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

Tritium concentration measurements in the JET divertor by optical spectroscopy of a Penning discharge

Obtaining precision measurements of the relative concentrations of hydrogen, deuterium, tritium, and helium in the divertor of a tokamak are an important task for nuclear fusion research. Control of the deuterium-tritium isotopic ratio while limiting the helium ash content in a fusion plasma are key factors for optimizing the fuel burn in a fusion reactor, like the International Tokamak Experimental Reactor (ITER). A diagnostic technique has been developed to measure the deuterium-tritium isotopic ratio in the divertor of the Joint European Torus (JET) with a species-selective Penning vacuum gauge. The Penning discharge provides a source of electrons to excite the neutral hydrogen isotopes in the pumping duct. Subsequently, the visible light from the hydrogen isotopes is collected in an optical fiber bundle, transferred away from the tokamak into a low radiation background area, and analyzed in a high resolution Czerny-Turner spectrometer, which is equipped with a fast charge coupled device (CCD) camera for optical detection. The intensity of the observed line emission (D{sub {alpha}} -- 6561.03 {angstrom}; and T{sub {alpha}} -- 6560.44 {angstrom}) is directly proportional to the partial pressure of each gas found in the divertor. The line intensity of each isotope is calibrated as a function of pressure. The ratio of the line intensities thus provides a direct measurement of the deuterium-tritium isotopic ratio. The lower limit for the determination of the deuterium-tritium isotopic ratio is about 0.5%. The applicable pressure range for this system is from 10{sup {minus}5} mbar to a few times 10{sup {minus}3} mbar