Quasi-plane waves

The quasi-plane waves have been introduced by Devaney [46], and for z > 0, they are defined as... 

As in (B.3), the expansion of quasi-plane waves in terms of radiating spherical wave functions is given by... 

Inserting the spherical wave expansion of the quasi-plane wave Q k,/3, a, ro) into (B.61) yields... 

L.W. Davis, Theory of electromagnetic beams, Phys. Rev. 19, 1177 (1979) A.J. Devaney, Quasi-plane waves and their use in radiation and scattering problems, Opt. Commun. 35, 1 (1980)... 

At this point it may seem as though we can conclude our discussion of optimization methods since we have defined an approach (Newton s method) that will rapidly converge to optimal solutions of multidimensional problems. Unfortunately, Newton s method simply cannot be applied to the DFT problem we set ourselves at the beginning of this section To apply Newton s method to minimize the total energy of a set of atoms in a supercell, E(x), requires calculating the matrix of second derivatives of the form SP E/dxi dxj. Unfortunately, it is very difficult to directly evaluate second derivatives of energy within plane-wave DFT, and most codes do not attempt to perform these calculations. The problem here is not just that Newton s method is numerically inefficient—it just is not practically feasible to evaluate the functions we need to use this method. As a result, we have to look for other approaches to minimize E(x). We will briefly discuss the two numerical methods that are most commonly used for this problem quasi-Newton and... 

For each section, the quantity p(w/k)2 is given for the pure shear wave, which is polarized perpendicular to the plane of the section, and for the quasi-shear and quasi-longitudinal waves, which each have particle motion in the plane of the section. The angle between the wavevector k and the lowest symmetry direction in the plane is denoted by 0. 

Note that expression (14.64) resembles formula (14.53) for quasi-linear approximation. However, the reflection coefficient A now has a much clearer physical meaning. It has been demonstrated by Bleistein et al. (2001), that to leading order (high frequency asymptotics), this reflection coefficient is the same as that derived for plane waves reflected by planar interfaces in the media with a piecewise-constant distribution of the parameters. Note also that to leading order in the frequency w, the... 

A much better choice is to use the length of s to define the spin density . In the non-relativistic limit, the absolute value of the spin density of a one-electron system equals the charge density. While one cannot exactly retain this property in the relativistic case because of the small component contribution (except for single-particle plane waves, where s can be parallel to the momentum everywhere), the length of s equals the charge density for one-electron systems in the weakly relativistic limit and in two-component quasi-relativistic approaches. The same holds if there is one electron outside a closed-shell core. The non-collinear approach is not too difficult to implement [27], it generates a spin-dependent exchange-correlation potential of the form... 

Chemical bonding in several transition metal carbides was theoretically investigated by quasi self consistent augmented-plane-wave (APW) calculations [71-73] and by the extended Hiickel method [74]. These calculation indicated a charge transfer from the early transition metal to the carbon atoms. A crystal orbital overlap population analysis (COOP) revealed strong bonding T—T and T-C... 

The first parameter appears as a result of quasi-viscous stresses in the dispersed phase affecting the development of initial plane waves. In fact, this parameter characterizes an influence on fluidized bed stability caused by dispersed phase viscosity. The occurrence of the second parameter is due to the restriction imposed from below on permissible wave numbers for these plane waves. Actually, the second parameter descibes a so-called scaling effect of the bed dimensions on bed stability. The curves in Figure 4 correspond to the Carnahan-Starling model, save for the dotted ones which have been drawn when using Equation 4.8 to represent the osmotic pressure correction function and the Enskog factor. 

For the special case of a quasi-monochromatic plane wave E = Eq X exp(icof — it r), an optical path difference (r2 —ri) causes a corresponding phase difference... 

One of the best known applications of pseudopotential theory, is in the theory of simple metals. The basis for its success lies in the large freedom in the form of the pseudo wave function. In simple metals, the electrons behave quasi free, and one might hope that the pseudo wave functions can be fairly well described by plane waves, upon which the pseudopotential acts as a perturbation. This possibility is a direct consequence of the smoothness of the pseudo wave function, in contrast to the true wave function, for which... 

We now consider a simple application for the scattering formalism introduced above. We consider a dielectric or a metal nanosphere in the vacuum, centered at the fq = 0 and excited by a a plane-wave with Sine given by Eq. (1.282). If the sphere is very small R -C A.) we have the so called Rayleigh scattering and we can use the quasi-static solution of Sec. 1.4.1. In particular we can assume that sphere can be modeled by a dipole... 

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