Mitsui model with diagonal strains: A unified description of external pressure effect and thermal expansion of Rochelle salt NaKC₄H₄O₆·4H₂O

Mitsui model with diagonal strains: A unified description of external pressure effect and thermal expansion of Rochelle salt NaKC₄H₄O₆·4H₂O

We elaborate a modification of the deformable two-sublattice Mitsui model of [Levitskii R.R. et al., Phys. Rev. B. 2003, 67, 174112] and [Levitskii R.R. et al., Condens. Matter Phys., 2005, 8, 881] that consistently takes into account diagonal components of the strain tensor, arising either due to e...

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Published in:Condensed Matter Physics
Date:2011
Main Authors: Moina, A.P., Levitskii, R.R., Zachek, I.R.
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Language:English
Published: Інститут фізики конденсованих систем НАН України 2011
Online Access:https://nasplib.isofts.kiev.ua/handle/123456789/120027
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Cite this:Mitsui model with diagonal strains: A unified description of external pressure effect and thermal expansion of Rochelle salt NaKC₄H₄O₆·4H₂O / A.P. Moina, R.R. Levitskii, I.R. Zachek // Condensed Matter Physics. — 2011. — Т. 14, № 4. — С. 43602:1-18. — Бібліогр.: 44 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine
id nasplib_isofts_kiev_ua-123456789-120027
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spelling Moina, A.P.
Levitskii, R.R.
Zachek, I.R.
2017-06-10T18:53:43Z
2017-06-10T18:53:43Z
2011
Mitsui model with diagonal strains: A unified description of external pressure effect and thermal expansion of Rochelle salt NaKC₄H₄O₆·4H₂O / A.P. Moina, R.R. Levitskii, I.R. Zachek // Condensed Matter Physics. — 2011. — Т. 14, № 4. — С. 43602:1-18. — Бібліогр.: 44 назв. — англ.
1607-324X
PACS: 65.40.De, 77.80.B-, 77.65.Bn, 65.40.Ba, 77.22.Ch
DOI:10.5488/CMP.14.43602
arXiv:1108.2160
https://nasplib.isofts.kiev.ua/handle/123456789/120027
We elaborate a modification of the deformable two-sublattice Mitsui model of [Levitskii R.R. et al., Phys. Rev. B. 2003, 67, 174112] and [Levitskii R.R. et al., Condens. Matter Phys., 2005, 8, 881] that consistently takes into account diagonal components of the strain tensor, arising either due to external pressures or due to thermal expansion. We calculate the related to those strains thermal, piezoelectric, and elastic characteristics of the system. Using the developed fitting procedure, a set of the model parameters is found for the case of Rochelle salt crystals, providing a satisfactory agreement with the available experimental data for the hydrostatic and uniaxial pressure dependences of the Curie temperatures, temperature dependences of spontaneous diagonal strains, linear thermal expansion coefficients, elastic constants cijE and ci₄E, piezoelectric coefficients d₁i and g₁i (i=1,2,3). The hydrostatic pressure variation of dielectric permittivity is described using a derived expression for the permittivity of a partially clamped crystal. The dipole moments and the asymmetry parameter of Rochelle salt are found to increase with hydrostatic pressure.
Запропоновано модифiкацiю деформiвної двопiдґраткової моделi Мiцуї робiт [Levitskii R.R. et al, Phys. Rev. B. 2003, 67, 174112] та [Levitskii R.R. et al., Condens. Matter Phys., 2005, 8, 881], яка послiдовно враховує дiагональнi компоненти тензора деформацiй, що виникають пiд дiєю зовнiшнiх тискiв чи внаслiдок теплового розширення. Розраховано пов’язанi з цими деформацiями тепловi, п’єзоелектричнi та пружнi характеристики системи. Використовуючи запропоновану схему, для кристалiв сеґнетової солi знайдено такий набiр параметрiв теорiї, що забезпечує задовiльне узгодження з експериментальними даними для залежностей температур Кюрi вiд гiдростатичного та одновiсних тискiв, а також температурних залежностей теплових деформацiй, лiнiйних коефiцiєнтiв теплового розширення, пружних сталих cEij i cEi₄, п’єзоелектричних коефiцiєнтiв d₁i i g₁i (i = 1, 2, 3). Залежностi дiелектричної проникностi вiд гiдростатичного тиску описано за допомогою отриманого в роботi виразу для проникностi частково затиснутого кристалу. Виявлено, що дипольнi моменти та параметр асиметрiї в сеґнетовiй солi зростають з гiдростатичним тиском.
en
Інститут фізики конденсованих систем НАН України
Condensed Matter Physics
Mitsui model with diagonal strains: A unified description of external pressure effect and thermal expansion of Rochelle salt NaKC₄H₄O₆·4H₂O
Модель Мiцуї з дiагональними деформацiями: об’єднаний опис впливу зовнiшнiх тискiв i теплового розширення в сеґнетовiй солi NaKC₄H₄O₆·4H₂O
Article
published earlier
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
title Mitsui model with diagonal strains: A unified description of external pressure effect and thermal expansion of Rochelle salt NaKC₄H₄O₆·4H₂O
spellingShingle Mitsui model with diagonal strains: A unified description of external pressure effect and thermal expansion of Rochelle salt NaKC₄H₄O₆·4H₂O
Moina, A.P.
Levitskii, R.R.
Zachek, I.R.
title_short Mitsui model with diagonal strains: A unified description of external pressure effect and thermal expansion of Rochelle salt NaKC₄H₄O₆·4H₂O
title_full Mitsui model with diagonal strains: A unified description of external pressure effect and thermal expansion of Rochelle salt NaKC₄H₄O₆·4H₂O
title_fullStr Mitsui model with diagonal strains: A unified description of external pressure effect and thermal expansion of Rochelle salt NaKC₄H₄O₆·4H₂O
title_full_unstemmed Mitsui model with diagonal strains: A unified description of external pressure effect and thermal expansion of Rochelle salt NaKC₄H₄O₆·4H₂O
title_sort mitsui model with diagonal strains: a unified description of external pressure effect and thermal expansion of rochelle salt nakc₄h₄o₆·4h₂o
author Moina, A.P.
Levitskii, R.R.
Zachek, I.R.
author_facet Moina, A.P.
Levitskii, R.R.
Zachek, I.R.
publishDate 2011
language English
container_title Condensed Matter Physics
publisher Інститут фізики конденсованих систем НАН України
format Article
title_alt Модель Мiцуї з дiагональними деформацiями: об’єднаний опис впливу зовнiшнiх тискiв i теплового розширення в сеґнетовiй солi NaKC₄H₄O₆·4H₂O
description We elaborate a modification of the deformable two-sublattice Mitsui model of [Levitskii R.R. et al., Phys. Rev. B. 2003, 67, 174112] and [Levitskii R.R. et al., Condens. Matter Phys., 2005, 8, 881] that consistently takes into account diagonal components of the strain tensor, arising either due to external pressures or due to thermal expansion. We calculate the related to those strains thermal, piezoelectric, and elastic characteristics of the system. Using the developed fitting procedure, a set of the model parameters is found for the case of Rochelle salt crystals, providing a satisfactory agreement with the available experimental data for the hydrostatic and uniaxial pressure dependences of the Curie temperatures, temperature dependences of spontaneous diagonal strains, linear thermal expansion coefficients, elastic constants cijE and ci₄E, piezoelectric coefficients d₁i and g₁i (i=1,2,3). The hydrostatic pressure variation of dielectric permittivity is described using a derived expression for the permittivity of a partially clamped crystal. The dipole moments and the asymmetry parameter of Rochelle salt are found to increase with hydrostatic pressure. Запропоновано модифiкацiю деформiвної двопiдґраткової моделi Мiцуї робiт [Levitskii R.R. et al, Phys. Rev. B. 2003, 67, 174112] та [Levitskii R.R. et al., Condens. Matter Phys., 2005, 8, 881], яка послiдовно враховує дiагональнi компоненти тензора деформацiй, що виникають пiд дiєю зовнiшнiх тискiв чи внаслiдок теплового розширення. Розраховано пов’язанi з цими деформацiями тепловi, п’єзоелектричнi та пружнi характеристики системи. Використовуючи запропоновану схему, для кристалiв сеґнетової солi знайдено такий набiр параметрiв теорiї, що забезпечує задовiльне узгодження з експериментальними даними для залежностей температур Кюрi вiд гiдростатичного та одновiсних тискiв, а також температурних залежностей теплових деформацiй, лiнiйних коефiцiєнтiв теплового розширення, пружних сталих cEij i cEi₄, п’єзоелектричних коефiцiєнтiв d₁i i g₁i (i = 1, 2, 3). Залежностi дiелектричної проникностi вiд гiдростатичного тиску описано за допомогою отриманого в роботi виразу для проникностi частково затиснутого кристалу. Виявлено, що дипольнi моменти та параметр асиметрiї в сеґнетовiй солi зростають з гiдростатичним тиском.
issn 1607-324X
url https://nasplib.isofts.kiev.ua/handle/123456789/120027
citation_txt Mitsui model with diagonal strains: A unified description of external pressure effect and thermal expansion of Rochelle salt NaKC₄H₄O₆·4H₂O / A.P. Moina, R.R. Levitskii, I.R. Zachek // Condensed Matter Physics. — 2011. — Т. 14, № 4. — С. 43602:1-18. — Бібліогр.: 44 назв. — англ.
