Lattice vibrations coupled rotational-translational

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... 

When the MF model is further separated by Eq. (42) the MF excitations are either translational (T) or librational (L). The indices k and k which label the rows and columns of the matrix 0(q) run over both types of excitations. The elements 0 q) and 0 q) correlate the translational and rotational vibrations of the individual molecules the resulting lattice vibrations are the translational phonons and the librons. These are coupled by the elements and (g), which contain the translation-... 

McKean 182> considered the matrix shifts and lattice contributions from a classical electrostatic point of view, using a multipole expansion of the electrostatic energy to represent the vibrating molecule and applied this to the XY4 molecules trapped in noble-gas matrices. Mann and Horrocks 183) discussed the environmental effects on the IR frequencies of polyatomic molecules, using the Buckingham potential 184>, and applied it to HCN in various liquid solvents. Decius, 8S) analyzed the problem of dipolar vibrational coupling in crystals composed of molecules or molecular ions, and applied the derived theory to anisotropic Bravais lattices the case of calcite (which introduces extra complications) is treated separately. Freedman, Shalom and Kimel, 86) discussed the problem of the rotation-translation levels of a tetrahedral molecule in an octahedral cell. 

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 molecular solids the molecules cannot move around freely, but they are trapped in relatively deep potential wells, caused by the intermolecular potential. In these wells they can vibrate and since the vibrations of individual molecules are coupled, again by the intermolecular potential, one obtains collective vibrations of all the molecules in the solid, called lattice vibrations or phonons. Phonons associated with the center of mass motions of the molecules are called translational phonons, phonons associated with their hindered rotations or librations are called librons. The degree of hindrance of the rotations may vary. If the molecules have well-defined equilibrium orientations and perform small amplitude librations about these, one speaks about ordered phases. If the molecular rotations are nearly free or if the molecules can oscillate in several orientational pockets and easily jump between these pockets, then the solid is called orientationally disordered or plastic. Several molecular solids may occur in each of these phases, depending on the temperature and pressure they undergo order/disorder phase transitions. Also the intramolecular vibrations are coupled by the intermolecular potential, via its dependence on the internal coordinates. The excitations of the solid associated with such vibrations are called vibrational excitons or vibrons. 

In the hydrate lattice structure, the water molecules are largely restricted from translation or rotation, but they do vibrate anharmonically about a fixed position. This anharmonicity provides a mechanism for the scattering of phonons (which normally transmit energy) providing a lower thermal conductivity. Tse et al. (1983, 1984) and Tse and Klein (1987) used molecular dynamics to show that frequencies of the guest molecule translational and rotational energies are similar to those of the low-frequency lattice (acoustic) modes. Tse and White (1988) indicate that a resonant coupling explains the low thermal conductivity. 

Figure 2.6-2 Variation of the frequencies by the incorporation of a tetraatomic molecule with two degenerate vibrational states ( ) in a crystal lattice, a spectrum of the free molecule, R = rotations, T = translations b static influence of the crystal lattice. The degenerate states split, the free rotations change into librations L c dynamic coupling of the vibrations of molecules within a primitive unit cell with z = 2 molecules. Each vibrational level of a molecule splits into z components and 3 z - 3 translational vibrations TS and 3 z librations L appear d dependence of the vibrational frequencies on the wave vector k of the coupled vibrations of all unit cells in the lattice. The three acoustic branches arise from the three free translations with = 0 (for k 0) of the unit cell all vibrations of the unit cells with / 0 (for k 0) give optical branches .

Up to now, lattice dynamics calculations have addressed the translational phonons and the rotons separately. For the translational phonons one has used an isotropic (i.e. orientationally averaged) H2-H2 potential. The SCP method has been applied to the anharmonic vibrations [96], but it appeared to be necessary to introduce an approximate Jastrov function into this method (with one adjustable parameter) in order to obtain realistic results. The roton frequencies, and their softening at higher pressures (smaller volume) which precedes the disorder/order phase transition, have been calculated by the MF [97] and RPA [98] methods. Only quadrupole-quadrupole interactions were taken into account, and translation-rotation coupling was neglected. 

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