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

Continuous Vortices with Broken Symmetry in Rotating Superfluid 3He-A

New NMR measurements are reported on continuous 3He-A vortices in tilted magnetic fields. We introduce a symmetry classification of the continuous vortices with broken axial symmetry. It is found that the discrete internal symmetry may in addition be broken in two inequivalent ways, producing two different continuous vortices. Although NMR may not distinguish between these two vortices, the observed vortex satellite peak is well accounted for by spin waves localized in the soft core of such vortices.Peer reviewe

Diatoms and acid lakes. Proceedings of a workshop of the 8th International Diatom Symposium, Paris, September 1984

Diatom analysis of lake sedimentshasbecome one of the most important techniques used in the contemporary debate on lake acidification. In recent years the relationship between diatom assemblages and pH has been quantified allowing the pH history of individual lakes to be reconstructed. Much reliance is placed on these reconstructions yet we have little understanding of the causal mechanisms that underlie the impressive statistics. There has been little research on the ecology and physiology of diatom taxa in acid and acidifying waters and we have little information on the role of either planktonic or non-planktonic diatoms in acid lake ecosystems. This Workshop, hence, was not only devoted to the exchange of information on current research projects but also to a discussion of some of the ecological questions that require resolution to improve our understanding of the diatom:pH relationship

Theory and practice in Finnish teacher education : a rhetorical analysis of changing values from the 1960s to the present day

Our aim in this article is to examine the relationship between theory and practice in Finnish teacher education from the 1960s to the present. We identify four different periods to represent this relationship based on our analysis of national committee and evaluation reports. Theory and practice gradually converged, culminating in a research-based agenda that reflects the current situation. This relationship between theory and practice also reflects the ideal teacher of different times and what a teacher's working orientation should be. We use rhetorical analysis and the concept of a 'philosophical pair' introduced by Perelman and Olbrechts-Tyteca to interpret this development.Peer reviewe

The ground state of the Lithium atom in strong magnetic fields

The ground and some excited states of the Li atom in external uniform magnetic fields are calculated by means of our 2D mesh Hartree-Fock method for field strengths ranging from zero up to 2.35 10^8 T. With increasing field strength the ground state undergoes two transitions involving three different electronic configurations: for weak fields the ground state configuration arises from the field-free 1s^22s configuration, for intermediate fields from the 1s^22p_{-1} configuration and in high fields the 1s2p_{-1}3d_{-2} electronic configuration is responsible for the properties of the atom. The transition field strengths are determined. Calculations on the ground state of the Li+ ion allow us to describe the field-dependent ionization energy of the Li atom. Some general arguments on the ground states of multi-electron atoms in strong magnetic fields are provided.Comment: 11 pages, 6 figures, submitted to Physical Review

The ground state of the carbon atom in strong magnetic fields

The ground and a few excited states of the carbon atom in external uniform magnetic fields are calculated by means of our 2D mesh Hartree-Fock method for field strengths ranging from zero up to 2.35 10^9 T. With increasing field strength the ground state undergoes six transitions involving seven different electronic configurations which belong to three groups with different spin projections S_z=-1,-2,-3. For weak fields the ground state configuration arises from the field-free 1s^2 2s^2 2p_0 2p_{-1}, S_z=-1 configuration. With increasing field strength the ground state involves the four S_z=-2 configurations 1s^22s2p_0 2p_{-1}2p_{+1}, 1s^22s2p_0 2p_{-1}3d_{-2}, 1s^22p_0 2p_{-1}3d_{-2}4f_{-3} and 1s^22p_{-1}3d_{-2}4f_{-3}5g_{-4}, followed by the two fully spin polarized S_z=-3 configurations 1s2p_02p_{-1}3d_{-2}4f_{-3}5g_{-4} and 1s2p_{-1}3d_{-2}4f_{-3}5g_{-4}6h_{-5}. The last configuration forms the ground state of the carbon atom in the high field regime \gamma>18.664. The above series of ground state configurations is extracted from the results of numerical calculations for more than twenty electronic configurations selected due to some general energetical arguments.Comment: 6 figures,acc. Phys.Rev.

Curriculum and Teacher Education Reforms in Finland That Support the Development of Competences for the Twenty-First Century

Abstract This chapter analyzes how learning twenty-first century competences has been implemented in the Finnish educational context through the enactment of national and local level curricula and the design of a teacher education development program in a decentralized education system, in which teachers, schools, municipalities, and universities have high autonomy. The curricula and development program emphasize learning twenty-first century competences. Both were designed in collaboration with Finnish teachers and teacher educators, representatives from the Ministry of Education and Culture, the Association of Finnish Local and Regional Authorities, the Teacher’s Union, the Student’s Unions, and the Principal Association. The major actions taken to implement these changes included piloting, seminars and conferences, having different support and local level collaborations, and networking. According to recent evaluations, both endeavors – the development of national and local level curricula and a teacher education development program – have resulted in progress towards implementing twenty-first century competences in schools and for teacher education.Peer reviewe