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fulltext Condensed Matter Physics, 2011, Vol. 14, No 4, 43602: 1–18 DOI: 10.5488/CMP.14.43602 http://www.icmp.lviv.ua/journal Mitsui model with diagonal strains: A unified description of external pressure effect and thermal expansion of Rochelle salt NaKC 4H4O6 · 4H2O A.P. Moina 1, R.R. Levitskii 1, I.R. Zachek 2 1 Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, 1 Svientsitskii Str., 79011 Lviv, Ukraine 2 Lviv Polytechnical National University, 12 Bandera Str., 79013 Lviv, Ukraine Received August 29, 2011, in final form November 9, 2011 We elaborate a modification of the deformable two-sublattice Mitsui model of [Levitskii R.R. et al., Phys. Rev. B. 2003, 67, 174112] and [Levitskii R.R. et al., Condens. Matter Phys., 2005, 8, 881] that consistently takes into account diagonal components of the strain tensor, arising either due to external pressures or due to thermal expansion. We calculate the related to those strains thermal, piezoelectric, and elastic characteristics of the system. Using the developed fitting procedure, a set of the model parameters is found for the case of Rochelle salt crystals, providing a satisfactory agreement with the available experimental data for the hydrostatic and uniaxial pressure dependences of the Curie temperatures, temperature dependences of spontaneous diago- nal strains, linear thermal expansion coefficients, elastic constants cEij and cEi4, piezoelectric coefficients d1i and g1i (i = 1, 2, 3). The hydrostatic pressure variation of dielectric permittivity is described using a derived expression for the permittivity of a partially clamped crystal. The dipole moments and the asymmetry param- eter of Rochelle salt are found to increase with hydrostatic pressure. Key words: Rochelle salt, thermal expansion, hydrostatic pressure, uniaxial pressure, Mitsui model PACS: 65.40.De, 77.80.B-, 77.65.Bn, 65.40.Ba, 77.22.Ch 1. Introduction The Mitsui model [1] (two-sublattice Ising model with asymmetric potentials) has been origi- nally proposed for description of the reentrant phase behavior in Rochelle salt crystals. It considers the motion of certain ordering units in two interpenetrating sublattices of asymmetric double-well potentials. The model with certain modifications is also applicable to several other ferroelectrics [2], like those of the Rochelle salt type (deuterated and ammonium-doped [3] Rochelle salt at least at low doping ), RbHSO4 type [4, 5], AgNa(NO2)2 (SSN [6, 7]), SASD type [8] (NaNH4SO4·2H2O and NaNH4SeO4·2H2O), etc. Very often, inclusion of deformational effects into the Mitsui model is indispensable for a proper description of the system behavior even at ambient pressure. Thus, a strong piezoelectricity asso- ciated with polarization P1 and shear strain ε4 essentially affects the dynamic dielectric response of Rochelle salt due to the effect of crystal clamping by the high-frequency measuring field. The conventional Mitsui model yields a qualitatively incorrect behavior of the relaxation time and dynamic dielectric permittivity near the Curie temperatures. This problem is resolved [9] by tak- ing into account the piezoelectric coupling with ε4, also permitting to describe the phenomena of piezoelectric resonance and sound attenuation [10]. The sequence of phase transitions observed in NH4HSO4 crystals can be described within the mean field approximation for the Mitsui model only with temperature dependent interaction constants [5]. It means that one must take into account the effect of thermal expansion, which is, ultimately, a deformational effect. Since high pressure studies are the only means to continuously vary the system geometrical parameters, such as the orientation angles of atomic groups that form the dipole moments, inter- c© A.P. Moina, R.R. Levitskii, I.R. Zachek, 2011 43602-1 http://dx.doi.org/10.5488/CMP.14.43602 http://www.icmp.lviv.ua/journal A.P. Moina, R.R. Levitskii, I.R. Zachek atomic distances, hydrogen bonds parameters, etc, as well as the interparticle interactions, and other parameters of the system, they can provide a valuable information on the mechanism of the phase transitions in ferroelectric crystals. A better insight is obtained if the effects of hydrostatic and various uniaxial and biaxial pressures are explored. Irrespectively of the crystal symmetry, these pressures produce diagonal components of the lattice strain tensor εi (i = 1, 2, 3). In low- symmetry systems, shear strains εj (j = 4, 5, 6) can be induced as well. The diagonal strains also arise due to the thermal expansion of the crystals. Nowadays, Rochelle salt [11–14] and other crystals [15] described by the Mitsui model often serve as test materials for experiments with nanosize phenomena. Properties of the ferroelectric nano- inclusions in a solid matrix are strongly affected by surface tension and thermal mismatch stresses. Thus, radial stresses in the plane perpendicular to spontaneous polarization arise in Rochelle salt nanorods grown in pores of alumina films [11, 12], inducing diagonal strains only. A phenomeno- logical theory of ferroelectric properties of such nanorods was presented in [16]. The simplest way to incorporate an external pressure into a spin model is to consider its param- eters (interaction constants) phenomenologically as linear functions of pressure. This approach has been used for different versions of the Mitsui model to describe the hydrostatic pressure variation of the transition temperatures in Rochelle salt [17] and SASD [18]. However, to describe a uniaxial stress dependence of TC in the same way, one would have to find new values of the fitting param- eters for each stress direction. A unified model description of hydrostatic and uniaxial pressure effects, and, at the same time, the crystal thermal expansion at ambient pressure, with a single set of the fitting parameters is possible if we include into the model the lattice diagonal strains, instead of the pressures. Such a microscopic-like model of bulk crystals that includes the diagonal strains will be very helpful when one develops a model description of the above mentioned nanocrystal behavior. The goal of the present paper is to develop a modification of the deformable Mitsui model with the shear strain ε4 [9], which would also take into account the diagonal components of the lattice strain tensor. The first attempt to create such a modification was made in [19]. The interaction constants and the asymmetry parameter were taken to be linear functions of the diagonal strains. Expressions for the piezoelectric and elastic characteristics of Rochelle salt, associated with these strains, have been obtained. However, all actual calculations were performed in the approximation of zero thermal strains; the external hydrostatic or uniaxial pressure effect was not considered, and the fitting procedure was inappropriate. Here we shall use the model of [19] and take into account the thermal expansion strains properly. We shall develop a consistent fitting procedure, free from the drawbacks of the previous [19] calculations, allowing us to obtain the above mentioned unified description of high pressure effects, thermal expansion, as well as physical properties of Rochelle salt associated with the diagonal strains. Bulk Rochelle salt undergoes two second-order phase transitions at TC1 = 255 K and TC2 = 297 K, with the intermediate ferroelectric phase. Spontaneous polarization P1 is directed along the a axis, accompanied by spontaneous shear strain ε4 in the bc plane. The crystal is orthorhombic (space group P212121) in the paraelectric phases and monoclinic (P2111) in the ferroelectric phase. As it follows from the analysis of symmetry elements of its point group 222 and of the uniaxial pressure point group ∞/mmm, no uniaxial or biaxial pressure applied along the orthorhombic crystallographic axes of a Rochelle salt crystal changes its symmetry. Neither do the hydrostatic pressure or thermal expansion of the crystal. The mechanism of phase transitions in Rochelle salt remains rather obscure. According to the most recent measurements [20, 21], the largest displacements at a ferroelectric phase transition are undergone by the O8, O9, O10 oxygens. It appears