The unified model description of order-disorder and displacive structural phase transitions

A series of co-authors’ studies [1-7] devoted to the unified model description of structural phase transitions (SPT) in ferroelectrics and related materials are reviewed and partly innovated. Starting from a general Hamiltonian of pair-coupled anharmonic (quartic) oscillators, together with the...

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Datum:	1998
1. Verfasser:	Stamenkovic, S.
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spelling	irk-123456789-1189302017-06-02T03:02:22Z The unified model description of order-disorder and displacive structural phase transitions Stamenkovic, S. A series of co-authors’ studies [1-7] devoted to the unified model description of structural phase transitions (SPT) in ferroelectrics and related materials are reviewed and partly innovated. Starting from a general Hamiltonian of pair-coupled anharmonic (quartic) oscillators, together with the concept of local normal coordinates, a unified model description of both order-disorder and displacive types of SPT-systems is proposed. Within the framework of the standard variational procedure, a hybridized pseudospin-phonon Hamiltonian is formulated by introducing variables corresponding to phonon, magnon-like(flipping) and nonlinear(domain–wall-like) displacements of atoms participating in SPT. This is achieved by representing the cooperative atomic motion onto several quasiequilibrium positions (in the simplest case, two) as slow tunnelling displacement (decomposed into magnon-like and soliton-like deviations), in addition to comparatively fast phonon oscillations around inhomogeneous momentary rest positions, in turn induced by domain–wall-like (soliton) excitations. The qualitative and quantitative analyses show that SPT (of the first or second order) can be either of a displacive (governed by a phonon soft mode), order-disorder (governed by a tunnelling–magnon-like soft mode) or of a mixed type, depending on both the coupling energy between atoms and their zero-point vibrational energy. In the critical temperature region, the domain–wall-like excitations bring on the formation of microdomains (precursor clusters of the ordered phase) which induce SPT of the Ising type universality class. The incomplete softening of the phonon or pseudomagnon mode occurs and a central peak due to slow cluster relaxation appears in the spectral density of excitations. Зроблено огляд і часткове оновлення серії робіт автора з співавторами [1-7], присвяченої уніфікованому модельному опису структурних фазових переходів (СФП) у сегнетоелектриках та споріднених матеріалах. Виходячи з загального гамільтоніану попарно зв’язаних ангармонічних (четвірних) осциляторів та враховуючи концепцію локальних нормальних координат, запропоновано уніфікований модельний опис систем з СФП обидвох типів: порядок-безпорядок і зміщення. В рамках стандартної варіаційної процедури формулюється гібридизований псевдоспін-фононний гамільтоніан введенням змінних, що відповідають фононним, магноно-подібним (з переворотом) і нелінійним (типу доменної стінки) зміщенням атомів, що беруть участь у СФП. Це досягається представленням колективного руху атомів через декілька квазірівноважних положень (в найпростішому випадку - двох) як повільного тунельного зміщення (розбитого на магноноподібні і солітоно-подібні відхилення) додатково до порівняно швидких фононних осциляцій навколо неоднорідних миттєвих положень спокою, в свою чергу індукованих збудженнями типу доменної стінки (солітонами). Якісний і кількісний аналізи показали, що СФП (першого чи другого роду) можуть бути типу зміщення (керованих м’якою фононною модою), порядок-безпорядок (керованих тунельною магноноподібною м’якою модою) чи змішаного типу залежно як від енергії зв’язку між атомами, так і від їхньої нульової точки коливної енергії. В області критичної температури збудження типу доменної стінки приводять до утворення мікродоменів (кластерів впорядкованої фази), які індукують СФП класу універсальності Ізінгівського типу. Має місце неповне пом’якшення фононної чи псевдомагнонної моди і у спектральній густині збуджень виникає центральний пік, викликаний повільною релаксацією кластерів. 1998 Article The unified model description of order-disorder and displacive structural phase transitions / S. Stamenkovic // Condensed Matter Physics. — 1998. — Т. 1, № 2(14). — С. 257-309. — Бібліогр.: 36 назв. — англ. 1607-324X DOI:10.5488/CMP.1.2.257 PACS: 63.70.+h, 77.80.Bh, 64.60.-i, 64.60.Cn http://dspace.nbuv.gov.ua/handle/123456789/118930 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	A series of co-authors’ studies [1-7] devoted to the unified model description of structural phase transitions (SPT) in ferroelectrics and related materials are reviewed and partly innovated. Starting from a general Hamiltonian of pair-coupled anharmonic (quartic) oscillators, together with the concept of local normal coordinates, a unified model description of both order-disorder and displacive types of SPT-systems is proposed. Within the framework of the standard variational procedure, a hybridized pseudospin-phonon Hamiltonian is formulated by introducing variables corresponding to phonon, magnon-like(flipping) and nonlinear(domain–wall-like) displacements of atoms participating in SPT. This is achieved by representing the cooperative atomic motion onto several quasiequilibrium positions (in the simplest case, two) as slow tunnelling displacement (decomposed into magnon-like and soliton-like deviations), in addition to comparatively fast phonon oscillations around inhomogeneous momentary rest positions, in turn induced by domain–wall-like (soliton) excitations. The qualitative and quantitative analyses show that SPT (of the first or second order) can be either of a displacive (governed by a phonon soft mode), order-disorder (governed by a tunnelling–magnon-like soft mode) or of a mixed type, depending on both the coupling energy between atoms and their zero-point vibrational energy. In the critical temperature region, the domain–wall-like excitations bring on the formation of microdomains (precursor clusters of the ordered phase) which induce SPT of the Ising type universality class. The incomplete softening of the phonon or pseudomagnon mode occurs and a central peak due to slow cluster relaxation appears in the spectral density of excitations.
format	Article
author	Stamenkovic, S.
spellingShingle	Stamenkovic, S. The unified model description of order-disorder and displacive structural phase transitions Condensed Matter Physics
author_facet	Stamenkovic, S.
author_sort	Stamenkovic, S.
title	The unified model description of order-disorder and displacive structural phase transitions
title_short	The unified model description of order-disorder and displacive structural phase transitions
title_full	The unified model description of order-disorder and displacive structural phase transitions
title_fullStr	The unified model description of order-disorder and displacive structural phase transitions
title_full_unstemmed	The unified model description of order-disorder and displacive structural phase transitions
title_sort	unified model description of order-disorder and displacive structural phase transitions
publisher	Інститут фізики конденсованих систем НАН України
publishDate	1998
url	http://dspace.nbuv.gov.ua/handle/123456789/118930
citation_txt	The unified model description of order-disorder and displacive structural phase transitions / S. Stamenkovic // Condensed Matter Physics. — 1998. — Т. 1, № 2(14). — С. 257-309. — Бібліогр.: 36 назв. — англ.
series	Condensed Matter Physics
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fulltext	Condensed Matter Physics, 1998, Vol. 1, No 2(14), p. 257–309 The unified model description of order-disorder and displacive structural phase transitions S.Stamenković Institute of Nuclear Sciences, Laboratory for Theoretical Physics and Condensed Matter Physics, Belgrade, P.O.Box 522, Yugoslavia Received July 7, 1998 A series of co-authors’ studies [1-7] devoted to the unified model descrip- tion of structural phase transitions (SPT) in ferroelectrics and related ma- terials are reviewed and partly innovated. Starting from a general Hamiltonian of pair-coupled anharmonic (quar- tic) oscillators, together with the concept of local normal coordinates, a unified model description of both order-disorder and displacive types of SPT-systems is proposed. Within the framework of the standard variational procedure, a hybridized pseudospin-phonon Hamiltonian is formulated by introducing variables corresponding to phonon, magnon-like(flipping) and nonlinear(domain–wall-like) displacements of atoms participating in SPT. This is achieved by representing the cooperative atomic motion onto sev- eral quasiequilibrium positions (in the simplest case, two) as slow tunnelling displacement (decomposed into magnon-like and soliton-like deviations), in addition to comparatively fast phonon oscillations around inhomogeneous momentary rest positions, in turn induced by domain–wall-like (soliton) ex- citations. The qualitative and quantitative analyses show that SPT (of the first or second order) can be either of a displacive (governed by a phonon soft mode), order-disorder (governed by a tunnelling–magnon-like soft mode) or of a mixed type, depending on both the coupling energy between atoms and their zero-point vibrational energy. In the critical temperature region, the domain–wall-like excitations bring on the formation of microdomains (precursor clusters of the ordered phase) which induce SPT of the Ising type universality class. The incomplete softening of the phonon or pseu- domagnon mode occurs and a central peak due to slow cluster relaxation appears in the spectral density of excitations. Key words: structural phase transitions, order-disorder,displacive transition PACS: 63.70.+h, 77.80.Bh, 64.60.-i, 64.60.Cn c© S.Stamenković 257 S.Stamenković 1. Introduction Among numerous attempts to develop a unified microscopic theory of both displacive and order-disorder structural phase transitions (SPTs), on the occasion of the 60th anniversary of Prof. Stasyuk’s birthday, our modest contribution [1-7] to the same problem in ferroelectric phase transitions is reviewed and partly inno- vated. To expose as clear as possible the basic proposals for a unified description of SPT-systems and to better comprehend this peculiar problem in connection with its prospective development, we shall try to keep mostly to the original studies of the co-authors’ group, with reference to other similar approaches cited in the co-authors’ publications. Accordingly, to acquire a rather contemporary unified description of SPT (i.e. by novelizing the previous, original ones) it is convenient to employ the representation of local normal coordinates (LNCs) involving co- operative displacements of all the atoms in a primitive cell participating in the given critical (soft) vibrational mode. The LNCs are chosen so that they take into account the symmetry properties of the system, in order to be transformed by an irreducible representation of the point group of the primitive cell. Since the corresponding order parameter of the system is transformed by the irreducible rep- resentation of the space group, such “conjunctive” coordinates can be considered as the appropriate basis for describing SPT in the case when a single channel tran- sition (corresponding to a single-relevant irreducible representation) takes place in the system we deal with throughout this review. With this substantial concept in mind, the grounds for the unified description of SPT become physically transpar- ent when one appreciates the natural generalization of the traditional concept of atomic (including cooperative-like, represented by LNC) equilibrium states. In the next step the time dependent LNC attributed to the cooperative atomic motion within the effective anharmonic multi-well potential is decomposed into a “slow” tunnelling displacement (being a momentary rest position, it consists of flipping– magnon-like and domain-wall–soliton-like deviations) and a comparatively “fast” superimposed deviation of the phonon type around the inhomogeneous cluster induced by quasi-equilibrium positions (in the simplest case, two) inside the ef- fective one-particle potential. Thus, the ordering process is characterized by two order parameters - one reflecting the degree of the local precursor order (the av- erage population of equilibrium positions σz) and the other controlling the degree of the long-range order (the average atomic displacement η). The underlying approach is developed by incorporating the already adopted LNC and the corresponding canonical momentum into the familiar model Hamil- tonian expressed in the form of effective pair-coupled anharmonic (in the simplest case, quartic) oscillators. The further procedure is completed by a variational adap- tation of the microscopic model Hamiltonian (in the reduced phase space of coor- dinates and momenta) in various hybridized forms of phonon - tunnelling (pseu- dospin) - domain-wall (soliton) types separated in corresponding variables. On the basis of a self-consistent phonon and a molecular-field approximations (namely, a combination of an independent-mode and independent-site evaluation schemes), a 258 Unified description of structural transitions complete set of coupled self-consistent equations is obtained which enables one to estimate or to calculate (mainly numerically) all the physical (collective or local) quantities of the SPT-system (in the first place, both order parameters included). The qualitative and quantitative analyses of such a system of equations show that the SPT (of the first or second order) can be of a displacive (governed by a phonon soft mode), order-disorder (governed by a tunnelling - magnon-like soft mode) or a mixed type – depending predominantly on the reduced lattice coupling strength and, in a lesser degree, on the ratio of the zero-point vibrational energy to the height of the single-particle potential barrier. The possibility for phase transitions of the both types occuring at zero temperature (quantum limits) is also outlined. In the case of a strongly anharmonic system the quantum limit of the displacive type is examined in both, the ordered (σz = 1) and disordered (σz = 0) lattices. Finally, it is analytically demonstrated that with the onset of criticality in all, order-disorder, displacive, as well as in mixed types of structural phase transitions, the domain-wall (soliton) excitations bring on the formation of microdomains (pre- cursor clusters of the ordered phase) which induce a phase transition of the Ising universality class. The incomplete softening of the phonon or/and pseudomagnon mode occurs and a central peak due to the slow cluster relaxation appears in the energy spectrum of excitations. 2. Preliminaries 2.1. The elements of lattice dynamics at a structural transi tion In agreement with the soft mode concept put forward independently by several scholars in this field of solid state physics∗, a lattice undergoing transition is not stable in the harmonic approximation (it has a purely imaginary frequency for the optical, i.e. polar mode) owing to the compensation of long-range attractive and short-range repulsive forces. In that case in the symmetric phase the anharmonic interaction, even if it is small, is necessary for the lattice vibrations to be stable. Therefore, in a consistent dynamical theory of SPT one must take into account the anharmonic interaction from the very beginning (already in the zero approx- imation), which requires the application of the methods of statistical mechanics, and depart from the simple mechanical approach adopted in the Born-Karman theory. Meantime, a self-consistent phonon theory was developed ( mainly in con- nection with the study of quantum helium crystals) which was also applied to the description of structural transitions. As it is well known, unlike the customary self-consistent (molecular) field ap- proximation, in the self-consistent phonon theory account is taken of fluctuations of the order parameter, which play an important role in the second-order phase transitions. It should be pointed out that the self-consistent phonon theory makes the renormalised (due to fluctuations) Landau expansion for the free energy pos- sible and allows a self-consistent determination of the range of applicability of this ∗For a historical outline, for example, see [4] in [7]. 