Three Point Boundary Value Problems on Time Scales

This work formulates existence theorems for solutions to three-point boundary value problems on time scales. The ideas are based on a relationship between the three point boundary conditions, lower and upper solutions and topological degree theory.     

Third-order right-focal multi-point problems on time scales

We are concerned with the existence and form of positive solutions to a third-order multi-point boundary-value problem on time scales with mixed derivatives. We find and utilize the Green function for the corresponding homogeneous right-focal problem as the kernel of an integral equation of Hammerstein-type. Two examples are included to illustrate the results.     

Parabolic microstrain-like line broadening induced by random twin faulting

The broadening of diffraction lines of layer faulted and, in particular, of microtwinned microstructures is analysed in detail. Thereby, stacked layer models turn out to be valuable to simulate the diffracted intensity if the metrics of the layers pertaining to the two types of domains are described by an average metrics plus domain-specific deviations from that metrics. Characteristic changes occur for the line-broadening effects if the fault probability and the extent of deviations from the average layer metrics vary. Emphasis is put, in particular, on a regime of line broadening characterised by merging of otherwise split peaks to yield single ones for sufficiently high twin probability. These merged peaks show unusual quadratic/parabolic increase of the integral breadths with the reflection order (parabolic microstrain broadening). This type of broadening has been predicted previously for special types of microstrain broadening, and indeed the (metrical) distortions of the domains with respect to the average structure allow perceiving even the twin-faulted microstructures as, in some sense, microstrained ones. Moreover, the special situation of superstructures is analysed, where the above mentioned merging is only observed for the fundamental but not for the superstructure reflections. Inconsistent (with respect to lattice metrics) peak positions can be observed for fundamental and superstructure reflections, which can affect lattice parameters determined from diffraction patterns.     

On the thickness of ferroelastic twin walls in lead phosphate Pb3(PO4)2 an X-ray diffraction study

The effective thickness of ferroelastic twin-walls (W-walls) in Pb₃(PO₄)₂ at room temperature (≈ 0.7 T _c ) was determined by X-ray diffraction. The diffraction profiles were analysed using a wall profile e=eotanh x/w where W is the effective wall thickness including the effects of wall bending and surface relaxations. The experimental value W is around 10 unit cells which is larger than expected from standard renormalization arguments. The limitations of such experimental studies are discussed.     

Unusual domain evolution in semiconducting ferroelectrics: A phase field study

The effect of electrical conductivity on the domain evolution of semiconducting ferroelectrics is investigated using a phase field model which includes the drift of space charges. Phase field simulations show that the tail-to-tail 90° charged domain wall appears during the domain formation in the semiconducting ferroelectrics at zero field, which is prohibited in common insulating ferroelectrics. Due to the screening of polarization charges, the domain switching takes place through the motion of head-to-head 180° charged domain wall in the semiconducting single-domain ferroelectrics subjected to an electric field. Comparing to the insulating ferroelectrics, the semiconducting ferroelectrics have a lower speed of domain evolution due to the decrease of mobility of charged domain walls. The response of semiconducting ferroelectrics to a mechanical load is also found different from that of insulating ferroelectrics.     

Influence on the phase transitions of barium titanate of the manganese valency,introduced as a substituant in the titanium site

A small amount of manganese introduced in BaTiO₃ modifies strongly the vibrational and electronic properties of the pure compound. Dielectric response and EPR measurements are reported in order to study the influence of the manganese valency on the ferroelectric-paraelectric phase transition.     

CH3NH3 + as a quantum and classical rotor

Spin-lattice relaxation time T ₁ is determined for protons in three polycrystals (CH₃NH₃)₅Bi₂Cl₁₁, (CD₃NH₃)₅Bi₂Cl₁₁ and (CH₃ND₃)₅Bi₂Cl₁₁. The temperature dependence of the relaxation times obtained for (CH₃NH₃)₅Bi₂Cl₁₁ and (CD₃NH₃)₅Bi₂Cl₁₁ are interpreted as a result of correlated motions of the three-proton groups of the monomethylammonium cation. ²H NMR lines of (CD₃NH₃)₃Sb₂Br₉ have been recorded between 5 K and 291 K using solid echo method. The ²H NMR line shape analysis shows that characteristic shape of tunnelling methyl group appears at about 25 K and coming down with temperature up to 5 K is more distinct. From theoretical calculation, it has been found that in the quadrupolar constants is 161.3 kHz and tunnelling frequency is above 3 MHz.     

Boundedness of the Hessian of a Biharmonic Function in a Convex Domain

We consider the Dirichlet problem for the biharmonic equation on an arbitrary convex domain and prove that the second derivatives of the variational solution are bounded in all dimensions.