Abstract: Aim of this chapter is analytical representation of one wide class of geometric figures (lines, surfaces and bodies) and their complicated displacements. The accurate estimation of physical characteristics (such as volume, surface area, length, or other specific parameters) relevant to human organs is of fundamental importance in medicine. One central idea of this article is, in this respect, to provide a general methodology for the evaluation, as a function of time, of the volume and center of gravity featured by moving of one class of bodies used of describe different human organs.

Abstract: We show that a superconducting quantum nanowire undergoes a new type of BCS-BEC crossover each time when an electron subband approaches the Fermi surface. In this case the longitudinal Cooper-pair size drops by two-three orders of magnitude down to a few nanometers. This unconventional BCS-BEC crossover is driven by quantum-size effects rather than by tuning the fermion-fermion interaction.

Abstract: We study the effect, of electron confinement on the superconducting-to-normal phase transition driven by a magnetic field and/or on the current-carrying state of the superconducting condensate in nanowires. Our investigation is based on a self-consistent. numerical solution of the Bogoliubov-de Gennes equations. We show that, in a parallel magnetic field and/or in the presence of supercurrent the transition from superconducting to normal phase occurs as a cascade of discontinuous jumps in the superconducting order parameter for diameters D < 10 divided by 15 nm at T = 0. The critical magnetic held exhibits quantum-size oscillations with pronounced resonant enhancements.

Abstract: Strain fields introduced by coherent Ni4Ti3 precipitates in austenitic Ni-Ti are believed to be a possible origin of why the R-phase transformation is introduced as an extra step before transforming to the B19'. The presence of this strain field was already confirmed in the past by conventional transmission electron microscopy (TEM) techniques and measured quantitatively by high resolution TEM (HRTEM). This time the geometrical phase method is applied on HRTEM micrographs to measure the full 3D strain tensor of the strain fields. Since each atomic resolution micrograph only results in a 2D measurement of the strain, observations in two different zone orientations are combined to retrieve the 3 x 3 strain tensor. In this work observations in a [1-1 1](B2) and [1 0-1](B2) zone orientation are used and this in case of precipitates with a diameter of around 50nm. In a next step the measured strain tensor is compared to the calculated eigenstrain of the R-phase in reference to the B2 matrix. This comparison shows that the introduced strain is very similar to the eigenstrain of one R-phase variant. Since for both structures, Ni4Ti3 and R-phase, four orientation variants are possible, each variant of the R-phase is thus able to accommodate the strain field of one of the Ni4Ti3 variants.

Abstract: The surface of commercial Nitinol wires with original oxides and a thickness in the 30-190 nm range was investigated by different state of art TEM techniques. The oxide surface layer was identified as a combination of TiO and TiO2 depending on the processing of the wire. Between the core of the wires and the oxidized surface, an interfacial Ni3Ti nanolayer was observed while Ni nanoparticles are found inside the original oxide. The particle sizes, their distribution in the surface and the Ti-O stoichiometry were deduced from the analysis of the obtained data. Molecular dynamics calculations performed for evaluation of the stability of Ni particles relative to the atomic state revealed that a pure Ni particle has a lower energy than free Ni atoms inside the TiO2 lattice. The obtained results are discussed with respect to surface stability and Ni release in the human body.

Abstract: Electron continement and its effect on the superconducting-to-normal phase transition driven by a magentic field and/or a current is studied in nanowires. Our investigation is based on a self-consistent numerical solution of the Bogoliubov-de Gennes equations. We find that in a parallel magneitc field and/or in the presence of a supercurrent the transition from the superconducting to the normal phase occurs as a cascade of discontinuous jumps in the superconducting order parameter for diameters D < 10 divided by 15 nm at T = 0. The critical magentic field exhibits quantum-size oscillations with pronounced resonant enhancements as a function of the wire radius.

Abstract: Responses of mesoscopic superconducting rings and disks to perpendicular magnetic fields are studied by using the multiple-small-tunnel-junction method, in which transport properties of several small tunnel junctions attached to the sample are measured simultaneously. This allows us for a direct experimental observation of the paramagnetic supercurrent, which is closely related to the paramagnetic Meissner effect. The results are compared with numerical results based on the nonlinear Ginzburg-Landau theory.

Abstract: Lattice deformations and concentration gradients surrounding Ni4Ti3 precipitates grown by appropriate annealing in a Ni51Ti49 B2 austenite matrix are determined by a combination of TEM techniques. Quantitative Fourier analysis of HRTEM images reveals a deformed nanoscale region with lattice deformations up to 2% while EELS and EDX indicate a Ni depleted zone up to 150 nm away from the matrix-precipitate interface.