Kinetic equations for the pseudospin model with barriers

A new formalization of Glauber method is developed and applied to the pseudospin model with barriers. Kinetic equations are derived for this model and numeric solutions in simplest approximations are obtained. Relaxation and kinetic properties of the model are shown to depend on the barrier value as...

Ausführliche Beschreibung

Gespeichert in:

Bibliographische Detailangaben
Datum:	2012
1. Verfasser:	Kovarskii, V.L.
Format:	Artikel
Sprache:	English
Veröffentlicht:	Донецький фізико-технічний інститут ім. О.О. Галкіна НАН України 2012
Schriftenreihe:	Физика и техника высоких давлений
Online Zugang:	http://dspace.nbuv.gov.ua/handle/123456789/69531
Tags:	Tag hinzufügen Keine Tags, Fügen Sie den ersten Tag hinzu!
Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Zitieren:	Kinetic equations for the pseudospin model with barriers / V.L. Kovarskii // Физика и техника высоких давлений. — 2012. — Т. 22, № 1. — С. 14-24. — Бібліогр.: 12 назв. — рос.

Institution

Digital Library of Periodicals of National Academy of Sciences of Ukraine

id	irk-123456789-69531
record_format	dspace
spelling	irk-123456789-695312014-10-17T03:01:46Z Kinetic equations for the pseudospin model with barriers Kovarskii, V.L. A new formalization of Glauber method is developed and applied to the pseudospin model with barriers. Kinetic equations are derived for this model and numeric solutions in simplest approximations are obtained. Relaxation and kinetic properties of the model are shown to depend on the barrier value as well on the heatingcooling rate. Heating-cooling cycles reveal hysteresis. The relaxation times are determined by the temperature and the barrier value. The relaxation time for the structural order parameter Sz possesses two vertical asymptotes: the first one caused by phase transition, and the second one determined by slowing kinetics at low temperatures. Разработана новая формулировка метода Глаубера, дано ее приложение к псевдоспиновой модели с барьерами. Получены кинетические уравнения для этой модели, а также численные решения в простейших приближениях. Показано, что релаксация и кинетические свойства модели зависят от энергетического барьера и скорости нагрева–охлаждения. Циклы нагрева–охлаждения обнаруживают гистерезис. Времена релаксации определяются температурой и величиной энергетического барьера. Время релакации для структурного параметра порядка Sz обладает двумя вертикальными асимптотами: первая обусловлена фазовым переходом, вторая – замедлением кинетики при низких температурах. Розроблено нове формулювання методу Глаубера, яке застосовано до псевдоспiнової моделі з бар’єрами. Отримано кінетичні рівняння для цієї моделі, а також чисельні рішення у найпростіших наближеннях. Показано, що релаксація й кінетичні властивості моделі залежать від енергетичного бар’єру й швидкості нагрівання–охолодження. Цикли нагрівання–охолодження виявляють гістерезіс. Часи релаксації визначаються температурою та величиною енергетичного бар’єру. Час релаксації для структурного параметра порядку Sz має дві вертикальні асимптоти: перша обумовлена фазовим переходом, друга – уповільненням кінетики за низьких температур. 2012 Article Kinetic equations for the pseudospin model with barriers / V.L. Kovarskii // Физика и техника высоких давлений. — 2012. — Т. 22, № 1. — С. 14-24. — Бібліогр.: 12 назв. — рос. 0868-5924 PACS: 05.20.Dd http://dspace.nbuv.gov.ua/handle/123456789/69531 en Физика и техника высоких давлений Донецький фізико-технічний інститут ім. О.О. Галкіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	A new formalization of Glauber method is developed and applied to the pseudospin model with barriers. Kinetic equations are derived for this model and numeric solutions in simplest approximations are obtained. Relaxation and kinetic properties of the model are shown to depend on the barrier value as well on the heatingcooling rate. Heating-cooling cycles reveal hysteresis. The relaxation times are determined by the temperature and the barrier value. The relaxation time for the structural order parameter Sz possesses two vertical asymptotes: the first one caused by phase transition, and the second one determined by slowing kinetics at low temperatures.