259 S.Stamenković method. The self-consistent phonon theory itself is generally considered for an ar- bitrary lattice (e.g., in [7,8]), where various theoretical applications to the concrete SPT model (we will also deal with them hereafter) are surveyed in detail. Such a theory is also adjusted to describe order-disorder SPT or transitions of the mixed type which differ from the displacive ones only by a significantly higher degree of unharmonicity and the related relaxational nature of critical-mode excitations (see Subsections 3.5 and 3.7). To highlight the basic physical consequences including the main quantitative features of the unified description of the proposed SPT, it is sufficient to use only the first-order self-consistent phonon approximation (also named the pseu- doharmonic or renormalised phonon approximation), favoured, in addition, by its simplicity. It is worthwhile to note that in the studies of strongly anharmonic SPT-systems the so-called improved first-order self-consistent phonon (namely, the improved pseudoharmonic) approximation is often used. Likewise, within the framework of such an approximation the interaction of pseudoharmonic phonons is then described through the self-energy operator which determines the frequency shift and the damping of self-consistent phonons. The self-energy operator itself is then calculated self-consistently in the simplest approximation with taking proper account of the uncorrelated propagation of renormalised (in the intermediate state) phonons ∗∗. 2.2. The microscopic model description of structural phase transitions It is well established that in spite of the self-consistent phonon theory capable of accommodating the exact relevant expressions, it is not always possible to carry out concrete or explicit calculations with their aid. Hence, for a model description including a proper interpretation of experimental data, it is instructive to single out from a complete microscopic picture of the phenomenon, only its most essential features rendered into a simple physical model. The corresponding well conceived models permit one to clarify the scope of computational methods and the limits of their applicability. In the case of SPT only low-lying soft modes (responsible for the phase transition) and those phonon modes permitted (by the system symmetry) to interact with them are important. Therefore, in a model description it is physically quite acceptable to consider only a small number of normal lattice vibrations and their anharmonic interactions. As most SPTs are accompanied by a lattice deformation, in discussing real systems in terms of LNCs, of particular interest is to take into account the coupling of the critical mode with the relevant acoustic mode. 2.2.1. The concept of the local normal coordinate The inherent ingredients of any physically founded approach in describing SPT- systems and searching for their unified description certainly are the symmetry ∗∗The sizable survey of various current self-consistent phonon approximations with their findings included is given in [7,8] and Refs. therein. 260 Unified description of structural transitions properties apparently associated with crystal structures of these compounds. Thus, the symmetry requirements imposed on SPT-systems (locally and on the whole crystal) have to be taken into consideration and built in properly into the adopted model, namely, into the Hamiltonian of one singled out (soft or critical) mode. This enables one to overcome the standard difficulties in solving the eigenvalue problem (to obtain eigen-frequencies and polarization vectors of normal lattice modes), as SPT compounds, as a rule, have complex structures. Namely, in dealing with the singled out soft mode, the dimension of the eigenvector space can be reduced significantly if one applies symmetry arguments. If the wave vector star at which a structural phase transition takes place (the critical wave vector) is known, then it is possible to determine the relevant irreducible representation for the phase transition. This approach was applied, for instance, in calculating the lattice vibration frequencies of ferroelectric KH2PO4 for critical wave vector qc = 0 [9]. As the result of a symmetry analysis, 48-dimensional eigenvector space is reduced to 7-dimensional space, in which case, one of the seven phonon modes is acoustic, while the rest of them are optical. The frequencies and displacement vectors of the six low-lying optical modes, as calculated in [9] (using a rather rough, i.e. quasiharmonic approximation), are in satisfactory agreement with the results obtained in experiments on light scattering. In this case the corresponding singled out (critical) LNC practically coincides with the displacements of hydrogen atoms in the cell, and one may consider only the motion of the so-called, “active” atoms responsible for the phase transition. It should be noted that the physical situation in ABX3 perovskites resembles that in hydrogen-bonded ferroelectrics. Namely, the principal simplification adopted in the model description (by singling out the R25 or M3 mode) consists in neglecting displacements of all the ions with the exception of X cations and taking account only of their displacements which lie in the face plane of the unit cell (cf. [7]). The next step towards the model description consists in introducing the normal coordinate corresponding to the singled out soft mode (index s) of the spectrum which turns out critical at the SPT-temperature (T = Tc): Qqs = 1√ N ∑ ndα eα qs(d) √ mdu α nde −iqxn . (2.1) The above notation is obvious from the coordinates of atoms defined herein in a more complete form (for a crystal considered in the adiabatic approximation): Rnd = xnd + und = x0 nd + ηnd + und, (2.2) where x0 nd = n+d represents the equilibrium position of the d-th atom in the n-th primitive cell in the symmetric phase; ηnd describes the change in the equilibrium positions of atoms when the crystal undergoes a transition to the nonsymmetric phase; und (or taken as Snd in Section 3) are dynamic (time dependent) displace- ments of atoms with respect to the equilibrium positions xnd = x0 nd + ηnd; mnd is the mass of the nd-th atom and eα qλ(d) are polarization vectors forming an or- thogonalized basis (labelled by index λ) in the 3rN-vector space, r – the number 261 S.Stamenković of atoms in each primitive cell, N is the total number of cells in the crystal of volume V , α = x, y, z. Henceforth, for definiteness we shall consider transitions at the centre of the Brillouin zone (qc = 0) and introduce LNCs for the critical (soft) mode [10], xns = ∑ dα √ mde α q=0,s(d)u α nd. (2.3) These coordinates are local, meaning that summation in (2.3) runs over all the atoms in the n-th primitive cell and that it characterizes the cell as a whole. As it has been already remarked, coordinates xns obey the symmetry transformations of the point group (while the order parameter – obeys those of the space group), thus, forming the basis for describing SPT. The application of coordinates xns (2.3) and Qqs(q ≃ qc = 0) (2.1) in the Brillouin zone is restricted by the range of a weak q- dependence of the polarization vectors. As usual, in the case of optical modes the polarization vectors do not vary too much with a change in q. The corresponding limiting value of the wave vector intensity qm depends on the dispersion of the Fourier-transformed force constant (in the dynamical matrix), and, consequently, on the effective interaction radius in the system R(qm ∼ 1/R) [10]. 2.2.2. Conceiving the model Hamiltonian In describing SPT systems we start with quite a general Hamiltonian: H = ∑ i [ P2 i 2mi + U(Ri) ] + 1 2 ∑ i 6=j V (Ri,Rj). (2.4) Here Pi is a canonical conjugate momentum to coordinate Ri referring to every active in SPT atom i ≡ (n, d) of the masmi; U(Ri) is a single-site potential and the pair interaction potentials V (Ri,Rj) define the critical dynamics of the system. Since the atomic displacements with respect to the centre of the cell (henceforth designated by vector li) are usually small, the single-site potential U(Ri) and the pair-potential V (Ri, Rj) can be expanded in terms of displacements uα i as follows: V (Rα i ) = ∞ ∑ n 1 n! ( uα i ∂ ∂li )n U(li), (2.5) V (Rα i −Rβ j ) = ∞ ∑ n 1 n! [ (uα i − uβ j ) ∂ ∂li ]n V (li − lj). (2.6) To the lowest approximation it suffices to keep only the first few terms in expansions (2.5) and (2.6), thus, writing a single-particle potential in the form U(Rα i ) = U(lαi ) + 1 2 Aα i (u α i ) 2 + 1 4 Bα i (u α i ) 4, (2.7) while in the pair interaction it is sufficient to retain only the harmonic term 262 Unified description of structural transitions V (Rα i −Rβ j ) = V (lαi − lαj ) + 1 2 ϕαβ ij (u α i − uβ j ) 2. (2.8) In the above expressions the linear and the third order terms disappear from the equilibrium conditions (by taking U ′(li) = U ′′′(li) = V ′(li − lj) = 0) and the following abbreviations are used: Aα i = V ′′(lαi ), B α i = 1 6 V ′v(lαi ) ; ϕαβ ij = V ′′(lαi − lβj ). (2.9) The single particle potential (2.7) may be thought of as arising from an un- derlying sublattice of atoms which do not participate actively in SPT. Moreover, we will assume that atomic displacements are found along a given crystal axis, although the full spectrum of lattice vibrations is referred to a three-dimensional case. From the condition (indisputable in itself) that the minimum of the total energy in the system at zero temperature must be attained for the square value of the order parameter 〈uid〉2T=0 = −A/B, there follows that it must be A < 0 for positive B. Thus, a necessary condition for the existence of SPT is apparently equivalent to the demand that the single-particle potential (2.7) should be double-welled. In effect, the single-particle potential, strongly anharmonic in general, acquires the form of a quartic (double-well) oscillator having two stable atomic configurations, one of which prevails in the system at T < Tc, and yielding a homogeneously ordered phase with the order parameter 〈uid〉T=0 = ±(\|A\|/B)1/2 at zero temperature. With all the aforegoing remarks, restricting ourselves to the singled out mode (λ ≡ s), namely, using the LNC-representation (2.3), we arrive at the effective model Hamiltonian in the form: Hs = ∑ i [ p2i 2m − Ai 2 x2i + Bi 4 x4i ] + 1 4 ∑ i 6=i′ ϕii′(xi − xi′) 2, (2.10) where, for convenience, we drop subscript s of the variable xis (2.3). In the above Hamiltonian (2.10) pi is a conjugated momentum to xi, m ≡ ms being the ef- fective mass corresponding to the critical mode; both the harmonic and quartic constants associated with a single-site double-well potential are supposed to be site-independent, i.e. Ai ≡ A and Bi ≡ B; ϕii′ describes the coupling between the displacements of a local normal mode in cells i and i′ and determines the dispersion of the mode, i.e. the q-dependence of its frequency ωqs ≡ ωq. The space dimension of the model depends on the type of a lattice chosen, on which the pair-interaction ϕii′ is given. In the general case the local normal coordinate xis ≡ xi, in accor- dance with definition (2.3), represents a multicomponent vector the dimension of which is determined by the dimension of the relevant irreducible representation (the number of components of polarization vector eq=qc,s). For a one-component (scalar) coordinate xi the Hamiltonian (2.10) describes SPT in a uniaxial system∗. ∗Henceforth, the vector notation for direct and inverse one-dimensional (d = 1) spaces is omit- ted. 263 S.Stamenković In addition, it should be remarked that the one-site (cell) potential in equation (2.10) ensures a simple stabilizing interaction, since in the harmonic (index h) case (Bi ≡ 0) the local normal mode is unstable for A > 0: ω2 h,q=0 = −A/m (cf. [9]). To avoid any confusion, it should be noted that the dimensionalities of parameters A,B and ϕii′ in equation (2.10) are modified in accord with the dimensions of variables xi and pi, which is otherwise immaterial for “the essentials” of the unified model description of SPT. Thus, the Hamiltonian (2.10) represents a dynamic microscopic model of SPT in the reduced space of coordinates related directly to the phase transition. It is the Hamiltonian of one singled out mode, namely, of the critical mode. As we have already mentioned, its interaction with the remaining modes, if allowed by the system symmetry, can also be taken into account in terms of local normal coordinates. 2.2.3. The soft-acoustic mode interaction At this stage it should be stressed once more that at many SPTs the size and shape of the unit cell alter with temperature. These changes are characterized by the infinitesimal strain tensor uαβ, provided that the atomic displacements brought about by the lattice deformation are represented as ηαnd = ∑ β uαβx β nd. (2.11) Using a general expression for the normal coordinate Qqλ such as (2.1) (with running index λ 6 3rN instead of the singled out one (s) indicating the soft mode) together with (2.11), one obtains uαβ = lim qβ→0 iqβ √ N ∑ dmd Qqβα, (2.12) where Qqβα is an amplitude of the acoustic mode with its eigenvector polarized in the α-direction and its wave-vector projected onto the β-direction. However, no particular combination of strain parameters can be expressed in terms of the ampli- tudes of the acoustic mode. Consequently, the terms containing a long-wavelength acoustic mode (namely, the strains) must be eliminated from the summation over indices qλ in the initial harmonic Hamiltonian (in terms of normal coordinates and momenta) and written as an additional term to the Hamiltonian (2.10) in the form showing the relationship between the critical mode (i.e. the LNC) and the lattice deformation (index D) (cf. [7,11]): HD = ∑ i,αβ uαβ[U αβ 1s xis + Uαβ 2s xis 2] + Vo 2 ∑ αβγδ uαβuγδ[Cαβγδ + ...]. (2.13) In the above expression (2.13) only three most important terms are written out of the double power series expansion around the equilibrium positions of the sym- 264 Unified description of structural transitions metric phase. The first term in equation (2.13) describes the coupling of the inter- nal homogeneous deformation (the optic mode of vibrations in the long-wavelength limit) to the external homogeneous deformation determined by the last term which contains the product of the usual elastic constants Cαβγδ and the strain variables multiplied by the volume of the unit cell Vo = V/N . In what follows we neglect variations both in the shape and size of the unit cell with temperature and pressure, i.e. we use the standard approximation of a “clamped” crystal. As it was pointed out in [11], even with this oversimplification including less drastic ones (such as the uniaxial anisotropy, the absence of the third- order “site”-anharmonicity, short-range forces, etc.), most of the basic features of the theory of SPT are still comprised in the simple model Hamiltonian (2.10) that does offer a unified approach in describing the order-disorder and displacive systems – a description which is to be regarded as an inherent expression of their universal (Ising-like) critical behaviour. 