Vector spherical wave functions distributed

In the following analysis we summarize the basic concepts of the null-field method with distributed sources. The distributed vector spherical wave functions are defined as... 

The use of distributed vector spherical wave functions is most effective for axisymmetric particles because, in this case, the T matrix is diagonal with respect to the azimuthal indices. For elongated particles, the sources are distributed on the axis of rotation, while for flattened particles, the sources are distributed in the complex plane (which is the dual of the symmetry plane). 

The expressions of the distributed vector spherical wave functions with the origins located in the complex plane are given by (B.31) and (B.32). 

For a two-layered particle as shown in Fig. 2.2, the null-field equations formulated in terms of distributed vector spherical wave functions (compare to (2.70), (2.71) and (2.72))... 

The expressions of the elements of the Q matrix are given by (2.84)-(2.87) with the localized vector spherical wave functions replaced by the distributed vector spherical wave functions. The transition matrix is... 

The use of distributed vector spherical wave functions improves the numerical stability of the null-field method for highly elongated and flattened layered particles. Although the above formalism is valid for nonaxisymmetric particles, the method is most effective for axisymmetric particles, in which case the 2 -axis of the particle coordinate system is the axis of rotation. Applications of the null-field method with distributed sources to axisymmetric layered spheroids with large aspect ratios have been given by Doicu and Wriedt [50]. 

For a composite particle with two constituents, the null-field equations formulated in terms of distributed vector spherical wave functions... 

In addition to the MMP code, the DSM (discrete sources method) code developed by Eremin and Orlov [61] will be used for computer simulations. This DSM code is devoted to the analysis of homogeneous, axisymmetric particles using distributed vector spherical wave functions. For highly elongated particles, the sources are distributed along the axis of symmetry of the particle, while for highly flattened particles, the sources are distributed in the complex plane (see Appendix B). 

The above system of vector functions is also known as the system of localized vector spherical wave functions. Another system of vector functions which is suitable for analyzing axisymmetric particles with extreme geometries is the system of distributed vector spherical wave functions [49]. For an axisymmetric particle with the axis of rotation along the z-axis, the distributed vector spherical wave functions are defined as... 

In the same manner we can prove the completeness and linear independence of the system of distributed vector spherical wave functions and... 

If the structure of the helium atom were exactly described by the symbol la9 and that of neon by 1 a22a 2p these atoms would have spherically symmetrical electron distributions.24 However, the mutual repulsion of the two electrons in the atom causes them to avoid one another the wave function for the atom corresponds to a larger probability for the two electrons to be on opposite sides of the nucleus than on the same side (for the same values of the distances of the two electrons from the nucleus, there is greater probability that the angle described at the nucleus by the vectors to the electrons is greater than 90° than that it is less than 90°). This effect, which is called correla-... 

The delta function corresponds to Einstein s equation, which says that the kinetic energy of the emitted electron Ef equals the difference of the photon energy h(a and the energy level of the initial state of the sample, The final state is a plane wave with wave vector k, which represents the electrons emitted in the direction of k. Apparently, the dependence of the matrix element 1 j) on the direction of the exit electron, k, contains information about the angular distribution of the initial state on the sample. For semiconductors and d band metals, the surface states are linear combinations of atomic orbitals. By expressing the atomic orbital in terms of spherical harmonics (Appendix A),... 

It is noted that the phonon wavefunction is a superposition of plane waves with q vectors centered at In the literature, several weighting functions such as Gaussian functions, sine, and exponential functions have been extensively used to describe the confinement functions. The choice of type of weighting function depends upon the material property of nanoparticles. Here, we present a brief review about calculated Raman spectra of spherical nanoparticle of diameter D based on these three confinement functions. In an effort to describe the realistic Raman spectrum more properly, particle size distribution is taken into account. Then the Raman intensity 7(co) can be calculated by ... 

The vector q is the difference between vectorial wave numbers of incident and scattered rays such that q = q, as defined above. The nonsubscript i has the usual significance of (-1) and F(ry) is the probability distribution function (probability density) for the vector r,y. Since the molecules in solution exhibit no preferential orientation in space, F(rji) is spherically symmetric, and P(q) assumes the form (7)... 

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