that the dipoles of the Mitsui model, moving in the double-well potentials, should plausibly be associated with the OH9 and OH10 groups. Their motion, coupled with displacive vibrations of OH8 groups seems to be responsible for the phase transitions, as well as for the spontaneous polarization. It would be interesting to elucidate the pressure variation of the dipole moments, as well as of the asymmetry parameters of the double-well potentials. This can shed some light on the details of the transition mechanism. The set of experimental data for Rochelle salt, used for verification of the present modification of the Mitsui model, includes the hydrostatic pressure dependences of transition temperatures 43602-2 Mitsui model with diagonal strains [22, 23] and dielectric permittivity [23], the uniaxial [24, 25] and biaxial [26] pressure dependences of the Curie temperatures. Related to the diagonal strains, components of piezoelectric (e.g. d1i, i = 1, 2, 3) and elastic (ci4) tensors [27–29], appearing in the ferroelectric phase only, should be also described by the model. The other characteristics, included into the fitting, are thermal expansion coefficients and dilatations [30, 31], as well as the diagonal-strain-related elastic constants[32] cij or compliances sij (i, j = 1, 2, 3). We introduce the diagonal strains ε1, ε2, ε3 into the two-sublattice Mitsui model with the shear strain ε4 [9] in the manner it was done in [19]. In section 2 we make a further modification of the model by taking into account the host lattice contributions into the thermal expansion. Section 3 contains the obtained expressions for thermal, elastic, dielectric, and piezoelectric characteristics. In section 4 we propose a new fitting procedure. Using the found set of the fitting parameters for the Rochelle salt crystals, we show that the developed theory is capable of describing the entire complex of the phenomena, related to the diagonal strains: thermal expansion, temperature behav- ior of monoclinic piezoelectric and elastic characteristics, hydrostatic, uniaxial, and biaxial pressure dependences of the Curie temperatures in Rochelle salt. The behavior of the dielectric permittiv- ity under hydrostatic pressure is described using the derived expression for the permittivity of a partially clamped crystal. 2. System thermodynamics in the presence of diagonal strain s We consider an orthorhombic piezoelectric crystal in the paraelectric phase, to which an external hydrostatic or uniaxial or biaxial (along the crystallographic axes) pressure can be applied. All these pressures produce diagonal strains εi (i = 1, 2, 3); these strains are also present due to the thermal expansion. Paraelectric piezoelectricity is associated with the shear strain ε4. The Rochelle salt symmetry is presumed having the spontaneous polarization directed along the a axis and coupled to the strain ε4. We start with the modified two-sublattice Mitsui model with a piezoelectric coupling to the shear strain ε4 and with the diagonal strains [9, 19]. A unit cell of the model consists of two dipoles (two sublattices), oppositely oriented along the a-axis (the axis of spontaneous polarization); they are compensated in the paraelectric phases and get non-compensated in the ferroelectric phase. The actual unit cell of a real Rochelle salt crystal is twice as large and contains four dipoles. At the transition to the monoclinic phase the angle between the b and c axes changes from π/2 to π/2− ε4. The diagonal strains ε1, ε2, ε3 describe the relative changes in the lattice constants a, b, and c, respectively, due to thermal expansion or under pressure. In the mean field approximation, the model Hamiltonian reads [9] Ĥm = NUseed + N 8 J(η21 + η22) + N 4 Kη1η2 − ∑ q [ E(1) σq1 2 − E(2) σq2 2 ] , (1) where ηf ≡ 〈σqf 〉; N is the number of the unit cells; J , K are the Fourier-transforms (at k = 0) of the constants of interaction between pseudospins belonging to the same and to different sublattices, respectively. The phenomenological part of the Hamiltonian Useed is a “seed” energy of the host lattice of heavy ions which forms the asymmetric double-well potentials for the pseudospins. For the case of Rochelle salt symmetry in presence of diagonal strains, shear strain ε4, and field E1, it reads Useed = v0 2 cE0 44 ε 2 4 − v0e 0 14ε4E1 − v0ε0 2 χε011E 2 1 + v0 2 3 ∑ i,j=1 cE0 ij εiεj − v0 3 ∑ ij=1 cE0 ij α 0 i (T − T 0 i )εj . (2) Here ε0 is the vacuum permittivity; v0 is the unit cell volume of the model. The three first terms in Useed are the elastic, piezoelectric, and electric contributions due to the shear strain ε4 and longitudinal electric field E1. Two last terms are related to the diagonal strains. Here cE0 44 , cE0 ij , e014 are the “seed” constants describing the phenomenological contributions of the crystal lattice into the corresponding observed quantities cE44, c E ij , and e14. In fact, the index E in cE0 ij is redundant, as 43602-3 A.P. Moina, R.R. Levitskii, I.R. Zachek the difference between the observed cEij and cPij at i, j = 1, 2, 3 is negligible. The “seed” quantities are zeros if the corresponding observed quantities are zeros in the most symmetric phase (orthorhombic in the case of Rochelle salt). The last term, absent in the earlier model [19], is the contribution of the host lattice into the energy of thermal expansion. α0 i are the “seed” thermal expansion coefficients; T 0 i are the temperatures at which the components of this contribution vanish. It is known that the thermal strains can be set to be equal to zero at any arbitrary temperature T0 (the reference point for thermal expansion). This can be achieved by choosing the values of T 0 i accordingly; they will differ from T0 due to the pseudospin system contributions to the thermal expansion. To take into account the “seed” contribution of the host lattice is indispensable for a proper description of thermal expansion. The coefficients E(1) = 1 2 Jη1 + 1 2 Kη2 +∆− 2ψ4ε4 + µ1E1, E(2) = 1 2 Jη2 + 1 2 Kη1 −∆− 2ψ4ε4 + µ1E1 (3) in (1) are the local mean fields acting on pseudospins of the first and second sublattices in the qth unit cell. The parameter ∆ describes the asymmetry of the double well potential; µ1 is the effective dipole moment. The model parameter ψ4 describes the internal field created by the piezoelectric coupling with ε4 and essentially determines the piezoelectric and elastic characteristics associated with the shear strain ε4 [9, 19]. It is also assumed that a longitudinal electric field E1 is applied. J ±K = J0 ±K0 + 2 3 ∑ i=1 ψ± i εi , (4) as well as the asymmetry parameter ∆ = ∆0 + 3 ∑ i=1 ψ3iεi (5) are taken to be linear functions of the diagonal strains [19]. Here ψ± 3i are introduced simply as the expansion coefficients. However, for J and K such an expansion is equivalent to taking into account the electrostrictive coupling with the diagonal strains. For ∆ it implicitly describes the changes in the asymmetry parameter due to the changes produced by external pressure or thermal expansion in the interatomic distances and in the geometric parameters of the potential, like the distance between the potential wells, etc. The parameters ψ3i, are analogous to the deformational potentials 2γ = ∂∆/∂ε introduced in the Anderson-Halperin-Varma-Phillips [33, 34] model of two- level systems with asymmetric double-well potentials in order to describe ultrasound attenuation and thermal conductivity in amorphous solids at very low temperatures. The thermodynamic potential of the considered model is obtained within the mean field ap- proximation in the following form g2E(pi, T ) = Useed + J +K 4 ξ2 + J −K 4 σ2 − 2 ln 2 β − 1 β ln cosh γ + δ 2 cosh γ − δ 2 + v0 4 ∑ i=1 piεi , (6) where β = 1/kBT , kB is the Boltzmann constant, and γ = β ( J +K 2 ξ − 2ψ4ε4 + µ1E1 ) , δ = β ( J −K 2 σ +∆ ) . Here p1 = p2 = p3 = ph for hydrostatic pressure; p1 6= 0 and p2 = p3 = 0 for the uniaxial pressure applied along the axis a, etc. The shear pressure p4 is introduced formally, in order to find the elastic and piezoelectric characteristics associated with it; after that it is put equal to zero. We introduced the following linear combinations of the mean pseudospin values ξ = 1 2 (η1 + η2), σ = 1 2 (η1 − η2); 43602-4 Mitsui model