Hankel

The LMTO method [58, 79] can be considered to be the linear version of the KKR teclmique. According to official LMTO historians, the method has now reached its third generation [79] the first starting with Andersen in 1975 [58], the second connnonly known as TB-LMTO. In the LMTO approach, the wavefimction is expanded in a basis of so-called muffin-tin orbitals. These orbitals are adapted to the potential by constmcting them from solutions of the radial Scln-ddinger equation so as to fomi a minimal basis set. Interstitial properties are represented by Hankel fiinctions, which means that, in contrast to the LAPW teclmique, the orbitals are localized in real space. The small basis set makes the method fast computationally, yet at the same time it restricts the accuracy. The localization of the basis fiinctions diminishes the quality of the description of the wavefimction in die interstitial region. 

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. 

This last integral is a special case of a more general integral which has before been found useful in quantum mechanical problems1). It was shown by Hankel and Gegenbauer that... 

Johann (John) von Neumann, American mathematician (1903-1957). f Hermann Hankel, Gentian mathematician (1839-1873). 

Hamilton, William Rowan 130n Hankel, Hermann 114n Heaviside, Oliver 63n Heisenberg, Werner 146n Helmholtz, Hermann von 38n Hermite, Charies L02n Hooke, Robert 90n Httckei, Erich 316n... 

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... 

I. is the identity matrix and z is defined by z (k-v)AT), is determined. In the second step, this model is transformed into a discrete-time state space model. This is achieved by making an approximate realization of the markov parameters (the impulse responses) of the ARX model ( ). The order of the state space model is determined by an evaluation of the singular values of the Hankel matrix (12.). 

I would like to thank my many colleagues and friends who have collaborated with me over many years for their essential part in developing the methods outlined in this chapter. In particular 1 thank Alex Brown, Richard N. Dixon, Laszlo Fusti-Molnar, Simon C. Givertz, Stephen K. Gray, MarUes Hankel, Clay C. Marston, Alison Offer, and Moshe Shapiro, for their long and close collaboration. 

There are two main methods [145] for the reconstruction of n( p) from the directional Compton profile. In the Fourier-Hankel method [145,162], the directional Compton profile is expanded as... 

The radial momental Uni p) is related to e radial orbital r fnt r) by a Hankel... 

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 crystals of potassium and sodium sulphates exhibit triboluminesccnce and crystalloluminescence.28 According to G. Meslin, the crystals of both salts are diamagnetic.29 W. G. Hankel has studied the pyroelectrical phenomenon exhibited by the crystals. 

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