Reference data on multicharged ions

Springer Series on At'-.IUS+PlasDlas Editor: Igor I. Sobel'man 16 Springer Series on At()lUs+Plas...as Editors: G. ...

Author: Vitaliĭ Gennadievich Palʹchikov | Vi︠a︡cheslav Petrovich Shevelʹko

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Springer Series on

At'-.IUS+PlasDlas Editor: Igor I. Sobel'man

16

Springer Series on

At()lUs+Plas...as Editors: G. Ecker

P. Lambropoulos

I. I. Sobel'man

H. Walther

Managing Editor: H. K. V. Lotsch Polarized Electrons 2nd Edition By J. Kessler

11

Resonance Phenomena in Electron-Atom Collisions By V. I. Lengyel, V. T. Navrotsky and E. P. Sabad

2

Multiphoton Processes Editors: P. Lambropoulos and S. J. Smith

12

3

Atomic Many-Body Theory 2nd Edition By I. Lindgren and J. Morrison

Atomic Spectra and Radiative Transitions 2nd Edition By I. I. Sobel'man

13

Multiphoton Processes in Atoms By N. B. Delone and V. P. Krainov

14

Atoms in Plasmas By V. S. Lisitsa

15

Pulsed Electrical Discharge in Vacuum By G. A. Mesyats and D. I. Proskurovsky

Excitation of Atoms and Broadening of Spectral Lines By I. I. Sobel' man, L. Vainshtein and E. Yukov

16

Atomic and Molecular Spectroscopy 2nd Edition Basic Aspects and Practical Applications By S. Svanberg

Reference Data on Multicharged Ions By V. G. Pal'chikov and V. Shevelko

17

Lectures on Nonlinear Plasma Kinetics By V. N. Tsytovich

4

5

6

Elementary Processes in Hydrogen-Helium Plasmas Cross Sections and Reaction Rate Coefficients By R. K. Janev, W. D. Langer, K. Evans, Jr. and D. E. Post, Jr.

7

Interference of Atomic States By E. B. Alexandrov, M. P. Chaika and G. I. Khvostenko

8

Plasma Physics 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. A. Oks

10

Film Deposition by Plasma Techniques ByM. Konuma

V.G. Pal'chikov

V.P. Shevelko

Reference Data

on Multicharged Ions

With 78 Figures and 92 Tables

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest

Professor Dr. Vitalij G. Pal'chikov

Dr. Vjatcheslav P. Shevelko

National Research Institute for Physical-Technical and Radiotechnical Measurements, Mendeleevo, 141570 Moscow Region, Russia

P.N. Lebedev Physics Institute Optical Oivision, Russian Academy of Science 117924 Moscow, Russia

Series Editors:

Professor Dr. Giinter Ecker Ruhr-Universitat Bochum, Institut fUr Theoretische Physik, Lehrstuhl I, Universitatsstrasse 150, 0-44801 Bochum-Querenburg, Germany

Professor Peter Lambropoulos, Ph.D. Max-Planck-Institut fUr Quantenoptik 0-85748 Garching, Germany, and Foundation of Research and Technology - Hellas (FO.R.T.H.) Institute of Electronic Structure and Laser (lESL) and University of Crete, PO Box 1527, Heraklion, Crete 71110, Greece

Professor Igor I. Sobel'man Lebedev Physical Institute, Russian Academy of Sciences, 117924 Leninsky Prospekt 53, Moscow, Russia

Professor Dr. Herbert Walther Sektion Physik der Universitat MUnchen, Am Coulombwall I, 0-85748 GarchingIMUnchen, Germany

Managing Editor: Dr. Helmut K.V. Lotsch Springer-Verlag, Tiergartenstrasse 17,0-69121 Heidelberg, Germany

ISBN 3-540-58259-2 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-58259-2 Springer-Verlag New York Berlin Heidelberg CIP data applied for This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned. specifically the rights of translation. reprinting. reuse of illustrations. recitation. broadcasting. reproduction on microfilm 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 1995

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Preface

This book provides the first comprehensive presentation of reference data on the radiative and collisional characteristics of multicharged positive ions: energy levels and transition probabilities, effective cross sections and corresponding rate coefficients of different elementary processes occurring in hot laboratory and astrophysical plasmas. Such data are required by plasma physicists and astrophysicists who deal with X -ray spectroscopy, thermonuclear fusion, laserproduced plasmas, modelling of plasma properties and the development of VUV and X-ray lasers, etc. The contents of the book can roughly be divided into two main parts. The first part contains the data on energy levels, wavelengths, Lamb shifts, transition probabilities and other spectroscopic characteristics of multicharged ions. In the second part the experimental and theoretical values of the cross sections and corresponding transition rate coefficients are given for excitation, ionization, charge transfer, dielectronic recombination and other processes involving multicharged ions. The book contains a large number of figures, tables and easily understood formulas which permit one to estimate atomic characteristics without complicated computer calculations. We would like to thank all those who contributed to the preparation of this volume. We especially thank L.P. Presnyakov, L.A Vainshtein, R. Schuch, E.A. Yukov, AM. Umov, D.B. Uskov, AA Papchenko, 0.1. Tolstikhin and Yu.P. Garbusov for useful discussions and for communicating the results of their calculations. It is a pleasure to thank E. Salzbom from the Giessen University (Germany) for providing useful experimental data on electron-ion-atom collisions. Thanks go also to N.S. Strusevitch for his skilful assistance in computer work and T.A Shergina for her valiant work in preparing the figures for this book. Mendeleevo Moscow February 1994

V.G. Pal'chikov V.P. Shevelko

Contents

1. Introduction 2.

....................................

1

Atomic Structure and Spectra .......................

3

2.1

Classification of Spectral Lines .................... 2.1.1 Notations .............................. 2.1.2 Satellites .............................. 2.2 Relativistic and Quantum Electrodynamical Corrections .. 2.2.1 The Dirac Energy of H-like Ions ............. 2.2.2 Self-Energy ............................ 2.2.3 Vacuum Polarization ...................... 2.2.4 Finite Nuclear-Mass and Nuclear-Size Corrections. 2.2.5 He-like Ions ............................ 2.2.6 Li-like Ions ............................ 2.3 Binding Energies of the Inner-Shell Electrons ......... 2.4 Multicharged Ions in Stationary External Fields ........ 2.4.1 Stark Effect. The Ground State . . . . . . . . . . . . . . . a) H-like Ions ........................... b) He-like Ions .......................... 2.4.2 Stark Effect. Excited States ................. a) The States of H-like Ions with j < n - 1/2 . . . . b) The States of H-like Ions with j = n - 1/2 ... c) He-like Ions .......................... 2.4.3 Stark Effect. Hyperfine Structure ............. 2.4.4 Zeeman Effect .......................... a) H-like Ions ........................... b) He-like Ions in the Ground State ........... 2.4.5 Multipole Electromagnetic Susceptibilities and Shielding Factors for Multicharged Ions ........ a) H-like Ions ........................... b) Few-Electron Ions ......................

3 3 5 6 6 7 11 16 20 43 48 58 58 58 60 62 62 64 64 66 67 67 68 69 70 72

VIII

3.

4.

S.

Contents

Transition Probabilities ............................ 3.1 Selection Rules ............................... 3.2 Allowed and Forbidden Transitions ................. 3.2.1 H-like Ions ............................. 3.2.2 Radiative Decays of the n = 2 States in He-like Ions 3.3 Two-Photon Transitions ......................... 3.3.1 Two-Photon Decay of the 2S1/2 State in H-like Ions 3.3.2 . Two-Photon Decay of the 2 I So States in He-like Ions .......................... 3.4 Semiempirical and Asymptotic Formulas for Oscillator Strengths and Transition Probabilities in H- and He-like Ions 3.4.1 H-like Ions ............................. 3.4.2 He-like Ions ............................ 3.5 Autoionization Probabilities ...................... 3.6 Branching Ratios of Inner-Shell Vacancies ............

74 74 76 76 89 94 94 97 100 100 101 103 116

Radiative Characteristics ........................... 4.1 Radiative Recombination ........................ 4.1.1 General Properties. Photoionization . . . . . . . . . . . . 4.1.2 The Kramers Formulas and the Gaunt Factor. . . . . 4.1.3 Theory and Experiment .................... 4.2 Dielectronic Recombination ...................... 4.2.1 DR Cross Sections and Rates ................ 4.2.2 Electric-Field (EF) and Electron-Density (ED) Effects ................................ 4.3 Bremsstrahlung ............................... 4.3.1 Basic Formulas .......................... 4.3.2 Screening Effects ........................ 4.4 Polarization of X-Ray Lines ...................... 4.5 Photon Polarization in Radiative Recombination .. . . . . . .

124 124 124 125 127 128 129

Electron-Ion Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Excitation ................................... 5.1.1 Excitation of Outer-Shell Electrons ............ 5.1.2 Excitation of Inner-Shell Electrons ............ 5.1.3 Resonant Excitation . . . . . . . . . . . . . . . . . . . . . . . 5.2 Ionization ................................... 5.2.1 Direct Ionization (01) ..................... 5.2.2 Excitation-Autoionization................... 5.2.3 Resonant Ionization . . . . . . . . . . . . . . . . . . . . . . . 5.3 Multiple Ionization ............................

147 147 147 149 153 156 156 159 160 162

133 136 137 138 140 144

Contents

IX

6.

Ion-Atom Collisions ............................... 6.1 Electron Capture .............................. 6.1.1 Collisions with H and He .................. 6.1.2 Collisions with Multielectron Atoms ........... 6.2 Ionization ................................... 6.3 Excitation ...................................

166 166 168 171 174 177

7.

Ion-Ion Collisions ................................ 7.1 Electron Capture .............................. 7.2 Ionization ................................... 7.3 Excitation ................................... 7.4 Collisions Involving H- Ions ..................... 7.4.1 H+ + H- Collisions . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 H- + H- Collisions. . . . . . . . . . . . . . . . . . . . . . . 7.4.3 Collisions of H- with Multicharged Ions . . . . . . . .

180 180 183 184 192 192 193 194

References

198

Subject Index

213

Glossary of Terms

Units The system of atomic units (a.u.) is used: e Length (Bohr radius) Energy Rydberg Time Velocity Cross section Fine structure constant Velocity of light

= m = h = 1.

ao = h 2 /me 2 = 0.529177249(24) x 10- 8 cm Eo = e 2 /ao = 27.2113961(81) eV = 2Ry lRy = me4 /2h 2 = 13.6056981(40) eV TO = h 3 /me 4 = ao/vo = 2.41888433(11) x 10- 17 s Vo = e 2 /h = 2.187691417(98) x 108 cm S-I = 0.8797356696(80) x 10- 16 cm2

1faJ

a = e 2 /hc = 1/137.0359895(61) c = l/a = 137.036 ... a.u. = 2.99792458 x 1010 cm

S-I

The values of the fundamental physical constants are given in a report of the CODATA Task Group on Fundamental Constants, CODATA Bulletin No. 63. E.R. Cohen, B.N. Taylor: Rev. Mod. Phys. 59, 1121 (1987).

List of Symbols A [A] al( BK bl(

E Ecm

EK

f

I I M

MK

Radiative transition probability Ions of the isoelectronic sequence of atom A, or A-like ions Electric shielding factor Branching ratio coefficient Magnetic shielding factor Incident particle energy Center-of-mass energy Electric 2K-pole transition Oscillator strength Binding energy, ionization potential Orbital quantum number Nuclear mass Magnetic 2K-pole transition

XII

m N

n q T v (va)

Xz Z

z fJK

r

ll.E E K

Kd

Kr A f.L

a a+ X

Glossary of Tenns

Electron mass Total number of atomic electrons Principal quantum number Number of equivalent electrons Electron or ion temperature Relative velocity Maxwellian rate coefficient Ion with a charge z-I:Xz = X(z-ll+ Nuclear charge Spectroscopic symbol: z = Z-N+ 1 Electric 2K-pole polarizability Autoionization transition probability Transition energy, energy shift Hyperpolarizability Multiplicity Dielectronic recombination rate coefficient Radiative recombination rate coefficient Wavelength Reduced mass Cross section Net ionization cross section Magnetic-dipole susceptibility

Abbreviations

Direct Ionization DI Excitation-autoionization EA Multielectron (Multiple) Ionization MI Resonant-Excitation RE Resonant -Excitation-Auto-Double-Ionization READI Resonant-Excitation-Double-Autoionization REDA Resonant-Excitation-Quadruple-Autoionization REQA Resonant-Excitation-Triple-Autoionization RETA Resonant Ionization RI TI Transfer Ionization Special mathematical functions used in the book can be found in Handbook of Mathematical Functions, ed. by N. Abramowitz, LA. Stegun (Constable, London 1970).

1 Introduction

The physics of multicharged ions (or highly ionized atoms) is one of the most dynamic areas in modem atomic physics. The spectra of these ions contain important information about plasma macroparameters (electron and ion temperature and density, charge-state distribution, etc.) and, therefore, provide an important diagnostic tool for the investigation of hot laboratory and astrophysical plasmas. As a diagnostic tool high-resolution spectroscopy of multicharged ions has proved to be an extremely useful, and sometimes, even the only possible method for measuring these parameters. The novel techniques, involving recoil ions from ion-atom collision experiments or very highly ionized atoms from modem ion sources (EBIS (Electron Beam Ion Source), EBIT (Electron Beam Ion Trap), ECR (Electron Cyclotron Resonance), storage rings, etc.) have been successfully tested. Recent advances in heavy-ion-beam technology make it possible to produce very highly ionized atoms up to fully stripped uranium U92 + in the very wide kinetic energy range from a few eV up to more than 20 Ge V. The spectra of multicharged ions possess a number of specific properties, essentially distinguishing them from the spectra of neutrals. Among these properties one should mention the following: (i) the shift of radiation spectra to the VUV and X-ray spectral region, (ii) the growth of the ionization potentials up to several hundred or even thousand eV, (iii) an increase of the multiplet splitting proportionally to Z4, where z is a spectroscopic symbol of an ion, (iv) a deviation from the LS-scheme of the angular momentum coupling towards an intermediate and j j -coupling scheme, (v) an increase of the spectral line intensities, corresponding to forbidden transitions due to relativistic effects, (vi) the presence of so-called satellite lines in spectra of multicharged ions, connected with the radiative decay of autoionizing states. The long-range Coulomb force of an ion influences also the collisional properties of multicharged ions. The cross sections are proportional to the factor za. For electron-ion and ion-ion collisions a < 0, while for ion-atom, ionmolecular and radiative processes a > O. In some cases the factor a can be even much larger than unity. Therefore, the cross sections involving multicharged ions strongly depend on the collision partner. V. G. Pal'chikov et al., Reference Data on Multicharged Ions © Spring-Verlag Berlin Heidelberg 1995

2

1 Introduction

Research on the spectroscopy of multicharged ions and collision mechanisms taking place in plasmas is strongly motivated by a number of reasons. Highly charged ions constitute a new regime for theory since the interactions, which are small in neutral atoms, dominate in these systems and relativistic and Quantum Electrodynamical (QED) effects become to be significantly large. Therefore, the usual perturbation approaches may become inadequate in theoretical treatments. High temperatures or intense photon fluxes create highly charged ions in astrophysical objects, such as solar corona, X-ray emitting binary stars, supernova remnants, etc. The atomic characteristics of these ions determine X-ray and UV spectra, as well as heating and cooling rates and opacities of the objects [1.1]. The presence of highly ionized species in plasmas presents a significant energy-loss mechanism in fusion devices, but, on the other hand, is used for diagnostic purposes [1.2]. The structure and spectra of highly charged ions in laboratory plasmas has been the subject of several reviews and monographs from an experimental [1.3-7] and a theoretical [1.8-10] point of view. The intensified spectroscopic research of some selected isoelectronic sequences are caused by the interest to develop X-ray and VUV lasers [1.11-13]. Atomic and interaction physics, including electron-ion, ion-atom and ionion collisions, has also dramatically developed. The experimental and theoretical problems of elementary processes, involving multicharged ions, have been considered in several monographs [1.14-18] and conference proceedings devoted to the physics of highly charged ions [1.19,20]. The need of recommended data for different physical applications has lead to the creation of several large atomic data banks organized for storage and exchange of radiative and collisional characteristics of multicharged ions. Among them are the NIST data banks [1.21], the Belfast Atomic data bank [1.22], the Opacity Project data bank [1.23], ALADDIN (IAEA) [1.24], AMSTORE [1.25] and others. The development and availability of such supercomputer facilities made it possible to create computer codes for calculation, with high accuracy, of the atomic wavefunctions, energy levels, transition probabilities, cross sections and rate coefficients: the codes SUPERSTRUCTURE [1.26], MCHF [1.27], AUTOSTRUCTURE [1.28], CATS [1.29], GRASP [1.30], AUTOLSJ [1.31], ATOM, MZ [1.18] and others. The aim of the present book is to try to cover the spectroscopic and collisional data of multicharged ions in a broad field (energy levels, transition probabilities, cross sections and rate coefficients for different elementary processes, etc.). The material is presented in a brief form giving the scaling laws for different characteristics as well as the universal figures and tables. The detailed description of the experimental techniques is outside the scope of this book, but necessary references are presented.

2 Atomic Structure and Spectra

2.1 Classification of Spectral Lines In this chapter, the spectral characteristics of few-electron highly charged ions are considered on the basis of relativistic and Quantum Electro Dynamic (QED) theories: energy levels, Lamb shifts, binding energies, transition probabilities and others. It comprises detailed tables of these characteristics obtained from experiment or sophisticated theoretical calculations. 2.1.1 Notations We will call multicharged ions those ions which have the spectroscopic symbol z > 5, where

z=Z-N+1.

(2.1.1)

Here, Z is the nuclear charge of the ion and N is the total number of electrons. The spectroscopic symbol coincides with the Coulomb charge of the ion at large distances U(r) ---+ -z/r,

r ---+ 00.

(2.1.2)

For positive ions z > 1, for neutrals z = 1. Ions are often designated as Xz

= X(z-l)+

(2.1.3)

so the difference between the spectroscopic symbol and the ion charge is unity. In spectroscopy the roman notations are also used for z. For example, the ion Fe 25 + is written as Fe XXVI, the neutral Fe atom is Fe I. The spectroscopic symbol is an important quantity used as a scaling factor for many atomic characteristics: wavelengths, transition probabilities, cross sections and rate coefficients. Ions with a given number of electrons N, arranged in increasing order of Z, belong to the isoelectronic sequence of the corresponding atom A and are termed A-like ions or [A]-ions. For example, for N == 1 one has hydrogen-like ions [H], for N = 2 helium-like ions, etc. The spectral line corresponds to the radiation of an ion which makes a transition from the excited state to the lower one. The energy terms are usually described by the LS-coupling scheme in the form (2.1.4) V. G. Pal'chikov et al., Reference Data on Multicharged Ions © Spring-Verlag Berlin Heidelberg 1995

4

2 Atomic Structure and Spectra

where L and S are the angular and spin momenta, and J is the total momentum of an ion. The LS-coupling scheme is used when the electrostatic interaction in an ion is much larger than the relativistic (spin-orbit and others) ones: Vel»

(2.1.5)

Vrel.

