Table Of Contents
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Series Editors: I. Appenzeller . M. Harwit . R. Kippenhahn
P. A. Strittmatter . V. Trimble
AS
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\S1 R()PI-[YSI( S IJBI{AI Y
LIBRARY
Series Editors: I. Appenzeller . M. Harwit . R. Kippenhahn
P. A. Strittmatter . V Trimble
The Solar System
By T. Encrenaz and J.-P. Bibring
Astrophysical Concepts 2nd Edition
By M. Harwit
Physics and Chemistry of Comets
Editor: W F. Huebner
Stellar Structure and Evolution
By R. Kippenhahn and A. Weigert
Moderu Astrometry
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By P. Lena
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Editor: A. Petschek
General Relativity, Astrophysics, and Cosmology
By A. K. Raychaudhuri, S. Banerji and A. Banerjee
Tools of Radio Astronomy
By K. Rohlfs
Atoms in Strong Magnetic Fields
Quantum Mechanical Treatment and Applications
in Astrophysics and Quantum Chaos
By H. Ruder, G. Wunner, H. Herold and F. Geyer
The Stars
By E. L. Schatzman and F. Praderie
Physics of the Galaxy and Interstellar Matter
By H. Scheffler and H. Elsasser
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Editors: G. L. Verschuur and K. 1. Kellermann
H. Ruder G. Wunner H. Herold
F. Geyer
Atoms in
Strong Magnetic Fields
Quantum Mechanical Treatment
and Applications
in Astrophysics and Quantum Chaos
With 93 Figures
Springer-Vedag
Berlin Heidelberg New York
London Paris Tokyo
Hong Kong Barcelona
Budapest
Professor Dr. Hanns Ruder
Professor Dr. Heinz Herold
Dr. Florian Geyer
Theoretische Astrophysik, UniversiHit Tiibingen, Auf der Morgenstelle 10
D-72076 Tiibingen, Germany
Professor Dr. Giinter Wunner
Theoretische Physik, Ruhr-UniversiHit Bochum, Lehrstuhl I
D-44780 Bochum, Germany
Series Editors
Immo Appenzeller Peter A. Strittmatter
Landessternwarte, Konigstuhl Steward Observatory
D-69117 Heidelberg, Germany The University of Arizona
Tucson, AZ 85721, USA
Martin Harwit
Virginia Trimble
The National Air and Space Museum
Smithsonian Institution Astronomy Program
7th St. and Independence Ave. S. W. University of Maryland
Washington, DC 20560, USA College Park, MD 20742, USA
and Department of Physics
Rudolf Kippenhahn University of California
Irvine, CA 92717, USA
Rautenbreite 2
D-37077 G6ttingen, Germany
ISBN-13: 978-3-642-78822-2 e-ISBN-13: 978-3-642-78820-8
DOl: 10.1007/978-3-642-78820-8
Library of Congress Cataloging·in·Publication Data. Atoms in strong magnetic fields: quantum mechanical
treatment and applications in astrophysics and quantum chaos 1 H. Ruder ... [et al.]. p. cm. Includes bibliographical
references and index.
\. Atomic transition probabilities. 2. Stars-Magnetic fields. 3. White dwarfs. 4. Neutron stars. 5. Quantum chaos.
I. Ruder, Hanns. QC454.N8A77 1994 539.7'23-dc20 94·23259
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Preface
In this book we summarize the essential results of our efforts over the years to
calculate energies, wave functions, and electromagnetic transitions of atoms as
functions of the magnetic field strength from laboratory fields up to neutron
star magnetic fields. Motivated by the observational evidence of huge magnetic
fields with strengths up to 105 T in the vicinity of white dwarf stars and of up
to 109 T in the vicinity of neutron stars the authors, together with coworkers
and candidates for doctor and diploma degrees, have investigated this ,fasci
nating quantum mechanical problem more or less continuously since 1978. The
extensive tables and figures in the appendices represent the most complete data
set to date in this field of research. For practical use all numbers are available
by "anonymous ftp" over Internet.
The first direct measurement of a neutron star magnetic field by Trum
per and his group, who observed a cyclotron feature at about 50 ke V in the
spectrum of the X-ray pulsar Hercules X-I corresponding to a field strength of
several 108 T, stimulated investigations of atoms within the framework of the
adiabatic approximation, which is well justified for such field strengths. This
method and its results are discussed in Chaps. 3, 5, and 6.
Angel's idea to explain mysterious lines in the spectra of highly magnetized
white dwarf stars in terms of "stationary" Zeeman components in the hydrogen
spectrum increased our efforts to also calculate with high accuracy atomic
properties in the complex transition regime where Coulomb and Lorentz forces
become comparable. In Chap. 7 we summarize the state of the art of this new
spectroscopy of stationary lines which became possible using our atomic data
and helped to pin down the field strengths of white dwarf stars in the range
from 3 x 103 T to 105 T.
