Phonon dispersion relation

The phonon dispersion relations for ( ,0) zigzag tubules have 4 X 3/j = 12/j degrees of freedom with 60 phonon branches, having the symmetry types (for n odd, and D j symmetry) ... 

In the following sections, we first show the phonon dispersion relation of CNTs, and then the calculated results for the Raman intensity of a CNT are shown as a function of the polarisation direction. We also show the Raman calculation for a finite length of CNT, which is relevant to the intermediate frequency region. The enhancement of the Raman intensity is observed as a function of laser frequency when the laser excitation frequency is close to a frequency of high optical absorption, and this effect is called the resonant Raman effect. The observed Raman spectra of SWCNTs show resonant-Raman effects [5, 8], which will be given in the last section. 

Fig. 22. Phonon dispersion relations for a (5,5) carbon nanotube. This armchair nanotube would be capped with a Cr,o hemisphere [194],...

Bilz, H. and Kress, W. (1979) Phonon Dispersion Relations in Insulators. Berlin Springer. 

The earliest applications of the shell model, as with the Born model, were to analytical studies of phonon dispersion relations in solids.These early applications have been well reviewed elsewhere.In general, lattice dynamics applications of the shell model do not attempt to account for the dynamics of the nuclei and typically use analytical techniques to describe the statistical mechanics of the shells. Although the shell model continues to be used in this fashion, lattice dynamics applications are beyond the scope of this chapter. In recent decades, the shell model has come into widespread use as a model Hamiltonian for use in molecular dynamics simulations it is these applications of the shell model that are of interest to us here. 

Saito R, Jorio A, Souza Filho AG, Dresselhaus G, Dresselhaus MS, Pimenta MA (2002) Probing phonon dispersion relations of graphite by double resonance Raman scattering. Phys Rev Lett 88 027401... 

The phonon dispersion relation around the K-point can be obtained from the G -band of the corresponding Raman spectra [57]. Some other bands of the second-order spectrum, namely, M, iTOLA, G, 2iTO, and 2G bands, have been assigned in References [37, 68, 69] (see definitions in the text below). 

Mafra DL, Samsonidze G, Malard LM, Elias DC, Brant JC, Plentz E, Alves ES, Pimenta MA (2007) Deterination of LA and TO phonon dispersion relations of grapehene near the Dirac point by double resonance Raman scattering. Phys Rev B 76 233407... 

By assuming harmonic forces and periodic boundary conditions, we can obtain a normal mode distribution function of the nuclear displacements at absolute zero temperature (under normal circumstances). The problem is then reduced to a classic system of coupled oscillators. The displacements of the coupled nuclei are the resultants of a series of monochromatic waves (the normal modes). The number of normal vibrational modes is determined by the number of degrees of freedom of the system (i.e. 3N, where N is the number of nuclei). Under these conditions the one-phonon dispersion relation can be evaluated and the DOS is obtained. Hence, the measured scattering intensities of equations (10) and (11) can be reconstructed. 

As mentioned in seetion 1, the MD approach can be utilized to obtain parameters sueh as frequency-dependent relaxation times and phonon dispersion relations (the dependence of the frequency w on the wave vector k). From the phonon dispersion curves, we can compute parameters such as the phonon group veloeity v , density of states D(w), and specific heat C . The phonon group velocities and relaxation times ean then be input into the BTE (Eq. 2.1). 

SOI and strained silicon transistors are comparable to or smaller than the phonon s mean free path (which, for silicon, has been estimated as 300 nm at 300K) [53], In this limit, the film surfaces alter the phonon dispersion relations [76], and the phonon-surface scattering may become the predominant scattering mechanism [3, 53], Since phonons are the main carriers of thermal energy in silicon, these effects alter the thermal conductivity, which differs from that of bulk silicon [10, 36, 77], Measurements of the thermal conductivities of silicon films of thicknesses down to 74 nm found a reduction of 50% with respect to the bulk value at 300K [53], This reduction depends on the temperature and the thickness of the film [3, 53],... 

Table III with the preceding columns. The new results calculated with the ab initio potential agree very well with the frequencies from inelastic neutron scattering (Kjems and Dolling, 1975) and from infrared and Raman spectroscopy (Thi6ry and Fabre, 1976 Fondire et al., 1981) for all types of modes. Also the phonon dispersion relations, displayed in Fig. 4, are in good agreement with the neutron-scattering data. Since most of the lattice modes are actually mixed libron-phonon modes, this indicates that the translation-rotation coupling is correctly included in the RPA formalism.

Figure 9-19 Phonon dispersion relation (angular frequency vs. relative wave vector) for the three-stripe phase of CH4 on the external surface of a bundle. LI, L2, and L3 are longitudinal branches, i.e., molecular motion parallel to the groove. The dotted curve is the result for a ID adsorbate at the same density. The remaining curves correspond to the dispersion relation of transverse modes. (Adapted from Ref. [89].)...

As a case study in the calculation of the relation between co and q (also known as phonon dispersion relations) for three-dimensional crystals, we consider the analysis of normal modes of vibration in fee Al. As will be evident repeatedly as coming chapters unfold, one of our principal themes will be to examine particular problems from the perspective of several different total energy schemes simultaneously. In the present context, our plan is to consider the dispersion relations in Al as computed using both empirical pair functional calculations as well as first-principles calculations. 

Fig. 5.6. Phonon dispersion relation for A1 as compnted nsing pair fnnctional approach (adapted from Mishin et al. (1999)). The lines correspond to the theoretical resnlts and the points are experimental data obtained from nentron scattering.

Fig. 5.7. Phonon dispersion relation for A1 as computed using first-principles methods (adapted from Quong and Klein (1992)).

We now consider a more quantitative model of the vibrational density of states which makes a remarkable linkage between continuum and discrete lattice descriptions. In particular, we undertake the Debye model in which the vibrational density of states is built in terms of an isotropic linear elastic reckoning of the phonon dispersions. Recall from above that in order to effect an accurate calculation of the true phonon dispersion relation, one must consider the dynamical matrix. Our approach here, on the other hand, is to produce a model representation of the phonon dispersions which is valid for long wavelengths and breaks down at... 

Obtain an analytic expression for the zone edge phonons in fee Cu using the Morse potential derived in the previous chapter. To do so, begin by deriving eqn (5.37) and then carry out the appropriate lattice sums explicitly for the Morse potential to obtain the force constant matrix. In addition, obtain a numerical solution for the phonon dispersion relation along the (100) direction. 

H.G. Smith in G. Venkataraman V.C. Sahni (1970). Rev. Mod. Phys., 42, 409-470. External vibrations in complex crystals also in H. Biltz W. Kress (1979). Phonon Dispersion Relations in Insulators. Springer tracts in modem physics, 10, Springer-Verlag, Berlin. 

Nipko JC, Loong CK, Loewenhaupt M, Braden M, Reichart W, Boatner LA (1997a) Lattice dynamics of xenotime The phonon dispersion relations and density of states of L11PO4. Phys Rev B 56 11584-11592... 

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