Spherical functions Hankel

Here h, (x) is the Hankel spherical function of the first kind, and the Cs are the Clebsch-Gordan coefficients [30],... 

Spherical Bessel (Hankel) functions of the third kind obey the relations ... 

The problem is not simplified by Eq. (15), since there exists a closed-form expression for the multi-scattering matrix for n spheres in terms of spherical Bessel and Hankel functions, spherical harmonics and 3j-symbols, where l, l and to, m are total angular momentum and z-projection quantum numbers, respectively (Henseler, Wirzba and Guhr, 1997) ... 

The functions PJT(cos 9) are associated Legendre functions of the first kind of degree n and order m, and z (kr) denotes any of four spherical Bessel functions. The choice of the spherical Bessel function depends on the domain of interest, that is, on whether we are looking for the solution inside the sphere (r < a) or outside the sphere (r > a). For the internal field we choose z (kr) = j (kr), where j (kr) is the spherical Bessel function of the first kind of order n. The solution for the external field can be written in terms of spherical Bessel functions j kr) and y kr), where the latter is the spherical Bessel function of the second kind, but it is more convenient to introduce the spherical Hankel function /i / (kr) to determine tj/ for the outer field. 

The superscript (3) signifies that the r-dependent eigenfunction is the spherical Hankel function /ij, (kr). The vectors M and N are called the normal modes of the sphere. 

These results can be put in a more useful and simpler form if kr is sufficiently large to permit asymptotic forms of the spherical Bessel functions and spherical Hankel functions to be applied. In this case the transverse components of the scattered electric vector are... 

Any linear combination of jn and yn is also a solution to (4.5). If the mood were to strike us, therefore, we could just as well take as fundamental solutions to (4.5) any two linearly independent combinations. Two such combinations deserve special attention, the spherical Bessel functions of the third kind (sometimes called spherical Hankel functions) ... 

In the region outside the sphere jn and yn are well behaved therefore, the expansion of the scattered field involves both of these functions. However, it is convenient if we now switch our allegiance to the spherical Hankel functions h[]) and h% We can show that only one of these functions is required by considering the asymptotic expansions of the Hankel functions of order v for large values of p (Watson, 1958, p. 198) ... 

The formalism of electron-atom scattering has been extensively dealt with elsewhere./27,28,29/ We shall only recall its main features here. Because of the assumed spherical symmetry, the partial-wave scattering approach is convenient. Namely, an incoming spherical wave h[2 kr) y/ (r), (—can scatter only into the outgoing spherical wave hf kr) Y/"(r) (here hf and hf2 are Hankel functions of the first and second kinds, k = 2n/h(2mE) i, E is the kinetic energy and r — r ). This occurs with amplitude t( (f is an element of the diagonal atomic <-matrix), which is related to the phase shifts 6l through... 

The solution of these equations is represented by certain combinations of spherical Bessel or Hankel functions and spherical harmonics [24—26]. 

Taking into account the properties of spherical harmonics [70], Clebsch-Gordon coefficients [71], and spherical Bessel and Hankel functions [70], it is possible to show that the mode functions in (18) obey the following condition of symmetry ... 

Our notation here essentially follows Refs. 4 and 5, which in turn closely followed that of Lebowitz, Stell, and Baer and of Stell, Lebowitz, Baer, and Theumann. There are minor differences from paper to paper, however. Our 2 and W here are the 2 and U fVof Ref. 4. In Ref. 6 n W is used to denote what we call IT here, whereas in Ref. 7, W is used to denote our 2. The 13(12) here is the <1> of Refs. 6 and 7. Moreover, the modified two-particle functions we denote here with the subscript S are denoted in the papers above with a caret, and their Fourier transforms carry a bar. (We reserve the caret and bar here to denote certain spherically symmetric functions and Hankel transforms that play a fundamental role in the mathematics of polar fluids.) Finally, in Refs. 4 and 5, p(l), p(12), and F 2) were written as p,(l), P2(12), and Fjfn), respectively. The subscripts are redundant when one exhibits the arguments, so we drop them here. 

There are also spherical analogs of the Hankel functions ... 

They are complemented by the Hankel functions of first and second kind (spherical Bessel function of third and fourth kind)... 

For the general description of wave fields it is important to note that the spherical Bessel functions of first kind j diverges at small arguments and vanishes at infinity, while the opposite applies to the spherical Hankel functions of first kind... 

For the translational function, we choose the spherical Hankel function of the second kind /ip (a ) [45]. properly normalized to give unit incoming flux ... 

The spherical Hankel function of the first kind has been chosen because of its asymptotic behavior as r oo, which is... 

In this appendix we recall the basic properties of the solutions to the Bessel and Legendre differential equations and discuss some computational aspects. Properties of spherical Bessel and Hankel functions and (associated) Legendre functions can be found in [1,40,215,238]. 

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