format	Article
author	Kovarskii, V.L.
spellingShingle	Kovarskii, V.L. Kinetic equations for the pseudospin model with barriers Физика и техника высоких давлений
author_facet	Kovarskii, V.L.
author_sort	Kovarskii, V.L.
title	Kinetic equations for the pseudospin model with barriers
title_short	Kinetic equations for the pseudospin model with barriers
title_full	Kinetic equations for the pseudospin model with barriers
title_fullStr	Kinetic equations for the pseudospin model with barriers
title_full_unstemmed	Kinetic equations for the pseudospin model with barriers
title_sort	kinetic equations for the pseudospin model with barriers
publisher	Донецький фізико-технічний інститут ім. О.О. Галкіна НАН України
publishDate	2012
url	http://dspace.nbuv.gov.ua/handle/123456789/69531
citation_txt	Kinetic equations for the pseudospin model with barriers / V.L. Kovarskii // Физика и техника высоких давлений. — 2012. — Т. 22, № 1. — С. 14-24. — Бібліогр.: 12 назв. — рос.
series	Физика и техника высоких давлений
work_keys_str_mv	AT kovarskiivl kineticequationsforthepseudospinmodelwithbarriers
first_indexed	2025-07-05T19:03:35Z
last_indexed	2025-07-05T19:03:35Z
_version_	1836834845411508224
fulltext	Физика и техника высоких давлений 2012, том 22, № 1 © В.Л. Коварский, 2012 PACS: 05.20.Dd V.L. Kovarskii KINETIC EQUATIONS FOR THE PSEUDOSPIN MODEL WITH BARRIERS A.A. Galkin Donetsk Physico-Technical Institute of the Ukrainian Academy of Sciences R. Luxembourg Str., 72, Donetsk, 83114, Ukraine E-mail: kvl29@mail.ru Received January 26, 2011 A new formalization of Glauber method is developed and applied to the pseudospin model with barriers. Kinetic equations are derived for this model and numeric so- lutions in simplest approximations are obtained. Relaxation and kinetic properties of the model are shown to depend on the barrier value as well on the heating- cooling rate. Heating-cooling cycles reveal hysteresis. The relaxation times are determined by the temperature and the barrier value. The relaxation time for the structural order parameter Sz possesses two vertical asymptotes: the first one caused by phase transition, and the second one determined by slowing kinetics at low temperatures. Keywords: structural disorder, kinetic equations, Ising model, pseudospin model, phase transitions Introduction The pseudospin model can be used for a wide class of the objects which repre- sent structural disorder and order-disorder transitions. These are Jahn–Teller crystals, hydrogen bonded crystals, molecular crystals, binary alloys, lattice gas and lattice model of fluid. A lot of papers have been published about this ap- proach since pioneer works of J.S. Slater [1] and P.G. De Gennes [2]. Unfortu- nately, this model regards only two states at the bottoms of the potential minima and ignores all others. It’s really correct as an approximation at the temperatures kT << U0, where U0 is the value of the energy barrier between the minima, but it excludes possibility of description of the kinetic properties and such phenomena as pretransition slowing of kinetics and metastable glass-like frozen disorder, be- cause the states of the over-barrier motion are essential in these cases. In order to include over-barrier states to the pseudospin framework, we developed the Ising model with barriers [3]. The main results of this work are: three-component pseu- Физика и