5. electrocalorical effect and dispersion of ferroelectric susseptibility icon

5. electrocalorical effect and dispersion of ferroelectric susseptibility



Название5. electrocalorical effect and dispersion of ferroelectric susseptibility
Дата конвертации10.08.2012
Размер10.46 Kb.
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5.ELECTROCALORICAL EFFECT AND DISPERSION OF

FERROELECTRIC SUSSEPTIBILITY.

The polarization of ferroelectric crystal under the action of variable electric field E(t) is proportional to the average coherent displacement X of ions of soft sublattice. These ions move under the action of a Lorenz - Wise field and under the action of near-range molecular field which creates force turning an ion to a balance place, which is displaced from neutral centre of an elementary cell on Õ2, with rigidity A = a (T-Tt), which is included adiabatically with t = -  by the generalized relaxation function of Heaviside and is switched off with Tt - T < 0 [12] as relaxation.

This relaxation is stabilized by the interaction with rigid sublattices with rigidity

B = b (T-Tc), which is included with Curie temperature Tc < Tt by the interaction with rigid sublattices, under action of a Lorenz - Wise field with "elasticity" L, and of the pushs of fluctuations with elasticity , and of the external field of intensity Eo (t).

For quasi-static processes the law of a movement [12] for coherent displacement X of ions of soft sublattice is established [12,13].

(72)

The solving of a return task of the mechanics for this law квазистатического of a movement gives the law of returning force, acting on an soft sublattice ion



(73)

This force is the sum of force returning an ion A(X-X2), of Lorentz force and of force of the pushs of fluctuations (f - ) X, and of force of stabilization B X. Except them on the ion acts the force of a feedback of polarization and Lorentz - Wise field f (X - Xs) dX/dt . The elasticity of soft sublattice depends on temperature of transition Tt (Ed), which in turn through dependence on intensity of a depolarizating field Ed depends on polarization, that depends on displacement of ion of soft sublattice X, so the equation of a movement is nonlinear and has variable factors owing to relaxation of elasticity with temperature of phase transition [12].



(74)

The transformation of this exhibitors expression gives

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The substitution of the law of force of a feedback and of quasi-elastic force in the basic law of dynamics gives the equation of a movement for coherent displacement X of soft sublattice ions or polarization P = q N S X with late factors. This equation conterminous with the dynamic equation of nonequilibrium thermodynamics (7.129) of [17]



(75)

With quasistatic process in a locally equilibrium system the elasticity depends on average time of delay  but not depends on time t.

With decrease of it a depolarizing field decreases. That results in increase of temperature of transition Tt (Ed) > T which becomes more than temperature Т. The spontaneous polarized state becomes equilibrium. The the ferroelectric comes in relaxation process to spontaneous polarization Рs and coherent displacement of ions X grow.

The intensity of a depolarized field Ed grows, that reduces temperature of transition Tt (Ed) up to T> Tt. Depolarized state Ps = 0 becomes equilibrium, the return process begins. The force of a feedback acts as compelling of auto oscillations of nonlinear oscillator if Т ~ Tt, P (0) ~ Po  Ps. Owing to inertia of a ions the polarization makes the pulsing oscillations from zero up to spontaneous polarization and more.

A feedback between a Lorenz field and polarization causes an establishment of auto -fluctuations according to the equation of a movement (67).

If temperature of a ferroelectric crystal Т is close to temperature of phase transition Tt, if a parameter of depth of a feedback of polarization and Lorenz field f is more than a parameter of attenuation g, if the amplitudes of elasticity of soft sublattice A and of elasticity of stabilization B are more than parameters of attenuation and of depth of a feedback and if parameter of relaxation of elasticity W much more than parameter of an establishment of stabilization V, in absence of external field E(t)=0 the autooscillations of polarization are established with decreasing of frequency. Their form becomes similar to the jumps up and down of elastic ball on the oscillating horizontal surface. The law of autooscillations is found by points representation method [13, 16] and numerically by the Runge - Kutta method [20]. The equation of a movement is represented in an initial kind Y " = F (t, y, z), y = X/m, z = y '

With Tt = 150 K, Tc = 130 K, W = 1, v = 0.00001, g = 0.001, f = 0.2, A = 2, B = 0.01,

(76)

With a step of integration h = 0.1 and with number of steps N = 2000 decision by a Runge - Kutta method is made on the formulas [20]

(77)

With initial polarization Y = 1, placed in a matrix of the entry conditions

(78)



Fig. 4 . The polarization avtooscillations.




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