Screening the Thomas-Fermi approximation

We will solve eqn (6.2) within the Thomas-Fermi approximation by linking the change in electron density, 5p(r), to the local potential, K(r). At equilibrium the chemical potential or Fermi energy must be constant everywhere as illustrated in Fig. 6.1, so that 

The Thomas-Fermi approximation assumes the variation in the potential, K(r), to be sufficiently slow that the local kinetic energy, 7T(r), is equal to that of an homogeneous free electron gas with the same density p(r) as seen locally, that is 

using the binomial expansion and substituting eqn (6.6) into (6.4), we have 

Poisson s equation (6.2) takes the simpler form 

Poisson s equation (6.9) can be solved directly by writing the potential, K(r), in terms of its Fourier transform, K(q), that is 

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