Lattice vibrations translational

Spectra of molecules in the crystalline state, i.e., of molecular crystals, are obtained from molecules which are at fixed positions (sites) in the lattice (Fig. 2.6-1C). Normal (first-order) infrared and Raman spectra can be seen as spectra of hyper molecules , the unit cells (Schneider 1974, Schneider et al., 197.5). As a consequence, any molecular vibration is split into as many components as there are molecules present in the unit cell. Their infrared and Raman activity is determined by the symmetry of the unit cell. In addition, the translational and rotational degrees of freedom of molecules at their sites are frozen to give rise to lattice vibrations translational vibrations of the molecules at their sites and rotational vibrations about their main inertial axes, so-called librations. 

Energy absorptions in the far-IR correspond to rotational and translational movement states, overlying lattice vibrations and differential... 

It is furthermore possible with the correlation method to divide the lattice vibrations into translational types and librationcd (rotational) types. The same results are obtained with this method as with that of Bhagavantam and Venkatara-yudu. Adams and Newton (28, 29) have recently pubhshed tables which make a simple factor group analysis possible [see also (30)]. 

The interpretation of the lattice vibrations for scheeUte type molybdates or tungstates with relatively light cations, Ca or Sr, has indicated that the lowest translational vibrations are produced by Mo—Mo or W—W motions respectively, while those at higher frequency are from cation-cation motions 98). This has not been found, however, in the case of the barium or lead compounds. The librational frequencies have been found to decrease hnearly with the ionic radius of the cation for AMO4 type compounds, where A = Ca, Sr, Ba, or Pb and M =Mo or W 98). 

Landau (26) proposed that an additive electron in a dielectric can be trapped by polarization of the dielectric medium induced by the electron itself. Applying the model to electrons in the conduction band of an ionic crystal is rather complicated since the translational symmetry of the solid must be considered and the interaction of the excess electron with the lattice vibrations must be treated properly (I, 13, 14). 

As previously mentioned, the primitive unit cell is the smallest unit of a crystal that reproduces itself by translations. Figure 1-37 illustrates the difference between a primitive and a centered or nonprimitive cell. The primitive cell can be defined by the lines a and c. Alternatively, we could have defined it by the lines a and c. Choosing the cell defined by the lines a" and c" gives us a nonprimitive cell or centered cell, which has twice the volume and two repeat units. Table 1-11 illustrates the symbolism used for the various types of lattices and records the number of repeat units in the cell for a primitive and a nonprimitive lattice. The spectroscopist is concerned with the primitive (Bravais) unit cell in dealing with lattice vibrations. For factor group selection rules, it is necessary to convert the number of molecules per crystallographic unit cell Z to Z, discussed later, which is the number of molecules per primitive cell. For example,... 

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. 

The temperature-dependent Raman spectra are depicted in Fig. 4-27a, b. Figure 4-27a shows the spectra of H2O-I (the water molecules in the inner coordination sphere) from 133-223 K. Figure 4-27b shows the spectra of H2O-II (the water molecules in the outer sphere). The spectra above 223 K are not shown because of the overlap with fluorescence that is observed with the 514.5 nm excitation. Plots of the variations of band frequency with temperature are illustrated in Fig. 4-28a, b for H2O-I and H2O-II. Two discontinuities are observed at 195 5K and 140 5K, indicative of three distinct phases occurring in the temperature range studied, as indicated in Fig. 4-28a. The higher-frequency OH stretch region, as shown in Fig. 4-28b does not show any discontinuities for H2O-I. A plot of full width at half maximum intensity (FWHM) vs. T for H2O-I shows a discontinuity at 140 K (Fig. 4-28c, d). Additional support for these phase transitions was found from the temperature dependences of the UO vibrational mode, lattice vibrations and the NO3 ion vibrations (translations and rotations). 

To study the reaction of molecules on surfaces in detail, it is important to consider the different degrees of freedom the system has. The reacting molecule s rotational, vibrational and translational degrees of freedom will affect its interaction as it hits the surface. In addition, the surface has lattice vibrations (phonons) and electronic degrees of freedom. Both the lattice vibrations and the electrons can act as efficient energy absorbers or energy sources in a chemical reaction. 

The special points method depends upon retention of the translational periodicity of a lattice, which is lost if we consider defects, surfaces, or lattice vibrations. (Even the special vibrational mode with frequency listed in Table 8-1 entailed a halving of the translational symmetry.) It is therefore extremely desirable to seek an approximate description in terms of bond orbitals, so that the energy can be summed bond by bond as discussed in Chapter 3. We proceed to that now. 