техника высоких давлений 2012, том 22, № 1 15 dospin 1 0 0 0 0 0 0 0 1 ziS ⎛ ⎞ ⎜ ⎟= ⎜ ⎟ ⎜ ⎟−⎝ ⎠ (instead of usual two-component one), where Szi = 0 corresponds to the states of the over-barrier motion, and the effective Hamilto- nian: 21eff 0 0 1 ln ln 2 zi zj ziij i j i zH J S S T S TN z z≠ ⎛ ⎞ = − − −⎜ ⎟ ⎝ ⎠ ∑ ∑ , (1) where 1 1 0 exp ( )dz z z g+ − ε⎛ ⎞= ≡ = − ε ε⎜ ⎟θ⎝ ⎠∫ , 0 1 exp ( )dz g ∞ ε⎛ ⎞= − ε ε⎜ ⎟θ⎝ ⎠∫ are partial sta- tistical integrals taken over the states within the potential wells and over the states of the cross-well motion respectively, 0 E U ε = is the reduced energy, U0 is the energy barrier between the minima, E is the energy, g(ε) is the density of states, 0 kTt U = is the reduced temperature, ijJ are the pair interaction con- stants, T is the absolute temperature, N is the number of lattice sites. Let’s notice that we are currently considering the classical mechanics case, so all matrix operators commute with each other and should be regarded just as pseudospin variables which can take values from the set of their eigenvalues. We have chosen such representation because of benefits of matrix represen- tation and in order to have a good start for extending the results to the quan- tum mechanics case. This model was shown by means of computer simula- tions really to reveal pretransition kinetics slowing and glass-like metastable state of the frozen disorder [3]. By the way, these results are in accordance with frustration-volumetric theory of glass by A.S. Bakai [4], where glassy state is also regarded as a metastable phase, but not as an unstable very slowly evolving one. Further investigations were performed on the four-sublattice model in order to apply the results to interpretation of the Mössbauer investigations of the Jahn– Teller crystal of Cu(H2O)6·SiF6 [5,6]. The state of metastable disorder and the equilibrium phase transition for a four-sublattice model have been studied in this work. These results were used for the theoretical explanation of the statics– dynamics transformation observed in the disordered phase of this crystal when the structural phase transition point is approached [7]. The next step of these investigations is studying of kinetic properties. So, the aim of this paper is obtaining of corresponding kinetic equation from the model (1). The basic approach was founded by Glauber [8]. But our case dif- fers from the original by two means: a) multicomponent pseudospin; b) transfer rules: direct jumps between wells are forbidden because of classical character of Физика и техника высоких давлений 2012, том 22, № 1 16 our model, only over-barrier – well transfers and vice versa are allowed. In order to fit these conditions, we have to reformulate Glauber approach. It should be no- ticed that the same approach was used by Vaks with coworkers in their investiga- tions of kinetic properties of alloys [9–12]. There are five steps, which we firstly illustrate on traditional two-component model and then apply to our model with barriers. 1. Ising model 1. We start from the master equation in a general form: { }( ) ( ) { }( ) ( ) { }( )d , d j zj zj j zj zj j p S w S p S S w S p S S⎡ ⎤= − + − −⎢ ⎥⎣ ⎦τ ∑ where { }S is a certain configuration of pseudospin values at each site { , , , }z i z jS S… … … , ( ), ( )j z j j z jw S w S− are the transition probabilities for the changes ,z j z j z j z jS S S S− → →− respectively at the site j, ({ }), ({ })z j z jp S S p S S− are the probabilities of the pseudospin values ,z j z jS S− respectively at the site j in the given configuration { }S . 