Decay Rates and Probability Estimates¶for Massive Dirac Particles¶in the Kerr–Newman Black Hole Geometry

The Cauchy problem is considered for the massive Dirac equation in the non-extreme Kerr–Newman geometry, for smooth initial data with compact support outside the event horizon and bounded angular momentum. We prove that the Dirac wave function decays in at least at the rate t ^−5/6. For generic initial data, this rate of decay is sharp. We derive a formula for the probability p that the Dirac particle escapes to infinity. For various conditions on the initial data, we show that p = 0, 1 or 0 < p < 1. The proofs are based on a refined analysis of the Dirac propagator constructed in [4]. Received: 20 August 2001 / Accepted: 22 January 2002 RID="*" ID="*"Present address: NWF I – Mathematik, Universit?t Regensburg, 93040 Regensburg, Germany.?E-mail: felix.finster@mathematik.uni-regensburg.de RID="**" ID="**"Research supported by NSERC grant # RGPIN 105490-1998. RID="***" ID="***"Research supported in part by the NSF, Grant No. DMS-0103998. RID="****" ID="****"Research supported in part by the NSF, Grant No. 33-585-7510-2-30.     

Hypersurfaces of prescribed Gauß curvature in exterior domains

We prove an existence theorem for convex hypersurfaces of prescribed Gau? curvature in the complement of a compact set in Euclidean space which are close to a cone. Received: 23 February 2001 / Accepted: 11 May 2001 / Published online: 19 October 2001     

Size control and catalytic activity of bio-supported palladium nanoparticles

The development of nanoparticles has greatly improved the catalytic properties of metals due to the higher surface to volume ratio of smaller particles. The production of nanoparticles is most commonly based on abiotic processes, but in the search for alternative protocols, bacterial cells have been identified as excellent scaffolds of nanoparticle nucleation, and bacteria have been successfully employed to recover and regenerate platinum group metals from industrial waste. We report on the formation of bio-supported palladium (Pd) nanoparticles on the surface of two bacterial species with distinctly different surfaces: the gram positive Staphylococcus sciuri and the gram negative Cupriavidus necator. We investigated how the type of bacterium and the amount of biomass affected the size and catalytic properties of the nanoparticles formed. By increasing the biomass:Pd ratio, we could produce bio-supported Pd nanoparticles smaller than 10nm in diameter, whereas lower biomass:Pd ratios resulted in particles ranging from few to hundreds of nm. The bio-supported Pd nanoparticle catalytic properties were investigated towards the Suzuki-Miyaura cross coupling reaction and hydrogenation reactions. Surprisingly, the smallest nanoparticles obtained at the highest biomass:Pd ratio showed no reactivity towards the test reactions. The lack of reactivity appears to be caused by thiol groups, which poison the catalyst by binding strongly to Pd. Different treatments intended to liberate particles from the biomass, such as burning or rinsing in acetone, did not re-establish their catalytic activity. Sulphur-free biomaterials should therefore be explored as more suitable scaffolds for Pd(0) nanoparticle formation.     

An Efficient Microwave-Assisted Synthesis of Structurally Diverse Styrylquinolines

Summary. Aromatic aldehydes undergo condensation with quinaldines under microwave irradiation to afford structurally diverse styrylquinolines in high yields under solvent-free conditions. A comparison with the conventional method clearly indicates the advantages of the new protocol.     

Non-Existence of Black Hole Solutions¶for a Spherically Symmetric, Static Einstein–Dirac–Maxwell System

We consider for j=?, … a spherically symmetric, static system of (2j+1) Dirac particles, each having total angular momentum j. The Dirac particles interact via a classical gravitational and electromagnetic field. The Einstein–Dirac–Maxwell equations for this system are derived. It is shown that, under weak regularity conditions on the form of the horizon, the only black hole solutions of the EDM equations are the Reissner–Nordstr?m solutions. In other words, the spinors must vanish identically. Applied to the gravitational collapse of a “cloud” of spin-?-particles to a black hole, our result indicates that the Dirac particles must eventually disappear inside the event horizon. Received: 2 November 1998 / Accepted: 23 February 1999     

Nonexistence of time‐periodic solutions of the Dirac equation in an axisymmetric black hole geometry

We prove that in the nonextreme Kerr‐Newman black hole geometry, the Dirac equation has no normalizable, time‐periodic solutions. A key tool is Chan‐drasekhar's separation of the Dirac equation in this geometry. A similar nonexistence theorem is established in a more general class of stationary, axisymmetric metrics in which the Dirac equation is known to be separable. These results indicate that, in contrast to the classical situation of massive particle orbits, a quantum mechanical Dirac particle must either disappear into the black hole or escape to infinity. © 2000 John Wiley & Sons, Inc.     

On the Support of Minimizers of Causal Variational Principles

A class of causal variational principles on a compact manifold is introduced and analyzed both numerically and analytically. It is proved under general assumptions that the support of a minimizing measure is either completely timelike, or it is singular in the sense that its interior is empty. In the examples of the circle, the sphere and certain flag manifolds, the general results are supplemented by a more detailed and explicit analysis of the minimizers. On the sphere, we get a connection to packing problems and the Tammes distribution. Moreover, the minimal action is estimated from above and below.     

Photoelectron intensity spectroscopy  analytical and structural information from peis applied to zeolites

X-ray photoelectron spectroscopy (XPS or ESCA) based on calculated photoionization cross sections was used to investigate the surface Si/Al ratio of different zeolites and the cation distribution in Ag- and Ca-exchanged NaA-zeolites. We find the same module at the surface as in the bulk, but a strong de-alumination of the surface after HCl treatment. The Ag⁺ ions are enriched and the Na⁺ ions depleted in the outer cavities of the zeolite crystals, both in strong dependence on the Ca content.     

Oxidation of etched silicon in air at room temperature; Measurements with ultrasoft X-ray photoelectron spectroscopy (ESCA) and neutron activation analysis

The oxide growth rate for p-type Si〈100〉 crystals after HF etching in air at room temperature was measured for exposures betwen 3 min and ≈ 10³ min after etching. The oxide thicknesses were determined with traditional Al Kα excited XPS and two nontraditional methods — especially suited for very thin layers: Zr Mζ excited XPS and neutron activation analysis (NAA). The oxide thicknesses (mainly SiO_x with x < 2) lie between 0.1 and 0.55 nm with a logarithmic growth rate of ≈ 0.2 nm/decade.