---
title: "Dielectrics, Polarization, and the Local Field"
module: Dielectrics and Ferroelectrics
moduleNumber: 8
lessonNumber: 1
order: 801
summary: >
  An insulator responds to an electric field by polarizing. This lesson builds
  the macroscopic polarization and the dielectric constant, sorts the microscopic
  response into electronic, ionic, and orientational polarizability, and corrects
  the field an atom actually feels to the Lorentz local field E + P/3 epsilon-0.
  The Clausius-Mossotti relation links the measured permittivity to the atomic
  polarizability, and the frequency dependence of each mechanism explains why the
  static and optical dielectric constants differ.
topics: [Dielectrics and Ferroelectrics]
sources:
  - book: Kittel
    ref: "Ch. 16 — Dielectrics and Ferroelectrics"
  - book: Ashcroft & Mermin
    ref: "Ch. 27 — Dielectric Properties of Insulators"
  - book: Hook & Hall
    ref: "Ch. 9 — Dielectrics"
---

A conductor screens an applied electric field by moving free charge until the
interior field is zero. An insulator has no free charge to move, so a field
penetrates it, but the bound charges shift: electron clouds distort, ions
displace, and permanent molecular dipoles reorient. Each of these responses
produces a **polarization**, an electric dipole moment per unit volume, and the
polarization in turn modifies the field. The measurable summary of all this
microscopic rearrangement is the **dielectric constant** $\varepsilon_r$. This
lesson connects that macroscopic number to the polarizability of a single atom or
molecule, with one subtlety that controls the whole subject: the field acting on
an atom inside the material is not the macroscopic field but a **local field**
that includes the contribution of every other dipole.

## Polarization and the dielectric constant

The **polarization** $\vec P$ is the electric dipole moment per unit volume. If
each of $N$ identical entities per unit volume carries an average induced moment
$\vec p$, then $\vec P = N\vec p$. Polarization is a source of field in exactly
the way bound charge is: a uniform $\vec P$ terminating on a surface leaves a
bound surface charge density $\sigma_b = \vec P\cdot\hat n$, and a nonuniform
$\vec P$ leaves a bound volume charge $\rho_b = -\nabla\cdot\vec P$.

To keep the bound charge out of Gauss's law one defines the **electric
displacement**

$$
\vec D = \varepsilon_0\vec E + \vec P,
$$

whose divergence counts only the free charge, $\nabla\cdot\vec D = \rho_{\text{free}}$.
For a linear, isotropic, homogeneous dielectric the polarization is proportional
to the field,

$$
\vec P = \varepsilon_0\chi_e\vec E, \qquad
\vec D = \varepsilon_0(1+\chi_e)\vec E \equiv \varepsilon_0\varepsilon_r\vec E,
$$

which defines the **electric susceptibility** $\chi_e$ and the **relative
permittivity** (dielectric constant) $\varepsilon_r = 1+\chi_e$. In a
parallel-plate capacitor filled with such a dielectric the capacitance rises by
the factor $\varepsilon_r$, because the polarization charge partly cancels the
free charge on the plates and lowers the field at fixed charge.

> **Definition (Polarization, susceptibility, dielectric constant).** The
> **polarization** $\vec P$ is the dipole moment per unit volume. The **electric
> susceptibility** $\chi_e$ and **relative permittivity** $\varepsilon_r$ relate
> it to the macroscopic field through $\vec P = \varepsilon_0\chi_e\vec E$ and
> $\varepsilon_r = 1+\chi_e$. In an anisotropic crystal $\chi_e$ and
> $\varepsilon_r$ are second-rank tensors; the scalar forms hold in cubic
> crystals and isotropic media.

The macroscopic field $\vec E$ appearing here is the spatial average over a region
large compared with the atomic spacing. Inside a finite sample $\vec E$ differs
from the externally applied field $\vec E_0$ by the **depolarization field** of
the sample's own surface charges: for an ellipsoid polarized along a principal
axis, $\vec E = \vec E_0 - N_d\vec P/\varepsilon_0$, with $N_d$ the depolarization
factor of that axis ($N_d = 1/3$ for a sphere, $1$ for a thin slab polarized
across its faces, $0$ along a long needle). The three factors of an ellipsoid sum
to $N_x+N_y+N_z = 1$.

