Page 249 - Electrical Properties of Materials

Polar and non-polar materials 231

which is also the velocity with which energy and information travel. Let us
relate it to the variation of the refractive index,

k
c
dn d(c/v) d –k dk
= = c = + , (10.21)
dω dω dω ω ω ω dω
whence
dω c
= . (10.22)
dk n + ω(dn/dω)
What is anomalous is that the group velocity may be negative provided the
refractive index varies fast enough with frequency. The phase velocity ω/k is
positive, so we have a situation where the phase and group velocities are in
opposite directions. The waves in this situation are called backward waves.
Phase travels in one direction and energy in the other direction. Is that a very
anomalous situation? Not really. One example of a backward wave will appear
later in the present chapter when we discuss the optical branch of acoustic
waves. Admittedly there are not many types around, but it is mostly a question
of getting used to them. Familiarity breeds comprehension.

10.8 Polar and non-polar materials

This is a distinction that is often made for semiconductors as well as dielectrics.
A non-polar material is one with no permanent dipoles. For example, Si, Ge,
and C (diamond) are non-polar. The somewhat analogous III–V compounds,
such as GaAs, InSb, and GaP, share their valency electrons, so that the ions
forming the lattice tend to be positive (group V) or negative (group III). Hence,
the lattice is a mass of permanent dipoles, whose moment changes when a field
is applied. As well as these ionic bonded materials, there are two other broad
classes of polar materials. There are compounds, such as the hydrocarbons
(C 6 H 6 and paraffins) that have permanent dipole arrangements but still have a
net dipole moment of zero (one can see this very easily for the benzene ring).
Then there are molecules such as water and many transformer oils that have
permanent dipole moments, and the total dipole moment is determined by their
orientational polarizability.
A characteristic of non-polar materials is that, as all the polarization is elec-
tronic, the refractive index at optical wavelengths is approximately equal to the
square root of the relative dielectric constant at low frequencies. This behaviour
is illustrated in Table 10.1.
From Table 10.1 (more comprehensive optics data would show the same
trend) you can see that most transparent dielectrics, polar or not, have a re-
fractive index of around 1.4–1.6; only extreme materials like liquid hydrogen,
diamond, and rutile (in our list) show appreciable deviation. Let us look for
an explanation of this remarkable coalescence of a physical property; starting
with our favourite (simplest) model of a solid, the cubic lattice of Fig. 1.1,
with a lattice spacing a. Suppose the atoms are closely packed, each having a
radius r, so that
a =2r. (10.23)