Anisotropy of critical indices of ferroelectric phase transition in TGS crystals by the optical interference investigation

Temperature dependences of the optical path difference, variable part of the refractive index and thickness of triglycine sulphate crystal for three crystallophysic directions are studied in the temperature range of 39–70 C, including the ferroelectric phase transition at Tc=49 C, using the Jamen type optical interferometer. Temperature dependences of the spontaneous changes of the characteristics studied in the range of 39–49 C are fitted by the power-like low Y ∼ τ with doubled averaged effective critical indices 2β=0.87–0.95. The 2β values being different from the unity is explained by the essential temperature dependence of the coefficients of electrooptic, inverse piezoelectric and electrostriction effects in the range close to the phase transition point.


Introduction
It is known, that the critical behaviour of the spontaneous polarization P s at the 2nd order phase transition (PT) in a crystal can be described by the critical index where T c is the PT temperature [1].Here, index β can be treated as asymptotic or effective one [2].The effective critical index β eff is determined to be the derivative taken at some temperature T < T c .Here, τ = (T c − T )/T c .The effective critical index β eff is becoming the asymptotic one, when the value τ is approaching zero (τ < 10 −3 ) [2].
On the other hand, experimental temperature dependences of crystal parameters in the range of PT can be also presented with the help of coefficients of thermodynamic potential expansion.According to the Landau theory, the elastic Gibbs energy G 1 in the case of uniaxial ferroelectric can be expressed in the polynomial form where D is electric displacement, A, B and C are coefficients [3].Here, the energy is measured from the non-polar phase and the polynomial is arbitrarily terminated at D 6 .Dielectric state equation E = ∂G 1 /∂D takes the following form where E is electric field.In the first approximation, which is in many cases satisfying, we assume: Coefficient A 0 is usually determined from the measurements of dielectric constant as a function of temperature in the paraelectric (PE) phase.In the ferroelectric (FE) phase the coefficients B and C can be determined from the temperature dependence of P s (E=0, D=P s ).The equation of dielectric state (3) can be now written as follows To characterize the order of PT, the parameter V = B 2 /A 0 C can be used [3].The physical sense of this parameter is clearly visible in the description of the first-order PT.The polarized state becomes stable at the temperature T 0 = T c +3/16(B 2 /A 0 C), whereas the upper limit of a superheating is T 1 = T c + B 2 /4A 0 C. When the firstorder PT comes near the second-order PT, the absolute value of V decreases at the tricritical point Both sets of parameters, index 2β in formulae (1), and coefficients A, B, and C in formulae (6), can be used for the approximation of temperature dependences of spontaneous polarization of ferroelectrics near the PT point.In the first five columns of table 1 the parameter V as well as the parameters A 0 , B, C of the equation of state for some ferroelectric crystals are presented [3].
Based on the values of coefficients A, B and C presented in table 1, we have calculated the temperature dependences of spontaneous polarization P s (T ) by the formula (6) in the range of 312-322 K.Then, using the dependence P s (T ) obtained, we have calculated the corresponding effective critical index β eff averaged in this temperature range using the formula (2).
Experimental temperature changes of the optical path difference determined by birefringence, D = l∆n, are frequentely identified in practice with the changes of the birefringence ∆n.But the thickness l and birefringence ∆n can have different temperature dependences.The temperature dependences of D(T ) and l(T ) are usually investigated in different experiments.This restricts the accuracy of determining

Crystal
A 0 the corresponding dependence of ∆n(T ) and l(T ), and makes it complicated to compare these different parameters of crystal in the region of PT.We have suggested the techniques of simultaneous determination of temperature dependences of the variable part of the refractive index (n − 1) and the thickness l of a sample.Temperature dependences of the refractive indices and the linear thermal expansion of TGS in the range of PT have already been studied [4][5][6], but the corresponding critical indices have not been determined.The goals of the present investigation were the precise measurements of the temperature dependences of optical path difference (OPD) determined by the refractive index for TGS crystal in the range of 2nd order PT at 322 K, calculating the temperature dependences of refractive indices and linear thermal expansion for the principal crystallophysic directions, as well as the study of these dependences using the corresponding effective critical indices 2β eff .

Experimental
Temperature dependences of OPD, D = l(n − 1), determeined by the variable part of refractive index (VPRI), (n − 1) = η, for two interfering beams, one of which has passed through a sample studied, and the other one through the air, were measured using the Jamen type home built interferometer (figure 1).In this case, the OPD D and its temperature dependence D(T ) can be written as follows where l is thickness of a sample.The laser light of the wavelength λ=632.8nm was used in the experiment. .
Proceeding from the relation ( 6), the temperature changes of relative OPD ∆D/D along three crystallophysic directions can be written in the form of a system of linear equations where index i denotes the direction of light propagation, index j denotes the direction of light polarization.Based on the six temperature dependences ∆D ij /D ij measured we have determined the relative temperature changes of the thickness ∆l i /l i and VPRI ∆η j /η j [7].The results of the corresponding computer calculations have shown that the relative errors of the temperature changes determination of thickness δl i /l i and VPRI δη j /η j caused by solving the system (8) did not exceed 5% of the respective maximum magnitudes ∆l i /l i and ∆η j /η j for the case of TGS crystal.The initial l i and η j values were measured independently at the initial temperature T 0 .