## The microscopic polarizability

At the level of one atom or molecule the response is captured by the
**polarizability** $\alpha$, defined so that the induced moment is proportional to
the field the entity actually experiences,

$$
\vec p = \alpha\,\vec E_{\text{loc}},
$$

with $\alpha$ in SI units of $\text{C}\,\text{m}^2\,\text{V}^{-1} = \text{F}\,\text{m}^2$.
(The often-quoted "polarizability volume" is $\alpha/4\pi\varepsilon_0$, which has
dimensions of volume and is comparable to the atomic volume.) Three physically
distinct mechanisms contribute, distinguished by what moves and how fast it can
follow an oscillating field.

- **Electronic polarizability** $\alpha_e$: the electron cloud of each atom shifts
  relative to its nucleus. Present in every atom, it is the only mechanism fast
  enough to follow optical frequencies and therefore sets the refractive index.
- **Ionic polarizability** $\alpha_i$: in an ionic crystal the positive and
  negative sublattices displace in opposite directions, stretching the bonds.
  This is a lattice motion, resonant at the infrared transverse-optical phonon
  frequency.
- **Orientational (dipolar) polarizability** $\alpha_d$: molecules carrying a
  permanent dipole (water, HCl) partially align with the field against thermal
  disorder. Present only in polar substances, it is the slowest response and the
  only one that depends strongly on temperature.

$$
% caption: The three polarization mechanisms. Electronic: the electron cloud
% displaces from the nucleus. Ionic: the cation and anion sublattices move apart.
% Orientational: a permanent molecular dipole rotates toward the field. Their
% characteristic response frequencies rise from left (dipolar, microwave) to
% right (electronic, ultraviolet).
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % field arrow legend
  \draw[black, ->, thick] (-0.2,3.0) -- (1.0,3.0) node[right, black] {$E$};
  % --- panel 1: orientational ---
  \begin{scope}[xshift=0cm]
    \node[black, anchor=south] at (1.2,2.2) {orientational};
    \draw[acc, ->, very thick] (0.7,0.2) -- (1.9,1.4);
    \fill[acc] (1.9,1.4) circle (2pt);
    \fill[black] (0.7,0.2) circle (2pt);
    \node[black, anchor=north] at (1.2,0.0) {dipole rotates};
    \node[black, anchor=north, font=\scriptsize] at (1.2,-0.45) {radio to microwave};
  \end{scope}
  % --- panel 2: ionic ---
  \begin{scope}[xshift=4.2cm]
    \node[black, anchor=south] at (1.2,2.2) {ionic};
    \fill[acc!14] (0.5,0.8) circle (7pt);
    \draw[acc] (0.5,0.8) circle (7pt);
    \node[acc] at (0.5,0.8) {$+$};
    \fill[black] (2.0,0.8) circle (7pt);
    \draw[black] (2.0,0.8) circle (7pt);
    \draw[acc, ->] (0.75,0.8) -- (1.15,0.8);
    \draw[black, ->] (1.75,0.8) -- (1.35,0.8);
    \node[black, anchor=north, font=\scriptsize] at (1.25,-0.1) {sublattices displace};
    \node[black, anchor=north, font=\scriptsize] at (1.25,-0.55) {infrared};
  \end{scope}
  % --- panel 3: electronic ---
  \begin{scope}[xshift=8.4cm]
    \node[black, anchor=south] at (1.2,2.2) {electronic};
    \draw[acc] (1.35,0.8) circle (9pt);
    \fill[acc!14] (1.35,0.8) circle (9pt);
    \fill[black] (1.0,0.8) circle (2.4pt);
    \node[black, anchor=east, font=\scriptsize] at (0.9,0.8) {nucleus};
    \node[acc, anchor=west, font=\scriptsize] at (1.7,1.35) {cloud};
    \node[black, anchor=north, font=\scriptsize] at (1.2,-0.1) {cloud shifts};
    \node[black, anchor=north, font=\scriptsize] at (1.2,-0.55) {ultraviolet};
  \end{scope}
\end{tikzpicture}
$$