2. Demand of detailed balance principle leads to a definite form of the transi- tion probabilities: 0 0( ) ({ }) ( ) ({ }) 0,j z j z j j z j z jw S p S S w S p S S− + − − = ⇒ 0( ) ({ }) exp( β )j z j z j j z jw S p S S h S⇒ ∼ − ∼ − , where β 1/ kT= , j j i z i j i h J S ≠ =∑ – molecular field at the site j . 3. Operators ˆ ˆ,1zS form the basis of the matrix algebra, and all functions of them are expressed through the linear combinations of this basis: 2ˆ ˆ ˆ ˆ ˆ1,exp( ) cosh sinh cosh (1 tanh )z z z zS aS a S a a S a= = + = + . 4. Using statements 2 and 3, one can express the transfer probabilities ( 0τ is a phenomenological constant): ( ) ( ) 0 1 1 tanh 2j zj zj zj jw S S S h′ ′→ = − β τ . 5. Using 4, one can calculate the mean values and finally obtain the well known kinetic equation: { } { }( ) ( ) 0 dd 1 tanh b d d t t zl zl zl j S p SS S S h= = − − τ ∑ . Физика и техника высоких давлений 2012, том 22, № 1 17 2. Ising model with barrier We can generalize now this approach on Ising model with barriers [3]. 1. Master equation: { }( ) ( ) { }( ) ( ) { }( ) d d zj j zj zj j zj zj j S p S w S S p S w S S p S ′ ⎡ ⎤′ ′ ′= − → + →⎣ ⎦τ ∑∑ where ( )j zj zjw S S ′→ is a probability of a transition from the spin value Szj to the spin value zjS ′ at the site j. 2. Demand of the detailed balance principle: ( ) 0 0 2 0 ( ) ({ }) ( ) ({ }) 0 ( ) ({ }) exp β β ( ) , j j j j j j j j S S j j j j z j z j w S S p S w S S p S w S S p S h S h S ′ ≠ ⎡ ⎤′ ′ ′− → + → =⎣ ⎦ ⇓ ′ ′ ′ ′→ ∼ ∼ + ∑ where j ji zi j i h J S ≠ =∑ is the molecular field at the site j, 1 0 ln zh T z ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ is the effec- tive energy barrier. 3. Rules of the matrix algebra: basis 2ˆ ˆ ˆ, ,1z zS S , operations: ( )22 2 2ˆ ˆ ˆ ˆ ˆ,z z z z zS S S S S= ⋅ = . 4. Form of the matrix of transfer probabilities: ( )21( ) 1 exp(β )sinh(β ) ( ) exp(β ) cosh(β ) 1 2τS S z j j z j j S S w j S h h S h h′ ′ ⎤⎡ ′ ′= + + −⎣ ⎦ , where phenomenological matrix 1 SS′τ defines accordingly transfer rules: 10 0 τ 1 1 10 τ τ τ 10 0 τ S S + ′ + − − ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠ , zero values of non-diagonal elements do satisfy our transfer rules here; finally one obtains: 10 0 τ 1( ) 0 2 τ τ 10 0 τ SS e ew j + + − ′ + − − ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠ , where expβ( ), expβ( )j je h h e h h+ −= + = − . Физика и техника высоких давлений 2012, том 22, № 1 18 5. Calculating the mean values with ( )SSw j′ from statement 4 yields: ( ) { } { } { } 2 2 d ({ }) d d ( ) ({ }) ( ) ({ }) ( ) ({ }) 1 1 1 1 1 1 1 , 2 τ τ 2 τ τ τ τ l l l z l z l S z l S S S S S j S S S l l S S z l z l z l S dp SS t t S w j p S w j p S w S S p S e eS S S ′ ′ ′ ′ ′ + − + − + − + − = = ⎡ ⎤′= − + =⎣ ⎦ ′= − + = ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ = − + − − + − −⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ∑ ∑ ∑∑ ∑∑ ( ) 2 2 { } 2 { } 2 2 { } 2 2 d ({ }) d d ( ) ({ }) ( ) ({ }) ( ) ({ }) 1 1 1 1 1 1 1 . 