Lattice vibrations spectrum

Herman F. (1959). Lattice vibrational spectrum of germanium. J. Phys. Chem. Solids, 8 405-418. 

Due to the translational symmetry to each intramolecular vibration there corresponds at least one optical branch in the lattice vibration spectrum (in practice, the number of branches is given by or, where a is the number of molecules per unit cell, and r is the multiplicity of degeneracy of intramolecular vibrations). Therefore in this approximation the excited states of the crystal are described by the wavefunction... 

KF, NaF, KI, and Nal) that accurate descriptions of relaxation as a function of temperature are not likely when one uses a poor description of the lattice vibrational spectrum.It was tentatively concluded that a reasonable (but not overly detailed) description of the optic vibrational branch is needed to describe the Ti behaviour as a function of temperature when the mass ratio between the metal and halogen deviates greatly (ca. > 3) from unity, but would not be required when the mass ratio was near one. 

To interpret these observations we must consider the spectrum of lattice vibrations in an ice crystal, since it is in deviations of this spectrum from that of a simple Debye continuum that their explanation will be found. Experimental information about the lattice vibration spectrum can be found from infrared and Raman... 

In addition to the Poole-Frenkel effect and the field-induced tunneling from traps to conduction band states, the Zener effect (field-induced transitions from valence band to conduction band) and various forms of avalanche breakdown effects, can give a bulk conductivity rising sharply with field. These effects are difficult to assess in the present systems, because little is known about the electronic states in amorphous oxides, the electronic transport process, or the lattice vibration spectrum. 

The absorption with the maximum at ca 80cm in the PS spectrum is mainly due to benzene ring librations [75]. A similar band at 75cm is observed in the spectrum of liquid benzene [76] and the absorption with a maximum at 60cm in the Raman spectrum of PS is assigned to this mechanism [36], Lattice vibration spectrum of crystalline benzene in the frequency range under discussion exhibits bands at 65, 74, 92 and 116 cm ... 

Fig. 29 Raman spectrum of p-S at high pressure and room temperature [109]. The wavenumbers indicated are given for the actual pressure. No signals of other allotropes have been detected. The line at 48 cm (ca. 25 cm atp 0 GPa) may arise from lattice vibrations, while the other lines resemble the typical pattern of internal vibrations of sulfur molecules...

As noted in the introduction, vibrations in molecules can be excited by interaction with waves and with particles. In electron energy loss spectroscopy (EELS, sometimes HREELS for high resolution EELS) a beam of monochromatic, low energy electrons falls on the surface, where it excites lattice vibrations of the substrate, molecular vibrations of adsorbed species and even electronic transitions. An energy spectrum of the scattered electrons reveals how much energy the electrons have lost to vibrations, according to the formula ... 

What is the structure of this Co-Mo-S phase A model system, prepared by impregnating a MoS2 crystal with a dilute solution of cobalt ions, such that the model contains ppms of cobalt only, appears to have the same Mossbauer spectrum as the Co-Mo-S phase. It has the same isomer shift (characteristic of the oxidation state), recoilfree fraction (characteristic of lattice vibrations) and almost the same quadrupole splitting (characteristic of symmetry) at all temperatures between 4 and 600 K [71]. Thus, the cobalt species in the ppm Co/MoS2 system provides a convenient model for the active site in a Co-Mo hydrodesulfurization catalyst. 

Vibrational properties can also be computed using the total energy formalism. Here, the atomic mass is required as input and the crystal is distorted to mimic a frozen-in lattice vibration. The total energy and forces on the atoms can be computed for the distorted configuration, and, through a comparison between the distorted and undistorted crystal, the lattice vibrational (or phonon) spectrum can be computed. Again, the agreement between theory and experiment is excellent. [22]... 

This Lorentzian line-shape function has been sketched in Figure 1.4(b). The natural broadening is a type of homogeneous broadening, in which all the absorbing atoms are assumed to be identical and then to contribute with identical line-shape functions to the spectrum. There are other homogeneous broadening mechanisms, such as that due to the dynamic distortions of the crystalline environment associated with lattice vibrations, which are partially discussed in Chapter 5. 

With further increase of the concentration (in p, phase range for H cLii cNb03) many new bands were observed. The fact that the low concentration boundary of the P phases is approximately x = 0.5 leads to the assumption for some kind of ordering of Li and as reported. On one hand, it can be assumed that the protons form a (nearly) ordered sub-lattice. Such a structure would have a phonon spectrum different from that of a pure LiNbOs, see . On the other hand, the PE probably leads to a reduction of the crystal symmetry, i.e. due to the incorporation of H, the two Li sites in the unit cell may become non-equivalent. In such case, the symmetry would be reduced from Csv to C3. As a result, the number of molecules per unit cell would remain the same, but new bands would appear in the vibration spectrum. 

Diamond is crystallized in cubic form (O ) with tetrahedral coordination of C-C bonds around each carbon atom. The mononuclear nature of the diamond crystal lattice combined with its high symmetry determines the simplicity of the vibrational spectrum. Diamond does not have IR active vibrations, while its Raman spectrum is characterized by one fundamental vibration at 1,332 cm . It was found that in kimberlite diamonds of gem quality this Raman band is very strong and narrow, hi defect varieties the spectral position does not change, but the band is slightly broader (Reshetnyak and Ezerskii 1990). 

A simple application of the multiple-oscillator theory is to fit measured reflectance data for MgO in the Reststrahlen region. In Section 9.1 we considered the electronic excitations of MgO, whereas we now turn our attention to its lattice vibrations. A glance at the far-infrared reflectance spectrum of MgO in Fig. 9.7 shows that it does not completely exhibit one-oscillator behavior there is an additional shoulder on the high-frequency side of the main reflectance peak, which signals a weaker, but still appreciable, second oscillator. The solid curves in Fig. 9.7 show the results of a two-oscillator calculation using (9.25) the reflectance data were taken from Jasperse et al. (1966), who give the following parameters for MgO at 295°K ... 

Similar methods have been used to integrate thermodynamic properties of harmonic lattice vibrations over the spectral density of lattice vibration frequencies.21,34 Very accurate error bounds are obtained for properties like the heat capacity,34 using just the moments of the lattice vibrational frequency spectrum.35 These moments are known35 in terms of the force constants and masses and lattice type, so that one need not actually solve the lattice equations of motion to obtain thermodynamic properties of the lattice. In this way, one can avoid the usual stochastic method36 in lattice dynamics, which solves a random sample of the (factored) secular determinants for the lattice vibration frequencies. Figure 3 gives a typical set of error bounds to the heat capacity of a lattice, derived from moments of the spectrum of lattice vibrations.34 Useful error bounds are obtained... 

This prescription could have been applied in determining directly the relaxation- and retardation-spectra, Eqs. (3.7) and (3.6), of the linear lattice, where the exact vibrational spectrum is well known. 

DEBYE THEORY OF SPECIFIC HEAT. The specific heal of solids is attributed to the excitation of thermal vibrations of the lattice, whose spectrum is taken to be similar to that of an elastic continuum, except that it is cut off at a maximum frequency in such a way that the total number of vibrational modes is equal to the total number of degrees of freedom of the lattice. 

---