### Electronic polarizability from a bound-electron model

Model an atomic electron as bound to its nucleus by a spring of natural frequency
$\omega_0$, driven by a field $E_{\text{loc}}(t) = E\,e^{-i\omega t}$. Its
displacement $x$ obeys

$$
m\ddot x + m\gamma\dot x + m\omega_0^2 x = -eE\,e^{-i\omega t},
$$

with $\gamma$ a damping rate. The steady-state solution
$x(t) = x_0 e^{-i\omega t}$ has

$$
x_0 = \frac{-eE}{m(\omega_0^2 - \omega^2 - i\gamma\omega)},
$$

so the induced moment $p = -ex_0 = \alpha_e(\omega)E$ gives a **frequency-dependent
electronic polarizability**

$$
\alpha_e(\omega) = \frac{e^2/m}{\omega_0^2 - \omega^2 - i\gamma\omega}.
$$

At zero frequency this is the static value $\alpha_e(0) = e^2/m\omega_0^2$;
dimensionally, with $\omega_0$ in the ultraviolet, $\alpha_e(0)/4\pi\varepsilon_0$
is of order the atomic volume, a few $\text{\AA}^3$. The response stays nearly
constant until $\omega$ approaches $\omega_0$, where it resonates and the
imaginary (absorptive) part peaks. Because $\omega_0$ is an optical/ultraviolet
frequency, the electronic response is flat across the entire visible range,
which is why an atom's contribution to the refractive index is nearly
dispersionless far from any absorption line.

### Orientational polarizability and the Langevin–Debye law

A molecule with permanent moment $p_0$ in a field has orientation energy
$U = -p_0 E\cos\theta$. In thermal equilibrium the mean projected moment follows
the Boltzmann average over solid angle,

$$
\langle p_0\cos\theta\rangle = p_0\,\frac{\int_0^\pi \cos\theta\,e^{p_0E\cos\theta/k_BT}\sin\theta\,\d\theta}
{\int_0^\pi e^{p_0E\cos\theta/k_BT}\sin\theta\,\d\theta}
= p_0\,L(a), \qquad a = \frac{p_0E}{k_BT},
$$

where $L(a) = \coth a - 1/a$ is the **Langevin function**. At ordinary fields
$a\ll 1$ (even $E \sim 10^7\ \text{V/m}$ gives $a \sim 10^{-3}$ at room
temperature), so $L(a)\approx a/3$ and

$$
\langle p\rangle = \frac{p_0^2}{3k_BT}\,E, \qquad
\alpha_d = \frac{p_0^2}{3k_BT}.
$$

The orientational polarizability falls as $1/T$: thermal agitation randomizes the
dipoles and weakens the alignment. This **Curie-law** temperature dependence is
the experimental signature that distinguishes a polar substance from a nonpolar
one, and measuring $\alpha_d(T)$ yields the permanent moment $p_0$.

$$
% caption: The Langevin function L(a) = coth(a) minus 1 over a, giving the mean
% aligned fraction of a permanent dipole versus a = p0 E over kB T. It rises
% linearly with slope one-third at small a (the Curie-law regime, dashed) and
% saturates at complete alignment for large a.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (6.6,0) node[below] {$a = p_0 E / k_B T$};
  \draw[->, black] (0,0) -- (0,3.0) node[left] {$L(a)$};
  \draw[black, dashed] (0,2.4) -- (6.3,2.4) node[right, black, font=\scriptsize] {saturation};
  % Langevin curve, saturating toward 2.4
  \draw[acc, very thick] (0,0)
    .. controls (1.2,1.05) and (2.6,2.05) .. (4.2,2.28)
    .. controls (5.0,2.36) and (5.6,2.39) .. (6.3,2.4);
  % small-a tangent slope 1/3 (scaled): value 2.4 at a where linear meets... draw slope
  \draw[black, dashed] (0,0) -- (2.9,2.4) node[midway, above left, font=\scriptsize] {slope $\tfrac{1}{3}$};
\end{tikzpicture}
$$

## The local field

The polarizability is defined through the field an atom actually experiences, and
in a dense material that is not the macroscopic $\vec E$. Every other dipole in
the medium contributes a field at the site of a given atom. Summing that
contribution is the central problem of dielectric theory, solved by H. A. Lorentz
with a geometric decomposition.

