Wave-like Modes Research Articles

Lattice dynamics of Hittorf's phosphorus and identification of structural groups and defects in amorphous red phosphorus

Using the force constants and bond polarisabilities of black phosphorus the dispersion, eigenvectors, and density of states for phonons, the sound velocity and the Raman spectra are calculated for polymeric violet Hittorf's phosphorus. Good agreement with Raman and Brillouin (v s,110,exp=6.95·10 5cm s −1) measurements reported for the first time on oriented single crystals is found. Thus we are able to assign specific vibrational modes, such as extended wave-like vibrations and modes localised on the P 8 and P 9 structural subunits. Comparison with vibrational spectra of amorphous red phosphorus shows that the latter contains tubes similar to those of Hittorf's P. The absence of the group of lines (around 373 cm −1) due to P 8 cages in a-Pproves that the P 8 cages are present in much reduced numbers only there, or that the weaker bonds are broken or distorted. The features at 400 cm −1 and 455 cm −1 in a-P might be explained by the occurrence of P 7 cages. Broken P 8 units may also be the origin of the metastable intrinsic defects giving rise to the photo-induced ESR and axial luminescence centers in amorphous phosphorus.

Stationary waves forced by topography in a vertically sheared, stratified, rotating fulid

AbstractThe effect of an isolated topographic bump in a two-layer fluid on a β-plane is investigated. An analytical solution is derived in terms of the appropriate Green's function for arbitrary topography of finite horizontal extent. It is found that the disturbances generated by the bump are composed of two fundamental modes which may be wave-like or evanescent. The wave-like modes are topographically induced Rossby waves which occur only when there is eastward flow in at least one of the layers. These waves are always confined to the downstream (eastward) side of the bump. Whereas previous studies of this type have concentrated on eastward flow over topography, the theory has been extended here to include a wide range of vertically sheared flows. Particularly important is the case of low level westward flow combined with upper level eastward flow, as it has direct application, for example, to the summertime atmospheric circulation over the sub-tropical regions of the continental land.masses. In this case a wave-like disturbance extends far downstream from the bump for sufficiently large shear, and is of smaller amplitude in the upper layer than in the lower layer because of the effects of the stratification. For small shears, the wave-like mode in the lower layer is small and the character of the disturbance is evanescent, confining it to the immediate neighbourhood of the bump. A stability analysis of the solutions shows that the disturbances may be baroclinically unstable for sufficiently large mean shear.

Use of the drone-fermion representation. I. Thermodynamic properties of an interacting spin-phonon system

The thermodynamic properties of an interacting spin-phonon system are studied by means of a diagrammatic expansion of the free energy which makes use of the drone-fermion representation for spin-1/2 operators. Not only do the diagrams systematize the evaluation of the traces of products of spin operators but they also assist in the physical interpretation of the results. For a model in which the spins interact independently with phonons it is found that the major effects are due to virtual phonon energy shifts. When the spins can exchange phonons coherently the excitations are coupled spin-phonon modes and the results can be correlated with the energy shifts of the spin- wave-like modes. By examining the contribution of the phonons to the thermal energy the enhancement of the phonon occupation number is calculated. For phonons with frequencies close to the spin precession frequency it is found that no anomalous peak as proposed by Becker and Clover for the independent spin mode. In the case of coupled modes agreement is obtained with the theory of Van Eekelen and Stevens.

Localization of normal modes in vitreous silica, germania and beryllium fluoride

The authors previously (1968) presented vibrational spectra for vitreous silica, germania and beryllium fluoride, calculated from atomic arrangements in physical models based on the random network theory. The present work describes the localization properties of the normal modes of vibration. It is shown that the vibrations are, on the whole, less extended in space than the plane wave-like modes which characterize regular crystals. Localization tends to be greatest at high frequencies and, especially, near band edges. If nonbridging oxygen atoms are present in the structure, the spectra of vitreous silica and germania exhibit an additional band of very intense localization; atomic motions associated with modes in this band are effectively restricted to no more than a few atoms.

Atomic vibrations in vitreous silica

Frequency spectra and normal modes of vibration have been computed for vitreous silica. They have been calculated from atomic arrangements in physical models based on the random network theory. The positions of bands in the computed spectra agree well with observed features in the experimental infra-red and Raman spectra of the glass. Detailed analysis of the normal modes indicates that the bands at 1050, 750 and 400 cm–1 are associated with bond-stretching, bending and rocking motions, respectively, of the oxygen atoms. Atomic vibrations in the glass are, on the whole, less extended in space than the plane wave-like modes which prevail in perfect crystals. The spatial localization tends to be greatest at high frequencies and near band edges. If non-bridging oxygen atoms are present in the structure, the frequenncy spectrum exhibits an additional band of very intense localization, associated with bond-stretching vibrations of the non-bridging atoms.