Phase diagrams of incommensurate ferroelectric NH₄HSeO₄: phenomenological treatment

The phenomenological analysis of the pressure-temperature (P −T ) phase diagram of NH₄HSeO₄ crystals is presented. It is shown that the disagreement between the experimental results and the theory may be removed assuming that the coefﬁcient of the Landau free-energy expansion κ at the gradient te...

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Datum:	2000
Hauptverfasser:	Kityk, A.V., Zadorozhna, A.V.
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Sprache:	English
Veröffentlicht:	Інститут фізики конденсованих систем НАН України 2000
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Zitieren:	Phase diagrams of incommensurate ferroelectric NH₄HSeO₄: phenomenological treatment / A.V. Kityk, A.V. Zadorozhna // Condensed Matter Physics. — 2000. — Т. 3, № 4(24). — С. 759-766. — Бібліогр.: 9 назв. — англ.

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spelling	irk-123456789-1209932017-06-14T03:06:09Z Phase diagrams of incommensurate ferroelectric NH₄HSeO₄: phenomenological treatment Kityk, A.V. Zadorozhna, A.V. The phenomenological analysis of the pressure-temperature (P −T ) phase diagram of NH₄HSeO₄ crystals is presented. It is shown that the disagreement between the experimental results and the theory may be removed assuming that the coefﬁcient of the Landau free-energy expansion κ at the gradient term (dq/dz)(dq∗/dz) changes the sign in the experimental range of pressures. According to the present model the triple point observed in NH₄HSeO₄ at PK ≈ 455 MPa, TK ≈ 236 K may be considered as artiﬁcial points which result from the limitation of experimental resolution. Therefore, even above PK there still exist two very close (unresolved) lines of the incommensurate phase transitions. Представлений феноменологічний аналіз фазової діаграми тисктемпература (P − T ) кристалів NH₄HSeO₄. Показано, що неузгодження між експериментальними результатами і теорією може бути зняте, припускаючи, що коефіцієнт κ при градієнтному члені (dq/dz)(dq∗/dz) в розкладi вільної енергії Ландау змінює знак в області прикладеного тиску. Відповідно до представленої моделі, потрійна точка, спостережена в NH₄HSeO₄ при PK≈ 455 MPa, TK≈ 236 K, може розглядатись як штучна точка, яка є результатом експериментального обмеження. Отже, навіть вище PK ще існують дві дуже близькі лінії неспівмірних фазових переходів. 2000 Article Phase diagrams of incommensurate ferroelectric NH₄HSeO₄: phenomenological treatment / A.V. Kityk, A.V. Zadorozhna // Condensed Matter Physics. — 2000. — Т. 3, № 4(24). — С. 759-766. — Бібліогр.: 9 назв. — англ. 1607-324X DOI:10.5488/CMP.3.4.759 PACS: 63.20.Dj, 64.70.Kb http://dspace.nbuv.gov.ua/handle/123456789/120993 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	The phenomenological analysis of the pressure-temperature (P −T ) phase diagram of NH₄HSeO₄ crystals is presented. It is shown that the disagreement between the experimental results and the theory may be removed assuming that the coefﬁcient of the Landau free-energy expansion κ at the gradient term (dq/dz)(dq∗/dz) changes the sign in the experimental range of pressures. According to the present model the triple point observed in NH₄HSeO₄ at PK ≈ 455 MPa, TK ≈ 236 K may be considered as artiﬁcial points which result from the limitation of experimental resolution. Therefore, even above PK there still exist two very close (unresolved) lines of the incommensurate phase transitions.
format	Article
author	Kityk, A.V. Zadorozhna, A.V.
spellingShingle	Kityk, A.V. Zadorozhna, A.V. Phase diagrams of incommensurate ferroelectric NH₄HSeO₄: phenomenological treatment Condensed Matter Physics
author_facet	Kityk, A.V. Zadorozhna, A.V.
author_sort	Kityk, A.V.
title	Phase diagrams of incommensurate ferroelectric NH₄HSeO₄: phenomenological treatment
title_short	Phase diagrams of incommensurate ferroelectric NH₄HSeO₄: phenomenological treatment
title_full	Phase diagrams of incommensurate ferroelectric NH₄HSeO₄: phenomenological treatment
title_fullStr	Phase diagrams of incommensurate ferroelectric NH₄HSeO₄: phenomenological treatment
title_full_unstemmed	Phase diagrams of incommensurate ferroelectric NH₄HSeO₄: phenomenological treatment
title_sort	phase diagrams of incommensurate ferroelectric nh₄hseo₄: phenomenological treatment
publisher	Інститут фізики конденсованих систем НАН України
publishDate	2000
url	http://dspace.nbuv.gov.ua/handle/123456789/120993
citation_txt	Phase diagrams of incommensurate ferroelectric NH₄HSeO₄: phenomenological treatment / A.V. Kityk, A.V. Zadorozhna // Condensed Matter Physics. — 2000. — Т. 3, № 4(24). — С. 759-766. — Бібліогр.: 9 назв. — англ.
