Electric susceptibility (${\chi }_e$)

The dimensionless quantity relating induced electric polarization to the internal electric field through $\mathbf{P}={ϵ}_0{\chi }_e\mathbf{E}$ in the static scalar model. It describes the material contribution; relative permittivity includes the vacuum term as well, ${ϵ}_r=1+{\chi }_e$. Zero electric susceptibility gives ${ϵ}_r=1$, not zero permittivity.

The scalar ${\chi }_e$ assumes a local response with linearity and isotropy, settled electric polarization, fixed material conditions, and no residual electric polarization at zero electric field. It relates $\mathbf{P}$ to the averaged total internal electric field, which already includes the field of the bound electric charges. It does not relate $\mathbf{P}$ to the field the external electrodes would produce in vacuum. The distinction is essential for a self-consistent calculation.^[ref]

Substitute the scalar relation into the definition of electric displacement field:Comparison with $\mathbf{D}=ϵ\mathbf{E}$ proves $ϵ={ϵ}_0(1+{\chi }_e)$ and ${ϵ}_r=1+{\chi }_e$. Both $\mathbf{D}$ and $\mathbf{P}$ have units $C/{\mathrm{m}}^2$, so ${\chi }_e$ is a dimensionless quantity. In a stipulated static model with ${ϵ}_r=3$, ${\chi }_e=2$: at a prescribed internal electric field, the material contribution $P$ is twice ${ϵ}_{0}E$. This is a hypothetical response, not a measured material constant. The electric susceptibility is a convenient bridge from microscopic dielectric polarization to a macroscopic capacitance measurement, but it is not a measure of electrical conductivity.

A bound charge obeying $m\ddot{u}+m\gamma \dot{u}+m{\omega }_0^{2}u=qE$ gives a resonant electric susceptibility. For carrier density $n$ and the $e^{-i\omega t}$ convention,Positive damping gives $\mathrm{Im}{ϵ}_r\ge 0$ for $\omega >0$, expressing passive absorption. Independent resonances add to the electric susceptibility. Setting ${\omega }_0=0$ gives the Drude frequency response.

For a single lossless polar optical oscillator with frequency-independent electronic background, ${ϵ}_r(\omega )={ϵ}_{\infty }({\omega }_{\mathrm{LO}}^2-{\omega }^2)/({\omega }_{\mathrm{TO}}^2-{\omega }^2)$. Setting frequency to zero yields the displayed static-to-background ratio. Multiple infrared-active branches require the corresponding product relation, rather than selecting an arbitrary pair of modes.

An optical oscillator strength is a dimensionless weight assigned to an optical resonance. In an independent-resonance Lorentz oscillator model using a reference electron mass $m_e$ and a specified participating number density $n$,Its normalization must be stated: assigning one unit of total strength per participating electron means ${\sum }_j{f}_j=1$. A one-resonance approximation with $f_1=1$ represents the whole chosen electron population, rather than asserting that the first allowed interband transition carries all optical strength. The smallest band gap therefore need not equal the fitted resonance energy.

For $\ddot{z}+2\gamma \dot{z}+{\omega }_0^{2}z=(q{E}_0/m)\cos \omega t$, the $e^{i\omega t}$ convention gives complex displacement amplitude $\tilde{z}=(q{E}_0/m)/({\omega }_0^2-{\omega }^2+2i\gamma \omega )$. If the real amplitude is defined nonnegative, the phase is $-\delta$ for $q>0$ but $\pi -\delta$ for $q<0$, where $\delta =atan2(2\gamma \omega ,{\omega }_0^2-{\omega }^2)$. An electron's low-frequency displacement is opposite to the electric field; its high-frequency displacement is in phase. The induced electric dipole moment $qz$ instead has phase $-\delta$ for either sign of charge. A signed amplitude can absorb the extra $\pi$, but that convention must be stated.

For a charge $q$ of mass $m$ bound by spring constant $k=m{\omega }_0^2$, the undamped equation $m\ddot{x}+kx=q{E}_0{e}^{-i\omega t}$ has complex amplitude $x_0=q{E}_0/[m({\omega }_0^2-{\omega }^2)]$. The induced electric dipole moment is $p_0=q{x}_0$. Averaged electric dipole radiation power, divided by incident light intensity, givesThe Rayleigh scattering limit $\omega \ll {\omega }_0$ gives $\sigma \sim {\sigma }_T(\omega /{\omega }_0{)}^4$, proportional to ${\lambda }^{-4}$. For ${\omega }_0\ll \omega \ll m{c}^2/\hslash$, the charge behaves as free and the Thomson cross-section is recovered. At resonance the undamped steady-state formula is invalid; a finite damping mechanism or radiation reaction must be included. The formula describes independent weakly driven scatterers, not arbitrary coherent scattering from an extended medium.