Mode density lattice vibrational

Here it is our intention to show that for a system constituted by substrate phonons and laterally interacting low-frequency adsorbate vibrations which are harmonically coupled with the substrate, the states can be subclassified into independent groups by die wave vector K referring to the first Brillouin zone of the adsorbate lattice.138 As the phonon state density of a substrate many-fold exceeds the vibrational mode density of an adsorbate, for each adsorption mode there is a quasicontinuous phonon spectrum in every group of states determined by K (see Fig. 4.1). Consequently, we can regard the low-frequency collectivized mode of the adsorbate, t /(K), as a resonance vibration with the renormalized frequency and the reciprocal lifetime 7k-... 

As a rule, the density of states for molecular lattice vibrations is negligible as compared to that for crystal phonons. Therefore, the K-mode of a molecular lattice is coupled with the crystal phonons specified by the same wave vector K. Besides, the low-frequency collective mode m of adsorbed molecules can be considered as a... 

The study of dynamics of a real polymer chain of finite length and containing some conformational defects represents a very difficult task. Due to the lack of symmetry and selection mles, the number of vibrational modes is enormous. In this case, instead of calculating the frequency of each mode, it is more convenient to determine the density of vibrational modes, that is, the number of frequencies that occur in a given spectral interval. The density diagram matches, apart from an intensity factor, the experimental spectmm. Conformational defects can produce resonance frequencies when the proper frequency of the defect is resonating with those of the perfect lattice (the ideal chain), or quasi-localized frequencies when the vibrational mode of the defect cannot be transmitted by the lattice. The number and distribution of the defects may be such... 

If, however, the transition is of a pure displacive nature, the fluctuation amplitude of the order parameter is critical and is by no means temperature-independent. Since the soft mode is an under-damped lattice vibration (at least outside the close vicinity of Tc), defined by its frequency a>s and damping constant Tj, the spectral density is a Lorentzian centred at s and the... 

Inelastic neutron scattering (INS) measurements have been successfully used to study dynamical phenomena such as molecular or lattice vibrations in pristine C60 [43] and a variety of fullerides [44-48]. When INS spectra are collected on instruments with a large energy window, it is possible to observe all phonon modes including the molecular vibrations and the generalised phonon density-of-states (GDOS) can be directly calculated. 

In Eq. (10), E nt s(u) and Es(in) are the s=x,y,z components of the internal electric field and the field in the dielectric, respectively, and p u is the Boltzmann density matrix for the set of initial states m. The parameter tmn is a measure of the line-width. While small molecules, N<pure solid show well-defined lattice-vibrational spectra, arising from intermolecular vibrations in the crystal, overlap among the vastly larger number of normal modes for large, polymeric systems, produces broad bands, even in the crystalline state. When the polymeric molecule experiences the molecular interactions operative in aqueous solution, a second feature further broadens the vibrational bands, since the line-width parameters, xmn, Eq. (10), reflect the increased molecular collisional effects in solution, as compared to those in the solid. These general considerations are borne out by experiment. The low-frequency Raman spectrum of the amino acid cystine (94) shows a line at 8.7 cm- -, in the crystalline solid, with a half-width of several cm-- -. In contrast, a careful study of the low frequency Raman spectra of lysozyme (92) shows a broad band (half-width 10 cm- -) at 25 cm- -,... 

It should be emphasized that although this success of the Debye model has made it a standard starting point for qualitative discussions of solid properties associated with lattice vibrations, it is only a qualitative model with little resemblance to real normal-mode spectra of solids. Figure 4.1 shows the numerically calculated density of modes of lead in comparison with the Debye model for this metal as obtained from the experimental speed of sound. Table 4.1 list the Debye temperature for a few selected solids. 

The g((o) can be used to provide a simple approximation to the lattice vibrations, the Einstein approximation. Let us begin by agreeing that single characteristic frequencies, could be chosen to individually represent each of the six types (three translational and three rotational) of external mode. The ( 6)y value is the density of states weighted mean value of all of the frequeneies over which that external mode, j, was dispersed. Normalising as discussed above ... 

The Einstein model gives a good qualitative agreement with the real behavior of solids, but the quantitative agreement is poor (Cranshaw et al. 1985). A more realistic representation of a solid is given by the Debye model. The model describes the lattice vibration of solids as a superposition of independent vibrational modes (i.e., collective wave motion of the lattice, associated with phonons ) with different frequencies. The (normalized) density function p(co) of the vibrational frequencies is monotonically increasing up to a characteristic maximum of cod, where it abruptly drops to zero (Kittel 1968) ... 

Further indirect information on the transport mechanism can be obtained by spectroscopic investigations. Venuti et al., using Raman spectroscopy in conjunction with lattice dynamics calculation, found evidence of an efficient coupling between the lattice vibrations and low energy intramolecular modes [237]. Information of an even more indirect nature is provided by analyzing the field effect transistor characteristics which allow one to estimate the tail of the density of states of the semiconductor [238]. Interestingly, it is possible to compare quantitatively the tail of the density of states evaluated experimentally with the theoretical value obtained from MD/QC procedure [239]. 

For molecular crystals in which the molecules are nearly spherical there will frequently be a high-temperature crystal phase of which the site symmetry is higher than the molecular symmetry. These crystals are referred to as plastic crystals and they have unusual properties such as very high temperatures of melting and low anisotropic properties similar to those of liquids. The theory of lattice vibrations of orientationally disordered solids has been addressed, and the results indicate that such disorder should lead to a broadening of the vibrational bands and that all modes may be active in both the infrared and Raman spectra. Spectra of this type are referred to as density-of-states spectra, since the bands correspond to the flat points in the dispersion curve. 

Iditional importance is that the vibrational modes are dependent upon the reciprocal e vector k. As with calculations of the electronic structure of periodic lattices these cal-ions are usually performed by selecting a suitable set of points from within the Brillouin. For periodic solids it is necessary to take this periodicity into account the effect on the id-derivative matrix is that each element x] needs to be multiplied by the phase factor k-r y). A phonon dispersion curve indicates how the phonon frequencies vary over tlie luin zone, an example being shown in Figure 5.37. The phonon density of states is ariation in the number of frequencies as a function of frequency. A purely transverse ition is one where the displacement of the atoms is perpendicular to the direction of on of the wave in a pmely longitudinal vibration tlie atomic displacements are in the ition of the wave motion. Such motions can be observed in simple systems (e.g. those contain just one or two atoms per unit cell) but for general three-dimensional lattices of the vibrations are a mixture of transverse and longitudinal motions, the exceptions... 

By scattering within molecular solids and at their surfaces, LEE can excite with considerable cross sections not only phonon modes of the lattice [35,36,83,84,87,90,98,99], but also individual vibrational levels of the molecular constituents [36,90,98-119] of the solid. These modes can be excited either by nonresonant or by resonant scattering prevailing at specific energies, but as will be seen, resonances can enhance this energy-loss process by orders of magnitude. We provide in the next two subsections specific examples of vibrational excitation induced by LEE in molecular solid films. The HREEL spectra of solid N2 illustrate well the enhancement of vibrational excitation due to a shape resonance. The other example with solid O2 and 02-doped Ar further shows the effect of the density of states on vibrational excitation. 

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