Isotopic effects in partially deuterated piezoelectric crystals of Rochelle salt

We develop a theory for dielectric, piezoelectric, and elastic properties of partially deuterated (quenched disorder) crystals of Rochelle salt with taking into account the piezoelectric coupling. Results of numerical calculations are presented for a completely deuterated Rochelle salt and compared...

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Datum:	2004
Hauptverfasser:	Levitskii, R.R., Zachek, I.R., Moina, A.P., Andrusyk, A.Ya.
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Sprache:	English
Veröffentlicht:	Інститут фізики конденсованих систем НАН України 2004
Schriftenreihe:	Condensed Matter Physics
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Zitieren:	Isotopic effects in partially deuterated piezoelectric crystals of Rochelle salt / R.R .Levitskii, I.R. Zachek, A.P. Moina, A.Ya. Andrusyk // Condensed Matter Physics. — 2004. — Т. 7, № 1(37). — С. 111-139. — Бібліогр.: 65 назв. — англ.

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spelling	nasplib_isofts_kiev_ua-123456789-1188922025-06-03T16:27:05Z Isotopic effects in partially deuterated piezoelectric crystals of Rochelle salt Ізотопічний ефект у частково дейтерованих п’єзоелектричних кристалах сегнетової солі Levitskii, R.R. Zachek, I.R. Moina, A.P. Andrusyk, A.Ya. We develop a theory for dielectric, piezoelectric, and elastic properties of partially deuterated (quenched disorder) crystals of Rochelle salt with taking into account the piezoelectric coupling. Results of numerical calculations are presented for a completely deuterated Rochelle salt and compared with available experimental data. Isotopic effect is explored within the mean crystal approximation, as well as within the developed theory using different trial sets of the fitting parameters. Theory predictions are given for the temperature and composition dependences of the calculated characteristics for partially deuterated crystals. Запропоновано теорію діелектричних, п’єзоелектричних і пружних властивостей частково дейтерованих (нерівноважний безлад з повним сортовим хаосом) кристалів сегнетової солі з врахуванням п’єзоелектричних взаємодій. Представлені результати числових розрахунків для повністю дейтерованої сегнетової солі порівнюються з наявними експериментальними даними. Ізотопічний ефект чисельно досліджується в наближенні середнього кристалу та в рамках запропонованої теорії при різних пробних значеннях параметрів моделі. Наведені передбачені теорією для частково дейтерованих систем залежності розрахованих характеристик від температури та рівня дейтерування. This work was supported by the State Foundation for Fundamental Research, project 02.07/00310. 2004 Article Isotopic effects in partially deuterated piezoelectric crystals of Rochelle salt / R.R .Levitskii, I.R. Zachek, A.P. Moina, A.Ya. Andrusyk // Condensed Matter Physics. — 2004. — Т. 7, № 1(37). — С. 111-139. — Бібліогр.: 65 назв. — англ. 1607-324X PACS: 77.22.Gm, 77.65.Bn, 77.80.Bh DOI:10.5488/CMP.7.1.111 https://nasplib.isofts.kiev.ua/handle/123456789/118892 en Condensed Matter Physics application/pdf Інститут фізики конденсованих систем НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We develop a theory for dielectric, piezoelectric, and elastic properties of partially deuterated (quenched disorder) crystals of Rochelle salt with taking into account the piezoelectric coupling. Results of numerical calculations are presented for a completely deuterated Rochelle salt and compared with available experimental data. Isotopic effect is explored within the mean crystal approximation, as well as within the developed theory using different trial sets of the fitting parameters. Theory predictions are given for the temperature and composition dependences of the calculated characteristics for partially deuterated crystals.
format	Article
author	Levitskii, R.R. Zachek, I.R. Moina, A.P. Andrusyk, A.Ya.
spellingShingle	Levitskii, R.R. Zachek, I.R. Moina, A.P. Andrusyk, A.Ya. Isotopic effects in partially deuterated piezoelectric crystals of Rochelle salt Condensed Matter Physics
author_facet	Levitskii, R.R. Zachek, I.R. Moina, A.P. Andrusyk, A.Ya.
author_sort	Levitskii, R.R.
title	Isotopic effects in partially deuterated piezoelectric crystals of Rochelle salt
title_short	Isotopic effects in partially deuterated piezoelectric crystals of Rochelle salt
title_full	Isotopic effects in partially deuterated piezoelectric crystals of Rochelle salt
title_fullStr	Isotopic effects in partially deuterated piezoelectric crystals of Rochelle salt
title_full_unstemmed	Isotopic effects in partially deuterated piezoelectric crystals of Rochelle salt
title_sort	isotopic effects in partially deuterated piezoelectric crystals of rochelle salt
publisher	Інститут фізики конденсованих систем НАН України
publishDate	2004
url	https://nasplib.isofts.kiev.ua/handle/123456789/118892
citation_txt	Isotopic effects in partially deuterated piezoelectric crystals of Rochelle salt / R.R .Levitskii, I.R. Zachek, A.P. Moina, A.Ya. Andrusyk // Condensed Matter Physics. — 2004. — Т. 7, № 1(37). — С. 111-139. — Бібліогр.: 65 назв. — англ.
series	Condensed Matter Physics
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first_indexed	2025-12-01T23:44:22Z
last_indexed	2025-12-01T23:44:22Z
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fulltext	Condensed Matter Physics, 2004, Vol. 7, No. 1(37), pp. 111–139 Isotopic effects in partially deuterated piezoelectric crystals of Rochelle salt R.R.Levitskii 1 , I.R.Zachek 2 , A.P.Moina 1 , A.Ya.Andrusyk 1 1 Institute for Condensed Matter Physics, National Academy of Sciences of Ukraine, 1 Svientsitskii Str., 79011 Lviv, Ukraine 2 State University “Lvivs’ka Politekhnika”, 12 Bandera Str., 79013, Lviv, Ukraine Received February 4, 2004 We develop a theory for dielectric, piezoelectric, and elastic properties of partially deuterated (quenched disorder) crystals of Rochelle salt with tak- ing into account the piezoelectric coupling. Results of numerical calcula- tions are presented for a completely deuterated Rochelle salt and com- pared with available experimental data. Isotopic effect is explored within the mean crystal approximation, as well as within the developed theory using different trial sets of the fitting parameters. Theory predictions are given for the temperature and composition dependences of the calculated characteristics for partially deuterated crystals. Key words: Rochelle salt, deuteration, piezoelectric effect PACS: 77.22.Gm, 77.65.Bn, 77.80.Bh 1. Introduction Rochelle salt (double sodium-potassium tartrate NaKC4H4O6 · 4H2O) has two Curie points. The ferroelectric phase exists in a rather narrow temperature interval from TC1 = 255 K to TC2 = 297 K (TC1 = 251 K, TC2 = 308 K in deuterated crys- tals dRs). Spontaneous polarization is directed along the a axis; it is accompanied by a spontaneous shear strain ε4. Crystal structure is monoclinic C2 2(P21) in the ferroelectric phase and orthorhombic D3 2(P212121) in the paraelectric phases. According to the classical concepts, based on structural data of Frazer et al. [1], the phase transitions in Rochelle salt are pure order-disorder ones. The ferroelec- tric polarization used to be attributed to rotation of hydroxyl groups of tartrate complexes OH5 between two equilibrium positions (see [2]). The