Vibrational modes longitudinal

Figure 12.13 Calculated total energy as a function of the displacement corresponding to the longitudinal vibration mode L(, ).

Comments are necessary concerning the significance of to and Oq in Eq, (8.21). Values of 10" to 10 s have been frequently determined for to , they have been assigned to longitudinal vibrational modes of the backbone chain. For PVC and PE extremely small values are obtained which certainly cannot be interpreted in the same manner. With flow processes it must be recognized that the activation energy is not temperature independent but decreases with temperature ... 

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... 

Figure 8.11 (a) Dispersion curve for CuCl(s) along [110] of the cubic unit cell, (b) Density of vibrational modes [3], Here L, T, A and O denote longitudinal, transverse, acoustic and optic. Reproduced by permission of B. Hennion and The Institute of Physics. 

In three dimensions, transverse and longitudinal optic and acoustic modes result. The dispersion curve for CuCl along [100] of the cubic unit cell [3] is shown in Figure 8.11(a) as an example. The number of discrete modes with frequencies in a defined interval can be displayed as a function of the frequency. This gives what is termed the density of vibrational modes or the vibrational density of states (DoS). The vibrational DoS of CuCl is given in Figure 8.11(b). 

For the particular case of longitudinal optical modes, we found in Eq. (9-27) the electrostatic electron-phonon interaction, which turns out to be the dominant interaction with these modes in polar crystals. Interaction with transverse optical modes is much weaker. There is also an electrostatic interaction with acoustic modes -both longitudinal and transverse which may be calculated in terms of the polarization generated through the piezoelectric effect. (The piezoelectric electron phonon interaction was first treated by Meijer and Polder, 1953, and subsequently, it was treated more completely by Harrison, 1956). Clearly this interaction potential is proportional to the strain that is due to the vibration, and it also contains a factor of l/k obtained by using the Poisson equation to go from polarizations to potentials. The piezoelectric contribution to the coupling tends to be dominated by other contributions to the electron -phonon interaction in semiconductors at ordinary temperatures but, as we shall see, these other contribu-... 

In the solid state, the polar phonons (those that are IR active) split into two components, the transverse optical mode (TO) and the longitudinal optical mode (LO). This TO/LO splitting occurs because the electric field associated with the transverse wave = 0 while that associated with the longitudinal wave is 0. The coupling of these modes with the electric fields associated with the vibration gives rise to Vlo > Vto- This factor is relevant in relation to the shape and interpretation of the IR spectra of solid materials and will be further considered below. 

Raman spectra of certain crystalline polymers and their oligomers show a series of identifiable LAM bands in the low-frequency region, corresponding to symmetrical longitudinal backbone vibration modes. In the simplest approach, the stem is considered as a continuous elastic rod, the frequency of the vibration being35... 

Schematic of the fourth vibration mode (with wavelength A = L/2) of a rigid rod of length L. The transverse oscillation with amplitude h. reduces the projected rod length along the x-axis. The amount that the rod length is reduced, per wavelength A, oscillates with longitudinal amplitude h .

The calculations show that there are two modes with the same frequency of 1569cm one of which is Raman active and the other one is IR active. The IR active mode corresponds to longitudinal vibrations. The Raman active 1569cm mode was not obtained in our simplified calculations. This mode corresponds to a vibration of the kinks, which are directed along the chemical bonds on the ends of linear fragments. The 2100 cm mode is directed along the chain axis. 

This is an example of LO-TO splitting of the vibrational modes according to their direction of motion at the gamma point ( 2.6.1.3, 4.2.6). For any general direction the polarisation of the waves is not strictly longitudinal or transverse and the polarisation vectors are dependent on k as well as k. Similarly, the acoustic modes split into a longitudinal and two transverse modes. These modes vary in frequency along different directions in the unit cell [16]. 

The substrate (or adsorbent) was modelled in one dimension by a linear chain of five carbon atoms, (We could have chosen any other atom.) A terminal carbon atom represents a surface atom and the remaining atoms the bulk. There is only one force constant, the C-C stretching force constant (C—C of C2 12.16 mdyn A" ). The dynamics were treated by the Wilson GF method. The substrate then had four vibrations similar to longitudinal acoustic modes (LAM) ( 10.1.2) LAM1,170 cm LAM2, 330 cm LAM3, 455 cm LAM4, 535 cm. The calculated INS spectrum comprised four bands of equal, weak intensity. 

Figure 17. Time-resolved fluorescence spectra of a solute with one vibrational mode in ethanol at 247 K.68 The various frames show the fluorescence spectrum measured at successively later times after the application of a 1 ps excitation pulse. Each spectrum is labeled with the observation time. The steady-state fluorescence spectrum is given by the dashed curve in the bottom frame. In the electronic ground state, the solute vibrational frequency is400cm 1, and in the excited state, the frequency is 380 cm 1. The dimensionless displacement is 1.4. The permanent dipole moment changes by 10 Debye upon electronic excitation. The Onsager radius is 3A. The longitudinal dielectric relaxation time, xL, is 150 ps.

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