With increasing nuclear charge (Z

~

(0), the opposite situation is realized

(2.1.6)

Vrel» Vel,

and the jj-coupling is used with another notation. In the case Vrel ~ Vel the states are described by the so-called intermediate coupling scheme (see [2.1, 2]). In this book we will mainly use the LS-coupling scheme. Figure 2.1 shows a smooth transformation from LS- to jj-coupling for low-lying configurations in Be-like ions. Transitions with I~SI = 1 are called intercombination transitions. Transitions which are not allowed by the selection rules (Sect. 3.1) are called forbidden transitions. Notations used for electric (EK) and magnetic (MK) 2K-pole transitions are given in Table 2.1.

jj

LS 1

2fJ~ -=::;;;:::~~

J=

5'r(P-3/./

3PO,1j{

.~~.....----:;;;;;;=~ }(PV2)(P?{t

1}(SV2)(PW

o

---1

(P1/)2 12

(f===--=;:;;'=o }(SV2)(PVt

o (S1f/

20

40

60

80

100 Z

Fig. 2.1. Calculated energy structure of the ground and excited states in Belike ions [2.3]

Table 2.1. Notations for electric and magnetic 2K-pole transitions Multipole

Remark

Example

E1 M1 ElM 1 E1M2 E2M1 E2 M2 2E1 2E2 2M1 2M2

Electric dipole transition Magnetic dipole Two-photon electro-magnetic Two-photon electro-magnetic Two-photon electro-magnetic Electric quadrupole Magnetic quadrupole Two-photon electric dipole Two-photon electric quadrupole Two-photon magnetic dipole Two-photon magnetic quadrupole

2 1PI- 11So 2 3 SI-1 1So 23PO-11S0

in [He] in [He] in [He]

2p3/2- 2PI/2 23P2-11S0 2 150-1 150

in [H] in [He] in [He]

2.1 Classification of Spectral Lines

5

2.1.2 Satellites The spectra of multicharged ions are much reacher as compared to those of neutrals because of the presence of so-called satellite lines arising from the radiation decay of autoionizing states, where two or more electrons are excited. Such states lie above the ionization limit and are created by either direct excitation of the inner-shell electron

Xz(ao)

+ e --+

X;(aI)

+e

(2.1.7)

or by a capture of a free electron

Xz(ao)+e--+ X;~I(Y)'

(2.1.8)

The radiative decay of an ion in reaction (2.1.8) leads to the dielectronic recombination process (Sect. 4.2): (2.1.9) Let ao and al denote the sets of quantum numbers of an ion X z in the initial and final states, respectively. The satellite to transition al-ao is called the line corresponding to transition aInl-aonl of an ion Xz-I, where nl are the quantum numbers of an electron-observer. For example, 1s2pnl-ls2nl transitions in Li-like ions are the satellites to the resonance line in He-like ions Is2p-ls2. The lines arising in dielectronic recombination (2.1.8,9) are called the dielectronic satellites. The number of excited electrons in autoionizing states can be more than two; for example, in Be-like ions the following autoionizing states are possible: Is2s 22p, Is2s2p2, Is2p 3, ... , Isnln'I'n"I", .... In highly charged ions the wavelengths of satellites are very close to the "parent" line and their intensities increase with increasing z. Therefore, in each spectral interval one has a large number of spectral lines of comparable intensity. Figure 2.2 shows a typical example of dielectronic satellite spectra of He-like Fe XXV ions observed in tokamak plasmas. The notations for basic spectral lines in H- and He-like ions and corresponding dielectronic satellites are given in Tables 2.2a and 2.2b. The spectral lines in He-like ions and their satellites are usually identified using the Gabriel's notations [2.5] .

•
--

....

0

~

x

2

ftI

~

!!l <:

::J

0

x t

Y q (k,r)

1J

1•

z

~

(.)

<:

.s0

.t::.

Q.

0

1.848

1.853

1.858

1.863

1.868 ). [AI

Fig. 2.2. Dielectronic satellite spectra of He-like Fe XXV ions observed in Tokamak plasmas at a temperature of 1.5 keV [2.4]. Notations of lines are given in Table 2.2; f3 corresponds to transitions Is22s2 IS -ls22s2p I PI in Be-like Fe ions

6

2 Atomic Structure and Spectra

Table 2.2a. Notations for the basic spectral lines in Hand He-like ions Ion sequence

Transition

Notation

H H He He He He

IS I/2 - 2PI/2 ISI/2-2P3/2 Is2 I So-ls2p I PI Is21So-1s2p3p2 Is2lSo-1s2p3pI Is2 I So-ls2s 3 SI

Lal La2

w x y Z

Table 2.2b. Notations for dielectronic satellites to lines in H-and He-like ions Transition

Notation

Transition

Notation

Is2p 3 P2 _2p2 3 P2 Is2p 3 PI _2 p2 3 P2 Is2p3P2-2p23PI Is2p 3 PI _2p2 3 PI Is2p 3 PO-2 p2 3 PI Is2p 3 PI _2p2 3 Po Is2p I PI_2 p2 3 P2 Is2p I PI_2 p2 3 PI Is2p I PI _2p2 3 Po Is2p I PI _2p2 I D2 Is2p 3 P2 _2p2 I D2 Is2p 3 PI_2 p2 I D2 Is2p I PI_2p2 I So Is2p 3 PI-2p 21 So Is2p I PI _2s2 I So Is2p 3 PI -2s 2 I So Is2s 3S1-2s2p 3P2 Is2s 3 SI-2s2p 3 PI Is2s 3 SI-2s2p 3 Po Is2s I So-2s2p I PI Is2s ISo-2s2p3 PI Is2s 3 SI-2s2p I PI

A

Is22p 2 P3/2 -ls2p2 2 P3/2 Is22p 2 PI/2 -ls2p2 2 P3/2 Is22p 2 P3/2 -ls2p2 2 PI/2 Is22p 2 PI/2 -ls2p2 2 PI/2 Is22p 2 P3/2 -ls2p2 4 PS/2 Is22p 2 P3/2 -ls2p2 4 P3/2 Is22p 2 PI/2 -ls2p2 4 P3/2 Is22p 2 P3/2 -ls2p2 4 PI/2 Is22p 2 PI/2 -ls2p2 4 PI/2 Is 22p 2 P3/2 -ls2p2 2 DS/2 Is22p 2 PI/2 -ls2p2 2 D3/2 Is22p 2 P3/2 -ls2p2 2 D3/2 Is22p 2 P3/2 -ls2p2 2 SI/2 Is22p 2 PI/2 -ls2p2 2 SI/2 Is22p 2 P3/2 -ls2s2 2 SI/2 Is22p 2 PI/2 -ls2s 2 2 SI/2 Is 22s 2S I/2-(2s2p 3 P)ls 2 P3/2 Is 22s 2S I/2-(2s2p 3 P)ls 2 PI/2 Is 22s 2S I /2 -(2s2p I P) Is 2 P3/2 Is 22s 2S I/2-(2s2p I P)ls 2 PI/2 Is 22s 2 SI/2 -(2s2p 3 P) Is 4 P3/2 Is 22s 2 SI/2 -(2s2p 3 P) Is 4 PI/2

a b c

B C

D E F G H I

J K L M N 0

P Q R S

T U V

d

e

f g

h j

k m

n 0

P q r

s U

v

In these notations w is the resonance line, y is the intercombination one, z is forbidden line and x corresponds to magnetic quadrupole (M2) line.

2.2 Relativistic and Quantum Electrodynamical Corrections 2.2.1 The Dirac Energy of H-like Ions The energy of the nlj level of a H-like ion can be presented in the form [2.6] E(nlj) = Eo(nj)

where

+ ER(n) + ENs(nl) + EM(nlj) + EQEo(nlj),

(2.2.1)

2.2 Relativistic and Quantum Electrodynamical Corrections

7

= the Dirac energy of a level with the principal quantum number n, j = I ± 1/2 is the total electron momentum with

Eo(nj)

account for spin, I is the orbital momentum; = the relativistic correction due to the non-separability of the

Dirac equation in terms of reduced mass;

ENS(nl) EM (nlj) EQEO(nlj)

= the correction due to the finite nuclear charge distribution; = the mass correction to the radiative Lamb shift; = the sum of the radiative corrections.

The main contribution to the energy (2.2.1) is given by the term Eo: Eo(nj) = [2z 2RY(JL/m)(aZ)-2]{[(aZ/{n - k

+ [k 2 -

(aZ)2]'/2})2

+ 1]-'/2 -

I},

k = j

+ 1/2,

(2.2.2)

where JL is the reduced mass and Z is the nuclear charge. If a Z « 1 one has from (2.2.2) the well-known expression: Eo(nj) = Eo Eo

+ E, + ... ,

= _Z2Ry/n2, E, = - a

2 Z 4 Ry

n3

(1+ j

3)

1/2 - 4n

(2.2.3) '

where Eo is the non-relativistic part and E, is the fine-structure splitting. This means that E, ex (aZ)2Eo. In the Dirac theory, the levels with different I = j ± 1/2 have equal energy according to (2.2.3). However, this degeneration is taken off due to the radiative corrections connected with QED effects: Lamb shift, vacuum polarization and others. The radiative corrections are of the order of a(lna)E, « E,. 2.2.2 Self-Energy The EQEO term consists of three parts: Self-Energy (SE), Vacuum Polarization (VP) and anomalous Magnetic Moment (MM). For multicharged ions, the dominant QED-corrections are given by the lowest-order (in a) self-energy and vacuum polarization terms. The lowest-order QED-shift has the form: EQEo(nlj) =

(~Rya3Z4/1fn3)

{8[,0 [In(aZ)-2 + 11/24 -

-In[L(nl)/Z2Ry]

3c[' } + (21; 1)

,

1/5] (2.2.4)

where

C . _ { 1/(21 + 1), IJ -1/1,

j = 1+1/2, j = 1- 1/2.

(2.2.5)

The terms 11124 and -1/5 are associated with the electron self-energy and vacuum polarization corrections, respectively, and the last term in (2.2.4) is the anomalous magnetic moment correction. The term In L(nl) is the Bethe logarithm, which represents the essentially non-relativistic part of the Lamb shift arising from lowest-order QED effects in H-like ions. The Bethe logarithm is

8

2 Atomic Structure and Spectra

obtained numerically. The recent accurate theoretical results forln[L (nl) / Z 2 Ry] are presented in [2.7]. In [2.8] the simple asymptotic expression was obtained: 0.515 In[L(nl)/Z2Ry] = 2.723 + 2 ± 0.0003 (for 1 = 0) (n + 0.405) 0.3 0.56 = -

(21

+ 1)(12 + 1 + 0.04)

± 0.0007

C:

+ ---,-,------,----,-----;:(21 + I)(n + 0.751 + 0.39)2

(for 1 #- 0).

3)

(2.2.6)

Equation (2.2.4) gives the Lamb shift for the 2S1/2-2Pl/2 splitting for neutral hydrogen with an accuracy of ~ 1%. The accuracy of the recent measurements is higher, for example, in [2.9] the measured Lamb shift is 1057.8514(19) MHz (the corresponding error ~ 2 x 10-4 %). The uncertainty of (2.2.4) rapidly increases with increasing Z. For low and intermediate Z, sophisticated methods [2.10-12] give EQED =

(~Rya3 Z4/7rn3) F(aZ),

Z

~ 40,

(2.2.7)

where F(aZ) is a dimensionless slowly varying function:

+ A 41 In(aZ)-2 + A50(aZ) + (aZ)2[A61In(aZ)-2 + A62In2(aZ)-2 + G(aZ)] + aB4Q/7r.

F(aZ) = A40

(2.2.8)

Each of the coefficients Ai) and B40 can be written as a the sum of SE, VP and MM contributions, i.e. Ai) = A~E + Ail + A~M. The SE and MM contributions are given in the Table 2.3. The corresponding values for the vacuum polarization contributions are presented in Sect. 2.2.3. The contribution of the binding effects of the higher orders in a Z is described by the function G(aZ) in (2.2.8). The dominant source of theoretical uncertainty (2.2.8) arises from the SE-part of the function G(aZ), i.e. G(aZ) = GSE(aZ)

+ Gvp(aZ),

(2.2.9)

where Gvp(aZ) is the VP-part of the function G(aZ) (Sect. 2.2.3). For example, in [2.10] the function GSE(aZ) is presented in the form (by analogy with the corresponding high-order terms in the vacuum polarization) GSE(aZ) = a

+ b(aZ) In(aZ)-2 + c(aZ) + ... ,

(2.2.10)

where the coefficients a, band c were determined by the interpolation method using accurate calculations for Z = 10,20 and 30 [2.10]. The coefficients a, b and c are given in Table 2.4. The corresponding value for 2S1/2-2Pl/2 splitting in H-like ions with 1 ~ Z ~ 40 is presented in the form GSE(aZ) = -24.1

+ 7.5aZ In(aZ)-2 + 15.3aZ ± 1.2.

(2.2.11)

The GSE(aZ)-value was calculated in [2.17] using the Green's function method in the non-relativistic form. The result obtained, GSE(aZ) = -24.9 ± 0.9 for aZ « I, agrees with the data in [2.10].

2.2 Relativistic and Quantum Electrodynamical Corrections

9

Table 2.3. The SE and MM contributions to the coefficients Aij and B40 [2.10-16]* Term

Value

A~

In[L(n/)/Z 2 Ry]

+ (11/24)8/,0

3C/j

AW

8(21

A~

3]l' (1

A~l'

8/,0 3

A~~

+ 1) + 11/128 -ln2/2) 8/,0

-4 8/,0 (71n2 - 3In(n) + 3

~~-

:

-

~:2) 8/,0 + (I -

n12)

(1~ + ~8j'I/2) 8/,0

+ 1)/n 2 ](1 - 8/,0) 1)21(21 + 1)(21 + 2)(21 + 3)

8[3 -I(l

+ (21 -

4819 49]l'2 [ -1728 - 144 3197 [ 4144 *~(z)

]l'2

+ 12 -

3]l'2

+T

]l'21n2 -2-

9] ln2 - 4~(3) 8/,0 3

3] 3C/j

+ 4~()

21

+1

is the Riemann zeta-function

Table 2.4. Numerical coefficients a, b· and c in (2.2.10) for the states in H-like ions [2.10,16] State

a

b

c

ISI/2 2S1/2 2Pl/2

-23.25 -24.74 -0.5895 -0.3752

5.033 7.330 -0.2183 0.09039

18.60 17.20 1.871 0.7336

2P3/2

A more extended five-parameter fit to ten calculated points Z = 10, 20, ... , 100 is described in [2.18]. The final results for the IS I/2, 2SI/2, 2PI/2 and 2P3/2 states are GSE(1SI/2; aZ)

= -23.419 + 5.852aZ In(aZ)-2 + 15.922aZ

+ 4.294(aZ)2 + 2.572(aZ) 15 , GSE(2SI/2; aZ) = -24.177 + 6.293aZln(aZ)-2 + 16.548aZ + 5.549(aZ)2 + 2.977(aZ)1l, GSE(2P1/2; aZ) = -0.715 + 0.170aZ In(aZ)-2 + 1. 195aZ + 0.357(aZ)2 + 1.502(aZ)8, GSE(2P3/2; aZ) = -0.417 + 0.233aZln(aZ)-2 + 0.449aZ + 0.191(aZ)2 - 0.032(aZ)7.

(2.2.12) (2.2.13) (2.2.14) (2.2.15)

10

2 Atomic Structure and Spectra

Table 2.S. GSE-factor for the transition 2SI/2-2PI/2 in Hlike ions* Ion

Experiment [2.18]

Theory [2.18]

H He 1+

-27.45 ± 1.25 ± 0.57 -22.47 ± 0.35 ± 0.02 -17.22 ± 4.0 ± 0.78 -22.92 ± 7.9 ± 0.03 -21.47 ± 9.1 ± 0.03 -19.63 ± 0.35 ± 0.004 -19.45 ± 0.52 ± 0.003 -19.22 ± 1.3 ± 0.013 -19.55 ± 1.6 ± 0.005 -7.83 ± 0.46 ± 0.02

-22.91 ± 0.40 -22.45 ± 0.38 -22.10 ± 0.33 -20.52 ± 0.22 -20.24 ± 0.20 -18.75±0.13 -18.53 ± 0.12 -18.31 ±0.12 -18.09 ± 0.10 -7.563 ± 0.04

Li2+ 07+ pH p14+

SI5+ C1 16+ ArI7+

{fl+

*The first uncertainty in the experimental data is due to the experimental uncertainty. and the second one is due to the uncertainty of the theoretical nuclear size

Since the main theoretical error in (2.2.8) arises from the self-energy part GSE of G(aZ), it is of interest to assume that the other terms in (2.2.8) are correct, and to extract an experimental value for GSE. The results are given in Table 2.5. The main sources of error in estimating GSE(aZ) are the uncertainty in the nuclear size corrections and the corresponding experimental error. The self-energy correction is strongly dependent on Z and has to be calculated non-perturbatively [without an expansion on parameter a Z in the general expression (2.2.7)] to achieve a good accuracy for high nuclear charge Z. The first exact self-energy calculations were performed in [2.19] (level shift of the ISI/2 state for Z = 80) and in [2.20] (the level shift of inner-shell electrons in the Coulomb and the screened Coulomb potentials for heavy atoms with Z = 70-90). Assuming point-like nuclei, the energy levels for some lowest states with principal quantum number n = 1 or n = 2 and Z in the range Z = 10,20, ... , 110 was calculated in [2.10,16,21]. High precision calculations [2.10,16] are valid of the order of a and of all orders of the parameter aZ. The numerical results are given in Table 2.6 in terms of the function FSE(aZ) which is defined by F(aZ) = FSE(aZ)

+ Fyp(aZ),

(2.2.16)

where Fvp(aZ) is the vacuum-polarization contribution (Sect. 2.2.3). A similar expression FSE(aZ) for other Z-values is given by [2.22]: FSE(aZ) =

<51.0~3 In(aZ)-2 + ti=1 (IT Z - Zj) Hi Zi - Zj x [FSE(aZi )

-<51.o~ln(aZi)-2],

(I

~ Z ~ 60).

Here, the points Zi are ZI, Z2, ... , Z6 = 10, 20, ... , 60, respectively.

(2.2.17)

2.2 Relativistic and Quantum Electrodynamical Corrections

II

Table 2.6. Calculated FSE(aZ) values for H-like ions [2.23] (Uncertainties are shown in parenthesis) Z

5 10 15 20 25 26 30 35 36 40 45 50 54 55 60 65 66 70 75 79 80 82 83 85 90 92 95 100 105 110

ISI/2

2S I /2

2PI/2

2P3/2

6.251627(8) 4.6541622(2) 3.8014108(1) 3.2462556(1 ) 2.8501042(1) 2.79393814(5) 2.5520151(1) 2.3199761(1) 2.27969676(8) 2.1352284(1 ) 1.9859437(2) 1.8642743(2) 1.7831410(2) 1.7648303(3) 1.6838358(3 ) 1.6186364(4) 1.6073191(2) 1.5674075(4) 1.5289841 (4) 1.5070511 (3) 1.5027775(4) 1.4956900(3) 1.4928824(3) 1.4887626(4) 1.4875419(4) 1.4909160(3) 1.5005122(4) 1.5301997(4) 1.5809122(7) 1.660063(1)

6.4848(2) 4.89445(6) 4.05088(1) 3.506648(2) 3.1229593(7) 3.0594657(6) 2.8388385(7) 2.6223358(7) 2.5853631(4) 2.4548292(7) 2.3246900(7) 2.2243377(7) 2.1620498(3 ) 2.1487267(7) 2.0945176(6) 2.0596107(6) 2.0548320(3) 2.0428911(6) 2.0441145(6) 2.0584011(3) 2.0639061(6) 2.0773446(3) 2.0853160(3) 2.1038757(6) 2.1668834(6) 2.1994938(3) 2.2575400(6) 2.3831222(6) 2.555298(2) 2.793592(5)

-0.1228(2) -0.11483(4) -0.104549(6) -0.092519(3) -0.079066(1) -0.0762190(5) -0.0643302(4) -0.0483423(4) -0.0449909(2) -0.0310500(4) -0.0123303(3) 0.0080122(2) 0.0256366(1 ) 0.0302529(2) 0.0547632(1) 0.0820375(1 ) 0.0878783(1) 0.1127325(1) 0.1477294(1) 0.1796152(2) 0.1882263(1 ) 0.2063172(2) 0.2158304(3) 0.2358858(4) 0.2930723(8) 0.3193408(4) 0.363253(2) 0.451711(4) 0.566883(5) 0.723101(6)

0.1256(1) 0.13036(2) 0.136567(4) 0.143839(2) 0.151921(1) 0.1536189(4) 0.160647(1) 0.1699009(8) 0.1718068(2) 0.1795949(4) 0.1896631(3) 0.2000537(2) 0.20856976(3) 0.2107246(2) 0.2216410(1) 0.2327729(1) 0.23502264(1) 0.2440925(1) 0.2555732(1 ) 0.26485443(1) 0.2671861 (I) 0.27186113(1) 0.27420388(1 ) 0.2788982(1) 0.2906678( I) 0.29537993( 1) 0.3024393(1 ) 0.3141342(1 ) 0.3256378(1) 0.3367752(1)

The corresponding errors for function FSE(a Z) in (2.2.17) are small and associated with uncertainty in the calculated points Zi. FSE(aZ) values for 1 ~ Z ~ 100 are determined by the least-squares fit with Zi = 10,20, ... , 100 for i = 1,2, ... , 10 in [2.18]. The exact calculations of the SE-contribution for states with n = 3 and higher are presented in [2.23-25]. Values for FSE(aZ) are plotted as a function of Z in Fig. 2.3. 2.2.3 Vacuum Polarization The level shift due to vacuum polarization of H-like ions in the lowest order on a is expressed as [2.26]:

E~~D = (~Rya3Z4/Jrn3) Fvp(aZ),

(2.2.18)

12

2 Atomic Structure and Spectra 7

Fig. 2.3. Self-energy for nSl/2 states of H-like ions with n = 3 and n = 5 [2.25]

6

n=3 n=5

........... N o 5 ........... w (/) LL

"

4

30

"

"

/

'-

/

10 20 30 40 50

60 70 80 90 100 110

z where Fvp(aZ) is a slowly varying function of Z. The function Fvp(aZ) can be written as Fvp(aZ)

=

F~~(aZ)

+ F~~+)(aZ),

(2.2.19)

where

F~~+)(aZ)

= F~~(aZ) + F~~(aZ) + ...

