Integral and Fourier representation

The solution procedure of the linearized Poisson-Boltzmann equation used above is not suited to include hard core effects of the ions, the most we can do is to give a size to the central ion, but that makes the pair distribution function asymmetric. To include the hard core effects in a symmetric way, we have to change the formalism. We notice, first, that Poisson s equation (1.8) relates the potential (r) to the charge distribution We can formally integrate this equation to yield  

We notice that if we multiply (6.7) by we get Poisson s equation for a point charge, the charge density being j,(r) =, ed(r). 

We would like to separate the contribution to the potential due to the central 

Equation (6.14) can be derived for a much more general class of distributions and is know under the name of Ornstein-Zernike (OZ) equation. 

In the discussion of the solution of the OZ equation it will be necessary to unify both descriptions of the Poisson equation this can be achieved by using the FT technique. Our discussion of the FT will also serve as an introduction to the mathematical techniques used in solving the MSA, 

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