actual situation is far more complicated, and the mechanism of the phase transitions in Rochelle salt remains rather obscure. More recent neutron scattering data indicate that the OH5 hydroxyl groups do not perform any orientational motion and therefore play little c© R.R.Levitskii, I.R.Zachek, A.P.Moina, A.Ya.Andrusyk 111 R.R.Levitskii et al. role in the phase transition, at least in deuterated Rochelle salt [3,4]. Furthermore, experimental facts suggest that the phase transitions in Rochelle salt are displacive [5,6] ones or of mixed order-disorder and displacive type [7,8]. According to X-ray scattering experiments [9], spontaneous polarization in Rochelle salt is created by cooperative displacements of tartrate molecules and water molecules in a frame of K and Na ions. Recently Hlinka, Petzelt et al. [10], based on their X-ray diffraction data, proposed that it is the order-disorder motion of OH9 and OH10 groups, cou- pled with the displacive vibrations of OH8 groups that is responsible for the phase transitions in Rochelle salt, as well as for the spontaneous polarization. So far it has not been definitively established the motion of which atoms is the order-disorder one. The most successful microscopic model for Rochelle salt was proposed by Mitsui in [11]. It is based on the assumption that ordering structure elements move in asymmetric double well potentials. The dipoles form two co-penetrating sublattices, with the local potentials which are the mirror reflections of each other. Therefore, even though the dipoles in each sublattice are always ordered (non-zero sublattice polarization), there may be no total polarization at certain temperatures. Later [12,13] the model was reformulated in terms of pseudospin operators; the model itself and its modifications were used for a description of Rs, dRs, RbHSO4, NH4HSO4 and other crystals. It should be mentioned here that crystals of RbHSO4 undergo a single second order phase transition into the ferroelectric phase, whereas in NH4HSO4 the ferroelectric phase exists in a narrow temperature interval but, in contrast to the situation with Rs, the lower phase transition is of the first order. Thermodynamic characteristics of the Mitsui model within the mean field ap- proximation (MFA) were calculated in [13–15]. Tunneling of ordering structure ele- ments was taken into account in [14,15]. Relaxation phenomena in the crystals de- scribed by the Mitsui model were explored within the stochastic Glauber model [16] in [17] and Bloch equation method in [18]. Within the MFA the relaxation times for Rs and dRs were calculated. It should be noted that in the mentioned papers only one or very few physical characteristics of Rs and dRs were fitted to experimental da- ta, whereas the other characteristics were not considered. Naturally, such a method cannot prove the adequacy of the Mitsui model to Rochelle salt crystals, since it is quite simple to get a good fit to experiment, when only selected characteristics are calculated, and a sufficient number of the free parameters is invoked. Ferroelectric crystals RbHSO4 and NH4HSO4 within the Mitsui model were stud- ied in [19–21]. In these crystals the phase transitions are associated with ordering of sulphate groups (see [19]). Thermodynamic characteristics were calculated with- in the MFA and two-particle appoximation and compared with experimental data. Overall a fair description of the data was obtained for RbHSO4. As for NH4HSO4, in [21] it was shown that within the MFA for the Mitsui model it is impossible to describe the low-temperature phase transition in it. At the same time in [20] the temperature curve of spontaneous polarization in NH4HSO4 was described, assuming that certain interaction parameters are temperature dependent. Possibility to describe several systems with different numbers of phase transitions 112 Isotopic effects in partially deuterated Rochelle salt within a Mitsui model indicates an importance of exploring the phase diagram of the model. One of the first approximate phase diagrams was presented in [14]; tunneling was not taken into account. A more precise phase diagram with zero tunneling was constructed in [22]. Regions with three phase transitions were shown to exist. In [22] a Mitsui model with tunneling was considered as well. Within the MFA it was shown that with non-zero tunneling there are such values of the model parameters when the system undergoes a low temperature first order transition and a high temperature second order transition, with the ferroelectric phase in between, as observed in NH4HSO4. The most thorough analysis of the phase diagram of the Mitsui model was obtained in [23] within the MFA both with and without tunneling. A more consistent attempt to describe the Rs, dRs and RbHSO4 crystals within the Mitsui model was undertaken in [24–26]. The free energy, spontaneous polar- ization, entropy, specific heat, static and dynamic dielectric permittivity (the latter within the stochastic Glauber model) of the Mitsui model were calculated. For the first time there have been found such values of the theory parameters which provid- ed a more or less satisfactory description of several characteristics of Rs, dRs, and RbHSO4. The remaining contradictions between theory and experiment and doubts in reliability of some experimental data for Rs, dRs, and RbHSO4 were approached in [27,28], where the fundamental dielectric dispersion in Rs, dRs, RbHSO4, and RbH0.3D0.7SO4 was thoroughly examined, both theoretically and experimentally, and the description of the experiment was much improved, especially for RbHSO4. Recently a new approach to the fitting procedure for RbHSO4 type crystals was ap- plied [22]. This approach was developed in [29–31] for the description of the KH2PO4 family ferroelectrics. It was shown that at choosing the values of the theory param- eters it is crucial to obtain a good fit to the experimental data for the contribu- tion of the ordering subsystem to the specific heat. Following this method, for the RbHSO4 and RbH0.3D0.7SO4 a good description of all calculated characteristics was obtained [22]. However, for Rs some qualitative discrepancies between theory and experiment persisted. Thus, it was impossible to simultaneously fit the data for the spontaneous polarization and static dielectric susceptibility; the temperature curves of polariza- tion relaxation time and dynamic dielectric permittivity in the vicinity of the phase transitions in Rs and dRs were qualitatively different from the experimental ones. Method of fitting to the specific heat is useless here, because the specific heat pecu- liarities at the Curie points in Rs crystals are very small, and no reliable experimental data exist. It should be noted (see also [32–34]) that the physical characteristics of Rs, RbHSO4, and NH4HSO4 can be essentially influenced by tunneling of the ordering structure units [14,15,17,35] and by their interaction with phonons [36–38]. Let us note that in [17,35] the isotopic effect in Rs1−xdRsx was attributed to the changes in tunneling only. In [38] a generalized pseudospin-phonon model of a partially deuter- ated order-disorder type ferroelectrics with asymmetric double-well potential was proposed and studied. Using a decoupling procedure for the Green’s functions, within which the pseudospin-phonon interaction can be