(2.2.20)

The subscript denotes the order of the external Coulomb field (only odd powers give non-zero contributions to E VP ). The first term F~~(aZ) is associated with the expectation value of the Uehling potential [2.6] and gives the main part of the vacuum-polarization correction (for all values of the nuclear charge Z [2.27 - 30]). The behavior of F~~ in the limit aZ « 1 was studied in [2.26,31]. In the important case of the Coulomb field of a point nucleus Z, F~~ can be represented in the form [2.31]: (1)

Fvp(aZ« 1)

= -3

(n + l)!(l!) + 3)[(21 + 1)!]2(n -1- 1)! (21 + 1)(1 + 1) 8. )

(2aZ) -n- 2n(21

x ( I +2 21 + 5

+

412

I,J+I/2·

(2.2.21)

The expectation values of the Uehling potential evaluated with the DiracCoulomb wave functions have been tabulated for ions with Z = 10, 20, ... , 100 in [2.16]. The corresponding results are given in Table 2.7 in terms of the function F~~, which is defined by VP

EQED

=

2Rya 3 z4 lfn

3

(1)

Fvp (aZ).

(2.2.22)

2.2 Relativistic and Quantum ElectrodynamicaI Corrections

13

Table 2.7. Calculated function F~~(aZ) (2.2.22) [2.19] for H-Iike ions

Z 10 20 30 40

50 60 70 80 90 100 110

ISI/2

2SI/2

2PI/2

2P3/2

-0.249449 -0.240914 -0.238684 -0.242087 -0.251334 -0.267463 -0.292650 -0.331025 -0.390972 -0.489026 -0.670610

-0.250399 -0.244587 -0.246936 -0.257197 -0.276418 -0.307130 -0.354174 -0.426679 -0.542896 -0.743531 -1.136909

-0.000335 -0.001370 -0.003257 -0.006323 -0.011150 -0.018773 -0.031087 -0.051786 -0.088728 -0.160837 -0.323108

-0.000069 -0.000255 -0.000536 -0.000902 -0.001349 -0.001876 -0.002490 -0.003199 -0.004019 -0.004969 -0.006075

The corrections of the order of a(az)n with n ~ 3 to the Uehling potential [second term in (2.2.19)] have been considered in [2.26]. Corrections F~t+) to energy levels of mesoatoms and multicharged H-like ions have been studied by several authors [2.32-33]. Vacuum-polarization contributions of the order of a 2(aZ) to the Lamb shift of K- and L-shell electrons in high-Z H-like ions are presented in [2.34]. For point-like nuclei, an explicit expression for the F~t+) in the lowest order (in a) was derived in [2.26-32]. The dependence of F(3+) on Z in the case aZ « 1 is conveniently presented in the form [2.31]: F~+)(aZ

«

1)

1 = 0,

a2Z2b23rr, a 4 Z 4 (n 2 -

l)rr

3n 2

=

[22~rr

+ Cn - C) + b4 + ~b281,j+l/2]' 32a4 Z 4(5n 2 - 3/ 2 - 31 + 1)(21 - 3)! 2 1 ~ 2, 75n (21 + 4)! (-In Z:Z

1 = 1,

(2.2.23) where b2 = 0.0045105564, b4 = 0.004252588, C2 = 0, C3 = -7/16, C4 = -147/200, Cs = -77/80, C = 0.57721566. The simple analytical formula (2.2.23) allows to obtain F~t+) with a relative in the accuracy better than 10-4 • The corresponding values of coefficients general expansion (2.2.7) are presented in Table 2.8. The vacuum-polarization part Gvp(aZ) of G(aZ) (2.2.8) has been calculated in the form [2.10]:

Ail

Gvp(aZ) = Gu(aZ)

+ GWK(aZ),

(2.2.24)

14

2 Atomic Structure and Spectra

Table 2.8. Coefficients Aij for the VP-type contributions [2.10] Tenns

Value

VP A 40

-81.0/5

A VP 50

(511"/64)81.0

VP A 61

-(1/10)81.0

VP B40

-(41/54)81.0

where Gu(aZ) is the contribution from the Uehling potential and GWK(aZ) is a contribution of the third order, obtained by Wichmann and Kroll (WK) [2.26]: (2.2.25) Using a calculation for the Uehling vacuum polarization [2.27] and calculations in [2.26] for the WK-contribution, the more accurate fit to Gu(aZ) and GWK(aZ) for 1 ~ Z ~ 100 was found [2.18]: GU(IS1/ 2; aZ)

= -0.475180 + 0.16399aZ In(aZ)-2 - 0.15552(aZ)2 - 1.06482(aZ)8,

GU(2S1/2; aZ)

= =

(2.2.27)

-0.048214 - 0.00553aZln(aZ)-2 - 0.05614aZ - 0.23203(aZ)2 - 1.15799(aZ)8,

GU(2P3/2; aZ)

(2.2.26)

= -0.619167 + 0.16875aZ In(aZ)-2 + 0.1l393aZ - 0.54118(aZ)2 - 2.46701 (aZ)8,

GU(2Pl/2; aZ)

0.02517aZ

-0.010714 + 0.00115aZ In(aZ)-2

(2.2.28)

+ 0.00829aZ

+ 0.00162(aZ)8, (2.2.29) 0.10305aZ + (aZ)2[0.04793In(aZ)-2

- 0.00560(aZ)2

= 0.04251 + 0.12930 - 0.06826aZ In(aZ)-2], (2.2.30) GWK(2S1/2; aZ) = 0.04251 - 0.10305aZ + (aZ)2[0.04463In(aZ)-2 + 0.17512 - 0.10396aZ In(aZ)-2], (2.2.31) GWK(2Pl/2; aZ) = (aZ)2[0.00113In(aZ)-2 + 0.05431 GWK(1S1/2; aZ)

- 0.05878aZ In(aZ)-2],

GWK(2P3/2; aZ)

(2.2.32)

= (aZ)2[0.00113ln(aZ)-2 + 0.00212 - 0.00098aZ In(aZ)-2].

(2.2.33)

Two independent calculations of the vacuum polarization were carried out with an exact account for F~~+) of the states with n = 1, 2 and with a finite size

2.2 Relativistic and Quantum Electrodynamical Corrections

15

Table 2.9. The contribution of F(3+)(aZ) to the Lamb shift of K - and L-electrons in H-like ions [2.35]

Ion

ISI/2

2SI/2

2PI/2

2P3/2

Zn29 +

0.0020 0.0027 0.0033 0.0041 0.0051 0.0059 0.0073 0.0084 0.0102 0.0116 0.0136 0.0150 0.0170 0.0207 0.0236 0.0269

0.0020 0.0028 0.0035 0.0044 0.0054 0.0064 0.0081 0.0094 0.0118 0.0137 0.0166 0.0185 0.0216 0.0272 0.0320 0.0377

0.0000 0.0001 0.0001 0.0002 0.0003 0.0004 0.0007 0.0010 0.0015 0.0020 0.0028 0.0035 0.0045 0.0068 0.0089 0.0118

0.0000 0.0000 0.0000 0.0001 0.0001 0.0001 0.0002 0.0002 0.0003 0.0003 0.0004 0.0005 0.0006 0.0007 0.0009 0.0010

KJ35+ Z~9+

Rh44 + Sn49 + Xe53+ Nd59+ Gd63+ Yb69 + W73+

Au 7S + Pbsl + Rn S5+

091+ Cm95 + Fm99 +

Table 2.10. Calculated vacuum-polarization correction F~~(aZ) for H-like multicharged ions [2.31] Z

nlj

10

20

82

100

110

130

137

ISI/2 2SI/2 3SI/2 4SI/2 5SI/2 2PI/2 3PI/2 4PI/2 5PI/2

0.4328 0.4336 0.4335 0.4334 0.4333 0.4825 0.4822 0.4820 0.4819 0.9550 0.9554 0.9555 0.9554 1.0942 1.1052 1.1096 1.0341 1.0385 1.0402

0.3896 0.3931 0.3929 0.3925 0.3921 0.4786 0.4779 0.4772 0.4768 0.9166 0.9187 0.9191 0.9191 1.3001 1.3355 1.3493 1.1020 1.1152 1.1205

0.3479 0.4315 0.4285 0.4205 0.4139 0.8359 0.8149 0.7936 0.7779 0.9081 0.9574 0.9682 0.9699 4.2302 4.6060 4.7415 1.7385 1.8570 1.9037

0.4073 0.5788 0.5709 0.5528 0.5382 1.2782 1.2205 1.1663 1.1279 0.9936 1.0778 1.0966 1.0996 5.9125 6.4957 6.6965 1.9898 2.1622 2.2301

0.4749 0.7464 0.7310 0.6997 0.6751 1.7866 1.6745 1.5750 1.5064 1.0708 1.1835 1.2091 1.2133 7.1949 7.9411 8.1904 2.1592 2.3707 2.4542

0.9135 1.9541 1.8360 1.6737 1.5585 5.9115 5.0967 4.5016 4.1289 1.3589 1.5717 1.6218 1.6302 11.319 12.611 13.014 2.6502 2.9804 3.1119

2.4351 6.7830 5.8460 4.9360 4.3697 27.862 20.575 16.536 14.319 1.6509 1.9485 2.0197 2.0321 14.594 16.342 16.874 3.0625 3.4865 3.6564

2P3/2 3 P3/2 4P3/2 5P3/2 3D3/2 4 D3/2 5D3/2 3 D 5/2 4D5/2 5D5/2

16

2 Atomic Structure and Spectra

of the nucleus [2.35,31] and for the states with n = 1 to 5 for a point charge Z ranging from 10 to 137. The calculations significantly reduce the ambiguity in the theoretical estimates of the energy levels of multicharged ions. An analytical expression for the asymptotic expansion of the polarization potential at large distances from the nucleus was obtained in [2.31]. The corresponding results are listed in Tables 2.9 and 2.10, respectively, in terms of the functions F~~+) and F~~ which are defined as: vp

EQED =

2Rya 3 Z4 (3+) , 3 Fvp (aZ), rrn

= (1

F,(3+)(aZ) VP

(2.2.34)

j ,l/2 + [1 _ 1J(aZ)2]1/2 )

F,(3)(aZ)F,(3+) VP

VP

(aZ« 1), (2.2.35)

where F~~+)(aZ « 1) is defined in (2.2.23). A comparison of the energy shift F.g,+)(aZ), calculated in [2.35] with nuclear size corrections and in [2.31] with a point-like nucleus, shows that the difference in F.g,+)(aZ) for Z = 80 to Z = 100 ranges from 6% to 10%.

2.2.4 Finite Nuclear-Mass and Nuclear-Size Corrections In the general formula (2.2.1) the correction ER(n) is required because the relativistic two-body problem does not reduce exactly to an equivalent reducedmass single-particle Coulomb problem. ER(n)is given approximately by [2.36]: JL2 a 4Z4

(2.2.36)

where m and M are the electronic and nuclear masses, respectively and JL is the reduced mass. This correction depends only on the principal quantum number n, and hence does not give a contribution to the Lamb shift. The contribution to the energy shift (2.2.1) of the effects associated with the finite size of the nucleus is determined by the non-relativistic formula [2.6,12]: ENS

= Ry

+

4Z4(r2) 64Z 5 4y 3n 3 1JI,O - Ry 9rrn 3 ((r2)1/2) 1JI,O

20a 2 Z4 y 3(5 + 2l)n3

R

[a-;;-Z]

21

2 1+1

(r)

(1

-

1J )

1,0 ,

(2.2.37)

obtained assuming a uniform distribution of the charge over the nuclear volume. Here (r2)1/2 is the root-mean-square (rms) radius of the nucleus, y = (aZ)2. The finite nuclear-size corrections to the s-state energy levels of H-like ions are calculated up to the order of a 4 Z6 in [2.37], where additional contributions of the orders of a 2Z4, a 3Z5, a 4Z6 and a 4Z 6 10g(a Z) have been determined. These corrections have been separated into terms of non-relativistic and relativistic orders.

VI -

2.2 Relativistic and Quantum Eiectrodynamicai Corrections

17

Compact formulas for the level shift of H-like ions with 10 ::;; Z ::;; 40 are given in [2.22]: ISI/2 : ENS = Ry

4Z2 Z2y (r2}y 3 [1

2SI /2

Z2 Z2y (r2}y 6 [1

Ry

: ENS =

2PI/2 :

ENS

= Ry

2P3/2 :

ENS

= O.

+ 0.50(aZ)2],

+ 1.38(aZ)2],

Z4 Z2y (r2) Ya 2 32 '

(2.2.38) (2.2.39) (2.2.40) (2.2.41)

The uncertainty in the nuclear radii and neglected terms in (2.2.37) lead to an uncertainty of the level shift given by [2.22]: ~(r2}1/2

~ENS = 2 «(r}2)1/2

+ [Z (r2}1/2 + (aZ)4]

4 x 3n 3 RyZ2 Z2Y (r 2)Y. (2.2.42)

The leading relativistic corrections in (2.2.38-41) agree with the calculation in [2.37]. For example, for the hydrogen atom (Z = 1) the calculation of the Lamb shift for the transition 2SI/2-2PI/2 using (2.2.6) with allowance for QED corrections of higher orders of a gives [2.38]: E2SI/2 -

E2PI/2

=

(r2) 1/2 = 0.805(11)fm, (r2) 1/2 = 0.862(12)fm.

{ I 057 .849(11)MHz,

1057.867(11)MHz,

The correction ER ~ 0.125 MHz for (r2) 1/2 = 0.805(11) fm is used. A larger value, (r2}1/2 = 0.862(12) fm, correspondingly increases the contribution of the correction ENS to the Lamb shift (~ 0.146 MHz). It follows from the foregoing that a 10 percent change in the rms proton radius affects the Lamb-shift value within the range of uncertainty of the best experimental measurements [2.9]. The corresponding numerical analysis for the H-like Arl7+ ion is presented below. There are also significant finite-size corrections to the self-energy and vacuum-polarization terms calculated for H-like ions with 1 ::;; Z ::;; 110 in [2.27]. Relativistic recoil corrections, the term EM in (2.2.1), are of the order of Zm/ M relative to the lowest-order radiative correction (self energy and vacuum polarization). In contrast to correction (2.2.36), the order (a Z)5 m 2/ M corrections do contribute to the Lamb shift in H-like ions and were obtained in the form [2.12,13]: 8a 3 Z 4 EM = Ry 31l"n 3 -0/,0/3

{(

3m) [SE VP - M A40 + A40

MM + A40 + (2 + 0/,0)/3

+ Alfln(aZ)-2] + (Zm/M) [2A~ + 2Alfln(aZ)-2

-0/,0] - 7 [0/,0 In(aZ)-2

+ An]

},

(2.2.43)

18

2 Atomic Structure and Spectra

where

1)

A - -2 ( In(2/n) + ' n" -I + 1 - n ~q 2n

8

[,0

+

1(1

1-8

+ 1)(21[,0+ 1) . (2.2.44)

An

Ail

M and The corresponding values for coefficients A~jE, are presented in Tables 2.3 and 2.8, respectively. The term An in (2.2.43) was originally calculated exactly only for n = 2, but, as was shown in [2.12], is also correct for all n. The additional contribution to the Lamb shift of the order of a 4 Z5 m / M was obtained in the form [2.38]:

2Rya 4 Z 5 m (35

4

n3M

+

ln2 - 8 +

2Rya4 Z 5 m n 3M

1

5" +

31 ) 192 8[,0

(-0.415 ± 0.004)8[,0'

(2.2.45)

In particular, the corrections EM for some states of H-like ions for the lowest order in aZ are [2.22]: IS: EM = Ry

2Z5 a 3 m [1 8 14 62] -In(aZ)-2 - -2.9841 + -ln2 + , rrM 3 3 3 9

Z5 a 3 m [1 8 187] 2S: EM = Ry 4rr M "3 ln (aZ)-2 - "3 ·2.8118 + 18 2P : EM =

Z5 a 3m Ry 4rrM

[8"3. 0 .0300 - 187]

(2.2.46)

The Lamb shift of the transition 2S I/ 2-2PI/2 is mainly associated with QED corrections because, as has been pointed out before, according to oneelectron Dirac theory, levels with the same principal quantum number n and equal total angular momentum j are degenerate. The theoretical contributions to the Lamb shift are self-energy, vacuum polarization, the nuclear-size effect, reduced mass and relativistic recoil effects. Including all contributions listed in the preceding sections, the 2SI/2-2PI/2 Lamb shift for H-like ions can be expressed as f:l.E2SI/2 -

[rr

f:l.E2PI/2

3 4 = Rya Z {(IL)3[ -m + In(aZ) -2 -2.207909 + aZ 3rr m 8IL

(~~~ - ~ln(2))] + (aZ)2 [-~ln2(aZ)-2+ (4ln2+

!!)

0.323)] + Z M m In(aZ)-2 ] + (aZ)2 [(-24.0± 1.2) +a ( --;1 ln ) 3mrr [( 35 31) (4 (aZ)-2 + 2.39977 + aZ 4M 4 1n2 - 539 + 192

2.2 Relativistic and Quantum Electrodynamical Corrections

+ (-0.415 ±0.004)]} + ~RyZ4

19

((r2)'/2f - 2~Rya2Z4(m/M)2. (2.2.47)

This expression is obtained from [2.12] with addition for a 4 Z 5 m/M corrections as well as a previously calculated a 2Z4 (m / M)2 correction [2.38]. The various theoretical contributions to the Lamb shift of the 2SI/2-2PI/2 transition for the hydrogen atom and H-like argon are given in Table 2.11. These values were obtained from [2.27]. Analogous results for the ground-state energy level of H-like uranium are given in Table 2.12. In Fig. 2.4 the various contributions to the 2S I / 2-2PI/2 Lamb shift as functions of nuclear charge Z are shown. As can be seen from Fig. 2.4 the selfenergy and the vacuum polarization are the dominant contributions to the Lamb Table 2.11. A summary of the different contributions to the Lamb shift for the transition 2S1/2-2P1/2 in H and in ArI8+. The function F(aZ) is presented. The corresponding energy shift follows from t:.E = (2a 3Z 4RY/1fn 3)F(aZ) Numerical values [2.27] Description Self-energy correction assuming point-like nuclei Nuclear-size effect on the self-energy correction Shift due to the Uehling potential assuming pointlike nuclei Nuclear-size effect on the Uehling potential correction Wichmann-Kroll terms, higher-order vacuumpolarization corrections Higher-order radiative corrections Nuclear-size effect on the Dirac energy Relativistic-recoil correction Sum of Lamb-shift effects

H

10.6732

3.7965 -0.0001

-0.2644

-0.2440

o o

o

o

0.0008 0.0010 0.0259 0.0011 3.5813

0.0010 0.0013 0.0036 10.4146

Table 2.12. Lamb shift for the 1SI/2 energy level in H-like uranium (Z [2.27] Self-energy for point-like nuclei Nuclear-size effect on the self-energy correction Uehling vacuum polarization Nuclear-size effect on the Uehling vacuumpolarization Higher-order vacuum-polarization correction Other higher-order radiative corrections Nuclear-size effect on the Dirac energy Relativistic-recoil correction Total Lamb shift

= 92) in units of Rya 3 Z4/1f

1.4894(7) -0.0266(15) -0.4064 0.0179 0.0194 0.0013(13) 0.8038(104) 0.0005(5) 1.9012(112) = 3.698(22) x 106 cm- I

20

~

~ I

~

cii

2 Atomic Structure and Spectra

6

5

4

C\I I

E o

o '".... ~ s::; rn

.a E

j

3

2

ol------=~~~~===== 5

-1

-2~O~~10~720~~370~4~O~5~O~6~O~~70~~870~9~O~1~OO~110 Nuclear charge Z

Fig. 2.4. Contributions to the Lamb shift in H-like ions as a function of nuclear charge Z. (l) finite nuclear size; (2) self-energy; (3) anomalous magnetic moment of the electron; (4) finite nuclear mass (radiative recoil); (5) vacuum polarization

shift for low and intermediate values of Z. However, for Z ~ 100 the nuclear finite-size corrections become as -important as the self-energy contribution. Experimental results and theoretical calculations for the Lamb shift for neutral hydrogen and H-like ions are compared in Tables 2.13 and 2.14, respectively. The theoretical uncertainty listed with each Lamb-shift value is the quadrature sum of the contribution uncertainties. The experimental Lamb-shift values for high-Z H-ions have a much larger error than those for hydrogen and helium (Fig. 2.5). The latest experimental information on the Lamb shift of 1 ~ Z ~ 18 H-like ions is also presented in this figure. There is good agreement between theory and experiment at the level of about 1% for H-like ions with Z > 3. In Table 2.15 the theoretical values of wavelengths and transition probabilities for the transitions nPl/2-1S1/2 and nP3/2-1S1/2 are given. A comparison of the wavelengths with the experimental results are given in Table 2.16. For these transitions, the main theoretical component is the energy separation predicted by the Dirac equation (2.2.1). The effects of quantum electrodynamics, analogous to the Lamb-shift corrections, are also significant at the modem level of experimental precision. In some of these measurements (for example, in H-like krypton [2.39]) the quantum-electrodynamical effects are increased to about the 10% level. Experimental results of wavelengths in H-like magnesium together with theoretical data are presented in Table 2.17.

