Long-wavelength dispersion equation

The Bragg equation for a fee crystal as assumed in the photonic crystal based on polystyrene nanoparticles [154] is expressed in Eq. (82). The spacing between (111) planes in the fee crystal is related to dm. The effective refractive index ( eff) was obtained by considering the long-wavelength limit of the photon-dispersion relation co(k), Eq. (83), where c is the speed of light. 

Under the long wavelength and quasistationary approximations and with the use of the linearized forms of the hydrodynamic and thermodynamic boundary conditions, first, we solve the Orr-Sommerfeld equation for the amplitude of perturbed part of the stream function from the Navier-Stokes equations. Second, we solve the equation for the amplitude of perturbed part of the temperature in the liquid film. The dispersion relation for the fluctuation of the solid-liquid interface is determined by the use of these solutions. From the real and imaginary part of this dispersion relation, we obtain the amplification rate cr and the phase velocity =-(7jk as follows ... 

The elastic constants determine the acoustic phonon dispersion for long wavelengths according to the equations of motion in the continuum limit ... 

This shows the anticipated fact that the instability of the uniform oscillation to long wavelength fluctuations corresponds precisely to the negative sign of the phase diffusion constant. The equality = —y may also be confirmed, where y is the quantity which appeared in (4.2.36) and is the abbreviation of —a> defined in (4.2.35). More generally, it is possible to prove that the dispersion curve of the phaselike branch has an exact correspondence to the linearized form of the phase diffusion equation (4.2.36), or one may possibly have... 

Here, the simple dispersion (oiq) v g was used for the acoustic phonons (if whole molecules vibrate with respect to each other, like the change of the stacking distance in a stack one speaks of acoustic phonons, because they have a long wavelength comparable to those of acoustic waves) and the sound velocity v, is determined by the relation o, = Ci/p), where c, is the longitudinal elastic constant and p is the mass density. One should remark that this very simple linear dispersion relation (o q) = v,g is not necessarily correct. With the help of the FG method described in Section 9.1 one can obtain more accurate dispersion curves. Equation (9.48) can now be used to calculate the charge carrier mobilities and free paths, defined in this case hy p= e xlm ) and A = (t ), respectively, where... 

A more detailed analysis of the dispersion of and An has been made [70, 73]. Whereby not only the long-wave electronic band but also two further bands are involved in the dispersion equation. Besides a cr—>CT transition in the vacuum UV region (at lq) two 7t n transitions at longer wavelengths A and A2 (UV or visible) are considered. In this way more exact... 

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