Results and discussion
Temperature dependences of the relative changes of OPD ∆D ij /D ij for TGS crystal are shown in figure 2. Refractive indices n j (T 0 ) of TGS crystal were taken from the paper [6].The forms of temperature dependences of the thickness ∆l i /l i and VPRI δη j /η j calculated from the system of equations ( 8) are characterized by the similar anomalies at PT temperature.Based on the known relation for the temperature changes of the order parameter p = P s for 2nd order PT in the range of T < T c , we have calculated the double effective critical indices 2β averaged in the range of 39-49 • C, replacing P 2 s value by the spontaneous increases of ∆Y s (T )/∆Y s (T min ) (Y =D, l and η).Here T c =49 • C is the temperature of PT, T min is the lower edge of the temperature range studied (T min =39 • C in our case), ∆Y s (T ) and ∆Y s (T min ) are spontaneous increments corresponding to the T c and T min temperatures.The double critical indices 2β of TGS in the range of 39-49 • C are shown in table 2.
where a(τ ) is the temperature dependent coefficient.It follows from the character of experimental dependences of spontaneous increases of ∆D s /D, ∆l s /l, and ∆η s /η, that the corresponding coefficients a(τ ) are maximal in the region of PT (figure 3).To obtain the additional proofs of the validity of this viewpoint, we have performed experimental study of the artificially induced electrooptic effect in TGS crystal in the temperature range of 30-65 • C.This investigation was carried out using the same optical scheme.The external electric field of the magnitude E ≈ 3.5 kV/cm was applied to the sample of TGS crystal at different temperatures along the [010]direction of spontaneous polarization P s , and the corresponding induced increments of the OPD ∆D e /D were measured.The maximum-like temperature dependence of ∆D e /D value (figure 4) correlates well with the temperature dependence of a(τ ) in the ferroelectric phase.This maximum-like character of the coefficient mentioned is connected with the inequality 2β < 1.
Taking into account that the temperature dependences of spontaneous increments of ∆Y s /Y parameters can be presented in two forms, Y (τ ) = τ 2β , and Y (τ ) = a(τ )τ , one can obtain the relation for the temperature dependence of a(τ In the cases of 2β = 1 and τ = 1, the decreasing temperature dependence of the coefficient a(τ ) takes place in the ferroelectric phase at the removal from the PT point τ = 0 (figure 3).Taking into account all the results obtained, we can summarize that the temperature dependences of the coefficients of quadratic electrooptic and electrostriction effects for TGS crystals take place (see relations (10) and ( 11)).The analysis of table 1 testifies to some segregation of the [010]-direction of spontaneous polarization.Really, among the temperature changes of spontaneous increments ∆l i and ∆η j (i, j = 1, 2, 3), the dependence ∆l 2 (τ ) is characterised by the least index 2β while the dependence ∆n 2 (τ ) is characterised by the greatest index (see table 2).On the other hand, a proximity of the values 2β 2 is observed (see table 1), whereas obvious inequalities of similar characteristics for the other two crystallophysic directions 2β 1,3 take place (see table 2 and figure 4).The latter features can be interpreted as a different rate of the ordering of two subsystems.One subsystem relates to the electrons forming the refractive index n while the other is connected with geometric parameters of the crystal unit cell for the directions [100] and [001].The equality 2β (l) 2 ≈ 2β η 2 for the direction of spontaneous polarization [010] can be interpreted as good correlation of the subsystems in TGS crystal mentioned above.From such a viewpoint, the observable inequalities of the indices 2β 3 .Such a crossing will take place in all cases if the experimental temperature dependence of the parameters studied (V = ∆D s /D, ∆l s /l, ∆η s /η) is described by different indices β.
We suppose that such a peculiarity in the temperature dependence of different parameters can be characteristic of the ordering of the other ferroelectric crystals.2. Deviation from the unity of the double effective critical index 2β for the temperature dependence of the optical path difference induced by a spontaneous polarization in TGS sample is explained by a significant temperature dependence of the maximum-like character of the coefficient of electrooptic, inverse piezooptic, and electrostriction effects.
3. An anisotropy of the critical indices 2β

Figure 2 .
Figure 2. Experimental temperature dependences of the relative changes of the optical path difference ∆D ij /D ij of TGS crystal (indices ij indicate the corresponding curves)

Figure 3 .
Figure 3. Temperature dependence of the derivative d(∆D s32 /D 32 )/dτ in the ferroelectric phase of TGS crystal

Figure 4 .
Figure 4. Temperature dependence of the relative optical path difference ∆D e32 /D 32 of TGS crystal induced by the constant electric field of 3.5 kV/cm along the [010]-direction

> 2β η 1 , 3
testify to various rates of the temperature changes of the corresponding subsystems of the crystal in the temperature range (∆T ∼ 10 • C) below PT point.It is seen in figure 5, in case of z-direction of the crystal studied.A crossing of the curves on figure 6 corresponds to two different indices β

Table 2 .
Effective critical indices 2β, corresponding to the temperature dependences of spontaneous increments of ∆D s /D, ∆l s /l and ∆η s /η for different crystallophysic direction of TGS crystal If these effects were displayed in the form indicated, the double critical index 2β would be equal to unity, 2β=1.Let us try to explain the experimental facts obtained.Analytical relation of the observed temperature dependence of OPD ∆D s /D induced by spontaneous polarization can be presented in the most common form