Frequency external mode

The presence of isotopic impurities causes clear effects in the vibrational spectra. Almost all modes studied so far show frequency shifts on S/ S substitution [81, 107]. The average shift of the internal modes is ca. 0.6 cm , and of the external modes it is 0.1-0.3 cm (Tables 3, 4 and 5). Furthermore, the isotopomers which are statistically distributed in crystals of natural composition can act as additional scattering centers for the phonon propagation. Therefore, in such crystals the lifetime of the phonons is shortened in comparison with isotopically pure crystals and, as a conse-... 

In the condensed phase the sum is over all 3n frequencies, but in the ideal vapor phase the six external (zero) frequencies do not contribute to the IE s, the sum is over the remaining 3n — 6 internals. For condensed rare gases the harmonic assumption is highly approximate, and this is also true for the lattice modes of polyatomics. However as molecular size increases the relative contribution of the external modes becomes less and less important relative to internals. 

As the oscillators of the OPP model vibrate independently of each other, the frequencies are dispersionless, that is, independent of a wavevector q. For the internal modes of a molecular crystal, this tends to be a very good approximation. For the external modes, the dispersion can be pronounced, as shown in Figs. 2.1 and 2.2. In order to obtain the mean-square vibrational amplitudes for the latter, a summation over all phonon branches in the Brillouin zone must be performed. 

In molecular crystals, the separation between internal and external modes is of importance. Except for torsional oscillations in some types of molecules, the internal modes have much higher frequencies than the external modes. According to expressions such as Eqs. (2.51) and (2.58), the latter are then the dominant... 

Fig. 4 Phonon density of states at 180 K for the low-frequency modes of C60 at atmospheric pressure and at 0.5 GPa. Reprinted with permission from H Schober and B Renker, Pressure dependence of the external mode spectrum of solid C60 , Phys. Rev. B vol. 59 (1999) 3287-90 [39]. Copyright 1999 The American Physical Society...

The optical spectral region consists of internal vibrations (discussed in Section 1.13) and lattice vibrations (external). The fundamental modes of vibration that show infrared and/or Raman activities are located in the center Brillouin zone where k = 0, and for a diatomic linear lattice, are the longwave limit. The lattice (external) modes are weak in energy and are found at lower frequencies (far infrared region). These modes are further classified as translations and rotations (or librations), and occur in ionic or molecular crystals. Acoustical and optical modes are often termed phonon modes because they involve wave motions in a crystal lattice chain (as demonstrated in Fig. l-38b) that are quantized in energy. 

Since the disjjersion in k space of the energy of internal modes (vibrons) is generally very small (on the order of a few wave numbers, but see Section IV C), the requirements of energy and wave-vector conservation make the decay of a vibron into two (or more) other vibrons a highly unlikely event. The most generally observed decay channel is thus observed to be a mixed process, where the initial vibron interacts with another vibron and a low-frequency (disjjersed) lattice phonon. (When the frequency separation of internal and external modes becomes less well defined, decay can also involve two external modes.)... 

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 Debye-Waller factor is smaller for adsorbed benzene than for solid benzene. This is due to a larger mean square amplitude of the external modes of adsorbed benzene (0.017 A ) than solid benzene (0.011 A ) because the benzene molecules are less strongly bound in the zeolite than in the solid. The external modes of adsorbed benzene are at lower frequency (see Table 7.20) and their contribution to the total displacement amplitude is increased. The intensity of the fundamentals is decreased and is transferred to multiphonon features with consequent... 

The presence of several molecules bound together in a molecular crystal generates an external force field and new sequences of vibrations are created, the external modes. These run down from about 80 cm", see below, and again the lowest attainable frequency is a function of the number of oscillators. The number of molecules in a crystal is generally very large and the dispersion curve of the external modes becomes truly continuous, the very lowest frequencies are available and the density-of-states is not composed of sharp individual transitions. However, the spectral features assoeiated with the TAMs remain sharp sinee the original wavevectors retain their significance. 

There are three translational dispersion curves and three rotational dispersion curves. In the absence of rotation-translation coupling these vibrations are also distinctly separate. However, external modes and internal modes of the same character will become mixed, the closer they are in frequency the greater will be the mixing. Atomic displacements are no longer controlled by exclusively internal or external forces but by both. Modes of a mostly internal (or mostly external) character now involve displacements of the centre of mass (or deformations of the molecular frame). This is a general feature of all molecular crystals but it is usual to ignore such complications when the frequency of the lowest... 

Brillouin zone. It should be noted that several symmetry-required degeneracies occur which have not been indicated in the figure and that the traditional spec-troscopist s frequency unit of cm (1 cm = 3 X 10 Hz) has been used. The recipe given above applied to this crystal (A = 8, w = 2) gives eight internal mode branches or dispersion curves and 16 external mode branches of which three are translational acoustic, four are rotatory optic (sometimes called librational), and nine are translatory optic. 

In the case of the vibrations, one distinguishes two types the intramolecular vibrations (molecular vibrations), which are also referred to as internal modes and whose frequencies differ from those of the free molecules only slightly or not at all and the external vibrations (lattice vibrations), which are also termed external modes, where the organic molecules as a whole oscillate around their equilibrium positions. Usually, only the external vibrations are called phonons. 

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