23.14.
THE INTERNAL FIELD IN SOLIDS
ngases the atoms are in constant random motion and are separated by sufficiently large
aistances. As such the interaction between the atoms can be neglected. When an extemal field E
will be equal to the
I5 applied, the
intensity of the electric field felt by, a given atom in the gás
applied field E. In solids and liquids, the atoms are so close that they touch each other leading
to
a strong interaction between them. As the atoms are surrounded orn al1 sides by other polarized
atoms. theinternail intensity of the electric field at a given point is,
of the material in general, not
equai to the intensity of the applied field E. The internal field E, which is defined electric
as the
field acting at the location of a given atom, is given by the sum of the electric fields created by the
neighbouring atoms and the applied field. bulk polarization, the additional
In evaludtion of the
effects of the surrounding polarized atoms is to be taken into account. The effective field intensity
E, in the dielectric is given by
E, =
E + E .(23.43)
The value of E' can be evaluted by the sun1mation of all the effects of the surTounding atoms.
To illustrate the method of evaluation, let us consider a one-dimensional solid consisting of a string
of equidistant identical atoms, each of polarizability a,, as depicted in Fig. 23.16.
E E E
-- e+Ze
- na
-a
Fig. 23.16. The clectric eld E, seen by the atoms is different ftom the extermal field E, which is, say giver
byE 1 A one-dimensional solid is considered for computation of local field.
680
to the string.
We shall
field E applied in a direction parallel
external
Let us consider an
by one of the atoms, say A.:The field seen by all
field E experienced
det minethe net internal and from the consideration of symmetry. E, will be parallel to
E
other atoms will also be the same,
is thus
moment induced in each of the atoms of the string
The dipole
ind E
(23.44)
induced in an atom located at a distance 'na' írom i is given
The field at A due to the dipole
by
Ze Ze (na +d)-(na)
E,4tE (na) (na +d) 4TE(na)' (na +d)
Ze 2nad+d 2Zed
(since d« na)
4ntglna)
4Te (na) (na +d)'
24 (Since , Zed)
4TE (na)3
or
2tE (na)
The total field E, at A is given by
EE+27
The factor 2 in the parenthesis takes into account thie atoms to the left and to the right of
atom A.
E-E+
E,E+ t,a° 12H 23.45)
Thus, the combined effect of induced dipoles of neighbouring atoms is to produce a net field
at the location of a given atom which is larger than the applied field. It is seen from the equation
(23.45) that the greater the polarizability a, or the smaller the intermolecular
spacing 'a', the larger
is the intemal field.
The local field in three
a
dirmensionl solid is determined by the structure of the solid. An
accurate calculation of the internal field in solids and liquids is in general very complicated. The
general expression for E, is very much similar to (23.45) where the number density N of atdms
replaces 1la. Since N 4=P, the general expression may be written as
.(23.46)
where Y is the proportionality constant which is known as the interoal field constant.
The value of Y is dependent on the internal
arangement of atoms in the dielectric. In
general, it is of order of unity. It is equal to l1.2/ in case of an infinite chain
of molecules. It can
DIELECIHIUS 31
becvaluatecd in other simple cases also. In case of crystals possessing cubic symmetry. the inierrai
field constant Y 1/3 and the intemal field is given by
E-E L.orentz filed (23.47
160
The field given by the above equat ion (23.47) is called Lorentz field.
23.16. FREQUENCYDEPENDENCE OFTHEDIELECTRICCONSTANT
In many practical situations, a dielectric
is subjected to an alternating electric field. An ac
field changes its direction with time. With each direction
reversal, the polarization components are
required to follow the field reversals in order to contribute to the total polarization of the dielectric.
It follows that the total polarization depends on the ability of dipoles to orient themselves in the
direction of the ficld during cach altcrnation of the field.
The relative permittivity which is a mea-
sureofthe polarization sbows markcd differences in behaviour at dífferent frequencies. The depe
dence of E, on frequency of th electric field is sketched, in Fig. 23.17, for a polar dielectric.
