Optical mode frequency

For finite C some analytically obtainable limits exist For large optical mode frequency, i.e. both g and/ are large such that / > C l(2f + g), the dynamics are determined by the proton tunnelUng, and the phase transition is driven by the proton mode condensation. In this case Tc is given by ... 

At infrared wavelengths extinction by the MgO particles of Fig. 11.2, including those with radius 1 jam, which can be made by grinding, is dominated by absorption. This is why the KBr pellet technique is commonly used for infrared absorption spectroscopy of powders. A small amount of the sample dispersed in KBr powder is pressed into a pellet, the transmission spectrum of which is readily obtained. Because extinction is dominated by absorption, this transmission spectrum should follow the undulations of the intrinsic absorption spectrum—but not always. Comparison of Figs. 10.1 and 11.2 reveals an interesting discrepancy calculated peak extinction occurs at 0.075 eV, whereas absorption in bulk MgO peaks at the transverse optic mode frequency, which is about 0.05 eV. This is a large discrepancy in light of the precision of modern infrared spectroscopy and could cause serious error if the extinction peak were assumed to lie at the position of a bulk absorption band. This is the first instance we have encountered where the properties of small particles deviate appreciably from those of the bulk solid. It is the result of surface mode excitation, which is such a dominant effect in small particles of some solids that we have devoted Chapter 12 to its fuller discussion. 

Comparison of measurements for particles dispersed on and in KBr is quite revealing. The extinction curve for particles on a KBr substrate shows a peak at approximately 400 cm-1, the transverse optical mode frequency for bulk MgO. This feature has been observed a number of times and it is discussed in some of the references already cited. Its explanation now appears to be the tendency of MgO cubes to link together into chains, which more closely... 

Dust around the carbon star shows an excess emission feature between about 10.2 and 11.6 jam, clearly distinguishable in both shape and position from the 9.7-jum feature of the oxygen star, which has been attributed to small SiC particles. These particles cannot be spherical, however. According to the discussion in Section 12.2, shape effects spread an absorption band in small particles of materials like SiC between the transverse (to,) and longitudinal (to,) optical mode frequencies these frequencies for SiC are indicated on the figure. This point was made by Treffers and Cohen (1974) using Gilra s unpublished calculations. To illustrate this further, calculations for a random distribution of... 

A linear relation between and T at the zone center is found (see Figure 1.12) suggesting that the temperature dependence of the optic mode frequency relates to the phase transition. In accordance with the Lyddane-Sachs-Teller relation... 

In the simple molecular lattice, at k = 0, the problems of finding the acoustical modes and the optical modes arc independent. Even at k 0 they would be independent problems if the ratio of the silicon to the oxygen mass were sufficiently large. In the end we shall see that the lowest optical mode frequencies overlap the acoustical mode frequencies, so to separate the problems is not entirely valid. However, for the present, let us separate the problems. 

We turn now to a matter of more direct physical relevance, the local electric dipole moment induced when a particular ion is displaced by some vector u. The transverse charge is defined to be the magnitude of that dipole moment divided by the displacement (and by the magnitude of the electronic charge). We saw in Eq. (9-22) that is directly related to an observable splitting between the longitudinal and transverse optical-mode frequencies, so that this is a quantity that can be compared with experiment. 

The extent of the contribution the second term in (2.45) is then decisive for the accuracy of the matrix formulation. Investigations have shown that this term affects the q = 0 or optic mode frequencies to greatly different degrees in different cases. For solid COj, it was found (Suzuki and Schnepp, 1971) that the translational modes are changed by less than 1 % except for the lowest frequency mode, which is changed by 5%. On the other hand, the librational modes are affected by between 3 and 8%. [Note that this separation between translations and librations is only valid at the F and R points of the Brillouin zone of space group Pai (7ft).] For a-N the error introduced in the translational modes is less than 2% in all cases but for the librations the error ranges from 25 to 44%. It should be noted that the classical harmonic treatment is in any case subject to question for librational modes of a-Ng but this is not the case for solid COg (see Section IIC.2). No accurate data are available for aromatics, but it may be assumed that the error in these cases is similar to those for solid COj. 

Brot et al. (1968) studied the far infrared spectra of liquid and solid tertiary butyl chloride. The study is of interest in connection with rotations in solid phases. The lattice vibrations of thiourea and its deuterated analog were studied by Takahashi and Schrader (1967), who also carried out calculations of the optical mode frequencies and achieved good agreement with infrared and Raman measurements. 

Table 4.1-157 Phonon frequencies/wavenumbers at symmetry points for cadmium compounds. Cadmium oxide (CdO), Fundamental optical-mode frequencies cadmium sulfide (CdS), 25 K cadmium selenide (CdSe), 300 K, from infrared and Raman spectroscopy cadmium tellmide (CdTe), 300 K, from inelastic neutron scattering...

Fig. tt.5-23 BaTiOs. Avq and F versus T, obtained from hyper-Raman scattering in the cubic phase. Avq and F are the optical mode frequency and damping constant, respectively. The different symbols brown and gray) show results from different authors. Avq decreases as the temperature decreases to the Curie point, showing the presence of mode softening, c ligth velocity... 

From only two calculations of force and stress one can determine the optic mode frequency (the restoring force fo the internal displacements), the internal strain parameter f, and the elastic constant. This may be compared with a very arduous task of fitting multiple parameter curves if one has only the energy, as was done by Harmon, et al. ... 

The composition dependence of the optical mode frequency can be calculated with the modified random-element isodisplacement (MREI) model [93-96] and similar models [97-99] or determined by measuring the Raman spectra of materials with well-defined compositions. Composition-dependence schemes can be also established from the Raman spectra of quaternary alloys like Cd ,Zn gi- yTe. The room-temperature Raman spectrum of the quaternary alloy Cdo.4Zno.3Mgo.3Te (Fig. 4) shows pairs of the LO-TO modes identified as CdTe-like, ZnTe-like, and MgTe-like modes [100]. Caused by the high degree of disorders in this system, additional features of the disorder-allowed LA and TA can be seen. The multimode behavior of this alloy can be constructed on the basis of the MREI model [100]. 

Improvements Further improvements have been introduced, as experimental data have become more reliable. For example, neutron scattering experiments have proved that the longitudinal optic mode frequencies at the zone centre are systematically lower than predicted. Various models result from the idea that this discrepancy may be assigned to the neglect of ionic polarization. 

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