taken into account more consistent- 113 R.R.Levitskii et al. ly than within the random phase approximation, the Green’s functions, longitudinal dielectric susceptibility, and coupled pseudospin-phonon vibrations of Rs1−xdRsx and N(H1−xDx)4H1−xDxSO4 ferroelectrics were calculated. Thermodynamic char- acteristics of the systems were calculated within the mean field appoximation. No numerical analysis of the results obtained in [38] was performed. The origin of the remaining qualitative discrepancies between theory and exper- iment for Rs and dRs was still unclear. Could it be that the MFA is a too poor ap- proximation? The presence of chain fragments of ordering structure elements most likely indicated the necessity of using a better approximation than the MFA. In [39,40] a model of two chains of Ising spins moving in an asymmetric double well potential was proposed; an interaction between the chains is taken into account within the MFA, whereas the interactions between nearest neighbors within the chains are taken into account exactly. In [41] the approach was further improved by exact accounting for the interactions between the pair of chains. These improve- ments were shown to “deform” the phase diagram (see [14,21]); temperature curve of polarization at different values of the model parameters was explored. Thus, the task to calculate the physical characteristics of the Mitsui model with- in the two-particle cluster approximation appeared quite natural. It was fulfulled in [42], where an original approach to the description of thermodynamic and dy- namic characteristics of mixed order-disorder type ferroelectrics with asymmetric double-well potential was proposed. Within the two-particle cluster approximation for the short-range interactions and mean field approximation for the long-range interactions, the thermodynamic potentials, static and dynamic Green’s functions for annealed and quenched systems were calculated. The obtained expressions for the thermodynamic and dynamic characteristics of Rs1−xdRsx, Rb(H1−xDx)SO4, and N(H1−xDx)4H1−xDxSO4 crystals contained an increased number of the fitting param- eters, which could potentially improve a description of experimental data, in particu- lar for Rs, dRs, RbHSO4. However, this has not removed the above mentioned quali- tative and quantitative discrepancies between theory and experiment for Rs and dRs. This problem was recently solved by considering the piezoelectric properties of Rs crystals. In [43–45] we modified the conventional Mitsui model for Rs by taking into account the piezoelectric coupling between the ordering structure elements and the shear strain ε4. It permitted us to calculate and obtain a good description of experimental data for the elastic and piezoelectric characteristics of Rochelle salt, to obtain dielectric permittivities of free and clamped crystals, and to properly describe the temperature curves of relaxation times and dynamic dielectric permittivities near the Curie points. In [45] the influence of shear stress σ4 on the physical characteristics of Rochelle salt was studied. In the present work we explore the temperature and composition dependences of thermodynamic, dielectric, elastic, and piezoelectric characteristics of the Rs1−xdRsx crystals. A disordered Mitsui model is considered with taking into account the piezo- electric coupling between the ordering structure units and shear strain ε4. Calcula- tions for the deuterated Rs within the model modified by piezoelectric effects have been also presented for the first time. 114 Isotopic effects in partially deuterated Rochelle salt 2. Thermodynamics of the system Let us consider the behavior of a disordered (partially deuterated) piezoelectric Rochelle salt crystal. So far it is unclear which structure elements of Rochelle salt lattice play the role of ordering units in a phase transition. However, since there is a certain (though rather weak) isotopic effect for the transition temperatures in these crystals, we may assume that the order-disorder motion in the system somehow involves the motion of hydrogens (protons or deuterons). We suppose that in mixed (partially deuterated) crystals there are two interpenetrating subsystems of ordering structure elements: one is associated with protons and the other with deuterons. Calculations are performed within the Mitsui model with taking into account the piezoelectric coupling. Then, we should introduce different constants for pair interactions between the structure ordering units: 1) when both units are associated with protons; 2) with deuterons; 3) when one unit is associated with a proton, while the other with a deuteron. The model Hamiltonian then reads H = N 2 vcE0 44 ε 2 4 −Nve0 14E1 − N 2 vχε0 11E 2 1 − 1 2 ∑ qfα q′f ′β Rqq′( αβ ff ′ )Sz qf(α)Sz q′f ′(β) − ∑ qfα [∆fα − (µαE1 − 2ψ4αε4)]S z qf (α). (2.1) Three first terms in (2.1) correspond to that part of elastic, piezoelectric, and elec- tric energies, which is attributed to the heavy ions lattice and independent of the arrangement of the ordering units (cE0 44 , e014, χ ε0 11 are the “seed” elastic constant, co- efficient of piezoelectric stress, and dielectric susceptibility, respectively); v is the unit cell volume. The lattice strain ε4 and the “seed” constants are assumed to be composition dependent and averaged over the crystal. The fourth term describes a direct interaction between the ordering structure units; Rqq′ ( αβ 11 ) = Rqq′ ( αβ 22 ) = Jqq′(αβ) and Rqq′ ( αβ 12 ) = Rqq′ ( αβ 21 ) = Kqq′(αβ) are the potentials of interaction between the ordering units belonging to the same and to different sublattices, respectively; indices f, f ′ = 1, 2 number the sublattices, whereas α, β = p, d correspond to subsystems of ordering units associated with protons (p) and deuterons (d). The fifth term is the energy associated with the assymetry of the potential profile (∆1α = −∆2α = ∆α). The sixth and seventh terms are the interaction of the ordering structure elements with external electric field and internal field created by the piezoelectric coupling [43], µα is the effective electric moment per unit cell. The operator of the internal degrees of freedom Sz qf(α) by which the state of the ordering structure elements is described can be written in the following form Sz qf(α) = Xαα qf S z qf , where Xpp qf = ( 1 0 0 0 ) , Xdd qf = ( 0 0 0 1 ) , Xpp qf +Xdd qf = 1 115 R.R.Levitskii et al. are the Hubbard operatores, obeying the following permutation relations [ Xαβ qf X α′β′ q′f ′ ] = [ Xαβ′ qf δβα′ −Xα′β qf δβ′α ] δff ′δqq′ . Hereafter, the thermodynamic and dynamic characteristics of the considered ferroelectric systems will be calculated within the mean field approximation. After an identity transformation of the quasispin operators Sz qf (α) = 〈 Sz qf(α) 〉 + [ Sz qf(α) − 〈 Sz qf(α) 〉] ≡ 1 2 η̄f(α) + ∆Sz qf(α), the initial Hamiltonian (2.1) can be presented as Ĥ = U + Ĥ ′ + Ĥ0 (2.2) where U = 1 8 ∑ q,f,α q′f ′β Rqq′ ( αβ ff ′ ) η̄f (α)η̄f ′(β) + N 2 vcE0 44 ε 2 4 −Nve0 14ε4E1 − N 2 vχε0 11E 2 1 , Ĥ ′ = − 1 2 ∑ q,f,α q′f ′β Rqq′ ( αβ ff ′ ) ∆Sz qf(α)∆Sz q′f ′(β), Ĥ0 = − ∑ qfα ε̄f(α)Sz qf(α), and ε̄f(α) is the local field acting on the quasispins associated with protons (p) or deuterons (d) in the fth sublattice ε̄1(p) = 1 2 ∑ β J0(pβ)η̄1(β) + 1 2 ∑ β K0(pβ)η̄2(β) + ∆p − 2ψ4pε4 + µpE1, ε̄1(d) = 1 2 ∑ α J0(αd)η̄1(α) + 1 2 ∑ α K0(αd)η̄2(α) + ∆d − 2ψ4dε4 + µdE1, ε̄2(p) = 1 2 ∑ β J0(pβ)η̄2(β) + 1 2 ∑ β K0(pβ)η̄1(β) − ∆p − 2ψ4pε4 + µpE1, ε̄2(d) = 1 2 ∑ α J0(αd)η̄2(α) + 1 2 ∑ α K0(αd)η̄1(α) − ∆d − 2ψ4dε4 + µdE1 . (2.3) In further calculations the term Ĥ ′ in the Hamiltonian will be neglected. To obtain the observable quantities, we should perform both thermodynami- cal and configurational (over sort configurations) averagings. We consider a case of quenched disorder, when distribution of quasispins associated with protons or deuterons over the lattice is fixed and temperature independent. Therefore, the thermodynamical averaging refers to the spin degrees of freedom only. For instance, for the single-particle distribution function one has η̄f (α) = 〈 SpSz qf(α)e−βĤ0 Sp e−βĤ0 〉 x = 〈Xαα qf tanh 1 2 βε̄f(α)〉x = xα tanh 1 2 βε̄f(α) (2.4) 116 Isotopic effects in partially deuterated Rochelle salt (xα is the concentration of the α – component). Let us introduce new variables ξ̄(α) and σ̄(α): ξ̄(p) = 1 2 [η̄1(p) + η̄2(p)] , ξ̄(d) = 1 2 [η̄1(d) + η̄2(d)] , σ̄(p) = 1 2 [η̄1(p) − η̄2(p)] , σ̄(d) = 1 2 [η̄1(d) − η̄2(d)] . (2.5) Then the system (2.4) can be presented as ξ̄(p) = 1 2 xp { tanh 1 2 [γ1 + δ1] + tanh 1 2 [γ1 − δ1] } = xp sinh γ1 cosh γ1 + cosh δ1 , ξ̄(d) = 1 2 xd { tanh 1 2 [γ2 + δ2] + tanh 1 2 [γ2 − δ2] } = xd sinh γ2 cosh γ2 + cosh δ2 , σ̄(p) = 1 2 xp { tanh 1 2 [γ1 + δ1] − tanh 1 2 [γ1 − δ1] } = xp sinh δ1 cosh γ1 + cosh δ1 , σ̄(d) = 1 2 xd { tanh 1 2 [γ2 + δ2] − tanh 1 2 [γ2 − δ2] } = xd sinh δ2 cosh γ2 + cosh δ2 , (2.6) where γ1 = R̃+(pp) 2T ξ̄(p) + R̃+(pd) 2T ξ̄(d) − 2 T ψ̃4pε4 + µpE1 kT , δ1 = − R̃−(pp) 2T σ̄(p) − R̃−(pd) 2T σ̄(d) + ∆̃p T , γ2 = R̃+(pd) 2T ξ̄(p) + R̃+(dd) 2T ξ̄(d) − 2 T ψ̃4dε4 + µdE1 kT , δ2 = − R̃−(pd) 2T σ̄(p) − R̃−(dd) 2T σ̄(d) + ∆̃d T , and R̃±(αβ) = 1 kB [K0(αβ) ± J0(αβ)]. To calculate piezoelectric, elastic, dielectric characteristics of the mixed Rochelle salt crystals we shall use the thermodynamic potential (calculated per one pair of quasispins) g1E(4) = G1E(4) NkB = = −v̄σ4ε4 + 1 2 v̄cE0 44 ε 2 4 − v̄e0 14ε4E1 − 1 2 v̄χε0 11E 2 1 + 1 4 ( R̃+(pp)x2 pξ 2(p) + R̃+(dd)x2 dξ 2(d) − R̃−(pp)x2 pσ 2(p) − R̃−(dd)x2 dσ 2(d) ) + 1 2 R̃+(pd)xpxdξ(p)ξ(d)− 1 2 R̃−(pd)xpxdσ(p)σ(d) − 2(xp + xd)T ln 2 − Txp ln ( cosh γ1 + δ1 2 cosh γ1 − δ1 2 ) − Txd ln ( cosh γ2 + δ2 2 cosh γ2 − δ2 2 ) , (2.7) 117 R.R.Levitskii et al. (v̄ = v kB ). From the conditions: 1 v̄ ( ∂g1E ∂ε4 ) E1,σ4 = 0, 1 v̄ ( ∂g1E ∂E1 ) σ4 = −P1 we obtain that σ4 = cE0 44 ε4 − e014E1 + xp ψ̃4p v̄ 2ξ(p) + xd ψ̃4d v̄ 2ξ(d), (2.8) P1 = e014ε4 + χε0 11E1 + µp v xpξ(p) + µd v xdξ(d). (2.9) Using expressions (2.6) and (2.9), let us calculate the static dielectric susceptibil- ity of mixed Rochell salt type crystal along its a-axis for the case of a mechanically clamped system χε 11(0) = ( ∂P1 ∂E1 ) ε4 = χε0 11 + µp v xp ( ∂ξ(p) ∂E1 ) ε4 + µd v xd ( ∂ξ(d) ∂E1 ) ε4 = = χε0 11 + 1 2T∆4 { v̄ µ2 p v2 xp(a0∆p1 + c0∆p3) + v̄ µ2 d v2 xd(b0∆d2 + d0∆d4) − v̄ µpµd v2 xp(b0∆p2 + d0∆p4) − v̄ µpµd v2 xd(a0∆d1 + c0∆d3) } . (2.10) The notations used here are given in Appendix. Similarly, for the coefficient of the piezoelectric stress we get e14 = ( ∂P1 ∂ε4 ) E1 = e014 + µp v ( ∂ξ(p) ∂ε4 ) E1 + µd v ( ∂ξ(d) ∂ε4 ) E1 = = e014 − 1 T∆4 { µp v ψ̃4pxp(a0∆p1 + c0∆p3) − µd v ψ̃4pxd(a0∆d1 + c0∆d3) + µd v ψ̃4dxd(b0∆d2 + d0∆d4) − µp v ψ̃4dxp(b0∆p2 + d0∆p4) } . And from (2.8) and (2.6) we obtain an expression for the elastic constant of the system at a constant electric field cE44 = ( ∂σ4 ∂ε4 ) E1 = = cE0 44 + 4 v̄T∆4 { ψ̃2 4pxp(a0∆p1 + c0∆p3) + ψ̃2 4dxd(b0∆d2 + d0∆d4) − ψ̃4pψ̃4dxp(b0∆p2 + d0∆p4) − ψ̃4pψ̃4dxd(a0∆d1 + c0∆d3) } . (2.11) The other piezoelectric (h14, d14, g14), dielectric (χσ 11) and elastic (cP44, s E 44) character- istics of disordered Rochelle salt type crystal can be obtained from the above found quantities, using known thermodynamic relations. To find the specific heat of the system we use the free energy equal to f(4) = g1E(4) + v̄P1E1 + v̄σ4ε4. (2.12) 118 Isotopic effects in partially deuterated Rochelle salt Respectively, molar entropy of the crystal associated with its quasispin subsystem is as follows: S4 = − R 2 ( ∂f(4) ∂T ) P1ε4 = = R { (xp + xd) ln 2 + xp 2 ln ( cosh 1 2 (γ1 + δ1) cosh 1 2 (γ1 − δ1) ) + xd 2 ln ( cosh 1 2 (γ2 + δ2) cosh 1 2 (γ2 − δ2) ) − xpγ1 sinh γ1 + xpδ1 sinh δ1 2(cosh γ1 + cosh δ1) − xdγ2 sinh γ2 + xdδ2 sinh δ2 2(cosh γ2 + cosh δ2) } , where R is the gas constant. Molar specific heat of the quasispin subsystem is ob- tained by numerical differentiation of entropy ∆Cσ 4 = T ( dS4 dT ) σ . (2.13) 3. Relaxation dynamics in partially deuterated Rs Dynamic dielectric characteristics of mixed ferroelectrics with the asymmetric double well potential will be considered within the Glauber model [16]. Similarly to [42,43] we obtain a system of equations for the single particle distribution functions of quasispins − ϕ d d t ξ̄(p) = ξ̄(p) − 1 2 xp [ tanh 1 2 (γ1 + δ1) + tanh 1 2 (γ1 − δ1) ] , − ϕ d d t ξ̄(d) = ξ̄(d) − 1 2 xd [ tanh 1 2 (γ2 + δ2) + tanh 1 2 (γ2 − δ2) ] , − ϕ d d t σ̄(p) = σ̄(p) − 1 2 xp [ tanh 1 2 (γ1 + δ1) − tanh 1 2 (γ1 − δ1) ] , − ϕ d d t σ̄(d) = σ̄(d) − 1 2 xd [ tanh 1 2 (γ2 + δ2) − tanh 1 2 (γ2 − δ2) ] . (3.1) The general form of the system (3.1) is quite complicated. Hereafter we shall restrict our consideration to the small deviations from equilibrium. Then ξ̄(α), σ̄(α) and the electric field E1 can be presented as sums of two terms each ξ̄(α) = ξ̄0(α) + ξ̄t(α), σ̄(α) = σ̄0(α) + σ̄t(α), E1 = E10 + E1t. (3.2) In the case of these small deviations, the expressions tanh(γi ± δi)/2 can be expanded in ξ̄t(α), σ̄t(α), Et up to the linear terms in a quite wide temperature 119 R.R.Levitskii et al. range. We obtain then systems of equations for the equilibrium functions (coinciding with (2.6)) and for the time-dependent parts − ϕ d d t ξ̄t(p) = a10ξ̄t(p) + a20ξ̄t(d) + a30σ̄t(p) + a40σ̄t(d) − a00 µpEt 2kT , − ϕ d d t ξ̄t(d) = b10ξ̄t(p) + b20ξ̄t(d) + b30σ̄t(p) + b40σ̄t(d) − b00 µdEt 2kT , − ϕ d d t σ̄t(p) = c10ξ̄t(p) + c20ξ̄t(d) + c30σ̄t(p) + c40σ̄t(d) − c00 µpEt 2kT , − ϕ d d t σ̄t(d) = d10ξ̄t(p) + d20ξ̄t(d) + d30σ̄t(p) + d40σ̄t(d) − d00 µdEt 2kT , (3.3) the quantities used here ai0, bi0, ci0, di0 (i = 0 − 3) are obtained from the given in Appendix ai, bi, ci, di (i = 0 − 3) by changing ξ(p) → ξ0(p), ξ(d) → ξ0(d), σ(p) → σ0(d), σ(d) → σ0(d). In the case Et = 0 the system of equations (3.3) can be reduced to a single differential equation for ξ̄t(p): d(4) ξ̄t(p) d t4 + n1 ϕ d(3) ξ̄t(p) d t3 + n2 ϕ2 d(2) ξ̄t(p) d t2 + n3 ϕ3 d ξ̄t(p) d t + n4 ϕ4 ξ̄t(p) = 0. (3.4) Here n1 = a10 + b20 + c30 + d40 n2 = ∣ ∣ ∣ ∣ a10 a20 b10 b20 ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ a10 a30 c10 c30 ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ a10 a40 d10 d40 ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ b20 b30 c20 c30 ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ b20 b40 d20 d40 ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ c30 c40 d30 d40 ∣ ∣ ∣ ∣ , n3 = ∣ ∣ ∣ ∣ ∣ ∣ ∣ a10 a20 a30 b10 b20 b30 c10 c20 c30 ∣ ∣ ∣ ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ ∣ ∣ ∣ a10 a20 a40 b10 b20 b40 d10 d20 d40 ∣ ∣ ∣ ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ ∣ ∣ ∣ a10 a30 a40 c10 c30 c40 d10 d30 d40 ∣ ∣ ∣ ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ ∣ ∣ ∣ b20 b30 b40 c20 c30 c40 d20 d30 d40 ∣ ∣ ∣ ∣ ∣ ∣ ∣ , n4 = ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ a10 a20 a30 a40 b10 b20 b30 b40 c10 c20 c30 c40 d10 d20 d30 d40 ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ . A general solution of (3.4) can be written as ξ̄t(p) = 4 ∑ i=1 Ci exp(− t τi ), (3.5) where Ci are constant coefficients, and τi are the relaxation times τ−i ϕ = − 1 q̃ ; (3.6) q̃ = ϕq are roots of the characteristic equation q4 + n1 ϕ q3 + n2 ϕ2 q2 + n3 ϕ3 q + n4 ϕ4 = 0. (3.7) 120 Isotopic effects in partially deuterated Rochelle salt Dynamic dielectric susceptibility is defined as χ11(ω) = µp v d ξ̄t(p) dE1t + µd v d ξ̄t(d) dE1t . (3.8) Let us solve the non-uniform system of equations (3.3) with respect to ξ̄t(p) and ξ̄t(d). Substituting these solutions into (3.8) and taking into account relations (3.6) and (3.7), we obtain the dynamic susceptibility χ11(ω) = χ0 11 + (iω)3n (1) 0 + (iω)2n (2) 0 + iωn (3) 0 + n (4) 0 vkBT 4 ∏ j=1 τj (1 + iωτ−j ) , (3.9) where ϕn (1) 0 = µ2 pa00 + µ2 db00 , ϕ2n (2) 0 = µ2 p(b20a00 + c30a00 + d40a00 − a30c00) + µ2 d(a10b00 + c30b00 + d40b00 − b40d00) − µpµd(b10a00 + a20b00 + b30c00 − a40d00), ϕ3n (3) 0 = µ2 p [(b20c30 − b30c20 + b20d40 − b40d20 + c30d40 − c40d30) a00 + (a20b30 − a30b20 − a30d40 + a40d30) c00] + µ2 d [(a10c30 − a30c10 + a10d40 − a40d10 + c30d40 − c40d30) b00 + (b30c40 − b40c30 − a10b40 + a40b10) d00] − µpµd[(b10c30 − b30c10 + b10d40 − b40d10)a00 + (a20c30 − a30c20 + a20d40 − a40d20)b00 + (a10b30 − a30b10 + b30d40 − b40d30)c00 − (a20b40 − a40b20 + a30c40 − a40c30)d00], ϕ4n (4) 0 = µ2 p   ∣ ∣ ∣ ∣ ∣ ∣ b20 b30 b40 c20 c30 c40 d20 d30 d40 ∣ ∣ ∣ ∣ ∣ ∣ a00 + ∣ ∣ ∣ ∣ ∣ ∣ a20 a30 a40 b20 b30 b40 d20 d30 d40 ∣ ∣ ∣ ∣ ∣ ∣ c00   + µ2 d   ∣ ∣ ∣ ∣ ∣ ∣ a10 a30 a40 c10 c30 c40 d20 d30 d40 ∣ ∣ ∣ ∣ ∣ ∣ b00 + ∣ ∣ ∣ ∣ ∣ ∣ a10 a30 a40 b10 b30 b40 c10 c30 c40 ∣ ∣ ∣ ∣ ∣ ∣ d00   − µpµd   ∣ ∣ ∣ ∣ ∣ ∣ b10 b30 b40 c10 c30 c40 d20 d30 d40 ∣ ∣ ∣ ∣ ∣ ∣ a00 + ∣ ∣ ∣ ∣ ∣ ∣ a20 a30 a40 c20 c30 c40 d20 d30 d40 ∣ ∣ ∣ ∣ ∣ ∣ b00 + ∣ ∣ ∣ ∣ ∣ ∣ a10 a30 a40 b10 b30 b40 d10 d30 d40 ∣ ∣ ∣ ∣ ∣ ∣ c00 + ∣ ∣ ∣ ∣ ∣ ∣ a20 a30 a40 b20 b30 b40 c20 c30 c40 ∣ ∣ ∣ ∣ ∣ ∣ d00   . Expression (3.9) can be presented as a sum of simple fractions χ11(ω) = χ1 1 + iωτ1 + χ2 1 + iωτ2 + χ3 1 + iωτ3 + χ4 1 + iωτ4 , (3.10) 121 R.R.Levitskii et al. whereas the system of equations for χi reads      n11 n12 n13 n14 n21 n22 n23 n24 n31 n32 n33 n34 n41 n42 n43 n44           χ1 χ2 χ3 χ4      =      n1 n2 n3 n4      , (3.11) where the following notations are used n11 = τ2τ3τ4, n12 = τ1τ3τ4, n13 = τ1τ2τ4, n14 = τ1τ2τ3, n21 = τ2τ3 + τ2τ4 + τ3τ4, n22 = τ1τ3 + τ1τ4 + τ3τ4, n23 = τ1τ2 + τ1τ4 + τ2τ4, n24 = τ1τ2 + τ1τ3 + τ2τ3, n31 = τ2 + τ3 + τ4, n32 = τ1 + τ3 + τ4, n33 = τ1 + τ2 + τ3, n34 = τ1 + τ3 + τ2, n41 = 1, n42 = 1, n43 = 1, n44 = 1, (3.12) n1 = 1 vakT τ1τ2τ3τ4n (1) 0 , n2 = 1 vakT τ1τ2τ3τ4n (2) 0 , n3 = 1 vakT τ1τ2τ3τ4n (3) 0 , n4 = 1 vakT τ1τ2τ3τ4n (4) 0 . Dielectric permittivity then is equal to ε11(ω) = 1 + 4πχ(ω) = ε′11(ω) − iε′′11(ω), (3.13) where ε′11(ω) = ε∞ + 4 ∑ i=1 4πχi 1 + (ωτi)2 , ε′′11(ω) = 4 ∑ i=1 4πχiωτi 1 + (ωτi)2 , (3.14) and ε∞ = 1+4πχ∞ is the high-frequency contribution to the dielectric permittivity. 4. Numerical analysis Within the above proposed theory we can calculate the physical characteristics of mixed Rochelle salt crystals with any deuteration level. The theory parameters that have to be set include the parameters for the limiting cases of pure and completely deuterated crystals (J0(αα), K0(αα), ∆α, ψ4α, µ1α, ϕα, cE0 44α, χε0 11 e 0 14) as well as the parameters relevant to partially deuterated systems (J0(pd), K0(pd)). The case of a pure crystal has been considered in [43]. In this paper we shall perform calculations for a completely deuterated Rochelle salt and later will attempt to describe the partially deuterated systems. 122 Isotopic effects in partially deuterated Rochelle salt 4.1. Completely deuterated Rochelle salt First, let us consider the case of a completely deuterated Rochelle salt (xp = 0, xd = 1). Fitting procedure for a pure Rochelle salt is described in detail in [43]. Here we follow the same procedure. For the unit cell volume (per two quasispins) we use the same values as for undeuterated Rs v = 0.5219[1 + 0.00013(T − 190)] · 10−21cm3. Following the method proposed in [43], in order to describe the dielectric, piezo- electric, and relaxational characteristics of Rs – second order derivatives of thermo- dynamic potential, we need to determine the values of the effective dipole moment µ1p by fitting to the values of dynamic permittivity εε 11(Tc1) and εε 11(Tc2) (data of [46] are used). It yields µ1d as a function slightly decreasing with temperature µ1d = [2.1 + 0.0066(308− T )] × 10−18esu · cm. It provides a good fit to the mentioned second derivatives of thermodynamic po- tential, but a rather poor description of the data for spontaneous polarization (see below). For µ1p(T ) of pure Rs in [43] we used µ1p = [2.52 + 0.0066(297 − T )] × 10−18esu · cm. Table 1 contains the used values of the parameters J , K, ∆, ψ4 for pure [43] and completely deuterated Rochelle salt crystals as well as the “seed” quantities, obtained by fitting the theory to experimental data. Results of the fitting for deuter- ated Rochelle salt are discussed below. Theoretical dependences of spontaneous polarization P1 and spontaneous strain ε4 of dRs are given in figure 4.1. As one can see, the maximal theoretical value of 240 260 280 300 0.0 0.1 0.2 0.3 0.4 P1 (µC/cm 2 ) T (K) 240 260 280 300 0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 T (K) ε4 Figure 4.1. Temperature dependences of spontaneous polarization P1 and spon- taneous strain of dRs at σ4 = 0; � – [47], 4 – ε4 = P1(χ σ 11 − χε 11)/(χ σ 11χ ε 11h14), where data for P1 are taken from [47], for χσ 11 – from [47], χε 11 – from [46], h14 – from [48]. 123 R.R.Levitskii et al. Table 4.1. Theory parameters for pure [43] and completely deuterated crystals of Rochelle salt Tc1 , Tc2 , J̃0 , K̃0 , ∆̃ , ψ4 , cE0 44 , e014 , K K K K K K dyn/cm2 esu/cm2 Rs 255 297 797.36 1468.83 737.33 –760 12.8 × 1010 1.d0 × 104 dRs 251 308 806.633 1499.532 751.861 –600 10.5 × 1010 0.15 × 104 P1 is by ∼ 50 % lower than the experimental one. So far we have no solution to this problem; it could be, however, that there is an error in the rather outdated experimental data. Further meausurements here are thus definitely needed. 240 260 280 300 0.0 0.1 0.2 0.3 -1 T (K) χ11 Figure 4.2. Temperature dependence on inverse static dielectric permittivity of free and clamped crystals of dRs. Experimental points are taken from: � – [47] (900 Hz), • – [51], � are recalculated from the obtained in [46] Cole-Cole curves, 5 – [48]. In figure 4.2 we showed the temperature dependences of inverse static dielectric susceptibilities of free kσ 11 = (χσ 11) −1 and clamped kε 11 = (χε 11) −1 crystals of deuterated Rochelle salt. As one can see, the experimental data for kε 11(T ) of [48] and [46] disagree. Experimental values of dielectric permittivity in the lower paraelectric phase should be also verified, since below 240 K the values of εσ 11 [47] become smaller than εε 11 obtained from the data of [46]. We get a satisfactory quantitative description of experimental data for kσ 11 of [47] and for kε 11 of [46] both in paraelectric phases and in the ferroelectric phase except for its middle part, where the calculated inverse susceptibilities are smaller than the experimental values. In figure 4.3 the temperature curves of elastic constants at constant field cE 44 and at constant polarization cP44 of deuterated Rochelle salt are depicted. The elastic constant cE44 is essentially temperature dependent, vanishing with the same rate at both Curie points. Theoretical results for cE44 are in a good agreement with the data obtained from the formula cE44 = (χε 11h14) 2/(χσ 11 − χε 11). The calculated cP44 is almost 124 Isotopic effects in partially deuterated Rochelle salt 240 260 280 300 320 0 2 4 6 8 10 12 14 cP cE c44 (1010dyn/cm2) T (K) Figure 4.3. Temperature dependence of elastic constants at constant field cE 44: 4 – cE 44 = ((χε 11h14) 2)/(χσ 11 − χε 11) and constant polarization cP 44: 4 – cP 44 = (χσ 11χ ε 11h 2 14)/(χ σ 11 − χε 11) of deuterated Rochelle salt crystal. Points for χσ 11 are taken from [47], for χε 11 from [46], for h14 from [48]. 