Chapter Five - Energy Levels of Light Atoms in Strong Magnetic Fields

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.

The energy levels of hydrogen and helium atoms in strong magnetic fields are calculated in this study. The current work contains estimates of the binding energies of the first few low-lying states of these systems that are improvements upon previous estimates. The methodology involves computing the eigenvalues and eigenvectors of the generalized two-dimensional Hartree-Fock partial differential equations for these one- and two-electron systems in a self-consistent manner. The method described herein is applicable to calculations of atomic structure in magnetic fields of arbitrary strength as it exploits the natural symmetries of the problem without assumptions of any basis functions for expressing the wave functions of the electrons or the commonly employed adiabatic approximation. The method is found to be readily extendable to systems with more than two electrons.

The energy levels of hydrogen and helium atoms in strong magnetic fields are calculated in this study. The current work contains estimates of the ground and first few excited states of these systems that are improvements upon previous estimates. The methodology involves computing the eigenvalues and eigenvectors of the generalized two-dimensional Hartree-Fock partial differential equations for these oneand two-electron systems in a self-consistent manner. The method described herein is applicable to calculations of atomic structure in magnetic fields of arbitrary strength as it exploits the natural symmetries of the problem without assumptions of any basis functions for expressing the wave functions of the electrons or the commonly employed adiabatic approximation. The method is found to be readily extendable to systems with more than two electrons.

Magnetic white dwarfs possess strong magnetic fields which can vary in the wide range 100 T–105T. Atoms and molecules of their atmospheres which are exposed to such extreme fields severely change their electronic structure and properties. In order to understand the observed spectra of magnetic cosmic objects it is indispensable to perform detailed investigations on atoms in strong magnetic fields. As a recent example, we discuss the explanation of the absorption features of GD229 through the transitions of the Helium atom in a field of the order of 5⋅104 T. The complicated scenario of the ground state crossovers for multi-electron atoms in strong fields is outlined for the example of the carbon atom.

view Abstract Citations (47) References (27) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Finite-Velocity Effects on Atoms in Strong Magnetic Fields and Implications for Neutron Star Atmospheres Pavlov, G. G. ; Meszaros, P. Abstract We consider the effects of a finite velocity on the properties of atoms in the strong magnetic field characteristic of neutron stars. Whereas in the absence of significant center-of-mass velocities the atomic structure is determined by the cylindrical symmetry, the electric field induced by the finite motion breaks this symmetry and distorts the atomic structure. The resulting dependence of the total energy on a generalized momentum of the atom can be interpreted in terms of a mass anisotropy the atom becomes "heavier" when it moves across the magnetic field, the transverse mass being higher for the more excited states. The field-dependent mass anisotropy, together with the field dependence of the binding energy of the atom, leads to a bending of the trajectories of neutral atoms in nonuniform magnetic fields, tending to channel and retain them in regions of high field. It also leads to a number of thermodynamic and spectroscopic effects. In particular, the mass anisotropy introduces both quantitative spectroscopic changes relative to the stationary magnetized atom, such as additional shifts and broadening of photoionization edges and lines, as well as qualitative changes, such as new selection rules for radiative processes and for the annihilation of magnetic positronium. The ionization balance of atoms and ions in pulsar atmospheres may also be strongly influenced, which together with the opacity changes could lead to effects of significant importance for the modeling of neutron star atmospheres in magnetic fields of strength B ≳109 G. Publication: The Astrophysical Journal Pub Date: October 1993 DOI: 10.1086/173274 Bibcode: 1993ApJ...416..752P Keywords: ATOMIC PROCESSES; MAGNETIC FIELDS; STARS: NEUTRON; STARS: PULSARS: GENERAL full text sources ADS | data products SIMBAD (2)

Recent years have seen tremendous progress in studies of the properties of hydrogen atoms in strong magnetic fields. Decisive stimulus came from the discovery of huge magnetic fields in astrophysical “laboratories”, viz. field strengths of order ~107–109 T in neutron stars and of order ~102–104 T in white dwarf stars. At these field strengths the magnetic forces acting on an atomic electron outweigh the Coulomb binding forces even in low-lying states, and thus atomic structure is completely changed. On the other hand, the rapid advancement of high-resolution laser spectroscopy has made it possible to produce atoms in highly excited states, with principal quantum numbers ranging up to n ≅ 520 (in Ba I) [1], and therefore Rydberg states can be used to investigate the effects of magnetic dominance on atomic structure also in terrestrial laboratories with magnetic fields of a few Tesla, or less.

The problem of a multi‐electron atom in a strong magnetic field is treated within the framework of a new statistical model. Here, the charge density n is required to have axial symmmetry. The partial factorization ansatz, n(ϱ, z) = nϱ(ϱ) · nz(ϱ;z), when introduced into Kadomtsev's equation, leads to a second‐order, non‐linear partial differential equation for nz that is valid over an important range of field strengths. Analytical solutions for two limiting approximations are consistent with earlier qualitative studies. Some illustrative, numerical solutions for a less restrictive approximation reveal atomic structures analogous to accurate hydrogenic results, already in the literature.

