Simple model and wave function for atoms in intense magnetic fields

Abstract

The structure of heavy atoms in strong magnetic fields of the order of ${10}^{12}$ G, such as are generally presumed to be present on pulsars, has been previously described in terms of a magnetic Thomas-Fermi (TF) model, suitably adapted from the normal TF model. We present here a model which is analogous to the well-known screening-theory approach for ordinary, "laboratory" atoms---the electrons occupy various orbits with a transverse dimension characterized by the radius of magnetic cyclotron orbits and an over-all dimension characterized by Bohr orbits. Not only does this model give results similar to, though quantitatively more reliable than, the magnetic TF model, but it provides more insight into the structure of atoms in strong magnetic fields. For instance, the number of electrons in the $n\mathrm{th}$ shell is seen to be proportional to ${n}^{4}$ instead of the usual $2{n}^{2}$ in the laboratory situation. This accounts for the tighter binding of such atoms, a result already derived in the magnetic TF model, but also goes further in suggesting a new kind of shell structure and a new Periodic Table for atoms in strong magnetic fields. Quantum mechanically, our model is based on energy minimization with a simple analytical wave function and this represents the first time a simple wave function has been suggested for heavy atoms in strong magnetic fields. The wave function consists of an antisymmetrized product of single-particle orbitals, each of which is a product of an ordinary hydrogenic radial function and a magnetic Landau orbital so that both the spherical Coulombic and cylindrical magnetic symmetry of the problem are taken into account. However, in analogy with Bohr theory, we motivate and discuss many of the results without explicit use of this wave function. An approximate electron density function is trivially established which is itself physically more reasonable than the corresponding magnetic TF function. Further, our model is valid over a broader range of magnetic-field values and its variational-bound character allows for both successive improvements in its predictions and for a knowledge of the sign of the error in the energy estimate. Useful scaling laws are presented so that, starting from the knowledge of the ground-state energy for some $Z$ and $B$, the energy for other combinations of $Z$ and $B$ can be estimated. A second major result of this paper is that the single-particle wave function that arises during the course of the development of the many-electron wave function is of interest in itself, combining as it does aspects of both spherical and cylindrical symmetry. This simple wave function seems to describe well the ground state of the hydrogen atom at all values of the magnetic field.