Wave function scattering radial

This transformed representation of the asymptotic wave functions can easily be verified for potential scattering. The asymptotic radial wave function in a given i-channel satisfies the identity... 

In order to test such an application we have calculated the spin and charge structure factors from a theoretical wave function of the iron(III)hexaaquo ion by Newton and coworkers ( ). This wave function is of double zeta quality and assumes a frozen core. Since the distribution of the a and the B electrons over the components of the split basis set is different, the calculation goes beyond the RHF approximation. A crystal was simulated by placing the complex ion in a lOxIOxlOA cubic unit cell. Atomic scattering factors appropriate for the radial dependence of the Gaussian basis set were calculated and used in the analysis. 

Fig. 4.3. Surfaces of equal phase (wave fronts) at intervals of n in the wave function for the elastic scattering of 200 eV electrons by the static potential of argon, plotted on a plane through the scattering axis. The radial scale is marked in a.u.

S-wave scattering is the only practical outcome since P-wave final neutron states are not accessible to thermal neutrons, because these wave functions have negligible amplitude at the small radial values that are typical of atomic nuclei. It is convenient to rewrite the equation as a dynamical structure factor (or Scattering Law), which emphasises the dynamics of the sample. 

For Eq. (11) S is the Bragg vector S = 2ttH, IT is the row vector (htk,l) and the scalar S - S = 4ir sin 0/A. The index / covers the N atoms in the unit cell. The atomic scattering factor f (S) is the Fourier-Bessel transform of the electronic, radial density function of the isolated atom. This density function is usually derived from a spin-restricted Hartree-Fock wave function for the atom in its ground state. The structure fac-... 

The coefficients Cm are determined from the boundary conditions which arises from the following considerations. A collimated beam of non-interacting particles may be represented by a plane wave (r) = elkz where z is the beam direction. If target molecules are present and some of the primary beam particles interact with them the scattered beam is represented by a radially outgoing wave,/(0) e kr/r where the angular coefficient,/(0), which is also a function of k, is the scattering amplitude. Since the distance of the particle detector from the point of interaction is effectively infinite compared to atomic dimensions the wave functions at the detector must be of the form... 

For elastic scattering the only role played by the potential is to shift the phase of the wave function. The magnitude of this shift is computed by solving for the radial wave function and determining S i from its definition by using the asymptotic form, Eq. (4.36). 

Where, /(k) is the sum over N back-scattering atoms i, where fi is the scattering amplitude term characteristic of the atom, cT is the Debye-Waller factor associated with the vibration of the atoms, r is the distance from the absorbing atom, X is the mean free path of the photoelectron, and is the phase shift of the spherical wave as it scatters from the back-scattering atoms. By talcing the Fourier transform of the amplitude of the fine structure (that is, X( )> real-space radial distribution function of the back-scattering atoms around the absorbing atom is produced. 

The conduction electrons are scattered by the alkali atoms, the coherence implicit in the radial distribution function. Unlike the case of the scattering of a single electron in a plane wave state by a liquid, discussed previously, in this case the structure factor S(k) must be known up to the Fermi energy (which is 0.5 e.v. — 1 e.v. in saturated metal ammonia solutions). 

Scattered waves from neighbouring atoms interfere in exactly the same way and unless the atoms are ordered as in a crystal, the total diffraction pattern is a function of the radial distribution of scattering density (atoms) only. This is the mechanism whereby diffraction patterns arise during gas-phase electron diffraction, scattering by amorphous materials, and diffraction... 

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