precursor order clusters at ferroelectric phase transitions

Physica 125B (1984) 53--62 North-Holland, Amsterdam

PRECURSOR ORDER CLUSTERS AT FERROELECTRIC PHASE TRANSITIONS

A. G O R D O N and J. G E N O S S A R Department of Physics, Technion- Israel Institute of Technology, Haifa 32000, Israel

Received 8 December 1983 Revised 14 February 1984

A model for the description of precursor order clusters at ferroelectric phase transitions is proposed. In this model the cluster walls separating the paraelectric and ferroelectric regions are obtained as kink solutions of the nonlinear dynamic equations including a damping term. The cases of first- and second-order phase transitions and of the tricritical point are considered. It is shown that the precursor order clusters of this type bring about the intrinsic central peak at the low-temperature first-order ferroelectric phase transition in proustite (Ag3AsS3). The analysis of the 75As nuclear quadrupole resonance and spin-lattice relaxation data for proustite indicates that the central peak for the hydrostatic pressures p ~< 3.2 kbar is apparently caused by pretransitional heterophase fluctuations which are described by the above-mentioned model. Beyond the pressure-induced tricritical point at p > 3.2 kbar the central peak is associated with the precursor order clusters of another type also described by this model. The features of spin-lattice relaxation near first-order phase transitions, in particular, in proustite are considered.

1. Introduction

According to the cluster picture of structural phase transitions [1], the homogeneously ordered phase is unstable against the formation of stron- gly nonlinear order parameter configurations associated with cluster walls within the temperature range Tc < T < Tt, where Tc is a phase transition temperature and T~ is a temperature at which the precursor order clusters appear. It leads to destruction of long-range order, but retains short-range order. At the temperature TI the crossover from the displacive phase transition behaviour to the order--disorder regime takes place. In the order--disorder region the atoms vibrate around positions displaced from the high-symmetry sites to form clusters of dynamic precursor order.

The experimental support for the cluster picture is provided by the observations of the central peak in inelastic neutron scattering [2, 3] and magnetic resonance studies [4-9]. Some evidence for the central peak at structural phase transitions is provided by the appearance of low- symmetry nuclear magnetic resonance and nuclear quadrupole resonance (NQR) spectra in

the high-symmetry phase [10]. The origin of the intrinsic central peak in such a case is associated with occurrence of precursor order clusters [11]. At the same time the precursor order clusters are shown to appear in the spin-lattice relaxation (SLR) time measurements by anomalous increase in SLR time on approaching the phase transition [12, 13], by the occurrence of SLR time fast component near the phase transition [8] and by a strong nuclear magnetic resonance frequency dependence [12, 13]. The results of calculations of the temperature dependence of the 35C1 N O R line intensity in ferroelectrics HCI and HCI-DCI have been also considered as an evidence of formation of pretransitional heterophase fluctuations [9].

A model for description of precursor order clusters at structural phase transitions has been proposed by some authors [14-17]. A modification of this model has been considered in [18] for tricritical phase transitions in ferroelectrics. In this model the solutions of the corresponding dynamic equations which have been derived as cluster walls were of domain wall type, i.e. they represent the walls which separate regions where the order parameter of the phase

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54 A. Gordon and J. Genossar / Precursor order clusters

transition has the same absolute value, but opposite sign. We assume that in the cluster picture one can deal with the walls separating the regions of ordered and disordered phases. Hence it is worthwhile to propose a model f rom which we can obtain cluster wall type solutions corresponding to the walls between the paraelectric and ferroelectric phases in the case of ferroelectric phase transitions.

In this paper we propose a model for the description of precursor order clusters at ferroelectric phase transitions. In this model the cluster walls separating the paraelectric and ferroelectric regions are obtained as kink solutions of the nonlinear dynamic equations including a damping term. The occurrence of the precursor order polarized clusters in the paraelectric phase of the ferroelectric proustite (mg3AsS3) leading to the central peak in the 75As N Q R and its SLR time is shown. The character of these clusters near the first-order phase transition and in the vicinity of the pressure-induced tricritical point is considered.

2. A model for precursor order clusters at ferroelectric phase transitions

We start with following classical Hamil tonian for a one-dimensional system which might undergo a phase transition at tempera ture T = 0 [14-171:

\ d t ] + V(u~)

K "{- "~- (Ui+I -- Ui)2] , (1)

where ui is the displacement of the ith particle of massa ml which moves in the double-well potential V(u~) and interacts with the nearest neigh- bour by the harmonic coupling and K is the positive constant of this interaction.

