Determination of exponent δ from measurements beyond the critical isotherm

Solid State Communications, Printed in Great Britain.

DETERMINATION

Vol. 73, No. 6, pp. 459-462,

OF EXPONENT

of Physics, (Received

Silesian

0038-1098190 $3.00 + 400 Pergamon Press plc

6 FROM MEASUREMENTS ISOTHERM

B. Fugiel Institute

1990.

BEYOND

THE CRITICAL

and B. Westwanski

University,

Uniwersytecka

16 June 1989; in revisedform

4, 40-007

11 October

Katowice,

Poland

1989 by P. Burlet)

The universal form of the susceptibility scaling function was analysed in the paraelectric phase for crystals of triglycine selenate (TGSe), triglycine sulphate (TGS) and also TGS crystals doped with chromium ions. A method is proposed by which, depending on whether the critical properties of the tested compounds are described according to the static scaling hypothesis or according to the Larkin-Khmel’nitski theory, it is possible either to determine the value of critical exponent 6 or to estimate the value of the non-universal constants.

x,, =

IN FERROELECTRICS the critical index 6 express the relation between the electric field E and polarization P, i.e. E N P’, on the critical isotherm T = T,. The exponent 6 has been measured for many different physical systems, i.e. magnetic systems, classical liquid-gas transitions, liquid-gas phase transitions in quantum liquids [ 1, 21 and for ferroelectrics [3]. Measured values of 6 lay in the interval from 2.8-5.75 [l], while the value most frequently reported was 6 = 4.2. Critical exponents (a, j?, y, 6, etc.) depend on symmetry, number of components of the order parameter and the dimensionality of the system. Theoretical studies (series expansions plus extrapolation by Pade approximants [4, 2]), of the Ising, Heisenberg model and the spherical model led to the conclusion that the exponent 6 differs from all the other critical exponents in that it is the most universal, i.e. it depends solely on the dimension d: thus for d = 2, 6 = 15 and for d = 3, 6 = 5. In this paper we wish to present a method for determining the critical exponent 6 basing on measurements performed beyond the critical isotherm in certain ferroelectrics. For this purpose it is necessary to take the assumption of the correctness of the static scaling hypothesis. According to the static scaling hypothesis, the singular part of the Gibbs potential in the paraelectric phase (T > T,) has the form - G,(t, E)

=

XX0

where y is the critical exponent x0, i.e.

1 + 6

+

(2)

KW~W$=l C%32

15 [G,/G2(2G2)4z](E&)4

where the exponent z =

(1) for initial susceptibility

T > T,

= f(Eti) =

rpy+zA [G, + G,(E/r”)*

t -y+2Ag(E/zA)

for

where A = gap exponent, r = (T - T&/T, is the reduced temperature, g = a certain function, and G, = constant coefficients. From formulae (1) and (2) it may be concluded that the relation between initial susceptibility x0 (for T > T,), electric susceptibility x in a nonzero electric field and the electric field strength E may be written in the form

+ G~(E/T~)~ + . . .] =

2G2C

S/(S -

1)

+ ...

(34

z has the form: (3b)

which follows from the known relation 6 = A/(A - y) between critical indices y, A, 6 and j(x) is the scaling function. It should be stressed that in the formulae (3a) there is a dependence on temperature (T > T,) due to the presence of the term x0. The explicit r dependence in equation (3a) may be reinstated when using the relation g = (2G,)‘r-“. It is noteworthy that equation (3a) has a universal form since it depends on magnitudes which are constant for a given class of universality. These constants are the exponents 6 and the Watson invariants [5]. Relation (3a) may be studied experimentally when knowing the values x0 and the dependence 1 vs. E at various temperatures (or pressures) in the vicinity of the critical point. In this case the experimental value of 6 is treated as a parameter which must be chosen such that the plot of the experimental function (3a) for various