2.2.5 He-like Ions The accurate calculations of characteristics of two-electron (He-like) ions are more complicated as compared to H-like ones because it is necessary to take

2.2 Relativistic and Quantum Electrodynamical Corrections

21

Table 2.13. A comparison between theory and experiment for the Lamb shift 2SI/2-2PI/2 for neutral hydrogen Theory (year)

Experiment (year)

1040 MHz (1947) Bethe's NR calc. 1051 (1949) Rya 3 Z4 1057.19 ± 0.16 (1952) Rya 3 Z5

1000 ± 100 MHz (1947) 1062 ± 5 (1949) 1058.3 ± 1 (1952) 1057.77±0.1O (1953) (Triebwasser et al. [2.40], rf resonance)

1058.01 ±0.13 (1957) Rya4Z4 Magn. mom. correction 1057.49 (1960) Rya 5Z6In2(aZ) 1057.77 (1961) Rya 5Z 6 In(aZ) 1057.64±0.21 (1964) Rya 5Z 6 [2.15] 1058.07 ± 0.10 (1965) (level anticrossing method [2.41]) 1057.64 ± 0.11 (1964) Rya 5Z6 1057.56 ± 0.08 (1967) a from 2e/ h

1057.86 ± 0.10 (1967) (motional field correction [2.42]) 1057.90 ± 0.10 (1969) (non-Maxwellian vel. distr. in an atomic beam [2.43])

1057.91 ±0.16 (1970) Rya 4Z 4 correction [2.44] 1057.910 ± 0.010 (1971) Rya 4Z 4 exact, Rya 5Z6 improved [2.45] 1057.864 ± 0.014 (1975) Rya 5Z6 (extrap. from high Z [2.10])

1057.893 ± 0.02 (1975) (separated oscillatory meth.[2.46]) 1058 ± 0.26 (1975) (rf resonance in fast atomic beam [2.47]) 1057.862 ± 0.02 (1976) [2.49]

1057.888 ± 0.013 (1976) (larger proton size [2.48]) 1057.860 ± 0.09 (1981) (with Rya 5Z6 improved [2.17])

1057.845 'f 0.009 (1981) [2.50] 1057.8583 ± 0.0022 (1982) [2.51] 1057.8514 ± 0.0019 (1983) [2.9]

1057.857 ± 0.011 (1984) if (r2) 1/2 = 0.805 ± 0.011 1057.875 ± 0.011 (1984) if (r2) 1/2 = 0.862 ± 0.012 (with revised new results for high order corrections [2.52]) 1057.852 ± 0.011 (1987) if (r2)1/2 = 0.805 ± 0.011 1057.870 ± 0.011 (1987) if (r2)1/2 = 0.862 ± 0.012 (with new Rya 3 Z 5 (m/M){lnaZ, 1] correction [2.38]) 1057.849 ± 0.011 (1988) if (r2)1/2 = 0.805 ± 0.011 1057.867 ± 0.011 (1988) if (r2)1/2 = 0.862 ± 0.012 (with new Rya4Z6 recoil correction [2.53]) 1057.859 ± 0.008 (1991) if (r2)1/2 = 0.805 ± 0.011 1057.878 ± 0.008 (1991) if (r2) 1/2 = 0.862 ± 0.012 (with new Rya 5Z6 improved [2.18])

fIn fIn

fm fm

fm fm

fIn fIn

22

2 Atomic Structure and Spectra

Table 2.14. A comparison of theoretical and experimental 2SI/2-2PI/2 Lamb shift in hydrogen-like ions in GHz

Ion

Experiment [2.18,39]

Theory [2.27]

He+

14.0462(12) 14.0402(18) 14.0402(29) 14.0419(15) 14.0420(12) 14.04222(35) 62.790(70) 62.765(21) 780.1(8.0) 2192(15) 2215.6(7.5) 2202.7(11.0) 3339(35) 3405(75) 20189(250) 20130(200) 20110(200) 25200(160) 25140(240) 31190(220) 38100(600) 37890(380)

14.04239(55)

Li2+ C5+ 07+

pH pl4+

SI5+ C1 16+ ArI7+

62.7379(66) 781.99(21) 2196.21(92)

3343.1(1.6) 20253(13)

25372(17) 31346(20) 38248(25)

2

t

~ ~

uf :::::. uf

-

0

I

+T

0.

~

-1

-2 2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18

Nuclear charge Z Fig. 2.S. Comparison between theoretical [2.27] and experimental [2.39] values for the Lamb shift of the 2SI/2 -2PI/2 transition in H-like ions

2.2 Relativistic and Quantum Electrodynamical Corrections

Table 2.15. Calculated ionization potentials I, wavelengths A, oscillator strengths f and radiative transition probabilities A for La- -transitions in H-like ions [2.54] H, Z = I, 1 = 13.59854564(7) eV Transition

A [A]

A [s-I]

2PI/2 -I SI/2 2P3/2 -I SI/2

1215.673646(8) 1215.668238(8) 1025.722966(6) 1025.721825(6)

6.265 6.265 1.673 1.673

3PI/2- ISI/2 3P3/2 -I SI/2

x x x x

f 108 108 108 108

0.1388 0.1388 0.0264 0.0264

Be3+, Z = 4,1 = 217.720361(1) eV Transition

A [A]

A [s-I]

2PI/2 -I SI/2

75.9313415(8) 75.9259362(8) 64.0655819(5) 64.0644417(5)

1.605 1.604 4.283 4.283

2P3/2-ISI/2 3PI/2- ISI/2 3P3/2-ISI/2

x x x x

f 1011 1011 1010 1010

0.1387 0.1386 0.0264 0.0264

07+, Z = 8,1 = 871.41700(2) eV Transition

A [A]

A [s-I]

2PI/2- ISI/2 21':l/2-ISI/2 3PI/2- IS I/2 3P3/2-ISI/2

18.9725134(8) 18.9671064(8) 16.0066566(5) 16.0055159(5)

2.568 2.565 6.850 6.852

x x x x

f 1012 1012 1011 1011

0.1386 0.1384 0.0263 0.0263

Ne9+, Z = 10, 1 = 1362.21029(6) eV Transition

A [A]

A [s-I]

2PI/2 -I SI/2 2P3/2 -I SI/2 3PI/2 -ISI/2

12.1374871(8) 12.1320790(8) 10.2396290(5) 10.2384880(5)

6.272 6.260 1.672 1.672

3P3/2-ISI/2

Mgll+, Z = 12,

x x x x

1012 1012 1012 1012

0.1385 0.1381 0.0263 0.0263

i= 1962.6798(1) eV

Transition

A [A]

A [s-I]

2PI/2 -I SI/2 2P3/2 -I SI/2

8.4246105(8) 8.4192009(8) 7.1 069096(5) 7.1057682(5)

1.301 1.297 3.466 3.467

3PI/2- ISI/2 3P3/2-ISI/2

f

x x x x

f 1013 1013 1012 1012

0.1384 0.1379 0.0262 0.0262

ArI7+, Z = 18, 1 = 4426.2588(7) eV Transition

A [A]

A [s-l]

2PI/2 -1S1/2

3.7365167(8) 3.7311015(8) 3.1513716(6) 3.1502287(6)

6.594 6.554 1.752 1.753

2P3/2 -I SI/2

3PI/2- I SI/2 3P3/2-ISI/2

x x x x

f 1013 1013 1013 1013

0.1380 0.1368 0.0261 0.0261

23

24

2 Atomic Structure and Spectra

Table 2.15. Continued Ti 21 +, Z = 22, I = 6625.861(1) eV Transition

A [A]

A [s-I]

2PI/2-ISI/2

2.4966181(8) 2.4911980(8) 2.1052251(5) 2.1040808(5)

1.473 1.460 3.904 3.909

2P3/2-ISI/2

3Plj2-ISI/2 3P3/2-ISI/2

x x x x

Transition

A [A]

A [s-I]

2PI/2- ISI/2

I. 7834425(9) 1.7780167(9) 1.5034965(6) 1.5023505(6)

2.877 2.840 7.602 7.618

Cu 28+, Z

x x x x

A [A]

A [s-I]

2PI/2 -I SI/2 2P3/2-ISI/2

1.4306952(8) 1.4252646(8) 1.2058735(5) 1.2047262(5)

4.457 4.387 1.175 1.178

3PI/2- ISI/2

Ion ~5+, Z

x x x x

A [A]

A [S-I]

2Plj2-ISI/2

0.923239(1) 0.917795(1) 0.7777193(9) 0.7765681(9)

1.061 1.036 2.778 2.790

Mo41+, Z

x x x x

10 14 10 14 1013 1013

0.1372 0.1346 0.0258 0.0258

A [A]

A [s-I]

2PI/2- ISI/2

0.674318(1) 0.668861(1) 0.567698(1) 0.566543(1 )

1.972 1.907 5.123 5.155

Xe 53+, Z

f 1014 10 14 10 14 1014

0.1368 0.1336 0.0256 0.0256

f 10 15 1015 10 14 10 14

0.1356 0.1308 0.0252 0.0252

= 42, I = 24572.31(4) eV

Transition

2P3/2 -I SI/2 3 PI/2 -I SI/2 3 P3/2 -I SI/2

f

= 36, I = 17936.34(1) eV

Transition

2P3/2 -I SI/2 3 PI/2 -I SI/2 3P3/2-ISI/2

0.1376 0.1358 0.0259 0.0259

= 29, I = 11567.702(5) eV

Transition

3P3/2-ISI/2

10 14 10 14 1013 1013

= 26, I = 9277.757(3) eV

Fe25+, Z

2P3/2-ISI/2 3PI/2 -I SI/2 3P3/2 -ISI/2

f

x x x x

f 10 15 10 15 10 14 10 14

0.1345 0.1279 0.0248 0.0248

= 54, I = 41299.7(1) eV

Transition

A [A]

A [s-I]

2PI/2 -1 SI/2 2P3/2-ISI/2 3PI/2 -I SI/2 3P3/2-ISI/2

0.401817(2) 0.396326(2) 0.337776( I) 0.336611 (I)

5.429 5.132 1.382 1.400

x x x x

f 1015 1015 10 15 1015

0.1314 0.1208 0.0236 0.0238

2.2 Relativistic and Quantum Electrodynamical Corrections

Table 2.15. Continued W73+, Z

= 74, I = 80749(1) eV

Transition

A [A]

A [s-I]

2Pl/2 -I SI/2 2P3/2 -I SI/2

0.206279(4) 0.200708(4) 0.172764(3) 0.171576(3)

1.945 1.743 4.701 4.861

3PI/2- I SI/2 3P3/2 -I SI/2

Au78 +, Z

x x x x

A [A]

A [s-I]

2PI/2 -I SI/2 2P3/2-ISI/2 3PI/2 -ISI/2 3P3/2 -ISI/2

0.178866(6) 0.173269(5) 0.149620(4) 0.148425(4)

2.538 2.236 6.024 6.284

x x x x

A [A]

A [s-I]

2PI/2 -I SI/2 2P3/2 -I SI/2

0.12712(1) 0.12143(1) 0.105900(8) 0.104682(8)

4.726 3.950 1.054 1.138

3PI/2- ISI/2

0.1241 0.1052 0.0210 0.0215

f 10 16 10 16 10 15 1015

0.1217 0.1006 0.0202 0.0208

= 92, I = 131745(9) eV

Transition

3P3/2 -ISI/2

10 16 1016 1015 10 15

= 79, I = 93241(2) eV

Transition

lfJI+, Z

f

f 1016

x x 10 16 x 10 16 x 10 16

0.1145 0.0873 0.0177 0.0187

Table 2.16. Wavelengths A in A for La-transitions in H-like ions Ion

Transition

Theory [2.27]

Experiment [2.12,39,54]

2PI/2- ISI/2 2P3/2-ISI/2 2PI/2 -I SI/2

10.026(3)

2P3/2-ISI/2 2PI/2- 1SI/2

10.0285898(26) 10.0231809(26) 8.42461205(23) 8.41920249(23) 7.17631871(34) 7.17090835(24) 6.18583954(21 ) 6.18042832(21) 5.38675756(22) 5.38134544(22) 4.73276714(22)

2P3/2-ISI/2

4.72735405(22)

2PI/2 -I SI/2

4.19074607(26)

2P3/2-1SI/2

A1 12+

2PI/2- ISI/2 2P3/2 -1 SI/2 2PI/2 -I SI/2

2P3/2-ISI/2 pl4+

2PI/2 -ISI/2

4.18533197(26)

8.4253(15) 8.4194(15) 7.1759(15) 7.1703(15) 6.1859(10) 6.1798(10) 5.3859(10) 5.3802(10) 4.7334(10) 4.73344(85) 4.7274(10) 4.72759(49) 4.19064(14) 4.1906(1) 4.19069(14) 4.19055(21 ) 4.18521(14) 4.1852(1) 4.18520(18) 4.18518(17)

25

26

2 Atomic Structure and Spectra

Table 2.16. Continued Ion

Transition

ArI7+

2PI/2- 1SI/2

3.73652002(36)

2P3/2 -1 SI/2

3.73110485(36)

cr23+

2PI/2 -1 SI/2 2P3/2 -1 SI/2 2PI/2 -1 SI/2 2P3/2 -1 SI/2

Fe25+

2PI/2- 1SI/2

2.49662292(22) 2.49120293(22) 2.0955728(41) 2.0901501(41) 1.7834491(50)

2P3/2 -I SI/2

1.7880234(50)

2PI/2- IS I/2 2P3/2-ISI/2 2PI/2- ISI/2 2P:l/2-ISI/2 2PI/2- ISI/2 21'3/2 -1 SI/2 2PI/2 -I SI/2

1.4307033(34) 1.4252729(34) 1.1724091 (94) 1.1669734(93) 0.9232508(10)

Ti 21 +

Cu28+ Ge31 + Kf35+

Mo41+

Theory [2.27]

2P3/2-1SI/2

Table 2.17. Wavelengths A, in Transition 4P3/2-2SI/2 4Ds/2- 2P3/2 3P3/2-2SI/2 3Ds/2 - 2P3/2 4Fs/2 -3D3/2 . 4F7/2-3Ds/2

Experiment [2.12,39,54] 3.7366(6) 3.7365(8) 3.73514(40) 3.736522(19) 3.7364(3) 3.7365(2) 3.7366(5) 3.7309(6) 3.7310(8) 3.731142(70) 3.731105(19) 3.7311(4) 3.7312(2) 3.7309(5) 2.501(5) 2.496(5) 2.08(1) 1.7835(15) 1.78364(19) 1.7779(15) 1.77815(19) 1.4305(10) 1.172326(15)

0.9178073(10)

0.917800(34)

0.6743336(12) 0.6688767(12)

0.6685(5)

A for transitions in Mg XII

Theory [2.54]

Experiment [2.55]

33.65357 33.73307 45.39718 45.53302 130.0405 130.1408

33.678 33.736 45.392 45.530 130.061 130.141

into account correlation, relativistic and QED effects simultaneously. The two latter corrections rapidly increase with increasing nuclear charge Z, for example, the Lamb shift is proportional to Z4. The relative contribution of the Lamb shift to the total non-relativistic energy is: ~ 0.002% for He (Z = 2), ~ 0.8% for Ar 16+ (Z = 18) and 9% for Xe52+ (Z = 54) [2.13]. For ions with low

2.2 Relativistic and Quantum Electrodynamical Corrections

27

and intennediate Z, a reasonable approach is to obtain the accurate solutions of the non-relativistic Schroedinger equation with inclusion of the relativistic corrections by perturbation theory. For He-like ions with Z ~ 10 the accurate variational calculations of the energy for the ground state ls2 1 So and for lsns 1,3 S, lsnp 1,3 P states (2 ~ n ~ 5) are presented in [2.56]. For high-Z ions the accurate results are obtained by the exact solution of the one-electron Dirac equation with inclusion of correlation and QED-corrections by perturbation theory. In particular, the method of expansion in the parameter 1/ Z provides the inclusion of the higher-order relativistic effects and radiative corrections [2.57,58]. The extensive numerical calculations using the expansion in l/Z and (aZ)2 are given in [2.59-65]. Numerical calculations based in the multiconfiguration Dirac-Fock approximation are given in [2.66-67]. Series of semiempirical formulas for S, P, D and F states of He-like spectra of Na X to Ar XVII ions are presented in [2.68]. The relativistic generalization of the random phase approximation is developed in [2.69]. This approach gives good results for ions with Z ) 30 which are described by the j j -coupling. In practice, all calculations use some expansion methods in which only a limited number of tenns is retained. The total energy of the He-like ions can be written as a double expansion in parameters A = l/Z and aZ [2.57,58]: 00

E

= Z2 L

00

L

E~mA -n(aZ)m

n=Om=O + ZOEg +a2Z2E~

+ ... + Z-I E~ +a 2 Z 1 Ej + ... ,

(2.2.48)

which excludes QED- and the nuclear-motion-(and size-)-type corrections. The following notations are used:

m

= 0, n

H

= (HNR + Bp)LS + R (HD + V12 + B)jj R- 1 -

) 0 : non-relativistic energy of the configuration including the correlation effect in all 1/ Z -orders; m ) 0, n = 0 : series expansion in the parameter (aZ)2 of the Dirac energy contributing to the energy of the two-electron configuration; m ) 0, n = 1 : correlation energy due to the one-photon exchange in the Breit interaction in all orders on (aZ)2, etc. As has been pointed out in [2.59], a double expansion in (2.2.48) is made by diagonalizing the matrix ~,

where HNR is the non-relativistic two-body Hamiltonian HNR

1

Z) 1

2 ( = -2 L V; + -rj +-, . 1 r12 1=

.

