Exact solution of the Zeeman effect in single-electron systems

Contrary to popular belief, the Zeeman effect can be treated exactly in single-electron systems, for arbitrary magnetic field strengths, as long as the term quadratic in the magnetic field can be ignored. These formulas were actually derived already around 1927 by Darwin, using the classical picture of angular momentum, and presented in their proper quantum-mechanical form in 1933 by Bethe, although without any proof. The expressions have since been more or less lost from the literature; instead, the conventional treatment nowadays is to present only the approximations for weak and strong fields, respectively. However, in fusion research and other plasma physics applications, the magnetic fields applied to control the shape and position of the plasma span the entire region from weak to strong fields, and there is a need for a unified treatment. In this paper we present the detailed quantum-mechanical derivation of the exact eigenenergies and eigenstates of hydrogen-like atoms and ions in a static magnetic field. Notably, these formulas are not much more complicated than the better-known approximations. Moreover, the derivation allows the value of the electron spin gyromagnetic ratio $g_s$ to be different from 2. For completeness, we then review the details of dipole transitions between two hydrogenic levels, and calculate the corresponding Zeeman spectrum. The various approximations made in the derivation are also discussed in details.Comment: 18 pages, 4 figures. Submitted to Physica Script

Rosenbrock time integration for unsteady flow simulations

This contribution compares the efficiency of Rosenbrock time integration schemes with ESDIRK schemes, applicable to unsteady flow and fluid-structure interaction simulations. Compared to non-linear ESDIRK schemes, the linear implicit Rosenbrock- Wanner schemes require subsequent solution of the same linear systems with different right hand sides. By solving the linear systems with the iterative solver GMRES, the preconditioner can be reused for the subsequent stages of the Rosenbrock-Wanner scheme. Unsteady flow simulations show a gain in computational efficiency of approximately factor three to five in comparison with ESDIRK

Determination of the trap-assisted recombination strength in polymer light emitting diodes

The recombination processes in poly(p-phenylene vinylene) based polymer light-emitting diodes (PLEDs) are investigated. Photogenerated current measurements on PLED device structures reveal that next to the known Langevin recombination also trap-assisted recombination is an important recombination channel in PLEDs, which has not been considered until now. The dependence of the open-circuit voltage on light intensity enables us to determine the strength of this process. Numerical modeling of the current-voltage characteristics incorporating both Langevin and trap-assisted recombination yields a correct and consistent description of the PLED, without the traditional correction of the Langevin prefactor. At low bias voltage the trap-assisted recombination rate is found to be dominant over the free carrier recombination rate.

Fokker-Planck equation with variable diffusion coefficient in the Stratonovich approach

We consider the Langevin equation with multiplicative noise term which depends on time and space. The corresponding Fokker-Planck equation in Stratonovich approach is investigated. Its formal solution is obtained for an arbitrary multiplicative noise term given by $g(x,t)=D(x)T(t)$, and the behaviors of probability distributions, for some specific functions of $D(x)$% , are analyzed. In particular, for $D(x)\sim | x| ^{-\theta /2}$, the physical solutions for the probability distribution in the Ito, Stratonovich and postpoint discretization approaches can be obtained and analyzed.Comment: 6 pages in LATEX cod

Confluence reduction for Markov automata

Markov automata are a novel formalism for specifying systems exhibiting nondeterminism, probabilistic choices and Markovian rates. Recently, the process algebra MAPA was introduced to efficiently model such systems. As always, the state space explosion threatens the analysability of the models generated by such specifications. We therefore introduce confluence reduction for Markov automata, a powerful reduction technique to keep these models small. We define the notion of confluence directly on Markov automata, and discuss how to syntactically detect confluence on the MAPA language as well. That way, Markov automata generated by MAPA specifications can be reduced on-the-fly while preserving divergence-sensitive branching bisimulation. Three case studies demonstrate the significance of our approach, with reductions in analysis time up to an order of magnitude

The Diffuse Gamma-Ray Background from Supernovae

The Cosmic Gamma-ray Background (CGB) in the MeV region is believed to be due to photons from radioactivity produced in SNe throughout the history of galaxies in the universe. In particular, gamma-ray line emission from the decay chain 56Ni-> 56Co->56Fe provides the dominant photon source. Although iron synthesis occurs in all types of SNe, the contribution to the CGB is dominated by SNIa events due to their higher photon escape probabilities. Estimates of the star formation history in the universe suggest a rapid increase by a factor \~ 10 from the present to a redshift z_p ~ 1.5, beyond which it either remains constant or decreases slowly. We integrate the observed star formation history to determine the CGB from the corresponding SN rate history. In addition to gamma-rays from short-lived radioactivity in SNIa and SNII/Ibc we also calculate the minor contributions from long-lived radioactivities (26Al, 44Ti, 60Co, and electron-positron pair annihilation). Although progenitor evolution for SNIa is not yet fully understood, various arguments suggest delays of order 1-2 Gy between star formation and the production of SNIa's. The effect of this delay on the CGB is discussed. We emphasize the value of gamma-ray observations of the CGB in the MeV range as an independent tool for studies of the cosmic star formation history. If the delay between star formation and SNIa activity exceeds 1 Gy substantially, and/or the peak of the cosmic star formation rate occurs at a redshift much larger than unity, the gamma-ray production of SNIa would be insufficient to explain the observed CGB. Alternatively, the cosmic star formation rate would have to be higher (by a factor 2-3) than commonly assumed, which is in accord with several upward revisions reported in the recent literature.Comment: Minor changes, 26 pages, 9 figures, Accepted by Ap