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

A study of the breakdown of the quasi-static approximation at high densities and its effect on the helium-like K ALPHA complex of nickel, iron, and calcium

The General Spectral Modeling (GSM) code employs the quasi-static approximation, a standard, low-density methodology that assumes the ionization balance is separable from a determination of the excited-state populations that give rise to the spectra. GSM also allows for some states to be treated only as contributions to effective rates. While these two approximations are known to be valid at low densities, this work investigates using such methods to model high-density, non-LTE emission spectra and determines at what point the approximations break down by comparing to spectra produced by the LANL code ATOMIC which makes no such approximations. As both approximations are used by other astrophysical and low-density modeling codes, the results should be of broad interest. He-like K$\alpha$ emission spectra are presented for Ni, Fe, and Ca, in order to gauge the effect of both approximations employed in GSM. This work confirms that at and above the temperature of maximum abundance of the He-like ionization stage, the range of validity for both approximations is sufficient for modeling the low- and moderate-density regimes one typically finds in astrophysical and magnetically confined fusion plasmas. However, a breakdown does occur for high densities; we obtain quantitative limits that are significantly higher than previous works. This work demonstrates that, while the range of validity for both approximations is sufficient to predict the density-dependent quenching of the z line, the approximations break down at higher densities. Thus these approximations should be used with greater care when modeling high-density plasmas such as those found in inertial confinement fusion and electromagnetic pinch devices.Comment: Accepted by Physical Review A (http://pra.aps.org/). 11 pages + LANL cover, 5 figures. Will update citation information as it becomes available. Abbreviated abstract is listed her

Momentum losses by charge exchange with neutral particles in H-mode discharges at JET

Introduction Extensive investigations both in theory and experiments have been done in recent years to identify the influence of rotation on plasma performance. Results have shown that a radial velocity gradient can play a role in the suppression of turbulent transport and profile stiffness [1]. It remains however largely unknown what processes determine the shape and magnitude of the observed rotation profile. With the presence of an inwards momentum pinch [2], the edge momentum density is observed to contribute significantly to the global confinement [3]. Therefore, in order to accurately predict the observed rotation profile, a better understanding of the processes that determine the edge rotation is needed. Several components play a role in momentum transport in the edge. The dominant source of torque at JET is provided by NBI, while radial transport by outwards diffusivity and an inwards convective pinch redistribute the momentum density. At the edge, a continuous sink is present in the form of charge-exchange (CX) interactions between plasma ions and a neutral particle background. Besides these direct losses, the penetration of low energy neutrals into the plasma periphery is also believed to play a role in several plasma models related to the pedestal shape [4] and the L-H transition [5]. In this paper, the results from a qualitative neutral transport model are used to assess the penetration of neutral atoms into the plasma edge. A forward model of the passive charge-exchange emission [6] is used in order to quantify the neutral density and to estimate the magnitude of momentum and energy losses by CX interactions