2.2.4. The essential features of the model The microscopic model Hamiltonian (2.10) is characterized by two energy pa- rameters: the depth of the potential well Uo = A2/4B, (2.14) and the relative binding energy of the particles Uc = ϕo \| A \| B , ϕo = ∑ n′ ϕnn′, (2.15) as well as by a parameter describing the quantum properties of the system, λ = ~ωo/4Uo, ω0 = √ A/m. (2.16) The parameter λ ∼ ~/ √ m determines a relative zero-point energy of vibrations. If the system is characterized by a large zero-point energy of vibrations, so that λ is greater than a certain critical value λc, then the effective one-particle potential in (2.10) turns into a harmonic potential with a single minimum (see subsection 4.5– 4.7). In this case no SPT occurs even at zero temperature (in strontium titanate, for example). When λ 6 λc, the Hamiltonian (2.10) describes two extreme cases of SPT: displacive and order-disorder, depending on the parameter f0 ≡ Uc/4Uo = ϕ0/ \| A \| . (2.17) The structural transformations are called displacive transitions under which in the non-symmetric phase atoms are displaced insignificantly (by several percent of the lattice constant) with respect to their equilibrium positions in the sym- metric phase (for example, at the ferroelectric transition in BaTiO3). The notion of a soft phonon mode was originally attributed to displacive transitions, which 265 S.Stamenković is based on extensive experimental observations (cf. in [7]). Displacive transitions are described by model (2.10) for f0 ≫ 1 or Tc ≪ U0. It is usual to consider the displacive transitions as being characterized by resonance dynamics, although in this case the critical mode may even undergo strong damping. The latest studies of the nonlinear properties of the model have revealed that in the critical region a displacive transition acquires the features of an order-disorder transition (f0 ≪ 1 or Tc ≪ U0) (see section 5). Order-disorder transitions are such transitions at which there arises a long- distance order owing to the ordering of certain atomic complexes in the nonsym- metric phase (for example, the orientational ordering of NO2 groups at a ferro- electric transition in NaNO2). Reorientations of NO2 groups involve quite large displacements of atoms (N ions) relative to the equilibrium positions, so no ex- pansion in these displacements can be performed. In such cases the notion of a soft mode does not represent the physical situation adequately, since the dynam- ics in the critical region is of a relaxational character. This is almost the situation envisaged in the Ising model of ferromagnetism. Otherwise, to an order-disorder transition there corresponds a phase transition in the Ising model which is ob- tained from model (2.10) when f0 → 0 (A,B → ∞ for A/B = const) [12]. Besides, it is this equivalency of the two models (model (2.10) – often called “φ4-model” and the Ising model with a transverse field – abbreviated as IMTF hereafter) which motivated the introduction of pseudospin formalism (i.e. various versions of the adopted pseudospin – phonon Hamiltonians) into the underlying unified SPT description. The division of SPTs into two types is based on quite sound thermodynamical arguments (cf. [7,11] and Refs. therein). However, in considering lattice dynamics such a division is only conventional. There exists a broad class of systems, for example, ferroelectrics of the KH2PO4 type, which are difficult to attribute to any of the limiting types, since from spectral measurements it is not clear whether the critical mode exhibits the character of an overdamped soft mode or of a relaxational mode. 3. The unified description of strongly anharmonic systems 3.1. Preamble: convenience of simple approximations As we have already mentioned, it is generally assumed that there are two basic kinds of SPTs (ferroelectric included), one being of the order-disorder type and the other being of the displacive type. In the former case the SPT results from a statistical or tunnelling induced disorder of atoms among several (in the simplest case, two) equilibrium positions. In the latter case SPT is caused by lattice insta- bility against a critical vibrational mode (soft mode). Nevertheless, it was shown about a quarter century ago that the both types of SPT can be described within a single model (2.10) and there are no essential differences between them (cf. [1]). The nature of the SPT described within such a model was examined by applying 266 Unified description of structural transitions both the Curie-Weiss (molecular-field) and the self-consistent phonon approxima- tions ([cf. [1,2,7]). By comparing the results of the both approximations it has been shown that for a weak lattice coupling (f0 ≪ 1) the character of the SPT is of the order-disorder type, which is more consistently described by the molecular-field approximation; for a strong lattice coupling (f0 ≫ 1) the SPT has to be related to the displacive type, which can be reasonably described by the self-consistent phonon approximation. Such a consistent description can be understood under the circumstances that in the order-disorder transition statistical or tunnelling fluc- tuations of atoms onto their equivalent equilibrium position play the main role, which is accurately enough described by a pseudospin model; while in the dis- placive transition the dynamical correlations of atomic displacements turn out to be more essential, so the self-consistent phonon approximation is more adequate. The structural transition itself can also have a mixed character, depending on the relations between the energy parameters introduced in the model. For a complete description of SPT one has to take into account the both above mentioned mechanisms simultaneously – within the framework of a universal model – as was originally proposed and done in [1,2]. In the next section we are going to present the main features and results of such a description. 3.2. The adopted model Hamiltonian and equilibrium conditi ons In the case of a relatively deep double-well local potential U(xi) (2.7) with a negligible tunnelling of atoms between the two minima, atomic (i.e. cell) oscilla- tions inside the wells are of finite frequency which cannot be ignored. Therefore, apart from two degenerate eigenstates per cell (one near the bottom of each site- potential) entering the statistical problem, one has to take into account these intra-well oscillations, too, so that the inter-well tunnelling is entirely neglected. Besides, the two equilibrium positions of active atoms (namely, in each cell) are randomly distributed in the lattice and apparently determined by a self-consistent account of the vibrational and configurational parts of free energy. For this reason it is convenient to represent the effective time-dependent displacement xi ≡ xis (2.3) as follows (compare with equation (2.2)): xi(t) = ri(t) + ui(t). (3.1) As dynamic displacements, ri have their particular value for each cell(i) and are practically quasiequilibriums, meaning that they can change with time. The dynamical displacement ui(t) is associated with the atomic vibrations around momentary rest positions ri(t). Below Tc there are two “stable” configurations rα = α〈ri〉(α = −,+), and above Tc the system is to be found in one as well as the other. However, in the case of strong anharmonic systems more appropriate is to introduce, right from the beginning, the projection operator σα i onto two states in the “left” hand (-) or the “right” hand (+) parts of one-site potential (2.7) and thus to represent the LNC (3.1) in the form: ∗ ∗One has to distinguish between state index (α = −,+) and coordinate index (α = x, y, z). 267 S.Stamenković xi = ∑ α=±1 xαi σ α i . (3.2) For convenience we also introduce variables Si and bi as alternatives to vari- ables xi and ri (in equation (3.1)), respectively. Then, representation (3.2) in new variables Si reads: Si = ∑ α=± Sα i σ α i . (3.3) The projection operators σα i themselves are expressed, as usual, through the operators of pseudospin (the Pauli operator) σz i , being independent variables which commute with coordinates xαi and the corresponding momenta pαi : σα i = 1 2 (1 + ασz i ). (3.4) The coordinate in state α, Sα i , can be written as a sum of static displacement bαi and thermal fluctuation uαi : Sα i = bαi + uαi ; bαi = 〈Sα i 〉 = bα, (3.5) where symbol 〈...〉 stands for a statistical average with the Hamiltonian (2.10). The representation of atomic distortions (3.3)–(3.5) enables one to take into account, firstly, the atomic random distribution over two equilibrium positions in the cell, using operator σα i , and, secondly, the thermal fluctuation uαi in the neighbourhood of a given equilibrium position. In describing order-disorder SPT the variables uαi are usually neglected, whereas in displacive SPT only one equilibrium position in the cells (α = +1 or α = −1) is assumed, meaning that operator σα i takes the same value at each lattice site i. In this generalized model we will be able to study the both types of SPT using full representation (3.3)–(3.5). It should be noted that a similar representation for atomic coordinates was proposed by Vaks and Larkin [12] in discussing the order- disorder type SPT. We generalize their representation to consider the displacive type SPT as well. Having inserted expression (3.2), i.e. (3.3), into the model Hamiltonian (2.10), it can be written in the form: H = ∑ iα=±1 σα i [ 1 2m (P α i ) 2 − A 2 (Sα i ) 2 + B 4 (Sα i ) 4 ] + 1 2 ∑ i,j ∑ α,β=±1 σα i σ β j ϕij (Sα i − Sβ j ) 2 2 , (3.6) where ϕαβ ij (2.9) is assumed to be independent of α and β, i.e. ϕαβ ij ≡ ϕij . The equilibrium positions of lattice atoms bα = 〈Sα i 〉 are determined from the equilibrium conditions 268 Unified description of structural transitions i(∂/∂t)〈P α i (t)〉 = 〈[P α i , H]〉 = 0, (3.7) which leads to the equation 〈 ∂ ∂Sα i U(Sα i ) 〉 + ∑ jβ 〈 σβ j ∂ ∂Sα i ϕij (Sα i − Sβ j ) 2 2 〉 = 0. (3.8) By using the molecular-field approximation for a pseudospin subsystem and assuming its independence of the phonon subsystem, ∑ β 〈σβ j (S α i − Sβ j )〉 = 〈Sα i 〉 − ∑ β σβ〈Sβ j 〉 = (bα − b−α)σ−α, (3.9) the equilibrium conditions (3.8) may be rewritten in the form: Abα +B〈(Sα i ) 3〉+ (bα − b−α)σ−α ∑ j ϕij = 0. (3.10) Having chosen the positive direction of the displacements along the mean value b+ and using the approximation 〈(Sα i ) 3〉 ≈ b3α + 3bα〈(uαi )2〉, (3.11) the equilibrium conditions (3.10) become η3α − (1− 3yα)ηα + (η+ + η−)f0σ−α = 0. (3.12) Here dimensionless quantities are introduced: η2α = (B/A)b2α, yα = (B/A)〈(uαi )2〉, fq = 1 A ∑ j ϕje iq(li−lj), f0 = fq=0, (3.13) and the average population of state α: σα = 〈σα i 〉 = 1 2 (1 + ασz), (3.14) in agreement with expression (3.4) The analysis of the equilibrium conditions (3.12) shows that in addition to the zero solution η+ = η− = 0, corresponding to the symmetric phase, the solutions ηα 6= 0 are also possible. In the case of small values, f0 ≪ 1, the two equilibrium positions can exist, the magnitudes of which are close to one another (η+ − η− ≃ σf0 ≪ 1), and there is also the solution σ = 0, corresponding to a complete disorder for an order-disorder phase transition. For values of the reduced coupling parameter f0 & 0.25, only one nonzero solution can exist at all temperatures, for example, η+ 6= 0 (for a complete atomic order, σ = +1). In this region of the coupling parameter only a displacive phase transition is possible. 269 S.Stamenković 3.3. The phonon subsystem The phonon spectrum and the average values of the atomic (cell) displacement correlation functions can be determined using the Green function method devel- oped in the theory of strongly anharmonic crystals (cf. [1,8,13] and Refs. therein). Consider a displacement operator Green function of the general type: Dij(t− t′) = 〈〈ui(t); uj(t′)〉〉 = ∫ ∞ −∞ dω 2π e−iω(t−t′)Dij(ω), (3.15) where ordinary notation is used. The above Green function describes the atomic displacement correlations at the lattice sites i and j, in arbitrary states, because here ui = σ+ i u + i +σ − i u − i . Let us also introduce a Green function for the fixed atomic state α at the site i by inserting σα i = 1: Di(α),j(t− t′) = 〈〈uαi (t); uj(t′)〉〉, (3.16) which is necessary for the definition of the average quadratic atomic displacement in state α, 〈(uαi )2〉 = ∫ ∞ 0 dω coth ω 2kBT { −1 π ImDi(α),i(ω + iε) } . (3.17) Taking into account the fact that the Green functions (3.15) and (3.16) contain the statistical average with the full Hamiltonian (3.6), thus also including the average over all the atomic states, and since those functions depend only on the difference between the atomic coordinates Si − Sj ≃ li − lj, we will write their Fourier expansion in terms of the reciprocal lattice vectors q as follows: Di(α),j(ω) = 1 NA ∑ q eiq(li−lj)Dα q (ω). (3.18) The equation of motion for the Green function (3.15), using Hamiltonian (3.6), has the form: −m d2 dt2 Di(α),j(t− t′) = δijδ(t− t′)− −(A− ∑ k ϕik)〈〈Sα i ; uj(t ′)〉〉+ B〈〈(Sα i ) 3; uj(t ′)〉〉 − − ∑ kγ ϕik〈〈σγ kS γ k ; uj(t ′)〉〉. (3.19) The Green function 〈〈σγ kS γ k ; uj(t ′)〉〉, on the right-hand side, describes the cor- relation of atomic displacements at sites k and j, under the condition that the atom at site i is in state α (for instance σα i = 1). However, since k 6= i, one can neglect the correlation between states γ and α for the atoms at sites k and i, thus annihilating the latter condition, i.e. it is possible to use the molecular-field 270 Unified description of structural transitions approximation for the pseudospin subsystem. In addition, the approximation of independence between the phonon and pseudospin subsystems yields: 〈〈σγ kS γ k ; uj〉〉\|σα i =1 ≃ 〈〈σγ kS γ k ; uj〉〉 ≃< σγ k > 〈〈Sγ k ; uj〉〉. (3.20) Now, having inserted Sα i (3.3) in the Green function and using the familiar pseudoharmonic approximation, 〈〈(uαi )3; uj〉〉 ≃ 3〈(uαi )2〉〈〈uαi ; uj〉〉, (3.21) for the Fourier (q, ω)- component of the Green function in equation (3.19), one obtains: Dα q (ν) = ν2 − ν2−α (ν2 + ν2q+)(ν 2 − ν2q−)− σ+σ−f 2 q = ν2 − ν2−α (ν2 − ν2q1)(ν 2 − ν2q2) , (3.22) where reduced frequencies are introduced: ν2 = ω2/(A/M); ν2α = ∆2 α + f0; ν2qα = ν2α − σαfq; (3.23) the gap appearing in the phonon spectrum ∆α is determined by a single-particle potential, so that ∆2 α = 3(η2α + yα)− 1; (3.24) the phonon frequencies νq(+,−) in expression (3.22) correspond to atomic vibrations in the “right-hand”(+) or “left-hand”(-) equilibrium positions. If disorder is present in the system, the phonon spectrum, as determined by the Green function poles in equation (3.22), has two branches, ν2q(1,2) = 1 2 (ν2q+ + ν2q−)± 1 2 [(ν2q+ − ν2q−) + (1− σz2)fq 2 ]1/2. (3.25) However, in the limiting case of a complete order (for instance, σ+ = 1 and σ− = 0), the Green function (3.22) becomes D+ q (ν) = [ν2 − (∆2 + + f0 − fq)] −1, (3.26) and has only one pole corresponding to the vibrations of all the atoms in the “right-hand” equilibrium positions. For σz = 0 the number of atoms (cells) in both states becomes equal, σ+ = σ− = 1 2 , so the average field at each site takes the same value: ∆2 + = ∆2 − = ∆2 0. Therefore, the phonon spectrum in this case is determined by a single frequency being the pole of the Green function Dα q (ν) = [ν2 − (∆2 0 + f0 − fq)] −1. (3.27) Hence, in both cases a soft mode emerges when the single-particle gap (3.24) vanishes, ∆2 α → 0. 