3. RELAXATION OF POLARIZATION.
The polarization of ferroelectric is decreased with heating due to action of a depolarising field in weak external fields with intensity Eo << Eds which much smaller than intensity of a depolarized field created by spontaneous polarization, despite of growth of a susceptibility before inclusion of stabilisation with Curie temperature Tc. In ferrophase the polarization is determined by substitution , the expressions (24), (39) and
Till achievement of Curie temperature the stabilization force reduce polarization faster.
The soft sublattice elasticity decreases. And the stabilization elasticity grows so that the total elasticity module is the small constant in some interval of temperatures.
With it the balance of a soft ion becomes almost indifferent, the returning forces, acting on it almost disappear and arise strong fluctuations of polarization. When they reach size of decreasing polarization, between its increment and intensity of a Lorentz field increment there is a positive feedback and the polarization is interrupted avalanchely.
. The elasticity of soft vibrones is switched off and polarization is reduced by avalanche jump. The phase transition with the temperature Tt is determined somewhat casually.
The soft sublattice susceptibility
is a sum of susceptibilities of pseudo-spin system (ordering of distribution of soft ions on potential holes) and of the soft oscillators system.
With temperature of ordering of pseudo-spins Tk ~ To to approximately is equal to temperature of stability loss of soft sublattice in ferrophase T < Tt < To and To < Tk pseudo-spin S ~ 1, dS/dE = 0.
The susceptibility of soft sublattice is equal to a sum of susceptibility of displacement and the susceptibility of pseudo-spin system. If the temperature of ordering of pseudo - spins Tk ~To it is equal to temperature of stability loss the susceptibility and the module of polarization elasticity ( invers susceptibility ) are equal to the module of displacement
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Ferroelectrics with small polarization of elementary cells, with weak Lorentz field with low temperature of pseudo-spins ordering Tk << Tc has a variable average pseudo-spin S < 1 near to phase transition. Polarization occurs at the expense of pseudo-spins ordering. Ferroelectric belongs to the order-disorder type.
In ferroelectrics with a strong Lorenyz field, with large shoulder of the pceudo-spin, with the temperature of ordering is greater than temperature of stability loss Tk >> To in ferrophase the average pceudo -spin S = 1. Ferroelectric belongs to a type of displacement.
In ferroelectric with a weak field of stabilization with small displacement Xo << D1, which much smaller than width of a stabilization potential hole, the strong fluctuations of polarization reach the sizes of the polarization and compensates polarization with temperatures of transition Tt close to Curie temperature Tc. Elasticity of soft vibrones together with elasticity of a Lorentz- Wise field are switched off when they are greater than stabilization elasticity . The susceptibility and the module of polarization elasticity grow by jump. There is an disintegration of polarization by avalanche and phase transition occurs. With the further heating in para-phase the susceptibility is decreased according the Curie - Wise law. In ferro-phase this law with Curie temperature Tc < Tt is broken, on the diagram of dependence of a susceptibility from temperature there is a break noticed for example in cesium digidrophosfate doped by deuterium . Ferroelectric appears own.
In ferroelectrics with a strong field of stabilization (weak screening) the potential energy of stabilization of soft vibrones with small displacement is represented by a parabolic hole.
Weak fluctuations of polarization suppressed by stabilization and strong Lorentz field, reaches the size of decreasing polarization and pushes the phase transition with high temperature Tt, close to stability loss temperature To, where the elasticity of soft vibrones almost is disappeared. The jump of polarization and polarization elasticity module occurs at the expense of disintegration of a Ljrentz -Wise field . The susceptibility decreases up to the susceptibility of para-phase by jump. The crystal has a properties of unown ferroelectrics.
Ferroelectric with temperature of pseudo-spins ordering Tk close to To, which approximately is equal to the temperature of stability loss, with T << Tk = To = Tt in ferrophase is similar to ferroelectric of displacement type .
With phase transition temperature Tt the Lorentz-Wisse field is switched off, average pceudo-spin of para-phase Sp << 1 becomes small, but the elasticity of pceudo-spin system is not switched off, bs = (T-Tk). The temperature dependence of elasticity module has a high-temperature break, that also is observed in experience . This ferroelectric with temperature T > Tk is similar to ferroelectric of order-disorder type and belong to mixed type as cesium digidriphosfat doped by deiterium. This ferroelectric belongs to mixed type
With the parameters of crystal obtained in [12,13]
- dielectric сщтыефте of vacuum, q - charge of ions, N - their concentration,
the Lorenz field elasticity
the elasticity of fluctuations of polarization
elasticity of soft vibroune
the stabilization elasticity
the pseudo - spin system elasticity
with the Curie - Wise constants
the polarizing elasticity module is equal
and a dielectric susceptibility is equal
Fig.4. Dependence of a polarization elasticity modulus of ferroelectrics from
Fig.5. Dependence of a dielectric susceptibility of ferroelectrics from temperature.
The spontaneous polarization of ferroelectric Ps arises owing to strong long-range interaction of ions of a crystal, which results in occurrence of an average molecular field consisting from all-round compression of crystal, which creates its potential hole, from screening returning fields of elementary cells, acting on ions of soft sublattice , and from long-range acting Lorenz-Wise field with intensity El, in which as in an external field, the ferroelectric is equivalent to paraelectric. Effective para-electric is a dense system of electric dipoles which are poorly connected with each other.
Effective para-electric is a dense system poorly cooperating electrical dipoles.
The movement of the polarization is determined by the kinetic equation for one-partical distribution function f of microstates probabilities of the locally equilibrium statistical system in relaxation approximation. The integral of the collisions is equal to the relation of a deviation of distribution function f from locally equilibrium function fo to time of relaxation tr which is the return probability of transition W of system in unit of time in a states closer to the locally equilibrium state. It the more, than more intensity of a Lorenz-Wise field [12,13].
