Phonon branch

In general, the number of phonon branches for a carbon nanotube is very large, since every nanotube has 6N vibrational degrees of freedom. The symmetry types of the phonon branches for a general chiral nanotube are obtained using a standard group theoretical analysis [194]... 

Because the ID unit cells for the symmorphic groups are relatively small in area, the number of phonon branches or the number of electronic energy bands associated with the ID dispersion relations is relatively small. Of course, for the chiral tubules the ID unit cells are very large, so that the number of phonon branches and electronic energy bands is also large. Using the transformation properties of the atoms within the unit cell transformation... 

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 above eqn, ID refers to the nanotubes whereas 2D refers to the graphene sheet, k is the ID wave vector, and t and Care unit vectors along the tubule axis and vector C, respectively, and p labels the tubule phonon branch. 

For this particular system, the phonon branches are not investigated as yet but, based on the accumulated knowledge on other B2 materials transforming to close packed structures, one would expect a low lying [110] TAi branch in the B2 range, possibly with a dip at 1/2... 

A. Nagasawa, R. Yamauchi, K. Kita, T. Makita and Y. Morii, [110)TA phonon branch and anomalous... 

The number of phonon modes are limited and have been described as "phonon branches" where two types are present, "optical " and "acoustical". (These names arose due to the original methods used to study them in solids). 

The band structure of bulk silicon, with possible optical transitions for (c) absorption and (d) emission of a photon, together with (e) the dispersion curves of phonon branches, is shown on the right. After [Kol5],... 

The calculated Rayleigh mode (SJ, the lowest lying phonon branch, is in good agreement with the experimental data of Harten et al. for all three metals. Due to symmetry selection rules the shear horizontal mode just below the transverse bulk band edge can not be observed by scattering methods. The mode denoted by Sg is the anomalous acoustic phonon branch discussed above. Jayanthi et al. ascribed this anomalous soft resonance to an increased Coulomb attraction at the surface, reducing the effective ion-ion repulsion of surface atoms. The Coulomb attraction term is similar for all three metals... 

For a small change in magnitude of q, the change in frequency to is small, and oj is a continuous function of q ( — 2n/X). The dependence of co on q is referred to as the dispersion relation. The number of phonon branches with continuously varying co equals 3n, but some of these may be degenerate due to the symmetry of the crystal. 

For a molecular crystal, the description can be simplified considerably by differentiating between internal and external modes. If there are M molecules in the cell, each with nM atoms, the number of external translational phonon branches will be 3M, as will the number of external rotational branches. When the molecules are linear, only 2M external rotational modes exist. For each molecule, there are 3nM — 6 (3nM — 5 for a linear molecule) internal modes, the wavelength of which is independent of q. Summing all modes gives a total number of N M(3nM — 6) + 6M = 3nN, as required, because each of the modes that have been constructed is a combination of the displacements of the individual atoms. 

As the oscillators of the OPP model vibrate independently of each other, the frequencies are dispersionless, that is, independent of a wavevector q. For the internal modes of a molecular crystal, this tends to be a very good approximation. For the external modes, the dispersion can be pronounced, as shown in Figs. 2.1 and 2.2. In order to obtain the mean-square vibrational amplitudes for the latter, a summation over all phonon branches in the Brillouin zone must be performed. 

The displacement of an atom j, in unit cell /, at time t, is obtained by the summation over all 3nN normal modes, combined into 3n phonon branches,... 

A very much simplified lattice-dynamical model is that of Debye. In the Debye approximation, discussed in the following section, a single phonon branch is assumed, with frequencies proportional to the magnitude of the wavevector q. 

The unit cell for the different polytypes will naturally vary, as will the number of atoms per unit cell. This will affect the number of electronic bands and the phonon branches for a given polytype. 

The phonon branch that is suspected to interact strongly with the charge by the ARPES measurement, namely the zone-boundary Cu-0 bondstretching LO phonon branch (Fig. 1), was found by inelastic neutron scattering to show unusual temperature dependence [8],... 

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