271 S.Stamenković A Green function of the general type (3.15) which can be obtained from equa- tion (3.22), using approximation (3.20), evidently has the same properties. The last self-consistent equation for the phonon subsystem represented by a phonon self-correlation function (3.17), in the high-temperature (classical) limit can readily be expressed in a simple form: yα = B A 〈(uαi )2〉 = = B NA2 ∑ q ∫ ∞ 0 dω coth ω 2kBT [ −1 π ImDα q (ω + iε) ] ≃ ≃ τ N ∑ q ∫ ∞ −∞ dν ν [ −1 π ImDα q (ν + iε) ] = = − τ N ∑ q ReDα q (0 + iε) = τ N ∑ q ν2−α ν2q1ν 2 q2 , (3.28) where the reduced temperature is given by τ = kBT (A2/B) . It is convenient to pass from the summation over q in the first Brillouin zone to the integration over frequencies by introducing the familiar frequency spectrum density g(ω2) = 1 N ∑ q δ(f0 − fq − ω2). (3.29) Taking into account expressions (3.23) and (3.25), equation (3.28) can be rewritten in the form: yα = τ ∫ ∞ 0 g(ω2)dω2 P +Qω2 [∆−α + f0], (3.30) where the following abbreviations are introduced: P = ∆2 +∆ 2 − + f0[∆ 2 + 1 2 (1 + σz) + ∆2 − 1 2 (1− σz)], Q = f0 +∆2 + 1 2 (1− σz) + ∆2 − 1 2 (1 + σz). (3.31) Finally, performing an integration in equation (3.28) in the case of zero tem- perature (when coth(ω/2kBT ) = 1) and taking into account equation (3.22), one obtains for the quantum limit (T = 0): yα = λ N ∑ q 1 2(νq1 + νq2) [ 1 + ∆2 −α + f0 νq1νq2 ] , (3.32) 272 Unified description of structural transitions where quantum parameter λ is defined by equation (2.16). Now, substituting yα (equation (3.30)) (or - in the quantum limit - equation (3.32)) and ∆2 α (3.24) in the equilibrium condition (3.12) we arrive at a self- consistent procedure to determine the order parameter η±, provided that the order parameter σz is to be independently found from the analysis of the pseudospin sub- system. 3.4. The pseudospin subsystem Besides phonon variables (un), the Hamiltonian (3.6) also contains configura- tion variables (3.4), therefore, the equilibrium values of pseudospin σz (3.14) can be directly calculated from the minimum of the total free energy. However, calcu- lation of the trace over the pseudospin and phonon variables for the density matrix with the Hamiltonian (3.6) is a rather difficult task, since the phonon and pseu- dospin variables cannot be separated. This fact is easily verified by substituting (3.4) into (3.6). As a result, we obtain the Hamiltonian: H = Hl +Hs, Hs = ∑ n hnσ z n − 1 2 ∑ n 6=n′ Jnn′σz nσ z n′ , (3.33) where Hl is independent of σz n, while the parameters hn and Jnn′ in Hs contain phonon variables in an explicit manner, as is entirely dictated by the Hamiltonian (3.6). We shall take advantage of the Bogolyubov variational principle for free energy, applying trial Hamiltonian for a pseudospin subsystem: H̃s = ∑ n h̃nσ z n − 1 2 ∑ n 6=n′ J̃nn′σz nσ z n′, (3.34) where the effective mean field h̃n and the exchange energy J̃nn, no longer depend explicitly on the phonon variables. Parameters h̃n and J̃nn′ are determined from the variational equations δF1/δJ̃nn′ = 0 and δF1/δh̃n = 0, where F1 = F0 + 〈H −H0〉H0 is the trial free energy with the Hamiltonian H0 = Hl + H̃s, and is of the form [7,14]: h̃n = ∑ α α [〈(pαn)2〉 4m + A 4 ∆α〈(uαn)2〉 ] + 1 8 ∑ αβn′ αϕnn′〈(uαn − uβn′) 2〉; (3.35) J̃nn′ = A 4B ϕnn′(η+ + η−) 2 − 1 8 ∑ αβ αβϕnn′〈(uαn − uβn′) 2〉. (3.36) The effective mean field h̃n depends on the vibrational energy difference be- tween the states (+) and (−). The effective exchange energy depends on the dis- tance (η+ + η−) between the equilibrium positions of the particles in a cell and also on the phonon correlation function. 273 S.Stamenković In the case of high temperatures, taking into account the solution of the Green function (3.22) and the approximative equality 〈P 2 α〉 ∼ (kBT )m, the effective field h̃n and the effective exchange energy, become, respectively: h̃n = A2 4B { 1 6 (∆4 + −∆4 −)− (η4+ − η4−)− 1 Q [P (y+ − y−) + τ(∆2 + −∆2 −)] } , (3.37) J̃nn′ = A 4B ϕnn′(η+ + η−) 2. (3.38) As it is easily seen, h̃ plays the role of a mean field caused by thermal atomic vibrations (when τ → 0, h̃ → 0) which tends to zero if σz → 0. The effective exchange energy (3.38) defined by the equilibrium atomic positions turns into zero above the critical (SPT) temperature, when η± = 0, leading to the unique solution σz ≡ 0. By using the molecular-field approximation, for the order parameter σz one obtains: σz = tanh( J̃0σ z − h̃ kBT ); J̃0 = ∑ j J̃ij, (3.39) or having taking into account definitions (3.37) and (3.38) the above equation (3.39) acquires the explicit form: σz = tanh { 1 4τ ( σzf0(η+ + η−) 2 + (η4+ + η4−)− 1 6 (∆4 + −∆4 −)+ + 1 Q [P (y+ − y−) + τ(∆2 + −∆2 −)] ) } . (3.40) Thus, the system of self-consistent equations for parameters η± (3.12) and σz (3.40) becomes complete, where quantities y± and ∆2 ± are defined by the suitable functions (3.30) and (3.24), respectively. In addition, the expression for the spontaneous polarization per atom (i.e. per cell) – the total order parameter of the system should be quoted. It is determined by both subsystem-order parameters, η± and σz. In dimensionless quantities the spontaneous polarization is given by Ps = 1 N ∑ n ( B A )1/2 (〈σ+ n S + n 〉 − 〈σ− n S − n 〉) = = 1 2 (η+ − η−) + 1 2 σz(η+ + η−). (3.41) If τ = 0, then it follows that σz = 1, so the polarization takes its maximum value Ps = 1; but if σz → 0, then it is clear that Ps → 0. 274 Unified description of structural transitions 3.5. The limiting types of SPT In a general case the system of self-consistent equations obtained for the order parameters η± and σz can only be solved numerically. Nevertheless, even a quanti- tative analysis of equations - in limiting cases - enables one to draw some definite general conclusions. At sufficiently low temperatures, τ ≪ τs (τs is a dimensionless lattice-instabili- ty temperature), it is possible to neglect the influence of lattice vibrations on the pseudospin subsystem and to consider only equation (3.40), with η+ ≃ η− = 1. In this case we have a well-known Ising model in which the phase transition of the order-disorder type (second order) takes place at temperature τk = f0 (using the molecular-field approximation, equation (3.40)). As we will see later, this result holds only if f0 ≪ 1. By neglecting the temperature dependence of parameter σz(τ), hereafter, we consider the limiting cases σz = 1 and σz = 0 (see subsection 3.6). If σz = 1, for quantity ∆2 + ≡ ∆2(τ), from equations (3.12), (3.24) and (3.30) (in the nonsymmetric phase η+ ≡ η 6= 0) one obtains the following equation: ∆2 = 2η2 = 2− 6τ ∫ ∞ 0 g(ω2)dω ∆2 + ω2 . (3.42) The solution of equation (3.42) was examined in a number of papers (cf. [1-3]), where it was shown that if a self-consistent phonon approximation is applied, then the phase transition becomes of the first order with two characteristic tempera- tures, one being soft-mode temperature τc (when ∆2 = 0), and the other being temperature τs at which the structural phase instability occurs (the overheating temperature). From equation (3.42) the soft-mode temperature is estimated as τ (1)c = f0 3µ−2 , (3.43) where index (1) corresponds to σz = 1. The constant µ−2 = ∫ ∞ 0 f0 ω2 g(ω2) ≡ 〈 f0 ω2 〉 ω , (3.44) depending on the type of the cubic lattice, is equal to 1.5 − 1.3. For the second characteristic temperature τs, in the case f0 ≪ 1 one finds the estimate τ (1)s ≃ 1 6 (1 + f0). (3.45) Note that the limiting value τs ≃ 1 6 , when f0 → 0, is not related to the phase transition, although it has an entirely defined physical meaning: the average kinetic energy of active atoms (i.e. cells) at this temperature is equal to the height of the effective potential barrier, 3 2 kBTs = A2/4B. In the case f0 ≫ 1, the estimate of temperature τs in the Debye model spectrum is given by 275 S.Stamenković τ (1)s ≃ τ (1)c ( 1 + 1 ω2 D ) ≃ τ (1)c ( 1 + 1 2f0 ) . (3.46) If σ = 0 the self-consistent equations, the quantities ∆2 + = ∆2 − ≡ ∆2 0(τ) and η2+ = η2− ≡ η20(τ) yield ∆2 0 = 2η20 − f0 = 2− 3f0 − 6τ ∫ ∞ 0 g(ω2)dω2 ∆2 0 + ω2 . (3.47) This equation can be solved only if f0 < 2 3 , whereas for f0 ≪ 1 the phase transition is of the same type as follows from equation (3.42). The characteristic soft-mode temperature in this case is estimated as τ (0)c = f0 3µ−2 ( 1− 3 2 f0 ) = τ (1)c ( 1− 3 2 f0 ) , (3.48) where index (0) corresponds to σz = 0. Similarly as in the case σz = 1, one obtains the following estimate for the instability temperature τ (0)s ≃ 1 6 (1− 2f0). (3.49) The estimates obtained show that in the general case of an arbitrary value for parameter σz the temperature τc(σ z) falls in the interval within the values (3.43), (3.48), and τs(σ z) is determined by relations (3.45), (3.46) and (3.49). Besides, a large hysteresis value (τs−τc)/τc > 1 corresponds to small values of f0 : f0 < 0.2 for σz = 0, and f0 < 0.25 for σz = 1; in the case f0 ≫ 1 the hysteresis value is small, Figure 1. Temperature dependence of the order parameters: η–average dis- placement and σz–average localization (pseudospin value) for dimensionless pa- rameter f0 = 0.10. in agreement with the estimates based on equation (3.46) and consistent with the results of other authors [15-17] (cf. also [1-3]). It follows from these es- timates that the order-disorder phase transition is possible only for small val- ues of f0: τk < τ (0)s , for f0 < 0.12, (3.50) if one estimates τk ≃ f0. At higher val- ues of f0 the lattice instability breaks down the order-disorder phase transi- tion, thus giving rise to the displacive phase transition: η±(τ → τs) → 0. At values f0 > 0.25, in agreement with equation (3.12), there is merely one sta- ble atomic equilibrium position at each lattice site, so the displacive-type phase transition is only possible, as described by equation (3.42). 276 Unified description of structural transitions A mixed-type phase transition, as described by all the three order parameters η+(τ), η−(τ) and σ(τ), may be expected only in a very narrow region: 0.11 < f0 < 0.25. (3.51) To confirm these general conclusions, the numerical solution of the self-consis- tent system of equations was obtained for the Debye model frequency spectrum (3.29), g(ω2) ∼ ω, ω < ωD, and the values of the coupling parameter f0 = 0.11, 0.12 and 0.15 were taken. The numerical results for σz(τ) and η±(τ) are presented in figures 1–3. It can be seen that the above estimates are in good agreement with the numerical calculations. The temperature dependence of the spontaneous polarization (3.41) for different f0 is shown in figure 4. Note, that in the region of the order-disorder phase transition (f0 < 0.11), as compared with an ordinary Ising model, the spontaneous polarization decreases more rapidly as temperature increases due to the temperature dependence of the effective exchange energy: J̃ = f0A 2/4B(η+ + η−) 2. We note that these features are also obtained by slightly improved calculations [14] based on the coherent potential approximation developed for the systems with disordered lattices (outlined hereafter in Subsection 3.7 for the case of mixed SPT). 3.6. The quantum limit in displacive SPT In this section we will consider only two cases, namely, a completely ordered (σz = 1) lattice and a completely disordered (σz = 0) one [3]. It is assumed that the right choice of a transverse field Ω (within IMTF) or Glauber-like relaxational dynamics ensure a transition from σz = 1 to σz = 0 at zero temperatures. 3.6.1. Displacive type phase transition in ordered lattice s Figure 2. Same as figure 1, for f0 = 0.12. In a completely ordered lattice all the atoms(cells) are in the same state, for example, α = +1 and σz = 1. In this case equation of self-consistency (3.32) takes the following form: y+ = λ 2N ∑ q 1 √ ∆2 + + f0 − fq = λ 2 ∫ ωD 0 g(ω2)dω2 √ ∆2 + + ω2 , (3.52) where the density of the phonon fre- quencies (3.29) is used. Taking into account the equilibrium condition (3.12), the self-consistent de- 277 S.Stamenković termination of the equilibrium displacement η (or the gap in the frequency spec- trum ∆2 + = 2η2) yields: η2 = 1− 3 2 λ ∫ ωD 0 g(ω2)dω2 √ 2η2 + ω2 . (3.53) As it can be seen, the solution of this equation for η 6= 0 exists only if λ < λc(1), where the critical value λc(1) is determined by the expression: λc(1) = 2 3 [ ∫ ωD 0 g(ω2)dω2 ω2 ]−1 = 2 3 √ f0 µ−1 ; (3.54) here µ−1 = ω−1 is the average of the inverse frequency provided that for the Debye spectrum, g(ω2) = 3ω/ω3 D, µ−1 = 3/2 √ 2 ≃ 1, if ω2 D = 2f0. Consequently, the displacive type transition in the ordered lattices can take place only if the lattice consists of sufficiently heavy ions, that is if m > mc = ( 2 3 √ ϕ0 µ−1 A B )−2 , (3.55) where the critical atomic (cell) mass (mc) is determined by model parameters (cf. also [15]). 3.6.2. Displacive type phase transition in disordered latt ices Let us discuss the effect of disordering on the displacive type phase transition. By putting in equations (3.12) and (3.32) the value σz = 0 which corresponds to an equal number of atoms in the states α = ±1 (meaning that ∆2 + = ∆2 − ≡ ∆2 0, y+ = y− ≡ y and η+ = η− ≡ η), the following system of equations is then obtained: η2 = 1− f0 − 3y, (3.56) y = λ 2 ∫ ωD 0 g(ω2)dω2 √ ∆2 0 + ω2 . (3.57) Therefore, a self-consistent equation for determining the gap ∆2 0 > 0 takes the form: ∆2 0 = 2η2 − f0 = 2− 3f0 − 3λ ∫ ωD 0 g(ω2)dω2 √ ∆2 0 + ω2 . (3.58) Hence, the displacive type phase transition (η > 0) can take place if λ < λc(0), where the critical value λc(0) is determined by the condition ∆2 0(λc(0)) = 0, where- from λc(0) = 2 3 √ f0 µ−1 ( 1− 3 2 f0 ) = λc(1) ( 1− 3 2 f0 ) . (3.59) Consequently, the occurence of the lattice disordering decreases both the limit- ing value of the allowed energy of zero-point fluctuations and the limiting value of the phase transition temperature (3.48). However, it has to be mentioned that the transition into the state σz = 0 can take place if f0 ≪ 1, only when formula (3.59) is valid. In the case f0 > 1, only the state with σz = 1 is possible, and formula (3.54) holds. 278 Unified description of structural transitions 3.7. Phase transition of the mixed type As we pointed out in the preceding subsection 3.5, the SPT a of mixed type can occur only in a very narrow region of the reduced lattice parameter f0(3.51) : 0.11 < f0 < 0.25. Besides various features arising from a considerable Figure 3. The same as figure 1, for f0 = 0.15. randomness of equilibrium positions ηn, for this type of SPT of particu- lar interest are the properties of the vibration(phonon)-frequency distribu- tion function. Namely, owing to the temperature dependence (explicit and through the parameter σα) of randomly distributed quantities ∆iα, the phonon spectrum resembles one of a perfect lat- tice, not only in a completely ordered phase, (σz = 1), but also in a com- pletely disordered one (σz = 0), as there is no perturbation in this last case (∆i+ −∆i− = 0). Owing to the disorder of the sys- tem, in the case considered a specific approach should be adopted, so that phonons are directly introduced into the Hamiltonian (3.6), i.e. (2.10). Since the disorder is due to single-site random potentials U({Sα i }), the quasiharmonic form (index qh) of expression (2.7), U(Sα i ) ≈ Uqh(S α i ) = A∆iα(S α i ) 2/2, is used in the ini- tial Hamiltonian separated in mutually orthogonal subspaces of eigenstates (3.6), with the gap-parameters ∆iα = 3(η2iα + yiα) − 1 ≈ 3(η2α + yα) − 1 ≡ ∆α (3.24), ηiα ≈ ηα (after equation (3.12)), being distributed arbitrarily in each cell. Then, Figure 4. Temperature dependence of the reduced polarization Ps for several values of the coupling parameter f0. within the framework of the coherent- potential method, by making use of a multiple scattering approach [18] and applying the usual one-site approxima- tion, both the ordinary, Dnn′(ω), and the conditional, Dn(α)n′(ω), Green func- tions (but for the given, i.e. fixed con- figuration {σα i } herein) are found (com- pare with expressions (3.15), (3.16) together with (3.18), (3.22)). These Green functions permit one to close a self-consistent set of equations for determining the parameters of the ef- fective Hamiltonian which corresponds to substituting potentials U(Sα i ) in equation (3.6) by quasiharmonic ones, Uqh(S α i ). 