A further stimulus came from laboratory experiments with highly excited
Rydberg atoms where the transition to the magnetically dominated regime
can already be performed with magnetic field strengths of just a few Tesla.
The almost enthusiastic study of such systems has two reasons: firstly, the
transition between quantum and classical physics can be investigated in detail
and, secondly, the hydrogen atom is the simplest physically real system which,
in the classical limit, changes from regular to chaotic behaviour with increasing
magnetic field strength. Therefore, it is an ideal object with which to study
phenomena of "quantum chaos" both experimentally and theoretically. A short
review of this new and wide field of research is given in Chap. 10.
While our knowledge of hydrogen-like systems in magnetic fields of arbi
trary strength is - apart from some minor points - fairly good and complete,
VI Preface
the situation for helium-like systems is far less satisfying. In Chap. 9 and in
Appendix 4 the results for helium available so far are documented.
This book contains the most comprehensive compilation to date of atomic
data for hydrogen, helium, and their isoelectronic sequences in strong mag
netic fields. Nevertheless, even more complete and accurate data are urgently
needed for detailed analyses of spectra of magnetic white dwarfs, which will
be obtained with increasing quality by present and future modern telescopes.
Moreover, such data are also prerequisite for quantitative studies of the ther
mal spectra of isolated neutron stars with strong cosmic magnetic fields. These
spectra have been observed by the ROSAT satellite, and will be observed even
more extensively and with higher accuracy by future X-ray satellites. Given
the progress of numerical methods and the still dramatically increasing power
of supercomputers, we feel well equipped to meet these challenges in the next
few years.
Finally, we would like to express our thanks to Wolfgang Schweizer, who has
done much work in this field together with us as a postdoc, and to our coworkers
Moritz Braun, Peter FaBbinder, Hans-Georg Forster, Susanne Friedrich, Roland
Niemeier, Heinrich Korbel, Peter Proschel, Wolfgang Rosner, Wolfgang Stru
pat, Giinther Thurner, Joachim Tworz, and Gudrun Zeller, each of whom has
added a more or less extensive mosaic piece to this book in form of a diploma
or PhD thesis, and last but not least to an army of students for their help in
preparing the figures, the tables, and the references.
Tiibingen and Bochum, July 1994 The authors
Contents
1. Introduction .............................................. 1
1.1 Magnetic Fields of Compact Cosmic Objects ............. 1
1. 2 Historical Review ..................................... 6
1.3 Notations and Abbreviations ........................... 11
2. Interacting Charged Particles in Uniform Magnetic Fields 13
2.1 The N-Body Problem ................................. 13
2.2 The Uncharged Two-Body Problem ..................... 14
2.2.1 Hamiltonian and Wave Functions
in Centre-of-Mass and Relative Coordinates ....... 15
2.2.2 Classification, Quantum Numbers and Degeneracy
in the Non-Interacting Case ..................... 15
2.2.3 Exact Solution for Harmonic Interaction .......... 17
2.3 Scaling Properties of the Coulomb Problem .............. 19
2.3.1 The General Case K I- 0 ....................... 19
2.3.2 The Special Case K = 0 ........................ 20
2.3.3 Nuclear Charge (Z) Scaling ..................... 21
3. Methods of Solution for the Magnetized Coulomb Problem 23
3.1 General Considerations ................................ 23
3.1.1 Characteristic Domains
of the Magnetic Field Strength .................. 23
3.1.2 Expansions of the Wave Functions
and Multiconfiguration Equations ................ 24
3.1.3 Low-Field High-Field Correspondence ............ 26
3.2 Numerical Treatment .................................. 29
3.2.1 Hartree-Fock-Like Methods ...................... 29
3.2.2 Coupled-Channels-Like Methods ................. 31
3.2.3 Diagonalization Methods ........................ 32
4. Results for Low-Lying States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.1 Energy Values ........................................ 35
4.2 Wavelengths of the Hydrogen Atom ..................... 39
4.3 Wave Functions of the Hydrogen Atom .................. 44
4.3.1 Graphic Representation of the Spatial
Probability Distribution of the Electron ........... 46
4.3.2 Pictorial Representation of the Spatial
Probability Distribution of the Electron ........... 49
VIII Contents
5. Energies for Arbitrarily Excited States
in Adiabatic Approximation . . .. . . . . . . .. . . . . . . . . . .. . . . .. . . . . 57
5.1 Asymptotic Property of the Effective Potentials ....... . . . 57
5.2 Numerical Results and Their Accuracy .................. 58
6. Electromagnetic Transition Probabilities . . . . . . . . . . . . . . . . . . . . . 69
6.1 The General Expressions ............................... 69
6.2 Effects of the Finite Proton Mass
