Dispersions relations

A dispersion relation relating co to k for photons propagating in metals can be obtained by starting with the index of refraction N for ta tUp (Equation 24.58). Since N is by definition N = c/v the phase velocity v=(ofk, = kv, 

Even ough tiae phase velocity j /k c, it can be seen that the group velocity d(ajdk approaches c as w cap but is always less than c as can be verified by differentiating Equation 24.62 with respect to k. 

Dispersion relation for radiation passing through a c nductive medium with frequency Wp. The dashed line represents the velocity of light. No propagation is possible for u Up. 

The optical properties of a material are characterized by the real and imaginary parts of the index of refraction N+iK) as a fimction of wavelength. These optical parameters are obtained from the square root of the dielectric function, which may also be complex. 

The optical behavior of conductive media is characterized by the plasma frequency, cOp = ne /meo which is a function only of the electron concentration. Plasma frequencies for metals are 0(10 /s) which corresponds to the vacuum UV. Below the plasma frequency, the N and K values are large making the material highly reflective (except for inner band transitions which are responsible for the color of Cu and Au). Above the plasma frequency, metals start to becomes transmissive, but they will still absorb strongly until the co 3 wp. 

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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... 

As already mentioned, the results in Section HI are based on dispersions relations in the complex time domain. A complex time is not a new concept. It features in wave optics [28] for complex analytic signals (which is an electromagnetic field with only positive frequencies) and in nondemolition measurements performed on photons [41]. For transitions between adiabatic states (which is also discussed in this chapter), it was previously intioduced in several works [42-45]. 

M. Floissart, in Dispersion Relations and their Connection with Causality, E. P. Wiguer, ed., Academie Press, New York, 1964, p, 1. 

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... 

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 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. 

In recent years there is a growing interest in the study of vibrational properties of both clean and adsorbate covered surfaces of metals. For several years two complementary experimental methods have been used to measure the dispersion relations of surface phonons on different crystal faces. These are the scattering of thermal helium beams" and the high-resolution electron-energy-loss-spectroscopy. ... 

Wightman, A. S., L invariance dans la Mdcamque Quantique Relativiste, in Dispersion Relations and Elementary Particles, C. de Witt and R. Omnes, ed., John Wiley and Sons, Inc., Hew York, 1960, and references listed in these lectures. 

By taking the real and imaginary parts of this last integral representation one obtains the Hilbert relations, which in physical applications have become known under the name dispersion relations ... 

We now apply these results to compute 1 v(2>) the Fourier transform of Kuv(x), in terms of its imaginary part Im OL p). Causality asserts that J uv(p) is an analytic function of p0 in Imp0 > 0, and hence that there exists a dispersion relation relating the real and imaginary parts of... 

Note that II(p2) approaches a constant asp2 —> oo so that if we wish to calculate Kip) from Im Kip) we must use the once subtracted dispersion relations (10-72). We, furthermore, note that due to the symmetry relation (10-86), and (10-87) we can write the unsubtracted dispersion relation as follows ... 

As 11(a) approaches a constant as a oo the function II(p2) as defined by Eq. (10-97) is logarithmically divergent. The once subtracted dispersion relation is well defined, and reads... 

For a mathematical discussion of the relation between causality and dispersion relations, see ... 

The application of dispersion relations (Chapter 10) to electrodynamic processes has received some attention in recent years. This subject, together with some of the more technical aspects (such as the computation of L, L", Am) not covered in these chapters, will be dealt with elsewhere. 

Calibration. Wavelength calibration to measure the dispersion relation between wavelength and position on the detector requires illumination... 

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 ... 

When both hydrodynamic and thermo-diffusive effects are simultaneously taken into account, it is found that the growth rate a of wrinkling is given by the roots of the dispersion relation [11,12] ... 

Example of dispersion relations calculated from Equation 5.1.5 for six lean propane-air flames. The wave number, k, is nondimensionalized by the flame thickness, S, and the growth rate, cr, is nondimensionalized by the transit time through the flame, r, = S/S. ... 

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