with diagonal strains ξ is the parameter of ferroelectric ordering in the system. The parameters ξ and σ are determined from the saddle point of the thermodynamic potential (6): a minimum of g2E with respect to ξ and a maximum with respect to σ are realized at equilibrium. The corresponding equations are ξ = sinh γ cosh γ + cosh δ , σ = sinh δ cosh γ + cosh δ . (7) 3. Physical characteristics of Rochelle salt related to dia gonal strains Using the following thermodynamic relations 1 v ( ∂g2E ∂εi ) E1 = 0, − 1 v ( ∂g2E ∂E1 ) σ = P1 (i = 1÷ 4), where v = v0(1 + ∑3 i=1 εi) is the pressure and temperature dependent unit cell volume, and retaining only linear in εi terms in the “seed” contributions, we obtain expressions for strains and polarization εi = − 3 ∑ j=1 sE0 ij pj + α0 i (T − T 0 i ) + 1 2v0 3 ∑ j=1 sE0 ij (ψ+ j ξ 2 + ψ− j σ 2 + 2ψ3jσ), (i = 1÷ 3), (8) ε4 = − p4 cE0 44 + e014 cE0 44 E1 − 2ψ4 v0cE0 44 ξ , (9) P1 = e014ε4 + ε0χ ε0 11E1 + µ1 v ξ . (10) Here sE0 ij are the elements of the matrix inverse to the matrix of “seed” elastic constants cE0 ij . Two first terms in equation (8) represent the host system contributions into the Hooke’s law and thermal expansion with regular pressure and temperature behavior. The sum in equation (8) gives the pseudospin subsystem contributions into the strains, having anomalous behavior in the ferroelectric phase. The second term in equation (8) was absent in the previous model [19]. From equations (8)–(10) we can derive expressions for other characteristics related to the diago- nal strains. Thus, the coefficients of linear thermal expansion are obtained in a rather cumbersome form αi = ( ∂εi ∂T ) p = 3 ∑ k=1 Bik { α0 k + 1 2v0T (ϕ2 − Λϕ3) × 3 ∑ j=1 sE0 kj [ ψ+ j ξ(λ2δ − ϕ3γ) + (ψ− j σ + ψ3j)(λ2γ − ϕ̃5δ) ] } , (11) where B̂ = [ Î + ŝE0ˆ̃c ]−1 , c̃jk = − β 2v0(ϕ2 − Λϕ3) { (ψ+ i ϕ4j + ψ+ j ϕ4i)ξ − ψ+ i ψ + j ξ 2ϕ3 + (ψ− i σ + ψ3i)(ψ − j σ + ψ3j)ϕ̃5 } , ϕ̃5 = λ1 − [ β(K + J) 4 + Λ ] (λ21 − λ22); Î is a unit matrix. The other notations are ϕ4i = ψ+ i ξϕ3 − ( ψ− i σ + ψ3i ) λ2 , ϕ2 = 1− βJ 2 λ1 − β2K 2 − J2 16 (λ21 − λ22), ϕ3 = λ1 + β(K − J) 4 (λ21 − λ22), λ1 = 1− ξ2 − σ2, λ2 = 2ξσ, Λ = 2βψ2 4 v0cE0 44 . 43602-5 A.P. Moina, R.R. Levitskii, I.R. Zachek Alternatively, the coefficients of thermal expansion can be found by numerical differentiation of equation (8) for the strains εi with respect to temperature; the results, of course, coincide. The coefficients αi are expected to have small anomalies at the Curie temperatures [30, 31]. The molar specific heat at constant pressure of the model is obtained from the molar entropy S=− NA 2 (∂g1E(εi, T ) ∂T ) εi = v0NA 2 3 ∑ ij=1 cE0 ij α 0 i εj + R 2 ( 2 ln 2 + ln cosh γ + δ 2 cosh γ − δ 2 − γξ − δσ ) , where g1E(εi, T ) = g2E(σi, T )−v0 ∑ i piεi; R is the universal gas constant, and NA is the Avogadro constant. Thus, the molar specific heat of the model is cp = T ( ∂S ∂T ) p = v0NAT 2 3 ∑ ij=1 cE0 ij α 0 iαj + R 4(ϕ2 − Λϕ3) { (λ2δ − ϕ3γ) ( γ + ξ 3 ∑ i=1 αi kB ψ+ i ) −(λ2γ − ϕ̃5δ) [ δ − 3 ∑ i=1 αi kB (ψ− i σ + ψ3i) ]} . (12) As we shall see, it has small anomalies at the transition points. To obtain the total specific heat of a crystal that can be compared to experimental data, we have to add to equation (12) a regular term linear in temperature (within the considered temperature range) that would correspond to a contribution of lattice vibrations not taken into account within our model. Thus, ctotp = cp + cvibr , cvibr = A+BT ; (13) the coefficients A and B will be specified by fitting equation (13) to experimental data. Often, an inverse procedure is performed, when a regular linear contribution is subtracted from the experi- mental data; the obtained result is then compared with the theoretical specific heat of the ordering subsystem. The other found characteristics are, in particular, the elastic constants at constant electric field (i, j = 1, 2, 3) cEij = − (∂pi ∂εj ) E,T = cE0 ij − β 2v0ϕ2 [ (ψ+ i ϕ4j + ψ+ j ϕ4i)ξ − ψ+ i ψ + j ξ 2ϕ3 + (ψ− i σ + ψ3i)(ψ − j σ + ψ3j)ϕ5 ] , (14) cEi4 = βψ4 v0 ϕ4i ϕ2 , (15) as well as the monoclinic piezoelectric coefficients e1i = ( ∂P1 ∂εi ) E1,T = µ1 v ( βϕ4i 2ϕ2 − ξ 1 + ∑3 i=1 εi ) , d1i = ( ∂εi ∂E1 ) pi,T = 4 ∑ j=1 sEije1j , (16) where sEij is the matrix of elastic compliances, inverse to the matrix of elastic constants cEij , and ϕ5 = λ1 − β(K + J) 4 ( λ21 − λ22 ) . The other piezoelectric and elastic characteristics are h1i = − ( ∂E1 ∂εi ) P1 = e1i ε0χε11 , g1i = − ( ∂E1 ∂pi ) P1 = d1i ε0χσ11 , cPij = ( ∂pi ∂εj ) P,T = cEij+e1ih1j . 43602-6 Mitsui model with diagonal strains Here, χε11 = 1 ε0 ( ∂P1 ∂E1 ) ε = χε011 + βµ2 1 2vε0 ϕ3 ϕ2 (17) is the dielectric susceptibility of a clamped crystal, and χσ11 = 1 ε0 ( ∂P1 ∂E1 ) p = χσ011 + β(µ′ 1) 2 2vε0 ϕ3 ϕ2 − Λϕ3 + 1 ε0 3 ∑ i=1 e1id1i (18) is the static dielectric susceptibility of a mechanically free crystal [19]. Here we introduce the following notations µ′ 1 = µ1 − 2ψ4d 0 14 , d014 = e014 cE0 44 , χσ011 = χε011 + 1 ε0 e014d 0 14 . In paraelectric phases, this expression for the free susceptibility coincides with that obtained within the modified Mitsui model without thermal strains [10]. The sum ε−1 0 ∑ e1id1i , different from zero in the ferroelectric phase and not exceeding 5% of the total susceptibility, was absent in the earlier model. As one can easily verify, monoclinic quantities e1i, d1i, h1i, g1i, cEi4, c P i4 (i = 1, 2, 3) differ from zero only at non-zero polarization, in agreement with the symmetry considerations. The temperature of the second order phase transition is determined from the condition that dielectric susceptibility of a free crystal χσ11 diverges at T → TC. From equation (18) and using equation (7) we obtain cosh2 ( J −K 4kBTC σc + ∆ 2kBTC ) = K + J 4kBTC + 2ψ2 4 v0cE0 44 kBTC , (19) where the model parameters J , K, ∆ are taken at TC, being renormalized by diagonal strains according to equation (4). Equation (19) is valid both for the ambient pressure case and for the stressed crystal. It can be rewritten in the two following convenient forms: σc = √ √ √ √1− kBTC K+J 4 + 2ψ2 4 v0cE0 44 , (20) which gives an explicit expression for σ at the transition points, and ∑ i δ3iεci + σc ∑ i ψ− i εci = −∆0 − J0 −K0 2 σc + 2kBTCArccosh √ K + J 4kBTC + 2ψ2 4 v0cE0 44 kBTC , useful in the fitting procedure. Here εci are the strains at the Curie temperature. 4. Numerical calculations 4.1. Fitting procedure The model parameters must provide a fit of the theory to the experimental data for the following characteristics: the Curie temperatures at ambient pressure TCk (k = 1, 2 in Rochelle salt) and their hydrostatic and uniaxial pressure slopes ∂TCk/∂ph and ∂TCk/∂pj, the temperature curves of thermal expansion strains εi, linear thermal expansion coefficients, monoclinic piezomodules g1i, and elastic constants cij and ci4 (i, j = 1, 3). Simultaneously we check for an agreement with experiment for the quantities previously described [9, 10] by the modified Mitsui model without thermal strains, such as spontaneous polarization, static free and clamped dielectric susceptibilities χσ,ε11 , piezomodule d14, specific heat, elastic constant at constant field cE44, as well as microwave 43602-7 A.P. Moina, R.R. Levitskii, I.R. Zachek dielectric permittivity ε11(ν, T ). A detailed analysis of the effect of diagonal strains on the physical characteristics of Rochelle salt associated with the shear strain ε4 will be given elsewhere. The adopted values of the model parameters are given in table 1; details of the fitting procedure are described below. Table 1. The model parameters used for description of Rochelle salt. ā0 b̄0 J0/kB K0/kB ∆0/kB ψ4/kB µ0 1 kT e014 χσ0 11 (K) (10−30 C·m) (K−1) C/m2 0.3162 0.662 764.63 1476.46 745.14 −750 8.7 −0.0008 0.033 10.1 ψ+ 1 /kB ψ+ 2 /kB ψ+ 3 /kB ψ− 1 /kB ψ− 2 /kB ψ− 3 /kB ψ31/kB ψ32/kB ψ33/kB (K) −13000 −12000 −9850 10614 13537 1080 −12125 −15993 −6043 α0 1 α0 2 α0 3 T 0 1 T 0 2 T 0 3 (10−5 K−1) (K) 5.800 3.353 4.333 238.35 185.32 311.71 cE0 11 cE0 12 cE0 13 cE0 22 cE0 23 cE0 33 cE0 44 (1010 N/m2) 2.842 1.794 1.541 4.219 2.031 3.987 1.18 The temperature variation of the order parameter ξ is determined by numerical minimization of the thermodynamic potential (6); σ is found from equation (7); the strains are determined from equations (8) and (9). As the reference point for thermal expansion (where εi = 0 at pi = 0) we chose the upper transition temperature TC2 = 297 K. This condition allows us to express T 0 i from equation (8) via α0 i , ψ − i , and ψ3i. Strictly speaking, T 0 i are not the fitting parameters of the model, since the reference point can be chosen arbitrarily. At 308 K and at ambient pressure, the lattice constants are [35] a = 11.927 Å, b = 14.292 Å, c = 6.225 Å. The “seed” linear thermal expansion coefficients α0 i were chosen to yield the thermal strains εi at TC1 equal to α275 i (TC1 − TC2), where α275 i are the experimental [31] values of the expansion coefficients in the middle of the ferroelectric phase (at 275 K). Special care has been taken that below 20 kbar for the hydrostatic pressure and below 200 bar for uniaxial or biaxial pressures and between 0 and 350 K, no additional phase transition takes place in the system, apart from those taking place at ambient pressure. The number and (if any) temperature and order of the phase transitions for the Mitsui model without thermal strains are usually analyzed in terms of the dimensionless variables ā and b̄ ā = K − J K + J + 8 v0 ψ2 4s E0 44 , b̄ = 8∆ K + J + 8 v0 ψ2 4s E0 44 (21) and the dimensionless transition temperature t̄0c = 4kBTC K + J + 8 v0 ψ2 4s E0 44 . (22) The phase diagram of the conventional (undeformable) Mitsui model in the (ā, b̄) plane [36–38] shows the regions with different numbers and types of the phase transitions; its topology is not changed by inclusion of the shear strain ε4. It has been found that only in a very narrow region of the (ā, b̄) plane, the system undergoes two second order phase transitions with the intermediate ferroelectric phase. In the presence of diagonal strains, ā and b̄ become functions of temperature and pressure. In the fitting procedure we shall deal with the values of ā and b̄ at the upper Curie temperature and at ambient pressure ā0 and b̄0. Absence of additional phase transitions at the chosen values of 43602-8 Mitsui model with diagonal strains the model parameters is verified directly, by calculating the order parameter at all temperatures between 0 and 350 K and at pressures below 25 kbar (hydrostatic) or 200 bar (uniaxial). The values of ā0 and b̄0 should be from the same region of the (ā, b̄) phase diagram of the undeformable Mitsui model that yield two second order phase transitions. With decreasing ā0, the maximal values of spontaneous polarization, spontaneous strain ε4, and anomalous parts of diagonal strains increase. We choose the value of ā0 that gives the best agreement with experiment for these characteristics. Once the values of ā0, b̄0, ψ4, and cE0 44 are chosen, we are in a position to find J0, K0, and ∆0, using equations (20), (21), and (22). The parameters cE0 44 , ψ4, and µ1 are varied around their values obtained in the previous study [9] in order to get the best fit for spontaneous polarization P1, piezoelectric coefficient d14, static free and clamped χσ,ε11 dielectric susceptibilities and dynamic dielectric permittivity ε11(ν, T ). The dipole moment µ1 is assumed to decrease linearly with an increasing temperature as µ1 = µ0 1 [1 + kT(T − TC2)] . The values of µ0 1 and kT are given in table 1. We require that the theoretical values of the elastic constants cEij (i, j = 1, 3) at TC2 should coincide with their experimental values [39], available for 307 K (this is a reasonable approximation due to a very weak temperature dependence of cEij). Thus, we can easily determine cE0 ij using equation (14). It is required that the best possible description of experimental [24, 25] uniaxial pressures dependence of the two Curie temperatures should be obtained. To this end, at the chosen a0, b0, ψ4, cE0 44 , ψ+ i , the six parameters ψ− i and ψ3i are determined from six linear equations (21) written at T iCk = T 0 Ck + (∂TCk/∂pi)pi (k = 1, 2 and i = 1, 2, 3) at pi = 100 bar and combined with equation (20). The slopes ∂TCk/∂pi were varied around their experimental values[24]. The strains at the Curie temperatures were approximated as εcl(T i Ck) = α275 l (T iCk − T 0 C2)− sE0 li pi during the fitting. The obtained values of δ−i and ψ3i were found to be independent of the used values of pi. One of ψ+ i parameters, say, ψ+ 3 , can be determined from the condition that the lower Curie temperature should be TC1 = 255 K. For the two remaining parameters ψ+ 1 and ψ+ 2 there is the condition that the two calculated transition temperatures at hydrostatic pressure of 1 kbar and the lower transition temperature at 20 kbar would be in agreement with the experimental data [22, 23]. However, the dependences of the transition temperatures on uniaxial pressures and on the hydrostatic pressure are not completely independent, (this will be discussed later). Therefore, normally ψ+ 1 and ψ+ 2 can be varied continuously in certain ranges and provide correct theoretical dependences TCk(ph) at given ā0, b̄0 and ψ4. From these ranges we should select the values which yield the best fit for the temperature curves of the piezoelectric coefficients g1i, anomalous parts of thermal strains εsi in the ferroelectric phase, and thermal expansion coefficients αi. It should be mentioned, however, that a perfect fit both for g1i ([27]) and εsi ([31]) cannot be obtained simultaneously, so a certain compromise has to be made. A criterion to make an unambiguous choice of the theory parameters hardly exists. Since the chosen set of ψ± 3i is not unique, it is not possible to precisely establish the temperature and pressure variation of the interaction constants. However, the overall tendency is such that the hydrostatic compression enhances the asymmetry parameter ∆, as well as the constants of interactions between the pseudospins within the same and in different sublattices. Pressure slopes of ∆, J , and K are very sensitive to the choice of b̄0 at given ā0, while the observable quantities are not that much sensitive. The average slopes are about 1–3%/kbar for J and 4.5–7%/kbar for ∆ and K. This issue will be explored in more detail elsewhere. 4.2. Thermal expansion and specific heat The temperature dependence of the diagonal strains εi caused by thermal expansion of a crystal in the absence of external pressures is plotted in the inset to figure 1. 43602-9 A.P. Moina, R.R. Levitskii, I.R. Zachek The experimental points, obtained from the data for thermal dilatations [31], are well described by the proposed theory. 260 280 300 320 340 -8 -4 0 4 8 240 260 280 300 320 -4 -2 0 T(K) 3 2 1 ε si , 10-5 T(K) ε i , 10-3 3 2 1 240 260 280 300 320 20 40 60 80 T (K) 3 2 1 α i (10-6 K-1) Figure 1. Anomalous parts of thermal diago- nal strains of Rochelle salt as functions of tem- perature: 1, �: εs1; 2, ©: εs2; 3, △: εs3. Inset: total strains as functions of temperature: 1, �: ε1; 2, ©: ε2; 3, △: ε3 Lines: a theory; symbols: experimental points taken from [31]. Figure 2. Coefficients of linear thermal expan- sion of Rochelle salt as functions of tempera- ture. 1, �, �: α1; 2, ©, •: α2; 3, △, N: α3. Lines: a theory. Open and closed symbols are experimental points taken from [31] and from [40], respectively. In the ferroelectric phase, the εi(T ) curves have small bucklings (anomalous parts) caused by electrostrictive coupling to spontaneous polarization. In order to extract these anomalous parts of the diagonal strains, Imai [31] extrapolated the measured temperature curves of the strains from the paraelectric phases onto the ferroelectric phase and subtracted them from the total measured strains. With the same purpose, we calculate some hypothetical paraelectric strains εhi (coinciding in the paraelectric phases with the actual strains εi) from equation (8) by putting ξ = 0 at all temperatures and determining σ from equation (7), and find the spontaneous strains as εsi = εi−ε h i . The obtained results are shown in the major part of figure 1. As one can see, at the adopted values of the model parameters, the theory well reproduces the asymmetric shape of the εs3(T ) curve, but underestimates the magnitude of εs1 and εs2. The agreement can be improved by choosing different values of the model parameters, but the agreement with g1i(T ) will be spoiled. The corresponding linear