Lattice dynamics translational 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... 

Depending on the character of the molecular motions, one can distinguish several physical situations. In most cases, the molecules are trapped in relatively deep potential wells. Then, they perform small translational and orientational oscillations around well-defined equilibrium positions and orientations. Such motions are reasonably well described by the harmonic approximation. The collective vibrational excitations of the crystal, which are considered as a set of harmonic oscillators, are called phonons. Those phonons that represent pure angular oscillations, or libra-tions, are called librons. For some properties it turns out to be necessary to look at the effects of anharmonicities. Anharmonic corrections to the harmonic model can be made by perturbation theory or by the self-consistent phonon method. These methods, which are summarized in Section III under the name quasi-harmonic theories, can be considered to be the standard tools in lattice dynamics calculations, in addition to the harmonic model. They are only applicable in the case of fairly small amplitude motions. Only the simple harmonic approximation is widely used the calculation of anharmonic corrections is often hard in practice. For detailed descriptions of these methods, we refer the reader to the books and reviews by Maradudin et al. (1968, 1971, 1974), Cochran and Cowley (1967), Barron and Klein (1974), Birman (1974), Wallace (1972), and Cali-fano et al. (1981). 

The dependence of the potential V,r on the translational degrees of freedom u, is rather intricate, and in order to use this potential in lattice dynamics calculations, we have to rewrite it in more tractable form. Beforehand, we make some remarks concerning the case when only rotational motions are considered. This is useful when the rotation-transla-... 

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... 

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. 

For the formulation of the lattice dynamics in a molecular crystal, only six coupled equations of motion are necessary three for translation and three for rotation of a single molecule, if all the molecules in the unit cell are connected by symmetry operations. (This assumption does not hold e.g. for dipolar disorder in dimethyl-naphthalene or dimethyl-anthracene crystals (see Sect. 5.7).) The translational-rotational displacements u(lk) of the kth molecule in the Ith unit cell are taken to be given by propagating plane waves u(lfe) = U(fk)e F Here, S2(K)... 

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. 

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. 

Lattice dynamics studies of the disordered j -phase are more scarce because, obviously, the standard harmonic method and the SCP method cannot be applied to this phase (although in some studies the harmonic method has still been used for the translational phonons, while neglecting the anisotropy of the potential.) Most calculations on have been made by classical Monte Carlo or Molecular Dynamics methods, using semiempirical atom-atom or quadrupole-quadrupole potentials. In our group [50, 52] we have investigated the motions in and the a — jS order/ disorder phase transition by means of the MF, RPA and TDH methods, using the same spherically expanded anisotropic ab initio potential which yields accurate properties for a-N2. 

The molecules treated in this chapter are indeed large systems with complex chemical structures. Moreover, in going from the oligomers to the polymers it becomes necessary to consider the systems as one-dimensional crystals. The optical transitions (both vibrational and electronic) are determined by one-dimensional periodicity and translational symmetry. Collective motions (phonons) that extend throughout the chain need to be considered and are characterized by the wave vector k, and their frequencies show dispersion with k. Lattice dynamics in the harmonic approximation are well developed [12,13J, and vibrational frequency spectroscopy has reached full maturity and has been widely applied in polymer science [8,9,14]. 

Before continuing with the discussion on the dynamics of SE s in crystals and their kinetic consequences, let us introduce the elementary modes of SE motion. In a periodic lattice, a vacant neighboring site is a necessary condition for transport since it allows the site exchange of individual atomic particles to take place. Rotational motion of molecular groups can also be regarded as an individual motion, but it has no macroscopic transport component. It may, however, promote (translational) diffusion of other SE s [M. Jansen (1991)]. 

Molecular motions in low molecular weight molecules are rather complex, involving different types of motion such as rotational diffusion (isotropic or anisotropic torsional oscillations or reorientations), translational diffusion and random Brownian motion. The basic NMR theory concerning relaxation phenomena (spin-spin and spin-lattice relaxation times) and molecular dynamics, was derived assuming Brownian motion by Bloembergen, Purcell and Pound (BPP theory) 46). This theory was later modified by Solomon 46) and Kubo and Tomita48 an additional theory for spin-lattice relaxation times in the rotating frame was also developed 49>. 

The photodissociation dynamics of F2 in solid argon were studied by Schwentner and Apkarian [1989] and Feld et al. [1990], The fluorine molecule exists as a substitutional impurity in the fee lattice of argon, where it can undergo rotational motion. A molecular dynamics simulation performed by Alimi et al. [1990] shows that this rotation is accompanied by a considerable displacement of the center of mass of F2 (Figure 9.6a), so the motion is strongly coupled with translation. This is similar to the... 

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. 

But what are possible dynamic consequences of such an equivariance breaking perturbation to a lattice group As many authors before us, we describe such consequences in terms of the tip motion of meandering spirals. We would like to keep in mind, however, that the prominently visible phenomenon of the tip motion only amounts to a visualization of the translational component of the SE 2) coordinates, which we will derive in section... 

In the following sections of this chapter, we first treat intramoleular vibrations then, in much more detail, phonons using typical examples and finally, very briefly, stochastic rotational motions ( reorientations ) and translational diffusion of molecules. Although the experimental methods for the characterisation of dynamics in molecular crystals are in principle no different from those used to investigate inorganic crystals, we shall briefly describe inelastic neutron diffraction, Raman scattering, infrared and far-infrared spectroscopy, as well as NMR spectroscopy to the extent necessary or useful for the specific understanding of molecular and lattice... 

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