459

460

DETERMINATION

OF 6 FROM THE CRITICAL

temperatures and various electric fields has the characteristic form of the scaling function. Hence the analysis of experimental data presented below may serve for determination of the value of 6. Measurements were conducted for the following ferroelectrics: triglycine selenate (TGSe), triglycine sulphate (TGS) and crystals grown from aqueous solutions containing pure TGS with a 0.1% and 1% addition of Cr, SO,. Mean dimensions of the samples were of the order of 10 x 10 x 1.5 mm3. In the case of TGSe the gold electrodes completely covered the relevant crystal surfaces (rectangular electrodes). For the samples doped with chromium ions there was only a partial degree of covering (circular electrodes). For TGS both types of electrodes were used. Prior to the tests the samples were heated for about fifty hours at a temperature of about 85’C (pure and doped TGS crystals) and about 30°C (TGSe crystals). The susceptibility at various temperatures (or pressures) and also in various electric fields E was determined on the basis of measurements of capacitance C of capacitor with sample according to the formulae x = (C/C,) - 1, where C, is the capacitance of the measurement capacitor without the sample (geometrical capacitance). The frequency of the measuring field was 1 kHz, while the amplitude did not exceed 1 kV/m. The value of susceptibility x for various externally applied constant electric fields E was determined by applying a constant voltage U to the measurement capacitor in such a way that the constant field E was parallel to the measurement field E,,,. The range of values of E (determined from formulae: E = U/d, where d is the distance between the electrodes) varied from zero up to about 800 kV/m and decreased on approaching to the critical point. The schematic diagram of measuring circuit used was similar to that described in [6], the only difference being the use of different capacitance meter. Figure 1 shows the curves of xxi’ as a function of Ed for TGS (for the sample with circular electrodes) for the paraelectric phase and for the various values of 6 [z = S/(S - I)]. The experimental points presented

ISOTHERM 1.

Vol. 73, No. 6 ._

F---. -‘..,

6= 3.2

‘..,

‘.

0,

‘.

i

‘.

.,\, rl i_

06

6 = 3.3

‘~. ‘..

-..

0

Ex@-c) 0

x

10-‘” (Vm-l)

Fig. 1. The experimental dependences xxi I versus Ex;, where z = S/(S - 1) for TGS (sample with the circular electrodes, see text). The experimental points have been obtained from measurements of relation x vs. E (a few values of E for each temperature) for nineteen temperatures in the interval corresponding to lo-3 < (T - T,)/T, < lo-2.

here were obtained from measurements of relation x(E) for nineteen temperatures (a few values of E for each temperature) in the interval corresponding to 10e3 < z < 10m2. As may be seen, the best concentration of measurement points around the curve (scaling type function) for various temperatures and various electric field strengths, is found for 6 = 3.2 + 0.1. On the figure are shown only the curves obtained for values near to 3.2. The further away from

Table 1. The values of exponent 6 obtained for five tested samples; c = circular electrodes, r = rectangular electrodes (see text). For TGSeCT’ and TGSe@’ the changes of x0 are governed by changes in temperature (T z 298 K, PC z 57 x lo6 Nmm2) of the critical points line (see 171). For pure and doped TGS crystals only the results of the temperature measurements (lo-’ Q (T - T,.)/T,. d 10e2) at atmospheric pressure have been presented

Sample

TGS

TGS I

TGS (0.1% Cr,SO,)

TGS (1% Cr2S0,)

TGSe”)

TGSe@’

Electrodes

c

r

C

c

r

r

6

3.20 + 0.1

3.15 f 0.1

3.20 f 0.1

3.15 f 0.1

3.20 f 0.1

3.20 f 0.1

Vol. 73, No. 6

DETERMINATION

OF 6 FROM THE CRITICAL

the value 3.2. the greater is the scatter of the measurement points. For the other compounds exponent 6 was determined in a similar manner. Values of 6 obtained for the tested samples are set out in the Table 1. As may be seen, for all the tested crystals values of 6 lay in the interval from 3.05 to 3.30 (taking into account that the experimental error of determining of 6 is equal to + 0.1). It was also possible to conclude that most probably stable additions of chromium ions do not exert any significant influence on the value of exponent 6. This method of determining 6 involves a simultaneous analysis of a considerable number of measurement points (in our case from one to four hundreds) and for this reason is relatively accurate. A further advantage is that it is not necessary to have a precise knowledge of the Curie temperature. This presented method of measurement of exponent 6 is, of course, an indirect method, requiring the assumption of the correctness of the static scaling hypothesis, as equation (3a) holds good in the case of a purely power type temperature dependence of initial susceptibility. The relation xx;’ as a function of Ed for TGS (Fig. 1) provides good confirmation of the assumption taken. It should be stressed, however, that the critical properties of TGS crystals are frequently described making use of the Larkin-Khmel’nitskii theory [8-lo]. In the approximation made by Natterman [lo] on the basis of this theory the singular part of the free energy dependent on polarizarion P has the form for b 6 1 and b* In (5,/r) $ 1: Fsing = 3 t~~r [I + 3b In (z~/~)]‘/~P* + + B. x [l + 3b In (r,/z)]-‘P4,