We assume that the Hamiltonian (1) can be replaced by a continuum representat ion

(2)

where ux, ut are the first derivatives of u with respect to the coordinate x and time t, respectively; ui+l -u i - -~dux , where d is the lattice spacing, c 2 = 2d2K.

Eq. (2) may be regarded as the energy func- tional of a classical scalar field u = u(x, t) with Lagrange density

m I~ = - f u ~, - u ~ - V ( u ) . (3)

The Euler -Lagrange equation of motion for the field u = u(x, t), which follows from eq. (2), is given by

d V mc2ou,~ - muu du - O, (4)

where u= and uu are the second derivatives of the displacement u with respect to the coordinate x and time t, respectively. We take into account a damping in the system adding the term with the first derivative of u with respect to t - ut:

dV mcgu,~, - mun + Tu, - ~ = 0 , (5)

where T is a damping constant. We take the four cases of the potential V(u):

a b 4 V = - 2 u2 + -4 u , (6a)

a u2 b c 4 v = ~ -~ u 3 +-~ u , (6b)

a u2_b 6 U 6 V=~- ~ u ' + , (6c)

a 2 b 6 V = - - ~ u + ~ u , (6d)

where a, b, c > 0. The potential (6a) provides a second-order phase transition, (6b) and (6c) give a first-order phase transition and (6d) leads to the tricritical phase transition.

The partial differential equation (5) in in-

A. Gordon and J. Genossar / Precursor order clusters 55

dependent variables x and t can be reduced to an ordinary differential equation in the variable s by the substitution

s = x - v t , (7)

type solution which we seek corresponds to a trajectory in the (u, y) phase plane, of the system (11). From (11a) and (11b) we can obtain a differential equation for the solution trajectory as

where v is a velocity in the direction x. This transformation implies the change of our coordinate system to a moving one. After the transformation (7) we obtain

( V02 ) d V 0 (8) mc2o 1 - Us, - yvu~ - d---ff = '

. . . . o . (12) m c 2 1 + v y Y

We recognize that (12) can be satisfied by a trajectory of the form

y = ~ ( n ) u ( u " - u ~ ) , (13)

where us, us` are the first and second derivatives of u with respect to s. Eq. (8) represents New- ton's second law for a particle of mass m c E ( 1 -

v2/c~) moving in a potential - V under influence of a frictional force proportional to - y v with time t replacing s, provided u is interpreted as displacement of a particle.

The solution which we seek is stationary in the moving coordinate system and it is of wave front type. We assume that it satisfies the following boundary conditions:

where n = 1 for (6a) and (6b) and n = 2 for (6c) and (6d), f l ( n ) is a constant which is determined upon substitution of (13) into (12). The substitution of (13) into (12) also gives the velocity v ( n ) . From (10) we have

1 f ( U " v ? ' ~ ) d u (14) s = /~(n)

The integration in (14) leads to the following solutions of (8):

u , ~ O , s ~ ---~, (9a)

u o u ~ f o r s - > + % u--*0 f o r s - > - o 0 (9b)

where u~ is the minimum of the potential V corresponding to a ferroelectric phase and u = 0 corresponds to a paraelectric phase (u2 is the second minimum of V).

To find the solution of eq. (8) satisfying the boundary conditions (9), we introduce the following notation

u, = - y . (10)

Then we arrive at the system of equations

us = - y , ( l l a )

m c 2 1 - Ys- vyy + V' = O, ( l l b )

Ul I TM u = l + e x p F s / A ( n ) ] J ' (15)

where A(n) is the thickness of the front. The solutions of this type are shown in the general case to be stable under small perturbations [19].

Thus the solution of (8) for the potential (6a) is given by

Ul u = 1 + exp( - s /A) ' (16a)

where

Ul = %/a---~, (16b)

and

x/ 2m A = Co a(1 + 9 a m / 2 y 2) (16c)

where V ' ~ d V / d u . Therefore the wave front provided


Co (16d) v = ~,/1 + 2 y 2 / 9 a m "

The kink solution (16a) describes the cluster wall separating the regions where u = V ' a ~ (a ferroelectric phase) and u = 0 (a paraelectric phase). Thus A is the thickness of the cluster wall and v is its velocity. The profile of the cluster wall given by eq. (16a) is shown in fig. 1.