series	Condensed Matter Physics
work_keys_str_mv	AT kitykav phasediagramsofincommensurateferroelectricnh4hseo4phenomenologicaltreatment AT zadorozhnaav phasediagramsofincommensurateferroelectricnh4hseo4phenomenologicaltreatment
first_indexed	2025-07-08T18:59:11Z
last_indexed	2025-07-08T18:59:11Z
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fulltext	Condensed Matter Physics, 2000, Vol. 3, No. 4(24), pp. 759–766 Phase diagrams of incommensurate ferroelectric NH4HSeO4: phenomenological treatment A.V.Kityk, A.V.Zadorozhna Institute of Physical Optics, 23 Dragomanov Str., 79005 Lviv, Ukraine Received April 17, 2000, in final form November 29, 2000 The phenomenological analysis of the pressure-temperature (P−T ) phase diagram of NH4HSeO4 crystals is presented. It is shown that the disagree- ment between the experimental results and the theory may be removed assuming that the coefficient of the Landau free-energy expansion κ at the gradient term (dq/dz)(dq∗/dz) changes the sign in the experimental range of pressures. According to the present model the triple point observed in NH4HSeO4 at PK ≈ 455 MPa, TK ≈ 236 K may be considered as artificial points which result from the limitation of experimental resolution. There- fore, even above PK there still exist two very close (unresolved) lines of the incommensurate phase transitions. Key words: phase diagram, critical point, incommensurate phase PACS: 63.20.Dj, 64.70.Kb External effects (electric field, mechanical stress, hydrostatic pressure, etc) es- sentially distort the structure of incommensurate (IC) phases. In many cases they lead to the appearance of a triple point in phase diagrams. Two lines of the IC phase transitions (refered usually as normal (N)-IC and IC-commensurate (C) tran- sition, respectively) merge in this point into one line of direct transitions from N- to C-phase. The similar point has been revealed recently in the pressure-temperature (P-T) phase diagram of the improper incommensurate ferroelectric NH4HSeO4 [1]. In these crystals the phase transformation from N- to C-phase is associated with a two- component order parameter (q1, q2) and Lifshitz invariant iδ(q1dq2/dz−q2dq1/dz) is allowed by the symmetry. In the plane wave approximation it gives the linear (with respect to wavevector k) contribution to the soft mode dispersion near a commen- surate point kC: ω2(k) = α+ 2δ(k − kC) + κ(k − kC) 2. (1) The minimum ω2(k), thus, corresponds to the point ki = k0 − kC = −δ/κ. Therefore if just only δ 6= 0, the direct second order phase transition N-C phase is c© A.V.Kityk, A.V.Zadorozhna 759 A.V.Kityk, A.V.Zadorozhna impossible. Accordingly, on the phase diagram only two types of polycritical points are expected to occur: (i) The condition α = δ = 0 defines the isolated point of direct second order phase transition from N- to C-phase in α − δ phase plane. Such a point appears in the intersection of two lines of phase transitions into IC-phase, i.e. it is the tetracritical point [2]. (ii) In principal, a direct first order phase transition from N- to C-phase is pos- sible. The corresponding phase diagram has been considered by Sannikov [3]. The direct first order N-C transition appears at α = anc for β < 0 if αnc = β2/4γ > αi (where β and γ are the free-energy coefficients at forth- and sixth-order terms, re- spectively), thus, the condition αnc = αi defines the coordinates of the triple point in the phase diagram. This point, however differs from the Lifshitz point, since the wavevector of the IC-modulation, as well as the angle between the tangents to the lines of IC phase transitions are finite in the triple point. Instead of this, only the lines of the first order transitions αnc(σ) and αic(σ) (σ = β2/4γ − δ2/2κ) have a common tangent in the triple point. 