DYNAMICS OF FERROELECTRIC ROCHELLE SALT
B. Žek, G. Shukla, R. Blinc
To cite this version:
B. Žek, G. Shukla, R. Blinc. DYNAMICS OF FERROELECTRIC ROCHELLE SALT. Journal
de Physique Colloques, 1972, 33 (C2), pp.C2-67-C2-68. <10.1051/jphyscol:1972218>. <jpa00214954>
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Submitted on 1 Jan 1972
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Colloque C2, supplkment au no 4, Tome 33, Avril 1972, page C2-67
JOURNAL DE PHYSIQUE
DYNAMICS OF FERROELECTRIC ROCHELLE SALT
B. ZEKS, G. C. SHUKLA and R. BLINC
Institute
((
J. Stefan D, University of Ljubljana, Ljubljana, Yugoslavia
R&um6. - Nous ktudions les propriCtCs dynamiques du modele de Mitsui gknCralis6. Le
spectre des fluctuations de polarisation peut &re a peu prh dkrit par un seul temps de relaxation
qui montre une attimuation. Ce temps de relaxation est de mkme ordre de grandeur pour le sel de
Rochelle deutCre ou non.
Abstract. - The dynamical properties of the extended Mitsui's model are investigated. The
spectrum of the polarization fluctuation can be approximately described by a single relaxation
time which exhibits the critical slowing down. This relaxation time is shown to be of the same
order of magnitude in deuterated and in undeuterated Rochelle salt.
Though Rochelle salt has been the first ferroelectric
crystal to be discovered, it is still not understood very
well from a microscopic point of view. The shifts of
the upper Curie point towards higher temperatures
and of the lower towards lower temperatures on deuteration demonstrate the role of the hydrogen atoms in
its ferroelectric behavior, but no theoretical explanation of these isotope shifts which increase the ferroelectric range by about 40 % has been proposed so
far.
We have extended the two-sublattice model of
Mitsui [l] in order to describe quantum effects in
Rochelle salt [2]. The model assumes that the ferroelectric dipoles move in asymmetric double-well
crystalline potentials and form two interpenetrating
sublattices, which are mirror images of each other.
The Hamiltonian of the problem can be expressed in
terms of quasi-spin-+ operators
X = - C [.Iij(s!t'
ij
- 2Q
siy + s::) s$)
+ K~~s!t>~!,3)]
ci (SLY + sgj) - A c (s::)- s$)
j
- 2 pE
(sly
+ s::)).
(1)
Here H, are the absolute values and Hz,, the z-components of the molecular fields acting on quasi-spin in
the sublattice a. One solution of the system (2) in the
absence of the external field is such, that
and the spontaneous polarization
equals zero. In one temperature region another solution exists with < sA2' > # - < sil)> and P f 0.
The transition temperature Tc is given by
At some values of parameters this equation has two
solutions : T,, and T,,. For T < Tc, or T > Tc, the
paraelectric phase is -stable, but fo; Tc, < T 2 Tc,
the system is spontaneously polarized. Using
A=873cm-l,
K = 1560cm-I,
i.
The indices (1) and (2) refer to the two sublattices, J
and K are the effective interaction constants of dipoles
belonging to the same and different sublattice, respectively, A is the measure of the asymmetry of the local
crystalline potential, p is the dipole moment interacting
with the external electric field E, and 52 is the tunneling
integral which measures the amount of delocalisation
of ferroelectric dipoles.
If we solve the problem in the molecular field approximation we obtain two coupled equations for the
two sublattice polarizations
J = 144cm-l,
52 = 0 and p = 4.9D
a rather good agreement between the experimental
and theoretical statical properties for deuterated
Rochelle salt can be obtained. The isotope shifts on
replacing hydrogen for deuterium are then obtained
by introducing a nonzero value of the tunneling integral Q x 30 cm-*.
We investigated [2]the dynamical properties of the
system described by the Hamiltonian (1) in the random
phase approximation only for the case of deuterated
Rochelle salt (0 = 0). The dipolar system was supposed to be in thermal contact with a large heat bath.
Following the treatment of the Ising model by Kubo
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972218
B.