279 S.Stamenković The explicit knowledge of the averaged over all configuration (index c) Green functions 〈Dnn′(ω)〉c and 〈Dn(α)n′(ω)〉c, as expressed by their Fourier expansions in the reciprocal lattice space, makes possible the calculation for the given values of parameter σz of the lattice phonon spectrum: ρ(ν2) = −(1/π)Im〈Dnn(ν + iε)〉c (3.60) and of the phonon density of states for vibrations of an atom(cell) in a fixed state α: ρα(ν 2) = −(1/π)Im〈Dn(α),n(ν + iε)〉c. (3.61) In figures 5 and 6 the results of such calculations [14] for the dimensionless coupling constant f0 = 0.1 are presented. In calculating the sum over the reciprocal lattice vectors q in (3.60) and (3.61) the model density of states utilized was of the form (compare with equation (3.29)): Figure 5. The dependence of the distribution function of the frequencies ρ(ν 2) on the order parameter σz for f0 = 0.1. ρ0(ν 2) = 1 N ∑ q δ(ν2 − f0 + fq) = 2 πf 2 0 √ ν2(2f0 − ν2) (3.62) for ν2 < 2f0. As one can see from figure 5, the spectrum behaves in different ways when σz → 1 (σ+ =1, σ−=0 ) and when σz → 0 (σ+=σ− =1/2) : in the first case, when all the particles are ordered in the state (+), the band corresponding to the state (-) vanishes, while, as σz → 0, the both bands merge into one. It should be noted that the calculations of the phonon spectrum in order-disorder SPTs are usually missing, and, therefore, the experimental verification of peculiar spectral features, as pointed in figures 5 and 6, is of particular interest. 280 Unified description of structural transitions Figure 6. The dependence of the vibration-frequency distribution function on the order parameter σz in the state α = ± for f0 = 0.1: the solid line represents ρ+, the dash line represents ρ−. 4. Generalized model of SPT: ordering, tunelling and phonon s 4.1. A need for the incorporation of the atomic tunnelling mo tion The previous studies of rather simple models [1-4] have shown that both the displacive and order-disorder type excitations play an essential role in the dynam- ics of SPTs. At the same time a considerable effort has also been made to describe the corresponding both types of SPTs (ferroelectric ones included) within a single universal model (cf.[5]). However, the single-particle tunnelling motion of active atoms has not been explicitly taken into account. The incorporation of the tun- nelling motion as an additional degree of freedom leads to collective excitations which may have a soft mode character [5] or cause the appearance of a central peak [5,6]. Since the tunnelling energies (of the order of the ground state quantum split- ting, Ω) are usually much smaller than the characteristic phonon energies, the role of such excitations is predominant at low temperatures (kBT ∼ Ω) in which case, the SPT can be viewed as a transition that is predominantly of the order-disorder type. On the other hand, in addition to the renormalization of pseudospin-energy parameters of the well-known De Gennes type, the higher phonon excitations can lead to the SPT of the displacive type (against a certain vibrational mode) at higher temperatures. In this section an interpolated way for improving the unified analytical proce- dure in the SPT dynamics is presented. For this purpose we take into account self- consistently (within the Bogolyubov variational approach) excitations of the both 281 S.Stamenković types (displacive and order-disorder) – in order to comprise both the tunnelling and higher phonon oscillations of active atoms (i.e. cells) within the framework of a hybridized pseudospin-phonon model Hamiltonian separated in the correspond- ing variables. This is achieved by representing the cooperative atomic motion as a slow tunnelling process among several (in the simplest case, two) equilibrium positions in addition to familiar phonon-like oscillations around some momentary rest position. On the basis of the self-consistent phonon and molecular-field ap- proximations a complete system of coupled equations for two order parameters (average displacement η ∼ rsa and average localization σz) is obtained. Both qual- itative and detailed numerical analyses show that the SPT (of the first or second order) can be either of an order-disorder, a displacive or of a mixed type, depend- ing predominantly on the ratio of a two-particle potential to a single-particle one and, in a lesser degree, on the ratio of the zero-point vibrational energy to the height of a single-particle potential barrier. The possible SPT of the both types at zero temperature (quantum limit), as well as the relation between these results and other relevant treatments are also discussed. 4.2. The model phonon-pseudospin Hamiltonian Our previous [1,2] coordinate representation si = ∑ α σiα(biα + uiα) (3.3) de- scribed the random atomic (cell) distribution (within the Ising model through the projection operator σiα = 1/2(1+ασz i )), over two (α=+,−) equilibrium positions (biα) at every site i, as well as thermal atomic fluctuations (uiα) around one or an- other momentary rest positions (biα) (“left” and “right” phonons) (see Subsection 3.2). However, to elucidate more profoundly such an additional pseudospin degree of freedom (σz i ), one has to take into account the inherent quantum mechanical effect manifested in a single-particle tunnelling motion of active atoms inside some (real or effective) double-well potential, which was fully missing in our previous investigations and rather implicit in the approaches of other authors (cf. [1,2] and Refs. therein). For this purpose we suggested [4] the “left-right” representation of the model Hamiltonian (3.6) in the non-orthogonal pseudospin (i.e. “left”-“right”) basis. However, in accordance with the exhaustive analyses of many authors (cf. [5]), a clearer physical picture (and a rather transparent procedure) can be intro- duced by natural generalization of the traditional concept of atomic equilibrium states. Thus, a time dependent local normal coordinate may be decomposed into a slow tunnelling-like displacement (ri) and comparatively fast superimposed de- viations of the phonon type (ui) [5-7] (compare with (3.1)): si = ri + ui; 〈ui〉 = 0. (4.1) Such a representation holds under the “adiabatic” condition Ω ≪ ω0, ω0, being a characteristic frequency of lattice vibrations. Having inserted definition (4.1) into the general Hamiltonian (2.4) (assuming one-component coordinate Ri ≡ xi and momentum Pi = −i~ ∂ ∂Ri ≡ −i~ ∂ ∂xi ) it can be written in a trial form separated in the corresponding variables: 282 Unified description of structural transitions H0 = Hph({ui}) +Ht({ri}), (4.2) where the phonon-like (index ph) and tunnelling-like (index t) Hamiltonians are given by Hph = ∑ i P 2 i 2m + 1 2 ∑ ij φijuiuj, (4.3) Ht = ∑ i [ p2i 2m + Ũ(ri) ] + 1 2 ∑ i 6=j Cij(ri − rj) 2; (4.4) φij, Cij and Ũ(ri) are variational parameters and Pi and pi – canonical conjugate momenta to ui and ri, respectively. For a strongly anharmonic motion described by equation (4.4) it is convenient to introduce the energy representation comprising single-particle ground doublet (symmetric (ψs) and antisymmetric (ψa)) states which satisfy the eigenvalue equation: [ P 2 i 2m + Ũ(ri) ] ψs,a(ri) = εs,aψs,a(ri). (4.5) Going over to the pseudospin representation, Ht (4.4) is presented in the well- known De Gennes form (e.g. IMTF) (cf. [5]): Hs = −Ω ∑ i σx i − 1 2 ∑ i Jijσ z i σ z i +E0 , (4.6) where the energy parameters Ω, Jij and Eo are simple functions of εα, Cij, and the matrix elements rαβ = 〈α \| ri \|β〉 and r2αα = 〈α \| r2i \|α〉 (α, β = s, a) calculated with the wave functions ψα in equation (4.5): Ω = 1 2 (εa − εs) + r2−C0, E0 = 1 2 (εa + εs) + r2+C0, Jij = 2r2saCij; r2± = 1 2 (r2aa ± r2ss); C0 = ∑ j Cij . (4.7) The variational parameters φij, Cij and Ũ(ri) are determined from the Bo- golyubov variational principle for the free energy of system F , namely, from the condition of stationarity of free energy: F1 = F0 + 〈H −H0〉0 > F, (4.8) 283 S.Stamenković with respect to variations over these parameters or, equivalently, over the cor- responding correlation functions (cf. [8]). In the above expression (4.8) symbol < ... >0 stands for the statistical averaging with the trial Hamiltonian H0 ((4.2)– (4.7)), to which free energy F0 corresponds. Computing free energy F1 (4.8) where F0 = −kBT ln Sp{e−H0/T}, (4.9) 〈H −H0〉0 = Sp{e F0−H0 T (H −H0)} = = ∑ i 〈U(ri + ui)〉0 + 1 2 ∑ ij 〈V (ri + ui, rj + uj)〉0 − − ∑ i 〈Ũ(ri)〉0 − 1 2 ∑ ij φij〈uiuj〉0 − 1 2 ∑ ij Cij〈(ri − rj) 2〉0 (4.10) under the assumption that condition 〈Pipi〉0 = 0 holds, for the variational param- eters of the trial Hamiltonians (4.3) and (4.4) (i.e. (4.6)) the following equations are obtained: φij = δ δ〈uiuj〉0 {2 ∑ i 〈U(ri + ui)〉0 + ∑ i 6=j 〈V (ri + ui, rj + uj)〉0}, (4.11) 2r2saCij = δ δ〈σz i σ z i 〉0 〈V (ri + ui, rj + uj)〉0, (4.12) and δ δ〈σx i 〉0 ∑ i 〈Ũ(ri)〉0 = δ δ < σx i >0 { ∑ i 〈U(ri + ui)〉0 + + 1 2 ∑ ij 〈V (ri + ui, rj + uj)〉0 − 1 2 ∑ ij Cij〈r2i + r2j 〉0}. (4.13) The self-consistent system of equations (4.3)–(4.7), (4.11)-(4.13) determines the phase transition of the model and describes the mutual influence of phonon and pseudospin subsystems if the potentials U(si) and V (si, sj) in equation (2.4) are appropriately modelled. 4.3. The structural phase transition in the model Having chosen a single-particle double-well potential U(si) in the convenient form (2.7) and a pair-potential V (si, sj) in the familiar harmonic approximation (2.8), we assume the model Hamiltonian of the system in the previous form (2.10): 284 Unified description of structural transitions H = ∑ i ( − ~ 2 2m ∂2 ∂s2i − A 2 s2i + B 4 s4i ) + 1 4 ∑ ij ϕij(si − sj) 2. (4.14) The variational approach for this Hamiltonian yields: φij = δij(A∆+ ϕ0)− ϕij(1− δij); 2Cij = ϕij, ϕ0 = ∑ j ϕij , (4.15) where ∆ = 3(B/A) < r2i >0 − a , a = 1− 3(B/A)〈u2i 〉 . (4.16) The effective (renormalised) single-particle (cell) potential in (4.4) can be writ- ten in the form: Ũ(ri) = −Ã 2 r2i + B 4 r4i ; Ã = A− 3B〈u2i 〉0 ≡ Aa . (4.17) The self-correlation displacement functions relevant to the nature of the SPT are determined by the following approximative equations (of the RPA type): 〈u2i 〉0 = 1 Nm ∑ k 1 2ωk coth ωk 2kBT , (4.18) 〈r2i 〉0 = 1 2 [(r2ss + r2aa) + (r2ss − r2aa) + (r2ss + r2aa)〈σx i 〉0], (4.19) where the phonon frequency ωk is given by the equation mω2 k = A∆+ ϕ0 − ϕk; ϕk = ∑ j ϕije ik(li−lj). (4.20) In the mean(molecular)-field approximation (equivalent to RPA for pseudospins) (see, e.g., [19]) one obtains: 〈σx,z i 〉 = σx,z = hx,z h tanh h kBT ; hx = Ω, hz = J0σ z, h2 = h2x + h2z; J0 = ∑ j Jij = r2saϕ0 . (4.21) Thus, the SPT-system is described by the solution of the above self-consistent system of equations which, owing to equation (4.5), can be obtained only numeri- cally. Nevertheless, a qualitative analysis for the limiting cases is possible. 285 S.Stamenković a) Order-disorder transition Analogously to the analysis in the previous sections, in the weak coupling limit of small ϕ0, i.e. in the temperature region when ϕ0 ≪ Ã = A− 3B〈u2i 〉0, (4.22) the order-disorder transition is possible in the pseudospin subsystem as charac- terized by the order parameter σz. In the molecular field approximation for the order-disorder transition temperature Tc one finds: Tc ≃ J0 2q ln 1+q 1−q ; q = Ω/J0 ≤ 1. (4.23) The estimates obtained in the case of weak tunnelling (Ω ≪ J0; r 2 ss ∼ r2aa ∼ r2sa ∼ r̃20 = Ã/B) correspond to the results of section 3 (for f0 ≪ 1, therein), namely, Tc ∼ J0 ∼ ϕ0r̃ 2 0 ∼ ϕ0 Ã B ≪ U0. (4.24) Note that phonon excitations do not play an essential role in this case, since 〈u2i 〉0 ≪ r̃20. b) Displacive transition When the temperature is raised, the atomic fluctuations 〈u2 i 〉0 cannot be ne- glected and the character of the coupling could be changed, i.e. ϕ0 ≫ Ã (even for ϕ0 ≪ A), thus, leading to the displacive phase transition: ∆(T0) → 0 and r̃20(T0) → 0. In the classical limit of high temperatures (T & T0>Tc) [1-7], Ã(T0) = 0, 〈u2i 〉0 = 1 3 A B , (4.25) while in the strong-coupling limit 〈u2i 〉 ∼ T/mω2 0 ∼ T/ϕ0 also holds and the dis- placive transition temperature is estimated as T0 ∼ 1 3 ϕ0(A/B), (4.26) provided that for the model considered the inequality Tc ∼ ϕ0Ã/B < T0 ∼ ϕ0A/B is constantly valid. For both limiting types of SPTs the similar results have been obtained in subsection 3.5. 4.4. Approximation of a double well by two truncated harmoni c oscillators For the trial wave functions in equation (4.5) one can assume the linear com- binations of the ground states, ψ− 0 and ψ+ 0 , referred to the “left” (-) and “right” (+) unperturbed harmonic oscillators, respectively, to be of the form: 286 Unified description of structural transitions ψs,a = [2(1± ρ)]−1/2[ψ+ 0 (r)± ψ− 0 (r)] , (4.27) where ψ± 0 (r) = ψ0(r ± r̃0) ; ψ0(r) = (a0 √ π)−1/2 exp(−r2/2a20) ; a20 = ~/mω ; ω2 = k̃/m , r̃0 = (Ã/B)1/2 . (4.28) Here ρ is the overlap integral of the “left” and “right” states; the harmonic force- constant k = 2A is renormalised to be k̃ = 2Ã (in the approximation of a strong particle (i.e. cell) localization). By performing the corresponding calculations one finds: ρ = ∫ ψ+ 0 (r)ψ − 0 (r)dr = exp{−Ã 2/B ω/2 } = exp(−1/λ) ; (4.29) λ = ω 2Ã2/B = a20/r̃ 2 0 = λ0√ 2(1− 3y)3/2 , (4.30) where the temperature independent quantum parameter (compare with equation (2.16)) characterizing the zero-point vibrations λ0 = ~ω0/(A 2/B), ω0 = √ A/m is introduced, and parameter y = (B/A)〈u2i〉 is the reduced average quadratic “fast” displacement ( after equation (4.18)). Consequently, for the parameters of the pseudospin Hamiltonian (4.6) the following expressions are obtained: Ω = A2 4 √ 2B λ0ρ √ 1− 3y 1− ρ2 (1 + 3 λ ) , (4.31) r2sa = r̃20/(1− ρ2); r̃20 = (A/B)(1− 3y) , (4.32) r2ss = r̃20 1± ρ [ 1 + λ 2 (1± ρ) ] , (4.33) where the reduced average quadratic “slow” displacement (4.19) becomes r ≡ B A 〈r2i 〉0 = 1− 3y 1− ρ2 [ 1 + λ(1− ρ2) 2 − ρσx ] . (4.34) To make explicit the reduced average quadratic “fast” displacement (4.18), the spectral density of phonon frequencies is introduced: g(ν) = 2 π √ 1− ν2; ν = ω A/m , (4.35) and then one finds y = λ0 ∫ 1 −1 dνg(ν) 2 √ ∆− f0ν coth λ0 √ ∆− f0ν 2θ , f0 = ϕ0/A; θ = kBT A2/B , (4.36) where 287 S.Stamenković ∆ = 3(y + r)− 1 , (4.37) is a gap in the phonon spectrum (4.20). Finally, owing to equation (4.32), the spontaneous polarization (per cell) of the system is simply expressed as a product of the displacive-like (η) and the order–disorder-like (σz) order parameters: Ps = 1 N ∑ i ( B A )1/2〈ri〉0 = ( B A )1/2 rsaσ z = ησz , (4.38) where the order parameters σz and η are given by the above self-consistent proce- dure: σz – by equation (4.21) and η ≡ (B/A)1/2rsa – by the expression η = { 1− 3y 1− ρ2 }1/2 . (4.39) Both the “order-disorder” (σz) and the “displacive” (η) order parameters are to be found as a self-consistent solution of equations (4.21), (4.29)-(4.37). For a given set of the reduced energy parameters (λ0, f0) the competition of these order parameters determines the character of the SPT. 4.5. The quantum limit of zero temperature At zero temperature equations (4.21) and (4.36) become σz = 1 J0 √ J2 0 − Ω2 , σx = Ω Jo ; (4.40) y = λ0 ∫ 1 −1 dν √ 1− ν2 π √ ∆− f0ν . (4.41) As it is easily seen from equation (4.40), σz > 0 if Ω 6 J0. Using equation (4.41) with equations (4.21), (4.32), one obtains the condition for λ0, ρc 4f0 (3 + λ0)(1− 3yc) = 1 , (4.42) which defines the maximum λ0(λ c 0) at which σ z > 0 is still possible. In a simplified case, when yc ≪ 1 3 , the graphical solution of equation (4.42) gives λc0 ≃ 1/ ln[(3/4)f0]. (4.43) Hence, if λ0 > λc0, then σ z = 0, even at zero temperature. The order parameter η can also vanish at zero temperature. Using equations (4.36), (4.37) and (4.39), one obtains λph0 ≃ 2 √ f0. (4.44) 288 Unified description of structural transitions In such a way the zero-point vibrations can destroy the ordered ground state at T =0 K, both in the pseudospin and the phonon subsystems. One can expect that Ps vanishes either in σz or in η, depending on the competition between λc 0 and λph0 , i.e. on the lesser of the two. The corresponding estimates of λc 0 and λph0 for various values of parameter f0 are given in table I. TABLE I f0 0.05 0.10 0.30 0.50 λc0 0.37 0.50 1.95 2.50 λph0 0.44 0.62 1.10 1.40 4.6. The numerical results 4.6.1. The case of the double-well potential modelled by two harmonic os- cillators Figure 7. The order parameters sigmaz and η, the polarization Ps and the overlapping pa- rameter ρ as functions of temperature (θ). Pa- rameters σz, η, Ps–upon θ-scale and ρ–upon above-scaled θ-abscissa correspond to λ0 = 0.5 and f0 = 0.5; the marked parameters σz I , ηI , P I s , etc., upon θI -scale, etc., correspond to λ0 = 0.1 and various f0 = 0.1, 0.2, 0.6, respec- tively. The system of self-consistent equations (4.21) and (4.29)– (4.39) is solved numerically and σz(θ), η(θ), Ps(θ) (for various f0 and λ0) and ρ(θ) (for f0 = λ0 = 0.5 as well as σz(λ0), η(λ0) (for f0 = 0.6 and θ = 0.001) and ρ(λ0) (for various f0 = 0.1, 0.2, 0.6 and θ = 0.001) are presented in fig- ures 7 and 8, respectively [5]. It can be observed that the quali- tative estimates in the preceding subsections (4.3–4.5) are in good agreement with the numerical cal- culations. For all marked param- eters (I to III) the overlapping in- tegral ρ, as calculated, is of the or- der 0.003 at T ≃ TSPT . The corre- sponding curves in figure 7 are in agreement with our previous ones [1] (cf. section 3). In the weak- coupling limit f0 = f 1 0 /(1−f 1 0 ) ≪ 1 (f 1 0 being the reduced coupling constant f0 in [1], i.e. in equa- tion (3.13)), the proper account of tunnelling effects is taken for ρ ≪ 1 (at least ρ < 0.5 and λ0 < 0.5), in agreement with the previous results for both the order parameter (here η) and the exact curves for σz [15]. 