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1 Springer Series on AtoIUs+PlasDlas 12 Editor: J. Peter Toennies Springer Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Santa Clara Singapore Tokyo

2 Springer Series on,atodls+plasdlas Editors: G. Ecker P. Lambropoulos I. I. Sobelman H. Walther Managing Editor: H. K. V. Lotsch Polarized Electrons 2nd Edition By J. Kessler 2 Multlpboton Processes Editors: P. Lambropoulos and S. J. Smith 3 Atomic Many-Body Theory 2nd Edition By I. Lindgren and 1. Morrison 4 Elementary Processes in Hydrogen-Helium Plasmas Cross Sections and Reaction Rate Coefficients By R. K. Janev, W. D. Langer, K. Evans, Jf. and D. E. Post, Jr. 5 Pulsed Electrical Discharge in Vacuum By G. A. Mesyats and D. I. ProskuJOvsky 6 Atomic and Molecular Spectroscopy 2nd Edition Basic Aspects and Practical Applications By S. Svanberg 7 Interference of Atomic States By E. B. Alexandrov, M. P. Chaika and G. I. Khvostenko II Resonance Phenomena in Electron-Atom Collisions By V. I. Lengyel, V. T. Navrotsky and E. P. Sabad 12 Atomic Spectra and Radiative Transitions 2nd Edition By Sobelman 13 Multipboton Processes in Atoms By N. B. Delone and V. P. Krainov 14 Atoms in Plasmas By V. S. Lisitsa IS Excitation of Atoms and Broadening of Spectral Lines By I. I. Sobelman, L. Vainshtein and E. Yukov 16 Reference Data on Multlcbarged Ions By V. G. Pal'chikov and V. P. Shevelko 17 Lectures on Non-linear Plasma Kinetics By V. N. Tsytovich 8 Plasma Pbyslcs 2nd Edition Basic Theory with Fusion Applications By K. Nishikawa and M. Wakatani 9 Plasma Spectroscopy The Influence of Microwave and Laser Fields By E. Oks 10 Film Deposition Tecbnlques by Plasma By M. Konuma

3 Igor I. Sobelman Atomic Spectra and Radiative Transitions Second Edition With 26 Figures Springer

4 Professor Dr. Igor I. Sobelman Lebedev Physical Institute, Russian Academy of Sciences, Leninsky Prospekt 53, Moscow, Russia Guest Editor: Professor Dr. J. Peter Toennies Max-Planck-Institut fur Stromungsforschung, Bunsenstrasse 10, Gottingen, Germany Managing Editor: Dr.-Ing. Helmut K. V. Lotsch Springer-Verlag, Tiergartenstrasse 17, Heidelberg, Germany Series Editors: Professor Dr. Gunter Ecker Ruhr-Universitiit Bochum, Fakultiit fur Physik und Astronomie, Lehrstuhl Theoretische Physik I,Universitiitsstra6e 150, D-4480 I Bochum, Germany Professor Peter Lambropoulos, Ph. D. Max-Planck-Institut fur Quantenoptik, Garching, Germany, and Foundation for Research and Technology -Hellas (FO.R.T.H.), Institute of Electronic Structure & Laser (IESL) University of Crete, PO Box 1527, Heraklion, Crete 71110, Greece Professor Igor I. Sobelman Lebedev Physical Institute,Russian Academy of Sciences, Leninsky Prospekt 53, Moscow, Russia Professor Dr. Herbert Walther Universitiit MUnchen, Sektion Physik, Am Coulombwall I, GarchinglMUnchen, Germany Library of Congress Cataloging-in-Publication Data. Sobelman, L L ([gor- llich), Atomic spectra and radiative transitions'l Igor L Sobelman. - 2nd ed. 1992, corr. print p. cm. - (Springer series on atoms + plasmas; 12) [ncludes bibliographical referencesand index. [SBN I 8-2 (Berlin : pbk. : alk. paper) J. Atomic spectroscopy. 2. Radiative transitions. L Title. II. Series QC454.A8S '028'7 - dc The first edition appeared as Springer Series in Chemical Physics, Vol. [ Second Edition 1992 Corrected Printing 1996 ISBN-13: e-isbn-13: DOl: / This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights oftranslation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg 1972, The use of general descriptive names. registered names. trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant. protective laws and regulations and therefore free for general use. SPIN / I 0 - Printed on acid-free paper

5 Preface to the Second Edition Recent years have witnessed increased activity in the investigation of the spectra of multiply charged ions. These ions are of special interest to many modern fields of research, such as X-ray space astronomy and astrophysics, controlled thermonuclear fusion, extreme ultraviolet and soft X-ray lasers. Studies of the structure of highly charged heavy ions are stimulated by the unique possibilities provided by modern accelerators and beam-foil spectroscopic techniques. This edition of the book includes two new chapters. In Chap. 10, basic information on the Dirac equation for an electron in a Coulomb field is given, together with the treatment of relativistic corrections in the theory of atomic spectra. The reader will gain an understanding of the basic concepts used in the relativistic theory of highly charged ions. Chapter 11 is devoted to an overview of the spectra of multiply charged ions. No attempt has been made here to reference the vast number of experimental investigations and calculations reported in the literature. Instead, attention is paid to the main features of the spectra of highly charged ions that distinguish them from the spectra of neutral atoms. Among them.are the so-called satellite structure and forbidden lines, which appear in the spectra due to violation of the selection rules for radiative transitions by relativistic effects. The author is very grateful to Dr. Helmut Lotsch. Without his efforts and encouragement this revised edition would not have been realized. Moscow, April 1991 l. I. Sobelman