thermal expansion coefficients are shown in figure 2. The theoretical and experimental values of their jumps at the Curie temperatures are summarized in table 2. Table 2. The calculated jumps of the thermal expansion coefficients and specific heat at Curie temperatures. The values in parentheses are experimental data of [31]. ∆α1 ∆α2 ∆α3 ∆cp (10−6K−1) (J/mol K) TC1 –6.0 (–7.0) 3.0 (3.5) 3.7 (4.5) 0.74 TC2 4.7 (7.1) –2.1 (–3.7) –1.1 (–2.0) 1.09 We have a fairly good agreement with experiment [31] for α1 and α3, both for the signs and for the values of the coefficient anomalies at the transition points, as well as for the temperature slope of the curves in the ferroelectric phase. The agreement is worse for α2. It should be noted that the experimental behavior of α2(T ) is somewhat different from that of α1 and α3, with a marked decrease below the lower Curie temperature and a large difference between the coefficient values in the two paraelectric phases. No perceptible temperature variation in the ferroelectric phase was experimentally detected, also in contrast with the α1 and α3 behavior. The theoretical temperature curve of α2(T ) is very much like that of α3 and qualitatively similar to that of α1. The theoretical 43602-10 Mitsui model with diagonal strains α2 in the upper paraelectric phase well accords with the single experimental value of [40] obtained from the synchrotron radiation Renninger scan, although the reported error of these measurements is so large that the values of [31] also fall in this error range (see figure 2). 252 255 258 261 344 348 352 294 297 300 381 384 387 T (K) c p (J/mol K) T (K) c p (J/mol K) Figure 3. Specific heat of Rochelle salt as a function of temperature. Lines: a theory. Solid line: this work; dashed line: the modified Mitsui model without the thermal strains [9]. Symbols: experimental points taken from [43]. Figure 3 shows that the present model yields a better agreement with experimental data for the small anomalies of specific heat of Rochelle salt at the Curie points than it was obtained with the earlier model [9], especially for the magnitude of the upper anomaly. The regular contribution of lattice vibrations was taken to be cvibr = 105.845+0.855T (J/mol K) for the present model and 103.456+ 0.944T (J/mol K) for the previous model. The calculated values of the specific heat jumps are given in table 2. We obtain positive anoma- lies at both transition points, in accordance with the most recent measurements [43]. The jumps were defined as the differences between the ferroelectric and paraelectric values of specific heat or expansion coefficients at the Curie points. 4.3. Diagonal-strain-related piezoelectric and elastic c onstants Figure 4 shows temperature dependences of piezoelectric constants g1i. As one can see, a fairly good agreement with experiment is obtained, including the opposite signs of g11 and of g12 and g13, asymmetric shape of g13(T ) dependence, as well as the interception of the g12 and g13 curves near the lower Curie point. The overall behavior of g1i(T ) is similar to that of εsi(T ). 240 260 280 300 -0.04 -0.02 0.00 0.02 0.04 g 12 g 13 g 11 g 1i (m2/C) T (K) 240 260 280 300 -0.2 -0.1 0.0 0.1 PP EE c 24 + c 34 c 24 + c 34 c 24 + c 34 (1010 N/m2) T (K) Figure 4. Piezoelectric constants of Ro- chelle salt as functions of temperature at E1 = 5 kV/cm: 1, �: g11; 2, ©: g12; 3, △: g13. Lines: a theory; symbols: experimental points taken from [27]. Figure 5. Elastic constants cE24+ c E 34 and cP24+ cP34 of Rochelle salt as functions of temperature at E1 = 500 V/cm. Lines: a theory; symbols: experimental points taken from [29]. 43602-11 A.P. Moina, R.R. Levitskii, I.R. Zachek We do not depict the calculated elastic constants cEij and cPij (i, j = 1, 2, 3), as they are practically temperature independent between 230 and 330 K. A very small variation can be detected in the ferroelectric phase for cEij , with the difference between cEij and constant cPij being less than 1% of cEij at most. The maximal values of monoclinic constants cEi4 are more than by one order of magnitude smaller than cEij . The shape of the experimental cE24+ c E 34 vs T curve is qualitatively reproduced by the proposed model, as seen in figure 5. A quantitative agreement with experiment is reasonable. The coefficients of piezoelectric strain d1i are shown in figure 6 at zero and high bias fields E1. The data of [27] were extracted from the given therein values of the d11g11 product and of g11. In absence of external field d1i actually diverges at the Curie temperatures, due to the term proportional to sE44 in equation (16); the bias field smears out these anomalies and lowers the peaks of the coefficients. A good agreement with the experimental points is obtained. 250 275 300 325 350 375 -200 -100 0 100 200 240 260 280 300 -90 -60 -30 0 d 1i (10-12C/N) T (K) d 12 d 13 d 11 321 d 11 (10-12 C/N) T(K) Figure 6. Piezoelectric coefficients d1i of Rochelle salt as functions of temperature. The inset: d11 as a function of temperature at different fields E1 (kV/cm): 1, 0; 2, △: 0.304; 3, �: 5. Lines: a theory; symbols: experimental points taken from [27] (�) and [28] (△). 4.4. Hydrostatic pressure effects Below we shall discuss how the high-pressure effects are described by the proposed modification of the Mitsui model. Figure 7 shows the calculated hydrostatic pressure dependence of the Curie temperatures. The proposed theory reproduces the experimentally observed linear increase of both Curie temperatures with pressure at its low values, as well as the increase of ∂TC1/∂ph at higher pressures. The calculated slopes are 3.7 K/kbar at low pressures and 4.3 K/kbar above 12 kbar. 0 5 10 15 20 250 275 300 325 350 p (kbar) T C (K) Figure 7. Hydrostatic pressure dependence of the Curie temperatures of Rochelle salt. Lines: a theory; open and closed symbols: experimental points taken from [22] and [23]. 43602-12 Mitsui model with diagonal strains To describe the pressure variation of the static permittivity we need to make some changes in the theory parameters. The above given value of µ1 provides a roughly equal fit to many different experimental data for χσ11 in paraelectric phases at atmospheric pressure. However, the sample to sample variation of permittivity at ambient pressure reaches 10% even in good samples [42] and is much larger in samples with defects. This is comparable with the changes in the Curie constants produced by hydrostatic pressure [9, 23] below 10 kbar. Thus, it seems impossible to try to describe these fine high-pressure effects, using for a non-deformed crystal the averaged value of µ1 given in table 1. Instead, we shall determine separate values of µ1, which provide the best fit to the experimental data for static permittivity obtained in [23] and in [41] at each pressure considered therein. Thus, the possible pressure variation of the effective dipole moment µ1 can be inferred. The measurements, reported both in [23] and in [41], revealed a decrease of the peak value of permittivity at a lower Curie temperature with increasing pressure. This is attributed [23] to a partial clamping of the samples due to the increased viscosity of the pressure-transmitting fluid and suppression of the piezoelectric shear strain ε4. To calculate the dielectric permittivity of a partially clamped crystal we assume that the clamping caused by the viscous fluid under hydrostatic pressure is uniform throughout the crystal sample. We assume that, at least, the part of shear strain ε4 induced by the measuring electric field (normally at 1 kHz) is smaller than the one given by equation (9). The total strain is then equal to ε4 = εs4 + k ( e014 cE0 44 E1 − 2ψ4 vcE0 44 ξi ) . (23) Here εs4 is a spontaneous part of the strain; ξi is the field-induced part of the order parameter. The introduced phenomenological coefficient 0 < k < 1 describes the extent to which the external pressure suppresses the shear strain ε4: the cases k = 0 and k = 1 correspond to the totally clamped and free crystals, respectively. Values of k are, naturally, pressure and temperature dependent and are different for experimental setups with different pressure-transmitting liquids. Thus, the level of clamping at TC1 was apparently much higher in