0 100 200 300 400 500 230 235 240 245 250 255 260 NH 4 HSeO 4 C-phase IC-phase N-phase T , K P, MPA Figure 1. The pressure-temperature phase diagram of NH4HSeO4 crystals [1]. It seems that none of these polycritical points have been revealed in P −T phase diagram of NH4HSeO4 (figure 1) [1]. In particular, above PK ≈ 455 MPa the direct phase transition from N- to C-phase is observed. However, the magnitude of the jump of the elastic constant at the first order phase transition from the IC- to C- phase (T = TC) critically decreases approaching PK [1], whereas the lines of N-C and IC-C phase transitions do not have a common tangent in the triple point (PK, TK) (figure 1). In fact, this contradicts to the prediction of the theory [3]. 760 Phase diagrams of incommensurate ferroelectric Considering the phase diagrams in the IC-systems I authors [2,3] have restricted the free energy expansion by the first order gradient terms of the order parameter. In this case only two types of the polycritical points are expected, which correspond to the tetracritical or triple points. Within these theories it was assumed that co- efficient κ at the gradient term (dq1/dz)(dq2/dz) is positive and not dependent on the external effects. Obviously such an assumption restricts the behaviour of IC– systems and results in a reduction of a number of possible types of polycritical points in their phase diagrams. In fact, there are no physical reasons for such a restriction and the coefficient κ for some real systems may change the sign at the experimental range of the applied pressures P . In order to stabilize the Landau free energy we must then include additional higher order gradient terms into the expansion. The corresponding phase diagram in α − κ phase plane has been recently considered[4] regarding TMATC-Cu crystals. It appears later that the application of this model exactly to this crystals is not quite successful. According to the recent X-ray results [5] the observed polycritical point in TMATC-Cu is indeed tetracritical point but not the triple point as it was assumed in our previous work [4]. Nevertheless, the appearance of the phase diagram described in this work is quite possible in other crystals. It seems that one of such crystals is NH4HSeO4. At least up to now there is no experimental evidence that the observed polycritical point in the P-T diagram of these crystals is a tetracritical point. In this case the application of the modified phenomenological model is inevitable. Let us consider the free-energy expansion for NH4HSeO4 which contains a sixth- order (n = 6) anisotropic invariant [6]: F = ∫ L/2 −L/2 φ(z)dz, φ(z) = α 2 qq∗ + β 4 (qq∗)2 + γ1 6 (qq∗)3 − γ2 12 (q6 + q∗6) + iδ 2 ( q∗ dq dz − q dq∗ dz ) + κ 2 dq dz dq∗ dz − iµ 2 ( q∗ d3q dz3 − q d3q∗ dz3 ) + λ 2 d2q dz2 d2q∗ dz2 − if(q3 − q∗3)P0 + 1 2χo P 2 0 , (2) where α = A0(T − T0) and β, δ, γ1, γ2, µ and λ are assumed to be positive. One must remember that in the case of NH4HSeO4 crystal, the IC-phase appears in the region Tc = 252 K < T < Ti = 262 K with the wavevector of IC modulation k0 = c∗(1/3 − ξ), where ξ ≈ 0.019 [7]. The free energy density functional φ(z) in equation (2) differs from [3,6] only by the presence of the two last terms, which contain third and second order derivatives of the inhomogeneous two-component order parameter (q(z), q∗(z)). For simplicity here we consider only one gradient invariant (µ-term) which produces the contribution to the free energy proportional to (k − kC) 3 (see below). Indeed, the free energy is stabilized by λ-term. Kind and Muralt [8] have used the similar free energy expansion to explain the sequence of the phase transitions in (NH3C3H7)2MnCl4. A trivial minimization procedure applied 761 A.V.Kityk, A.V.Zadorozhna to equation (2) leads us to the following expressions for the free energy in the IC- and C-phases: FIC = αk 2 qkq ∗ k + β 4 (qkq ∗ k) 2 + γ1 6 (qkq ∗ k) 3 − γ2 2 (q5kQK ′′ + q∗5k Q ∗ K ′′) + αQ(K ′′) 2 QK ′′Q∗ K ′′ − if(q3kPK ′ − q∗3k P ∗ K ′) + 1 χo(K ′) PK ′P ∗ K ′, αk = α + 2δ(k − kC) + κ(k − kC) 2 + 2µ(k − kC) 3 + λ(k − kC) 4, qk = qei(k−kc)z, q∗k = q∗e−i(k−kc)z, K ′ = c∗ − 3k, K ′′ = 2c∗ − 5k; (3) FC = α 2 qq∗ + β 4 (qq∗)2 + γ′1 6 (qq∗)3 − γ′2 12 (q6 + q∗6), q = reiψ, q∗ = re−iψ, γ′1 = γ1 − 6χof 2, γ′2 = γ2 + 6χof 2, ψ = (2m+ 1)π/6, m = 0, 1, 2... (4) The N-IC transition occurs at αi = A0(T − Ti) = αk(ki) = 0. The equilibrium value of the incommensurate wavevector ki = \|k0 − kC\| = ξic ∗ at this