Dielectric

Dielectric basics

Macroscopic properties

Electric field $E$
Polarization $\vec P$
Displacement $\vec D = \epsilon_0 \vec E + \vec P$

Susceptibility and dielectric constants

$\vec P = \epsilon_0 \chi_e \vec E$
$\vec D = \epsilon_0 (1+\chi_e) \vec E = \epsilon_0\epsilon_r \vec E$
Relative dielectric constant $\epsilon_r = \kappa = 1 + \chi_e$

Dielectric and matter

Under an external electronic field, the redistribution of electrons in matter causes depolarization and also contributes to the electric field in space.

Depolarization field and redistribution of electrons in a material. via Kittle (p.459)

Inside material

The depolarization field inside a material has the form $\vec E = -\frac{N}{\epsilon_0}\vec P$ , in the case of a sphere, $N = 1/3$ , which gives

$\vec E_\text{depol} = -\frac{1}{3\epsilon_0}\vec P$

Outside of material

The external field is

$\vec{E}(\vec{r})=\frac{3(\vec{p} \cdot \vec{r}) \vec{r}-r^{2} \vec{p}}{4 \pi \epsilon_{0} r^{5}}$

Clausius–Mossotti relation

We consider the field

$E = E_\text{sphere} + E_\text{cavity}$

Then

$E_\text{cavity} = E_\text{external} + 1/3\epsilon_0 \vec P$

$\vec P = N\alpha E = N\alpha(1+1/3\epsilon_0 \vec P)$

Polarizablity $\alpha$ is defined by $\vec P = N \alpha \vec E$ , which gives

$\frac{\alpha}{4\pi\epsilon_0} = 3V \left(\frac{\kappa - 1}{\kappa + 1}\right)$

And actually

Classical theory start from

$m\ddot x + m\omega_0x^2= eE = e E_0\cos\omega t$

treat it as a harmonic oscillator with a frequency $\omega_0$ under AC field $\alpha$ is a function of frequency $\omega$ , see Resonators, general susceptibility, and AC effects for the classical theory

Where the polarizability can be obtained

$\alpha = p / E = ex / E$

And we found $x = e^2 / m(\omega_0^2 - \omega^2)$

In fact we can see […]

Polarizability $\alpha$ as a function of frequency $\omega$ . via Kittle (p.464)

References

Kittel, C., & McEuen, P. (2018). Kittel’s Introduction to Solid State Physics. John Wiley & Sons.