(2.2.49)

28

2 Atomic Structure and Spectra 2

HD = LHD(i). i=1

Here HD(i) is the Dirac single-particle Hamiltonian [2.6]. Bp consists of the electron-electron interaction in the Breit-Pauli approximation [2.6], and two terms Vl 2 and B are the Coulomb interaction and the relativistic form of the Breit interaction including retardation, respectively. The subscripts LS and jj in (2.2.49) denote the coupling scheme in which the corresponding matrix elements are evaluated. R is the jj -+ LS recoupling-transformation matrix. The matrix t::.. is the correction and is counted twice in the first two terms. For example, the matrix R for Is2p 3PI and Is2p I PI is:

11S2 p3 PI)) _ R (11s1/22PI/2, 1)) ( 11s2p 3PI) 11s1/2 2 p3/2, 1) , where 1

R=,Jj

(-Ji 1

-I)

(2.2.50)

(2.2.51)

-Ji'

The terms of the t::..-matrix for this case are: t::..1,1 =

_~Z2 + 0.225727785Z - ::4'~2Z4 + 0.1304287a 2Z 3;

t::..2.2 = -

~ Z2 + 0.259868922Z -

-Ji

::4 a 2Z4 + 0.0554030a 2Z3;

24 + 0.0288508a 23 Z .

t::..12 . = t::..2. I = --6 9 a Z

(2.2.52) (2.2.53) (2.2.54)

The eigenvectors of the full Hamiltonian H in (2.2.49) (labeled by E).., ).. = 1, 2) are presented by

_T(11S2p3PI)) ( lEI)) I E 2) 11s2p I PI) '

(2.2.55)

where T is the matrix

T = (

co~ 0

- SID 0

sin 0 ) cosO

(2 2 56) ..

and 0 is the singlet-triplet mixing angle. For low Z, T -+ 1 (LS-coupling) since HNR» Bp; for high Z, T -+ R- I (jj-coupling) since HD» V12+B. For the states non-degenerate in zero order, such as the ground state Is2 I So and the excited states Is2p 3Po, Is2s 3SI, etc., T = 1. Thus, the final diagonalization of the matrix (2.2.49) produces an intermediate coupling scheme of the two states. Numerical calculations of the mixing angle 0 for some states in He-like ions are presented in Table 2.18 and in Fig. 2.6, showing that the states Is2p 3PI and Is2p I PI are progressively mixed by the spin-dependent interactions in H (2.2.49) as Z increases until ultimately the jj-coupling limit is reached at high Z [2.13]. The leading contributions to the matrix elements of the Hamiltonian (2.2.49) calculated in LS-coupling are presented in Table 2.19. The numerical

2.2 Relativistic and Quantum Electrodynamical Corrections

Table 2.18. Mixing angle () for the states of He-like ions [2.70] Z

2 3PI-2 I PI

3 3PI-3 1PI

33D2-31D2

33S1-33DI

2 3 4 5 6 7 8 9 10

0.000279 0.000771 0.001710 0.003220 0.005440 0.008510 0.012603 0.017711 0.024082 0.031775 0.040891 0.054827 0.063400 0.077317 0.092521 0.109318 0.127344 0.146726 0.167178 0.188413 0.210246 0.232489 0.255780 0.276510 0.785389

0.000319 0.000775 0.001740 0.003320 0.005680 0.008960 0.013310 0.018851 0.025723 0.034047 0.043894 0.055358 0.068473 0.083236 0.099625 0.117571 0.136827 0.157450 0.179904 0.201256 0.223968 0.246901 0.269656 0.292138 0.785389

0.017101 0.018901 0.029404 0.045316 0.067050 0.094742 0.128453 0.167583 0.210859 0.256607 0.302806 0.347661 0.389580 0.427723 0.461509 0.490991 0.516454 0.538443 0.557186 0.573291 0.587041 0.598859 0.609064 0.617870 0.684777

3.0 x 10-6 1.2 x 10-5 2.7 x 10- 5 4.7 x 10- 5 7.3 x 10-5 1.05 x 10-4 1.41 x 10-4 1.83 x 10-4 2.29 x 10-4 2.79 x 10-4 3.33 x 10-4 3.91 x 10-4 4.51 x 10-4 5.14 x 10-4 5.78 x 10-4 6.43 x 10-4 7.08 x 10-4 7.73 x 10-4 8.37 x 10-4 9.00 x 10-4 9.60 x 10-4 1.01 x 10- 3 1.07 x 10- 3 2.11 x 10- 3 0

11

12 13 14 15 16 17 18 19 20 21 22 23 24 25 jj limit

0.7 0.6 'U

!

0.5

CD ~

Cl

c:

0.4

as

Cl

c:

'x 0.2 ~

0.1 0.0

0

10

20

30

40

50

60

70

80

90

100

Nuclear charge Z

Fig. 2.6. Mixing angle () for the 2 3PI and 21 PI states of He-like ions [2.70]

29

30

2 Atomic Structure and Spectra

Table 2.19. Leading terms in Z-I and (aZ)2 expansions (a.u.) of the Hamiltonian matrix (2.2.49) in the LS-coupling scheme [2.70] State

Z2

Z

a 2Z 4

a 2Z 3

a4Z6

a4Z 5

Is2 I So Is2s I So Is2s 3SI Is2 p 3 Po Is2 p 3PI Is2p I PI 2 3 PI-2 1PI Is2 p 3 P2 Is3s I So Is3s 3SI Is3p 3Po Is3 p 3PI Is3p Ipi 3 3 PI-3 1PI Is3 p 3p2 Is3d 3DI Is3d 3D2 Is3d I D2 33D2-31D2 Is3d 3 D3 3 3S1-3 3 DI

-1 -5/8 -5/8 -5/8 -5/8 -5/8 0 -5/8 -5/9 -5/9 -5/9 -5/9 -5/9 0 -5/9 -5/9 -5/9 -5/9 0 -5/9 0

o.625()()()()()()

-1/4 -21/128 -21/128 -21/128 -59/384 -55/384

0.4801396 0.1694782 0.0769352 0.2197682 0.1304287 0.0554030 0.0288508 0.0406387 0.0581620 0.0331906 0.0715549 0.0460209 0.0269123 0.0088266 0.0189765 0.0238621 0.0140516 0.0108856 0.0030447 0.0044280 -0.00090986

-1/8 -85/1024 -85/1024 -85/1024 -235/3072 -215/3072

0.2197 0.1052 0.04223 0.1430 0.05693 0.03874 0.03695 -0.001877 0.0324 0.0172 0.0408 0.0190 0.0141 0.0106 0.00189 0.00431 0.00227 0.00122 0.00181 0.00118 6.4 x 10-4

0.231824417 0.187928669 0.225727785 0.225727785 0.259868922 0 0.225727785 0.105255127 0.093719482 0.104293823 0.104293823 0.113357543 0 0.104293823 0.110775757 0.110775757 0.111270142 0 0.110775757 0

-.fi/96

-17/128 -5/36 -5/36 -5/36 -11/81 -43/324 -.fi/324

-7/54 -7/54 -52/405 -23/180 -v'6/1620 -41/324 0

-5.fi/768

-65/1024 -3224/66 -3224/66 -3224/66 -3131/66 -3038/66 -93.fi/66

-2945/66 -2945/66 -2945/66 -2945/66 -5v'6/66 -2920/66 0

results for Is 21 So-ls2p 3 PI and Is2 I So-ls2p I PI transitions with 2 :( Z :( 120 are given in [2.59], for the Is2s 3 SI -ls2p 3 Po and Is2s 3 SI-ls2p 3 P2 transitions and Z :( 50 in [2.60] and for transition energies including Is3d I D2 and Is3d 3 DO.I.2 states with high Z in [2.70]. The precision variational eigenvalues in neutral He are given in [2.71,72]. As mentioned before, the general expression (2.2.49) does not conclude the set of corrections associated with the nuclear-size correction, mass-polarization and QED effects. The mass-polarization correction EM (analogously to the ER-correction in H-like ion) is connected with the non-separability of the wave equation for a two-electron ion. This correction is defined by m

EM = - M (PI' P2),

where (PI' P2) is the expectation value for the product of the linear momenta of the two electrons. The mass -polarization correction was calculated for the isotopes corresponding to each Z in the He-like isoelectronic sequence. The precision results for EM are tabulated in [2.56] for S and P states in He-like ions up to Z = 10; high values up to Z = 100 are considered in [2.73]. The finite nuclear-size correction, constructed on the H-like approximation together with relativistic effects and two-electron corrections, can be written in the form [2.73]

2.2 Relativistic and Quantum Eiectrodynamicai Corrections

ENS

4Jl" Z

(2 1/2)2

= -3-Ry (r)

{8 (rl)

31

+ 8 (r2») + 4Z2 3n 3 Ry

{[8/,0+8j ,I/2 Cn/(aZ)2] Z2>-. ({r2) 1/2) 2>-' -8>-.,oZ2 ({r2)1/2)2}

+ 4Z;RY

{[I + ~ (aZ)2] Z2>-. «r2)1/2r>-. _ Z2 «r2)1/2r} ,

A = ( 1 - a 2 Z 2)

1/2 ,

CIO = 0.5,

C 20 = 1.38,

C21 = 0.0234,

where {8 (rl») is the expectation value for the hydrogenic electron density. The QED-correction formulas for He-like ions are based on the work [2.74] for one- and two-electron contributions. The results can be expressed in the form [2.75] (I)

EQED

(2)

(3)

= EQED + EQED + E QED ,

(2.2.57)

where Eg~D represents the one-electron part of the order of ~ a 3Z4 of the Lamb shift:

Eg~D

8 19 = 3Rya3Z{8(rt> + 8 (r2») { Ln(aZ)-2 + 30

- Ln [L (lsnL;2S+1 L)

+aB4Q/Jl"

+ (aZ)2

/Z 2 Ry] + A4ILn(aZ)-2 + AsoaZ + A60(aZ)2

[A6ILn(az)-2

+ A62Ln2(aZ)-2 + G(aZ)]}. (2.2.58)

The function G(aZ) and coefficients Aij and B40 were presented in Sect. 2.2.2. The term Ln[L(1snL;2S+1 L)/Z 2Ry] is the Z-independent part of the two-electron Bethe logarithm. The shift (2.2.58) is the main QED contribution. If (8(r») in (2.2.58) is replaced by the H-like value (i.e. the leading term in the 1/ Z expansion), then (2.2.58) reduces to the corresponding expression for H-like ions [(2.2.7) in Sect. 2.2.2]. The second term in (2.2.59) is the J -independent part of the two-electron QED correction [2.76,77]: (2) = a 3Ry [ (8 (r12») (28 3 Lna

EQED

178 + 15

40 . S2 ) 3s1

+ Q]

,

(2.2.59)

where Q is defined by

Q = -(7/3Jl") lim [(ri/({J») f3~0

+ 4Jl"(y + Ln{J) (8(rn»)]

.

Here y is the Euler constant and {J is the radius of a sphere centered at that is excluded from the integration over r12, i.e. < 0, r12? O.

rl2

(2.2.60) rl2

=0

(2.2.61)

32

2 Atomic Structure and Spectra

A relatively simple closed expression for the Q function in the form of the Z expansion was derived in [2.78]: Q = -(8 (r12)) { LnZ

+ ~Ln

[~ (1 + *) 2]- I}

+ Z3 (Qo + QIZ-I + Q2Z-2 + ... ).

(2.2.62)

The expression (2.2.62) is simpler than (2.2.60) and is useful for applications. The first few Qi coefficients in (2.2.62) for He-like ions are given in [2.73]. The last term in (2.2.57) is the two-electron magnetic-moment correction [2.79], i.e. (2.2.63)

EgJD = (Hint),

where Hint is the spin-dependent part of the Breit interaction [2.6]

a 3 Ry

[Z

Hint = - - 3" [rl xpJ] rr rl X (SI

Z[r2 x P2] + 3" r

+ S2)] + 2a3~y rrr l2

-

2

[SI . S2 -

-i r l2

2 [r12x (PI - P2)]

-3

r l2

(r12 . SI) (r12 . S2)] .

(2.2.64)

The main uncertainty in calculating (2.2.58) arises from the two-electron Bethe logarithm. The accurate variational calculations for the ground and exited states of He and He-like ions up to Z = 10 has been calculated in [2.77]. The corresponding numerical results for the ground state of the He-like ions are given in Table 2.20. The Bethe logarithm was calculated from the screened H -like approximation in [2.80]: Ln [L (lsnL;2S+1 L)

/Z 2Ry]

~ Ln [L°(1snL)/Ry] - 2a/Z + 0(Z-2) = Ln [L°(1snl)(Z - a)2/Z 2Ry]

+ 0(Z-2),

Table 2.20. Values of Ln[L(l s 2)/Z 2Ry] for the ground state of the helium isoelectronic sequence [2.77] Z

Ln[L(ls 2)/Z2Ry]

Z

Ln[L(ls2)/Z2Ry]

2 3 4 5 6

4.37 ±0.01 5.21 ±0.01 5.777 ± 0.003 6.214 ± 0.002 6.565 ± 0.002

7 8 9 10

6.864 ± 0.002 7.115±0.002 7.334 ± 0.002 7.525 ± 0.002

(2.2.65)

2.2 Relativistic and Quantum Electrodynamical Corrections

33

°

where Ln [L (lsnL) IRy] is determined from the hydrogeoic Bethe logarithm Ln [L°(1snL)/RY] = {Ln[L(1s)]

+ Ln[L(nl)]ln 3} I (1 + n-3~I.o) (2.2.66)

and the screening constants (1 can be expressed in terms of the Z-expansion theory. The hydrogenic Bethe logarithms are presented in [2.7]. The numerical values for some low-lying S and P-states of He-like ions have been obtained in [2.80] in the form: Ln [L(I I S)/RY] = Ln [19.7693(Z - 0.00615)2] ,

(2.2.67)

Ln [L(2 IS) IRy] = Ln [ 19.3943(Z + 0.02040)2] ,

(2.2.68)

Ln [L(2 3S)/RY] = Ln [19.3943(Z + 0.01388)2] ,

(2.2.69)

Ln [L(2 1 P)/RY] = Ln [19.6952(Z + 0.00600)2] ,

(2.2.70)

Ln [L(2 3S)/RY] = Ln [19.6952(Z +0.00475)2].

(2.2.71)

These values are in good agreement with the precision theoretical variational data [2.77]. The_leading term in the Z-expansion of the vacuum-polarization correction has been calculated in [2.54,81]: Evp = _Ry_8_ a3z4 (1 + 15rr

~I .oln 3)

+ Rya 3Z 3 (Cl1 + CI2).

(2.2.72)

The coefficients Cl1 and C12 in (2.2.72) are state-dependent. The numerical values C 12 and C 11 for the S, P, D and F states are presented in Table 2.21.

Table 2.21. Calculated values for the coefficients

el1

and

el2 in (2.2.72) [2.81]

Configuration

singlet term

triplet term

el1

Is2

1.13342 X 10- 1 2.3881 X 10- 2 -1.8543 X 10-3 7.3429 X 10-3 -1.2815 X 10-4 -7.8032 X 10-5 3.149 x 10-3 -6.840 X 10-6 -3.748 X 10-5 -1.019 X 10-6 1.630 X 10- 3

o

1.06104 X 10- 2 2.09587 X 10- 3 6.98623 X 10-4 1.50244 X 10-4 2.17595 X 10-4 1.03617 X 10-5 6.9657 X 10-6 2.3273 X 10-5 5.8236 X 10-6 8.6920 X 10- 8 2.2179 X 10-7

Is2s Is2p Is3s 1s3p Is3d Is4s Is4p Is4d

Is4f Is5s

1.7951 x 10-2 7.2957 x 10-3 6.0723 x 10- 3 1.7742 X 10- 3 1.8342 X 10-4 2.702 x 10-3 7.047 x 10-4 9.594 X 10-5

2.105 x 10-6 1.428 x 10-3

34

2 Atomic Structure and Spectra

The various contributions to the energy of the ground and exited states in He-like ions are shown in Tables 2.22-24. The significant corrections (absolute values) to the ground state of He-like ion~ are presented in Fig. 2.7. for illustrating the Z-dependence and for comparing the relative importance of the various contributions. The theoretical wavelengths for levels with n = 2 are given in Tables 2.25-26 for 2:::; Z :::; 92. A comparison between measured and calculated wavelengths for He-like ions is presented in Tables 2.27-34.

Table 2.22. Contributions (in cm- I ) of the different corrections to the ionization energies of He-like ions [2.73]; Z = 2

State

Non-relativistic Relativistic energies corrections

IISo 198317.386 21So 32033.208 38453.132 2 3 S1 21pI 27176.690 23Po 29222.156 29222.156 2 3PI 29222.156 2 3P2

-0.565 0.399 1.922 0.467 -0.303 -0.315 -0.237

Mass-polarisation corrections

QED corrections

Nuclear-size corrections

Total energies

-4.785 -0.286 -0.224 -0.385 1.943 1.943 1.943

-1.379 -0.090 -0.135 0.000 0.041 0.043 0.043

-0.001 0.000 0.000 0.000 0.000 0.000 0.000

198310.655 32033.230 38454.695 27175.772 29222.837 29223.826 29223.903

Table 2.23. The same as in Table 2.19, for Z = 26 Non-relativistic Relativistic Mass-polarization State energies corrections corrections

QED corrections

Nuclear-size corrections

Total energies

IISo 2 1So 2 3 S1 21pI 2 3Po 2 3PI 2 3P2

-30020.43 -4010.18 -4085.73 -184.27 190.89 172.38 -100.48

-403.78 -50.41 -51.45 -0.22 0.95 0.84 1.33

71203851.34 17422714.80 17676224.08 17161466.92 17442620.16 17426438.72 17307479.49

70649800.23 17247619.69 17483497.29 17096779.65 17273522.16 17273522.16 17273522.16

584482.54 179156.36 196864.55 64958.08 168800.51 152655.85 33950.83

-7.21 -0.66 -0.58 -86.33 105.65 87.49 105.65

Table 2.24. The same as in Table 2.19 for Z = 92 Non-relativistic Relativistic State energies corrections IISo 21So 2 3 S1 2 I PI 2 3Po 2 3PI 2 3P2

916229320.0 227547820.0 228419457.0 226990852.0 227661876.0 227661876.0 227661876.0

Mass-polarization QED corrections corrections

-6.0 132452542.0 -1.0 41598109.0 42810672.0 -1.0 6648233.0 -110.0 40898465.0 328.0 41806363.0 111.0 6571628.0 328.0

Nuclear-size Total corrections energies

-2119664.0 -1540446.0 1045021745.0 -394728.0 -291935.0 268459266.0 -393182.0 -293159.0 270543788.0 -69664.0 238.0 233569548.0 -53517.0 -33240.0 268473913.0 -53547.0 -33853.0 269380951.0 -67169.0 1438.0 234168102.0

