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arXiv:physics/9710046 · Journal of Physics B Atomic Molecular and Optical Physics

Hydrogen atom moving across a strong magnetic field: analytical approximations

Alexander Y. Potekhin

Published Jan 14, 1998• 50 citations
arXivHTMLPDF

Abstract

Analytical approximations are constructed for binding energies, quantum-mechanical sizes and oscillator strengths of main radiative transitions of hydrogen atoms arbitrarily moving in magnetic fields - . Examples of using the obtained approximations for determination of maximum transverse velocity of an atom and for the evaluation of absorption spectra in magnetic neutron-star atmospheres are presented.

DOI
10.1088/0953-4075/31/1/008

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Physical Review A•Apr 10, 2006•31 citations
Analytical approximations are constructed for binding energies, quantum-mechanical sizes and oscillator strengths of main radiative transitions of hydrogen atoms arbitrarily moving in magnetic fields - . Examples of using the obtained approximations for determination of maximum transverse velocity of an atom and for the evaluation of absorption spectra in magnetic neutron-star atmospheres are presented.
For pt.I see ibid., vol.17, p.29-52 (1984). Wavelengths, dipole strengths, oscillator strengths and transition probabilities for electromagnetic transitions between low-lying states of the hydrogen atom embedded in a magnetic field of arbitrary strength (0<B<or=4.70*10 8 T) are calculated with high numerical accuracy for a sufficiently dense mesh of B values using the wavefunctions and energy values determined in paper I by expanding the wavefunctions in terms of spherical harmonics (weak and moderate fields), or in terms of Landau states (strong and very strong fields). Effects caused by the finiteness of the proton mass are taken into account, and the scaling laws are quoted by the aid of which the results presented can be utilised to cover quantitatively the whole hydrogenic sequence in intense magnetic fields; finally, the quality of the results of previous calculations is checked.
A hydrogen atom moving across a magnetic field is considered in a wide region of magnitudes of magnetic field and atom momentum. We solve the Schr\"odinger equation of the system numerically using an imaginary time method and find wave functions of the lowest states of atom. We calculate the energy and the mean electron-nucleus separation as a function of atom momentum and magnetic field. All the results obtained could be summarized as a phase diagram on the ``atom-momentum -- magnetic-field'' plane. There are transformations of wave-function structure at critical values of atom momentum and magnetic field that result in a specific behavior of dependencies of energy and mean interparticle separation on the atom momentum P.P.P. We discuss a transition from the Zeeman regime to the high magnetic field regime. A qualitative analysis of the complicated behavior of wave functions vs PPP based on the effective potential examination is given. We analyze a sharp transition at the critical momentum from a Coulomb-type state polarized due to atom motion to a strongly decentered (Landau-type) state at low magnetic fields. A crossover occurring at intermediate magnetic fields is also studied.
Using the numerical techniques that have been successful in elucidating the low-lying part of the level scheme of atoms in magnetic fields pertaining to white dwarf stars and neutron stars, we have computed the energies and wave functions of ``circular'' Rydberg states (i.e., with \ensuremath{\Vert}m\ensuremath{\Vert}=n-1) with \ensuremath{\Vert}m\ensuremath{\Vert}=24--35 for laboratory fields up to several tens of teslas. We can, therefore, follow a number of physical properties, such as transition frequencies and radiative decay times, of circular states as functions of the magnetic field strength and make predictions that should be verifiable experimentally.
Abstract The problem of the hydrogen atom in a magnetic field can be treated satisfactorily by approximate methods which are quite accurate for very low or very high magnetic field strengths. Before this paper the only useful treatments for intermediate field strengths were variational methods giving upper bounds to the energies. As described in the paper the diffusion QMC method was applied in the same way as in earlier solutions for the hydrogen atom, but with additional magnetic terms added to the Hamiltonian expression. These include complex expressions and the required wavefunction includes a corresponding factor. With the magnetic field aligned along one axis, integration over the angle ϕ in (r, θ, ϕ) coordinates reduces the additional terms to real functions of r and z as well as the magnetic quantum number m. In the diffusion/multiplication processes of a random walk procedure these may be simply included with the potential energy term. In this study the energy at each step was determined from the average multiplication factor for the walkers. Importance sampling was not used. Results were obtained for quantum numbers m = 0, 1 at each of three magnetic field strengths. A comparison of results showed good agreement with the lowest values obtained in several of the earlier analytic variational calculations.
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.
We have studied the radiative decay of atomic hydrogen in strong magnetic fields of up to 4T4\phantom{\rule{0.3em}{0ex}}\mathrm{T}4T. We have followed the radiative cascade from completely l,ml,ml,m mixed distributions of highly excited states as well as from distributions that involve highly excited states with $\ensuremath{\mid}m\ensuremath{\mid}\ensuremath{\sim}n$. We have found that the time it takes to populate the ground state is not affected by the magnetic field for the initial states with $n\ensuremath{\lesssim}20$. For higher nnn manifolds, the electrons in the most negative mmm states are substantially slowed down by the magnetic field resulting in a much longer lifetime. We show that less than 10% of the antihydrogen atoms with $n\ensuremath{\sim}35$ generated in antihydrogen experiments at 4K4\phantom{\rule{0.3em}{0ex}}\mathrm{K}4K will decay to their ground states before they hit the wall of the vacuum container unless they are trapped. We have also found that the decay time is mainly determined by the fraction of atoms that were initially in highest negative mmm states due to the fact that only $\ensuremath{\mid}\ensuremath{\Delta}m\ensuremath{\mid}+\ensuremath{\mid}\ensuremath{\Delta}\ensuremath{\pi}\ensuremath{\mid}=1$ transitions are allowed in the magnetic field. We give a semiclassical method for calculating the decay rates for circular states and show that when the initial states have high-$\ensuremath{\mid}m\ensuremath{\mid}$, semiclassical rates agree with the full quantum mechanical rates within a couple of percent for states with effective $n\ensuremath{\gtrsim}20$.