The spherical atom kappa formalism

A simple modification of the IAM model, referred to as the K-formalism, makes it possible to allow for charge transfer between atoms. By separating the scattering of the valence electrons from that of the inner shells, it becomes possible to adjust the population and radial dependence of the valence shell. In practice, two charge-density variables, P , the valence shell population parameter, and k, a parameter which allows expansion and contraction of the valence shell, are added to the conventional parameters of structure analysis (Coppens et al. 1979). For consistency, Pv and k must be introduced simultaneously, as a change in the number of electrons affects the electron-electron repulsions, and therefore the radial dependence of the electron distribution (Coulson 1961). 

The parameter k scales the radial coordinate r when k 1, the same density is obtained at a smaller value of r, and the valence shell is therefore contracted. Conversely, for k 1, the valence shell is expanded. The k3 factor satisfies the normalization requirement 

It is assumed in Eq. (3.16) that the inner or core electrons are not perturbed. There is abundant support for this approximation (Bentley and Stewart 1974), 

We note that Eq. (3.19) again illustrates the inverse relation between direct and scattering space, a contraction of charge density corresponding to an expansion in scattering space, and vice versa. Equation (3.19) implies that the /c-modified scattering factor can be obtained directly from the unperturbed IAM scattering factors tabulated in the literature. 

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