We obtain the following solution of eq. (8) for the first-order phase transition case with the potential (6b):

ul (17a) u - 1 + exp(s/A) '

where

b ~ /bc 2 a Ul,2 = --I- ¢ '

and

provided

v - Co (17e) V ' I + 1/L"

For the first-order phase transition case (6c) we have

u - Ul (18a) V'I + exp(-s/A) '

where

a u],2 = --- (18b)

C '

and

_ Co ~/( 3m A - ~ 1 + M ) ( b u 2 - a ) ' (18c)

2m (17c) where A = co ( l + L ) ( b u l - a ) '

M = m (bu21- 4a2 ) 2 (18d) where 3"), 2 (bUZl- a ) '

L = m ( b u l - 3 a ) 2 (17d) 2 y 2 ( b u l - a ) '

provided

Co (18e) v - ~ /1 + 1 / M "

u

1.0

/ ~ 0 . 5

i ~---,-----"~ I ~ l f i I I I I -4 -2 0 2 4 S ~

Fig. 1. The profile of the cluster wall separating ferroelectric and paraelectric regions describing by the kink solution (16) of the nonlinear equation (5) for a second-order phase transition.

For the tricritical phase transition (6d) we derive

ul (19a) u = X/1 + e x p ( - s / A ) '

where

u~ = ~ /a -~ , (19b)

and

Co 4 3m A = ~- a(1 + 1 6 m a / 3 y 2 ) ' (19c)


provided

C0

v = X/1 + 372/16ma (19d)

3. Precursor order clusters at the ferroelectric first-order phase transition in proustite

Proustite (mg3msS3) undergoes a structural phase transition at T = 26K [20,21,23]. This phase transition is ferroelectric [24]. It has been shown that in proustite the spontaneous polarization is not reversible, but is reoriented by the electric field through a crystallografically determined angle, which differs from 180 ° [24]. Structural aspects of this ferroelectric phase transition have been investigated in [21, 25, 26]. This phase transition is accompanied by a C3v~ Cs or C~ change of symmetry. The asymmetry parameter of the 75As electric field gradient tensor determined from the Zeeman-effect of N Q R spectra is equal to 6q = 0.053 --4-- 0.009 at T < 54 K and is equal to zero at T > 54 K [25].

Fig. 2 shows the temperature dependence of the 7~As (spin I = 3/2) N Q R frequency v within the temperature range 4.2 K < T < 100 K [21]. It is seen that there is a single line of the spectrum at T > 5 4 K and T < 2 6 K . Between 2 6 K < T < 54 K eight lines are observed. Their symmetry is characteristic of the ferroelectric phase of proustite (6q# 0), although they are observed in the parae!ectric phase. The measurements are done on powdered samples with different concen- trations of impurities. The occurrence of the eight

v,MHz I

68.5 F

67.01--

6 6 . 5 ~ l ~ I I I 0 20 40 60 80 I00 T°K

Fig. 2. The temperature dependence of the 75As N Q R frequency v within the tempera ture range 4.2 K < T < 100 K in prousti te [21].

N Q R lines does not depend on the impurity concentration in a sample.

The phase transition at T = 2 6 K is accompanied by a discontinuous change of the 75As NQR spectrum and by a thermal hysteresis with the value of 4 K. These features are characteristic of first-order phase transitions. The change of the multiplicity of the 75As N Q R spectrum at T = 54 K has been considered in [20, 21] as a phase transition. However the X-ray in- vestigations [26] have not confirmed the occurrence of a phase transition.

The temperature dependence of the 75As SLR time TI (fig. 3) does not contain the usual critical singularity near the phase transition up to the value of hydrostatic pressure p = 3.2 kbar [22]. In fig. 4 (the pressure dependences of the phase transition Tc and of the magnitude of thermal hysteresis and pressure change of the region of multiplicity within the range 0 k b a r < p < 5.8kbar are shown. The region of the temperature hysteresis of Tc is shaded. The hysteresis approaches zero as p ~ 3 . 2 k b a r . The number of lines decreases, as the hydrostatic pressure increases. At p > 3.2 kbar there are only two lines in the spectrum. At p i> 4.3 kbar only one N Q R line is observed.

