parameters of uniaxially oriented polymers. This approach, called "cylindrical" method, requires a smaller number of spectra that are easily measurable, but it assumes that the Raman tensor associated with the vibrational mode under study is cylindrical. In such case, at — a2 — a so that the depolarization ratio R so of an... 

Finally, Frisk et al. have proposed a third method, the "depolarization" method, that does not need the assumption of a cylindrical Raman tensor [56]. A randomly oriented sample is again necessary but the depolarization ratio takes the general form... 

Frisk et al. have compared the three basic methods on the same polyfpropylene terephthalate) (PPT) sample [56]. It appears that the assumption of a cylindrical Raman tensor may be too crude in some cases. Unless the hypothesis of a cylindrical tensor has been ascertained, the two other methods should be used. The "general" method is intrinsically the most accurate one but it... 

Because of their relative complexity, few Raman studies have dealt with biaxial systems, although valuable structural information can be obtained. Assuming a cylindrical Raman tensor, it has been shown that there are six independent quantities J,y = 70 (a ) given by [3]... 

Silk fibers, a basic system with a uniaxial symmetry, have also been investigated by Raman spectromicroscopy [63] that is one of the rare techniques capable of providing molecular data on such small (3-10 pm diameter) single filaments. The amide I band of the silk proteins has been particularly studied to determine the molecular orientation using the cylindrical Raman tensor approximation. In this work, it was assumed that Co Ci, C2 and the a parameter was determined from an isotropic sample using the following expression of the depolarization ratio... 

Experimental verification of the ISRS generation can be primarily given by the pump polarization dependence. The coherent phonons driven by ISRS (second order process) should follow the symmetry of the Raman tensor, while those mediated by photoexcited carriers should obey the polarization dependence of the optical absorption (first order process). It is possible, however, that both ISRS and carrier-mediated generations contribute to the generation of a single phonon mode. The polarization dependence is then described by the sum of the first- and second-order processes [20-22], as shown in Fig. 2.3. 

The SH intensity is proportional to P 2. Experimentally, the oscillatory part of the total SH is so small that one can ignore its second-order term. If coherent surface phonons are created by ISRS, the whole process including excitation and detection is the coherent time-domain analogue of stimulated hyper Raman scattering (y(4) process) [14]. The cross section of the SHG process is then proportional to the product of a Raman tensor in the pump transition and a hyper-Raman tensor dx k/dQn in the probe transition. 

It would be best perhaps if this technique were illustrated by an example. In the case of single crystal measurements for a compound which crystallizes in the -K2S04 form (D ji-Pnma] factor group the Raman tensor for the Ag... 

Table 36. Relative magnitude of the changes in the ay-components of the Raman tensor for K2Cr04 (79)...

The spectra at 5 K are shown for all available polarizations in Fig.3. As follows from comparison of XX and X Y spectra there is a pronounced anisotropy of Raman tensor in ab-plane which can be connected to a strong polarization of phonon modes at 162 and 596 cm 1. The XY spectra consist of 3Big modes, which are formed by motions of heavy ions and pyramidal -octahedral shared oxygen ions along x-direction. We attribute the mode at 307 cm 1 to a clockwise and anti-clockwise rotation of pyramids and octahedra along the c-direction. Our preliminary calculations show that a mode at 61 cm 1 consists mainly of Gd vibrations. 

The most prominent feature of the diagonal spectra of the Gd2 sample is a remarkable redistribution of Raman scattering intensities of some lines compared with the XX spectra of the single crystal Gd3. To explain this behavior we have to note that vibrations along the c-axis direction are mainly given by Ag lines. It is expected that Raman tensor elements of respective lines should demonstrate a strong anisotropy of the zz... 

General Form of the Raman Tensor for Transversely Isotropic Systems... 

This expression is identical in form to equation (5.20). In the case of Raman scattering, however, it is necessary to compute the average Raman tensor, (. For a transversely isotropic system, the segment is free to spin about the r. axis, and the vector ni is averaged over the unit circle normal to r . In addition to (n() = 0 and equation (5.22), we require the result,... 

The Raman tensor for a chemical bond oriented along the unit vector u, but attached to a segment oriented along a vector r, can be computed using the analysis given in section 5.3. For a flexible polymer chain, a procedure similar to the Kuhn and Grun analysis of section 7.1.3 can be used to provide a connection between the Raman tensor and the orientation of the end-to-end vector, R. We first express the Raman tensor of an individual segment,... 

The optical apparatus used in this work was described in section 8.6 and has the capability of providing both Raman scattering and birefringence measurements simultaneously. The Fourier expansion of the overall Raman scattering signal is given by equation (8.51), and the coefficients are given by equations (8.52) to (8.54). In these expression, a simple, uniaxial form for the Raman tensor was assumed. From these coefficients, the anisotropies in the second and fourth moments of the orientation distribution can be solved as... 

Scattering configuration Allowed Raman modes Corresponding Raman tensor elements... 

Single crystal Raman spectra of 123 compounds have defined selection rules and prove that in unpolarized spectra Ag modes dominate. Fig. 4.8-22 shows polarized Raman spectra of these compounds. The strongest spectra are obtained with a (zz) scattering geometry, except for the as-0 2,3) mode at 340 cm, which has a nearly zero Raman tensor component in this direction. It is therefore occasionally referred to as quasi Big. 

Symmetry considerations under TERS conditions have recently been considered as a method for obtaining molecular orientations [93, 94]. Berweger and Raschke considered using the polar phonon mode selection rules for TERS to determine nanocrystallographic information fi om solids [93]. TERS enables control of experimental parameters required to extract information from the Raman tensor such as the polarization- and k-vector-dependent field enhancement relative to the surface. The selection mles were demonstrated for both near-field TERS and far-field scatting from bulk and nanocrystalline LiNb03. 

The e term refers to a unit polarization vector, is the first-order derivative of the susceptibility tensor with respect to the phonon amplitude, also known as the Raman tensor, and rij is the phonon occupation number for the /th mode, given by ... 

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