Imagine the atom of interest at the center of a small **spherical cavity** cut out
of the polarized medium, small on the macroscopic scale but large compared with
the atomic spacing. The field at the center is written as a sum of four pieces,

$$
\vec E_{\text{loc}} = \vec E_0 + \vec E_1 + \vec E_2 + \vec E_3,
$$

- $\vec E_0$: the field from the free charges on the external plates.
- $\vec E_1$: the depolarization field from bound charge on the outer surface of
  the sample. Together they build the macroscopic field: $\vec E_0 + \vec E_1 = \vec E$.
- $\vec E_2$: the field from bound charge on the surface of the Lorentz cavity,
  the **Lorentz field**.
- $\vec E_3$: the field of the individual dipoles inside the cavity, treated
  discretely.

$$
% caption: The Lorentz construction. A spherical cavity is cut from the uniformly
% polarized medium; the polarization charge on the cavity wall, of surface density
% P cos(theta) (positive plus signs on the near wall, negative shown as gray dots
% on the far wall), produces the Lorentz field P over 3 epsilon-0 at the center,
% directed along P. Dipoles inside the cavity are summed separately and vanish for
% a site of cubic symmetry.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % polarized medium block
  \fill[acc!8] (-3.0,-2.2) rectangle (3.0,2.2);
  \draw[black] (-3.0,-2.2) rectangle (3.0,2.2);
  % polarization arrows in the medium
  \foreach \y in {-1.7,-0.5,0.7,1.9}{
    \draw[black, ->] (-2.6,\y) -- (-2.0,\y);
    \draw[black, ->] (2.0,\y) -- (2.6,\y);
  }
  \node[black, anchor=south west] at (-2.95,1.9) {polarized medium};
  % cavity
  \fill[white] (0,0) circle (1.35);
  \draw[acc, very thick] (0,0) circle (1.35);
  % cavity surface charge: + on left, - on right (P points right)
  \node[acc] at (-1.15,0) {$+$};
  \node[acc] at (-0.95,0.6) {$+$};
  \node[acc] at (-0.95,-0.6) {$+$};
  \fill[black] (1.15,0) circle (2.2pt);
  \fill[black] (0.95,0.6) circle (2.2pt);
  \fill[black] (0.95,-0.6) circle (2.2pt);
  % central atom
  \fill[black] (0,0) circle (2.4pt);
  % Lorentz field arrow at center (points along P, to the right)
  \draw[acc, ->, very thick] (0,0) -- (0.85,0);
  \node[acc, anchor=south, font=\scriptsize] at (0.5,0.08) {Lorentz E};
  % macroscopic P label
  \draw[black, ->, thick] (1.7,-1.85) -- (2.5,-1.85) node[right, black] {$P$};
\end{tikzpicture}
$$

The Lorentz field is a clean integral. The cavity wall carries bound charge of
surface density $\sigma_b = \vec P\cdot\hat n = -P\cos\theta$ (the inward normal of
the cavity points opposite $\hat r$). Integrating the field this charged spherical
shell produces at its center,

$$
E_2 = -\frac{1}{4\pi\varepsilon_0}\int \frac{\sigma_b\cos\theta}{a^2}\,\d A
= \frac{P}{4\pi\varepsilon_0 a^2}\int_0^\pi \cos^2\theta\,(2\pi a^2\sin\theta)\,\d\theta
= \frac{P}{2\varepsilon_0}\int_0^\pi\cos^2\theta\sin\theta\,\d\theta = \frac{P}{3\varepsilon_0},
$$

directed along $\vec P$. The near-dipole sum $\vec E_3$ depends on the crystal
structure; for a site with cubic symmetry (or for a completely random
arrangement) the contributions of the discrete dipoles cancel exactly, so
$\vec E_3 = 0$. Collecting the pieces gives the **Lorentz local field**.

