Page 256 - Electrical Properties of Materials

238 Dielectric materials

single-element case. There is, however, an upper branch, known as the op-
tical branch. The values of ω at the points k = 0 and π/2a can be obtained
from eqn (10.55), and may be seen in Fig. 10.12. The upper branch represents
a backward wave. Phase and group velocities are in opposite directions. The
highest point in the dispersion curve is at k = 0. The corresponding wavelength
is typically tens of micrometres in the middle of the THz region.
An interesting effect discovered in the 1920s (known as the Reststrahl
effect, or sometimes as residual radiation) is that these lattice vibrations
may interact with electromagnetic waves. The effect is manifested in large
absorption at one wavelength and in large reflection at a slightly different
∗
∗ Note that in the presence of absorption wavelength. The corresponding values are given in Table 10.2 for a few ma-
the frequency of maximum reflection, f r , terials. The fact that significant reflection occurs only within a narrow band has
need not coincide with that of minimum been used to provide monochromatic sources.
transmission, f t .
Another remarkable property of the optical branch is that it can provide,
within a certain frequency range, a negative effective dielectric constant. This
Table 10.2 Frequencies of maxi- is quite a rare phenomenon. We have come across it only once before, in
mum reflection (f r ) and minimum Chapter 1, when discussing the critical frequency of transparency of metals,
transmission (f t ) for a number of as part of the theory of plasmas.
alkali halides In order to discuss the interaction with electromagnetic waves we shall use
a model which is less general in one sense, in that it is valid only in the vicinity
Crystal f (THz) f (THz) of k = 0, but is more general in another sense. We shall assume that the restor-
r
t
ing force is electrical. To simplify the mathematics we shall not consider two
NaF 7.39 8.38
separate atoms but shall write the equation of motion for a single atom which
TlF 4.44 6.17
NaCl 4.90 5.76 has a charge e and a reduced mass given by
KCl 4.24 4.73
RbCl 3.54 4.06 1 1 1
TlCl 2.56 3.26 ∗ = + . (10.56)
m m 1 m 2
KBr 3.40 3.68
KI 2.94 3.19
The equation of motion may then be written as
2
d x T 2 e
+ ω x T =– E , (10.57)
T
dt 2 m ∗
where x T is the displacement relative to the centre of gravity of the two atoms,
2
and ω is a restoring force.
T
Up to now the two different atoms could be of any kind, provided they make
up a solid. A look at Table 10.2 will reveal that all those materials have ionic
bonds so we may legitimately assume that one of the atoms has a positive
charge and the other one has a negative charge.
We are interested in what happens in the infrared region. Whatever hap-
pens can be regarded as a small perturbation of the stationary state. We may
assume that the density of atoms will have a component, n 1 , that varies at a
frequency, ω, in the infrared range. Hence the total density may be written
as N o + n 1 exp(–iωt), where N o is the unperturbed density and n I N I .The
†
† Its physical meaning is that the cur- continuity equation is then
rent flowing out of an element dx can
be different from the current flowing in,
provided the charge density within that e dn I + dJ I = 0, (10.58)
interval has increased or decreased. dt dx