2 τ τ 2 τ τ τ τ l l l z l z l S z l S S S S S j S S S z l z l S S z l z l z l S dp SS S w j p S w j p S w S S p S e eS S S ′ ′ ′ ′ ′ + − + − + − + − = = τ τ ⎡ ⎤′= − + =⎣ ⎦ ′= − + = ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ = − + − − + + −⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ∑ ∑ ∑∑ ∑∑ As a result, in the molecular field approximation and assuming 0τ τ τ− += ≡ , we obtain the kinetic equations: ( ) ( ) ( )( ) ( ) ( ) ( )( ) 2 0 2 2 2 0 1 2 exp β sinh β 1 τ τ 1 2 exp β cosh β 1 τ τ zi zi i zi zi zi i zi S S h h S S S h h S ∂ ⎫ = − − − ⎪∂ ⎪ ⎬ ∂ ⎪ = − − − ⎪∂ ⎭ , (2) where τ is a time in relative units, 0/t kT U= is the reduced temperature. Let’s consider the limit passage. At the big barrier limit we have 1hβ >> , hence ( ) ( ) 2 exp β 1 2cosh βzi i h S h − ≈ − and ( )( ) 0 1 tanh β τ τ zi zi i S S h ∂ = − − ∂ , that is the well known Glauber result for the two-component model. So, the correspondence principle is valid in this case. 3. Three-component model with barrier Static behaviour of this model was investigated in our previous works [5,6]. The structure parameter is represented by the vector matrix (1) Физика и техника высоких давлений 2012, том 22, № 1 19 3 2 0 0 0 0 0 0 0ˆ ˆ ˆ(Q , Q ) 0 0 0 0 0 0 x y z ⎛ ⎞ ⎜ ⎟ ⎜ ⎟≡ = ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠ Q Q Q Q , (3) and over-barrier motion is described by the matrix 0 0 0 0 0 1 0 0ˆ 0 0 1 0 0 0 0 1 ⎛ ⎞ ⎜ ⎟ ⎜ ⎟= ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠ 2Q , (4) where 1 3( , ) 2 2 1 3( , ) 2 2 (1, 0) ⎫ = − ⎪ ⎪⎪ ⎬ = − − ⎪ ⎪ = ⎪⎭ x y z Q Q Q (5) are three equivalent minima in the 2d space of the normal modes 3 2(Q , Q ) , as it is illustrated in Fig. 1. Passing through 5 steps described above, one can obtain the following kinetic equations for this model: 3 2 3 0 0 2 2 2 0 0 2 2 2 0 0 d Q 1 1Q ( 2 ) (1 Q ) d τ τ 2τ d Q 1 3Q ( ) (1 Q ) d τ τ 2τ Q 1 1Q ( ) (1 Q ) d τ τ 2τ j j j x j y j z u j j j j x j y u j j j j x j y j z u j e e e e e e e d e e e e ⎫ ⎪= − + − − + − ⎪ ⎪ ⎪= − + − − ⎬ ⎪ ⎪ ⎪= − + + + − ⎪ ⎭ , (6) Fig. 1. Three-minima model potential U(ϕ) Физика и техника высоких давлений 2012, том 22, № 1 20 where j x j y j z u exp(β ) exp(β ) exp(β ) exp(β ) j x j y j z e e e e h = ⎫ ⎪ = ⎪ ⎬ = ⎪ ⎪= ⎭ h Q h Q h Q , j j jv=h Q is the molecular field at the site j, jv is the pair interaction constant, 1 0 ln zh T z ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ is the effective energy barrier with partial statistical integrals z0, z1 the same as for (1), but taken over three-minima potential (Fig. 1). 4. Numeric results for Ising model with barrier So, we have a toy – why not to play with it for a while? Results obtained from (2) are represented below. 1. Relaxation below phase transition temperature is shown in Fig. 2,a and re- laxation above the critical point is depictured in Fig. 2,b. In Fig. 2,а, we start from the fully disordered static state 20, 1z j z jS S= = and can observe how it evolves to the ordered state which is stable at this temperature. Some unusual (but expected) feature is unfreezing of over-barrier motions during this process (tem- poral decrease of 2 z jS ). In Fig. 2,b we start from the fully ordered static state 2 1z j z jS S= = and can observe relaxation to the stable disordered state and regular unfreezing of dynamics expressed in regular decrease of 2 z jS . 