Molecular polarisability and the low-frequency dielectric constant

It is relatively easy to calculate the observed polarisation of an assembly of non-interacting molecules for a given field l at each molecule at a temperature T. Assume that each molecule has a permanent dipole p and that there is no change in its magnitude on application of the field. Consider a molecule for which the dipole makes the angle 6 with l. The energy u of the dipole is then given by 

If there are dn dipoles within any small solid angle dco at angle 6 to the applied field, Boltzmann s law shows that 

For attainable electric fields, iiEi / kT) = x is very small and the exponentials can be expanded as e = 1 x -I- x jl v /6 + 0(x ). Substitution then shows that the Langevin function reduces approximately to L x) = v/3, so that finally 

In developing this equation, no account has been taken of any possible interactions between the molecular dipoles. It is therefore expected to be 

At optical frequencies a = 0 , because the molecules cannot reorient sufficiently quickly in the oscillatory electric field of the light wave for the orientational polarisability to contribute, as discussed further in section 9.2.4. Setting a = a in equation (9.9) gives in terms of the optical refractive index and, if this expression for is inserted, equation (9.16) can then be rearranged as follows  

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