Spheroidal wave function

This property is well-known for prolate spheroidal wave functions (the basic SVD functions in Fresnel or far-field approximation [2,7]), but, as it was shown [8], the double-orthogonality property is quite common for different physical Green fimctions. This property can be used for simple estimation of the noise (or stray light) impact on resolution enhancement [9]. 

The eigenfunctions of the free electron confined in the same prolate spheroids are expressed as products of regular radial and angular spheroidal wave functions [16] Chapter 21, in the respective coordinates u and v, and the eigenfunctions of Equations (34) and (35). The radial functions are expressed as infinite series of spherical Bessel functions of order m + s and argument kfu. Its eigenvalues are determined by the boundary condition on the radial factor,... 

C. Flammer, Spheroidal Wave Functions, Stanford Univ. 

In terms of the Spheroidal wave function, the order parameter may be re-expressed as... 

If the Spheroidal wave function Spo(cos9) is expanded in terms of the Legendre polynomials as... 

L.-W. Li, X.-K. Kang, M.-S. Leong, Spheroidal Wave Functions in Electromagnetic Theory (Wiley, New York 2002)... 

The gamma functions Ak(p) and Bj(pt) may be obtained by the use of recursion formulas an extensive tabulation is due to Flodmark (141). In the case of Slater orbitals of principal quantum number 4 or 6, application of Slater s rules leads to nonintegral powers of r in the radial wave function consequently, changing to spheroidal coordinates introduces A and B functions of nonintegral k values, that is, incomplete gamma functions. These functions can, however, be computed (56, 57) and the overlap... 

The confining boundary is defined by a fixed value of u = mo and a chosen value of / in Equation (76). The wave functions have the form of Equation (89) and must satisfy the boundary condition of vanishing at the position of the confining spheroid. This is equivalent to the vanishing of the factor in Equation (85) ... 

The free electron confined by a hyperboloid has the same type of wave function described in the paragraph of Equation (116). The difference consists in the boundary condition applied to the angular spheroidal functions, which are expressed as infinite series of associated Legendre polynomials of order m + r, degree m, and argument v ... 

Clemenger s model is, however, non-selfconsistent. Ekardt and Penzar [41,4.3] have extended the jellium mode to account for spheroidal deformations. In this model the ionic background is represented by a distribution of positive charge with constant density and a distorted, spheroidal, shape. The advantage with respect to Clemenger s model is that the spheroidal jellium model is parameter-free and that the calculation of the electronic wave functions is performed self-consistently using the density functional formalism. The distortion parameter is determined by solving the Kohn-Sham equations for different... 

In the case of spheroidal nuclei the a-particle formation may depend on the direction (in a body-fixed coordinate system). Furthermore, the noncentral electrostatic field of the deformed daughter nucleus may interact with the emitted a particle, causing an increase or decrease of its energy. These problems were discussed by several authors (Hyde et al. 1964). On the basis of these works, one can conclude that the a-particle wave function on the nuclear surface maybe quite different for different nuclei. Furthermore, the electric quadrupole field of the strongly deformed daughter nucleus plays an important role in the formation of the intensity of different a-particle groups. 

The introduction of 8 makes the external potential axial symmetric. Consequently, the spherical symmetry of the wave functions is replaced with a spheroidal symmetry. As before, the radius R is determined by the particle number N and the Wigner-Seitz radius rs as follows ... 

The Hamiltonian and the coordinates are discretized by means of the generalized pseudospectral (GPS) method in prolate spheroidal coordinates [44-47], allowing optimal and nonuniform spatial grid distribution and accurate solution of the wave functions. The time-dependent Kohn-Sham Equation 3.5 can be solved accurately and efficiently by means of the split-operator method in the energy representation with spectral expansion of the propagator matrices [44-46,48]. We employ the following split operator, second-order short-time propagation formula [40] ... 

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]. 

In Figs. 3.82 and 3.83, we show calculations of Re ifg and Im Jfs as functions of the size parameter x = kg max a, b for oblate and prolate spheroids. The spheroids are assumed to be oriented with their axis of symmetry along the Z-axis or to be randomly oriented. Since the incident wave is also assumed to propagate along the Z-axis, the medium is not anisotropic and is characterized by a single wave number Kg. The axial ratios are a/b = 0.66 for oblate spheroids and a/b =1.5 for prolate spheroids. As before, the fractional... 

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