Plane-wave decomposition

A complete set of standard time-harmonic solutions to Maxwell s equations usually involve the plane wave decomposition of the field into transverse electric... 

Plane-wave decomposition of the modal fields 12-13 Multilayered and elliptical fibers... 

Characteristics of leaky modes 24-4 Modal parameters 24-5 Modal fields 24-6 Radiation caustic 24-7 Classification of leaky modes 24-8 Plane-wave decomposition 24-9 Weakly guiding waveguides 24-10 Number of leaky modes... 

Fig. 36-1 Local plane wave decomposition of the modal fields within (a) the core of a step-profile planar waveguide and (b) the core cross-section of a step-profile fiber.

Figure 1 .Graphical decomposition of the plane wave icos9 into clockwise and counterclockwise rotating components. Construction and projection of only the left circulating component i cos 8 + j sin 9 is shown.

Figure 13 Decomposition of a plane wave into spherical waves (a) real part. The plane wave result is shown in the upper right-hand comer, the individual partial wave contributions with a given ( value in the left-hand column, the sum of partial wave contributions up to the value ( = N in the right-hand column. From The picture book of quantum mechanics, S. Brandt and H. D. Dahmen, 1st edition 1985, John Wiley Sons, Inc, 1985 John Wiley and Sons Inc.

Figure 7.3 (contd) Decomposition of a plane wave into spherical waves (b) imaginary part. 

The quantum theory of scattering also known as the method of partial wave analysis provides the frame work for this and we give here the main results of the theory. The basic step in this theory is the decomposition of the incident plane wave into various partial waves each with a well-defined angular momentum ... 

Let us consider first isovalent cations. MgO can dilute ions of similar size, as for instance Ni or Co forming NiO-MgO and CoO-MgO solid solutions with an infinite range of composition. The effect of progressively replacing Mg by Ni or similar cations (Co ", Cu ) on the surface properites has been investigated both experimentally [83,177,178] and theoretically [179,180]. The presence of Ni cations diluted in the MgO matrix results in an efficient catalyst for nitrous oxide, N2O, decomposition this has been attributed to the different bond strength of the Ni-0 and Mg-0 bonds at the surface [177]. Plane wave calculations on Ni-doped MgO have shown that the presence of Ni atoms on the... 

For basis sets derived from a cellular decomposition of space, e.g. the muflSn-tin decomposition used in LAPW and LMTO methods, the difficulty can be sidestepped by constructing the basis in terms of fully relativistic atomic-like functions determined by the miiflfin-tin potential. By restricting the selection to the electron states, the desired decomposition is automatically enforced. This is the essence of the Koelling-Harmon scalar relativistic scheme widely used in cellular-based modified plane-wave basis methods [45]. 

In the following, we restrict our attention to the early stages of spinodal decomposition. In the analysis of experiments one often uses the Landau-de Gennes functional (Eq. 96) which results in the Cahn-HilUard-Cook theory (105] for the early stages of phase separation. This treatment predicts that Fourier modes of the composition independently evolve and increase exponentially in time with a wavevector-dependent rate, 4>A(q, t) exp[it(q)fj. Therefore, it is beneficial to expand the spatial dependence of the composition in our dynamic SCF or EP calculations in a Fourier basis of plane waves. As the linearized theory suggests a decoupling of the Fourier modes at early stages, we can describe our system by a rather small number of Fourier modes. 

Phase multipliers at matrices U(k) need to be chosen so that decomposition coefficients Vj s, k) of functions Wj(r, k) on plane waves ... 

The calculation schemes usually used for point defect include 1. the choice of the model of the defective crystal 2. the choice of the Hamiltonian (Hartree-Fock, DFT or hybrid, semiempirical) 3. the choice of the basis for the one-electron Bloch functions decomposition hnear combination of atomic orbitals (LCAO) or plane waves (PW). 

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