the experimental setup of [41] than in [23], possibly because the corresponding temperatures are lower, and the viscosity of pressure-transmitting liquid is higher (benzine and silicone oil [41] vs pentane isopentane mixture [23]). Substituting equation (23) into equation (10) we obtain the corresponding polarization. Dif- ferentiating it with respect to E1, taking into account equation (7) and neglecting variation of diagonal strains with the field, we get the dielectric susceptibility of a partially uniformly clamped crystal χk11 = χk011 + β(µk1) 2 2vε0 ϕ3 ϕ2 − kΛϕ3 , (24) where χk011 = χε011 + k ε0 e014d 0 14 , µk1 = µ1 − 2kψ4d 0 14 . From equation (24) in the limiting cases k = 0 and k = 1 we obtain susceptibilities of totally clamped and free (in the paraelectric phases) crystals. In the ferroelectric phase, a more accurate expression for susceptibility should also contain terms like ∑ e1id1i, albeit small, produced by contributions of diagonal strains. In this subsection these contributions will be neglected, and permittivity will be calculated using equation (24). The temperature curve of dielectric permittivity near the lower Curie point at atmospheric pressure presented in [23] appears to be drawn qualitatively. In the fitting procedure we relied on the data of [3] for these temperatures. Near the upper Curie point, the data of [23] and [3] agree fairly well. Comparison of the calculated temperature dependences of the permittivity with experimental data is given in figures 8 and 9. A fairly good description of the experiment in paraelectric phases is obtained at a proper choice of k and µ1 values. The disagreement at 2.2 and 4.1 kbar in figure 9 is due to the mismatch between the calculated and experimental Curie temperatures at these 43602-13 A.P. Moina, R.R. Levitskii, I.R. Zachek pressures. The disagreement observed in the ferroelectric phase is due to the essential domain contributions to the permittivity, not included into the present model. 240 260 280 300 320 0.000 0.005 0.010 0.015 0.020 5 4 3 2 1 -1ε 11 T(K) Figure 8. The temperature dependence of the inverse static permittivity of Rochelle salt at different values of hydrostatic pressure ph (kbar): 1, �: 0; 2, ©: 0.5; 3, △: 1.2; 4, ∇: 2; 5, ⊲: 3.2. Lines: a theory; symbols: exper- imental points taken from [41]. To fit the data of [41] the coefficient k is taken to be 1 at 297 K (a free crystal). At 255 K we use the following values of k: 0.65, 0.4, 0.25, 0.2 at 0, 0.5, 1.2, 2.0 kbar, respectively. A linear interpola- tion between these values at 255 K and 1 at 297 K is used. For 3.2 kbar we use k = 0 at 265 K and 1 at 297 K, also with a linear interpolation for the intermediate temperatures. For the data of [23] we use k = 0.9 above 11.7 kbar for the lower Curie temperature. The pressure and temperature variation of the dipole moment µ1 is more complicated and contra- dictory. First, to fit the experimental data of [41] for permittivity at ambient pressure we have to as- sume that µ1 increases with temperature; whereas for the data of [23], [3] it should be assumed to decrease with increasing temperature, but slower than it was given in table 1. To fit the permittivity points at high pressures we need to assume that the pressure dependence of µ1 is opposite to its temperature dependence: if it decreases with increasing temperature, then it increases with pressure and vice versa. Thus, for the data of [23] we use the following dependence µ1 = µ0 1 [ 1 + kT ( T − T 0 C2 )] (1 + kpp) with µ0 1 = 8.94 · 10−30 C·m and kT = −0.001 K−1. The pressure coefficient kp was 0.03 kbar−1 for pressures below 5 kbar and 0 above 10 kbar. 300 320 340 360 0.00 0.01 0.02 5 4 3 21-1ε 11 T(K) 240 260 280 300 320 340 360 0.00 0.01 0.02 0.03 9876 1 -1ε 11 T(K) (a) (b) Figure 9. (Color online) The temperature dependence of the inverse static permittivity of Rochelle salt near the upper (left) and lower (right) Curie points. The values of hydrostatic pressure are: 1, ⋆, �: 0; 2, △: 2.2; 3, ©: 3; 4, ▽ : 4.1; 5, ⊲: 5; 6, ⋄: 11.6; 7, ⊳: 14; 8, ⊕ : 16.6; 9, ⊞: 20. Symbols are experimental points taken from [3] (⋆) and [23] (other symbols). Lines: a theory. This behavior is quite unusual since the dipole moments are expected to be reduced by hydro- static pressure or by decreasing temperature due to the overall reduction of interatomic distances. The increase of µ1 with hydrostatic pressure can be explained with the help of a certain assump- tion used in constructing the spatial four-sublattice model of Rochelle salt [44]. It states that the dipole moments in it are actually the 3D vectors, which are not oriented along the a-axis like in 43602-14 Mitsui model with diagonal strains the two-sublattice model. Their projections on the b and c axes are different from zero, but com- pensated at all temperatures, unless an electric field perpendicular to spontaneous polarization is applied. The a-projections are the dipole moments of a two-sublattice Mitsui model, compensated in paraelectric phases. It may be assumed that hydrostatic pressure rotates the spatial dipoles in such a way that their projections on the a-axis (and µ1) increase. This increase is fast at low pressures and slows down above 10 kbar, possibly because the dipoles are already oriented along the a-axis. To fit the data of [41] we use µ0 1 = 1.0 · 10−30 C·m, kT = 0.003 K−1, and kp = −0.012 kbar−1. This is consistent with the picture when pressure and temperature affect the dipole moment µ1 in the opposite ways. The measurements [41] of humidity (extreme drying and wetting) effect on the permittivity of Rochelle salt have shown that in strongly wet samples the permittivity increases significantly as compared to the permittivity of normal samples, especially in the upper paraelectric phase. It appears that the samples used in the hydrostatic pressure studies [41] were moderately wet. On the other hand, a decrease of µ1 with increasing temperature (like in [3, 23, 42]) seems to be an intrinsic behavior of normal Rochelle salt samples. To explain the increase of µ1 with temperature in wet samples we can assume that due to the excess of water, the crystal conductivity increases, thus increasing the dielectric permittivity, especially at high temperatures where the mobility of charge carriers is very high. 4.5. Uniaxial pressures effects To get a clearer picture of the uniaxial pressure effect on the Curie temperature and dielectric permittivity, it is useful to analyze it within the phenomenological approach. We start with the thermodynamic potential G1 V = − 1 2 3 ∑ ij=1 sijpipj − sP44 2 p24 − P1 3 ∑ i=1 M1i4pip4 + g14P1p4 + P 2 1 3 ∑ i=1 Q1ipi + 1 2 αP 2 1 + 1 4 βP 4 1 , (where sPi4 = M1i4P1 are the elastic compliances, and the M1i4 are simply the proportionality coefficients between sPi4 and polarization. Also, Q1i are the electrostriction constants; α = αT(T − T 0 C2) or α = αT(T 0 C1 − T ), β are the coefficients of the Landau expansion). Hence, one gets the following expressions for the temperature and magnitude of the permittivity maxima Tmax 1,2 = T 0 C1,2 ± 2 αT 3 ∑ i=1 Q1ipi + 3 4 (4β)1/3 αT  E1 − 3 ∑ ij=1 M1i4pip4 + g14p4   2/3 , ε−1 max = 3 2 (4β)1/3  E1 − 3 ∑ ij=1 M1i4pip4 + g14p4   2/3 . (25) In the absence of the fields conjugate to the order parameter (E1 and p4), the Tmax 1,2 correspond to the phase transition temperatures. As one can see, in this approximation the uniaxial and hydrostatic (ph = p1 = p2 = p3) pressures lead to linear shifts of the Curie temperatures, where the permittivity still diverges. In combination with the shear pressure p4 they smear the transitions, shift the permittivity maxima, and lower the peaks height. It also follows from equation (25) that if p4 = 0 ∂TCk ∂ph = 3 ∑ j=1 ∂TCk ∂pj . (26) In table 3 we summarize the experimental data on the uniaxial pressure slopes of the transi- tion temperatures of Rochelle salt and present our results obtained within the herein developed modification of the Mitsui model. The data of [25] are estimated from the presented therein ε11(T ) curves, each being measured at a single value of pi near the upper Curie temperature. 