Librational modes

In the crystal, the total number of vibrations is determined by the number of atoms per molecule, N, and the nmnber of molecules per primitive cell, Z, multiplied by the degrees of freedom of each atom 3ZN. In the case of a-Sg (Z =4, N =8) this gives a total of 96 vibrations ( ) which can be separated in (3N-6)—Z = 72 intramolecular or "internal" vibrations and 6Z = 24 intermo-lecular vibrations or lattice phonons ("external" vibrations). The total of the external vibrations consists of 3Z = 12 librational modes due to the molecular rotations, 3Z-3 = 9 translational modes, and 3 acoustic phonons, respectively. 

In the case of translational modes simply the mass of the rings has to be considered, while in the case of librational modes the moment of inertia has to be taken into account... 

A noteworthy feature of the photoacoustic spectra shown in Figure 2 Is the presence of water librations. These are frustrated rotations and have been observed for ice (24) by infrared spectroscopy, as well as for water adsorbed on Ft and Ag surfaces by electron energy loss spectroscopy (25-27). The three libration modes have been associated with the bands at 600, 538 and 468 cm" > this set of peaks occurs for water adsorbed on both the hydroxylated and methoxylated silica. 

R. Wugt Larsen and M. A. Suhm, Cooperative organic hydrogen bonds The librational modes of cyclic methanol clusters. J. Chem. Phys. 125, 154314 (2006). 

With 4o/44Ca- and 92/ioojio- data the differentiation of the translational and librational modes was obtained for the molybdates and tungstates with scheelite structure 98, 100, 101). Tables 29 and 30 reproduce these results. In contrast to early findings, the lower frequency bands were found to be librations and not translations. Furthermore, the band associated with Vi for CaMo04 and CaW04 cannot be expressed as a pure deformational vibration of the MOl group, but this mode is coupled with the translational vibrations, as indicated in Table 29. The relationship between the translational vibration, E, and the square root of the mass of the cation for compounds of the type AMO 4 (A = Ca, Sr, Ba, Pb M = Mo, W) was determined to be linear (100). 

An interesting aspect of many structural phase transitions is the coupling of the primary order parameter to a secondary order parameter. In transitions of molecular crystals, the order parameter is coupled with reorientational or libration modes. In Jahn-Teller as well as ferroelastic transitions, an optical phonon or an electronic excitation is coupled with strain (acoustic phonon). In antiferrodistortive transitions, a zone-boundary phonon (primary order parameter) can induce spontaneous polarization (secondary order parameter). Magnetic resonance and vibrational spectroscopic methods provide valuable information on static as well as dynamic processes occurring during a transition (Owens et ai, 1979 Iqbal Owens, 1984 Rao, 1993). Complementary information is provided by diffraction methods. 

The infra-red spectrum of Li-6 hydroxide was discussed by Decius and Lilley 69> who report that the major feature at 3681 400 cm"1 in the LiOH spectrum (which was previously assumed to involve a librational mode) is partly due to the absorption at 3681 290 cm 1 which involves a translator mode in which there is a significant amount of Li7-participation the rest may be a two-phonon process (see Oehler and Gunthard S6>). 

This band is a combination of deformation and librational modes, and is a very broad band. The band appears in a region typically free of interference in many agri-food samples. In some cases, the band can be used to monitor water content. 

The microscopic origin of the collective modes has been identified since a long time. They are reported here with the corresponding typical correlation times (CT) reorientation modes (this is the so-called Debye region, CT > 10-12s), libration modes (rotations impeded by collisions, CT = 10 13s), atomic motions (vibrations, CT = 10-14s), electronic motions (CT = 10 16s). When the frequency of the external field increases, the various components of the polarization we have introduced here become progressively no longer active, because the corresponding motions of the solute lag behind the variation of the electric field. 

Work by Nandi et al. [50] and by Hsu et al. [48] illustrated the importance of including high-frequency data into the expression for or the representation of s(cu) that is used as input into the continuum theory for S(t). They investigated SD in water and found that very good agreement with experimental SD data could be obtained using a representation of (o>) that includes the librational modes of the hydrogen bond network in addition to the exponential decay of the collective dipole time correlation. 

In the anthracene crystal, the center of the molecule is the center of symmetry. Since we have two molecules per unit cell, Raman spectroscopy allows us to observe six libration modes, three symmetric (Ag) and three antisymmetric (Bg), with respect to the b axis of the crystal.47 The Raman spectra obtained in our laboratory will be discussed in Section II.D.2. We may estimate, with very good approximation, that the librations occur around the three inertial principal axes of the molecule, with the following ordering 49.4 and 56.9cm-1 around the normal (N) axis, 70.7 and 81.4cm"1 around the mean (M) axis, and 131.7 and 140.1 cm"1 around the long axis (with weight between 60 and 70% for each of these axes47). 

Dispersion of vibrations (i.e. of phonons) is neglected. This approximation is well adapted to the intramolecular vibrations and, to a less degree, to the libration modes, particularly to optical modes. 

This is the most puzzling requirement. It is not known why certain crystals—for example, HMTSF TCNQ or Cu(DMDCNQI)2—conduct very well at low temperatures but do not form Cooper pairs. One may wonder whether certain intramolecular or intermolecular vibrations or rigid-body librational modes must be "right" for superconductivity. 

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