Relative permittivity (${ϵ}_r=ϵ/{ϵ}_0$)

The dimensionless ratio of a medium's permittivity to vacuum permittivity. The constant scalar value used in a Wannier-Mott exciton is an approximation to the relevant material response.

The generally complex, frequency- and wavevector-dependent relative permittivity describing linear polarization response. Its imaginary part represents dissipative response with the sign fixed by the time convention; optical prefactors connect it to an absorption coefficient.

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, ${\kappa }^2=m_e{e}^2{k}_F/({ϵ}_0{\pi }^2{\hslash }^2)$ andHere $F(0)=1$, $F(1)=1/2$ by continuous extension, and $F(x)\sim 1/(3x^2)$ 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.

The three-dimensional static screening function is finite at $q=2k_F$, but its derivative diverges logarithmically. Expanding the explicit static Lindhard dielectric function gives the title's nonanalytic term. The singularity comes from particle–hole excitations connecting opposite parts of the Fermi surface. It produces a Friedel density tail from a derivative singularity; analogous electronic response can cause a Kohn anomaly in a coupled lattice vibration.

Modification of the interaction between electric charges by a medium's electrical response. In the simplest Wannier-Mott exciton model, the Coulomb potential is reduced by a constant relative permittivity; actual screening can depend on spatial and temporal scales.

For a point source $Ze$ and the Fourier transform convention $\rho (\mathbf{q})=\int e^{-i\mathbf{q}\cdot \mathbf{r}}\rho (\mathbf{r})d^{3}r$, self-consistent screening gives ${\rho }_{\mathrm{ind}}(q)=Ze[ϵ(q{)}^{-1}-1]$. Angular inversion yields ${\rho }_{\mathrm{ind}}(r)=-(Ze/r){\int }_0^{\infty }g(q)\sin (qr)dq$. The factor $1/(2{\pi }^2)$ includes the inverse-transform normalization and spherical angular integral. Perfect long-wavelength screening implies total induced charge $-Ze$.

The sharp Fermi surface gives a nonlocal oscillatory electronic response around an impurity. In a three-dimensional isotropic metal its far-field oscillations have characteristic wavevector $2k_F$ and algebraically decaying amplitude. They are absent from the purely exponential Thomas-Fermi screening potential. The local screening formula is a useful long-wavelength model, not an exact asymptotic expression at every distance.

A single isotropic three-dimensional valley with two equal spin populations has $n=k_F^3/(3{\pi }^2)$. A density oscillation at wavevector $2k_F$ therefore has the wavelength in the title. This depends on carrier density, independently of effective mass and dielectric screening strength. Extra valley degeneracy, spin polarization or reduced dimensionality changes the state-counting relation.

For the radial induced-charge screening kernel, integrate the sine transform twice by parts, with $g(0)=g(\infty )=0$ and vanishing remaining endpoint terms. This gives ${\rho }_{\mathrm{ind}}=Ze\int g^{\prime \prime }(q)\sin (qr)dq/r^3$. A Cauchy principal value singularity at $Q>0$ contributes $C\pi \cos (Qr)$ to that integral: set $s=q-Q$ and use ${\int }_{-\infty }^{\infty }\sin (sr)ds/s=\pi$ for $r>0$. Smoothly localizing around $Q$ justifies the large-$r$ extension of the limits. In a three-dimensional isotropic Fermi gas, $Q=2k_F$.

In the long-wavelength static response of a degenerate conductor, a small electric potential $\varphi$ changes the electron density by $\delta n=eg(E_F)\varphi$, where $g(E_F)$ is the spin-summed density of states per volume per energy. The induced charge is therefore $-e^{2}g(E_F)\varphi$. With background permittivity ${\epsilon }_b$, Poisson equation becomesA point charge $Q$ then has $\varphi (r)=Q{e}^{-k_{\mathrm{TF}}r}/(4\pi {\epsilon }_{b}r)$. Larger electronic density of states gives stronger screening. This local response approximation does not capture the microscopic oscillatory tail of Friedel oscillations.^[ref]

The Thomas-Fermi screening length is the exponential decay scale of a statically screened potential in the local degenerate-electron approximation. For a three-dimensional parabolic free electron model, $g(E_F)=3n/(2E_F)$, so ${\lambda }_{\mathrm{TF}}^2=2{\epsilon }_b{E}_F/(3n{e}^2)$. At fixed mass and background permittivity it scales as $n^{-1/6}$, rather than the classical Debye density power at fixed temperature.