> **Theorem (Lorentz local field).** At an atomic site of cubic symmetry in a
> uniformly polarized medium, the local field is
> $$
> \vec E_{\text{loc}} = \vec E + \frac{\vec P}{3\varepsilon_0},
> $$
> the macroscopic field augmented by the Lorentz cavity field $\vec P/3\varepsilon_0$.
> The discrete near-neighbor dipole sum vanishes by cubic symmetry.

The correction is not small. In a solid with $\varepsilon_r$ of a few, the
Lorentz term $P/3\varepsilon_0$ is comparable to $E$ itself, so an atom in a dense
dielectric feels substantially more field than the macroscopic average. Ignoring
it gives qualitatively wrong permittivities.

## The Clausius–Mossotti relation

Combine the microscopic and macroscopic descriptions. With $N$ polarizable
entities per unit volume, each acquiring $\vec p = \alpha\vec E_{\text{loc}}$,

$$
\vec P = N\alpha\vec E_{\text{loc}} = N\alpha\left(\vec E + \frac{\vec P}{3\varepsilon_0}\right).
$$

Solving for $\vec P$,

$$
\vec P = \frac{N\alpha}{1 - N\alpha/3\varepsilon_0}\,\vec E,
$$

and comparing with $\vec P = \varepsilon_0(\varepsilon_r-1)\vec E$ gives, after
rearrangement, the **Clausius–Mossotti relation**:

$$
\frac{\varepsilon_r - 1}{\varepsilon_r + 2} = \frac{N\alpha}{3\varepsilon_0}.
$$

> **Theorem (Clausius–Mossotti relation).** For a cubic or isotropic dielectric of
> number density $N$ and polarizability $\alpha$, the dielectric constant satisfies
> $$
> \frac{\varepsilon_r-1}{\varepsilon_r+2} = \frac{N\alpha}{3\varepsilon_0}.
> $$
> The left side is a measurable macroscopic quantity; the right side is
> microscopic. The relation follows entirely from the Lorentz local field and
> holds only for materials in which that field applies.

For a mixture of several polarizable species the right side becomes a sum
$\sum_j N_j\alpha_j/3\varepsilon_0$. Written for the optical range, where
$\varepsilon_r = n^2$ with $n$ the refractive index, the same relation is the
**Lorentz–Lorenz equation**,

$$
\frac{n^2-1}{n^2+2} = \frac{N\alpha_e}{3\varepsilon_0},
$$

which, multiplied by the molar volume, defines the **molar refractivity**, a
nearly additive and density-independent property used to infer bond
polarizabilities from measured refractive indices.

$$
% caption: The dielectric constant from Clausius-Mossotti as the density-times-
% polarizability parameter y = N alpha over 3 epsilon-0 grows. Eps-r = (1 + 2y) /
% (1 - y) rises slowly at first, then diverges as y approaches 1: the
% polarization catastrophe that anticipates the ferroelectric instability.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (6.4,0) node[below] {$y$};
  \draw[->, black] (0,0) -- (0,4.2) node[left] {permittivity};
  % vertical asymptote at y=1 (x=5)
  \draw[black, dashed] (5.0,0) -- (5.0,4.1) node[above, black, font=\scriptsize] {$y=1$};
  % curve eps = (1+2y)/(1-y), map y in [0,1] to x in [0,5]; eps from 1 upward
  % sample points: y=0 ->1, .3->1.6/.7=2.29, .5->2/.5=4(cap), .6->2.2/.4=5.5(cap)
  \draw[acc, very thick] (0,0.5)
    .. controls (1.6,0.85) and (3.0,1.5) .. (4.0,2.6)
    .. controls (4.5,3.3) and (4.75,3.9) .. (4.9,4.1);
  \fill[acc] (0,0.5) circle (1.6pt);
  \node[black, anchor=north east, font=\scriptsize] at (0.0,0.5) {$1$};
  \node[acc, anchor=west, font=\scriptsize] at (3.55,1.7) {diverges};
\end{tikzpicture}
$$