(4)

where a,, BOYto and b are constants. Hence, for example, the temperature dependence of initial susceptibility is not of purely power form. However, according to [lo] (see also references in [lo]) for TGS the value b is very small, equal to 0.036. Therefore, verification of the Larkin-Khmel’nitskii theory, and in particular to decide whether for such small values of b, the initial susceptibility has a experimental temperature dependences of logarithmic-power type r-’ [I + 3b In (ro/r)]“3 or a purely power type dependence r-(‘+*), presents unusually difficult problems and is not the object of this present paper. It is worth adding, however, that when we take the LarkinKhmel’nitskii theory as the basis for interpretation of our results, the method presented here may serve for estimation of the parameter b instead of exponent 6. In this case the exponent z = S/(S - 1) on the right side of equation (3a) may, for small values of b, be replaced with fair accuracy by 1.5/(1 + b), and then from equation S/(S - 1) = 1S/(1 + b), b, may be

ISOTHERM

461

estimated for 6 shown in Table 1 (for example b = 0.03 if 6 = 3.2). The main object of the present paper was to give a method for determination of exponent 6 without the necessity of performing the relatively inaccurate, and at the same time very difficult measurements on the critical isotherm, at the expense, however, of taking assumption of the correctness of the scaling hypothesis in the case of measured compounds. The proposed method may be used not only for ferroelectrics but also in magnetic systems (for the appropriate magnetic magnitudes). As regards the experimental values in our investigations it should be noted that irrespective of whether the exponent 6 is treated as in the scaling hypothesis or as a certain effective magnitude (when there are logarithmic corrections), the critical properties of the tested compounds cannot be described in terms of the mean field theory, for which 6 = 3 (i.e. b = 0). The values of 6 obtained by us for TGS are, within the limits of the experimental errors, in good agreement with the value 6 = 3.1 reported in [3] obtained on the basis of measurements of polarization as a function of electric field at a temperature T = T, + 2 x 10m3K. According to the authors of [3] deviations of 0.1 from the value 6 = 3 resulting from the mean field theory are due to the fact that measurements were not performed exactly on the critical isotherm. However, on the basis of later experimental data given for instance, in paper [lo], it may be concluded, as earlier suggested, that the mean field theory does not accurately describe the critical properties of TGS. Thus it would be difficult to assume that for this compound the value of exponent 6 is exactly 3. Results obtained from our experiments confirm the occurrence of differences between experimental results and mean field theory not only for TGS but also for the remaining tested samples. Theoretical results [9] obtained by the renormalization group method actually appear to support the existence in uniaxial ferroelectrics with dipolar interactions of corrections of logarithmic type, and not power type, supplementing the temperature dependences of various thermodynamic magnitudes resulting from the molecular field approximation. But as regards experimental verification of this fact, to the best of our knowledge no convincing empirical proof has yet been presented. Acknowledgements - The authors wish to express their thanks to Professor Jadwiga Stankowska of the Adam Mickiewicz University in Poznari for making available a TGSe single crystal. This work was supported by The Institute for Low Temperature and Structure Research of The Polish Academy of Science from the funds of The Central

DETERMINATION

462

OF 6 FROM THE CRITICAL

Programme for Basic Research on “Structure, Phase Transitions and Properties of Molecular Systems and Condensed Phases” (CPBP 01. 12). REFERENCES 1.

2. 3.

L.P. Kadanoff, W. Giitze, D. Hamblen, R. Hecht, E.A.S. Lewis, V.V. Palciauskas, M. Rayl, J. Swift, D. Aspnes & J. Kane, Rev. Mod. Phys. 39, 395 (1967). H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena, Pergamon, Oxford (1971). K. Deguchi & E. Nakamura, Phys. Rev. B5, 1072 (1972).

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ISOTHERM

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