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诚实是什么意思Fluctuations and Landau-Devonshire expansion for Barium Titanate A.I.Sokolov 1,A.K.Tagantv 21Saint Petersburg Electrotechnical University,St.Petersburg 197376,Russia;e-mail:ais@u.spb.ru 2Laboratoire de C´e ramique,EPFL,CH-1015Lausanne,Switzerland;e-mail:Alexander.Tagantv@epfl.ch (February 20,2002)Abstract The experimentally obrved temperature dependence of the quartic coeffi-cients in the Landau-Devonshire expansion for BaTiO 3is naturally accounted for within a proper fluctuation model.It is explained,in particular,why one of the quartic coefficients varies with temperature above T c while the cond is constant.The tetragonal pha in BaTiO 3is argued to exist esntially due to the thermal fluctuations,while the tru
e Landau-Devonshire expansion with temperature-independent coefficients favours the rhombohedral ferroelectric pha.Certain conclusions concerning the temperature dependence of the xtic Landau-Devonshire coefficients are also made.PACS numbers:77.80.Bh,77.84.Bw
Typet using REVT E X
The pha diagram of the barium titanate contains three lines of pha transitions and its structure is known to be properly reproduced by the phenomenological Landau-Devonshire theory[1].It is accepted within the phenomenological approach that all the coefficients in the Landau free energy expansion should be either constant or weak functions of temperature, pressure,etc.,apart from the quadratic term that changes sign crossing the cond-order transition line or the low-temperature spinodal.For the barium titanate,however,the matching with the theory of the experimentally obrved temperature dependences of the nonlinear dielectric susceptibility and spontaneous polarization forces one to allow for the strong temperature dependence of some higher-order coefficients[2–4].Being in conflict with the spirit of the Landau theory itlf,rapid temperature variation of the quartic and xtic coefficients is also quite unexpected from the microscopic point of view.Indeed,the barium titanate is a displacive ferroelectric with weak anharmonicity that can only results in rather slow temperature dependence of the macroscopic parameters[5].Moreover,there is
an extra point showing how unsatisfactory the real situation is:becau of the fast temperature variation of the certain coefficient,the six-order Landau-Devonshire expansion for BaTiO3 with updraded coefficients[6,7]turns out to loo its global stability at T=at the temperature that exceeds the cubic-tetragonal transition point T c by only50K.脸部起皮
In this Letter,we will show that the temperature dependence of the quartic Landau-Devonshire coefficients,experimentally obrved in a paraelectric pha of BaTiO3,can be attributed to the thermalfluctuations of polarization and naturally accounted for within a properfluctuation model.It will be explained,in particular,why in experiments one of the quartic coefficients demonstrates well-pronounced variation with temperature while the cond is temperature independent.The conjecture will be put forward concerning the structure of the true Landau-Devonshire form for BaTiO3with the constant(temperature independent)coefficients,and thefirst-orderfluctuation corrections to the xtic Landau-Devonshire coefficients for T>T c will be found.
Barium titanate is a displacive ferroelectric undergoing thefirst-order ferroelectric pha transitions,with the six-order anharmonicity and electrostriction playing an esntial role in
forming its pha diagram.The electrostriction is known to convert the cond-order transi-tion,appro
priate to the clamped crystal,into thefirst-order one.Strong dipole-dipole inter-action affects the vibrational spectrum of BaTiO3,resulting in a big gap between transversal (soft)and longitudinal polarization modes.The main features are properly described by the effective Hamiltonian that is the natural generalization of the Landau-Devonshire free-energy expansion:
H=1
Cǫ0
+s2q2 δαβ+∆2nαnβ φαqφ∗βq+1
q
,C is the Curie constant,∆2∼ǫ−10∼sa−1
is the dipole gap in thefluctuation spectrum,a being a lattice constant.Contrary to the original Landau-Devonshire expansion,the Hamiltonian(1-3)accounts for the inhomoge-neousfluctuations of the polarization and elastic strains making it possible to explore the fluctuation effects in BaTiO3.
Dealing with thefirst-order pha transition,we are in a position to study thefluctuation effects in the r
egion where thermalfluctuations of the order parameter are weak.Hence,in what follows we limit ourlves by the calculation of thefirst-orderfluctuation corrections to the quantities of interest.The quantities to be found are the full four-leg and six-leg vertices, reducing,under zero external momenta,to the effective(”dresd”)quartic B i and xtic Γi coefficients,entering the Landau-Devonshire expansion.Amongfive Feynman diagrams reprenting the lowest-order non-trivial terms in the perturbative expansions of the vertices
mentioned,only two one-loop graphs give contributions that rapidly,as (T −T 0)−1/2,grow up approaching the transition point.The graphs have an obvious structure and may be calculated in a standart way using the propagator
G αβ(q )=k
B T (δαβ−n αn β)
Cǫ0
T −T 0(24β21+4β1β2+6β2
2),
B 2=β2−k
B T √
10πs 3√
Since in the vicinity of T c the elastic and electrostrictive moduli weakly depend on tem-perature,similar relations should be valid for thefluctuation modified(dresd)quartic coefficients B1,B2,B f1,and B f2.
At the transition point(T c=393K),B f1=−2.0·108V m5C−3,B f2=1.6·108V m5C−3 [6,7]and,hence,B1=5.4·108V m5C−3,B2=−0.7·108V m5C−3.It is easy to e that for B1>B2the rhombohedral pha has lower free energy than the tetragonal one,provided the six-order form is isotropic,i. e.does not influence their competition.Hence,in the clamped crystal the quartic form of the Landau-Devonshire expansion strongly favours the pha transition into the rhombohedral pha.The same is true for higher temperatures T=415K and T=423K,where B f1=−1.3·108V m5C−3[2],B1=6.1·108V m5C−3 and B f1=−1.0·108V m5C−3[2],B1=6.4·108V m5C−3,respectively,with B f2and B2kept unchanged.
Now we are ready to estimate to what extend thefluctuations can modify the behaviour of barium titanate in the vicinity of T c.As is en from Eqs.(5),the quartic Landau-Devonshire coefficients should vary with temperature in a similar way,provided thefluctuation correc-tions to them are of the same order of magnitude.Let us compare the magnitudes of the fluctuation terms B(1)2and B(1)1within the domain where parametersβ1andβ2have relevant values and signs.The quantity characterizing relative weights of B(1)2and B(1)1is their ratio, that can be found directly from Eqs.(5).The plot of the ratio R(1)=B(1)2
is shown in Fig.1.Analyzing this function,one canfind,in particular,that B(1)2β2
lies between−28and−2.3.
β2
Let us estimate furtherβ1
quite reasonable.Another quartic coefficient,B2,does not depend on T and the estimate β2=B2=−0.7·108V m5C−3may be considered as an accurate one.Since,in any ca,β1>B1(T=393K)=5.4·108V m5C−3,we e that−14<β1
