= Static Lindhard dielectric function
{title2=$\epsilon(q,0)=1+\kappa^2F(q/2k_F)/q^2$}
The static independent-electron density response of a uniform three-dimensional, two-spin <Fermi gas>, combined with self-consistent <Poisson equation> screening, gives the displayed <dielectric function>. For bare electron mass and vacuum background, $\kappa^2=m_ee^2k_F/(\epsilon_0\pi^2\hbar^2)$ and
$$
F(x)=\frac12+\frac{1-x^2}{4x}\ln\left|\frac{1+x}{1-x}\right|.
$$
Here $F(0)=1$, $F(1)=1/2$ by continuous extension, and $F(x)\sim1/(3x^2)$ at large $x$. A host dielectric background and <band effective mass> require replacing the vacuum parameters appropriately. This nonlocal response contains <Friedel oscillations>, which its long-wavelength <Thomas-Fermi screening> limit misses.
Back to article page