second or- der phase transition corresponds to the absolute minimum of αk in equation (3), therefore it should be found from the condition: ∂αk/∂k = δ + κ(k − kC) + 3µ(k − kC) 2 + 2λ(k − kC) 3 = 0. (5) The analytical solution of equation (5) is rather complicated, therefore we solved it numerically for the case of NH4HSeO4 crystals. It is convenient to use the nor- malized coupling constants δ ′ = δc∗, κ′ = κc∗2, µ′ = µc∗3 and λ′ = λc∗4 presenting thus the incommensurate modulation by the dimensionless wave number ξ. Their magnitude has been found in order to adjust the experimental value ξ i ≈ 0.019 for NH4HSeO4 at P = 0.1 MPa [7]. An interesting case occurs when the coeffi- cient κ′ changes in the limits −κ′ m 6 κ′ 6 κ′m, which are defined by the conditions \|κ′m\| ≈ λ′ξ2m and \|κ′m\| ≫ δ′/ξm; here ξm is the equilibrium value of the wave number ξ at κ′ = −κ′m. The corresponding dependences αi(κ ′) = A0(T − Ti) and ξi(κ ′) are presented in figure 2. As the parameter κ′ arises to its certain value, the wave number ξi(κ ′) initially decreases rapidly and then gradually tends to zero; indeed it approaches zero-value at κ′ → ∞. The phase transition point αic = A0(T − Tc) from IC- to C-phase cannot be exactly determined since it usually depends on a chosen approximation. In the sim- ple soliton model, that is a second order, transition never occurs in a real crystal. The inclusion of just only the mechanisms of the soliton-defects or soliton-lattice interaction makes this transition discontinuous [9]. Accordingly, the magnitude of αic can be approximately estimated using the condition F IC(αic) = FC(αic). Indeed the free-energy expansion for the IC-phase (see equation (3)) can be reduced to a more suitable form. The forth and the fifth terms in equation (3) are of the order of (qq∗)5 and thus they can be neglected. Then eliminating the polarization PK ′ from equation (3) one obtains: FIC = αk 2 qkq ∗ k + β 4 (qkq ∗ k) 2 + γ′1 6 (qkq ∗ k) 3, (6) 762 Phase diagrams of incommensurate ferroelectric -4 -2 0 2 4 0 5 10 15 a) -20 -15 -10 -5 0 5 b) αi αic C-phase IC-phase N-phase κ'×10 -3 , arb.units ξ i× 10 3 , a rb .u ni ts α, a rb .u ni ts µ' = 3×104, λ' = 9.82×106 β = 2×10-3, γ 1 ' =2×10-5, γ 2 ' =3×10-6, δ'= 6 , Figure 2. The calculated equilibriumwavenumber ξi of the incommensurate wave on the αi-line (a) and calculated phase diagrams in the α − κ′ coordinate plane (b). The values of the free-energy coefficients have been found in order to adjust the experimental values ξi = 0.019 [7] and αic/αi ≈ 20 [1] for NH4HSeO4 at P = 0.1 MPa. where γ′1 is defined by equation (4). The values of the free energy coefficients γ ′ 1, γ′2 and β have been chosen to adjust the ratio αic/αi ≈ 20 which follows from the experiment for NH4HSeO4 crystals [1]. Obviously in this case γ ′ 1 ≫ γ′2. The line αic(κ ′) was calculated numerically (figure 2) for the given set of parameters. As κ′ arises both the lines of N-IC and IC-C phase transitions approach and merge at κ′ → ∞. It is evident, that in the infinity these lines have a common tangent, whereas the wavevector of IC-modulation ki = ξic ∗ approaches a zero value. The unusual phase diagram considered above does not contain any finite triple point, which seems to be observed in the experiment (figure 1). The appearance of this point can be attributed to the limitation of experimental resolution. There is always a certain limit up to which both N-IC and IC-C lines can be observed sepa- rately. It immediately becomes obvious if we present the phase diagrams in the P−T coordinate plane (figure 3), which has been obtained assuming that the free energy coefficients β, γ ′1, γ ′ 2, λ ′, δ′, µ′ are nearly pressure independent. Some justification for 763 A.V.Kityk, A.V.Zadorozhna 0 100 200 300 400 500 600 230 240 250 260 κ'=a(P-P), a=1×10-5, P=460 MPa NH 4 HSeO 4 IC-phase C-phase N-phase T , K P, MPa Figure 3. The phase diagrams (figure 2) presented in the P −T coordinate plane for NH4HSeO4 crystals. At sufficiently high pressures the lines of N-IC and IC-C phase transitions cannot be experimentally resolved. such an assumption follows from the