Dispersion relation for

Surface waves at an interface between two innniscible fluids involve effects due to gravity (g) and surface tension (a) forces. (In this section, o denotes surface tension and a denotes the stress tensor. The two should not be coiifiised with one another.) In a hydrodynamic approach, the interface is treated as a sharp boundary and the two bulk phases as incompressible. The Navier-Stokes equations for the two bulk phases (balance of macroscopic forces is the mgredient) along with the boundary condition at the interface (surface tension o enters here) are solved for possible hamionic oscillations of the interface of the fomi, exp [-(iu + s)t + i V-.r], where m is the frequency, is the damping coefficient, s tlie 2-d wavevector of the periodic oscillation and. ra 2-d vector parallel to the surface. For a liquid-vapour interface which we consider, away from the critical point, the vapour density is negligible compared to the liquid density and one obtains the hydrodynamic dispersion relation for surface waves + s>tf. The temi gq in the dispersion relation arises from... 

It can be shown that this fonn leads to an unphysical dispersion relation for capillary waves t/,7 -- f/", rather... 

A good discussion of plasma waves and a tabulation of their characteristics is available (12). Useful plots of the dispersion relations for various frequencies, field conditions, geometries, and detailed mathematical relationships are given in Reference 13. 

Fig. 18. One-dimensional energy dispersion relations for (a) armchair (5,5) nanotubes, (b) zigzag (9,0) nanotubes, and (c) zigzag (10,0) nano tubes. The energy bands with a symmetry arc non-degenerate, while the e-bands are doubly degenerate at a general wave vector k [169,175,176].

Closely related to the ID dispersion relations for the carbon nanotubes is the ID density of states shown in Fig. 20 for (a) a semiconducting (10,0) zigzag carbon nanotube, and (b) a metallic (9,0) zigzag carbon nanotube. The results show that the metallic nanotubes have a small, but non-vanishing 1D density of states, whereas for a 2D graphene sheet (dashed curve) the density of states... 

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) ... 

Clavin and Garcia [15] have obtained a more general dispersion relation for an arbitrary temperature dependence of the diffusion coefficients. Their nondimen-sional result is qualitatively the same as Equation 5.1.4, but the coefficients contain information on the temperature dependence of the diffusivities ... 

The complex wave frequency Q (= ico — F) is related to k via a dispersion relation. For an inviscid liquid, Lamb s equation is well-known as a classical approximation for the dispersion relation [10]... 

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

Figure 8.7 Experimental dispersion relations for acoustic modes for lead at 100 K [2], Reproduced by permission of B. N. Brockhouse and the American Physical Society.

Smith, D. Y., 1976. Comments on the dispersion relations for the complex refractive index of circularly and elliptically polarized light, J. Opt. Soc. Am., 66, 454-460. 

In some cases this equation will become useful for the analysis, but it does not introduce more information than that already contained in Eq. (45). As will be shown later, Eq. (46) leads to the same dispersion relation for divE / 0 as Eq. (45) for the wave as a whole. 

For branch 1 of a vanishing electric field divergence, the corresponding axisymmetric EM mode is obtained from Eqs. (45) and (74)-(76). Since no dispersion relation for such a mode is available at this point of the deductions, we first introduce the notation... 

Figure 2. Dispersion relation for uppermost energy levels for the isx(k), Ey k) bands discussed in the text for a given k = (k ky). The energy surface has fourfold symmetry about the origin. The result is compared with the photoemission data of Shen et al [7] also shown in the lower part of the figure.

In contrast to the just discussed classical models [43,44], authors of works [38, 39] treated the problem of the dynamical polarizability ay(carbon cage quantum-mechanically, utilizing the 5-potential model concept, where the C60 cage is simulated by the Dirac 5-potential, V(r) = —Vo8(Rc—r). However, instead of calculating cy( >) directly, the latter was determined from experimental data on the C60 photoabsorption cross section a (o>) [60, 61] with the help of the dispersion relations for the real Re ay and imaginary Imay parts of the dipole polarizability ay ( >) ... 

Once we have identified the elements of the transition matrix and normalized this matrix to satisfy the unitary condition, the dispersion relation for a full multilayer is immediately obtained. Then the Nth power of this matrix is... 

Figure 1-38 (a) Dispersion relation for the optical and acoustic branches in solids, (b) Wave motion in an infinite diatomic lattice. (Reproduced with permission from Ref. 49.)... 

For uniform spacing shown in Fig. 2.2(a) the energy dispersion relation for the it band is [14]... 

Parabolic energy dispersion relation for free particle E(k)=Ti2k2/ 2m in one dimension, ignoring Brillouin zones. 

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