Spectra of samples in the liquid state (Fig. 2.6-lB) are given by molecules which may have any orientation with respect to the beam of the spectrometer. Like in gases, flexible molecules in a liquid may assume any of the possible conformations. Some bands are broad, since they are the sum of spectra due to different complexes of interacting molecules. In the low frequency region spectra often show wings due to hindered translational and rotational motions of randomly oriented molecules in associates. These are analogous to the lattice vibrations in molecular crystals, which, however, give rise to sharp and well-defined bands. The depolarization ratio p of a Raman spectrum of molecules in the liquid state (Eqs. 2.4-11... 13) characterizes the symmetry of the vibrations, i.e., it allows to differ between totally symmetric and all other vibrations (see Sec. 2.7.3.4). 

According to rules (1) and (2) hexamethylenetetramine C6H12N4 with the space group and one molecule per primitive cell cannot show any optically allowed lattice vibrations. Indeed, the Raman and IR spectra do not show any bands in the low-frequency region. Oxamide with one molecule per primitive cell Cj at sites C, shows three Raman bands, i.e., the three librations. Naphthalene with two molecules per primitive cell C2/, at sites C, shows six librations in the Raman spectrum. Three translational vibrations are allowed in the IR spectrum. 

The dispersion relationships of lattice waves may be simply described within the first Brillouin zone of the crystal. When all unit cells are in phase, the wavelength of the lattice vibration tends to infinity and k approaches zero. Such zero-phonon modes are present at the center of the Brillouin zone. The variation in phonon frequency as reciprocal k) space is traversed is what is meant by dispersion, and each set of vibrational modes related by dispersion is a branch. For each unit cell, three modes correspond to translation of all the atoms in the same direction. A lattice wave resulting from such displacements is similar to propagation of a sound wave hence these are acoustic branches (Fig. 2.28). The remaining 3N-3 branches involve relative displacements of atoms within each cell and are known as optical branches, since only vibrations of this type may interact with light. 

If the solid is molecular, the molecules (considered to be formed by M atoms, where M = N/r and r is the number of molecules in the smallest Bravais cell) can be treated as for the gas phase, so giving rise to 3M- 6 (or 3M- 5 if linear) vibrations for each molecule. The degrees of freedom associated with the external modes of every molecular unit (6r for non-linear molecules and 5r for linear molecules) give rise to lattice vibrations ( frustrated translations and rotations ) and to three acoustic modes. On the other hand, the internal vibrations of each molecules should in principle give rise to r-fold splitting, owing to the coupling of the vibrations within its primitive unit cell as a whole. 

In the case of crystaUine sohds, more than one equivalent structural unit may be present in the primitive cell. This results in sphttings of the fundamental vibrational modes of these units. In the case of many crystalline solid materials covalent units (e.g. oxo-anions for oxo-salts) are present, together with other groups bonded by ionic bonds (e.g. the cations in the oxo-salts). According to the above group approximation, the internal vibrations of the covalent units can be considered separately from their external vibrations hindered rotations and translations of the group that finally contribute to the lattice vibrations and to the acoustic modes of the unit cell) and those of the other units. The presence of a number of covalent structural units in the primitive cell, causes their internal modes to spHt... 

E. Whalley and J. E. Bertie. Optical spectra of orientationally disordered crystals I. Theory for translational lattice vibrations, J. Chem. Phys., 46 1264-1270 (1966). 

The calculations of TSM have recently been extended to include the effects of intermolecular forces by Tasumi and Shimanouchi 35). Estimates for the magnitude of intermolecular force constants for these calculations were obtained from the small splitting observed for higher-frequency modes. It was shown that intermolecular forces split every mode into two components belonging to different symmetry species. The acoustic modes vj and of TSM were also affected by intermolecular forces. For an isolated chain, these correspond to deformation and torsional vibrations respectively, but in crystals, they are mixed. Further, the zero and n phases of the acoustic modes predicted for an isolated chain correspond to zero frequency. In the crystal, non-zero values corresponding to rotary and translational lattice vibrations are obtained. 

Because we are dealing with molecules, two types of lattice vibrations can be distinguished translational and rotational. In order to describe these motions we have to know the potential energy of the crystal, expressed as a function of the center of mass positions and the orientations of all molecules. In Section II, we give a fairly detailed description of the different ways in which the potential can be expressed, each way having its own merits, depending on the subsequent calculations in which it has to be used. 

A scheme as described here is indispensable for a quantum dynamical treatment of strongly delocalized systems, such as solid hydrogen (van Kranendonk, 1983) or the plastic phases of other molecular crystals. We have shown, however (Jansen et al., 1984), that it is also very suitable to treat the anharmonic librations in ordered phases. Moreover, the RPA method yields the exact result in the limit of a harmonic crystal Hamiltonian, which makes it appropriate to describe the weakly anharmonic translational vibrations, too. We have extended the theory (Briels et al., 1984) in order to include these translational motions, as well as the coupled rotational-translational lattice vibrations. In this section, we outline the general theory and present the relevant formulas for the coupled... 

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