289 S.Stamenković Figure 8. The order parameters σz, η and the overlapping parameter ρ as functions of the quantum parameter λ0. σz and η are presented for f0 = 0.6; ρ1, ρ2, ρ3 correspond to various f0 = 0.1, 0.6, 1.0, respectively. The appearance of imaginary solu- tions in both figures, 7 and 8, is as- signed by cross-points. A discontinuity itself in the displacive SPT (but in the case f0 ≪ 1) is a well-known character- istic of the self-consistent phonon ap- proximation (cf. [1,7]). In the strong- coupling limit (f0 ≫ 1) (when the tunnelling can be disregarded) the dis- placive SPT does occur and is properly described as previously [1]. As it can be observed in figure 7, the case “III”(f0 = 0.6) corresponds to a mixed SPT. In fig- ure 8 η(λ0) has a minimum indicated as λ ph(1) 0 ; λ ph(2) 0 denotes a value at which η becomes imaginary and λc 0 is a critical λ0 for the pseudospin subsystem (for λ0 > λc0, σ z = 0). 4.6.2. The exact results based on the model of quartic oscill ator The self-consistent numerical calculations using the renormalized double-well potential (4.17) are also performed, provided that the overlap of the exact (index exc) wave functions ψs and ψa, ρexc = 〈s \| a〉exc, is now equal to zero [5]. The dependence of all the relevant parameters for various f0, λ0 and temperatures fairly agrees with the numerical calculations (using trial doublet states) when ρ < 0.5 (figures 7 and 8). As an illustration, only some truncated results for the dimensionless parameters are presented in tables II and III. As an indicative corollary we also note that for θ close to zero an appreciable λ0 – dependence of all the parameters is found, while the f0 – dependence of these parameters (except for J0) can be ignored (cl.[5]). TABLE II λ0 = 0.1; f0 = 0.1 θ Ω 10−3 J0 η y σz σx Ps 0.020 0.190 0.075 0.866 0.041 0.999 0.002 0.865 0.040 0.220 0.074 0.858 0.045 0.939 0.003 0.805 0.060 0.320 0.070 0.837 0.056 0.617 0.004 0.516 0.080 0.530 0.064 0.803 0.072 0.1 · 10−6 0.006 0.8 · 10−7 0.100 0.110 0.055 0.744 0.098 0.1 · 10−6 0.011 0.8 · 10−7 290 Unified description of structural transitions TABLE III θ = 0.001; f0 = 0.1 λ0 Ω 10−3 J0 η y σz σx Ps 0.100 0.19 · 10−3 0.075 0.866 0.041 1.000 0.002 0.866 0.200 0.18 · 10−1 0.043 0.653 0.095 0.903 0.430 0.590 0.300 0.57 · 10−1 0.037 0.609 0.122 0.1 · 10−6 1.000 0.61 · 10−7 0.400 0.106 0.036 0.599 0.151 0.1 · 10−6 1.000 0.60 · 10−7 0.500 0.164 0.037 0.606 0.177 0.1 · 10−6 1.000 0.61 · 10−7 4.7. Remarks on the applicability of the model The model dealt with essentially represents a generalization of the De Gennes model (cf. [5-7]). Here, both the dependence of the model double-well potential on temperature T and the quantum properties of the system (λ2 0 ∼ ~ 2/m) have been considered. The principal approximation consists in the fact that the total spectrum is divided into two parts: a low-lying part of a doubly split level due to tunnelling and the other part of higher excited states. The mutual influence of these two parts of the spectrum is, then, taken into account in the frames of mean-field type approximations. The parameters of the pseudospin Hamilto- nian (4.6), Ω and Jii′, depend essentially on the state of the phonon subsystem (the populations of the higher levels) through the effective one-particle potential (4.17). And vice versa, the state of the phonon subsystem depends on the state of the pseudospin subsystem, since the phonon frequency (4.20) depends on the mean square displacement of the equilibrium positions of vibrations 〈r2i 〉0 (4.19), depending otherwise on σx. For describing a phase transition in the system it is necessary to solve a set of equations (4.15)-(4.21). A numerical solution was obtained in Ref. [5]. At λ0 = 0.1 there exists a double-minimum potential with two closely situated levels below the potential barrier. Degeneracy of the ground state is removed by a quantum tunnelling of the particle (cell). When λ0 = 0.3, the kinetic energy of the particle (or cell) is so high that the energy levels are situated above the potential bar- rier. In this case, evidently, the representation of the effective displacement in the form (4.1) loses sense. Moreover, since the energy spectrum of a particle in a local (double-well) potential has quite a complex structure [5,20], such a separation has merely an interposal character, i.e. being physically inapplicable for the temper- ature region, when ~Ω ∼ ~ω0. In particular, one could expect a more complex renormalization of the pseudospin parameters in real order-disorder compounds, especially when the excited atomic states lie in the critical temperature region, kBTc ∼ J0 ∼ ~ω0. Summing up the theoretical and numerical analyses presented, it should be pointed out that the generalised model satisfactorily reveals the essential features of both the order-disorder and the displacive types of SPTs at finite and zero temperatures. The true quantum-mechanical situation, i.e. the tunnelling (local or effective) motion of active atoms (or cells), is treated properly with respect to the approximations applied and the limiting cases considered. In particular, 291 S.Stamenković Figure 9. The dependence of Ps on the quantum parameter λ0 for T = 0. the numerical results show [5] that in the approximation of strong anhar- monicity (f0 ≪ 1), as in the case of the displacive limit (see Subsection 3.5), zero vibrations can violate the ordered state, thus, not arising even at T = 0. In other words, quantum fluctuations may suppress the phase transition in the system, as one can see from figure 9 in which the dependence of the “to- tal” order parameter Ps (4.38) on λ0 is shown for two values of the coupling constant f0. Concluding this section, we believe that the present model, as analysed by means of a more accurate self- consistent procedure, in addition to its extension in the spirit of the central peak dynamics and solitary waves, can complete the analytic description of the critical dynamics and yield a deeper insight into the nature of SPTs in general. 5. Nonlinear effects and the incomplete softening of critic al modes 5.1. A need for domain-wall soft mode dynamics The principal role of nonlinear effects at STPs was originally pointed out in [21]. Thus, the exact solution for a one–dimensional (d=1) lattice shows that be- sides phonons (the softening of which causes a displacive phase transition) there are also nonlinear (soliton–like) excitations which independently of (or additionally to) phonons can lead to the instability of the lattice. This fundamental inference is also consistent with molecular dynamics simulations (from d=1 to d=4) [22], as well as with the results of the exact dynamics of a few coupled quartic oscillators [23]. In a series of subsequent theoretical works (cf. [7,11]) it was envisaged that with the onset of criticality a crossover from a “displacive” to an “order–disorder” regime could occur manifested in the formation of long–lived clusters of the precursor or- der. The existence and dynamics of such clusters give rise to the central peak (i.e. the nonclassical critical behaviour) and to the fact that the soft mode is suppressed due to the local symmetry violation (see, e.g., [24]). While such a physical picture is conceptually quite acceptable, it is still not analytically substantiated and also lacks an indisputable experimental verification. Nevertheless, the authors [24-26] interpreting EPR–experiments on monondomain–transforming SrTiO3 (through motional narrowing, using Fe3+–V0 centres) drew attention to the challenging accord between theoretical and experimental estimates of a non–vanishing soft– 292 Unified description of structural transitions mode frequency at the phase transition temperature (Tc), thus, prompting further progress in comprehensive SPT investigations. Apparently, such a coincidence cor- roborates the model description [1-5,21-26] by which the local normal coordinate si(t) is decomposed into a slow momentary quasirest position (ri(t)) and a com- paratively fast deviation ui(t) (around ri(t)) of the phonon (quasiharmonic) type. In the frames of such ideas, our preceding works [1-5,7] were devoted to analytical studies of the SPT description within a single universal model, using in addition the Bogolyubov variational approach. Thus, a novel interpolar scheme attempting to include all the intriguing features of the SPT, i.e. statistical order–disorder, tunnelling (real and effective) and phonon excitations has been proposed (see Sec- tion 3 and 4). To describe a possible order–disorder behaviour of the traditionally displacive SPT (revealed in an incomplete phonon softening) the adjustment of the previous models is accomplished by employing both the solitary–waves and renormalization–group (RG) techniques (cf. [5-7]). Note that the RG–method has been used in this sense only for order–disorder systems [27]. 5.2. The critical region The unified theory of SPTs afforded by the model studies which were surveyed in the preceding sections can be qualified as applicable within the same limits as the Landau theory. Such a description provides a qualitative picture of SPTs, particularly of traditionally limiting types, as well as for some model parameters of mixed transitions which do not belong to either of two extreme cases. However, like the previous theories, it does not permit investigation of their properties in the critical region (cf. also [7,11]). Investigation of the phase transition (for instance, in strontium titanate SrTiO3, with the aid of inelastic neutron scattering (cf. [7])) reveals that the excitation spectrum in the critical region is more complex than that which fol- lows from the classical theory. The scattering intensity exhibits, besides two side peaks corresponding to the soft mode, a narrow peak at zeroth energy transfer. As T → Tc the intensity of this central peak increases sharply, while its width decreases. The frequency of the soft mode decreases up to a certain finite value at T =Tc. Subsequent experiments demonstrate that this pattern is characteristic of all SPTs. At present the nature of the central peak still remains to be clarified. The reasons for its appearance are divided into two sorts: intrinsic (fluctuations of the phonon density, nonlinear effects, etc.) and extrinsic (defects, impurities, surface). The results of numerous studies of this question, as well as an extensive bibliogra- phy, can be found in the reviews (cf. [7,11]). It can be concluded from the results of these works that a system in the critical region is characterized by two time scales, instead of one, as in the conventional theory of the soft mode. In addition to the relatively short (noncritical) relaxation time of the system characterizing the soft phonon, there exists a large (critical) relaxation time characterizing the narrow central peak. In this section, on the basis of model (2.10), the rise of a double-time evolution at a displacive or an order-disorder phase transition in a 293 S.Stamenković perfect (defectless) crystal is considered. In accordance with the universality hypothesis, model (2.10) belongs to the Ising universal class, i.e. it shows the same critical behaviour as the Ising model. This means that for f0 ≫ 1, at least in the critical region, when T >Tc, a short- range order should arise in the system leading to a non-Gaussian distribution of the order parameter. Such a result already follows from the first applications of the RG-method to model [11]. Utilization of this method for calculating the coordinate distribution function for blocks shows [28] that in a two-dimensional case (d=2) it has a pronounced structure with two maximums. When d=3, the distribution has a single maximum but of a non-Gaussian type. From calculations [28] it follows that a phase transition in model (2.10) in the limit f0 ≫ 1 exhibits clearly pronounced features of an order-disorder transition at d=2 and close to the features of such a transition at d=3. A detailed investigation of the transition from the displacive regime to the order-disorder regime in model (2.10) for d=1 and d=2 was carried out in [29] by applying the RG-method in the direct space. It was shown that on the phase diagram (0 6 T 6 ∞, 0 6 f0 6 ∞) there exists a crossover line T0(f0) with the finite points T = 0 and f0 ∼ 0.5. Below this line the system goes over from the displacive regime to the Ising one in which the depth of potential minima increases after each iteration of the RG- transformation. In this region of parameters T and f0 the system has a fixed Ising point. When d = 2, it is also established that, at least for d 6 2, the system described by equation (2.10) undergoes, in the displacive limit (f0 ≫ 1) and in the temperature region Tc < T < T0, a transition to the order-disorder regime exhibiting the Ising critical behaviour. However, the RG-method does not permit a description of the excitation spectrum in this region. For this purpose other methods should be applied (see subsection 5.3). Investigation of model (2.10) by analytical methods [21] for d = 1 and by the method of molecular dynamics [22] (cf. also [6,7,11]) for d ≤ 4 reveals that a transition to the order-disorder regime is accompanied by the appearance of short-range order clusters, or microdomains, with different polarizations of the order parameter. A natural conjecture was to assume that such clusters appear at T =T0(f0) [29]. When T >T0(f0), the system is characterized by small vibrations relative to the position of the hump in the effective one-particle potential. As the temperature drops lower than T0, clusters walls are formed which put bounds for the regions of dimensions of the correlation length. At T = Tc(f0) the system undergoes a continuous phase transition to an ordered state with Ising critical exponents. When f0 6 0.5, the excitations most significant for the system are the ones (soliton waves-like) which appear due to the formation of walls. The influence of phonons on this process is of no principal importance, since the potential minima are so deep that jumps (i.e. tunnelling “crossings”) from one well to another are extremely rare. The system is Ising-like at all temperatures. When f0 ≫ 1, the system is Ising-like within the region Tc<T <T0. In this temperature region the principal excitations are the ones related to the formation of walls between clusters with different polarizations of the order parameter. At T > T0 the walls between 294 Unified description of structural transitions the clusters start vanishing, and phonon excitations become the most significant ones. Now one can envisage a qualitative picture of the behaviour of the excitation spectrum. When T > T0 (the displacive regime), the particles vibrate relatively to their symmetric equilibrium positions; the spectrum exhibits two peaks (phonon emission and absorption in scattering intensities) corresponding to the frequency of soft phonons and determined by the poles of the standard Green function, Dq(ω) = [ω2 − ω2 q −Mq(ω)] −1 (unlike, e.g., equation (3.27), self-energy operator Mq(ω) is introduced herein). At Tc < T 6 T0 the system passes to the order- disorder regime in which the particles vibrate around quasiequilibrium positions displaced relatively to the lattice sites in the symmetric phase: there arise clusters of different polarizations serving as the initial nuclei of the new phase; under the stabilizing influence of the clusters the high-temperature phonon evolves into its low-temperature counterpart without exhibiting a complete softening. The critical slowing down of the system is provided by the relaxation motion of the cluster walls contributing to the energy (scattering) spectrum and the central quasielastic component [7,11]. 