The scattering potential

A basic difference between both potentials, VionCf) and is that while the ionic component remains essentially frozen when the ion velocity increases (besides possible changes due to the increased stripping), the outer screening component changes with velocity since the electrons readjust themselves in a dynamical way to the field of the moving ion. 

This property is included in our approach, where the screening component Vs(r) is adjusted for each velocity in order to satisfy the extended sum rule. 

In a previous study [39] various ion models were tested and a new model was proposed, which was called the Moliere-ion potential . Due to its simplicity and good comparison with other models (including Dedkov s model for ions [43] and Moliere and Thomas-Fermi models for neutral atoms [44]) we have used this model throughout this study. 

In this model, the ionic component, Vion(0 = is determined by the Moliere-ion function ( Mi(O (i-e., 0ion — which -following the form of the original Moliere function - is expressed as 

To represent ions with increasing degrees of ionization i = q/Z the coefficients in are modified as follows first the value of the third 

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This equation describes the Fourier transfonn of the scattering potential V r). It should be noted that, in the Bom approximation the scattering amplitude/(0) is a real quantity and the additional phase shift q(9) is zero. For atoms with high atomic number this is no longer tme. For a rigorous discussion on the effects of the different approximations see [2] or [5]. 

Friedel used the local perturbation f/—as a model for the scattering potential, reduced it even to one spherically symmetric scatterer and calculated the scattering by free electrons. By that, for the scatterer at the origin labeled by j = 0,... 

To a first order approximation, the scattering potential of a crystal may be represented as a sum of contributions from isolated atoms, having charge distributions of spherical symmetry around their nuclei. In a real crystal the charge distribution deviates from the spherical symmetry around the nucleus and the difference reflects the charge redistribution or bonding in the crystal. The problem of experimental measurement of crystal bonding is therefore a problem of structure factor refinement, i.e. accurate determination of the difference between the true crystal structure factors... 

Calculations have been performed where this bond distance has been varied over several angstroms in order to find the best fit with experiment. 8.9 These BtudieB have also shown that there iB a sensitivity of the polar angle distribution to the effective size of the adsorbed atom. Thus, it is important to know more about the scattering potential parameters if this distance is to be determined accurately. It appears, however, that the type of adsorption site may be determined in a reasonably straightforward manner. 

The scattering-potential difference between real and effective bonds is a 2x2 matrix... 

Special cases For an isotropic order parameter, t/m+4/ 2 has to be replaced by um and pair breaking from defect scattering is absent, irrespective of the exact form of the scattering potential. For 5-functions scatterers, only uo 0. Then the pair breaking is equally effective whatever the exact form of the dxi-yi order parameter k4e 2 =... 

XANES spectra are more sensitive to the scattering potential than EXAFS ... 

Here P( R ) is the scattering potential for the ensemble of scatters at locations R G is the free-space electron propagator and T0 the f-matrix for multiple scattering of the electron by the surface. 

Fig. 9.2. Lifetime function T e) for the scattering potential shown in Fig. 9.1. (From Bliimel (1993b).)...

When the scattering potential F(r) is not zero, the only change is the radial equation, which becomes... 

The cross sections for scattering of the two beams with opposite polarisation differ from each other because of the spin—orbit part of the scattering potential, which is proportional to the scalar product L S of the... 

It was Fermi who realized that it was possible to invoke an equivalent potential, which can be used to calculate the changes in the wavefunction outside the interaction by perturbation theory [13]. The unknown form of the strong nuclear interaction can be replaced by a new potential, which gives the same scattered wavefunction as the square well potential. In the derivation of Fermi s equivalent or pseudo potential [14] it is seen that the magnitude of the scattering potential depends on the scattering length of the nucleus and the mass of the neutron, m ... 

As indicated before, the non-linear approach to the energy loss of ions in solids is based on the following methods (a) the transport cross section (TCS) method, and (b) the extended Friedel sum rule (EFSR). In addition, particular models will be used to represent the scattering potential by a sum of (i) the ion potential and (ii) the screening potential. The scattering potential will be adjusted in a self-consistent way, for each ion velocity, by the constraint of the EFSR. 

We describe now the basic ingredients used in this approach, and in Section 4 we consider the determination of the scattering potential. 

Figure 7.Decomposition of continuum MS-Xa cross section by symmetry of continuum electron for the scattering potential given by occupation of the 3t2 orbital (eg and channels give aOOA and are not shown)/ (from ref. 9).

We invoke the Bom approximation, which views the scattering potential, V(r), as so weak a perturbation that, for incident, i, and final, f, conditions ... 

There are four terms in Eq. (A2.26), first is the ratio of the incident and final neutron momenta. The second term groups the fundamental constants and the final term ensures that the difference between the incident and final neutron energies equals the difference between quantised energy states of the system (or zero for elastic scattering). The third term describes how the initial states are related to the final states through the scattering potential, V(r). 

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