The divergence of $\varepsilon_r$ as $N\alpha \to 3\varepsilon_0$ is the
**polarization catastrophe**. As the right side approaches $1$, the denominator
$1 - N\alpha/3\varepsilon_0$ vanishes and the dielectric constant runs away: the
local field of the already-polarized dipoles is by itself enough to sustain the
polarization with no applied field. That runaway is precisely the microscopic
origin of spontaneous polarization in a
[ferroelectric](/condensed-matter/dielectrics-and-ferroelectrics/ferroelectrics-and-piezoelectrics),
where the ionic polarizability grows with falling temperature until the catastrophe
condition is met at the Curie point.

## Frequency dependence and dispersion

Each mechanism responds on its own timescale, so the dielectric constant depends
on the frequency of the applied field. As $\omega$ rises past the characteristic
frequency of a mechanism, that contribution can no longer follow the field and
drops out, stepping $\varepsilon_r$ down. Two kinds of dynamics appear.

**Resonant mechanisms** (electronic, ionic) are driven oscillators. Their
contribution follows the bound-charge form derived above,
$\alpha(\omega) \propto (\omega_0^2 - \omega^2 - i\gamma\omega)^{-1}$: the real
part shows normal dispersion (rising with $\omega$) far from $\omega_0$ and
anomalous dispersion near it, while the imaginary part is an absorption peak of
width $\gamma$ centered on $\omega_0$. Ionic resonances lie in the infrared, near
the transverse-optical phonon frequency; electronic resonances lie in the
ultraviolet.

**Relaxational mechanisms** (orientational) are not oscillators but overdamped
reorientations. A permanent dipole cannot flip instantaneously; it relaxes toward
equilibrium with a time constant $\tau$. The response is the **Debye
relaxation** form

$$
\varepsilon_r(\omega) = \varepsilon_\infty + \frac{\varepsilon_s - \varepsilon_\infty}{1 - i\omega\tau},
$$

whose real and imaginary parts are

$$
\varepsilon'(\omega) = \varepsilon_\infty + \frac{\varepsilon_s - \varepsilon_\infty}{1+\omega^2\tau^2},
\qquad
\varepsilon''(\omega) = \frac{(\varepsilon_s - \varepsilon_\infty)\,\omega\tau}{1+\omega^2\tau^2}.
$$

Here $\varepsilon_s$ is the static (low-frequency) constant and $\varepsilon_\infty$
the value once the dipolar term has dropped out. The loss $\varepsilon''$ peaks at
$\omega\tau = 1$; for water at room temperature $\tau \approx 10^{-11}\ \text{s}$,
placing the dipolar relaxation in the microwave band, the physical basis of
microwave heating.

$$
% caption: The dielectric function across the spectrum. The real part (top)
% steps down as each mechanism freezes out: dipolar relaxation in the microwave,
% the ionic resonance in the infrared, and the electronic resonance in the
% ultraviolet. Each step in the real part is accompanied by a loss peak in the
% imaginary part (bottom). Above the electronic resonance the medium hardly
% polarizes and the response approaches unity.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % ---- real part (upper) ----
  \begin{scope}[yshift=2.7cm]
    \draw[->, black] (0,0) -- (9.6,0) node[below right, font=\scriptsize] {frequency};
    \draw[->, black] (0,0) -- (0,2.5) node[left] {real part};
    % staircase down: start high, three drops
    \draw[acc, very thick]
      (0.2,2.1) -- (2.0,2.1)
      .. controls (2.5,2.1) and (2.7,1.55) .. (3.2,1.55) -- (4.6,1.55)
      .. controls (5.1,1.55) and (5.3,0.95) .. (5.8,0.95) -- (7.2,0.95)
      .. controls (7.7,0.95) and (7.9,0.35) .. (8.4,0.35) -- (9.3,0.35);
    \node[black, anchor=south, font=\scriptsize] at (1.1,2.12) {static};
    \node[black, anchor=south, font=\scriptsize] at (9.0,0.37) {unity};
  \end{scope}
  % ---- imaginary part (lower) ----
  \begin{scope}
    \draw[->, black] (0,0) -- (9.6,0) node[below right, font=\scriptsize] {frequency};
    \draw[->, black] (0,0) -- (0,2.2) node[left] {loss};
    % three peaks at the three transition regions
    \draw[acc, very thick]
      (0.2,0.05)
      .. controls (2.3,0.05) and (2.6,1.5) .. (2.9,1.5)
      .. controls (3.2,1.5) and (3.5,0.1) .. (4.3,0.1)
      .. controls (5.0,0.1) and (5.2,1.7) .. (5.5,1.7)
      .. controls (5.8,1.7) and (6.1,0.1) .. (6.9,0.1)
      .. controls (7.6,0.1) and (7.8,1.8) .. (8.1,1.8)
      .. controls (8.4,1.8) and (8.7,0.1) .. (9.3,0.1);
    \node[black, anchor=north, font=\scriptsize] at (2.9,-0.05) {dipolar};
    \node[black, anchor=north, font=\scriptsize] at (5.5,-0.05) {ionic};
    \node[black, anchor=north, font=\scriptsize] at (8.1,-0.05) {electronic};
  \end{scope}
\end{tikzpicture}
$$

