Bessel terms

That T is a series of Bessel rather than trigonometric functions is merely a consequence of using cylindrical polar coordinates (r, j, cz ) for atoms in real space and (R, iji, i/a for points in reciprocal space. Not only is this a convenient framework for describing a helical molecule, but it can lead to economies in computing T. For helices, only Bessel terms with... 

What (xvii) shows is that the Bessel function terms provide single-crystal-like diffraction and all other Bessel terms continuous intensity. Although this kind of array is not the only one for which Bragg and continuous intensity both occur in the same pattern, it is mainly for this that the unfortunate term "semi-crystalline" has sometimes been used. Examples are C-DNA (9,10), ribosomal RNA fragments (11). Fig. 4 shows an example of the diffraction from such an array of molecules. The helices in this case are 12-fold therefore has large values only for small values of Si. It follows that the Bragg diffractid is confined to the center of the pattern. 

For the near formula, one has separate error estimates for the sum of Bessel terms, the complex sum, and the polygamma sum. For the Bessel smn. 

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. 

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. 

The analytical solution of this equation Is Known (9) (10) In terms of modified Bessel functions of the first kind. AccorxUngly, the dlstrltutlon of the active chains In the particles with volume V, fn(v)/f(v), and the average nunher of active chains In the same — 00... 

Using the valence profiles of the 10 measured directions per sample it is now possible to reconstruct as a first step the Ml three-dimensional momentum space density. According to the Fourier Bessel method [8] one starts with the calculation of the Fourier transform of the Compton profiles which is the reciprocal form factor B(z) in the direction of the scattering vector q. The Ml B(r) function is then expanded in terms of cubic lattice harmonics up to the 12th order, which is to take into account the first 6 terms in the series expansion. These expansion coefficients can be determined by a least square fit to the 10 experimental B(z) curves. Then the inverse Fourier transform of the expanded B(r) function corresponds to a series expansion of the momentum density, whose coefficients can be calculated from the coefficients of the B(r) expansion. 

This expression can be put in terms of modified Bessel functions. Applying the method of steepest descent to Eq. (3.52) yields... 

Here, a represents the set of all combinations of 1,2, possessing w elements and Kv(x) is the Bessel function of the third kind. This is an analytical extension of the multivariable Epstein-Hurwitz eta-function to the whole complex /x-plane (A.P.C. Malbouisson et.al., 2002). The first term in Eq. (71) leads to a contribution for Ed which is divergent for even dimensions D >2 due to the pole of the T-function. We renormalize Ed by subtracting this contribution, corresponding to a finite renormalization when D is odd. 

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) ... 

Therefore the Casimir energy for the two spherical cavities inside a non-relativistic non-interacting fermion background can be approximated in terms of a spherical Bessel function j as... 

The procedure described in Example 8-4 may be used to obtain analytical solutions for concentration profiles and tj for other shapes of particles, such as spherical and cylindrical shapes indicated in Figure 8.9. Spherical shape is explored in problem 8-13. The solution for a cylinder is more cumbersome, requiring a series solution in terms of certain Bessel junctions, details of which we omit here. The results for the dimensionless... 

Sometimes an equation out of this classification can be altered to fit by change of variable. The equations with separable variables are solved with a table of integrals or by numerical means. Higher order linear equations with constant coefficients are solvable with the aid of Laplace Transforms. Some complex equations may be solvable by series expansions or in terms of higher functions, for instance the Bessel equation encountered in problem P7.02.07, or the equations of problem P2.02.17. In most cases a numerical solution Is possible. 

Equation (8.7) may be written more explicitly in terms of spherical Bessel functions ... 

Here, Jo(x) is the Bessel function of zero order, r7- kf-ki is the momentum transfer, which depends on the scattering angle 6, and 6 b) is the semiclassical phase shift, which is given in terms of the deflection function as b) = dd b)Hk , db. 

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. 

Denoting p = r — ro, the Green s function can be written in terms of a spherical modified Bessel function of the second kind. 

Similarly, for the sample wavefunction, up to the lowest significant term in the power expansion of the spherical modified Bessel function of the first kind, ,(kp),... 

Vp(fO is peaked at the surface. Many collective oscillations manifest themselves as predominantly surface modes. As a result, already one separable term generating by (74) usually delivers a quite good description of collective excitations like plasmons in atomic clusters and giant resonances in atomic nuclei. The detailed distributions depends on a subtle interplay of surface and volume vibrations. This can be resolved by taking into account the nuclear interior. For this aim, the radial parts with larger powers and spherical Bessel functions can be used, much similar as in the local RPA [24]. This results in the shift of the maxima of the operators (If), (12) and (65) to the interior. Exploring different conceivable combinations, one may found a most efficient set of the initial operators. 

This is called a point-spread function, because it describes how what should be a point focus by geometrical optics is spread out by diffraction. The expression in the curly brackets is the one that is of interest. The other terms are phase and overall amplitude terms, as are usual with Fraunhofer diffraction expressions. The function Ji is a Bessel function of the first kind of order one, whose values can be looked up in mathematical tables. 2Ji(x)/x, the function in the curly brackets, is known as jinc(x). It is the axially symmetric equivalent of the more familiar sinc(x) = sin(x)/x (Hecht 2002), the diffraction pattern of a single slit, usually plotted in its squared form to represent intensity. Just as sinc(x) has a large central maximum, and then a series of zeros, so does jinc(x). Ji(x) = 0, but by L Hospital s rule the value of Ji(x)/x is then the ratio of the gradients, and jinc(0) = 1. The next zero in Ji(x) occurs when x = 3.832, and so that gives the first zero in jinc(x). This occurs at r = (3.832/n) x (q/2a)Xo in (3.2), which is the origin of the numerical factor in (3.1). 

In the usual texts a multipole expansion involving spherical Bessel functions and spherical vector harmonics is also introduced [16,23,23,26]. The fields from electric and magnetic dipoles correspond to the lowest-order terms ( =1) in the expansion. If we define dipole by this expansion then our toroidal antenna is an electric dipole. In any event, the fields away from the source are the same. This is perhaps a matter of consistency in definitions. 

This is the mass transfer analog of the L v6que problem in heat transfer (S9, p. 81), and the solution is expressible in terms of Bessel functions (C2, p. 175) ... 

Bamford and Tompa (93) considered the effects of branching on MWD in batch polymerizations, using Laplace Transforms to obtain analytical solutions in terms of modified Bessel functions of the first kind for a reaction scheme restricted to termination by disproportionation and mono-radicals. They also used another procedure which was to set up equations for the moments of the distribution that could be solved numerically the MWD was approximated as a sum of a number of Laguerre functions, the coefficients of which could be obtained from the moments. In some cases as many as 10 moments had to be computed in order to obtain a satisfactory representation of the MWD. The assumption that the distribution function decreases exponentially for large DP is built into this method this would not be true of the Beasley distribution (7.3), for instance. 

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