Problem
A linear chain consists of polarizable molecules which are separated by lattice spacing a. The molecules are fixed to their position, but they have an internal degree of freedom described by the equation of motion
where p is the electric dipole moment of the molecule (assumed to be
parallel to the chain), E is the local electric field, and 
is
the polarizability. Each molecule feels the electric field of the others.
The system is at zero temperature, and quantum effects are small.
Calculate and plot the dispersion curve 
for small amplitude
polarization waves (optical phonons). Discuss the behavior of 
as
a function of ?
Solution
Assume that site m has a polarization given by
The electric field at site n due to site m is
where 
, the distance between m and n.
The total field at site n due to all other sites is therefore
The sum can be converted to
Therefore we obtain the local field at point n to be
Figure 2.1: The function C(k), defined by Eq. 2.7,
in the 
limit.
From the equation of motion we obtain
The function C(k), plotted in Figure 2.1,
looks like a distorted cosine curve.
Assuming C(k) is known, we can solve for 
:
The result is plotted in Figure 2.2. For small polarizability the
square root can be expanded and the dispersion is small: 
. As the coefficient 
increases
the k=0 mode frequency approaches zero.
If the polarizability exceeds a critical value, 
,
the solution for small k becomes imaginary. In this
range of parameters the system becomes ferroelectric: A spontaneous
polarization develops along the chain, and the inversion symmetry is
broken.
In a more general context,
symmetry breaking soft modes are called Goldstone bosons.
Our solution relies on the assumption that the polarization is oscillating
around p=0 (see Eq. 2.2). For 
the anharmonic terms in Eq. 2.1 has to be taken into
account,
and the vibrational frequencies can be derived in terms of expanding
p around the new equilibrium polarization.
Figure 2.2:
The frequency as a function of wavenumber for several values of
.
The softening of an optical
phonon mode is a typical precursor of the ferroelectric transition, and
the soft optical phonons has been observed in
ferroelectric materials like SrTiO 
.