Tt,m $ i I I I

400 i

80

2O

0 I I I I ~T IP 20 30 40 50 60 T.=K

Fig. 3. The temperature dependence of the 75As S L R t ime T] at var ious hydrostat ic pressures p in proust i te ; A atmosphere pressure, × p = 1.7; [ ] 3.0; O 3.6; + 4.0; • 5.0; • 5.8 kbar [22].


o K ! Tc,

6 0

50

40

I I l 1 I 0 1 2 3 4 5 P,kbor

Fig. 4. The pressure dependences of the phase transition temperature To, of the magnitude of thermal hysteresis (b) and the pressure change of the region of multiplicity (a) within the range 0 < p < 5.8 kbar measured by the 75As NQR in proustite [22].

The presence of the ferroelectric phase 75As NQR lines observed in measurements within the temperature-pressure range of the paraelectric phase of proustite may be regarded as evidence for the existence of an intrinsic central peak [10]. These lines can be interpreted by the presence of completely polarized, highly mobile clusters [4, 6], which locally interrupt the fast 75As electric field gradient averaging due to the soft mode fluctuations. Consequently, the absence of the usual critical singularity in the 75As SLR rate T~ 1 in the field of existence of eight lines at p-< 3.2 kbar may have arisen from the slow central peak fluctuations. In fact, the singularity in Ti 1 is caused by the dominating, condensing high frequency (to~-,~ 1, where v = to/2~" is the NQR frequency and z is a relaxation time of the order parameter fluctuations) soft mode [27], whereas the case to~" >> 1 brings about the vanishing of the critical singularity (see below). The latter may be explained, for instance, in proustite by the presence of slow fluctuations related to formation of the precursor order clusters. These polarized clusters give rise to the intrinsic central peak [14-17]. Within the pressure range 3.2 kbar < p < 4.3 kbar the central peak and the soft mode are observed together on the time scale of the experiment. The application of hydrostatic pressure results in an appearance of the singularity in T71 (fig. 3), i.e. in the change of

the order parameter fluctuations rate on approaching the tricritical point.

To analyze the behaviour of the 75As TI in mg3AsS3 near the phase transition we use the following approach. The probability of an SLR transition W,,,,.+~, (m is the spin quantum number, /z = - 1 , _-+2) may be expressed in terms of the spectral density of fluctuations of the order parameter 8rt and derivatives A of the electric field gradient tensor with respect to the order parameter 77 [27, 28]:

W m, ra +/.t Ioh.712 Z A t A r II'

x f exp(-il~oot)(Sn~(O)Snt,(t))dt, (20)

where the summation is over unit cells and O~,. are the matrix elements of a nuclear quadrupole moment operator. Let us introduce collective coordinates 6r/q [27] to take into account the space correlations of order parameter fluctuations:

6r/t(t) = ~ ~ 6r/a(t) exp(iqrt). (21) q

In a random phase approximation we derive

2 = QIg_ . W,.,=+~, h 2 z.J A~A~-qS(q, lZt°),

q

where

(22)

A~Aeq = ~'~ A~A~," exp[iq(rl - rr)], l l '

and

(23)

+co

S(q, tzto) = f (&q,(O)Srl-,(t)) exp(-i/zcot) dt. - o o

We assume that

(24)

( Sn,(O)Sn-,(t)) = (lSnql2) exp( - ~ ) . (25)


Taking into account (25), one obtains for S(q, tzto) the following expression:

S(q, I~to) = 1 + U2to2"r 2" (26)

The factor A q A _ q is usually a slowly varying function of a wave vector q and its dependence on q may be neglected in the first approximation [28]. The calculation of <lsn01 > gives [29]

/caT (27) <lsn, I ~> ~,. + aq 2,

where a is the coefficient of the inhomogeneity term in the expansion of the thermodynamical potential ~b in powers of the order parameter r/ [30], ~bn, is the second derivative of ~b with respect to r/ and kB is the Boltzman constant. The calculation of 7.q gives [22]

7.# = [ F ( , / , . . + a q 2 ) l - ' , (28)

where F is the Landau-Khalatnikov coefficient. Substituting (26), (27) and (28) into (22) we obtain

Wm,m+t~ [Q~ml2knT ~ 1

+ o t q 2 ) 2 + / . £ 2 ( . 0 2 / / - ' 2 '

(29)

In the case of slow fluctuations of the order parameter (to%)2-> 1, the critical singularity in Wm,.+~, ~ T i ~ (22), (29) is absent. Replacing in (29) the summation by integration in q up to the cut-off wave number qm and using the corresponding integral value [30], we derive

k B T " f 1 T~ ~ ~ W m ,.+~, 3a ~ " /~ - -~ ( .0 + 1) 1/2 In

• a 4 , , , , F ~ o . , 7 .