The stepwise structure explains the two dielectric constants a material carries.
The **static** dielectric constant $\varepsilon_s$ includes every mechanism; in
water $\varepsilon_s \approx 80$, dominated by orientational alignment of the polar
molecules. The **optical** dielectric constant $\varepsilon_\infty = n^2$ includes
only the electronic response, since at $\sim 10^{15}\ \text{Hz}$ neither the ions
nor the molecular dipoles can follow; for water $n^2 \approx 1.77$. The enormous
gap between $80$ and $1.77$ is the orientational and vibrational contribution that
has frozen out by optical frequencies.

| Mechanism | What moves | Resonance / relaxation | Frequency range | Temperature dependence |
| --- | --- | --- | --- | --- |
| Electronic | electron cloud | resonant, $\omega_0$ | ultraviolet, $\sim 10^{15}\ \text{Hz}$ | weak |
| Ionic | ion sublattices | resonant, TO phonon | infrared, $\sim 10^{13}\ \text{Hz}$ | weak |
| Orientational | permanent dipoles | relaxational, $\tau$ | radio–microwave, $\lesssim 10^{11}\ \text{Hz}$ | $\propto 1/T$ (Curie) |

> **Worked example.** Solid argon has no permanent dipole and no ions, so only
> electronic polarizability contributes. With number density
> $N = 2.66\times10^{28}\ \text{m}^{-3}$ and atomic polarizability
> $\alpha_e/4\pi\varepsilon_0 = 1.64\ \text{\AA}^3$, the Clausius–Mossotti
> parameter is
> $$
> \frac{N\alpha_e}{3\varepsilon_0} = \frac{4\pi}{3}N\left(\frac{\alpha_e}{4\pi\varepsilon_0}\right)
> = \frac{4\pi}{3}(2.66\times10^{28})(1.64\times10^{-30}) = 0.183.
> $$
> Then $\varepsilon_r = (1+2y)/(1-y) = 1.366/0.817 = 1.67$, and the refractive
> index $n = \sqrt{\varepsilon_r} = 1.29$, close to the measured value for solid
> argon. The small parameter $y = 0.183$ sits far from the catastrophe at $y=1$,
> as expected for a weakly polarizable noble-gas solid.

The dielectric constant is thus a compact record of every way an insulator's bound
charge can move, weighted by how fast the field asks it to. Reducing the
temperature strengthens the orientational term without bound in the Curie law and
can push the ionic term toward the polarization catastrophe. When a crystal's
soft ionic mode drives $N\alpha \to 3\varepsilon_0$, the dielectric constant
diverges and the material polarizes spontaneously — the
[ferroelectric transition](/condensed-matter/dielectrics-and-ferroelectrics/ferroelectrics-and-piezoelectrics)
that the next lesson develops through Landau theory and the soft-mode picture.