AUTO-OSCILLATIONS OF POLARIZATION AND DYNAMIC
ELECTROCALORIC EFFECT IN FERROELECTRICS
MokejevA. A., MokejevAn. A.
Private laboratory, Vitebsk, Byelorussia. AO Тreigon, Moscow, Russia.
e-mail: avtoferelrheo narod.ru
ABSTRACT. The ferroelectric phase transition of displacement type represents thereorganization of structures of crystal lattice -relaxation of soft sublattices elasticity of an initial meta-stable polarised phase of a crystal to elasticity of steady final phase and simultaneously avalanche disintegration of polarization under action of a feedback between polari-
zation and Lorentz - Wisse field. This feedback is expressed as negative friction in the equation of a movement of ions soft sub-lattice. Together with nonlinear dependence of its elasticity on displacement of these ions it results in occurrence of auto -oscillations of polarization near to phase transition. The power, tacked away by oscillations of polarization from a sample, is allocated in shunting loading, the sample is cooled. The decrease of tempe-
rature of transition owing to amplification of a depolarized field stabilizes cooling so, that in a significant part of a period of oscillations abnormal heat capacity appears small, and decrease of temperature - significant.
INTRODUCTION. The dispertion of dielectric permeability of ferroelectrics near to phase transition [1-6], has both of Debay relaxationof property and Tomson resonant properties, so the ferroelectric phase transition is relaxation of elasticity of soft sublattice and is the oscilla-
tions of polarization, when the complete disintegration of soft sub-lattice is prevented by increase of stabilizing elasticity others sub-lattices. The phase transition occurs before the sub-lattices becomes absolute soft. These phenomena have not an explanation in phenomeno-
logical thermodynamics and in the microscopic theory [5,6]. According to works [1,3] the smooth decrease of soft sub-lattice elasticity of a crystal is interrupt by spasmodic its destruc-
tion with temperature of transition Tt < To, smaller than temperature of loss of stability, with which the elasticity of soft sub-lattice disappears. Owing to quantum transitions of ions soft sub-lattice in a state with delta - figurative peak of number of micro-states , which arises as imposing of a "rich" bottom site of a spectrum of stationary states of these ions at bottom of a wide rectangular potential hole of stabilization ,on a rich top site of its spectrum in Gaussian hole of soft sublattice. There is an avalanche disintegration of polarization owing to an estab-
lishment of a positive feedback between it and Lorentz-Wise field . With it breaks up a depolarizing field, as a return external field reducing temperature of transition. The disinteg-
ration of polarization is replaced by growth and there are auto-oscillations of polarization.
The condensation of a spectrum of oscillations of soft sub-lattice with heating to temperature of phase transition leads to the abnormal fast growth of heat capacity up to temperature of transition and to spasmodic reduction with large temperatures.. The comparison with experi-
enced temperature dependence of heat capacity shows, that the Landau jump is not observed. The abnormal fast growth of heat capacity caused by fluctuations of polarization is not significant The electrical current of polarization with negative attenuation of auto-oscillations absorbs the heat, but does not give off.The increase of a depolarizing field with polarization reduces temperature of phase transition. The voltage on the shunting resistor makes an average deviation of temperature from temperature of phase transition by such, that heat capacity appears rather small for essential significant electrocalorical effect.
1. SWITCH ON OF POLARISATION. The polarization is determined in density of an
electrical current j (r, t) which depends on distance x ' of charges and from delay t ' of propagation of perturbation to a point of supervision with coordinates x in the moment t,
and from intensity of a molecular field created in the late moments, t' by charges in the removed points x ' through causality function f [1].
(1)
The causality function f is represented by the first components of series expansion in a number under the small relation of energy in a field Å to energy of coupling,
/Ug < 1. It leads to decomposition of density of a current on polarization density of a current and density of a current of screening and corresponding - to charge density
, ,
,
Quasi - static processes in a crystal are determined by the Macswell equations in locally equilibrum system for vector and scalar potentials
, (2)
,
which is transformed to a individual kind for quasi-static processes [2]
, , (3)
(4)
The decision last from them
(5)
determines processes of relaxation for parameters of a crystal.
The Gauss theorem after replacement in it of complete density of charges gives energy of ions in a screening molecular electrical field in an elementary cell.
, (6)
With energy = ê T> Uo the movement of a soft sub-lattice ion becomes infinitely and the interaction of stabilization with rigid sublattices with energy of stabilization should join.