5.3. Domain walls and phonons in an inhomogeneous medium Now let us consider in greater detail the range of temperatures Tc < T < T0 within which the system becomes inhomogeneous with respect to the distribution of equilibrium positions ri. These equilibrium positions have for each cell their par- ticular value and are practically quasiequilibriums, meaning that they can change with time. We consider the widely used model Hamiltonian ((2.10), i.e. (4.14)): H = ∑ i ( P 2 i 2m − A 2 s2i + B 4 s4i ) + ∑ ij 1 4 ϕij(si − sj) 2 , (5.1) with the local normal coordinate si represented in the form ((3.1), i.e. (4.1)): si = ri + ui ; (5.2) ri is now defined by ri = 〈si〉ph = 1 Zph Sp{ui}(e −H/T si) , (5.3) where symbol 〈...〉ph (i.e. indication “Sp{ui}”) stands for the quantum-statistical averaging with respect to quasiharmonic deviations ui (around quasiequilibrium positions ri). Note that representation (5.2) is based on the fact that variables ri(t) and ui(t) have distinct time scales: the large (i.e. “slow”) ri(t) and small (i.e. “fast”) ui(t) displacements correspond to the quasi-soliton (domain-wall) and phonon type excitations, respectively. 295 S.Stamenković For further consideration, similarly to the ansatz in subsection 4.2, it is conve- nient to represent the Hamiltonian (5.1) as a sum of two Hamiltonians describing the configurational (index c) and phonon (index ph) parts: H0 = Hc +Hph , (5.4) Hc = ∑ i [ p2i 2m − ai A 2 r2i + B 4 r4i ] + 1 4 ∑ ij ϕij(ri − rj) 2 , (5.5) Hph = ∑ i [−~ 2 2m ∂2 ∂u2i −∆i A 2 u2i ] + 1 4 ∑ ϕij(ui − uj) 2 . (5.6) The variational parameters ai and ∆i which take into account the coupling of two subsystems are determined (in the spirit of the Bogolyubov variational approach): ai = 1− 3(B/A)〈u2i〉0 , ∆i = 3(B/A)〈r2i〉0 − ai . (5.7) By the same variational approach, or directly from the equilibrium condition (for the system described by the initial model Hamiltonian (5.1)), 〈 i d dt Pi(t) 〉 ph = 〈[Pi, H]〉 = 0 , (5.8) equivalent to minimization of the free energy for a given configuration {ri}, ∂ ∂ri F ({ri}) = 0 ; F ({ri}) = −kBT ln Sp{ui}e −H/T , (5.9) one obtains the difference equation (compare with equation (3.12)): ηi[η 2 i − ai(T )] + ∑ j fij(ηi − ηj) = 0 ; fij = ϕij/A , (5.10) by which the reduced quasiequilibrium positions (i.e. the local order parameters) ηi = (B/A)ri are determined as site(i)-dependent functions of temperature. Thus, besides the zero solution ηi = 0, equation (5.10) can have a solution varying in space if the boundary conditions are chosen appropriately. To introduce phonons into an inhomogeneous system, it is necessary to perform the structural averaging over all the configurations {ri}. We shall restrict ourselves to the virtual (average)-crystal approximation in which the random one-particle potential is substituted by the average perturbation potential in the effective lattice invariant under translations: ∆ = 3(B/A)〈η2i〉 − a, a = 1− 3(B/A)〈u2i 〉 , (5.11) where the bar denotes the configurational average and index “0” being omitted here and further; the correlation function 〈u2〉 of displacements of the effective (“average”) lattice with the vibration frequency mω2 q = A(∆ + f0 − fq) ; fq = ∑ j fije iq(li−lj) , (5.12) 296 Unified description of structural transitions in accordance with equation (4.18) is given by 〈u2〉 = 1 mN ∑ q 1 2ωq coth ωq 2T . (5.13) Thus, the behaviour of phonons in the region Tc < T . T0 is determined by a self-consistent set of equations (5.10)-(5.13). We shall consider a quasi-one-dimensional case when the reduced displacement ηi depends only on the coordinate along certain chains, but the phonons in this case are three-dimensional. Such a model is quite specific in the sense that, as it will be demonstrated below, no ordered phase exists in it at finite temperatures. However, T = 0 can be considered as the critical temperature, while taking the temperature range T >0 to be a symmetric-phase region. a) The case of a weak coupling limit∗ In a weak–coupling limit (f0 ≪ 1), a transition of the order-disorder type at critical temperature Tc can be described by the effective Ising model [12] via pseu- dospin variables: ηi = σi √ a, σi = ±1. As the role of fluctuations is not significant in this case [5], i.e. < u2 >/(A/B) ∼ T/(A2/B) ≪ 1 at T ∼ Tc ∼ f0(A 2/B) ≪ A2/B, one estimates a ≈ 1, and, consequently, Tc ≪ T 0 c , T 0 c to be the critical temperature of a displacive SPT at which a(T 0 c ) ≡ 0 (compare Tc and T 0 c with equations (4.24) and (4.26), respectively). Furthermore, when the ground–state– quantum splitting within a local double–well potential plays a predominant role in the dynamics at low temperatures (e.g., when T ∼ Tc ≈ f0A 2/B ≪ U/kB), then the Ising model with a transverse field (IMTF) is more adequate, accommodating itself the additional “intra–well” degree of freedom [6]. b) The case of a strong coupling limit In a strong coupling limit (f0 ≫ 1) (we are mainly interested in throughout this subsection), a slow varying ηi evolves, e.g., \|ηi − ηj\| ∼ 1/f0 ≪ 1 (cf. equation (5.10)), so a the version ηi → η(x) in equation (5.10), is possible: η(x)[η2(x)− a(x, T )] + ∑ αβ cαβ∂ 2η(x)/∂xα∂xβ = 0 , (5.14) where elasticity moduli are introduced cαβ = ∑ j fij 1 2 (xαi − xαj )(x β i − xβj ) . (5.15) In a pseudo one–dimensional case (d = 1) (the only tractable analytically), the above equation (5.14) becomes c d2η(x) dx2 + η3(x)− a(x, T )η(x) = 0 , (5.16) ∗Henceforth, for convenience, the designations by bars are omitted 297 S.Stamenković where in the mean–field approximation a(x, T ) is determined by a(x) = 1− 3y(x) ; y(x, T ) = (B/A)〈u2〉 = (λ/2 √ ∆(x) + f0 ) coth(λ √ ∆(x) + f0 /2τ) , ∆(x) = 3 [ η2(x) + y(x) ]− 1 = 3η2(x)− a(x) , (5.17) where λ = ~ √ A/m/(A2/B) is a quantum parameter (equation (2.16)) and τ = T/(A2/B) is a reduced temperature. The system of equations (5.16),(5.17) differs from the corresponding ones in [30] as thermal (in addition to quantum) effects are taken into account herein. Hence, assuming homogeneous boundary equilibrium positions, η0 = η(x→ ±∞), one obtains a soliton solution: η(x) ≈ ± √ a(T ) tanh x ( √ 2ξ0)/ √ a(T ) , ξ0 = √ c = c0 √ m/Al2 , (5.18) where c is the elasticity modulus and c0 = l √ ϕ0/2m is the maximum soliton velocity (the velocity of sound in the system); η(x) describes a fixed wall between clusters, e.g. the transition region between stable particle (cell) positions r0 = ± √ aA/B. It is easily seen that phonon fluctuations renormalise (through a(T )) the soliton amplitude, η0 = √ a(T ) < 1, and its width, √ c/a(T ) > ξ0. The variation within a domain wall (x < ξ0) can be easily (iteratively) accounted for [30] but the form of the solution (5.18) is not substantially altered. From the demand for the homoge- neous solution η(x) = η (being the order parameter actually) to vanish identically and using equation (5.16) one estimates the phase transition temperature (i.e. of the soft–mode condensation, T 0 c ∼ (f0/3)A 2/B, since y ∼ TB/f0A 2). The inhomogeneous solution (5.18) also admits another type of a phase tran- sition corresponding to instability of the whole system with respect to differently oriented clusters (η(x) ∼ ±√ a). In that case the displacive SPT can be viewed as a transition of the order–disorder type, i.e. at T = Tc < T 0 c : η = 〈η(x)〉c = 0, but 〈η2(x)〉c ≈ a(Tc) 6= 0, symbol 〈...〉c denoting the average over all the clusters. This picture is dictated by non-linearity of the model. In addition, it is consistent with the results of the molecular dynamics methods from d=1 to d =4 dimensions (cf. [6,7,11]). To corroborate it more profoundly, in the next section we will consider the behaviour of the probability distribution function of quasiequilibrium positions P(η) in the limit T→Tc. 5.4. The (quasi)soliton correlations The time-dependent solution for η(y) is obtained from (5.18) with the aid of the Lorenz transformation, i.e. by the substitution y → (x− vt)/ √ 1− v2/c20, and corresponds to quasisoliton propagating along a chain with the velocity 0 6 v 6 c0. In the non-relativistic limit (v ≪ c0) the total energy E of such a soliton is composed [21] (see also [7]) of the potential (E0) and the kinetic (Ek = m∗v2/2) energies, respectively: 298 Unified description of structural transitions E = E0 + m∗v2 2 , E0 = A2 B 2 √ 2ξ0 3l a3/2 ; m∗ = 2 √ 2 3 m A/B lξ0 a3/2 , (5.19) where m∗ is the effective mass of the quasisoliton (the cluster wall) and l – the lattice constant. When \|x0i−x0,i+1\| ≫ ξ0, x0i is the domain-wall centre, i.e. when the density of the cluster walls is low, equation (5.16) has the approximate partial solution: ηNc (x, t; {x0i}, {vi}) = √ a Nc ∏ i=1 tanh x− x0i ∓ vit√ 2ξ/ √ a , (5.20) where ξ = ξ0 √ 1− v2/c20. Solution (5.20) describes a configuration consisting of Nc clusters. In the low temperature region T ≪ E0 the partition function of the system of cluster walls (5.20) assumed as a “gas consisting of quasiparticles (quasisolitons)” is given by (cf. [7]): Zc = exp [ L ( 2πT m∗D2 )1/2 e(µ−E0)/T ] , (5.21) where D is a normalizing factor in the phase space of the positions of cluster walls x0i and their velocities vi. By making use of the fact that the chemical potential of a quasisoliton gas µ = 0 and of expression (5.21), the density of the cluster walls is of the form: nc = Nc L = T L ∂ ∂µ lnZc = ( 2πT m∗ D2 )1/2 e−E0/T . (5.22) Now, let us consider the configurational averaging of quasiequilibrium posi- tions, after [31]. As to the initial condition at T =0, it is assumed that there exists one domain with η(x, 0) = η0 = √ a. At temperatures differing from zero, clusters with quasiequilibrium positions of different polarizations begin to appear in the system, so one finds: 〈η(x, 0)〉c = Z−1 c √ a = { √ a, T = 0 ; 0, T > 0. (5.23) Thus, the average value (thermodynamic and configurational) of the equilib- rium position of a particle in a one-dimensional model (5.1) is equal to zero at any nonzero temperature. The average squared values of the equilibrium positions are calculated likewise: 〈η2(x, 0)〉c = ae−2ncξ0 √ 2/ √ a . (5.24) The adopted procedure for configurational averaging also permits one to obtain the dynamic correlation function as follows (cf. [7,31]) : 299 S.Stamenković 〈η(x, t)η(0, 0)〉c = a exp [ − 2e E0 T ∫ ∞ −∞ dv D e− m∗v2 2T (x∓ vt) coth x∓ vt√ 2ξ0/ √ a ] . (5.25) The static correlation function that follows from (5.25) at t = 0 upon integra- tion over the velocities acquires the form: 〈η(x, 0)η(0, 0)〉c = a exp [ −nc2x coth x√ 2ξ0/ √ a ] . (5.26) When x≫ ξ0/ √ a, expression (5.26) can be presented in the form obtained by the transfer-matrix method [21], 〈η(x, 0)η(0, 0)〉c = ae−x/ξc , (5.27) where ξc = 1/2nc is nothing but the correlation length in the system (as T→Tc=0, ξc→∞). 5.5. The soft mode and the central peak In the preceding sections lattice vibrations were mainly considered in the clas- sical (high-temperature) limit. At low temperatures (or small effective mass) it is necessary to take into account zero-point (quantum) vibrations. The quantum properties of the model are characterized by the parameter λ ∼ ~/ √ m (see equa- tions (2.16), (4.30)), which plays the same role as temperature: when the coupling constant f0 is fixed, the system may become unstable with respect to the prop- agation of both phonons (ω2 qs < 0) and quasisolitons (E < 0), if λ > λc, even if T =0. The effect consisting in the suppression of a phase transition by quantum fluctuations is well-known. For example, the absence of a ferroelectric transition in SrTiO3 at normal pressures is most likely due to the above said effect. Let us find the critical parameters determining the limit imposed on the dy- namic stability of the system, which requires the phonon energy (5.12) and the soliton energy (5.19) to turn to zero. We introduce the dimensionless variables: Ω 2 q = ω2 q A/m ; ε0 = E0 A2/B ; θ = T A2/B ; λ = ~ √ A/m A2/B (5.28) and rewrite equation (5.13) in the form: B A 〈u2〉 = λ 2 ∫ ω2 m 0 g(ω2)dω2 √ ∆+ ω2 coth λ √ ∆+ ω2 2θ , (5.29) where the phonon frequency density g(ω2) = (1/N) ∑ q δ(f0 − fq − ω2) (5.30) is introduced for ω2 6 ω2 m = 2f0(A/m). 