6 Preface to the First Edition My previous book on the theory of atomic spectra was published in Russian about fifteen years ago. Besides the traditional problems usually included in a book on atomic spectroscopy, some other problems arising in various applications of spectroscopic methods were also discussed in the book. These include, for example, continuous spectrum radiation, excitation of atoms, and spectral line broadening. Extensive revisions were made in the English version of the book published by Pergamon Press in 1972, especially in the chapter devoted to the problem of excitation of atoms. This book is intended as the first part of a two-volume presentation of the theory of atomic spectra, atomic radiative transitions, excitation of atoms, and spectral line broadening. The aim in preparing these new books has been to stress the problems connected with the most interesting applications of atomic spec.troscopy to plasma diagnostics, astrophysics, laser physics, and other fields, which have been developed very intensively in recent years. The content of this first volume, devoted to the systematics of atomic spectra and radiative transitions, is similar to that of Chapters 1-6, 8 and 9 of the old book, but considerable revision has been made. Some sections, such as those on the Hartree-Fock method, the Dirac equation, and relativistic corrections, have been deleted. At the same time, more attention is paid to radiative transitions. More extensive tables of oscillator strengths, probabilities, and effective cross sections of radiative transitions in discrete and continuous spectra are given. The book is based on the courses of lectures on atomic spectroscopy and connected problems which the author and L.A. Vainshtein gave at the Moscow Physics and Technology Institute, and reflects the changes in these courses in recent years. As a rule, references are made only to monographs, reviews, and papers whose results are used in the text. In conclusion, I wish to express sincere thanks to Dr. V. I. Kogan, who prepared Sect. 9.5, and to Dr. E.A. Yukov, who helped me to prepare Sects. 9.6 and 9.7. Moscow, November Sobelman

7 Contents Part I Elementary Information on Atomic Spectra Chapter 1 The Hydrogen Spectrum 1.1 SchrMinger's Equation for the Hydrogen Atom Energy Levels Wave Functions Series Regularities Radiative Transition Selection Rules Spectral Series of the Hydrogen Atom Hydrogenlike Ions Fine Structure Velocity Dependence of Electron Mass Spin-Orbit Interaction Fine Structure. Selection Rules Lamb Shift Chapter 2 Systematics of the Spectra of Multielectron Atoms 2.1 Central Field Central Field Approximation Parity of States Systematics of Electron States in a Central Field Electrostatic and Spin-Orbit Splitting in the LS Coupling Approximation Spectral Terms. LS Quantum Numbers Fine Structure of Terms Finding the Terms of Multielectron Configurations Radiative Transitions jj Coupling Approximation Various Coupling Schemes Systematics of Electron States with jj Coupling

8 X Contents Chapter 3 Spectra of Multielectron Atoms 3.1 Periodic System of Elements Spectra of the Alkali Elements Term Scheme Series Regularities Fine Structure Copper, Silver, and Gold Spectra Spectra of the Alkaline Earth Elements He Spectrum Spectra of the Alkaline Earth Elements Zinc, Cadmium, and Mercury Spectra Spectra of Elements with p Valence Electrons One p Electron Outside Filled Shells Configuration p Configuration p Configuration p Configuration ps Configuration p Spectra of Elements with Unfilled d and f Shells Elements with Unfilled d Shells Elements with Unfilled f Shells Part II Theory of Atomic Spectra Chapter 4 Angular Momenta 4.1 Angular Momentum Operator. Addition of Angular Momenta Angular Momentum Operator Orbital Angular Momentum Electron Spin Addition of Two Angular Momenta Addition of Three or More Angular Momenta Angular Momentum Vector Addition Coefficients Clebsch-Gordan and Associated Coefficients Summary of Formulas for 3j Symbols Racah W Coefficients and 6j Symbols :4 Summary of Formulas for 6j Symbols j Symbols Irreducible Tensor Operators

