Bessel

Brit. Pat. Appl. 2,136,013 (Sept. 1984), D. Seddon and S. Bessell (to Broken HiU Pty. Co. Ltd., Commonwealth Scientific and Industrial Research Organization. 

The diffracted amphtude from illuminating such a grating with a unit plane wave normal to the surface is easily calculated again by resolving equation 9 into complex exponentials (as in eq. 10) where is the mUi Bessel function. 

The series converge for all x. Much of the importance of Bessel s equation and Bessel functions lies in the fact that the solutions of numerous linear differential equations can be expressed in terms of them. 

A short table (Table 3-1) of very common Laplace transforms and inverse transforms follows. The references include more detailed tables. NOTE F(/i -1- 1) = Iq x e dx (gamma function) /(f) = Bessel function of the first land of order n. 

Iq = zero-order Bessel function of an imaginary argument. For large u, 7 (t/) -e /V2. Hence for large n,... 

The notation of the Bessel functions is that of Jahnke and Emde Tables of Functions with Formulas and Cuiv/cs, Dover, 1945 Teub-ner, 1960). 

In Eq. (26), M is the hydrogen mass, X labels the mode, is the atomic eigenvector for hydrogen / in mode X, and co, is the mode angular frequency. is the number of quanta of energy Ao>, exchanged between the neutron and mode X. is a modified Bessel function. 

The metal cluster will be modeled as an infinitely deep spherical potential well with the represented by an infinitely high spherical barrier. Let us place this barrier in the center of the spherical cluster to simplify the calculations. The simple Schrodinger equation, containing only the interaction of the electrons with the static potential and the kinetic energy term and neglecting any electron-electron interaction, can then be solved analytically, the solutions for the radial wave functions being linear combinations of spherical Bessel and Neumann functions. 

Fig. 11. Simulated diffraction space of a chiral (40, 5) SWCNT. (a) Normal incidence diffraction pattern with 2mm symmetry (b),(c),(d) and (e) four sections of diffraction space at the levels indicated by arrows. Note the absence of azimuthal dependence of the intensity. The radii of the dark circles are given by the zeros of the sums of Bessel functions [17].

Several sections of the diffraction space of a chiral SWCNT (40, 5) are reproduced in Fig. 11. In Fig. 11(a) the normal incidence pattern is shown note the 2mm symmetry. The sections = constant exhibit bright circles having radii corresponding to the maxima of the Bessel functions in Eq.(7). The absence of azimuthal dependence of the intensity is consistent with the point group symmetry of diffraction space, which reflects the symmetry of direct space i.e. the infinite chiral tube as well as the corresponding diffraction space exhibit a rotation axis of infinite multiplicity parallel to the tube axis. 

Some useful exaet relations ean be obtained for speeifie fluid-wall interaetions. With this aim, we now eonsider a one-dimensional external potential, Va z). First, let us introduee the abbreviation zx,Z2,k), for the Fourier-Bessel transform of a funetion 0(zi,22, 12) where i i2 +... 

Jq denotes the Bessel funetion of the first kind and 0 stands for c p ap-With this notation, Eqs. (10), (14), and (16) ean be rewritten as follows... 

Using the two-dimensional Fourier-Bessel transform, the PYl equation (7) becomes (cf. Refs. 30,31)... 

The funetion Jq x) denotes the Bessel funetion of the first kind. 

A numerical algorithm for the solution of the system of Eqs. (15), (19) and (51) consists of the expansion of the two-particle functions into a Fourier-Bessel series. We omit all the details of the numerical method they can be found in Refs. 55-58, 85, 86. In Fig. 3 we show a comparison of the total... 

In the numerical solution the matrix structure is evaluated from Eqs. (44)-(46). Then Eqs. (47)-(49) with corresponding closure approximations are solved. Details of the solution have been presented in Refs. 32 and 33. Briefly, the numerical algorithm uses an expansion of the two-particle functions into a Fourier-Bessel series. The three-fold integrations are then reduced to sums of one-dimensional integrations. In the case of hard-sphere potentials, the BGY equation contains the delta function due to the derivative of the pair interactions. Therefore, the integrals in Eqs. (48) and (49) are onefold and contain the contact values of the functions... 

We have to calculate the integral /(/c) of Eq.(2) which depends on the spherical Bessel functions and is expressed as ... 

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