43602-15 A.P. Moina, R.R. Levitskii, I.R. Zachek Table 3. Uniaxial pressure derivatives of the transition temperatures of Rochelle salt (in K/kbar). [24] [25] [26] this work ∂TC1/∂p1 –29 –26.6 ∂TC1/∂p2 15 14.8 ∂TC1/∂p3 17 18.0 ∂TC1/∂(p2 + p3) 30± 2 34.0 ∂TC2/∂p1 35 35.6 32.9 ∂TC2/∂p2 –16 –18 –16.5 ∂TC2/∂p3 –8 –6.7 –8.5 ∂TC2/∂(p2 + p3) −22± 1 –26.2 The theory and experiment for the uniaxial pressures agree within 10%, which is close to the experimental error [24]. The theoretical dependence of the Curie temperatures on the biaxial pres- sure p2+p3 is a little stronger than the experimental one. Also it can be noticed that equation (26) is not fulfilled. Independent measurements [25, 26, 41] of dielectric susceptibility of Rochelle salt under different uniaxial and biaxial pressures revealed a decrease of the peak values of susceptibility at the Curie points as well as smearing out of the peaks. These effects are enhanced with increasing pressures. A uniform partial clamping of samples by an apparatus creating the uniaxial pressures can explain the observed lowering of the peaks, but not the smearing of the transitions. As an intrinsic phenomenon, both these effects can be caused by application of an external field conjugate to the order parameter: the electric field E1 directed along the axis of spontaneous polarization or the shear stress σ4, that is, by the field which induces polarization P1 in the paraelectric phases. In an ideal experiment, no uniaxial or biaxial pressure applied along the orthorhombic crystallographic axes should act in this way, because piezoelectric coefficients associated with these pressures are zeros outside the ferroelectric phase. It appears that the observed smearing of the anomalies is an artefact caused by experimental errors. We can think of the following factors that in a real experiment can lead to the smearing. (i) Stress inhomogeneity. Even a weak inhomogeneity of the applied pressure, partial clamp- ing/sample end constraints, surface irregularity result in a non-uniform strain distribution over the crystal sample; thus, in different parts of the sample the phase transition is shifted to different temperatures. The higher is pressure, the larger is the difference between these temperatures, and the more diffuse is the transition, exactly as observed in [41]. (ii) Stray shear stress σ4, whose effect would be enhanced by its combination with the uniaxial pressures pi, as it follows from equation (25). The stress σ4 can arise even at a slight misori- entation of the samples, as a component of the uniaxial loading intended to create p2 or p3 pressures. There can also occur built-in local shear stresses σ4 caused by sample defects, e.g. dislocations. There will be too much uncertainty if we try to take the effect of these factors into account in the theory. Thus, we shall not attempt to describe the behavior of the static dielectric susceptibility in uniaxially stressed Rochelle salt crystals. 5. Concluding remarks We proposed a generalization of the deformable Mitsui model [9], which along with the piezo- electric shear strain ε4 takes into account the diagonal strains ε1, ε2, and ε3 as well. In contrast to the previous attempt [19], in order to incorporate the diagonal strains into this model, the thermal expansion is consistently taken into account. 43602-16 Mitsui model with diagonal strains In the mean field approximation we find polarization and the strains, as well as thermal, elastic, and piezoelectric characteristics related to diagonal strains. For the case of Rochelle salt we suggest an elaborated fitting procedure and choose the set of the model parameter values, providing as good as possible consistent description of all these characteristics, as well as of external hydrostatic and uniaxial pressure effects. The expression for susceptibility of a partially clamped crystal is derived in order to describe the behavior of the observed susceptibility of Rochelle salt under hydrostatic pressure near the lower Curie point. By fitting the theoretical curves to the experimental points, the pressure variation of the effective dipole moment µ1 is estimated. An increase of µ1 with hydrostatic pressure at low pressures and a decrease with increasing temperature seem to be an intrinsic behavior for this crystal. The interaction constants and the asymmetry parameter were found to increase with hydrostatic pressure too. This is consistent with the picture, where the dipole moments in Rochelle salt are 3D vectors [44], assuming they rotate under pressure in such a way that their projection on the a-axis increases. Apart from Rochelle salt, the developed modification of the model with appropriate changes can be used for consideration of the pressure effects and thermal expansion in other ferroelectric crystals described by the Mitsui model. The fitting procedure, however, for each such crystal will require extensive experimental data; further measurements will, therefore, be necessary. The presented model and the found values of its parameters are a good starting point for developing a model description of ferroelectricity in nanosize inclusions of compounds, to which the Mitsui model is applicable, grown in a porous matrix [11, 13, 14]. References 1. Mitsui T., Phys. Rev., 1958, 111, 1259; doi:10.1103/PhysRev.111.1259. 2. Vaks V.G., Zinenko V.I., Schneider V.E., Sov. Phys. Uspekhi, 1983, 26, 1059; doi:10.1070/PU1983v026n12ABEH004584. 3. Schneider U., Lunkenheimer P., Hemberger J., Loidl A., Ferroelectrics, 2000, 242, 71; doi:10.1080/00150190008228404. 4. 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Solids, 1978, 39, 427; doi:10.1016/0022-3697(78)90084-7. 44. Stasyuk I.V., Velychko O.V., Ferroelectrics, 2005, 316, 51; doi:10.1080/00150190590963138. Модель Мiцуї з дiагональними деформацiями: об’єднаний опис впливу зовнiшнiх тискiв i теплового розширення в сеґнетовiй солi NaKC4H4O6 · 4H2O А.П. Моїна1, Р.Р. Левицький1, I.Р. Зачек2 1 Iнститут фiзики конденсованих систем НАН України, 79011 Львiв, вул. Свєнцiцького, 1 2 Нацiональний унiверситет “Львiвська полiтехнiка”, 79013 Львiв, вул. С. Бандери, 12 Запропоновано модифiкацiю деформiвної двопiдґраткової моделi Мiцуї робiт [Levitskii R.R. et al, Phys. Rev. B. 2003, 67, 174112] та [Levitskii R.R. et al., Condens. Matter Phys., 2005, 8, 881], яка по- слiдовно враховує дiагональнi компоненти тензора деформацiй, що виникають пiд дiєю зовнiшнiх тискiв чи внаслiдок теплового розширення. Розраховано пов’язанi з цими деформацiями тепло- вi, п’єзоелектричнi та пружнi характеристики системи. Використовуючи запропоновану схему, для кристалiв сеґнетової солi знайдено такий набiр параметрiв теорiї, що забезпечує задовiльне узго- дження з експериментальними даними для залежностей температур Кюрi вiд гiдростатичного та одновiсних тискiв, а також температурних залежностей теплових деформацiй, лiнiйних коефiцiєнтiв теплового розширення, пружних сталих cEij i cEi4, п’єзоелектричних коефiцiєнтiв d1i i g1i (i = 1, 2, 3). Залежностi дiелектричної проникностi вiд гiдростатичного тиску описано за допомогою отриманого в роботi виразу для проникностi частково затиснутого кристалу. Виявлено, що дипольнi моменти та параметр асиметрiї в сеґнетовiй солi зростають з гiдростатичним тиском. Ключовi слова: сеґнетова сiль, теплове розширення, гiдростатичний тиск, одновiсний тиск, модель Мiцуї 43602-18 http://dx.doi.org/10.1016/0022-3697(65)90079-X http://dx.doi.org/10.1143/JPSJ.39.868 http://dx.doi.org/10.1002/andp.19674740105 http://dx.doi.org/10.1143/JPSJ.33.1396 http://dx.doi.org/10.1007/BF01341554 http://dx.doi.org/10.1080/00150197608241452 http://dx.doi.org/10.1080/07315178808200706 http://dx.doi.org/10.1143/JPSJ.41.2005 http://dx.doi.org/10.1080/14786437208229210 http://dx.doi.org/10.1007/BF00660072 http://dx.doi.org/10.1107/S0021889881009114 http://dx.doi.org/10.5488/CMP.14.23603 http://dx.doi.org/10.1103/PhysRev.168.481 http://dx.doi.org/10.1016/0022-3697(78)90084-7 http://dx.doi.org/10.1080/00150190590963138 Introduction System thermodynamics in the presence of diagonal strains Physical characteristics of Rochelle salt related to diagonal strains Numerical calculations Fitting procedure Thermal expansion and specific heat Diagonal-strain-related piezoelectric and elastic constants Hydrostatic pressure effects Uniaxial pressures effects Concluding remarks

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