DYNAMICS OF FERROELECTRIC ROCHELLE SALT

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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�. �jpa-00214954�

JOURNAL DE PHYSIQUE Colloque C2, supplkment au no 4, Tome 33, Avril 1972, page C2-67

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 p r h 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 deute- ration demonstrate the role of the hydrogen atoms in its ferroelectric behavior, but no theoretical explana- tion of these isotope shifts which increase the ferro- electric 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 ferro- electric 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

Here H, are the absolute values and Hz,, the z-compo- nents 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 solu- tion exists with

< sA2' >

^{# -}

< sil) >

and P f 0.

The transition temperature Tc is given by

X = -

C

[.Iij(s!t'

siy + s::)

^s$)

+

K~~

s!t> ~!,3)]

At some values of parameters this equation has two

i j solutions : T,, and T,,. For T < Tc, or T > Tc, the - 2 Q

c (SLY ^{+ sgj)}

^{- A}

c ^(s::)

^-

s$)

paraelectric phase is -stable, but fo; Tc, < T

2

Tc,

i j the system is spontaneously polarized. Using

- 2 pE ( s l y

+ s::)) .

(1) A = 8 7 3 c m - l , K = 1 5 6 0 c m - I ,

.i

The indices (1) and (2) refer to the two sublattices, J J = 1 4 4 c m - l , 52 = 0 and p = 4.9D and K are the effective interaction constants of dipoles

belonging to the same and different sublattice, respec- tively, 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 appro- ximation we obtain two coupled equations for the two sublattice polarizations

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 inte- gral 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 sup- posed 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

C2-68 B. ZEKS, G. C. SHUKLA AND R. BLINC and Suzuki [3] we introduced the transition probabi-

lity for quasi-spin SZJj as

1 - 2 S , , tg h - 2 1

fl

H j ) ( 5 )

where 7, is the correlation time of a noninteracting dipole. In such an approximation a one-mode polari- zation 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 depen- dent 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?' >, ~ H P )

the fluctuating parts of the expec- tation values of spins and molecular fields, respecti- vely. 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 eigen- frequencies 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 spin- lattice relaxation terms to our equation of motion.

We assume that each average moment S

< s,!a) >

relaxes towards the (time-dependent) equilibrium value 6

< SSP' >

belonging to the instantaneous

molecular field. We assume two different relaxation times for the relaxation of the parallel and perpendi- cular components. We thus take as our equation of motion

P?) and

~ f )

are 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 relaxa- tion time 112 is proportional to the longitudinal spin- lattice 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 eigen- vector, we can approximately describe the dynamical properties by the single relaxation time 117. Neglecting the transversal relaxation (117, = 0) we get the appro- ximate 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. [4] THOMAS (H.), Phase Transitions in Magnetic Systems, Conference on Magnetism, Chania, Crete, 1969.

[2] ^{Z E K ~} (B.), SHUKLA (G. C.) and BLINC (R.), Phys. Rev., [5] KESSENIKH (G. G.), SHIROKOV (A. M.), SHUVALOV 1971, B3, 2306. (L. A.), Kristallografiya, 1968, 13, 452.

[6] KESSENLKH (G. G.), SHIROKOV ^(A. M.), SHUVALOV [3] SUZUKI (M.) and Kuso (R.), J. Phys. Soc. Japan, (L. A.) and SHCHAGINA ^(N. M.), Kristallografya,

1968, 24, 51. 1970, ^15, 1254.