Electrons Lindhard dielectric response

We wish to derive the Lindhard dielectric response function for the free-electron gas, using perturbation theory. The charge density is defined in terms of the single-particle wavefunctions as... 

The current density approach is shown to be effective in the analysis of the stopping phenomenon and also presents itself as a method for realistic description of experimental results. In the case of a uniform electron gas, the combination of free-electron and dielectric models provides a possibility to take into account both the higher order effects and the screening of the projectile field. In addition, the free-electron model provides a possibility to analyze, in detail, the role of quantum effects. It has been shown that origin of the Bloch correction is closely related to this problem. The results for uniform electron gas can be used to describe the energy loss in atomic substances if, additionally, the local response approach is applied. The Lindhard-Sharff model can be considered as a particular case where electron dynamics is treated in the linear response approach. 

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