In this talk I will discuss recent results on the magnetisation/current of large atoms in strong magnetic fields. It is known from the work (E. Lieb, J.P. Solovej, and J. Yngvason, “Asymptotics of heavy atoms in high magnetic fields: II. Semiclassical regions”, Commun. Math. Phys. (1994), no. 161, 77-124) of Lieb, Solovej and Yngvason that the energy and density of atoms in strong magnetic fields are given to highest order by a Magnetic Thomas Fermi theory (MTF-theory) when the magnetic field strength B and nuclear charge Z satisfy BZ -3 →0. It is, however, equally interesting to establish whether MTF-theory also gives the right asymptotic current. In this talk we will prove that this is indeed the case, at least for moderate magnetic fields. However, we will also prove that approximate ground states do not in general give the right asymptotics for the current.

We present a variational calculation, using Slater-type trial wavefunctions of the energy spectrum and ionization energy of the hydrogen atom in strong magnetic fields ranging from 10^ Gauss (typical of magnetic white dwarf surfaces) to 10 12 Gauss (typical of pulsar surfaces). In addition, we have calculated bound-bound transition probabilities. Furthermore, a method is developed to generalize the energy level results to any one-electron atom. In particular, results for He 11 are presented. We also present an exact calculation of the quadratic Zeeman wavelength displacements of the Balmer lines from DA white dwarfs in view of which the estimate of the surface magnetic fields is higher than that obtained by previous, less accurate calculations. Finally, the variational calculation is extended to two-electron atoms in strong magnetic fields: we present the energy spectrum and ionization energy of He I and the e l e c t r o n affinity of If- .

view Abstract Citations (51) References (2) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Energy Spectrum of Hydrogen-Like Atoms in a Strong Magnetic Field Surmelian, G. L. ; O'Conner, R. F. Abstract An expression for the energy spectrum of hydrogen-like atoms in strong magnetic fields is derived in terms of the energy spectrum of hydrogen. The development of a parametric expression for the latter is thus motivated. We present such an expression for the ground-state energy of hydrogen in magnetic fields B gauss, obtained by least-squares fittings of numerical results obtained previously from a variational calculation. Subject headings: magnetic fields - Zeeman effect Publication: The Astrophysical Journal Pub Date: June 1974 DOI: 10.1086/152934 Bibcode: 1974ApJ...190..741S full text sources ADS | Related Materials (1) Erratum: 1976ApJ...204..311S

A finite-basis-set method is introduced for calculations involving an electron in the presence of a finite-size nucleus. It is found that the inclusion of powers of the form ${\mathit{r}}^{\mathrm{\ensuremath{-}}\ensuremath{\gamma}+\mathit{n}}$ in the basis set, where \ensuremath{\gamma} is a real number and n an integer, is of fundamental importance, increasing the convergence in the energy eigenvalues by several orders of magnitude in the case of a large nuclear charge. This conclusion applies also to one-electron potential calculations such as Dirac-Hartree-Fock. The method is applied to calculations of the low-lying levels of hydrogenic atoms in strong magnetic fields. For B\ensuremath{\lesssim}${10}^{9}$${\mathit{Z}}^{2}$ G, the finite-nuclear-size correction is of the same order as that for B=0. For B\ensuremath{\gtrsim}${10}^{9}$${\mathit{Z}}^{2}$ G, the correction increases linearly with B for the ground state, and assumes a more complicated dependence for the excited states due to the presence of more than one peak in each radial wave function for B=0.

A variational basis with cylindrical symmetry is shown to provide highly accurate results for the lowest m=0 to -4 levels of a hydrogen atom in a strong magnetic field (B>2.35*105T). The simplicity of the calculation and formulae summarising the behaviour of the variational parameters make the results easy to reproduce.

Structure, dynamics and quantum chaos in atoms and molecules under strong magnetic fields

We present a new pseudospectral algorithm for the calculation of the structure of atoms in strong magnetic fields. We have verified this technique for one, two and three-electron atoms in zero magnetic fields against laboratory results and find typically better than one-percent accuracy. We further verify this technique against the state-of-the-art calculations of hydrogen, helium and lithium in strong magnetic fields (up to about $2\times 10^{6}$ T) and find a similar level of agreement. The key enabling advantages of the algorithm are its simplicity (about 130 lines of commented code) and its speed (about $10^2-10^5$ times faster than finite-element methods to achieve similar accuracy).

White Dwarfs for Physicists D. Koester. Magnetic White Dwarfs Observations in Cosmic Laboratories S. Jordan. Hydrogen in Strong Electric and Magnetic Fields and Its Application to Magnetic White Dwarfs S. Friedrich, et al. Helium Data for Strong Magnetic Fields Obtained by Finite Element Calculations M. Braum, et al.. The Spectrum of Atomic Hydrogen in Magnetic and Electric Fields of White Dwarf Stars P. Fassbinder, W. Schweizer. Neutron Star Atmospheres G. Pavlov. Hydrogen Atoms in Neutron Star Atmospheres: Analytical Approximations for Binding Energies A.Y. Potekhin. Absorption of Normal Modes in a Strongly Magnetized Hydrogen Gas T. Bulik, G. Pavlov. Electronic Structure of Light Elements in Strong Magnetic Fields P. Pouree, et al. From Field-Free Atoms to Finite Molecular Chains in Very Strong Magnetic Fields M.R. Godefroid. The National High Magnetic Field Laboratory - a Precis J.E. Crow. Self-Adaptive Finite Element Techniques for Stable Bound Matter-Antimatter Systems in Crossed Electric and Magnetic Fields J. Ackermann. A Computational Method for Quantum Dynamics of a Three-Dimensional Atom in Strong Fields V.S. Melezhik. 25 Additional Articles. Index.