Here V-local velosity, F - an external force field, Х- average coherent displacement of ions soft sublattice from balance displaced on Х2 from neutral centre of an elementary cell. Term by term multiplication of the equation (1) on the dipole moment of cell and on the dipoles concentration and integration on phase space. In homogeneous system in absence of external force fields it gives the Onzager equation of polarization relaxation of ferroelectric . In this equation the kinetic factor W (El) is proportional El in the first approximation
Change of temperature Т to values T> Tt of larger temperature of phase transition a state with spontaneous polarization Р = Ps is made nonequilibrium, that causes the relaxation to new depolarization equilibrium state under action of a feedback of polarization Р and a Lorenz -Wise field after a push by fluctuation Р = Ро with probability of transition W (El), that greater, than more intensity of a Lorenz -Wise field El and than more deviation of polarization from equilibrium polarization Ps.
In the relaxation equation with a feedback
the signum of a feedback varies on opposite, when P> Ps
The solution of a return task of the mechanics for this law of movement (definition of the law of force under the law of a movement) gives the force law of feedback between P and El
because the polarization is proportional to average displacement Х ions of soft sublattice from balance
gives the law of force of feedback a as forces of negative friction with X < Xs and of positive friction otherwise.
On a soft ion acts the linear force of friction. The complete force of "friction" is
The direct task of the mechanics for relaxation with a feedback (3) is solved by integration
By replacements integral is reduced to tabulared 
With P> Ps
With the initial condition t = 0, P = Po >Ps
With P < Ps
With the initial condition t = 0, P = Po
With t = ¥ , P = Ps =const.
The thermal properties of a crystal close to ferroelectric phase transition with heating to temperatures Т ~ Tt, close to phase transition temperature, are determined by the properties of free energy of its soft sublattice which is a function of the dynamic order parameter h = P ~ Xo = X. In own ferroelectrics this parameter is the polarization Р which proportional to coherrent displacement Xo of soft sublattice ions from neutral centres of elementary cells. The free energy of soft sublattice is a sum of ordering energy, of fluctuations energy, of pceudo-spin energy
Heat capacity of f ferroelectric with constant pressure р and intensity of an electrical field Е is expressed through free energy
It is the sum of heat capacity of displacements order of soft sublattice ions Су, of heat capacity of pceudo-spin system Cs, and of fluctuations heat capacity Cf. which is determined by free energy of the fluctuations , in which from the energy of fluctuations of soft sublattice the part depending on the displacement Xo is selected . This part is allocated in ordering heat capacity as expression of influence of fluctuations on heat capacity Cy of displacement ordering, which is determined by free energy of ordering
The substitution in this formular of expression for Xo gives
In absence of an external field Е = 0
In ferroelectrics of the displacement type Тк >> Tt, S = 1, j = const, f = const,
a weak external electric field influences on a heat capacity only through elasticity a, b ,f
With temperatures T < Tc, b = 0, j = 0, da/dt = - a, so
grows proportionally Т and in inverse proportionally (To - T). With temperatures close to the Curie temperature the stabilizing interaction is included, so b = b (Т - Tc).
The elasticity of soft sublattice becomes less elasticity of stabilization. The growth of heat capacity slowes .
. With temperatures Т @ Tt, a = 0, f = 0, j = 0 ferrоelasticity and Lorenz - Wise field are switched off , and the fluctuations of polarization are small, the appropriate degrees of freedom are switched off, so heat capacity jump to size
.With heating the heat capacity of para-phase decreases up to normal.
Fig.3. Dependence of heat capacity from temperature.
With temperatures which are near to Curie temperature, a stabilizing field almost is absent, b = 0, the fluctuations are small, the difference (Tt - Т) is great, the heat capacity grows with increase of temperature as (Tt-Т).
The decay of polarization (the growth of a difference T - Tc in a denominator of last expression ) begins long before of achievement of transition temperature Tt,To and the growth of heat capacity is slowed . To this the growth of fluctuations, which is expressed by function , is added. The growth of heat capacity at the expense of increase of stationary statesdensity with approach to Tt is replaced on its decrease in an almost rectangular potential hole of stabilizing earlier, than there is its jump. The phase transition is washed away. On experrience this phenomenon is observed even in such typical ferroelectric of order-disorder type as threeglicinsulfat .
The fluctuations of soft sublattice polarization give the direct contribution in heat capacity Cf = Cfo + Cf1 + Cf2, which are corresponds to three components in free energy of fluctuations
The fluctuation heat capacity is determined by second derivative from fluctuations free energy
In this expression the exponent index is significantly smaller than unit, so
the sum is replaced by integral approximately
Where is a wave vector of the top border of an oscillatory spectrum of soft sublattice , a sound velocity and lattice constant
The fluctuations of soft sublattice polarization give the direct contribution in heat capacity Cf = Cfo + Cf1 + Cf2, consisting from three component:
the constant component, proportional to a wave vector of the top border of an oscillatory spectrum of soft sublattice km, the component monotonously growing with temperature growth and abnormal component.
The given accounts show that the fluctuations of polarization bring in insignificant small contribution to the heat capacity of ferroelectric of displacement type .
The abnormal growth of heat capacity is determined by fast growth of density of a spectrum of stationary states of nonlinear Gauss oscillarors of soft sublattice with approach to temperature of loss of stability and Curie temperature and phase transition temperature. Thus the jump capacity, given by the Landau theory, is absent but heat capacity dependence from temperature is observed on experience [1,2,3] .
The given accounts displays properties similar to properties of ordering heat capacity of displacement, but they depends from temperature under the radical law.
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