0 10 20 30 40 50 60 0 0.2 0.4 0.6 0.8 1.0 S, S 2 τ 0 2 4 6 8 0 0.2 0.4 0.6 0.8 1.0 S, S 2 τ а b Fig. 2. Relaxation of zjS (dashed line) and 2 zjS (solid line) at t = 1 below the critical poind (a) and at t = 3 above the critical point (b) (reduced pair interaction 0 5,Jj U = = tc = 2.2), τ is time measured in relative units Физика и техника высоких давлений 2012, том 22, № 1 21 2. Kinetic behaviour is shown in Fig. 3. These results are obtained from the ki- netic equations (2) under condition that temperature linearly depends on time. On heating, we start from the fully disordered state, same as in Fig. 2,а. One can ob- serve evolution of this state to the ordered one firstly (increasing z jS ), and then – to the fully disordered state (decreasing z jS to 0). Then, cooling from the last state leads again to the static state: 2 1z jS = (frozen over-barrier states), but some long range order remains frozen, what is indicated by non-zero z jS . So that, we can see a hysteresis without any additional assumptions but kinetic equations (2) as is. These processes are also accompanied with unfreezing-freezing of the over- barrier motions which can be traced by the temperature dependence of 2 z jS . 0 1 2 3 0 0.2 0.4 0.6 0.8 1.0 S, S 2 t 3. Relaxation time for z jS is shown in Fig. 4,а and relaxation time for 2 z jS is shown in Fig. 4,b. The relaxation time for the structural order parameter z jS possesses two vertical asymptotes: the first one caused by phase transition, as usual, but the second one reveals slowing kinetics at low temperatures caused by 0 1 2 3 0 50 100 150 τ1 t 0 1 2 3 0 0.5 1.0 τ2 t а b Fig. 4. Relaxation time for zjS (а) and 2 zjS (b) in relative units versus temperature t (the reduced pair interaction 0 5,Jj U = = tc = 2.2) Fig. 3. Kinetic properties of zjS (dashed lines) and 2 zjS (solid lines) (the reduced pair interaction 0 5,Jj U = = tc = 2.2) at heating (thin lines), d 0.1 d t = τ and at cool- ing (bold lines) d 0.1 d t = − τ ; t is the re- duced temperature Физика и техника высоких давлений 2012, том 22, № 1 22 the energy barrier. The relaxation time for the «dynamics» parameter 2 z jS indi- cates the phase transition too, but not so dramatically: it undergoes only the dis- continuity of its first derivative at the critical point. In contrast with computer modelling results [3], we did not consider here a metastable glass-like state because we investigate here only the simplest spatially homogeneous structure, but the glassy state needs to take into account spatial de- pendence of parameters 2,z j z jS S in order to describe short range order which is essential for this state. We are planning to include inhomogeneous states just as the next step of these investigations. 5. Conclusions We explored our numeric solutions in a wide range of parameters, so it allows us to make the following conclusions. 1. A new formalization of Glauber method is developed and is applied to the pseudospin model with barriers. 2. Kinetic equations have been derived for this model and numeric solutions in the simplest approximations have been obtained. 3. Relaxation and kinetic properties of the model has been shown to depend on the barrier value as well on heating-cooling rate. 4. Heating-cooling cycles reveal hysteresis. 5. The relaxation times are determined by the temperature and the barrier value. 6. The relaxation time for the structural order parameter zjS possesses two vertical asymptotes: the first one caused by phase transition, and the second one determined by slowing kinetics at low temperatures. 1. J.S. Slater, J. Chem. Phys. 9, 16 (1941). 2. P.G. De Gennes, Solid State Commun. 1, 132 (1963). 