Polarization and Clausius-Mossotti equation

Polarization $\vec P$ is the net induced dipole moment per unit volume of a dielectric: $P = Np$, where $N$ is the number of molecules per m³ and $p$ the average dipole moment of each. It is related to the field by $P = \varepsilon_0(\varepsilon_r - 1)E$.

Step 1: Polarization in terms of $\varepsilon_r$

From $D = \varepsilon_0 E + P = \varepsilon_0\varepsilon_r E$:

$$P = \varepsilon_0(\varepsilon_r - 1)E \quad (1)$$

Step 2: Polarization in terms of polarizability

A molecule feels the local field $E_{loc}$, not the applied field, so

$$P = N\alpha_e E_{loc} \quad (2)$$

Step 3: Local (Lorentz) field

Imagine a small spherical cavity around a molecule. The field at its centre is

$$E_{loc} = E_1 + E_2 + E_3 + E_4$$

• $E_1$: field of the free charges on the plates
• $E_2$: field of the polarization charges on the dielectric surfaces facing the plates; together $E_1 + E_2 = E$, the applied macroscopic field
• $E_3$: field of the polarization charges on the cavity surface $= \dfrac{P}{3\varepsilon_0}$ (Lorentz field)
• $E_4$: field of the dipoles inside the cavity $= 0$ for cubic symmetry

$$E_{loc} = E + \frac{P}{3\varepsilon_0} \quad (3)$$

   +  +  +  +  +  +   positive plate      |
       + + + +                            |  E
     (   mol   )      spherical cavity    v
       - - - -        (+ on upper wall, - on lower wall;
   -  -  -  -  -  -   negative plate      field P/(3 e0) adds to E)

Step 4: Combine

Put (1) in (3):

$$\begin{aligned} E_{loc} &= E + \frac{\varepsilon_0(\varepsilon_r - 1)E}{3\varepsilon_0} = E\,\frac{\varepsilon_r + 2}{3} \end{aligned}$$

Equate (1) and (2):

$$\begin{aligned} \varepsilon_0(\varepsilon_r - 1)E &= N\alpha_e\, E\,\frac{\varepsilon_r + 2}{3} \\ \frac{\varepsilon_r - 1}{\varepsilon_r + 2} &= \frac{N\alpha_e}{3\varepsilon_0} \end{aligned}$$

This is the Clausius–Mossotti equation. It links the macroscopic quantity $\varepsilon_r$ with the microscopic quantity $\alpha_e$.

Notes

• Valid for non-polar materials with cubic symmetry (gases, elemental solids like Si, Ge, diamond).
• If $N\alpha_e$ is known, $\varepsilon_r = \dfrac{1 + 2N\alpha_e/3\varepsilon_0}{1 - N\alpha_e/3\varepsilon_0}$.
• At optical frequency $\varepsilon_r = n^2$, giving the Lorentz–Lorenz equation.