Parabolic Hole Research Articles

Elastic fields for a parabolic hole endowed with surface effects

Elastic fields for a parabolic hole endowed with surface effects

Transmission and Reflection of Wave Packets by Asymmetric Semi-Harmonic Potential Barriers

We study the scattering of a particle by a square barrier potential that has a parabolic hole in between. The barrier is parameterized in such a form that the hole can be partially or completely removed. This can be also chosen to be symmetric or asymmetric with respect to the hole. Some expressions for the phase time are given and the time spent by the particle in the interaction zone of the barrier is calculated. It is shown that the related time delay depends on the symmetry operations that one can do on the potential.

Mobility spectrum analytical approach for intrinsic band picture of Ba(FeAs)2

Unconventional high temperature superconductivity as well as three-dimensional bulk Dirac cone quantum states arising from the unique d-orbital topology have comprised an intriguing research area in physics. Here we apply a special analytical approach using a mobility spectrum, in which the carrier number is conveniently described as a function of mobility without any hypothesis, both on the types and the numbers of carriers, for the interpretations of longitudinal and transverse electric transport of high quality single crystal Ba(FeAs)2 in a wide range of magnetic fields. We show that the majority carriers are accommodated in large parabolic hole and electron pockets with very different topology as well as remarkably different mobility spectra, whereas the minority carriers reside in Dirac quantum states with the largest mobility as high as 70,000 cm2(Vs). The deduced mobility spectra are discussed and compared to the reported sophisticated first principle band calculations.

Broadband antireflective structures with hole for highly transparent glasses

Optimal sub-wavelength structures (SWS) with parabolic hole on glass substrate are designed using effective medium theory (EMT), which will reduce the reflectance over wavelength range from 300 to 1200nm and 0° to 80° angle of incidence. The performance of SWS integrated glass is verified by genetic algorithm. The optimized SWS with the cross-sectional profile of parabolic hole have parameters with ∼120nm diameter and 300nm height. The average reflectance over wavelength range from 300 to 1200nm and 0° to 80° angle of incidence could be reduced to only 0.5% by applying the optimized SWS onto glass.

X-ray collimation by crystals with precise parabolic holes based on diffractive–refractive optics

Two crystals with precise parabolic holes were used to demonstrate sagittal beam collimation by means of a diffractive-refractive double-crystal monochromator. A new approach is introduced and beam collimation is demonstrated. Two Si(333) crystals with an asymmetry angle of α = 15° were prepared and arranged in a dispersive position (+,-,-,+). Based on theoretical calculations, this double-crystal set-up should provide tunable beam collimation within an energy range of 6.3-18.8 keV (Θ(B) = 71-18.4°). An experiment study was performed on BM05 at ESRF. Using 8.97 keV energy, the beam profile at various distances was measured. The experimental results are in good agreement with theoretical predictions. Owing to insufficient harmonic suppression, the collimated (333) beam was overlapped by horizontally diverging (444) and (555) beams.

Effect of thermal treatment on hardness and fracture toughness of a poly-para-phenylene-copolymer

Up to now, poly-para-phenylene (PPP) can be considered as the stiffest and strongest polymer under ambient temperature conditions. As an amorphous thermoplastic, a thermal treatment will not change the material’s morphology, but may influence its inner free volume (physical aging) and corresponding mechanical properties. This was investigated in this study with regard to the hardness and fracture toughness of PPP. It turned out that under the thermal conditions chosen, only moderate effects on these properties could be detected. Besides it was of interest how PPP compares in these properties relative to other advanced thermoplastics (neat and short glass fiber reinforced). Various features, such as striations and parabolic holes, on the fracture surfaces of PPP were also discussed.

Diffractive-refractive optics: X-ray collimator

Diffractive-refractive optics are x-ray focusing monochromators based on the diffraction on profiled crystal surface. Diffraction on longitudinal parabolic groove machined in crystal surface forms a sagittaly focused synchrotron radiation beam. Such kind of monochromator may be realized as a crystal with parabolic hole, where the beam is diffracted on the inner wall of the hole. Two such asymmetrically cut crystals set into antiparallel position, creating a dispersive (+,-,-,+) arrangement, form a sagittaly focusing x-ray monochromator which should be practically aberration-free. The focusing properties of such kind of monochromator are discussed in detail and it is shown for the first time that it can be used not only for focusing but also for creating highly parallel monochromatic beam in the broad region of the Bragg angles. This device with parabolic hole has not been tested experimentally yet.

Optimization of compound refractive lenses for X-rays

The theory of compound refractive lenses for the focusing of Synchrotron radiation X-ray beams is reviewed. The optimization of the transmission and minimization of the image size is discussed for both circular and parabolic holes. A comparison is made with ray tracing. The small-angle scattering expected from the surface roughness and the density fluctuation of the material is addressed. A successful comparison is made with the transmission and spot sizes measured on focusing lenses made of beryllium and Pyrocarbone between 20 and 30 keV on the ID6 undulator beamline of the ESRF. A rms spot size of 6.8 μm×100 μm has been recorded at a photon energy of 23.8 keV in a two plane focusing geometry. The peak spectral flux per unit surface in the focused beam is 26 times higher than that observed without lens. The expected transmission of such lenses placed at a distance of 24 m from an undulator or a bending magnet source point of the ESRF is given in the 1–100 keV energy range for focal length varying between 1 and 24 m. These lenses are inexpensive, very flexible and insensitive to heat load. They are particularly well suited to the undulator beamlines of the third generation sources in the 1–200 keV range. Future developments are discussed.

XII.—Paraboloidal Co-ordinates and Laplace's Equation

SynopsisIn this paper we examine the general paraboloidal co-ordinate system, in which the normal surfaces are elliptic or hyperbolic paraboloids, including as special cases the “parabolic plate” and the “plate with a parabolic hole”. We then show that normal solutions of Laplace's equation in these co-ordinates are given as products of three Mathieu functions, and apply this to the solution of boundary-value problems for Laplace's equation in these co-ordinates. In a subsequent paper the corresponding treatment of the wave equation will be given.