240 260 280 300 320 0 40 80 120 160 e14 (104esu/cm2) T (K) 240 260 280 300 320 102 103 104 105 T (K) d14 (10-8esu/dyn) 240 260 280 300 320 0 2 4 6 8 10 12 h 14 (104 dyn/esu) T (K) 240 260 280 300 320 0 20 40 60 80 100 120 T (K) g 14 (10-8 cm2/esu) Figure 4.4. Temperature dependences of piezoelectric characteristics of deuter- ated Rochelle salt. Experimental points are taken from: 5 – [48], ♦ – [49], 4 – d14 = (χσ 11 − χε 11)/(χ ε 11h14), e14 = χε 11h14, g14 = (χσ 11 − χε 11)/(χ σ 11χ ε 11h14), values of χσ 11 taken from [47], χε 11 from [46], h14 from [48]. 125 R.R.Levitskii et al. temperature independent in all phases and accords with the data obtain from the relation cP44 = χσ 11χ ε 11h 2 14/(χ σ 11 − χε 11). Figure 4.4 contains theoretical temperature dependences of piezoelectric charac- teristics of dRs. The obtained curve for d14(T ) well agrees with the data of [48,49] as well as with recalculated via the given in caption formula in the entire explored temperature range, except for the low-temperature paraelectric phase. 240 260 280 300 320 2.4 2.8 3.2 3.6 ∆C σ (J/ mol K) T (K) Figure 4.5. Temperature dependence of ∆Cσ for deuterated Rochelle salt. The temperature dependence of the contribution from the ordering units to the specific heat ∆Cσ of deuterated Rochelle salt is given in figure 4.5. The theory predicts two positive anomalies of specific heat at both Curie points. Let us now consider the dielectric relaxation in crystals of dRs. Figure 4.6 con- tains the calculated temperature dependences of inverse relaxation times τ−1 1 and τ−1 2 , as well as the values of τ−1 1 obtained in [46] from experimental data for ε∗11(ν, T ). The latter points are well described by the proposed theory. Peculiar to the tempe- 240 260 280 300 320 0 4 8 12 T (K) 1τ -1 (10 10 c -1 ) 240 260 280 300 320 6.2 6.3 6.4 6.5 6.6 2τ -1 (10 13 c -1 ) T (K) (a) (b) Figure 4.6. Temperature dependence of inverse relaxation times (τ1) −1 and (τ2) −1: � – [46]. 126 Isotopic effects in partially deuterated Rochelle salt rature curve of τ−1 1 (T ) is the presence of two finite minima at the transition points. Let us note that any theory for Rochelle salt crystals which does not take into account the piezoelectric effects yields zero values of τ−1 1 (T ) at these points and, thereby, incorrect temperature dependence of dynamic permittivity in their vicinity. The values of τ2 are three orders smaller than those of τ1. 220 240 260 280 300 0 100 200 300 'ε 11 T (K) 220 240 260 280 300 0 50 100 150 "ε 11 T (K) (a) (b) Figure 4.7. Temperature dependences of real and imaginary parts of dynamic dielectric permittivity ε∗11 of deuterated Rochelle salt crystal at different frequen- cies ν (GHz): � – 0.6; ◦ – 2.8; 4 – 4.29; 5 – 9.3; ♦ – 24.0. Experimental points are taken from [46]. 150 175 200 225 250 275 0 5 10 15 20 'ε11 T (K) 150 175 200 225 250 275 0 5 10 15 20 '' T (K) ε11 (a) (b) Figure 4.8. Temperature dependences of real and imaginary parts of dynamic dielectric permittivity ε∗11 of deuterated Rochelle salt crystal at different frequen- cies ν (GHz): � – 102; • – 141; N – 180. Experimental points are taken from [50]. Temperature dependences of real and imaginary parts of dynamic dielectric per- mittivity for deuterated Rochelle salt at different frequencies are presented in figu- re 4.7. Overall, a good description of experimental data [46] is obtained, except for 127 R.R.Levitskii et al. ε′11 at 0.06, 1.5, and 2.8 GHz in the middle part of the ferroelectric phase as well as for ε′′11 at ν = 9.3 GHz at all temperatures studied. Dynamic permittivity of deuterated Rochelle salt at very high frequencies in the ferroelectric and lower paraelectric phases along with the experimental points of [50] are shown in figure 4.8. As one can see, at these frequencies the theory only qualitatively reproduces the temperature curves for ε′11 and ε′′11 ε′′11(T ). Figure 4.9 contains the frequency dependences of dynamic permittivity at differ- ent temperatures. As one can see, the agreement between the theory and experiment [46] is particularly good in the upper paraelectric phase. 10 8 10 9 10 10 10 110 50 100 150 200 250 300 350 1 2 ν (Hz) 'ε11 10 8 10 9 10 10 10 110 25 50 75 100 125 150 175 2 1 ν (Hz) "ε11 10 8 10 9 10 10 10 110 25 50 75 100 125 150 175 c) b) a) 'ε11 2 1 ν (Hz) 10 8 10 9 10 10 10 110 25 50 75 100 125 150 175 21 ε11 " ν (Hz) 10 8 10 9 10 10 10 110 50 100 150 200 250 300 350 2 1 ' ν (Hz) ε11 10 8 10 9 10 10 10 110 50 100 150 200 250 300 350 21 'ε11 ν (Hz) Figure 4.9. Frequency dependences of real and imaginary parts of dynamic dielec- tric permittivity ε∗11 of deuterated Rochelle salt crystal at different temperatures (K): a – 308(1), 298(2); b – 263(1), 298(2); c – 251(1), 243(2). Experimental points are taken from [46]. 128 Isotopic effects in partially deuterated Rochelle salt 240 260 280 300 0 40 80 120 160 200 240 240 260 280 300 4 6 8 10 12 240 260 280 300 40 60 80 100 240 260 280 300 102 103 104 105 240 260 280 300 0.0 0.1 0.2 0.3 0.4 240 260 280 300 0.0 0.1 0.2 0.3 240 260 280 300 0.0 0.2 0.4 0.6 0.8 1.0 T (K) e 14 (104esu/cm2) T (K) h 14 (104dyn/esu) T (K) g 14 (10-8 cm2/esu) T (K) d 14 (10-8esu/dyn) T (K) P 1 (µC/cm2) T (K) σ 1/χ 11 T (K) ε 4 (10-3) 240 260 280 300 0 4 8 12 T (K) c 44 (1010dyn/cm2) Figure 4.10. Temperature dependences of piezoelectric, dielectric, and elastic characteristics of pure (solid line, solid symbols), partially (x = 0.5, dashed line) and completely deuterated (dotted line, open symbols) Rochelle salt. Theory: mean crystal approximation; experimental points are: P1: �, � – [47], ε4: � – [55], • – [49], � – ε4 = P1(χ σ 11 − χε 11)/(χ σ 11χ ε 11h14); χσ 11: � – [56]; � – [47]; cE 44: � – [52], � – cE 44 = ( (χε 11h14) 2 ) /(χσ 11 − χε 11); cP 44: � – [54], � – cP 44 = ( χσ 11χ ε 11h 2 14 ) /(χσ 11 − χε 11); e14: � – [53], � – e14 = χε 11h14; d14: � – [53], N – [57], � – d14 = (χσ 11 − χε 11)/(χ ε 11h14), h14: � – [54], � – [48]; g14: � – [53], � – g14 = (χσ 11 − χε 11)/(χ σ 11χ ε 11h14). Values for P1, χ11, h14 used to calculate the points for dRs are taken from [47] (P1, χσ 11), [46] (χε 11), [48] (h14). 