/2 + x~ + X/2x,,,(g2 - 1) '/2 X

/2 + x 2 - X/2x. (/2 - 1) ~/2

..lrz V 2 x , ( O + I) 1:2) - V'2(12 - x) arctg _ 2 ~ (30)

11 - - X m J

where

/2 = ~ 1 + I./.20927 "2, 7. = ( ~ F ) -1, ol llZ q,. (31)

Xm -- ¢h 1/2 " "#" "¢/rl

According to (30) the SLR rate T~ 1 is the com- plicated function of to. If (/xwT.) 2 >> 1, iz~oT. >> 1 and tZWT.>>Xm, then T~ ~ - oJ -~, the frequency dependence of this type has been indeed observed in the nuclear magnetic resonance in the presence of the central peak in KH3(SeO3~h [12] and in KH2PO4 [13]. The dependence Tl~(W) in pure NQR, where the external magnetic field is zero, can be observed if I > 3/2. In proustite the dispersion of T~ ~ is not possible for the nucleus 75As as its spin I = 3/2 and there is only one resonance transition.

In the case of fast fluctuations (/ztoT.) 2,~ 1, we obtain

Xra k B T (arctg x,. 1 + x2 ) (32)

For x,, ~oo, eq. (32) becomes

k B T (33) T ] ' l - _~ .3 /21r" . t . l /2 •

8JJt~ J q~nn

In the homogeneous case x,, ~ 0

k B T T~ ~ - 6~.2F~b2" (34)

Near the temperature of the second-order phase transition we obtain the singularities [30]: T11 -- IT - T01-% where 1/2 ~< e <~ 2.

The decrease of the relaxation time 7. of the order parameter arising from the pressure increase is explained in the following two ways. In [31] the temperature dependence of r has been found from the 1 2 1 S b N Q R measurements near the low-temperature phase transition in pyrar- gyrite (Ag3SbS3) which is isomorphous to proustite. It turns out that the temperature dependence of the rate of silver ion jumps between the two wells of the double-minimum potential of the S-Ag-S bond 1/T is not exponential: 1/7.- T.


This form may be the result of the phonon- assisted tunnelling caused by the one-phonon process in low temperatures [32]. It was shown [33, 34] that the increase of hydrostatic pressure brings about the increase of the tunnelling rate leading to the decrease of the time ~-. The second reason for the decrease may be explained within the framework of the soft-mode theory. As it is known [27], the frequency square of the soft mode to].m, is proportional to a .difference between the short-range restoring force and the long-range Coulombe force. It is clear that the restoring force contribution increases faster than the Coulombe force contribution as the hydrostatic pressure increases, thereby leading to the increase of the soft-mode frequency [27]. This gives rise to the fulfillment of the criterion (to/to .... )2,~1. Therefore the condition [to~.(p)]2,~ 1 for 75As in AgaAsS3 is fulfilled and the singularity of T~ ~ appears at p I> 3.2kbar. At p/> 4.3 kbar the pressure increase results in the averaging of the 75As electric field gradient and hence in only one 75As NQR line in the spectrum. One can deduce from the condition t o T - 1 that the width of the central peak in Ag3AsS3 should be approximately 4.3 × 108 Hz. The pressure dependence of the 75As T1 (fig. 3) obeys the following form: 7"1 (T¢(p)- T] 2. It is in accordance with the critical singularity T~ 1 - ( T - T¢) -2- Eq. (34).

The crystal which undergoes a second-order phase transition exhibits pretransitional phenomena in the form of large amplitude lattice fluctuations. However, these fluctuations are labile, they have short lifetimes and give rise to the critical singularity in SLR rates. At first- order phase transitions the metastable fluctuations of competing phases (heterophase fluctuations [31-33]) occur. They should give rise to an intrinsic central peak [34]. When the two phases are both fluids only the interfacial free energy between two phases acts to suppress these fluctuations near the phase transition. In solids the volume and shape change associated with the phase transition leads to a strain energy contribution to the free energy which also limits the formation of heterophase fluctuations [33]. The negligible change of volume at the phase tran-

