By A S Sidorkin
The e-book examines area structuring as a result of lack of the preliminary part balance in fabrics of finite measurement. It additionally covers facets similar to the behaviour of area limitations in the course of their interplay with lattice defects, their constitution in actual ferroelectrically ordered fabrics, the impression of the lattice capability aid on their flow, and the flexural and translational elements in their dynamics in ferroelectric crystals. The contribution of the area barriers to the dielectric homes of ferroelectrics and elastic homes of ferroelectric elastomers is evaluated.
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Additional info for Domain Structure in Ferroelectrics and Related Materials
5) ∫ ⎨⎩ ∫ ⎩⎨ dx dx ⎭ dx 2 ⎭ Since the variation δP is an arbitrary small function, the identical equality to zero of the integral is possible only if the expression in the braces is equal to zero. From this we find the equation describing distribution of polarization in the boundary: d 2 P dϕ = = −α P + β P 3 . 6) 2 dx dP The sign of the second variation makes it possible to evaluate the stability of the corresponding solution. 5) 30 2. Structure of Domain and Interphase Boundaries 2 ⎧⎪ϕ "( P ) ⎫ ⎡ d (δ P ) ⎤ ⎪ 2 δ Φ = ∫⎨ (δ P ) + ⎢ ⎥ ⎬ dx = 2!
5) P = − P0 . At γ > β the pattern of the stable states is shown in Fig. 8. 6) IV , q1 = 0, q2 = − q0 , PI = PII = PIII = PIV = 0. 6) is determined by comparing the values of the thermodynamic potential in points ⎛ α2 ⎞ ⎛ α2 ⎞ A ⎜⎜ Φ = − ⎟⎟ and points B ⎜ Φ = − ⎟ , respectively. 6) represents domain walls in the material under consideration. These walls correspond to the lines (the sides, the diagonals of the square or other curves) on the graphs. 5), all consecutive transitions A I ⇔A II , A II ⇔A III, A III ⇔A IV , A IV ⇔A I represent ferroelectric domain walls whereas the transitions A I ⇔A III , A I ⇔A IV are antiphase boundaries.
3) ∇ iσ ik = 0 we discover the equation for displacement u of the following type ∂ 2u = 0. 4) has to meet specific conditions at the boundary of the material. e. it is assumed that at z = 0 there is a contact with the absolutely rigid material, and the second boundary of the material z = L is assumed to be free. 5): (α − ) k 2 k 2u + c u = B sin π z. 7) ⎟ . ⎝ 2kL ⎠ The value of k corresponding to the minimum of dependence α(k) α= 14 ⎛ cπ 2 ⎞ km = ⎜ ⎟ . 8) ⎝4 L⎠ At usual c ~ 10 10 erg· cm–3 , ~ c · a 2, a 2 ~ 10 –15 cm 2, L ~ 10 –1 cm, the period of the resultant structure d = π /k m has the order of 10 –4 cm which corresponds to the experimentally observed domain dimensions [2, 12, 16].