9 Contents XI Spherical Tensors Matrix Elements Some Examples of Calculation of Reduced Matrix Elements Tensor Product of Operators Matrix Elements with Coupled Angular Momenta Direct Product of Operators Chapter 5 Systematics of the States of MultielectroD Atoms 5.1 Wave Functions Central Field Approximation Two-Electron Wave Functions in LSMLMs Representation Two-Electron Wave Functions in mm'sms Representation Multielectron Wave Functions in a Parentage Scheme Approximation Fractional Parentage Coefficients Classification of Identical Terms of [R Configuration According to Seniority (Seniority Number) Matrix Elements of Symmetric Operators Statement of the Problem F Matrix Elements. Parentage Scheme Approximation F Matrix Elements. Equivalent Electrons Q Matrix Elements. Parentage Scheme Approximation Q Matrix Elements. Equivalent Electrons Summary of Results Electrostatic Interaction in LS Coupling. Two-Electron Configuration Coulomb and Exchange Integrals Configuration Mixing Electrostatic Interaction in LS Coupling. Multielectron Configuration Configurations In and [RI' More Than Half Filled Shells Filled (Closed) Shells Applicability of the Single-Configuration Approximation Multiplet Splitting in LS Coupling Preliminary Remarks Lande Interval Rule One Electron Outside Closed Shells Configuration In " Parentage Scheme Approximation Fine-Structure Splitting of Levels of He Spin-Spin and Spin-Other Orbit Interactions jj Coupling Wave Functions Spin-Orbit and Electrostatic Interactions

10 XII Contents 5.7 Intermediate Coupling and Other Types of Coupling Transformations Between LS and jj Coupling Schemes Intermediate Coupling jf Coupling Experimental Date Other Types of Coupling Chapter 6 Hyperfine Structure of Spectral Lines 6.1 Nuclear Magnetic Dipole and Electric Quadrupole Moments Magnetic Moments Quadrupole Moments Hyperfine Splitting General Character of the Splitting Calculation of the Hyperfine Splitting Constant A Calculation of the Hyperfine Splitting Constant B Radiative Transitions Between Hyperfine-Structure Components Isotope Shift of the Atomic Levels Chapter 7 The Atom in an External Electric Field 7.1 Quadratic Stark Effect Hydrogenlike Levels. Linear Stark Effect Inhomogeneous Field. Quadrupole Splitting Time-Dependent Field... " Amplitude Modulation... '" The Hydrogen Atom in a Rotating Electric Field Chapter 8 The Atom in an External Magnetic Field 8.1 Zeeman Effect Paschen-Back Effect Strong Field Splitting of Hyperfine Structure Components in a Magnetic Field 198 Chapter 9 Radiative Transitions 9.1 Electromagnetic Radiation Quantization of the Radiation Field Radiative Transition Probabilities Correspondence Principle for Spontaneous Emission Dipole Radiation

11 Contents XIII Stimulated Emission and Absorption Effective Cross Sections of Absorption and Stimulated Emission Electric Dipole Radiation Selection Rules, Polarization and Angular Distribution Oscillator Strengths and Line Strengths LS Coupling Approximation. Relative Intensities of Multiplet Components One Electron Outside Closed Shell Multielectron Configurations. Different Coupling Schemes Relative Intensities of Zeeman and Stark Components of Lines Multipole Radiation Fields of Electric and Magnetic Multipole Moments Intensity of Multipole Radiation Selection Rules Electric Multipole Radiation Magnetic Dipole Radiation Transitions Between Hyperfine Structure Components. Radio Emission from Hydrogen Calculation of Radiative Transition Probabilities Approximate Methods Three Ways of Writing Formulas for Transition Probabilities Theorems for Sums of Oscillator Strengths Semiempirical Methods of Calculating Oscillator Strengths Electric Dipole Transition Probabilities in the Coulomb Approximation Intercombination Transitions Continuous Spectrum Classification of Processes Photorecombination and Photoionization: General Expressions for Effective Cross Sections Bremsstrahlung: General Expressions for Effective Cross Sections Radiation and Absorption Coefficients Photorecombination and Photoionization: Hydrogenlike Atoms Photorecombination and Photoionization: Nonhydrogenlike Atoms Bremsstrahlung in a Coulomb Field Formulas for Q Factors Symmetry and Sum Rules LS Coupling. Allowed Transitions jl Coupling Tables of Oscillator Strengths and Radiative Transition Probabilities Transition Probabilities for the Hydrogen Atom Radiative Transition Probabilities in the Bates-Damgaard Approximation

12 XIV Contents Oscillator Strengths and Probabilities of Some Selected Transitions Effective Cross Sections and Rates of Photorecombination Chapter 10 Relativistic Corrections in the Spectroscopy of Multicbarged Ions to.l Dirac Equation. Pauli Equation Dirac Equation Electron Spin to.1.3 Non-Relativistic Approximation. Pauli Equation Central Field to.2.1 Non-Relativistic Approximation to.2.2 Second Approximation with Respect to vic. Fine Splitting Dirac Equation for a Central Field to.2.4 Coulomb Field. Energy Levels, Fine Splitting to.2.5 Coulomb Field. Radial Functions Relativistic Corrections Calculation of Some Radial Integrals to.3.2 Hyperfine Splitting Constant A ,.3.3 Hyperfine Splitting Constant B Nucleus Finite-Size Correction to.3.5 Radiative Corrections. Lamb Shift Chapter 11 Spectra of Multicbarged Ions 11.1 Energy Levels Forbidden Transitions H-like Ions He-like Ions Satellite Structure References List of Symbols Subject Index