C2-68
ZEKS, G. C. SHUKLA AND R. BLINC
and Suzuki [3] we introduced the transition probability for quasi-spin SZJjas
1
1 - 2 S , , tg h 2 fl H j )
(5)
where 7, is the correlation time of a noninteracting
dipole. In such an approximation a one-mode polarization relaxation results in the paraelectric phase and
two-mode relaxation in the ferroelectric phase. One of
these two relaxation times exhibits a critical slowing
down at Tc, and Tc,.
Let us now investigate the dynamical properties if
Q # 0. We will apply here the methods known from
the theory of ferromagnetism [4]. We now consider
the dynamic response of the system to a time dependent external field.
As a consequence, expectation values of observables
become in general time dependent. For the time rate
of change of the average spins < SF' >, we obtain
from the Heisenberg equations of motion, linearizing
them in the random phase approximation
Here < S(")>, Ha are the constant parts and
6 < s?' >,
the fluctuating parts of the expectation values of spins and molecular fields, respectively. We solve this system of equations for the wave
vector q = 0 in the paraelectric phase and obtain six
eigenfrequencies, only three of them contributing to
the polarization relaxation. One of them identically
equals zero and corresponds to the motion in the
direction of the molecular field. The other two eigenfrequencies are real and temperature dependent, but
they do not have any critical temperature dependence.
This approximation does not describe the dynamical
properties of the system properly. The eq. (6) allows
only the motion of spins in the plane perpendicular to
the molecular field, but the direction of the instability
of the paraelectric phase is not in this plane.
We must allow also the motion in the molecular
field direction. In order to take this effects into account
in a qualitative way, we add phenomenological spinlattice relaxation terms to our equation of motion.
We assume that each average moment S < s,!a)>
relaxes towards the (time-dependent) equilibrium
> belonging to the instantaneous
value 6 <
~HP)
SSP'
molecular field. We assume two different relaxation
times for the relaxation of the parallel and perpendicular components. We thus take as our equation of
motion
P?) and ~ fare) projection operators parallel and
perpendicular to the molecular field. If we linearize
these equations and solve them for the paraelectric
phase, we find that the fluctuations of polarization
with q = 0 are determined by three eigenfrequencies.
One of them is imaginary. The corresponding relaxation time 112 is proportional to the longitudinal spinlattice relaxation time 117, and exhibits the critical
slowing down. The direction of the corresponding
eigenvector is temperature dependent and at Tcl and
Tc, agrees with the direction of the instability of the
paraelectric phase. The other two eigenfrequencies
are complex and correspond to the motion of spins
in the plane perpendicular to the first eigenvector.
Because of the critical slowing down and the small
angle between the polarization axis and the first eigenvector, we can approximately describe the dynamical
properties by the single relaxation time 117. Neglecting
the transversal relaxation (117, = 0) we get the approximate expression
This relaxation time does not depend strongly on S2
and is of the same order of magnitude in deuterated
and undeuterated Rochelle salt. The relatively small
change in the tunneling integral S2 gives rise to the
relatively large change of static properties but does
not effect the dynamical properties very much. This is
in agreement with experiments which show that the
polarization relaxation time is in Rochelle salt [5]
and in deuterated Rochelle salt of the same order of
magnitude on the contrary to the situation in the
KDP-type crystals.
References
[I] MITSUI(T.), Phys. Rev., 1958, 111, 1259.
[2] Z E K ~(B.), SHUKLA
(G. C.) and BLINC(R.), Phys. Rev.,
1971, B3, 2306.
[3] SUZUKI(M.) and Kuso (R.), J. Phys. Soc. Japan,
1968, 24, 51.
(H.), Phase Transitions in Magnetic Systems,
[4] THOMAS
Conference on Magnetism, Chania, Crete, 1969.
[5] KESSENIKH
(G. G.), SHIROKOV
(A. M.), SHUVALOV
(L. A.), Kristallografiya, 1968, 13, 452.
[6] KESSENLKH
(G. G.), SHIROKOV
(A. M.), SHUVALOV
(L. A.) and SHCHAGINA
(N. M.), Kristallografya,
1970, 15, 1254.