The Difference between Thomas-Fermi Screening and Lindhard Screening Assuming the general theory of screening related to electron-electron interactions, I was wondering if anyone could provide a clear, yet conceptually complete explanation of the differences between the Thomas-Fermi and the Lindhard theories?
Following the treatment by Ashcroft and Mermin (1976),I get the impression that the main difference is that the Thomas-Fermi model assumes that the electrostatic potential is slowly-varying in $\vec{r}$: $$E_i(\vec{k}) = \frac{\hbar^2k^2}{2m}-e\phi(\vec{r})$$, but I'm having trouble grasping the full physical significance of this statement.
 A: First, the Thomas-Fermi screening is a semiclassical static theory which assumes that the total potential $\phi(\mathbf{r})$ varies slowly in the scale of the Fermi length $l_{\text{F}}$, the chemical potential $\mu$ is constant and that $T$ is low. In principle, it does not rely on linear response theory.
The condition of slowly varying potential is a general condition of validity of semiclassical models. Physically, if the particle [electron] is represented by a wave packet, what is tellying us is that all the waves in the wavepacket will see the same potential and the particle will suffer [or enjoy!] a force as if it was point-like ["classical"] because such potentials gives rise to ordinary forces in the equation of motion describing the evolution of the position and wavevector of the packet. The wavepacket must have a well-defined wavevector on scale of the Brillouin zone [thus $\Delta k \simeq k_{\text{F}}$] and therefore can be spread in the real space over many primitive cells.
Mathematically, the assumption that your potential is a slowly varying function of the position implies that the theory is not valid for $|\mathbf{q}| \gg k_{\text{F}}$ [and therefore for $|\mathbf{r}| \ll l_{\text{F}}$]. 
On the other hand, the static Lindhard dielectric function is a fully quantum treatment of the problem and it is valid for all the ranges of $\mathbf{q}$. It includes, in the limit $\mathbf{q} \rightarrow 0$, the linearized Thomas-Fermi dielectric function. It only assumes linear response, that is, the induced density of charge is proportional to the total potential $\phi(\mathbf{r})$. 
Note also that the Lindhard treatment is far more general than the Thomas-Fermi in the sense that it can describe both dynamic and static screening.
