The inverse interior Dirichlet problem

Abstract

An infinite number of rotationally symmetrical functions 'IJ(r ,z) can be found which approximate a given functionf(z) on the optical axis Oz (r and z being cylindrical polar coordinates). The 'IJ (r,z) are given in the form of a finite Fourier-Bessel series and it is shown how a solution that can be used in electron optical design can be selected by carefully assigning values to the following parameters: the number of terms in the series, the period of the series, and the amount of smoothing introduced. In this way a compromise is established between the quality of the axial approximation, the magnitude of potential gradients to be contended with in zonal regions, and the sizes of apertures allowed. Two-foil, one-foil, and open lenses can be modeled, and given axial potential functions can be accommodated which are either (i) analytic, (ii) continuous but with piecewise continuous z derivatives, or (iii) in the form of a set of experimentally determined values. The method requires no matrix inversion, the computer programming is of a simple nature, and memory requirements are modest enough to allow implementation of the program on small desk-top computers.

Magnetic properties of La0.6Sr0.4MnO3 thin films on SrTiO3 and buffered Si substrates with varying thickness

“…6 The film grows coherently, storing the biaxial strain energy below a critical thickness ͑t c ͒, above which the transition temperature of the film reaches the bulk transition temperature, such as thick films 19 and bulk samples. Above t c , the elastic energy ͑due to the strain in the film, especially at the interface͒ in the strained film exceeds that in the core of misfit dislocation which would be introduced at the filmsubstrate interface.…”

“…On the other hand, the strain relaxation of lattice mismatch between the film and the substrate at the interfaces occurs as the film thickness increases beyond a certain critical thickness of the film. 6 The film grows coherently, storing the biaxial strain energy below a critical thickness ͑t c ͒. Above t c , the elastic energy ͑due to the strain in the film, especially at the interface͒ in the strained film exceeds that in the core of misfit dislocation which would be introduced at the film-substrate interface.…”

Remanence breakdown in granular alloys at magnetic percolation

“…In fact, strains are stronger nearer the substrate than further away from it ͑because of the stress relaxation associated with plastic deformations as the film grows͒. 28 An axial distortion larger than 0.01% is at least needed to induce significant uniaxial anisotropy. 24 5 either through the metallic matrix aided by the diluted paramagnetic atoms 29 or because the matrix itself becomes a weak ferromagnet 30 acting to couple the FM particles.…”

Exact Analytical Solution for the Critical Layer Thickness of a Lattice-Mismatched Heteroepitaxial Layer

“…Among the most celebrated are the Matthews and Blakeslee, 1 Van der Merwe, 2 People and Beam, 3,4 and Fischer et al 5 models for the equilibrium theory. Owing to its clarity and reasonably good agreement with experiments, the model established by Fischer et al 5 has been widely adopted.…”