Hankel functions

Calculations were done with a full-potential version of the LMTO method with nonoverlapping spheres. The contributions from the interstitial region were accounted for by expanding the products of Hankel functions in a series of atom-ce- -ered Hankels of three different kinetic energies. The corrected tetrahedron method was used for Brillouin zone integration. Electronic exchange and correlation contributions to the total energy were obtained from the local-density functional calculated by Ceperley and Alder " and parametrized by Vosko, Wilk, and Nusair. ... 

Each Hankel function product tends exponentially fast to zero on the upper half-circle.Therefore the contribution of the integral over Cm tends to 0 when M +oo. 

F(x) can readily be evaluated in terms of Hankel functions. The salient property of F(x) is that it decreases exponentially for x ft/wc. 

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) ... 

Figure 12.3b shows the widths of the Bragg layers as a function of the layer number of the CBNL depicted in Fig. 12.3a. There are two notable properties of the Bragg layers (1) the width of the high-index layers is smaller than the width of the low-index layers, and (2) the width of the layers decreases exponentially as a function of the radius, converging asymptotically to a constant value. The first property exists in conventional DBRs as well and stems from the dependence of the spatial oscillation period, or the wavelength, on the index of refraction. The second property is unique to the cylindrical geometry and arises from the nonperiodic nature of the solutions of the wave equation (Bessel or Hankel functions) in this geometry.

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... 

For a short-range potential added to a Coulomb potential, such as we might have in a one-electron model of an atom, the external Ricatti-Hankel function of (4.22) is replaced by a solution of the bound Coulomb problem, discussed in the next section. 

Here, // refers to the Hankel function of the second kind and prime denotes differentiation with respect to the argument. The values of and are given by... 

The order I of the Riccati-Hankel functions hjgiven in Eq. (26) can be defined from the coupling matrix in Eq. (25) as... 

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 ... 

Here p = /E( 1 + E/c2) is the momentum, the functions h (pr) are the relativistic Hankel functions of the first and second kind (Rose 1961) and [ ]r denotes the relativistic form of the Wronskian evaluated at r outside the potential well. Finally, the single-site r-matrix t(E) is obtained from the expression (Ebert and Gyorffy 1988) ... 

In the outer region the wave field is a superposition of the incident wave (1) and the scattered wave. The latter is described by special cylindrical functions (Hankel functions). The cylindrical functions of another type (Bessel functions) also describe the wave motion in the inner region. The conditions at the boundary line between two-dimensional phases allow us to sew together the solutions of the hydrodynamic equations in the inner and outer regions. The wave motion in the transitional region can be rather complicated. However, if we are not interested in the details of the liquid dynamics in the transitional region, we can continue the solutions, which were obtained at a distance from this region, up to the boundary line. 

The / and g functions can in their turn be expanded in terms of Bessel and Neumann/Hankel functions [41] as... 

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