2.2 Relativistic and Quantum Electrodynamical Corrections

35

10' 10

-

7

10 •

I

E

~

10

J

w

5 10 10 -. 10

-J

~~~~~~~~~~~~~~~~~~~~

o

10

20

30

40

50

60

70

80

90 100 110

Z Fig. 2.7. Contributions to the ionization energy of the Is2 ISO state of He-like ions [2.73]: (1) nonrelativistic energy; (2) relativistic corrections; (3) QED-corrections; (4) nuclear-size corrections; (5) mass-polarization correction

Table 2.25. Theoretical wavelengths in (A) for He-like ions [2.73] Z

23Po-23S1

2 3 PI_2 3 S1

23P2-23S1

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

10832.055 5485.0687 3722.3657 2825.3613 2277.9583 1907.6334 1639.8914 1436.9372 1277.6955 1149.1731 1043.2598 954.3196 878.5819 813.2072 756.2141 706.0143 661.4381 621.6255 585.7829 553.3134 523.7849 496.7912 472.0195 449.1863 428.0750

10833.216 5486.6351 3723.9732 2826.6633 2278.6195 1907.3339 1638.3326 1433.8477 1272.8369 1142.3470 1034.3131 943.1505 865.1404 797.5012 738.3054 686.0216 639.5304 598.0097 560.7084 527.0598 496.6439 469.0644 444.0055 421.1772 400.3437

10833.306 5486.0070 3721.9111 2822.4650 2271.5944 1896.8111 1623.6671 1414.4312 1248.1029 1111.7859 997.4777 899.6683 814.7100 739.9090 673.4147 613.7854 559.9823 511.2483 466.9150 426.4841 389.5826 355.8673 325.0598 296.9057 271.1904

36

2 Atomic Structure and Spectra

Table 2.25. Continued Z

27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76

23PO-23S1

23PI-23S1

408.4818 390.2590 373.2453 357.3379 342.4127 328.3949 315.1968 302.7468 290.9846 279.8530 269.3019 259.2824 249.7624 240.7070 232.0808 223.8614 216.0053 208.5039 201.3304 194.4609 187.8807 181.5813 175.5289 169.7244 164.1555 158.8153 153.6460 148.7049 143.9245 139.3358 134.9119 130.6516 126.5429 122.5834 118.7741 115.1089 111.5499 108.1309 104.8141 101.5975 98.5647 95.5249 92.6702 89.9013 87.2064 84.6241 82.1080 79.7120 77.3509 75.0930

381.2830 363.8172 347.7661 332.9916 319.3495 306.7342 295.0348 284.1606 274.0275 264.5673 255.7103 247.3981 239.5855 232.2290 225.2869 218.7300 212.5072 206.6105 201.0036 195.6648 190.5746 185.7261 181.0799 176.6397 172.3909 168.3306 164.3900 160.6405 157.0034 153.5188 150.1553 146.9146 143.7847 140.7652 137.8613 135.0688 132.3373 129.7184 127.1606 124.6637 122.3575 119.9546 117.7556 115.6102 113.5005 111.4957 109.5198 107.6726 105.8043 104.0433

2 3 P2- 23S1

247.7104 226.2904 206.7552 188.9557 172.7414 157.9826 144.5519 132.3330 121.2188 111.1096 101.9139 93.5473 85.9341 79.0040 72.6928 66.9430 61.7002 56.9179 52.5526 48.5646 44.9189 41.5840 38.5300 35.7316 33.1653 30.8101 28.6449 26.6550 24.8229 23.1357 21.5800 20.1446 18.8189 17.5935 16.4601 15.4108 14.4379 13.5360 12.6985 11.9203 11.1979 10.5242 9.89805 9.31436 8.76983 8.26189 7.78732 7.34413 6.92925 6.54120

2.2 Relativistic and Quantum Electrodynamical Corrections

Thble 2.25. Continued Z

77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92

23PO-23SI

2 3PI-2 3SI

72.9130 70.8018 68.7650 66.8160 64.9094 63.0831 61.3270 59.6168 58.0030 56.4701 54.9465 53.5010 52.0944 50.7874 58.0030 56.4701

102.3478 100.7044 99.1288 97.6548 96.1911 94.8238 93.5377 92.2820 91.1840 90.2194 89.2116 88.3437 87.5192 86.9202 91.1840 90.2194

Thble 2.26. Theoretical wavelengths (in

2 3 Pz-2 3SI

6.17782 5.83726 5.51795 5.21847 4.93715 4.67295 4.42465 4.19102 3.97135 3.76462 3.56961 3.38589 3.21254 3.04915 3.97135 3.76462

A) for He-like ions [2.73]

Z

2 1PI- 11So

2 3PI-l I So

2 3 Pz-l 1So

2 3 SI- 11So

2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

584.33440 199.27954 100.25373 60.313810 40.267365 28.786966 21.601504 16.806371 13.447306 11.002899 9.1687500 7.7573015 6.6479474 5.7602007 5.0387265 4.4444296 3.9490747 3.5318569 3.1771533 2.8730641 2.6103979 2.3819542 2.1820344 2.0060773 1.8503995 1.7119984 1.5884088 1.4775888

591.41212 202.32250 101.69124 61.089670 40.730230 29.084321 21.803639 16.950004 13.553110 11.083173 9.2312083 7.8069569 6.6881869 5.7933673 5.0664923 4.4680068 3.9693631 3.5495387 3.1927472 2.8869718 2.6229366 2.3933744 2.1925366 2.0158219 1.8595168 1.7205939 1.5965691 1.4853841

591.41238 202.32165 101.68970 61.087706 40.727979 29.081861 21.801018 16.947254 13.550251 11.080218 9.2281668 7.8038348 6.6849879 5.7900934 5.0631443 4.4645847 3.9658664 3.5459669 3.1890994 2.8832474 2.6191353 2.3894962 2.1885819 2.0117917 1.8554125 1.7164172 1.5923224 1.4810697

625.56316 210.06890 104.54610 62.439099 41.471532 29.534686 22.097725 17.152773 13.698976 11.191756 9.3143384 7.8721185 6.7402944 5.8357606 5.1015005 4.4972974 3.9941535 3.5707331 3.2110312 2.9028723 2.6368627 2.4056491 2.2034172 2.0255163 1.8681941 1.7283935 1.6036063 1.4917557

37

38

2 Atomic Structure and Spectra

Table 2.26. Continued Z

30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80

21 Pl-l ISO 1.3778378 1.2877279 1.2060561 1.1317998 1.0640876 1.0021720 0.9454100 0.8932442 0.8451911 0.8008297 0.7597910 0.7217511 0.6864248 0.6535585 0.6229301 0.5943399 0.5676109 0.5425847 0.5191201 0.4970885 0.4763760 0.4568794 0.4385053 0.4211665 0.4047899 0.3893025 0.3746428 0.3607516 0.3475764 0.3350683 0.3231831 0.3118801 0.3011219 0.2908722 0.2811011 0.2717775 0.2628744 0.2543705 0.2462342 0.2384525 0.2310011 0.2238609 0.2170165 0.2104496 0.2041476 0.1980929 0.1922754 0.1866817 0.1813000 0.1761198 0.1711317

2 3 PI-l ISO 1.3853264 1.2949575 1.2130664 1.1386237 1.0707524 1.0087006 0.9518215 0.8995547 0.8514142 0.8069770 0.7658723 0.7277746 0.6923978 0.6594871 0.6288196 0.6001950 0.5734356 0.5483825 0.5248941 0.5028413 0.4821101 0.4625969 0.4442081 0.4268562 0.4104680 0.3949704 0.3803016 0.3664025 0.3532204 0.3407063 0.3288158 0.3175084 0.3067464 0.2964936 0.2867199 0.2773942 0.2684896 0.2599847 0.2518477 0.2440659 0.2366147 0.2294750 0.2226315 0.2160658 0.2097654 0.2037125 0.1978970 0.1923057 0.1869267 0.1817494 0.1767645

23P2-11S0

1.3809471 1.2905162 1.2085660 1.1340674 1.0661432 1.0040416 0.9471154 0.8948044 0.8466224 0.8021461 0.7610047 0.7228726 0.6874634 0.6545224 0.6238263 0.5951748 0.5683900 0.5433130 0.5198020 0.4977278 0.4769764 0.4574439 0.4390368 0.4216676 0.4052628 0.3897493 0.3750654 0.3611519 0.3479558 0.3354283 0.3235250 0.3122051 0.3014311 0.2911666 0.2813816 0.2720450 0.2631296 0.2546143 0.2464671 0.2386753 0.2312143 0.2240651 0.2172120 0.2106370 0.2043274 0.1982654 0.1924410 0.1868407 0.1814528 0.1762668 0.1712730

2 3 SI- lISo 1.3911138 1.3002299 1.2178828 1.1430350 1.0748024 1.0124274 0.9552581 0.9027304 0.8543545 0.8097042 0.7684064 0.7301332 0.6945965 0.6615401 0.6307393 0.6019926 0.5751211 0.5499651 0.5263817 0.5042415 0.4834296 0.4638415 0.4453834 0.4279675 0.4115195 0.3959665 0.3812460 0.3672988 0.3540717 0.3415155 0.3295857 0.3182413 0.3074446 0.2971593 0.2873550 0.2780007 0.2690691 0.2605383 0.2523776 0.2445728 0.2370999 0.2299399 0.2230769 0.2164929 0.2101748 0.2041054 0.1982742 0.1926677 0.1872743 0.1820833 0.1770850

2.2 Relativistic and Quantum ElectrodynamicaI Corrections

39

Table 2.26. Continued Z 81 82 83 84 85 86 87 88 89 90 91 92

2 I PI-I I So 0.1663243 0.1616903 0.1572210 0.1529075 0.1487446 0.1447245 0.1408380 0.1370815 0.1334476 0.1299332 0.1487446 0.1447245

2 3PI-l I So

2 3 P:z- 1IS0

2 3SI-l I So

0.1664603 0.1618212 0.1573470 0.1530289 0.1488616 0.1448373 0.1409467 0.1371863 0.1335487 0.1300308 0.1488616 0.1448373

0.1719605 0.1673301 0.1628647 0.1585554 0.1543969 0.1503815 0.14(i4998 0.1427484 0.1391198 0.1356111 0.1543969 0.1503815

0.1722684 0.1676259 0.1631488 0.1588283 0.1546587 0.1506326 0.1467408 0.1429794 0.1393413 0.1358230 0.1546587 0.1506326

Table 2.27. Theoretical and experimental wavelengths (inA.) for Is2 I So-ls2s 3SI, 1s 2 I So-lsnp 1.3 PJ transitions in He-like ions

Ion

Transition

Ne8+

Is2 lSo-ls2p I PI

All 1+

Ti2O+

-ls2p 3PI -ls3plpl -ls4p I PI -ls5plpl -ls6p I PI -ls7p I PI Is2 I So-ls2s 3SI -ls2 p 3P:z -ls2p 3PI -ls2p I PI -1s3p 3PI -ls3p I PI -ls4p I PI -ls5plpl -ls6plpl -ls7p I PI -ls8p I PI -ls9plpl Is2 I So-ls2s 3SI

-ls2p 3P:z -ls2p 3PI -ls2s 3S1 V21+

Is 21 So-1s2p I PI

Experiment [2.54,68,73]

Theory

13.447 13.55 11.547 11.000 10.76 10.64 10.56

13.448 13.553 11.546 11.000 10.764 10.640 10.567

7.872 7.8067(8) 7.7575(8) 6.6449(20) 6.6351(8) 6.3142(8) 6.1750(15) 6.1010(10) 6.0595(20) 6.0299(10) 6.0100(15) 2.6099(1) 2.6097 2.61104(4) 2.6187(1) 2.6184 2.6223(1) 2.6223 2.6355(1) 2.3820(4) 2.38185(7)

[2.54] [2.54] [2.54] [2.54] [2.54] [2.54] [2.54]

13.447 13.553 11.547 11.000 10.764

[2.73] [2.73] [2.83] [2.83] [2.83]

13.448 [2.83] 13.554 [2.83]

7.87721 [2.54] 7.8038 [2.54] 7.80068 [2.54] 7.7574 [2.54] 6.6451 [2.54] 6.6343 [2.54] 6.31370 [2.54] 6.17542 [2.54] 6.10277 [2.54] 6.05976 [2.54] 6.03219 [2.54] 6.01341 [2.54]

7.8721 [2.73] 7.8038 [2.73] 7.8070 [2.73] 7.7573 [2.73] 6.6449 [2.83] 6.6349 [2.83] 6.31400 [2.83] 6.17562 [2.83]

7.8724 [2.83] 7.804 [2.83] 7.8080 [2.83] 7.7576 [2.83]

2.610209 [2.54]

2.61039 [2.73]

2.61105 [2.83]

2.619140 [2.54]

2.61914 [2.73]

2.6192 [2.83]

2.622925 [2.54]

2.62294 [2.73]

2.62230 [2.83]

2.636878 [2.54]

2.63686 [2.73]

2.6369 [2.83]

2.381774 [2.54]

2.38949 [2.73]

2.38193 [2.82]

40

2 Atomic Structure and Spectra

Table 2.27. Continued Ion

Transition

Fe24 + Is2 I So-ls2p I PI

Experiment [2.54,68,73]

Theory

1.85046(6) 1.8504(4) 1.85035(5) 1.85542(8) 1.85956(22) 1.859546(8) 1.86822( 10) 1.5738(6) 1.4948(6) 1.4605(6) 1.44433(6)

1.850280 [2.54]

1.85039 [2.73]

1.85038 [2.82]

1.855428 [2.54] 1.859548 [2.54]

1.85541 [2.73] 1.85952 [2.73]

1.8555 [2.83] 1.8596 [2.83]

1.868218 1.572949 1.494519 1.460790 1.443094

1.86819 [2.73] 1.5732 [2.83] 1.4946 [2.83] 1.46083 [2.83]

1.8692 [2.83]

Co25 + Is2 I So-ls2p I PI

1.7122(6) 1.71197(5)

1.71191 [2.54]

1.71199 [2.73]

1.71198 [2.82]

Ni26+ Is2 I So-ls2p I PI

1.5884(4) 1.58840(5)

1.588350 [2.54]

1.58841 [2.73]

1.58839 [2.82]

Cu27 + Is2 I So-ls2p I PI

1.47761(5)

1.477563 [2.54]

1.47759 [2.73]

1.47757 [2.82]

Zn28 + Is2 I So-ls2p I PI

1.3783(5)

1.377847 [2.54]

1.37783 [2.73]

1.37782 [2.82]

Ga29 + Is2 I So-ls2p I PI

1.28782(5)

1.287775 [2.54]

1.28773 [2.73]

1.28771 [2.82]

-ls2p 3P2 -ls2p 3PI -ls2s 3S1 -ls3plpl -ls4p I PI -ls5p l PI -ls6p I PI

KJ34+

[2.54] [2.54] [2.54] [2.54] [2.54]

Is2 I So-ls2p I PI 0.0945337(20) 0.09456722 [2.54] 0.0945409 [2.73] 0.09454 [2.83] -ls2p 3PI 0.095177(20) 0.09518348 [2.54] 0.0951821 [2.73] 0.095203 [2.83] 0.0951770(20)

sr36+ Is2 I So-ls2p I PI 0.084549(14)

0.08455507 [2.54] 0.0845191 [2.73] 0.084517 [2.82]

y37+

Is2 ISo-ls2p I PI

0.080117(16)

0.08012404 [2.54] 0.080830 [2.73]

Xe52+

IS2 I So-ls2p I PI

0.0410417(50) 0.04098112 [2.54] 0.0404789 [2.73] 0.04104 [2.83]

Table 2.28. Theoretical and experimental wavelengths (in A) in Al XII Transition

Experiment [2.84]

Theory [2.54]

Is2p 3 Po-ls4d 3 DI Is2p 3 PI -ls4d 3 D2 Is2p 3 P2 -ls4d 3 D3 Is2p I PI -ls4d I D2 1s2p 3 SI -1s3d 3 P2 Is2p 3 SI -ls3d 3 PI Is2p 3 Po-ls3d 3 D2 1s2p 3 P2 -ls3d 3 D3 Is2p I PI -ls3d I D2 1s3p 3 PI-ls4d 3 D2 Is3p 3 ~-ls4d3 D3

3.2989 3.3020 3.3081 3.3945 4.2591 4.2619 4.4279 4.4378 4.5929 12.726 12.7432

3.3031 3.3007 3.3091 3.3987 4.2588 4.2632 4.4235 4.4380 4.5994 12.7052 12.7373

0.080081 [2.82]

2.2 Relativistic and Quantum Electrodynamical Corrections

Thble 2.29. Theoretical and experimental energies (in em-I) for the Is2p3Po-ls2s3SI transition in He-like ions

Z

2 3 5 6 7 8 9 10 12 13 14 15 16 17 18 26 92

Experiment [2.85-87] 9231.85640(50) 18231.303(1) 18231.30188(19) 35393.2(0.6) 43899.0(0.1) 52413.9(1.4) 52420.0( 1.1) 60978.2( 1.5) 60978.4(0.6) 69586.0(3.0) 69590.9(3.5) 78266.9(2.4) 78265.0(1.2) 95851(8) 104778(11) 113815(4) 122940(30) 132198(10) 132219(5) 141643(40) 151204(9) 232558(550) 260.0(7.9)eV

Theory [2.73] 9231.857(32) 18231.312( 10)

Theory [2.85] 9231.803

35393.70(8) 43898.96(16) 52420.97(29)

35394.07 43899.70 52421.87

60979.65(49)

60980.89

69592.5(0.8)

69594.08

78265.9(1.2)

78268.19

95853(2) 104787(3) 113820(5) 122970(6) 132238(8)

95857.86 104789.65 113824.91 122975.10 132245.45

141640(10) 151186(13) 233604(57) 256.6(1.0)eV

141649.49 151196.81 233654.56

Thble 2.30. Theoretical and experimental energies (in em-I) for Is2 p 3 PI-ls2s 3 SI transition in He-like ions

Z

3 4 5 6 7 8 9 10 12 13

Experiment [2.85-86] 18226.108(1) 18226.11206(21) 26853.1 (0.2) 35377.2(0.6) 43866.1 (0.1) 52429.0(0.6) 52428.2( 1.1) 61036.6(3.0) 61037.6(0.9) 69743.8(3.0) 69739.9(3.5) 78566.3(2.4) 78565.7(1.8) 96683(3) 106023(7)

Theory [2.73]

Theory [2.85]

18226.107(10)

18226.16

26853.039(30) 35377 .40(8) 43866.22(16) 52429.20(29)

26853.42 35377.89 43886.98 52430.07

61037.67(49)

61038.91

69742.4(0.8)

69744.20

78564.7(1.2)

78567.63

96683(2) 106028(3)

96689.28 106036.13

41

42

2 Atomic Structure and Spectra

Table 2.31. Theoretical and experimental energies (in em-I) for Is2p 3 P2 -ls2s 3 SI transition in He-like ions Z

3 4 5 6 7 8 9

10 12 13 14 15 16 17 18 20 22 26 28 29 36

Experiment [2.85-87] 18226.198(1 ) 18228.19935(25) 26867.9(0.2) 35429.5(0.6) 44021.6(0.1 ) 52719.5(0.6) 52720.2(0.7) 61588.3( 1.5) 61589.7(0.6) 70700.4(3.0) 70697.9(3.5) 70705.8(3.0) 80120.5(1.3) 80123.3(0.8) 100263(6) 111157(6) 122746(3) 135153(18) 148493(5) 148494(4) 162923(6) 178591 (31) 214225(45) 256746(46) 368960(125) 441950(78) 483910(200) 900034(160)

Theory [2.73]

Theory [2.85]

18228.194(10)

18228.16

26867.917(30) 35430.02(8) 44021.94(16) 52720.06(29)

26867.98 35429.79 44021.48 52719.03

61588.98(49)

61587.43

70699.8(0.8)

70697.63

80121.6(1.2)

80119.08

100253(2) 111152(3) 122743(5) 135152(6) 148497(8)

100249.10 111146.73 122738.85 135146.65 148493.65

162923(10) 178577(13) 214172(19) 256685(28) 368745(47) 441910(74) 483664(85) 900012(202)

162921.07 178576.12 214176.76 256699.20 368795.08 441991.04 483761.42

Table 2.32. Theoretical and experimental fine-structure intervals Is2p 3 Po-Is2p 3 PI (in em-I) in He-like ions Z

Experiment [2.85,88]

Theory [2.73]

2 3 4 5 6 7 8 9 10 12 14 16 18 20

-0.987919685 -5.1948(0.0012) -14.4(0.2) -16.0(0.6) -12.9(0.6) 15.1(1.4) 58.5(3.0) 157.8(3.0) 299.4(2.4)

-0.91 -5.03 -11.39 -16.10 -12.53 8.51 58.50 150.86 300.48 834.33 1781.51 3235.84 5233.11 7727.03

Theory [2.85]

-5.2048 -16.29 -12.72 8.26 58.07 150.05 298.94

2.2 Relativistic and Quantum Electrodynarnical Corrections

43

Table 2.33. Theoretical and experimental fine-structure intervals I s2 P 3 PI Is2p 3 P2 (in cm- I ) in He-like ions Z

2 3 4 5 6 7 8 9 10 12 14 16 18 20

Measurement [2.85-88]

Theory [2.73]

-0.076426648 2.09 (0.001) 14.8 (0.2) 52.3 (0.6) 135.5 (1.0) 290.5 (0.6) 551.5 (3.0) 957.88 (0.003) 956.6 (3.0) 1554.2 (3.4)

-0.0732 1.995 14.8829 52.6134 135.632 290.525 550.394 955.26

Theory [2.85] 2.0871 14.89

290.92 957.51

1551.478 3559.93 7135.96 13018.19 22155.51 35735.26

Table 2.34. Theoretical and experimental fine-structure intervals I s2 p 3 P2 Is2p 3 Po (in cm- I ) in He-like ions

z

Measurement [2.85,88]

5 6 7 8 9 10 13 14

36.3(8) 122.6(1.4) 305.6(1.4) 610.1 1114.4(3.0) 1853.6(2.4) 6180(150) 8931(5) 8955(29) 8913(20) 16295(12) 16280(44) 21280(40) 27150(400)

16 17 18

Theory [2.73]

8917.47

Theory [2.85] 36.35 123.04 299.18 609.48 1107.56 1856.05 6366.00 8924.38

16254.03

16261.5

21278.96

21286.2 27395.5

2.2.6 Li-Iike Ions The perturbation-theory approach described in the previous section has an extension to the case of few-electron ions that is a generalization of non-relativistic many-electron perturbation theory. Consider the resonance transitions ls22s l / 2-1s 22PI/2.3/2 in Li-like ions. These transitions have been observed for all ions with Z ~ 20 and for ions with Z = 22, 24-26, 28, 29, 32, 34, 36, 42, 54, 92 (solar spectra [2.89,90], Tokamak plasma [2.91,92], laser-produced plasmas [2.93]). Due to the volume limitation other isoelectronic sequences are not presented here. Theory and experiments in highly ionized atoms containing four

44

2 Atomic Structure and Spectra

