Estimation of Specific Heat of BaTiO 3 Crystals Derived from Relationship Between Uniaxial Pressure and Electric Field

2021 Shifting the Curie temperature in dependence on both uniaxial pressure and electric field in BaTiO 3 crystals was studied based on literature data. It was shown that both these dependences perfectly coincide when adjusting the scale. Based on coincidence of these dependencies a relationship between both an uniaxial pressure and an electric field when shifting the Curie temperature was established. The specific heat is calculated using this


Introduction
Barium titanate, BaTiO 3 , is a well-known ferroelectric, has been discovered in 1945 and since is extensively studied. It is well documented that upon heating BaTiO 3 undergoes a structural transition from ferroelectric to paraelectric phase pointed out by sharp peak of dielectric permittivity at the Curie temperature, T c . An effect of both electric field, E, and mechanical pressure, p, on T c shift is studied as well.
Shifting the T c on 8.5°C, detected from the hysteresis loops of BaTiO 3 crystals grown by the Remeika method, was found to be a linear as E enhances up to 6 kV/cm at the fixed temperatures up to 116°C 1 . Meanwhile, shifting the T c on 15°C, detected from the birefringence of BaTiO 3 crystals grown by the Remeika method, was found to be a nonlinear as E enhances up to 12 kV/cm and a linear as E weakens down to 0 kV/cm at the fixed temperatures up to 136°C 2 . However, shifting the T c on 3°C, detected by the acoustic emission of BaTiO 3 crystals grown by the melt-grown method, was found to be a linear upon heating at the fixed fields up to 2 kV/cm 3 . Recently, shifting the T c on 8.5ºC, detected from the electrocaloric effect of BaTiO 3 crystals grown by the melt-grown method, was found to be a nonlinear upon heating at the fixed fields up to 10 kV/cm 4 . While the dependencies between T c and E in Merz 1 and Dul'kin et al. 3 . coincide well, the dependencies between T c and E in Meyrhofer 2 and Bai et al. 4 . are not consistent.
Shifting the T c on 3°C, detected by the dielectric permittivity of BaTiO 3 crystals grown by the melt-grown method, upon heating up to 410°C at the fixed uniaxial pressures up 1000 bar was measured previously 5 . Shifting the T c in dependence on p was approximated to be a linear, but one can clearly see that it is a very rough approximation. In fact the T c (p) dependence is a nonlinear and visibly trends to saturation as the p enhances. Also a relationship between p and E, p/E one can calculate to be 23.8 bar·cm/kV at room temperature, not at T c , as it might be expected.
The goal of the present paper to derive the relationship between both p and E within the T c shifting region based on comparison the data of above cited works and check it for usefulness in practical application.

Material
In this paper a consideration is devoted to comparison the data of BaTiO 3 crystals, used in Bai et al. 4 and Suchanicz et al. 5 , because they were grown by the same melt-grown method, and exhibit the same T c ≈ 407 K, and studied at the same conditions: upon heating under fixed pressure and field up to their higher values, and, thus, can be compared truly. Figure 1 presents the T c shifting in dependence on both p and E reconstructed from the corresponding data of Bai et al. 4 and Suchanicz et al. 5 , respectively. Accurate reconstructed the T c shifting in dependence on p is indeed a nonlinear in contrast to that declared in Suchanicz et al. 5 . One can see that both these dependencies perfectly coincide when adjusting the scale. Such the perfect coincidence unambiguously proves that both p and E shift the T c in the same manner.

Results and Discussion
Both these T c (p) and T c (E) dependencies are approximated the following equations: for uniaxial pressure From these equations one can establish the relationship between both p and E values at the same T c . For example, to shift the T c on 1 K, i.e. up to 408 K one need apply or the equivalent p = 335 bar or the equivalent E = 1 kV/cm, and, consequently, the relationship between p and E, dp dE is about 335 bar·cm/kV = 3.35·10 2 N/mV at T c .
Let's now check this relationship for usefulness in practical application. For example, let us apply it to calculate the specific *e-mail: evgeniy.dulkin@mail.huji.ac.il heat, L, during the phase transition in BaTiO 3 crystals due to essential contradiction in their L values: ~ 2.37 J/kg 4 and ~ 0.54 J/kg 6 .
When multiplying Equation 3 by Equation 4 we obtain the relation: from which the L is calculated to be ≈ 6.42 J/kg. This data is obviously lies closer to L ≈ 2.37 J/kg 4 , not to L ≈ 0.54 J/kg 6 , and the former is believed to be really true.
Note that our L value is calculated at dp dE = 335 bar·cm/kV of ΔT=1 K. Unfortunately, L value varies in dependence on ΔT due to some nonlinearity of both T(p) and T(E) dependencies.
The error is approximately to be 14% for dp dE = 382 bar·cm/kV of ΔT = 2.5 K in relation to dp dE = 335 bar·cm/kV of ΔT = 1 K, that is satisfactory for estimation of the specific heat. Thus, this relationship between p and E is proved to be useful for practical applications.

Conclusions
In summary, we have compared the Curie temperature shifting in dependence on both uniaxial pressure and electric field based on literature data and found their acting proportionally the same. Based on this proportionality we have established the relationship is equal to be 335 bar·cm/ kV between both uniaxial pressure and electric field when shifting the Curie temperature. Using this relationship we estimated the specific heat is equal to be 6.42 J/kg during the phase transition in BaTiO 3 crystals.