(a)() (c) (
-----|
radio or
microwavc
infra-red
ultra-violet
. 23.17. Schematic illustration of variation of dielectric constant as a function of angular frequency of
alternating electric field applied across a dielectric. In the optical region (d) only electonic
polarization contributes to E,; in the microwave region (c) both electronic and ionic polariza-
tions contribute to E, and in radio frequency region (a) orientation polarization also contribules
In audio frequency region, all types of polarization are possible and the dielectric is characterized
by a polarizabiBity a = a, + a , +a and the polatization P= P+ P, + Pa: At low frequencies, the
dipoies Till get sufficient time to orient themselves completely along the instantaneous direction of
the field. This orientation occurs first in one direction and then in the
other, following the changes
in the direction of the field, as shown in
Fig. 23.18. The average time taken by the dipoles to
No
O NaCA
wwww.O
E
Owwwwwo
(o) (6) (c
Fig. 23.18. The behaviour of(a) permanent; and (b) and (c) induced dipoles ín an alternating clectric field.
The orientation of dipoles occur first in one direction and then in the other following the changes
in the direction of the electric field.
reorient in the field direction is known as the relaxation time t. The reciprocal of the reiaxation
time is called the telaxation frequency T.ifthe frequency of the appied electric field is much
higher than the relaxation frequency of the dipoles, the dipoles cannot reverse fast enough. If the
dipole relaxation timee tis less than half the period of the electric field T ( « Tn), the dipole ean-
easily follow electric field alternations and contribute to orientation polarization. Consequently, the
orientation polarization, which is effective at low frequencies, is damped out for higher frequencies,
reru Sretr). Usually in the radiofrequency or microwave band region, the pernanent dipoles faii to
follow the field reversals and the polarization falls to a value corresponding to (P, + P). As a
result, e, decreases considerably
Again, typically in the infrared region the ionic polarization fails to follow the field reversals
due to the inertia of the system and the contribution of ionic polarizability ceases. In this region,
only electronic polarizatión contributes to the total polarization. Therefore P= P, ln the optical
region, the electron cloud follows the field variations and the material exhibits an electronic polar-
izability a,. The relative permittivity in the optical region will be equal to the square of the
refractive index 'n' of the dielectric. Thus,
| [e, optical regon 23.52)
In the ultravoilet region, the electron cloud too fails to follow the field alternations and
electronic contribution' to the polarization ceases. Consequently, the total polarization becomes
zero. It follows from equation (23.32) that the relative permitivity approaches unity at frequencies
above the ultaviolet range. Thus,
, Ixay 1 (23.53)
To cite the example of water, the Jow frequency dielectric constant, generally referred to as
static dielectric constant, at room temperature is about 80. It falls to about 1.8 in the optical region.
Example 23.5. The value oftrfor glass is 6.75 atfrequencies 10° H. Whar mechanisms are
contributing towards dielectric aonstant? What percentage may be atributed to ionic polariz-
ability? Givn refractive index of'glàss is 1.5. ":
electrostriction. In this case the materials becomes strained in an elect..C field
Ebut in a manner depending on E and there is no inverse of
this effect.
Fig.
2.27. shows the difference between the two effects.
A pyroelectric material is
one which exhibits a spontaneous
the absence of an electric field polarizatiorn in
and changes its polarization on heating. The
effect is represented by
equation
AP AT (2.92)
AP is change of polarization on raising the temperature AT and ^ is the
pyroelectric coefficient.
A ferroelectric material is one which exhibits a spontaneous
the absence of polarization
electric field which may be switched in direction
an
in
by the
application of a field. Ferroelectrics show a hysteresis in the
direction like a ferromagnetic material. polarization-field
All the above mentioned properties are
possible only ir inherently asym-
metric materials. It has been observed that in some
materi-ls, the existence of
a polar axis in a
crystal allows the appearance of spontaneous electrical
polarization and these materials are pyroelectric. A restricted group of
pyroelectrics have the further property of being ferroelectric. Thus, all fer
roelectrics are pyroelectric and piezoelectric. All pyroelectrics are
piezoelectric, but the converse is not true. All piezoelectrics are not
pyroelectrics and all pyroelectrics are not ferroelectrics.