or more electrons are the subject of several reviews (for example, [2.94-97] and references therein). The precision results of the variational calculations for neutral lithium and Li-like ions up to Z = 8 are given in [2.98,99]. The liZ-expansion perturbation method is applied in [2.100-102] to obtain the resonance transition energies. Numerical Dirac-Fock and multi-configuration Dirac-Fock calculations are presented in [2.103,104], respectively. The semiempirical fit to the experimental data is presented in [2.105], the screening approximations are discussed in [2.106]. A more sophisticated method is Many-Body Perturbation Theory (MBPT) [2.107,108]. The accurate calculations for transitions in ions with Z = 3-92 were performed in [2.108], including the second- and third-order correlation effects, the second-order correlation corrections to the Breit interaction, and corrections for reduced mass and mass polarization. The QED contributions were not included. The estimated theoretical uncertainty is less than that of the experimentally determined energies. The QED contributions to the resonance transitions have been calculated in [2.109] using Grant's multi-configuration Dirac-Fock computer package for ions with Z = 24-54. The self-energy contributions is derived from H-like contributions calculated in [2.10]. The resonance-transition wavelengths have been computed in [2.110] for Z = 3-92 using the theoretical calculations of [2.108,111] and the screening parameterization of the H-like Lamb shift.

0.05

2p3/2 -

25 1/2

2p1/2 -

2S1/ 2

-0.01 ::i

~ c: 0

-0.07

:s.c .;:

"E 0

I.)

-0.13

0

w

0

-0.19

-0.25 '-_-'--_-':-_ooL-_-':-_-'-_-=':-_--'-_-:' 20 30 40 50 60 Z

Fig. 2.8. The total QED contribution (self-energy and vacuum polarization) to the resonance wavelengths in Li-like ions [2.109]

2.2 Relativistic and Quantum Electrodynamical Corrections

Table 2.35. Comparison between theoretical and experimental wavelengths (in

45

A) for the resonance

transition Is22s1/2 -ls2pl/2 in Li-like ions Z

3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 19 20 21 22 23 24 24

Theory [2.1 10]

1550.63 1242.74 1037.59 890.765 780.332 694.142 624.960 568.160 520.668 480.351 445.682 415.537 389.075 365.663 344.777 326.029 309.106 293.749 279.745

Theory [2.108]

Theory [2.1 09]

6707.59 3130.64 2067.21 1550.25 1242.32 1037.09 890.153 779.634 693.337

Experiment 6709.764 [2.105,112] 3131.973 [2.105,112] 2067.893 [2.105,112] 1550.772(24) [2.105,112] 1242.807(15) [2.105,112] 1037.613(11) [2.105,112] 890.781(16) [2.105,112] 780.329(12) [2.105,112] 694.146(14) [2.105,112] 624.953(12) [2.105,112] 568.143(13) [2.105,112] 530.668(8) [2.105,112] 480.377(28) [2.105,112] 445.686(12) [2.105,112] 415.526(21) [2.105,112] 389.075(11) [2.105,112]

567.139 479.105 414.058

342.937

344.771(10) [2.113]

307.024

309.065(15) [2.113]

277.421

279.728

25

266.919

26

255.125

252.560

255.112

27 28

244.240 234.160

231.357

244.227 234.147

29

224.797

30 31 32

216.073 207.923 200.294

213.485

33 34

193.129 186.390

189.751

35 36

180.043 174.042

37 38 39 40

168.368 163.000 157.904 153.048

266.904

224.783

170.340

216.059 207.910 200.278 193.114 186.376 180.024 174.026 168.351 162.973 157.867 153.014

279.729(20) 279.69(2) [2.89] 279.70(3) [2.90] 266.88(2) [2.89] 266.86(3) [2.90] 255.094(10) [2.113] 255.10(2) [2.89] 255.11(3) [2.90] 255.11(2) [2.91,92] 234.155(10) [2.113] 234.140(20) [2.91,92] 224.795(10) [2.113] 224.80(2) [2.91,92]

200.290(10) [2.113] 200.30(2) [2.91,92] 186.375(15) [2.113] 186.36(2) [2.91,92] 174.036(26) [2.113] 174.03(3) [2.91,92]

46

2 Atomic Structure and Spectra

Table 2.35. Continued Z

41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 60 70 74 80

90 92

Theory [2.110] 148.406 143.985 139.791 135.799 131.987 128.339 124.838 121.469 118.221 115.086 112.051 109.110 106.256 103.483 100.804 98.231 88.854 70.178 64.091 56.188 45.837 44.112

Theory [2.108]

Theory [2.109]

144.206

148.393 143.989 139.784 135.765 131.919 128.235 124.701 121.310 118.050 114.915 111.896 108.987 106.182 103.474

98.301

Experiment

143.998(20) [2.113]

58.366

44.170 [2.112]

38.368 38.460 [2.111]

Table 2.36. Comparison between theoretical and experimental wavelengths (in A) for the resonance transition ls22s1/2 -ls22P3/2 in Li-like ions Z

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

Theory [2.110]

1548.04 1238.74 1031.89 883.082 770.397 681.705 609.785 550.032 499.395 455.768 417.651 383.952 353.863 326.780 302.217 279.817 259.297

Theory [2.108] 6707.50 3130.13 2065.79 1547.71 1238.33 1031.40 882.504 769.741 680.962 549.121 454.705 382.760

Theory [2.109]

Experiment 6709.613 [2.105,112] 3131.327 [2.105,112] 2066.436 [2.105,112] 1548.196(24) [2.105,112] 1238.820(15) [2.105,112] 1031.928(11) [2.105,112] 883.104(16) [2.105,112] 770.416(12) [2.105,112] 681.696(23) [2.105,112] 609.804(11) [2.105,112] 550.031(15) [2.105,112] 499.406(10) [2.105,112] 455.741(31) [2.105,112] 417.648(9) [2.105,112] 383.982(22) [2.105,112] 353.867(13) [2.105,112]

300.891

302.208(9) [2.105,112]

257.927

259.300(20) [2.113]

2.2 Relativistic and Quantum Electrodynamical Corrections

47

Table 2.36. Continued Z

Theory [2.110]

Theory [2.108]

Theory [2.109]

23 24 24

240.428 223.034

25

206.935

26

192.035

190.685

192.029

27 28

178.225 165.411

164.112

178.218 165.403

29

153.517

30 31 32

142.474 132.222 122.707

141.250

33 34

\13.876 105.685

112.790

35 36

98.092 91.052

37 38 39 40 41 42 43 44 45 46 47 48 50 51 52 53 54 55 56 60 70 74 80 90 92

84.530 78.492 72.901 67.723 62.926 58.488 54.390 50.602 47.100 43.859 40.859 38.082 35.508 30.913 28.862 26.958 25.192 23.554 22.028 16.999 9.225 7.3\1 5.219 3.076 2.780

221.647

223.013

206.926

153.509

90.\15

62.228

24.904

7.237

2.755 2.756 [2.\11]

142.466 132.215 122.699 \13.869 105.678 98.084 91.044 84.522 78.481 72.888 67.711 62.921 58.488 54.387 50.593 47.084 43.837 40.832 38.052 35.478 30.888 28.842 26.947 25.187

Experiment 223.010(20) [2.1 13] 223.00(2) [2.89] 222.99(3) [2.90] 206.90(2) [2.89] 206.87(3) [2.90] 192.012(20) [2.113] 192.04(2) [2.89] 192.03(3) [2.90] 192.05(2) [2.91,92] 165.396(10) [2.113] 165.396(20) [2.91,92] 153.507(20) [2.113] 153.51(2) [2.90]

122.705(20) [2.\13] 122.73(2) [2.90] 105.686(20) [2. \13] 105.70(2) [2.90] 91.049(25) [2.1 13] 91.06(3) [2.90]

58.499(20) [2. \13]

48

2 Atomic Structure and Spectra

Figure 2.8 exhibits the result of a semiempirical calculation of the total QED contributions (self-energy and vacuum polarization). As can be seen, the QED contributions (especially for the 1s22s1/2-1s22pl/2 transition) rapidly increase with increasing Z. The uncertainty in the observed wavelengths in the range 26 ~ Z ~ 54 is much smaller than the QED contribution to the wavelengths. Tables 2.35 and 2.36 list theoretical wavelengths for resonance transitions in Li-like ions and the corresponding experimental data. We note that there is a larger deviation between the experiment and theoretical calculations [2.109,110] for results without QED contributions [2.108] in the range Z > 20.

2.3 Binding Energies of the Inner-Shell Electrons In this section, we give a brief description of the results for the atomic innershell energy levels in heavy atoms. Details of the theoretical methods as well as experiments can be found in [2.114-122]. For heavy atoms the main electromagnetic effect in inner-electron binding is the nuclear Coulomb field. The Coulomb binding energies are affected by electrostatic screening. These effects are accurately described by the DiracHartree-Fock (DHF) method. The self-energy shift can be expressed as [2.118] ESE

= 2Ry

a 3(Z*)4 :rrn

3

F(aZ),

800~----------------------------------------~

K shell 600

Breit interaction

400

>" ~

1;; Gic:

energy

200

w

0t::==============-~ Totol 15 energy· (1/500)

-200

-400+----r---r---.--~---,----r_--,---._--._--~

55

60

65

70

75

80 Z

85

90

95

100

105

Fig. 2.9. Various contributions to the binding energy of K -shell electrons in heavy atoms

Kr

As Se Br

Ga Ge

Ni Cu Zn

Co

Cr Mn Fe

V

Ca Sc Ti

K

Si P S C1 Ar

Al

13.60 24.59 58 115 192 288 403 538 694 870.1 1075 1308 1564 1844 2148 2476 2829 3206.3 3610 4041 4494 4970 5470 5995 6544 7117 7715 8338 8986 9663 10371 11107 11871 12662 13481 14327

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

H He Li Be B C N 0 F Ne Na Mg

K

z

5.392 9.322 12.93 16.59 20.33 28.48 37.85 48.47 66 92 121 154 191 232 277 326.5 381 441 503 567 633 702 775 851 931 1015 1103 1198 1302 1413 1531 1656 1787 1927

L)

8.298 11.26 14.53 13.62 17.42 21.66 34 54 77 104 135 170 208 250.6 299 353 408 465 525 589 656 726 800 877 958 1047 1146 1251 1362 1479 1602 1731

Lz

21.56 34 54 77 104 134 168 206 248.5 296 349 403 459 518 580 645 713 785 860 938 1024 1119 1220 1327 1439 1556 1678

L3

10.62 13.46 16.15 20.20 24.54 29.24 37 46 55 64 72 80 89 98 107 117 127 141 162 184 208 234 262 292

7.646

5.139

M)

49 55 61 68 75 82 94 111 130 151 173 197 222

44

5.986 8.151 10.49 10.36 12.97 15.94 19 28 33 39

Mz

15.76 18.7 28 33 38 43 48 53 59 66 73 80 91 107 125 145 166 189 214

M3

8.25 9 9 9 10 11 12 21 33 46 61 77 95.0

M4

11.2 20 32 45 60 76 93.8

10.4

10

M5

Nz

N3

6.820 6.740 6.765 7.434 7.870 7.864 7.635 7.726 9.394 11 6.00 14.3 7.90 17 9.81 20.15 9.75 23.80 11.85 27.51 14.67 14.00

6.540

4.341 6.113

N)

N4

N5

N6

N7

0)

Oz

03

04

05

06 07 p)

Pz P3

P4

Q)

Table 2.37. Semiempirical binding energies (in eV) of electrons in free neutral atoms [2.115]. If a digit is thought to be uncertain by more than 10 units, it is printed in italics. (Notation: K = ISI/2, L1 = 2S1/2, L2 = 2p1/2, L3 = 2p3/2, M1 = 3S1/2, M2 = 3p1/2, M3 = 3p3/2, M4 = 3d3/2, M5 = 3d5/2, etc.)

~

01>\0

'"

~

trl

[

:r ~

Vl

~ 7

9'

g

g,

,,''"

oa

trl

(JQ

8:5'

tll

i..J

N

37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74

z

Rb 15203 Sr 16108 Y 17041 Zr 18002 Nb 18990 Mo 20006 Tc 21050 Ru 22123 Rb 23225 Pd 24357 Ag 25520 Cd 26715 In 27944 Sn 29204 Sb 30496 Te 31820 I 33176 Xe 34565 Cs 35987 Ba 37442 La 38928 Ce 40446 Pr 41995 Nd 43575 Pm 45188 Sm 46837 Eu 48522 Gd 50243 Tb 51999 Dy 53792 Ho 55622 Er 57489 Tm 59393 Yb 61335 Lu 63320 Hf 65350 Ta 67419 W 69529

K

2068 2219 2375 2536 2702 2872 3048 3230 3418 3611 3812 4022 4242 4469 4703 4945 5195 5452 5717 5991 6269 6552 6839 7432 7432 7740 8056 8380 8711 9050 9398 9754 10118 10490 10876 11275 11684 12103

LI

1867 2010 2158 2311 2469 2632 2800 2973 3152 3337 3530 3732 3943 4160 4385 4618 4858 5106 5362 5626 5894 6167 6444 6727 7017 7315 7621 7935 8256 8585 8922 9267 9620 9981 10355 10742 11139 11546

L2

Table 2.37. Continued

1807 1943 2083 2227 2375 2527 2683 2844 3010 3180 3357 3542 3735 3933 4137 4347 4563 4785 5014 5249 5486 5726 5968 6213 6464 6720 6981 7247 7518 7794 8075 8361 8651 8946 9250 9564 9884 10209

L3

325 361 397 434 472 511 551 592 634 677 724 775 830 888 949 1012 1078 1149 1220 1293 1365 1437 1509 1580 1653 1728 1805 1884 1965 2048 2133 2220 2309 2401 2499 2604 2712 2823

MI

251 283 315 348 382 416 451 488 526 565 608 655 707 761 817 876 937 1001 1068 1138 1207 1275 1342 1408 1476 1546 1618 1692 1768 1846 1926 2008 2092 2178 2270 2369 2472 2577

M2

242 273 304 335 367 399 432 466 501 537 577 621 669 719 771 825 881 939 1000 1063 1124 1184 1244 1303 1362 1422 1484 1547 1612 1678 1746 1815 1885 1956 2032 2113 2197 2283

M3

116 139 163 187 212 237 263 290 318 347 379 415 455 497 542 589 638 689 742 797 851 903 954 1005 1057 1110 1164 1220 1277 1335 1395 1456 1518 1580 1647 1720 1796 1874

M4

114 137 161 185 209 234 259 286 313 342 373 408 447 489 533 578 626 676 728 782 834 885 934 983 1032 1083 1135 1189 1243 1298 1354 1412 1471 1531 1596 1665 1737 1811

Ms

32 40 48 56 62 68 74 81 87 93 101 112 126 141 157 174 193 213 233 254 273 291 307 321 335 349 364 380 398 416 434 452 471 490 514 542 570 599

NI

16 23 30 35 40 45 49 53 58 63 69 78 90 102 114 127 141 157 174 193 210 225 238 250 261 273 286 300 315 331 348 365 382 399 420 444 469 495

N2

15.3 22 29 33 38 42 45 49 53 57 63 71 82 93 104 117 131 147 164 181 196 209 220 230 240 251 262 273 285 297 310 323 336 349 366 386 407 428

N3

6.38 8.61 7.17 8.56 8.6 8.50 9.56 8.78 11 14 21 29 38 48 58 69.5 81 94 105 114 121 126 131 137 143 150 157 164 172 181 190 200 213 229 245 261

N4

8.34 10 13 20 28 37 46 56 67.5 79 92 103 111 117 122 127 132 137 143 150 157 164 172 181 190 202 217 232 248

NS

13 21 30 38

6 6 6 6 6 6 6 6 6 6 6

N6

12 20 28 36

N7

7.58 8.99 10 12 15 17.84 20.61 23.40 25 31 36 39 41 42 43 44 45 46 48 50 52 54 56 58 62 68 74 80

4.18 5.69 6.48 6.84 6.88 7.10 7.28 7.37 7.46

01

5.79 7.34 8.64 9.01 10.45 13.44 14 18 22 25 27 28 28 29 30 31 32 33 34 35 36 37 39 43 47 51

02

12.13 12.3 16 19 22 24 25 25 25 26 27 28 28 29 30 30 31 32 35 38 41

03

6.6 7.0 8.3 9.0

6 6

5.75 6

04

Os

06 07

3.89 5.21 5.58 5.65 5.42 5.49 5.55 5.63 5.68 6.16 5.85 5.93 6.02 6.10 6.18 6.25 7.0 7.5 7.9 8.0

PI P2

P3 P4 QI

~.

gj

~

c:>v.>

§

~ ~

v.>

~

tv

VI

o

Re 71681 73876 76115 PI 78399 Au 80729 Hg 83108 TI 85536 Pb 88011 Bi 90534 Po 93106 At 95729 Rn 98404 Fr 101134 Ra 103919 Ac 106759 Th 109654 Pa 112604 U 115611 Np 118676 Pu 121800 Am 124984 Cm 128229 Bk 131536 Cf 135906 Es 139340 Fm 142839 Md 146404 No 150036 Lw 153736 Ku 157500 161340 165250 169240 173290

75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108

Os Tr

K

z

12532 12972 13422 13883 14356 14845 15350 15867 16396 16937 17490 18055 18637 19237 19850 20475 21112 21762 22427 23109 23808 24524 25257 26007 26774 27559 28361 29181 30023 30887 31770 32670 33600 34540

LI

11963 12390 12828 13277 13738 14214 14704 15206 15719 16244 16782 17334 17903 18488 19086 19696 20318 20953 21602 22267 22949 23648 24361 25097 25847 26614 27399 28202 29027 29874 30740 31630 32530 33460

L2

Table 2.37. Continued

10540 10876 11219 11567 11923 12288 12662 13041 13426 13816 14212 14615 15028 15449 15874 16303 16735 17171 17612 18059 18512 18971 19435 19905 20380 20860 21345 21835 22332 22837 23346 23860 24380 24900

L3

8717 8995 9280

8446

2937 3054 3175 3300 3430 3567 3710 3857 4007 4161 4320 4483 4652 4827 5005 5185 5368 5553 5742 5936 6135 6339 6548 6762 6982 720B 7440 7678 7925 8182

MI

2686 2797 2912 3030 3153 3283 3420 3560 3704 3852 4005 4162 4324 4491 4661 4833 5008 5187 5370 5557 5748 5944 6145 6351 6562 6779 7002 7231 7469 7716 7970 8231 8499 8770

M2

4574 4710 4848 4989 5132 5278 5426 5577 5731 5891 6057 6227 6401 6579 6760

4440

2371 2461 2554 2649 2748 2852 2961 3072 3185 3301 3420 3542 3666 3793 3921 4049 4178 4308

M3

5048 5196 5350 5507 5667 5830 5996

4906

1953 2035 2119 2206 2295 2390 2490 2592 2696 2802 2910 3019 3134 3254 3374 3494 3613 3733 3854 3977 4102 4230 4360 4493 4628 4766

M4

1887 1964 2044 2126 2210 2300 2394 2490 2588 2687 2788 2890 2998 3111 3223 3335 3446 3557 3669 3783 3898 4016 4136 4258 4382 4508 4636 4766 4901 5042 5185 5330 5478 5628

M5

2074 2147 2225 2304 2385 2468 2553

2006

1044 1096 1153 1214 1274 1333 1390 1446 1504 1563 1623 1684 1746 1809 1873 1939

994

693 727 764 806 852 899 946

660

629

NI

951 1003 1060 1116 1171 1225 1278 1331 1384 1439 1495 1553 1613 1675 1739 1805 1873 1946 2024 2105 2189 2276 2366

904

522 551 581 612 645 683 726 769 813 858

N2

450 473 497 522 548 579 615 651 687 724 761 798 839 884 928 970 1011 1050 1089 1128 1167 1207 1248 1289 1331 1373 1416 1459 1506 1557 1609 1662 1716 1771

N3

278 295 314 335 357 382 411 441 472 503 535 567 603 642 680 717 752 785 819 853 888 923 959 995 1032 1069 1107 1146 1188 1233 1279 1326 1374 1422

N4

1002 1036 1071 1109 1151 1193 1236 1279 1323

968

645 679 712 743 774 805 837 869 902 935

609

264 280 298 318 339 363 391 419 448 478 508 538 572

N5