129 R.R.Levitskii et al. 4.2. Partially deuterated crystals. Mean crystal appoximation Let us now consider the isotopic effect in Rochelle salt crystals. First we shall explore a pure model ([43]; the same formulas are obtained from the ones presented in previous section by putting xp = 0 or xd = 0). We use the mean crystal ap- proximation, attributing the isotopic effect to the monotonic changes in interaction parameters. Thus, for a partially deuterated crystal with deuteration level x = xd we take J = J0(pp)(1 − xd) + J0(dd)xd, K = K0(pp)(1 − xd) +K0(dd)xd, etc, (4.15) where the parameters for pure (pp) and completely deuterated (dd) crystals of Rochelle salt are given in table 1. Results of such calculations are given in figure 4.10. As one can see, the increase of deuteration leads to the widening of the ferroelectric phase, to the increase of the maximal values of spontaneous polarization P1 and strain ε4, and of constant of piezoelectric strain g14 and to the decrease of static dielectric susceptibilities of free χσ 11 and clamped χε 11 crystals in the ferroelectric phase. In the paraelectric phases the slopes of the temperature curves for (χσ 11) −1 and (χε 11) −1 are almost independent of deuteration level. Maximal value of the elastic constant cE 44 in the ferroelectric phase is practically the same for Rs and dRs, as well as the rate of changes in cE 44 with temperature on approaching the Curie points. Elastic constant cP 44 with the increasing x decreases from 13·1010 dyn/cm2 for Rs to 10·1010 dyn/cm2 for dRs. 240 260 280 300 0 2 4 6 8 10 12 1τ -1 (10 10 c -1 ) T (K) Figure 4.11. Temperature dependence of inverse relaxation time τ−1 1 of Rs and dRs crystals. Experimental points are taken from � – [46], � – [12]. Experimental points for the constant of piezoelectric stress h14 for Rs and dRs practically coincide at all temperatures, whereas the theoretical values of h14 for dRs are somewhat smaller that for Rs. The theory predicts that on increasing the 130 Isotopic effects in partially deuterated Rochelle salt deuteration level, the minimal values of the coefficients of piezoelectric stress e14 and strain d14 decrease in the ferroelectric phase; the peak values of e14 at the transition points decrease as well. In figure 4.11 the temperature dependences of inverse relaxation time (τ1) −1 for Rs and dRs are compared. Deuteration decreases with values of τ−1 1 at the transition points and in the paraelectric phases. In the middle of the ferroelectric phase, τ−1 1 in Rs is smaller than in dRs. 4.3. Partially deuterated crystals. A general theory Let us now consider a more general case and attempt to describe the deuteration effects in Rochelle salt crystals within the framework of the above developed (“com- plete”) theory. At the absence of experimental data even for transition temperatures of partially deuterated crystals of Rochelle salt, we cannot determine the values of J(pd) and K(pd). Therefore, we shall perform calculations at different trial values of these parameters, in order to find out in what way the changes of these trial values affect the calculated physical characteristics of partially deuterated system. �� X T � (K) � � �� ! "$#�% ! &(' "$#�% ! &(' )�% ! &(' " #�% )$' "$#�% &(' )�% ) ' ! "$#�% )$' )�% &$' ! "$#�% &(' * +�,.- /10 Figure 4.12. Composition dependence (x = xd) of transition temperatures of mixed Rochelle salt crystals (deuteration levels indicated) calculated within a complete theory. First, let us define the arithmetic mean set values of the pd-parameters as J̃arth 0 (pd) = J̃0(pp) + J̃0(dd) 2 , J̃arth 0 (pd) = 802.0115 K, K̃arth 0 (pd) = K̃0(pp) + K̃0(dd) 2 , K̃arth 0 (pd) = 1484.181 K, (4.16) The sets of pd-parameters used in calculations are determined by a relative deviation from the arithmetic mean set of the pd-parameters, denoted by two numbers (u, v). u = J̃0(pd) − J̃arth 0 (pd) \|J̃0(pp) − J̃0(dd)\| · 100%, 131 R.R.Levitskii et al. v = K̃0(pd) − K̃arth 0 (pd) \|K̃0(pp) − K̃0(dd)\| · 100%. (4.17) Thus (−15; 2) stands for J̃0(pd) = J̃arth 0 (pd) − 0.15 · \|J̃0(pp) − J̃0(dd)\| = 800.61605 K, K̃0(pd) = K̃arth 0 (pd) + 0.02 · \|K̃0(pp) − K̃0(dd)\| = 1484.79504 K, whereas (0; 0) denotes the arithmetic mean set of the parameters. Figure 4.12 illustrates the composition dependence of transition temperatures in mixed Rochelle salt crystals calculated within a complete theory at different sets of parameters (u, v) given by (4.17). The arithmetic mean set (4.16) yields very weakly non-linear dependences Tci(x). The more the parameters J̃0(pd) and K̃0(pd) deviate from the arithmetic mean set, the more the composition dependence of transition temperatures deviate from the linear one. 0.3 0.4 0.5 253.0 253.2 253.4 253.6 253.8 254.0 0.0 0.2 0.4 0.6 0.8 251.5 252.0 252.5 253.0 253.5 254.0 254.5 255.0 geometric mean set arithmetic mean set mean crystal approximation x T C1 (K) 0.40 0.45 0.50 301.0 301.5 302.0 302.5 303.0 0.0 0.2 0.4 0.6 0.8 298 300 302 304 306 308 geometric mean set arithmetic mean set mean crystal approximation x T C2 (K) Figure 4.13. Composition dependence (x = xd) of transition temperatures of mixed Rochelle salt crystals (deuteration levels indicated) calculated within a complete theory with the arithmetic (4.16) and geometric (4.18) mean sets of the theory parameters as well as within the mean crystal approximation (4.15). In figure 4.13 we show the composition dependence of transition temperatures in mixed Rochelle salt crystals calculated within a complete theory with the arithmetic (4.16) and geometric J̃geo 0 (pd) = √ J̃0(pp)J̃0(dd), K̃geo 0 (pd) = √ K̃0(pp)K̃0(dd), (4.18) mean sets of the theory parameters as well as within the mean crystal approximation (4.15). As one can see, all these methods predict nearly the same slightly non-linear dependences of Tci(x). As one can see in figure 4.14, deuteration effects for the other physical charac- teristics of Rochelle salt predicted by the complete theory qualitatively agree with those given by the mean crystal approximation. 132 Isotopic effects in partially deuterated Rochelle salt 2�3 452 6�652 7�452�8�6:9�4�4;9 <(6 4 2 4 3 4 = 4 8�4 <(4�4 <$2 4 < 3 4 <$= 4 <(8�4 = =>@?$A B(B >CB�A D(E = = >CB�A E(B >CB�A F(E =>CB�A B(B e14, 104esu/cm2 T, K 2�3 452 6�652 7�452�8�6:9�4�4;9 <(6 4 G 4�6 4 G <(4 4 G <(6 4 G 2 4 4 G 2 6 = =>H?$A B$B >CB�A D$E = = >CB�A E(B >CB�A F�E =>CB�A B(B 1/χε 11 T, K 2�3 4I2�6�6J2 7�4J2 8�6K9�4�4K9�<(6 4 G 4�4 4 G 4(3 4 G 4�8 4 G < 2 4 G < = 4 G 2�4 1/χσ 11 = = = = =>CB�A F�E>CB�A B(B >CB�A E(B >CB�A D$E >@?$A B(B T, K 2�3 452 6�6;2 7�452 8�6;9�4�4;9 <(6 < <$4 = =>@? A B$B >CB�A D(E = = >CB�A E(B >CB�A F�E=>CB�A B$B d14, 10-5esu/dyn T, K L�M NOL P�NQL�R NQL S�NOL T�NOL U�NOV�N�NWV�X(N N L M R T X(N c44 E, 1010dyn/cm2 = = = = =YCZ�[ \�] YCZ�[ Z(Z Y^Z�[ ](Z Y^Z�[ _(] Ya`$[ Z(Z T, K L�M NOL P�NQL�R NQL S�NOL T�NOL U�NOV�N�NWV�X(N M�b P P b N P b P R�b N R�b P S b N S b P ∆C4 σE, J/(mol K) = = = = =Ŷ Z�[ \�] Ŷ Z�[ Z(Z YCZ�[ ](Z Ŷ Z�[ _(] Y@̀$[ Z(Z T, K L�L NKL�M NKL�R NKL T�NcV�N�NcV�L N P b N P b P R�b N R�b P S b N S b P T b N T b P U b N g14, 10-7cm2/esu = = = = =Ŷ Z�[ \�] Ŷ Z�[ Z(Z Ŷ Z�[ ](Z Ŷ Z�[ _(] Y@̀$[ Z(Z T, K L�L NKL�M NKL�R NKL T�NcV�N�NcV�L N P b N P b P R�b N R�b P S b N S b P T b N T b P U b N = =Ya`$[ Z(Z Y^Z�[ _(] = = Y^Z�[ ](Z Y^Z�[ \�] =Y^Z�[ Z(Z h14, 104dyn/esu T, K Figure 4.14. Temperature dependences of the physical characteristics of mixed Rochelle salt crystals (deuteration levels x = xd indicated). Lines: a complete theory. Experimental points for 1/χε 11(T ) are taken from: � – [12], � – [46], O – [48]; The rest of the points are the same as in figure 4.10. 