3. V.L. Kovarskii, A.Yu. Kuznetsov, A.V. Khristov, Low Temperature Physics 26, 348 (2000). 4. A.S. Bakai, J. Chem. Phys. 125, 064503 (2006). 5. V.L. Kovarskii, A.Yu. Kuznetsov, High Pressure Physics and Technology 14, 49 (2004). 6. V.L. Kovarskii, A.Yu. Kuznetsov, Low Temperature Physics 34, 216 (2008). 7. B.Ya Sukharevskii, V.G. Ksenofontov, V.L. Kovarskii, A.N. Ul’yanov, I.V. Vilkova, Sov. Phys. JETP. 60, 767 (1984). 8. R.J. Glauber, J. Math. Phys. 4, 294 (1963). 9. V.G. Vaks, S.V. Beiden, V.Yu. Dobretsov, Pis’ma v ZhETF 61, 65 (1995) [JETP Lett. 61, 68 (1995)]. 10. V.G. Vaks, Pis’ma v ZhETF, 63, 447 (1996) [JETP Lett. 63, 471 (1996)]. 11. V.G. Vaks, Phys. Reports 391, 157 (2004). Физика и техника высоких давлений 2012, том 22, № 1 23 12. V.Yu. Dobretsov, I.R. Pankratov, V.G. Vaks, Pis’ma v ZhETF 80, 703 (2004). В.Л. Коварський КІНЕТИЧНІ РIВНЯННЯ ДЛЯ ПСЕВДОСПIНОВОЇ МОДЕЛІ З БАР’ЄРАМИ Розроблено нове формулювання методу Глаубера, яке застосовано до псевдо- спiнової моделі з бар’єрами. Отримано кінетичні рівняння для цієї моделі, а також чисельні рішення у найпростіших наближеннях. Показано, що релаксація й кінетичні властивості моделі залежать від енергетичного бар’єру й швидкості нагрівання–охолодження. Цикли нагрівання–охолодження виявляють гістерезіс. Часи релаксації визначаються температурою та величиною енергетичного бар’єру. Час релаксації для структурного параметра порядку Sz має дві вертикальні асим- птоти: перша обумовлена фазовим переходом, друга – уповільненням кінетики за низьких температур. Ключовi слова: структурний безлад, кiнетичнi рiвняння, модель Iзинга, псевдо- спiнова модель, фазовi перетворення В.Л. Коварский КИНЕТИЧЕСКИЕ УРАВНЕНИЯ ДЛЯ ПСЕВДОСПИНОВОЙ МОДЕЛИ С БАРЬЕРАМИ Разработана новая формулировка метода Глаубера, дано ее приложение к псевдо- спиновой модели с барьерами. Получены кинетические уравнения для этой модели, а также численные решения в простейших приближениях. Показано, что релакса- ция и кинетические свойства модели зависят от энергетического барьера и скоро- сти нагрева–охлаждения. Циклы нагрева–охлаждения обнаруживают гистерезис. Времена релаксации определяются температурой и величиной энергетического барьера. Время релакации для структурного параметра порядка Sz обладает двумя вертикальными асимптотами: первая обусловлена фазовым переходом, вторая – замедлением кинетики при низких температурах. Ключевые слова: структурный беспорядок, кинетические уравнения, модель Изинга, псевдоспиновая модель, фазовые переходы Рис. 1. Трехминимумный модельный потенциал U(ϕ) Рис. 2. Релаксация zjS (штриховая линия) и 2 zjS (сплошная) при t = 1 ниже критической точки (а) и при t = 3 выше критической точки (b) (приведенная кон- станта парного взаимодействия 0 5,Jj U = = tc = 2.2), t – время в относительных единицах Физика и техника высоких давлений 2012, том 22, № 1 24 Рис. 3. Кинетические свойства zjS (штриховые линии) и 2 zjS (сплошные) (приведенная константа парного взаимодействия 0 5,Jj U = = tc = 2.2) при нагрева- нии (тонкие линии), d 0.1 d t = τ и охлаждении (жирные линии), d 0.1 d t = − τ ; t – при- веденная температура Рис. 4. Время релаксации для zjS (а) и 2 zjS (b) в относительных единицах в зависимости от приведенной температуры t (приведенная константа парного взаи- модействия 0 5,Jj U = = tc = 2.2)

Kinetic equations for the pseudospin model with barriers

Institution

Ähnliche Einträge