Lattice dynamics librational motions

When a body undergoes vibrations, the displacements vary with time, so time averages must be taken to derive the mean-square displacements, as we did to obtain the lattice-dynamical expression of Eq. (2.58). If the librational and translational motions are independent, the cross products between the two terms in Eq. (2.69) average to zero, and the elements of the mean-square displacement tensor of atom n, U"j, are given by... 

The chain dynamics of the MGIMx copolymers has been studied in deuter-ated chloroform solutions through spin-lattice relaxation time, T y or more precisely through the Ti/T min ratio, in order to avoid the contribution of libration motions (Sect. 8.2.3.2). The temperature range considered extends from - 50 to 50 °C. 

Since it became clear from various observations that the librational motions of the molecules, even in the ordered a and y phases of nitrogen at low temperature, have too large amplitudes to be described correctly by (quasi-) harmonic models, we have resorted to the alternative lattice dynamics theories that were described in Section IV. Most of these theories have been developed for large-amplitude rotational oscillations, hindered or even free rotations, and remain valid when the molecular orientations become more and more localized. 

This potential was subsequently used in self-consistent phonon lattice dynamics calculations [115] for a and y nitrogen crystals. And although the potential—and its fit— were crude by present day standards, lattice constants, cohesion energy and frequencies of translational phonon modes agreed well with experimental values. The frequencies of the librational modes were less well reproduced, but this turned out to be a shortcoming of the self-consistent phonon method. When, later [ 116,117], a method was developed to deal properly with the large amplitude librational motions, also the librational frequencies agreed well with experiment. 

The lattice dynamics of solids consisting of atoms or ions is a well developed field in physics. The work of Born and Huang (1954) is the classic which deals with the theory of small vibrations as applied to periodic solids. Another standard reference on the subject is by Ziman (1960). However, a new element is introduced when molecular solids are treated. Librational motions occur only in these solids and in some ionic solids which contain polyatomic ions. 

The lattice dynamics of librational motions is much less develop d than that for translational motions. The first full treatments are due to Cochran and Pawley (1964) and Pawley (1967), although the treatment of these degrees of freedom at q = 0 was included in the matrix formulation of Shimanouchi, Tsuboi, and Miyazawa (1961) and the work on solid CO2 by Walmsley and Pople (1964). More recently Schnepp and Ron (1969) and Kuan, Warshel, and Schnepp (1969) carried out calculations on solid a-Na, and Suzuki and Schnepp (1971) on solid CO2. 

Such displacement coordinates have been used as a basis in the analysis of the lattice dynamics of librational motions (Cochran and Pawley, 1963 Pawley, 1967). Oliver and Walmsley (1968) have pointed out that u, Uy are independent to first order only and are often called infinitesimal rotation coordinates for that reason. We shall here discuss the use of these coordinates in some detail as applied to a linear molecule having two angular degrees of freedom. 

Richardson and Nixon (1968) have measured the relative infrared and Raman intensities of the lattice modes of cyanogen (NCCN), a linear molecule. The six infrared active modes are all translational motions and the quadrupole-induced intensities were calculated using the theory of Schnepp (1967) adapted for a different crystal structure. It was found that the experimental intensities of individual lines could be fitted to the theory within factors of two to three by adjusting the sample thickness, i.e., one parameter. However, if suras of intensities for each symmetry type are compared, to separate the intensity problem from the lattice dynamics problem, much better agreement is obtained, well within a factor of two. The relative Raman intensities of the librational lattice modes were also found to give good agreement between theory and experiment. 

In a crystal, displacements of atomic nuclei from equilibrium occur under the joint influence of the intramolecular and intermolecular force fields. X-ray structure analysis encodes this thermal motion information in the so-called anisotropic atomic displacement parameters (ADPs), a refinement of the simple isotropic Debye-Waller treatment (equation 5.33), whereby the isotropic parameter B is substituted by six parameters that describe a libration ellipsoid for each atom. When these ellipsoids are plotted [5], a nice representation of atomic and molecular motion is obtained at a glance (Fig. 11.3), and a collective examination sometimes suggests the characteristics of rigid-body molecular motion in the crystal, like rotation in the molecular plane for flat molecules. Lattice vibrations can be simulated by the static simulation methods of harmonic lattice dynamics described in Section 6.3, and, from them, ADPs can also be estimated [6]. 

Dynamic processes such as fluctuational motions of water molecules can be studied from the spin-lattice relaxation times of proton nuclear resonance . In the presence of highly mobile H atoms, the separation of the proton resonance disappears and the remaining signal sharpens as for isotropic liquids. Furthermore, the frequencies of H2O librations (see Sect. 4.4) can be computed . ... 

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... 

To obtain information on the role of dynamics of molecular motions in the reactive systems, the approach of phonon spectroscopy is used. Phonons are low-frequency cooperative lattice vibrations of a solid and, therefore, probe the lattice interactions and dynamics directly. Phonons can be observed as optical transitions in the Raman spectra and in the electronic spectra (in the latter as a phonon side band). Some information regarding averaged librational and translational phonon motions can also be obtained from the rigid-motion analysis of the thermal parameters of x-ray diffraction studies. 

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