on the 'fransition Matrix Element ...................... . 72
6.3 Results ............................................. . 74
6.4 Expressions for Electromagnetic 'fransitions
in Adiabatic Approximation ........................... . 78
6.4.1 Dipole Strengths .............................. . 78
6.4.2 Selection Rules ............................... . 79
6.4.3 Sum Rules ................................... . 79
6.4.4 Asymptotic Formulae .......................... . 80
6.4.5 Results and Discussion ........................ . 82
7. Stationary Lines and White Dwarf Spectra . . . . . . . . . . . . . . . . . . . 89
7.1 Stationary Lines ...................................... 89
7.2 Spectra of Selected Magnetic White Dwarfs .............. 95
7.2.1 Grw+7008247 ................................. 95
7.2.2 PG 1031+234 ................................. 99
7.2.3 SBS 1349+5434 ................................ 100
7.2.4 PG 1015+014 ................................. 101
7.2.5 G227-35·. . . . . . .. . .. . . . . . . . . . .. . . . . . . . . . . . . . . . 101
7.2.6 MR Serpentis .................................. 104
7.2.7 LB 11146 (PG 0945+245) ....................... 105
7.3 Table of Magnetic White Dwarfs ........................ 106
7.4 Future Work ......................................... 107
8. Relativistic Effects, Nuclear Mass Effects,
and Landau-Excited States ... , ... . . . ... . . . . . . .. . . . . . . .. . . . . 109
8.1 Spin-Orbit Coupling .................................. 109
8.2 Effects of the Finite Proton Mass
and of Motion Perpendicular to the Magnetic Field ....... 113
8.3 Landau-Excited States ................................ 116
9. Helium-Like Atoms in Magnetic Fields
of Arbitrary Strengths ..................................... 123
9.1 Correspondence Diagrams ............................. 123
9.1.1 Short Review of the One-Electron Problem ........ 123
9.1.2 The Two-Electron System in a Magnetic Field .... . 124
9.1.3 The Correspondence for 1/Z = 0 ................ . 126
9.1.4 The Correspondence for the General Case ........ . 127
Contents IX
9.2 Method of Solution ................................... 129
9.2.1 The Hartree-Fock Method
for Low to Intermediate Field Strengths Ulz ;:S 1) 130
9.2.2 The Hartree-Fock Method
for High Field Strengths Ulz ~ 1) ................ 132
9.3 Dipole Strengths, Oscillator Strengths,
and Transition Probabilities ............................ 134
9.3.1 Selection Rules for the Spherical Ansatz .......... 135
9.3.2 Selection Rules for the Cylindrical Ansatz ......... 135
9.4 Results for the Two-Electron Problem ................... 136
9.4.1 Energy Levels Calculated
with a Single-Configuration Ansatz ............... 136
9.4.2 Influence of Configuration Mixing ................ 137
9.4.3 Comparison of Energy Values Obtained
by Different Methods ........................... 140
9.4.4 Wavelengths, Dipole Strengths, Oscillator Strengths,
and Transition Probabilities ..................... 143
10. Highly Excited States ...................................... 153
10.1 Results .............................................. 154
10.1.1 Energy Levels ................................. 154
10.1.2 Transition Probabilities
and Comparison with Experiments ............... 155
10.2 Is There Chaos in Quantum Mechanics? ................. 165
10.2.1 Introduction ................................... 165
10.2.2 Microwave Ionisation of Rydberg States
of the Hydrogen Atom .......................... 167
10.2.3 Statistical Analysis of Energy-Level Sequences ..... 169
10.2.4 Order and Chaos in the Hydrogen Atom
in a Magnetic Field ............................ 172
10.2.5 Level Statistics for the Hydrogen Atom
in Magnetic Fields ............................. 174
10.2.6 Resonances in Chaos - the Role of Periodic Orbits . 175
10.2.7 "Scarring" of Wave Functions 177
Outlook 183
AI. Energy Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
ALl Tables of the Energy Values ............................ 185
A1.2 Figures of the Energy Values ........................... 217
A2. Wavelengths .............................................. 225
A2.1 Figures of the Wavelengths ............................. 225
A2.2 Tables of Stationary Wavelengths ....................... 231
A2.3 Figures of Stationary Wavelengths ...................... 236
x
Contents
A3. Electromagnetic Transition Probabilities . . . . . . . . . . . . . . . . . . . . . 241
A3.1 Wavelengths, Dipole Strengths,
Oscillator Strengths, and Transition Rates ............... 241
A3.2 Oscillator Strengths and Transition Probabilities
in Adiabatic Approximation ............................ 253
A3.3 Dipole Strengths of Stationary Transitions ............... 259
A4. Helium and Helium-Like Atoms ............................. 263
A4.1 Tables of the Energy Values ............................ 263
A4.2 Wavelengths, Dipole Strengths,
Oscillator Strengths, and Transition Rates ............... 283
References .................................................... 293
Subject Index ................................................. 307