, D1 > D (7)
With energy T < Uo the ion of soft sublattice moves in one of potential holes concerning one of poles of pceudo-spin poles Sn = 1 and forms dipole moment dn and polarization Ð, creating a Lorentz-Wise field with intensity El = P/o l (l - the Lorentz factor) and a depolarizig field Ed = -P/o d. In this field the soft sub-lattice receives energy.
(8)
The ions of soft sub-lattice with energy = kB T < Uc, smaller than average energy in potential holes, oscillates in one of holes about one of pceudo-spin poles Sn = 1, between points of turn with coordinates Xon (amplitude), determined according to conservation law,
Xon increasesindefinite with approach of energy to a threshold of loss of stability of a movement Uo, with effective rigidity a(Xn) and own frequency c , which decreases with growth of displacement Xn
, (9)
With energy = k T> Uc a movement of ions of soft sublattice is free oscillations between collisions with walls of an almost rectangular potential hole of stabilisating interaction with kinetic energy
, D1 = 2 D
with average speed V and with frequency of collisions c , and with effective elasticity b
, , (10)
The electromagnetic interaction between soft vibrons and optical oscillations causes transitions between their stationary states /n >, /m > with probability [3].
(11)
With energy < Uo in the sum on q are allocated components, which correspond to transitions from stationary states with energy equal to a thermal energy, in states with threshold energy and in states with energy appropriate to transition in a new phase .
With energy in the sum on q are allocated components, which correspond to transitions from stationary states with a thermal energy, In states with average energy . These transitions create effective dissipation force causing resistance to internal current in vibrones, which determines parameter of speed of relaxation
The modules of elasticity of soft sub-lattice, of stabilization and ofLorenz - Wise field are included adiabatically with t = - and are switched off with t> 0 T> Tt, To, Tc by relaxation function
(12)
In quasi-static process, this function becomes the generalized function of Heaviside.
(13)
The displacement of ions of soft sub-lattice is a superposition of flat waves with wave vectors k both amplitudes Xk and frequencies k [5,6]. With k = 0 they form coherent displacement of all ions of sublattice with energy of ordering [1,2].
(14)
and fluctuation energy ,
(15)
which contains energy of influence of fluctuations on coherent displacement Xo with factor ,
energy of stabilization
(16)
energy soft sub-lattice in a Lorentz-Wise field
(17)
energy soft sub-lattice in a external and depolarizating field
(18)
energy of oscillations of soft sub-lattice
(19)
energy of pceudo-spin system
(20)
2.VARIABILITY OF PHASE TRANSITION TEMPERATURE. The polarization of a ferroelectric crystal P (in the consent with fluctuation - dissipation theorem) is a special case of locally average value on locally equilibrum distribution f (Ðî, P, t) of statistical system and represents process in dense system of poorly connected soft vibrones [2]. The electrical and thermodynamic properties of ferro-electic are determined by the statistical sum Z of soft sublattice, which is designed in phase space with coordinates: displacement of ions of soft sublatticern, their pulses pn, and components a pceudo-spin Sn.. Excess of energy of a movement alongtheofcoherent displacement above thermal leads to selection of product of number of microstates with this energy (n) = exp(Nn/kBT) and ofthe factor, containing latent and kinetic energies, depending from amplitude of pulses of ions of soft sublattice. According to the theorem of affinity of canonical distribution to micro-canonical their product is proportional to Dirac -function from average energy of soft sublattice ions
(Xo) = < ( xon>)> = kBT, whichcorresponds to their average displacement Xo=< Xon >.
(21)
.
(22)
The statistical sum becomes the product of the pceudo-spin sum Zs and sum of system of effective linear oscillators Ze. On coordinate q = Xn the energy is more thermal, so the statistical sum and free energy becomes the functions of dynamic variable Xo (incomplete thermodynamic potential [4])
Soft sub-lattice is broken into system soft vibrones and pceudo-spin subsystem. The free energy in locally equilibrum state is equal to the sum of free energy of system of nonlinear oscillators, and energy of pceudo-spin system.
(23)
In it are allocated energy of ordering eo and energy of fluctuation f . From f the component is allocated which expresses about influence of fluctuations of polarization on the ordered displacement of ions of soft sub-lattice. It is displaced to free energy of ordering.
(24)
The probability of locally equilibrum fluctuations near transition is equal [4]
(25)
The polarization of a crystal is equal average dipole moment of nonlinear oscillators system of all soft sub-lattice ions.