300 Unified description of structural transitions Consider a phase transition in the centre of the Brillouin zone at q=0. Then, in the symmetric phase, in the absence of clusters (when 〈η2〉c = 0) the condition for determining the critical parameters has the form 3B〈u2ra/A = 1, from which for the Debye spectrum g(ω2) = 3ω/ω3 m one obtains: λ0(T =0) ≈ √ f0 ; θ0 = T0(λ→0) A2/B ≈ f0 9 ; ε0(λ→0, T→0) ≈ √ f0 . (5.31) Thus, application of the presented approach is limited by the following values of the model parameters λ0 and f0 and the reduced temperature: f0 ≫ 1 ; λ0 ≪ √ f0 ; 0.05 √ f0 < θ ≪ √ f0 . (5.32) The temperature behaviour of the phonon energy is determined by a self- consistent set of equations (5.11)-(5.13), (5.19), (5.22), (5.24), and (5.26). The numerical solution of this set of equations for the model parameters f0 = 10 and λ0=0.1 [7,31] shows that for θ>θ0≈0.5, the soft mode behaves as in the classical theory. When θ ≈ θ0, the “high-temperature” phonons (phonons in the symmetric phase) become unstable, which is related to the appearance of clusters (quasisoli- tons) in the system. In the presence of clusters the phonons behave like “low tem- perature” phonons (phonons in the nonsymmetric phase). As θ→θc=0, instead of decreasing as in the classical theory, the soft mode frequency increases and tends to a finite value. A qualitative result provides the observed incomplete softening of the soft(critical) mode as a consequence of inhomogeneity of the system which leads to a nonzero value of the averaged square equilibrium positions (〈η2〉c 6= 0). At the same time the order parameter (for example, spontaneous polarization in the case of ferroelectrics) that is proportional to the average displacement < η >c remains equal to zero till θc = 0. Now let us consider the behaviour of the central peak as an intrinsic feature of the critical (SPT) dynamics apart from the revealed in the spectral density func- tion. In various scattering experiments the scattering function S(q, ω) is measured as one being expressed through the dynamic correlation function. In the case of quasisolitons it is represented as the double (in q and ω) Fourier transformation of the correlation function (5.25). Within the framework of the variational approach and approximations applied, the scattering function of the SPT-system involves three singled peaks–two sharp side peaks corresponding to phonon energies (5.12) and a central one at ω = 0, due to the dynamics of the cluster walls (quasisoli- tons). As to the quasisolitons (assumed as a “gas of quasiparticles”), the two limit regimes of their behaviour are feasible: collisionless and diffusive. We shall not comment on the collisionless regime (otherwise corresponding to a free motion of quasisolitons) as it is of a less physical interest. Diffusive behaviour itself corre- sponds to the asymptotic t→ ∞ in the Fourier (q, ω) - transformed correlation function (5.25). Assuming the correlation function (5.25) to be independent of the coordinate within the limits of the correlation length ξc and integrating (in x and 301 S.Stamenković t) its Fourier(qω) component, the scattering function in the critical region (index cr) is estimated as follows [7,31]: Scr(q, ω) ≈ a A B σλ(q) π γ ω2 + γ2 , γ = 4T m∗D e−E0/T , (5.33) where σλ(q) is an approximate space transformation of the correlation function within the correlation-length limits. Thus, in the case of a simple diffusion of the cluster walls there should be observed, in the scattering function, a central peak of the Lorentzian form and half-width Γ0 ≈ γ (Γ0 ≈ v/ζ , where v is the mean velocity of the walls and ζ is the mean distance between them). As the temperature decreases, the height of the central peak increases, while its half-width decreases. In the limit, as T→0, Scr(q, ω) (5.33) transforms into a δ-function. Being analytically intractable, the proper scattering function for (quasi)solitons (index c) Sc(q, ω) versus temperature is obtained by a numerical integration (cf. [7]). For the chosen parameters of the model (f0 = 10, λ = 0.1) and q 6= 0(q/l = 0.016), the intensity of the central peak first increases as T decreases (starting from T0) and then starts decreasing at T → 0. At q=0 the central peak has the same form as at finite q, but its intensity increases as T → Tc, while its half-width(Γ0) decreases. Therefore, it follows that for finite q, in an ideal (defectless) lattice (in the symmetric phase, T >0), the contribution of moving cluster walls to the central peak decreases as T→Tc=0. But the peak becomes a central (intensive) one when q→ 0. Thus, the analytical investigation of the one-dimensional model dynamics presented in this subsection confirms the qualitative picture and gives details of the phase transition which follow from the analysis based on the renormalization group method, as described above. Below we outline the application of the RG- method to explain, rather analytically and in part quantitatively, the pronounced influence of cluster excitations on phonons and pseudomagnons, i.e. their impact on critical (SPT) dynamics in all. 5.6. RG - predictions and domain-wall relaxation As it is evident from equation (5.10), the probability distribution function of quasiequilibrium positions, P (η), is described by the configuration-potential energy of the system (see equation (5.1)). In the RG–analysis (index rg) it is convenient to use the dimensionless quantities (r = a/c, g = 1/4c), so in the continuum version Prg(η) = Z−1 exp[−Hrg(η)] , (5.34) where Hrg(η) = βUrg(η) = 1 2 ∫ ddx{rη2(x) + (∇η(x))2 + 2gη4(x)}, β = c/kBT, (5.35) is the standard Landau–Wilson Hamiltonian. As it is well known, at the critical point (Tc) the stochastic field {η(x)} (and correspondingly its distribution P (η)) is scaling–invariant, thus describing a dis- tribution of large–size clusters. It is defined by the fixed point (r∗, g∗) of the 302 Unified description of structural transitions RG–transformation (R), RPr∗g∗(η) = Pr∗g∗(η) , (5.36) wherefrom the first order in ǫ ≡ (4− d) expansion yields (cf. [6] and Refs. therein) r∗ = −n+ 2 n+ 8 ε+ O(ε2) ; g∗ = 8π2 n+ 8 ε+ O(ε2) ; n = d, (5.37) where n is the number of components of the order parameter. For d 6 3, a∗ = a(Tc) =−r∗c > 0 and P(η) is non–Gaussian (as distinct from [28], only due to the relaxation dynamics of ηi). Keeping in mind that the soft– mode frequency Ωq in equations (5.12), (5.28) is given by the pole of the ordinary phonon Green function (ω→0)(cl.[7]), Gq(ω) = A ∑ j eiq(li−lj)〈〈ui\|uj〉〉ω = {(m/A)ω2 − [∆ + f0 − fq +Mq(ω)]}−1 , (5.38) where Mq(ω) is a self–energy operator, a straightforward inspection of limits T→ T 0 c , q→0 yields Ω2 q ≈ Ω2 0 + c2q2 ; Ω2 0 = ∆+M0(0) . (5.39) Thus, while in the homogeneous case, as usual Ω0(T → T 0 c ) → 0, for d 6 3 – on account of nonlinear effects (equations (5.16), (5.37)) – one finds Ω2 0(T → Tc) → 2〈η2〉c = 2a → −2r∗c > 0, Tc<T 0 c . As a consequence, when T → Tc the displacive phase transition turns to be of the order–disorder (Ising) type governed by the relaxation cluster dynamics, i.e. 〈ηi〉c = 0, but 〈η2i 〉c ∼ −r∗c 6= 0, and a complete phonon softening, i.e. a long–homogenous order (ηi = η ≡ 〈ηi〉 = 0), fails to occur. To outline the critical cluster dynamics we use the phenomenological equation of Landau–Khalatnikov in the coordinate space (see, e.g., [32]), ∂τη −∇2η + rη + 4gη3 = 0, (5.40) where τ = Γt, Γ is a “kinetic” phenomenological parameter. For d=1 this equation has a soliton solution: η = f(x−vτ, τ) in the form (\| v \|≫ 1 or \|△η\| ≪ 1; ζ = x−vτ) f 2(ζ, τ) = −r/[4g + exp(−2ζτ/v)] (5.41) which describes the destruction of clusters arising as fluctuations around Tc. The spectrum of ηq–fluctuations in the “harmonic” approximation (if g = 0) is given by [33]: Iq(ω) = ∫ +∞ −∞ dte−iωt〈ηq(0)η−q(t)〉 = kBT r + q2 2τq 1 + (ωτq)2 = 2Γ(kBT ) τ−2 q + ω2 , (5.42) where the correlation time, τq = [Γ(r + q2)]−1; r ∼ a ∼ (T − T 0 c ), (5.43) and the reduced wave vector q are introduced. 303 S.Stamenković By performing calculations for nonlinear effects (g 6= 0), using ansatz [32], the asymptotic spectral response (i.e. when q→ 0, ω ≪ 1, T → Tc) exhibits a central peak behaviour, thus reflecting a slow relaxation of cluster walls: I(q=0, ω) ∼ kBTc π Γ ω1+(2−η)/∆ω . (5.44) The combination of the small correlation critical (η) and dynamical (∆ω) in- dices in equation (5.44) is given by the fixed point–parameter (g∗), i.e. by the ε–expansion (cf. [6,32] and Refs. therein): 2− η ∆ω = 1− (g∗)2 8π4 9 ln(4/3) ≈ 1− ε2 2(n+ 8) ln(4/3) . (5.45) To conclude, our theoretical findings based on the decomposing picture on “slow” and “fast” dynamical variables (equation (5.2)) show that (in accord with the previous results, i.e. with various above mentioned approaches and suitable experiments (cf. [7,11])) at some (“true”) critical temperature, Tc = T 0 c − δTc, δTc ∼ εT 0 c , the traditionally displacive system becomes structurally unstable against nonlinearly induced precursor clusters (i.e. by their order–disorder de- scribed by Ising–like variables ηi), so that 〈ηi(T→Tc)〉 → 0 without a complete phonon softening: Ωq(q→0, T→Tc)→ 2〈η2i (Tc)〉 = 2a∗ ∼ ε. These clusters relax in a peculiar (dynamical) fashion: Iq(ω→0, q→0, T→Tc) ∼ ω−λ, λ . 2. It should be gratifying that the present description can, at least in part, con- tribute to a better understanding of the SPT-dynamics, and to corroborate the appealing idea about the intrinsical “order–disorder origin” of universality. 5.7. The pseudospin dynamics: the limiting order–disorder transition and universal features of SPT So far in the unified model description of SPT, i.e. in the corresponding self- consistent procedure exposed in the preceding sections, the dynamics of a pseu- dospin subsystem as described by the transverse Ising model (equation (4.6)) has not been taken into account unless intermediary through the average pseudospin components σx and σz(equation (4.21)) in the mean-field approximation. In the case of the order-disorder SPT the soft-phonon picture of critical dynamics has to be replaced by the concept of a tunnelling mode to a great extent resembling the phonon soft mode as the instability (critical) temperature is approached. Conse- quently, to be properly considered, the searching for the true, universal nature of the SPT, i.e. the real physical systems undergoing structural transitions, has to take into account, in an equal manner, all the intriguing features of critical dy- namics comprising phonons, tunnelling(magnon-like) and domain-wall excitations. Thus, it has surfaced an important conclusion that the system can be considered to have not one, but three characteristic frequencies: the pseudoharmonic frequency within either side of a double well, the tunnelling frequency characterizing inter- well motion and a third one corresponding to domain-wall excitations (moving 304 Unified description of structural transitions domain walls), by which the order parameter changes from one potential mini- mum to another inside the well defined distance [34,35]. Moreover, inside these slow moving clusters the particles participate in a quantum-mechanical flippings between their momentary rest positions ri(t) (equation (5.3)) which are described by the pseudospin (i.e. transverse Ising) Hamiltonian (4.6). Within the framework of a classical-quantum approach proposed recently [35], the pseudospin dynamics (as a counterpart of the displacive type dynamics) can be treated in an almost analogous way. In the spirit of the preceding approach and definitions (section 4 and subsection 5.6), as well as introducing the representation of decomposed variables of the form (5.2), the pseudospin components can be represented as follows: Sα i = σα i + χα i ; (5.46) σx i = 〈Sζ i 〉 sin θi cosϕi ;χ x i = δSζ i sin θi cosϕi + Sξ i cos θi cosϕi − Sη i sinϕi , σy i = 〈Sζ i 〉 sin θi sinϕi ;χ y i = δSζ i sin θi sinϕi + Sξ i cos θi sinϕi + Sη i cosϕi , σz i = 〈Sζ i 〉 cos θi ; χz i = δSζ i cos θi − Sξ i sin θi . (5.47) Here ϕi, ψi = 0 and θi are the Euler angles after the unitary transformation in the pseudospin space Oxyz. Apparently, new temperature dependent axes 0ξηζ are such that the mean value 〈Si〉{χα i } is directed along 0ζ-axis∗. In the scheme of the local (site dependent) mean field approximation (LMFA) [35] angle θi is given by (see subsection 4.3) sin θi = 2Ω/Hi, H2 i = (2Ω)2 +H2 iz, H2 iz ≃ (4J0σ z i ) 2, (5.48) while angle ϕi is expressed via θi [36], cosϕi = 1 2λ sin θi {β2 + [(1− β2)(4λ2 sin2 θi − β2)]1/2} ; λ = 1 2 sin θ0 , (5.49) where sin θ0 corresponds to a monodomain at given temperature and β = v/c0 is the velocity of the domain-wall (in units of its maximum value c0 = 2Ωl/~ √ 2, l- the lattice constant) which evolves in the pseudospin space. The corresponding quasisoliton solution σz(x, t) is of the same form obtained for η(x, t)(5.18) (by using the Lorentz transformation, x→y) as a consequence of the equilibrium condition i.e. minimization of the free energy of the pseudospin subsystem for a given configuration {σα i }: ∗For simplicity we omitted the averaging index associated with magnon-like variables {χα i } in equation (5.47). 305 S.Stamenković ∂ ∂σz i F ({σα i }) = 0 ;F ({σα i }) = −kBT ln Sp{χα i }e −βHIMTF (5.50) where HIMTF is the pseudospin Hamiltonian (4.6). In the critical temperature region (T ∼ Tc), in the LMFA-approximation, the above equation (5.50) acquires the same form (5.16) in which the following iden- tifications have to be carried out [35] (cf. also [11]): η(x, t) ≡ 2σz(x, t) ; A ≡ 2 [ βcΩ sinh(βcΩ) ]2 kBTc , B ≡ J3 0 Ω2 [1− βcJ0 cosh2(βcΩ) ] ; βc = 1/kBTc ; Tc ∼ J0 . (5.51) By introducing the quantum (flipping) effects in the effective (average-crystal) lattice (the bar symbol being associated with such an approximation)-through the boson-like representation, δSζ i = −a+i a−i , S± i = √ 2 〈Sζ i〉 a∓i , (5.52) or by directly using the adopted pseudospin Green function [7,19], Gξξ q (ω) = − H 〈Sζ〉 ω2 − ω2 q , (5.53) the frequency of the pseudospin (tunnelling) mode is expressed in the well-known but slightly modified form [7,11,33-35]; ωq 2 = H2(1− Jqsin 2 θ/J0) ; H2 = (2Ω)2 + (4J0) 2〈(σz(x))2〉c . (5.54) It is easily seen that the results analogous to the case of displacive SPT (elab- orated in the previous subsection) are obtained: with the onset of criticality an incomplete softening of the pseudospin (tunnelling) mode occurs, ω2(q→0, T→Tc) ≃ (4J0) 2〈(σz)2〉c 6= 0 , (5.55) while in the spectral density of excitations a central peak of the form (5.44), S(q = 0, ω) ∼ ω−λ, λ . 2, (5.56) evolves as a consequence of the pseudospin domain-wall relaxation. Instead of extended conclusions usually presented at the end of each section, it is gratifying that the present contribution can, at least in part, complete a better understanding of the SPT dynamics and to corroborate, the more and more appealing idea about the intrinsical order-disorder nature of SPT as its essential universal feature. Although invalid in quantitative predictions, our unified model 306 Unified description of structural transitions description of SPTs (bearing in mind simple approximations used) is illuminating, as Drs. Bruce and Cowley, great specialists in this field, stated in [11]. We hope that the theoretical results presented in this review have “displaced” and contributed to the “order” of some aspects of knowledge about the univer- sal nature of SPTs, as well as to a prompt further progress in all-inclusive SPT investigations. 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C: Solid State Phys. 1981, vol. 14, 3231. 308 Unified description of structural transitions Уніфікований модельний опис структурних фазових переходів типу порядок-безпорядок та типу зміщення С.Стаменкович Інститут ядерних наук, Лабораторія теоретичної фізики та фізики конденсованої речовини, Бєлград, Пошт. Скр. 522, Югославія Отримано 7 липня 1998 р. Зроблено огляд і часткове оновлення серії робіт автора з співавтора- ми [1-7], присвяченої уніфікованому модельному опису структурних фазових переходів (СФП) у сегнетоелектриках та споріднених ма- теріалах. Виходячи з загального гамільтоніану попарно зв’язаних ангармоніч- них (четвірних) осциляторів та враховуючи концепцію локальних нор- мальних координат, запропоновано уніфікований модельний опис систем з СФП обидвох типів: порядок-безпорядок і зміщення. В рамках стандартної варіаційної процедури формулюється гібри- дизований псевдоспін-фононний гамільтоніан введенням змінних, що відповідають фононним, магноно-подібним (з переворотом) і не- лінійним (типу доменної стінки) зміщенням атомів, що беруть участь у СФП. Це досягається представленням колективного руху атомів че- рез декілька квазірівноважних положень (в найпростішому випадку - двох) як повільного тунельного зміщення (розбитого на магноно- подібні і солітоно-подібні відхилення) додатково до порівняно швид- ких фононних осциляцій навколо неоднорідних миттєвих положень спокою, в свою чергу індукованих збудженнями типу доменної стінки (солітонами). Якісний і кількісний аналізи показали, що СФП (першого чи дру- гого роду) можуть бути типу зміщення (керованих м’якою фонон- ною модою), порядок-безпорядок (керованих тунельною магноно- подібною м’якою модою) чи змішаного типу залежно як від енергії зв’язку між атомами, так і від їхньої нульової точки коливної енергії. В області критичної температури збудження типу доменної стінки при- водять до утворення мікродоменів (кластерів впорядкованої фази), які індукують СФП класу універсальності Ізінгівського типу. Має міс- це неповне пом’якшення фононної чи псевдомагнонної моди і у спек- тральній густині збуджень виникає центральний пік, викликаний по- вільною релаксацією кластерів. Ключові слова: структурні фазові переходи, порядок-безпорядок, перехід зміщення PACS: 63.70.+h, 77.80.Bh, 64.60.-i, 64.60.Cn 309 310

The unified model description of order-disorder and displacive structural phase transitions

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