133 R.R.Levitskii et al. 5. Concluding remarks Despite the fact that Rochelle salt appears to be the first known ferroelectrics, the mechanism of ferroelectricity in it has not been completely established. In this section we shall analyse the major achievements and miscalculations made in the studies of this type of crystals. It would be useful to compare the history of inves- tigation of ferroelectric crystals with double asymmetric potential with that of the KH2PO4 family crystals, since both types of crystals are also piezoelectric. The most prominent peculiarity of the study of the KH2PO4 family ferroelectrics is a close connection between theory and experiment. Ferroelectricity in KH2PO4 was discovered in 1938, and already in 1941 the first microscopic theory (Slater mod- el) of the phase transition in it was proposed. Despite certain doubts in its validity, the most important results for these crystals were obtained in the framework of the proton ordering model (see [58]), yielding within a cluster approximation a good description of experimental data for thermodynamic and dynamic characteristics. of the deuterated members of the family. Later, a microscopic theory of deformed deuterated crystals of this type was developed [63], within which a proper description of pressure and electric field effects on their physical characteristics was obtained [59–62]. Essential point in the theoretical studies was a development of the well- grounded fitting procedure; especially we would like to mention the papers [29–31], where it was shown that in the fitting it is important to have reliable experimental data for the specific heat of the crystals. In our recents works [64,65] we modified the proton ordering model in order to take into account the contributions of piezo- electric coupling, which allowed us to improve the theory and properly describe the experimental data for the elastic, piezoelectric, and dielectric characteristics of the crystals. There was no such a close connection between theory and experiment in the in- vestigation of the Rochelle salt type ferroelectrics. Due to complexity of the crystal structure of Rochelle salt, there is no much data about the microscopic mechanism of the phase transitions in it. Neither there has been much theoretical effort after the Mitsui model was proposed. The existing theoretical works were either incomplete (only one or very few characteristics were calculated, which does not permit to test the adequacy of the used model) or contradictory in a sense that, for instance, it was impossible to simultaneously obtain a good description of spontaneous polar- ization and static dielectric permittivity. Qualitatively incorrect were the calculated temperature curves of relaxation times and dynamic dielectric permittivity in the microwave region near the Curie points. Such a situation was to a great extent due to the fact that strong piezoelectric coupling in Rochelle salt was not taken into ac- count and, in fact, the models corresponded to a mechanically free crystal, whereas at microwave frequencies a crystal is effectively clamped (theories of the KH2PO4 type crystals which do not take into account the piezoelectric effect, yield at least qualitatively correct results due to the first order phase transition in the crystals). It was fruitful to use an experience obtained in the study of the KH2PO4 family crystals. Because of a much simpler crystal structure and more detailed experimental 134 Isotopic effects in partially deuterated Rochelle salt structural data, the microscopic Hamiltonian with taking into account the piezoelec- tric coupling for these crystals could be and was derived ab initio [63]. Performing analogous calculations for the Rochelle salt type crystals, we came to a model which allowed us to obtain elastic and piezoelectric characteristics of the crystals, calculate dielectric characteristics of free and clamped crystals, and get a qualitatively (and quantitatively) correct system dynamics near the Curie points [43,44]. An essential insight into the problem of the phase transitions mechanisms in Rochelle salt type crystals can be obtained by investigation of the disordered (par- tially deuterated) crystals, as it was for the partially deuterated crystals of the KH2PO4 family, where the essential role in obtaining a proper description of exper- imental data was played by taking into account the tunneling effects. Availability of extensive experimental data for these crystals for several deuteration levels was strongly helpful. Experimental measurements of the deuteration effects on the physi- cal characteristics of the Rochelle salt type crystals have not been performed yet and are therefore necessary. They would help to elucidate the role played by hydrogen bonds subsystem in the phase transitions in the crystals. 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Appendix ∆p1 = ∣ ∣ ∣ ∣ ∣ ∣ b2 b3 b4 c2 c3 c4 d2 d3 d4 ∣ ∣ ∣ ∣ ∣ ∣ , ∆p3 = ∣ ∣ ∣ ∣ ∣ ∣ a2 a3 a4 b2 b3 b4 d2 d3 d4 ∣ ∣ ∣ ∣ ∣ ∣ , ∆p2 = ∣ ∣ ∣ ∣ ∣ ∣ a2 a3 a4 c2 c3 c4 d2 d3 d4 ∣ ∣ ∣ ∣ ∣ ∣ , ∆p4 = ∣ ∣ ∣ ∣ ∣ ∣ a2 a3 a4 b2 b3 b4 c2 c3 c4 ∣ ∣ ∣ ∣ ∣ ∣ , ∆d1 = ∣ ∣ ∣ ∣ ∣ ∣ ∣ b1 b3 b4 c1 c3 c4 d1 d3 d4 ∣ ∣ ∣ ∣ ∣ ∣ ∣ , ∆d3 = ∣ ∣ ∣ ∣ ∣ ∣ ∣ a1 a3 a4 b1 b3 b4 d1 d3 d4 ∣ ∣ ∣ ∣ ∣ ∣ ∣ , ∆d2 = ∣ ∣ ∣ ∣ ∣ ∣ ∣ a1 a3 a4 c1 c3 c4 d1 d3 d4 ∣ ∣ ∣ ∣ ∣ ∣ ∣ , ∆d4 = ∣ ∣ ∣ ∣ ∣ ∣ a1 a3 a4 b1 b3 b4 c1 c3 c4 ∣ ∣ ∣ ∣ ∣ ∣ , ∆4 = ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ a1 a2 a3 a4 b1 b2 b3 b4 c1 c2 c3 c4 d1 d2 d3 d4 ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ , a0 = 1 − ξ2(p) − σ2(p), c0 = −2ξ(p)σ(p), b0 = 1 − ξ2(d) − σ2(d), d0 = −2ξ(d)σ(d), a1 = 1 − 1 4T R̃+(pp)xp[1 − ξ2(p) − σ2(p)], c1 = 1 4T R̃+(pp)xp2ξ(p)σ(p)], b1 = − 1 4T R̃+(pd)xp[1 − ξ2(d) − σ2(d)], d1 = 1 4T R̃+(pd)xp2ξ(d)σ(d)], a2 = − 1 4T R̃+(pd)xd[1 − ξ2(p) − σ2(p)], c2 = 1 4T R̃+(pd)xd2ξ(p)σ(p)], b2 = 1 − 1 4T R̃+(dd)xd[1 − ξ2(d) − σ2(d)], d2 = 1 4T R̃+(dd)xd2ξ(d)σ(d)], a3 = − 1 4T R̃−(pp)xp2ξ(p)σ(p)], c3 = 1 + 1 4T R̃−(pp)xp[1 − ξ2(p) − σ2(p)], b3 = − 1 4T R̃−(pd)xp2ξ(d)σ(d)], d3 = 1 4T R̃−(pd)xp[1 − ξ2(p) − σ2(p)], a4 = − 1 4T R̃−(pd)xd2ξ(p)σ(p)], c4 = 1 4T R̃−(pd)xd[1 − ξ2(p) − σ2(p)], b4 = − 1 4T R̃−(dd)xd2ξ(d)σ(d)], d4 = 1 + 1 4T R̃−(dd)xd[1 − ξ2(d) − σ2(d)]. 138 Isotopic effects in partially deuterated Rochelle salt Ізотопічний ефект у частково дейтерованих п’єзоелектричних кристалах сегнетової солі Р.Р.Левицький 1 , І.Р.Зачек 2 , А.П.Моїна 1 , А.Я.Андрусик 1 1 Інститут фізики конденсованих систем, 79011, Львів, вул. Свєнціцького,1 2 Національний університет “Львівська політехніка”, 79013, Львів, вул. С.Бандери, 12 Отримано 4 лютого 2004 р. Запропоновано теорію діелектричних, п’єзоелектричних і пружних властивостей частково дейтерованих (нерівноважний безлад з пов- ним сортовим хаосом) кристалів сегнетової солі з врахуванням п’є- зоелектричних взаємодій. Представлені результати числових роз- рахунків для повністю дейтерованої сегнетової солі порівнюються з наявними експериментальними даними. Ізотопічний ефект чисе- льно досліджується в наближенні середнього кристалу та в рамках запропонованої теорії при різних пробних значеннях параметрів мо- делі. Наведені передбачені теорією для частково дейтерованих сис- тем залежності розрахованих характеристик від температури та рів- ня дейтерування. Ключові слова: сегнетова сіль, дейтерування, п’єзоелектричний ефект PACS: 77.22.Gm, 77.65.Bn, 77.80.Bh 139 140

Isotopic effects in partially deuterated piezoelectric crystals of Rochelle salt

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