(26)
It is submitted by product an average pceudo-spin and of the moment of vibrons and is determined by average displacement of ions soft sub-lattice Xo. Coherent displacement Xo of ions of soft sub-lattice and average pceudo-spin S in locally thermodynamic balance or in quasi-static process are determined by the equations of local balance through free energy,
(27)
, P = q S N Xo (28)
(29)
Temperature of ordering of pseudo-spins, is determined by expression [5]
Tk (T) = q h El /kB
With temperatures Ò < Tc of so smaller Curie temperature, that the stabilization is switched off, the fluctuations of polarization are small, free energy and the module of polarising elasticity are decrease linearly with growth of temperature .
With T> Tc the stabilization is switched on also diagram of temperature dependence of free energy and module of elasticity demonstrates the gap, which is equivalent to change of temperature of loss of stability. The free energy of a ferro-phase in a Lorentz-Wisse field
El > Ed and in the external field Eo and in depoplarizating field Ed = -P/o d [4]is compared to free energy of para - phase, where a = 0, E = Ep = Eo-Epd with the further heating
(30)
The phase transition occurs, when the free energy of para - phase is equalled the energy of a ferro-phase
(31)
Temperature of transition is determined by this equation. In the first approximation of a method succesive approximation intensity of depolarizing field Ed and the intensity ofLorenz - Wise El are accepted of equal intensity created by the polarization P with temperature of the transition Tt (T, E) = 200 K. Intensity of a depolarizing field Ed and the Lorenz - Wise field are given by equality
(32)
(33)
The intensity of the Lorentz - Wise field is determined
(34)
The replacement in it of expressions for a, b Xo and Xop in ferro- and para - phase gives the equation of temperature of transition in the following approximation
(35)
(36)
Temperature of phase transition is displaced from a position between Curie temperature Tc and temperature of loss of stability by Lorentz-Wisse field and external field to greater value, both by a fluctuations and the depolarizing field to smaller. Known intensity of a Lorentz-Wisse field determines temperature of ordering of pseudo-spins Tk. [5].
3.RELAXATION AND AUTO-OSCILLATION OF POLARIZATION. The movement of polarization is determined by the kinetic equation for one-partical function of distribution f of probabilities of microstates for locally equilibrum statistical system in relaxation approxima-
tion. Integral of collisions is equal to the relation of a deviation of function of distribution f from locally equilibrum function fo to time of relaxationtr, which is return probability of transition W of system in unit of time in states close to a locally equilibrum state. It the more, than more intensity of a Lorenz-Wise field [1,2].
(37)
In homogeneous system in absence of external forces it gives the Onzager equation of polarization relaxation of ferroelectric [3]. In it the factor W (El) is proportional El in the first approximation
(38)
Its decision gives the law of relaxation after a push by the fluctuation Ðî
. (39)
The decision of a return task of the mechanics (the definition of the law of force under the law of a movement) gives the law of force of a feedback between P and El
(40)
The relaxation of polarization and of elasticity determines the abnormal heat capacity of ferroelectrics. It develops from heat capacity of soft sub-lattice ionsdisplacements order Су, ofheat capacity of pceudo-spin system Cs, and of fluctuationsheat capacity Cf.. Only the heat capacity of the ordering of displacements of soft sublattice ions Суhas significant value .
(41)
The abnormal growth of heat capacity is determined by fast growth of density of a spectrum of stationary states of nonlinear Gauss oscillators of soft sub-lattice with the approach to temperature of loss of stability, Curie temperature and temperature of phase transition. The Landau jump is absent. Just such dependence of heat capacity from temperature is observed on experience [3,4]. The polarization of a ferro-electric crystal with action of a variable electrical field E (t) in approximation of a molecular field is proportional to average coherrent displacement Õ (t) of ions of soft sub-lattice. The solution of a return task of the mechanics for this law of a quasi-static movement gives the law of coming back force, working on an ion soft sub-lattice
(42)
In it the elasticity of soft sub-lattice depends on temperature of transition Tt (Ed), which depends on polarization through dependence on intensity of a depolarizating field Ed. Ed depends on displacement of a soft ion X, so the equation of a movement non-linear and has variable factors owing to relaxation of elasticity with temperature of phase transition [2]. The replacement of the law of force of a feedback and quasi-elastic force in the basic law of dynamics gives the equation of a movement for coherent displacement X of ions of soft sub-lattice or polarization P = q N S X.
