Elements of Modern Physics
306
p =
cos
cos exp
cos
cos
exp
cos
θ
θ
θ
θ
θ
∫
∫
pE
d
kT
p
pE
d
kT
(8.135)
=
ln
−
=
−
a
a
pE
a kT
d
e e
p da
a
For pE << kT, p ≈ p
2
E/3kT which leads to
α = p
2
/3kT
(8.136)
which at room temperatures is of the order of 10
–39
F.m
2
, comparable to the
electronic polarizability. It is distinguished by its temperature dependence, and
suggests a relation
α tot = α 0 + p
2
/3kT
(8.137)
This expression is quite useful in determining dipole moments of dipolar
substances, e.g., HCl (p = 1.1 debyes, 1 debye = 10
–39
C.m), by looking at the
temperature dependence of α tot . It is tempting to substitute Eq. (8.136) in
Eq. (8.124) to obtain
ε =
2
3
1
,
9
+
=
−
ε
c
c
c
o
T
Np
T
T T
k
(8.138)
which would imply that spontaneous polarization (E = 0) sets in at T = T c . It has
however been shown by Onsager that Eqs. (8.123) and (8.124) are not valid for
permanent electric dipoles. The theory of Onsager for permanent dipoles does
not imply the existence of a critical temperature for such dipoles.
Ferroelectric Crystals
The phenomenon of spontaneous polarization (E = 0), known as ferroelectricity
is observed in (i) Rochelle salt and some of the associated salts, (ii) some crystals
with hydrogen bonds, in which the motion of protons gives rise to ferroelectric
behaviour (e.g., KH 2 PO 4 , RbH 2 PO 4 , etc.) and (iii) ionic crystals with perovskite
(CaTiO 3 ) and ilmenite (FeTiO 3 ) structures. The perovskite structure illustrated
by BaTiO 3 is the simplest structure which exhibits ferroelectricity—it has a
cubic structure with Ba at the corners, oxygen at the face centres and Ti at the
body centre. Ferroelectricity in barium titanate (BaTiO 3 ) is briefly described
here.
Barium titanate becomes ferroelectric at 380 K and exhibits hysteresis curves
for T < T c , in the plot of D against E. The ferroelectricity in BaTiO 3 is due to
induced electronic and ionic dipole moments. From Eq. (8.124)
ε =
0
0
2
1 3
1
1 3
+
α
ε
−
α
ε
∑
∑
i i
i
i i
i
N
N
(8.139)
306
p =
cos
cos exp
cos
cos
exp
cos
θ
θ
θ
θ
θ
∫
∫
pE
d
kT
p
pE
d
kT
(8.135)
=
ln
−
=
−
a
a
pE
a kT
d
e e
p da
a
For pE << kT, p ≈ p
2
E/3kT which leads to
α = p
2
/3kT
(8.136)
which at room temperatures is of the order of 10
–39
F.m
2
, comparable to the
electronic polarizability. It is distinguished by its temperature dependence, and
suggests a relation
α tot = α 0 + p
2
/3kT
(8.137)
This expression is quite useful in determining dipole moments of dipolar
substances, e.g., HCl (p = 1.1 debyes, 1 debye = 10
–39
C.m), by looking at the
temperature dependence of α tot . It is tempting to substitute Eq. (8.136) in
Eq. (8.124) to obtain
ε =
2
3
1
,
9
+
=
−
ε
c
c
c
o
T
Np
T
T T
k
(8.138)
which would imply that spontaneous polarization (E = 0) sets in at T = T c . It has
however been shown by Onsager that Eqs. (8.123) and (8.124) are not valid for
permanent electric dipoles. The theory of Onsager for permanent dipoles does
not imply the existence of a critical temperature for such dipoles.
Ferroelectric Crystals
The phenomenon of spontaneous polarization (E = 0), known as ferroelectricity
is observed in (i) Rochelle salt and some of the associated salts, (ii) some crystals
with hydrogen bonds, in which the motion of protons gives rise to ferroelectric
behaviour (e.g., KH 2 PO 4 , RbH 2 PO 4 , etc.) and (iii) ionic crystals with perovskite
(CaTiO 3 ) and ilmenite (FeTiO 3 ) structures. The perovskite structure illustrated
by BaTiO 3 is the simplest structure which exhibits ferroelectricity—it has a
cubic structure with Ba at the corners, oxygen at the face centres and Ti at the
body centre. Ferroelectricity in barium titanate (BaTiO 3 ) is briefly described
here.
Barium titanate becomes ferroelectric at 380 K and exhibits hysteresis curves
for T < T c , in the plot of D against E. The ferroelectricity in BaTiO 3 is due to
induced electronic and ionic dipole moments. From Eq. (8.124)
ε =
0
0
2
1 3
1
1 3
+
α
ε
−
α
ε
∑
∑
i i
i
i i
i
N
N
(8.139)