Bessel function potentials

ANG AO ATA BF CB CF CNDO CPA DBA DOS FL GF HFA LDOS LMTO MO NN TBA VB VCA WSL Anderson-Newns-Grimley atomic orbital average t-matrix approximation Bessel function conduction band continued fraction complete neglect of differential overlap coherent-potential approximation disordered binary alloy density of states Fermi level Green function Flartree-Fock approximation local density of states linear muffin-tin orbital molecular orbital nearest neighbour tight-binding approximation valence band virtual crystal approximation Wannier-Stark ladder... 

As an example of the use of Bessel functions in potential theory we shall consider the problem of determining a function ip[g, z) lor the half-space n Si 0, z 0 satisfying the differential equation... 

The solution to this problem is to transform, or half-transform, the S matrix from the body-fixed to the space-fixed axis system then to use the known analytic properties of the spherical Bessel functions, which are the solutions to the potential-free scattering problem in the space-fixed axes and finally to transform back to the body-fixed axes and then to use Eq. (4.46) to calculate the differential cross section. 

TABLE 11.2 Bessel Functions K0(kx) and Kx(kx) for Computing Potentials of Long Cylinders3... 

The modification of the radial functions is obvious because the atomic potential V(r) will modify the spherical Bessel functions j/jcr) which belong to a free plane wave. Also, the dependence on products of k and r is lost. The RK/r) functions follow as regular solutions from the time-independent Schrodinger equation ... 

The effect of a non-vanishing potential on the radial function is illustrated in Fig. 7.4 for the example of a repulsive potential of rectangular shape. It can be seen that the selected s-wave radial function RK0(r) strongly differs in the region of potential from the corresponding spherical Bessel function j0(fcr). However, far away from the influence of the potential, the function Rk0(t) behaves like the asymptotic spherical Bessel function j0(Kr), except that it is shifted in phase. A repulsive potential pushes out, and an attractive potential pulls in the radial functions RKf(r) as compared to j/xr). This behaviour is expressed in the asymptotic forms of these radial functions (for the general case with ( and an attractive potential) ... 

Figure 7.4 Definition of the phase shift A as introduced by a potential. The solution of the radial function RKAr) of a wave with energy e = k2/2 (in atomic units) and with ( = 0 is shown for two situations under the influence of a repulsive potential V(r) as indicated by the shaded region (top), and for vanishing potential (bottom). In the first case one has RK((r) = FK0(r), and in the second case the radial function is equal to the spherical Bessel function, i.e., RKAr) = j0(fcr). Asymptotically, both solutions, FK0(r) and j0(Kr), differ only by a constant distance A in the r coordinate which is related to the phase shift A( as indicated. From The picture book of quantum mechanics, S. Brandt and H. D. Dahmen, 1st edition, 1985, John Wiley Sons Inc., NY. 1985 John Wiley Sons Inc.

In Eq. (6) the modified spherical Bessel functions of third kind k,(z)10 are used. The total potential in the surrounding media can be written as follows... 

The modified Bessel function, 70, is always positive increasing, the free energy is therefore always negative. That is, the free energy of interacting surfaces is always attractive irrespective of whether the surfaces are held at constant charge or constant potential. 

This series arises naturally, when expressing the Coulomb potential of a charge separated by a distance s from the origin in terms of spherical coordinates. The positive powers result when r < s, while for r > s the potential is described by the negative powers. Similarly the solutions of the linearized Poisson-Boltzmann equation are generated by the analogous expansion of the shielded Coulomb potential exp[fix]/r of a non-centered point charge. Now the expansion for r > s involves the modified spherical Bessel-functions fo (x), while lor r < s the functions are the same as for the unshielded Coulomb potential,... 

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