# Clausius Mossotti equation

**STATIC DIELECTRIC CONSTANTS OF SOLIDS**

As discussed in the previous articles,the dielectric polarization can be considered to arise from three major sources:

Electronic ,ionic and orientational (dipolar) polarizations ,that is

P_{total} =P_{e}+P_{i}+P_{0}

Thus, the materials may be classified into three types according to the dielectric behaviour of solids is concerned :

**I. **Solids exhibits only electronic polarizability **Element Solid** **Dielectrics.**

**II. **Solids which possess electronic and ionic polarizabilities **Ionic non-polar solid dielectrics (Ionic dielectrics without permanent dipoles).**

**III. **Solids which possess orientational ,electronic and ionic polarizabilities –**Polar solids (Solids containing permanent dipole moments)**

Let us discuss them one by one:

**(i) ****Elemental solid Dielectics**

These are the materials consisting of single type of atoms such as diamond, germanium. These materials contain neither ions nor permanent dipoles and therefore exhibit only electronic polarization.

Under the influence of the internal field E_{i}the dielectric solid displays an electric dipole moment P

That is P=α_{e}E_{i }(1)

Where α_{e} is electronic polarizability.

If n is the number of molecules per unit volume of the dielectric, then polarization (p) is given by

P=np

By substituiting equation (1) in above equation we get

P=nα_{e}E_{i } (2)

As E_{i}=E+P/3ε_{0 }(3)

By putting equation (3) in equation (2) we get

P=nα_{e}[E+P/3 ε_{0}] (4)

If £is the permittivity of the dielectric, the displacement D is given as

D= εE= ε_{0}E+P

Or P={ ε- ε_{0})E

Or E=P/ε- ε_{0} (5)

By putting equation (5) in equation (4), we get

P=nα_{e}[P/( ε- ε_{0)}+ P/3 ε_{0}]

= nα_{e}P[2 ε_{0}+ ε/3 ε_{0} (/ε- ε_{0})

Or 3 ε_{0}/nα_{e}= (ε+2 ε_{0})/( ε- ε_{0})

Or (ε- ε_{0})/(ε+2 ε_{0})=nα_{e}/3 ε_{0 }(6)_{ }

As ε_{0}= ε_{r} ε_{0} (7)

By putting equation(7) in equation (6) we get

(ε_{r} ε_{0}– ε_{0)}/ (ε_{r} ε_{0}+2 ε_{0)}=nα_{e}/3 ε_{0}

Or ε_{0} (ε_{r}-1)/ ε_{0} (ε_{r}+2)= nα_{e}/3 ε_{0}

Or (ε_{r}-1)/ (ε_{r}+2) = nα_{e}/3 ε_{0}

This equation is called **Clausius –Mossotti equation**.

This equation shows that the dielectric constant is determined by n, α_{e }and γ (Here γ=1/3)

α_{e }is not the same as the polarizability of the free atoms because the binding between the atoms effect the various electrons. The distance between the atoms in the solid is affected only slightly by temperature and thus n, α_{e}, γ and relative dielectric constant £_{r} are independent of temperature for the dielectric material under discussion.