sition in Ag3AsS3 [26] offers favourable conditions for formation of heterophase fluctuations. In such a case the precursor order clusters, which result in the intrinsic central peak in proustite are apparently heterophase fluctuations. Upon application of a hydrostatic pressure the value of thermal hysteresis decreases and the tricritical point is reached at p = 3.2 kbar and T = 55.5 K. At the tricritical point the heterophase fluctuations must vanish and the critical singularity in T~ 1 must appear. Indeed, the number of the 75As NQR lines at the tricritical point decreases and the 74As T~ 1 exhibits the critical singularity. Two lines appear, instead of eight. Above p = 3.2 kbar there is a singularity in T~ 1. The existence of the doublet and the intrinsic central peak beyond the tricritical point shows that the precursor order polarized clusters are of nonlinear or soliton origin. In the temperature- pressure region of the first-order phase transition the precursor order clusters are apparently heterophase fluctuations described by the potential (6b) or (6c), giving the kink excitations (16a) and (17a) correspondingly. In the tricritical point the precursor order clusters are described by the potential (6d), giving the kink excitation (19a).

The possible explanation of this experiment associated with the occurrence of heterophase fluctuations has been presented in [22]. The temperature dependence of the kinetic coefficient F was assumed to depend exponen- tially on temperature [22, 35]:

( F = F0 exp -

where A~ is the barrier of the thermodynamical potential, separating the stable and metastable phases. For second-order phase transitions A~b = 0 and F = F0. The exponential factor determines the crossover from the soft mode regime of fast critical fluctuations (~to%)2,~ 1 in which the critical singularity of TT ~ takes place in the regime of slow heterophase fluctuations (tzto%)2~>l, where the singularity in (22) vanishes. The condition


may be fulfilled in the metastable region at the first-order phase transition, because the heterophase flucturations are slow compared to the critical fluctuations, as pointed out in [33]. The increase of %(p), which brings about the vanishing of the critical singularity of T~', takes place due to the appearance of the exponential factor exp(Aqb(p)/kBT) in the metastable region near the first-order phase transition.

4. Summary

A model for description of precursor order clusters at ferroelectric phase transitions has been proposed. In this model the cluster walls separating the regions of paraelectric and ferroelectric phases have been obtained as kink solutions of the nonlinear dynamic equations including a damping term. The cases of the precursor order clusters for the first-order and second- order phase transitions and for the tricritical point have been considered.

We have shown that in the ferroelectric proustite (Ag3AsS3) under the hydrostatic pressure p ~< 4.3 kbar a narrow central peak with the width - 4 . 3 × 108 Hz is observed on a time scale of the 75As NQR. The central peak gives rise to the appearance of the ferroelectric phase 75AsNQR lines in the paraelectric phase. The additional new evidence of the central peak in SLR obtained here is absence of the critical singularity in T~ ~ near the phase transition. At the pressure-induced tricritical point (p = 3.2 kbar and T = 55.5 K) and beyond it at p/> 3.2 kbar the critical singularity caused by the soft mode fluctuations occurs. Within the range 3.2 kbar ~< p ~< 4.3 kbar the central peak and the soft mode are only observed at the same time, demonstrating a usual situation [10]. At p<~ 3.2kbar the central peak is only observed, at p >t 4.3 kbar the soft mode is only observed. It is pointed out that the crossover from the fast soft mode fluctuations into the slow central peak fluctuations, related to the polarized precursor order clusters, takes place. A reason for this crossover is a pressure change of the order parameter relaxation time. When the hydrostatic

pressure increases the order parameter relaxation time r decreases. Two possible reasons of this decrease are considered: 1) the phonon- assisted tunnelling rate and 2) the dominant contribution of the short-range restoring force to the soft-mode frequency increase. It leads to the appearance of the critical singularity of the 75As SLR rate T~ 1. An absence of the critical singularity in T~' at p~< 3.2 kbar is caused by pretransitional heterophase fluctuations which bring about the intrinsic central peak in proustite up to a reaching of the tricritical point. The negligible change of volume at the phase transition in Ag3AsS3 offers favourable conditions for formation of heterophase fluctuations. Within the range 3.2 kbar ~< p ~< 4.3 kbar, i.e. in the region of a coexistence of the central peak and the soft mode on a time scale of NQR, where the tricritical point is reached, heterophase fluctuations must vanish. However, the intrinsic narrow central peak is observed. Hence these polarized precursor order clusters leading to the central peak are apparently of nonlinear nature. It is suggested to describe the heterophase fluctuations and the precursor order clusters in the tricritical point as kink excitations.

Acknowledgements

The authors are grateful to Dr. J. Adler for helpful discussions. One of us (A.G.) is indebted to Drs. D. Baisa, A. Bondar and S. Maltsev for the fruitful collaboration in the works [21, 22].

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precursor order clusters at ferroelectric phase transitions

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