(43)
In absence of an external field the solution of this equation gives the law of a movement as auto-oscillations on frequency, which is determined under the law of dispersion [6,7]. With T> Tt a depolarizing state is equilibrum. The polarization decreases. With its reduction decreases a field. It causes increase of temperature of transition Tt (Ed) > T, which becomes more than temperature Ò. The spontaneously polarized state becomes equilibrum. The rerroelectric comes in relaxation process to spontaneous polarization Рs. The intensity of a depolarized field Ed grows, that reduces temperature of transition Tt (Ed) up to T> Tt. Depolarizedstate Ps = 0 becomes equilibrium becomes equilibrum, the return process begins.. The feedback between a Lorentz-Wisse field and polarization causes an establishment of auto-oscillations according to the equation of a movement (43) with reduction of frequency. If Ò ~ Tt, P (0) ~ Ðî < Ps, the polarization makes pulsing oscillations from a zero before spontaneous polarization similar jumps of an elastic sphere on a oscillating horizontal surface.
Fig.4. Oscillogram of an establishment of auto oscillations of polarization with reduction
frequencies owing to growth of amplitude.
4. DYNAMIC ELECTROCALORIC EFFECT. The polarization of a crystal is an electrical current of density
(44)
which allocates, and with negative friction absorbs quantity of heat dQ under action of Lorenz - Wise feild with intensity El
(45)
The polarization under action of an external electrical field is accompanied by absorption of heat
The change of temperature is equal
(46)
The power absorbed from of a ferroelectric sample, shunted by loading with resistance R, by auto-oscillations of polarization P (t) or compelled oscillations, which are oscillations of an electromagnetic field in the condenser generated by surfaces of a sample, capacity C, is allocated on resistance of loading R. These oscillations in a circuit of a sample and the loading create electro-moving force E equal to the sum of voltage on loading and on internal resistance of the sample Ri, and current j, equal to a current of polarization
E = Ri dP/dt = j (R + Ri), (47)
which transfers through a surface of a sample a charge
(48)
to the moment of supervision t the intensity of depolarizing field Ed decreases with relation on its intensity in absence of loading resistor
, (49)
The reduction Ed by shunting loading increases temperature of phase transition Tt, and the increase - reduces it. The cooling of a sample increases polarization, and with it reduces Tt after reduction of temperature Ò, that supports conditions of occurrence of auto-oscllations. The choice of loading stabilizes these conditions with lowered temperature. Change of temperature in inverse proportion to heat capacity [8]. The choice of resistance of loading provides an average deviation of temperature from temperature of transition < Tt -T > with such, that in a significant part of a period of oscillations theheat capacity appears rather small, that the effect was essential.
THE CONCLUSION.
1. The ferroelectric phase transition - process of relaxation of elasticity of soft sub-lattice and polarization of a crystal from a unstable initial phase .
2. The de-polarizing field with intensity proportional to polarization, reduces temperature of phase transition, but the disintegration of polarization causes its growth. There are auto-oscillations of polarization
3. The law of polarization relaxation owing to a positive feedback of polarization and Lorentz -Wise field with small polarization determines force of negative friction, and with polarization exceeding spontaneous polarization, determines force of positive friction in the equation of a movement.
4. The decision of this equation shows existence auto - oscillations of polarization with temperatures close to temperature of phase transition.
5. The shunting of a ferroelectric sample by resistance of loading results to dynamic electro-caloricaffect . The choice of resistance of loading provides change of temperature from temperature of transition < Tt -T > with such, that in an essential part of a period of fluctuations the heat capacity appears rather small, that the electro-caloricaffect was essential.
THE LITERATURE
1. Mokejåv A.A, Mokejåv. Àn. A. Procedings of IV international conferences: " Crystals: growth, form, application ", Alksandrov, VNIISIMS,1999.
2. . Mokejåv A.A, Mokejåv. Àn. A. Procedings of V international conferences: " Crystals: growth, form, application ", Alksandrov, VNIISIMS, 2001.
3. . Mokejåv A.A, Mokejåv. Àn. A. Procedings of IV international conferences: " Crystals: growth, form, application ", Alksandrov, VNIISIMS ,2003.
4. Patashinski A.Z., Pokrovski V.L. Fluctuatig theory of phase transitions. М., Science, 1975.
5. . StrukovB.A., LevanukkA.P.. Physical bases of the ferroelectrics phenomena in crystals. М., Science, 1983.
6. LinesM.E., GlussA.M.. Ferroelectricsand related materials.М., Science, 1985.
7. Acoustic crystals. The editor Shаskolskaja Ì. P., Ì., Science, 1982.
8. Kubo R. Termodynamics, M. Mir, 1970/
9. Mokejev A.,A.