Ph.D. thesis (pdf) - dirac

50 Experimental techniques and observables
A more general problem with dielectric spectroscopy is the so-called local field problem,
which we shall shortly discuss in the following. The problem has minor importance
for the present work, we therefore refer to textbooks for more details (e.g.
[Böttcher, 1973]). The specific implications of the local field problem on the interpretation
of dielectric relaxation in viscous liquids is moreover discussed in [Niss and
Jakobsen, 2003; Fatuzzo and Mason, 1967; Daz-Calleja et al., 1993].
The microscopic response corresponding to the dielectric susceptibility is the polarizability,
α, given by 3 p = αE l , (4.2.4)
where p is polarization of a molecule and E l is the field strength applied over the
dipole. The macroscopic polarization is given by the sum of the microscopic polarizations,
∑ p = P. However, the relation between the macroscopic field in the
sample and the field “seen” by a molecule is more complicated. This is because
the field of the molecule itself is part of the macroscopic field while it is not part
of the local field. The most common description of the local field is the Lorentz
field, which leads to the Clausius-Mossotti approximation. The Lorentz field is the
field in a spherical imaginary vacancy in the liquid. By imaginary is meant that the
polarization of the rest of the liquid is calculated as if the dipole was there. This
description includes the polarization of the surroundings due to the dipole, and the
fact that this polarization gives rise to a field acting back on the dipole. Only the
field from the dipole itself is excluded from the calculation of the local field.
The Lorentz field is given by
E l =
(
1 + χ )
E m = ǫ + 2
3 3 E m. (4.2.5)
and inserting this in equation 4.2.4, combined with ∑ p = P and equation 4.2.3
gives the Clausius-Mossotti relation
χ
χ + 3 = ǫ − 1
ǫ + 2 = Nα
3ǫ 0
. (4.2.6)
In deriving the Lorentz field it is assumed that the macroscopic field and the polarization
of each molecule are always parallel. A general field, without this assumption
is given by Onsager [1936]. However the Onsager field is not adequate when the response
is frequency dependent, because the derivation assumes that the field and
3 The polarizability should in general be a tensor quantity as field and polarization are not
necessarily parallel. However, the average polarization 〈p〉 will in isotropic material be parallel to
the average local field 〈E l 〉, and this leads to a meaningful definition of α as a scalar.