fact, that the pressure behaviour of these coef- ficients is at least not critical in the experimental range of pressures. Otherwise, the appearance of the polycritical points of other types would be inevitable. One must remember that P − T phase diagram of NH4HSeO4 crystals has been determined from the ultrasonic measurements [1]. Taking into account a slightly smeared char- acter of the elastic anomalies in the vicinity of both N-IC and IC-C transitions one can estimate the limit of experimental resolution for T i and Tc as about 0.5 K. Since above 455 MPa the lines of N-IC and IC-C phase transitions are separated by a tem- perature interval smaller than 0.5 K, they cannot be experimentally resolved. Thus, the triple point in the phase diagram (figure 1) can be considered as an artificial point. In conclusion, we have presented here the phenomenological analysis of the press- ure-temperature phase diagram of NH4HSeO4 crystals. A serious disagreement be- tween the experimental results and the theory can be removed assuming that this compound shows the P − T phase diagram with an infinite Lifshitz point. An al- ternative explanation for this phase diagram can be obtained assuming the exis- tence of the tetracritical point at about 455 MPa. Precise X-ray experiments in this temperature-pressure range would be quite desirable. 764 Phase diagrams of incommensurate ferroelectric References 1. Kityk A.V., Vlokh O.G., Zadorozhna A.V., Czapla Z. On the pressure-temperature phase diagram of NH4HSeO4 crystals: acoustical treatment. // Ferroelectrics Lett., 1994, vol. 17, No. 1–2, p. 1–4. 2. Bruce A.D., Cowley R.A., Murray A.F. The theory of structurally incommensurate systems. II. Commensurate-incommensurate phase transitions. // J. Phys. C.: Solid State Phys., 1978, vol. 11, No. 17, p. 3591–3608. 3. Sannikov D.G. Lifshitz points for bidimensional representations. // Pisma JETP, 1979, vol. 30, No. 3, p. 173–175. 4. Kityk A.V., Zadorozhna A.V., Mokry O.M., Sahraoui B. Infinite Lifshitz point in in- commensurate type-I dielectrics. // Phys. Rev. B., 1999, vol. 60, No. 1, p. 10–13. 5. Shimomura S., Terauchi H., Hamaya N., Fujii Y. Multicritical point in structurally incommensurate [N(CH3)4]2CuCl4. // Phys. Rev. B., 1996, vol. 54, No. 10, p. 6915– 6920. 6. Ishibashi Y. Phenomenology of incommensurate phases in the A2BX4 family. – In: Incommensurate Phases in Dielectrics 2 (ed.by R. Blinc and A. P. Levanyuk), Elsevier Science Publishers B.V., 1986, p. 49–69. 7. Denoyer F., Rozycki A., Parlinski K., More M. Neutron investigation of incommensu- rability and metastability in NH4HSeO4 and ND4DSeO4. // Phys. Rev., 1989, vol. 39, No. 1, p. 405–415. 8. Kind R., Muralt P. Unique Incommensurate-commensurate phase transitions in layer- structure perovskite. – In: Incommensurate Phases in Dielectrics 2 (ed.by R. Blinc and A. P. Levanyuk), Elsevier Science Publishers B.V., 1986, p. 301–318. 9. Cummins H.Z. Experimental studies of structurally incommensurate crystal phases. // Phys. Rep., 1990, vol. 185, No. 5–6, p. 211–409. 765 A.V.Kityk, A.V.Zadorozhna Фазові діаграми неспівмірного сегнетоелектрика NH4HSeO4 А.В.Кітик, А.В.Задорожна Інститут фізичної оптики, 79005 Львів, вул. Драгоманова, 23 Отримано 17 квiтня 2000 р., в остаточному виглядi – 29 листопада 2000 р. Представлений феноменологічний аналіз фазової діаграми тиск- температура (P − T ) кристалів NH4HSeO4. Показано, що неуз- годження між експериментальними результатами і теорією може бути зняте, припускаючи, що коефіцієнт κ при градієнтному члені (dq/dz)(dq∗/dz)в розкладi вільної енергії Ландау змінює знак в області прикладеного тиску. Відповідно до представленої моделі, потрійна точка, спостережена в NH4HSeO4 приPK≈ 455 MPa, TK≈ 236 K, може розглядатись як штучна точка, яка є результатом експериментально- го обмеження. Отже, навіть вище PK ще існують дві дуже близькі лінії неспівмірних фазових переходів. Ключові слова: фазова діаграма, потрійна точка, неспівмірна фаза PACS: 63.20.Dj, 64.70.Kb 766

Phase diagrams of incommensurate ferroelectric NH₄HSeO₄: phenomenological treatment

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