47 56 67 78 91 107 127 148 170 193 217 242 268 296 322 347 372 396 421 446 471 497 525 554 584 615 647 680 716 755 795 836 878 921

N6

45 54 64 75 87 103 123 144 165 187 211 235 260 287 313 338 362 386 410 434 458 484 517 539 569 599 630 662 697 735 774 814 855 897

N7

611 634

565 588

86 92 99 106 114 125 139 153 167 181 196 212 231 253 274 293 312 329 346 363 380 397 414 431 448 465 482 499 519 542

01

125 139 153 167 183 201 218 233 248 261 274 287 301 315 330 345 361 377 393 410 430 454 479 505 529 554

III

56 61 66 71 76 85 98

02

45 49 53 57 61 68 79 90 101 112 123 134 147 161 174 185 195 203 211 219 227 235 243 252 261 270 279 288 300 315 330 345 360 375

03

9.6 9.6 9.6 9.6 12.5 14 21 27 34 41 48 55 65 77 88 97 104 110 116 122 128 134 141 149 157 166 175 185 198 213 229 246 263 281

04

06 07

11.1 12 19 25 32 38 44 51 61 73 83 91 97 6 101 6 106 6 III 6 116 6 121 6 1266 131 6 137 6 143 6 149 7 155 8 163 13 12 173 21 20 183 29 27 193 37 35 203 46 44 213 56 54

05

114 120 126 134 144 154 164 174 184

lOB

9.1 9.0 9.23 10.4 8 10 12 15 19 24 33 46 56 64 70 74 78 83 87 92 97 103

8.5

7.9

PI

93 99 105 113 123 133 143 153 163

88

14 17 28 39 48 54 57 61 65 69 73 78 83

II

6.11 7.42 7.29 8.43

P2

76 81 86 93 102 III 120 129 138

72

68

9.3 10.7 13 23 33 41 46 48 51 54 57 60 64

P3

6

6

8 9 9 10 10

6

6 6 6 6

6

6.1

9

6 6 6

6 6 6 6

6 6.0

6.3

6 6

6 6

4.0 5.28

QI

5.7

P4

VI

'"

g

tTl

[

:r ~

r;n

ii:';'

9"

if

s,

l-

0
e: :l

5'

til

tv

W

9 10 II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

5 6

4

2

I

zV

13.6 24.5 58 115 192 288 403 538 694 870 1075 1308 1564 1844 2148 2476 2829 3206 3610 4041 4494 4970 5470 5995 6544 7117 7715 8336 8986 9663

54.4 15.6 126 206 306 426 565 724 903 1101 1333 1590 1872 2178 2508 2862 3241 3644 4075 4530 5008 5511 6038 6589 7164 7763 8397 9035 9708

II

122 153 221 325 448 592 754 937 ll39 1360 1618 1901 2208 2540 2896 3276 3681 4110 4567 5047 5551 6080 6633 7210 7811 8436 9086 9760

III

217 259 344 471 618 785 971 1177 1402 1646 1930 2238 2571 2929 33ll 3718 4149 4604 5086 5592 6122 6677 7256 7859 8486 9136 9813

IV

340 392 494 645 815 1005 1215 1444 1692 1959 2269 2603 2962 3346 3754 4187 4644 5125 5633 6165 6721 7301 7906 8535 9188 9866

V

490 552 671 846 1039 1253 1486 1738 2009 2300 2636 2996 3381 3791 4225 4684 5167 5674 6208 6766 7348 7954 8585 9240 9919

VI

667 739 876 1074 1291 1528 1784 2059 2354 2668 3030 3417 3828 4264 4724 5209 5717 6250 6810 7394 8002 8635 9292 9973

VII

871 953 ll08 1329 1570 1830 2109 2408 2726 3064 3452 3866 4303 4764 5251 5761 6295 6854 7440 8050 8685 9344 10028

VIII

1103 ll95 1367 1612 1876 2160 2462 2785 3126 3488 3902 4342 4805 5292 5805 6341 6901 7486 8098 8735 9396 10082

IX

1362 1465 1654 1922 2210 2517 2843 3189 3554 3940 4380 4846 5335 5848 6387 6949 7535 8146 8785 9448 10136

X

1648 1761 1968 2260 2571 2902 3252 3621 4010 4420 4886 5370 5893 6432 6997 7585 8198 8835 9500 10190

XI

1963 2086 2310 2625 2960 3315 3688 4881 4494 4928 5420 5938 6479 7045 7636 8250 8889 9552 10244

XII

2304 2437 2679 3018 3377 3755 4152 4569 5005 5464 5982 6526 7094 7686 8303 8943 9609 10298

XIII

2673 2816 3076 3439 3821 4223 4644 5085 5546 6028 6572 7143 7737 8355 8998 9665 10357

XIV

3069.8 3223.8 3501 3887 4293 4719 5164 5629 6114 6621 7191 7788 8408 9053 9722 10415

xv

3494 3658 3953 4363 4793 5243 5712 6201 6711 7242 7838 8461 9108 9779 10474

XVI

3946 4121 4433 4867 5321 5795 6288 6802 7336 7891 8513 9163 9836 10533

XVII

4426 46ll 4941 5399 5877 6375 6893 7431 7999 8563 9217 9893 10592

4934 5129 5477 5959 6461 6984 7526 8088 8671 9275 9949 10651

XVIII XIX

5470 5675 6041 6547 7074 7621 8187 8774 9381 10009 10709

xx

6034 6249 6633 7164 7715 8286 8877 9488 IOll9 10772

XXI

6626 6851 7254 7809 8384 8980 9595 10230 10886

XXII

7246 7482 7903 8482 9082 9702 10341 llool

XXIII

7895 8141 8580 9184 9808 10452 Ill16

XXN

8572 8828 9286 9914 10562 ll231

xxv

Table 2.38. Semiempiricai binding energies (in eV) for the Is shell in atoms and ions [2.116]; Z is the nuclear charge, z is the spectroscopic symbol

9278 9544 10029 10672 11345

XXVI

U1

~

]

en

P-

§

I

en

~.

~

IV

IV

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

II

III

IV

V

VI

VII VIII IX

X

XI

XII XIII XIV XV

1034 1087 1128 1165 1203 1267 1330 1397 1485 1592 1702 1816

1157 1213 1256 1296 1336 1404 1471 1541 1633 1745 1860

1288 1346 1392 1434 1476 1548 1619 1694 1788 1905

1425 1486 1569 1535 1634 1721 1579 1685 1788 1622 1731 1842 1699 1776 1890 1775 1858 1938 1853 19382023 195020202108

1880 1950 2007 2056 2106 2196

2045 2119 2178 2229 2281

2218 2295 2393 2357 2478 2410 2543

XVI XVII XVIII XIX XX XXI XXII XXIII XXIV XXV XXVI

5.39 9.32 18.2 12.9 25.2 37.9 16.6 30.9 47.9 64.5 20.3 36.7 55.8 77.5 97.9 28.5 42.6 63.8 87.6 113 138 37.9 53.8 71.9 97.8 126 157 185 48.5 66.4 86.2 108 139 171 207 239 67 80.1 102 126 151 186 224 264 299 92 104 119 144 172 201 241 283 328 367 121 134 148 164 193 225 257 302 349 398 441 154 169 183 198 217 250 285 321 371 422 476 523 191 207 223 239 255 277 313 352 392 446 502 561 611 232 249 267 284 302 319 444 384 426 468 527 589 651 706 277 295 314 333 352 371 390 417 460 506 553 617 683 750 809.0 326 345 365 386 406 426 447 468 498 544 594 643 712 783 855.3 918 381 399 420 442 464 485 507 530 553 585 635 688 741 815 891.3968 441 459 478 501 525 548 571 595 620 645 680 733 790 847 925.1 1006 503 523 543 564 589 614 639 664 690 717 744 781 839 899 958.4 1042 567 589 611 633 656 683 710 737 764 792 821 850 890 951 1015 1077 633 658 682 706 730 733 784 813 842 871 901 932 964 1006 1070 1137 702 730 756 782 808 834 861 892 923 954 985 1017 10501083 1129 1197 775 804 833 861 889 917 945 974 1007 1040 1073 1106 1140 1175 1210 1259 851 882 913 943 973 1003 1033 1063 1094 1129 1164 1199 1234 1270 1307 1344 931 963 996 1028 1060 1092 1124 1156 1188 1221 1258 1295 1332 1369 1407 1446 1015 1048 1082 1116 1150 1184 1218 1252 1286 1320 1355 1394 1433 1472 1511 1551 1103 1136 1171 1207 1243 1279 1315 1351 1387 1423 1459 1496 1537 1578 1619 1660 1198 1228 1264 1301 1339 1377 1415 1453 1491 1529 1567 1605 1644 1687 1730 1773

zVI

Table 2.39. Semiempirical binding energies (in eV) for the 2s shell in atoms and ions [2.116]; Z is the nuclear charge, z is the spectroscopic symbol

VI

...,

'"

::I

§

~

~

til

~

5"'

S. ~

'"

oa"ii'

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t"r:I

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5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

II

III

N

V

VI

VII VIII IX

X

XI

656 685 755 714 786 861.1 726 818 894.5 757 831 927.5 788 863 941.9 860 897 976 940 974 1013 1023 1059 1095 1110 1147 1185 1200 1239 1278 1293 13341375 1390 1432 1475 1491 15341578

XII XIII XIV XV

8.30 11.3 24.4 14.5 29.6 47.4 13.6 35.1 54.9 77.4 17.4 35.0 62.7 87.1 114 21.6 41.1 63.4 97.1 126 158 34 47.3 71.7 98.9 138 172 209 54 65 80.1 109 141 186 225 265 77 90 103 120 153 190 241 284 330 104 118 133 148 166 205 247 303 351 401 134 150 166 183 200 220 263 309 371 424 479 168 185 203 221 240 258 281 328 379 447 504 565 206 224 243 263 283 303 323 348 400 455 529 591 248 267 287 308 330 351 373 395 422 479 539 618 296 314 335 357 380 403 426 450 474 503 564 629 349 366 386 409 433 458 483 508 534 560 592 657 403 423 443 465 490 516 543 570 597 625 653 686 459 482 504 527 551 578 606 635 664 693 723 751 518 543 568 593 618 644 673 703 734 765 796 828 580 607 634 662 689 716 744 775 807 840 873 906 645 674 703 733 762 792 821 851 884 918 953 988 713 744 775 807 839 871 902 933 965 1000 1036 1073 785 818 851 884 918 952 986 1019 1052 1086 1123 1161 860 895 930 965 1000 1036 1072 1108 1143 1178 1214 1253 938 975 1012 1049 1086 1123 1161 1199 1237 1274 1311 1349 1024 1058 1097 1136 1175 1214 1253 1293 1333 1373 1412 1451

ZVI

974 1009 1044 1060 1097 1136 1223 1318 1416 1518 1623 1094 1131 1168 1185 1224 1265 1358 1458 1561 1668

1221 1260 1299 1317 1358 1402 1500 1605 1713

1355 1396 1437 1456 1500 1546 1649 1759

1496 1539 1582 1603 1648 1697 1805

1644 1689 1735 1756 1804 1854

1799 1846 1894 1916 1966

1962 2011 2131 2060 2182 2308 2084 2234 2361 2493

XVI XVII XVIII XIX XX XXI XXII XXIII XXIV XXV XXVI

1Bble 2.40. Semiempirical binding energies (in eV) for the 2p shell in atoms and ions [2.116]; Z is the nuclear charge, z is the spectroscopic symbol

VI

~

()

'"g

(I)

c::>.

~

~

()

~

(I)

~.

i1: 0

N

"""

11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

zV

5.14 7.65 10.6 13.5 16.1 20.2 24.5 29.2 37 46 55 64 72 80 89 98 107 117 127 141

15.0 18.8 22.9 26.8 30.7 36.0 41.7 47.9 58 68 79 89 99 109 119 130 141 153 165

II

28.4 33.5 38.6 43.8 48.9 55.5 62.4 70 82 93 106 117 129 141 153 166 179 192

III

45.1 51.5 57.6 64.1 70.4 78.0 86.4 95.6 109 122 136 149 162 176 190 205 220

IV

65.0 72.7 79.8 87.6 95.1 103 114 124 139 154 169 184 199 215 231 247

V

88.0 97.0 105 114 123 132 144 156 172 189 205 222 239 257 275

VI

144 154 164 177 191 209 227 245 264 283 303

133

114 124

VII

143 155 165 176 188 200 214 229 249 268 288 309 330

VIII

175 188 200 212 225 238 254 270 292 313 335 357

IX

211 226 237 251 265 279 297 315 338 361 385

X

250 265 278 294 309 324 343 363 387 412

XI

292 308 322 339 355 372 393 414 439

XII

423 446 468

406

336 354 370 387

XIII

420 439 458 477 502

404

384

XIV

436 457 474 494 514 535

XV

490 512 531 552 574

XVI

547 571 591 613

XVII

608 633 654

XVIII

672 698

XIX

Table 2.41. Semiempirical binding energies (in eV) for the 3s shell in atoms and ions [2.116]; Z is the nuclear charge, z is the spectroscopic symbol

739

XX

N

Ul Ul

.,g

n

(>

til

~ =-~ m

9'

~

0

...

"..,

oa

:s (>

rn

:s (JQ

S· 9:

III

i.>

13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

zV

73 80 91

66

5.99 8.15 10.5 10.4 12.9 15.8 18.7 28 33 38 43 48 53 59

liS

67 74 81 89 97 106

60

16.3 19.7 23.3 23.8 27.6 31.7 38 46 53

II

104

liS

125 135 146 158 170

103 112 122 132 142

94

94

47.3 53.5 59.7 61.1 67.3 73.9 83

IV

30.3 35.0 39.9 40.9 45.8 51.2 59 68 77 86

III

123 135 147 159 171 184 197

III

67.6 75.2 82.7 84.5 91.9 99.9

V

120 129 142 155 169 182 196 210 225

III

91.2 100 109

VI

118 128 138 140 150 161 176 190 206 221 236 252

VII

148 159 170 173 184 197 213 229 246 263 280

VIII

181 193 206 209 221 234 253 271 289 308

IX

217 230 244 248 262 276 296 316 336

X

256 271 286 290 305 321 342 364

XI

298 314 330 336 352 368 391

XII

343 361 379 384 401 419

XIII

392 41l 430 435 454

XIV

484 490

464

444

XV

499 520 542

XVI

557 579

XVII

Table 2.42. Semiempirical binding energies (in eV) for the 3p shell in atoms and ions [2.116]; Z is the nuclear charge, z is the spectroscopic symbol

619

XVIII

~

~

en

Q.

~

..=

;:

()

2'

en

()

e.

> is

N

VI

0-

2.3 Binding Energies of the Inner-Shell Electrons

57

Table 2.43. The screened self-energy (in eV) for atomic inner-shell energy levels [2.118] in heavy neutral atoms Z

K

L,

L2

L3

70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106

121.300 127.707 134.346 141.305 148.529 156.073 163.937 172.075 180.613 189.468 198.671 208.345 218.355 228.795 239.629 250.880 262.591 274.788 287.440 300.691 314.536 328.810 343.789 359.457 375.616 392.621 410.263 428.831 448.230 468.258 489.490 511.603 534.623 558.750 583.773 610.211 637.099

15.754 16.711 17.718 18.779 19.893 21.063 22.290 23.583 24.940 26.367 27.867 29.456 31.115 32.861 34.689 36.608 38.615 40.730 42.951 45.281 47.733 50.321 53.027 55.906 58.914 62.100 65.451 69.004 72.746 76.705 80.881 85.303 89.980 94.918 100.141 105.656 111.411

0.810 0.911 1.021 1.142 1.275 1.421 1.581 1.755 1.946 2.154 2.381 2.637 2.913 3.210 3.529 3.874 4.246 4.649 5.085 5.558 6.071 6.630 7.235 7.896 8.614 9.397 10.249 11.178 12.191 13.294 14.495 15.804 17.231 18.783 20.469 22.303 24.287

1.734 1.859 1.991 2.130 2.277 2.432 2.596 2.767 2.948 3.138 3.337 3.546 3.766 3.997 4.237 4.491 4.755 5.031 5.320 5.623 5.939 6.269 6.613 6.971 7.345 7.732 8.139 8.563 9.001 9.455 9.932 10.418 10.934 11.458 12.004 12.574 13.158

where n is the principal quantum number, Z the nuclear charge and Z* the effective nuclear charge, defined by Z*

= (r)H/(r)OHF.

Here (r)H is the expectation value of the radius of the inner-shell orbital in the H-like ion with a nuclear charge Z, (r)OHF is the radius of this orbital in the DHF approximation, the function F(aZ) was defined in Sect. 2.2.2. The screening and nuclear finite-size effects (in the form of the Fermi model for

58

2 Atomic Structure and Spectra

the nuclear-charge distribution) in ab initio methods are included in the se1fenergy calculation by replacing the Coulomb wave functions and potentials by numerically determined DHF wave functions and potentials [2.114,119]. As mentioned in [2.114], the effects of screening by the nuclear finite-size effect on the electron self-energy can be compensated by reducing the Coulomb-field value by about 2% for K -shell electrons in the range 70 ~ Z ~ 90. Figure 2.9 shows the relative importance of the frequency-dependent Breit interaction for K -shell electrons in heavy atoms, compared with QED corrections (self-energy and vacuum polarization in the screening approximation) and with the total binding energy. The binding energies for K, L, and M shells in neutral and ionized atoms are given in Tables 2.37 -42. The self-energy corrections in a screened Coulomb field as a function of the nuclear charge Z are given in Table 2.43. The theoretical and experimental data for atoms with Z ~ 70 are presented in [2.115, 120-122].

2.4 Multicharged Ions in Stationary External Fields The study of the interaction of ions with external fields gives rise to two big problems: the influence of external fields (mainly stationary fields) on spectra and radiative transition probabilities, and the electrodynamic effects of the interaction with variable (monochromatic) fields, followed by multiphoton absorption and photon radiation. In this sections we restrict ourselves to the consideration of the behavior of the H- and He-like ions in external stationary fields, with a brief listing of the main results for the few-electron ions. These results are also important for the case of monochromatic external fields, as sufficiently intensive radiation sources (lasers) are available only in the optical and near ultraviolet range of frequencies, which are small as compared to the spectral line frequencies of ions with large Z. Under these conditions the influence of a laser field (with neglect of multiphoton ionization) is equivalent to that of stationary fields.

2.4.1 Stark Effect. The Ground State a) H-Uke Ions An electric dipole moment d induced in a H-like ion by a uniform electric field F can be expressed by d = fJF

+ sF 3 /6 + ... ,

where fJ is the electric dipole polarizability, s is the hyperpolarizability. The polarizability fJ describes a linear polarization law for small field strengths F, while the hyperpolarizability s corresponds to deviations from this law for large F. The standard treatment of the Stark effect by perturbation theory leads to an expression for the Stark shift as a power series of the field strength in the

2.4 Multicharged Ions in Stationary External Fields

59

fonn [2.6]:

E = Eo - f3F2/2! - BF4/4! - ... ,

(2.4.1)

where Eo is the field-free energy of the H-like ion. For the ground states only the coefficients of even powers in F are non-vanishing. The non-relativistic expansion coefficients for the ground state of the hydrogen atom are given up to the lO-th order in [2.123] Eis

= -1/2 -

9F 2/4 - 3555F4/64 - 4908F 6 -794237F 8 (2.4.2)

Besides a purely theoretical interest the relativistic generalization of this results can be applied in practice, e.g. for the determination of the Stark shift of X-ray lines in the inner-shells of heavy atoms. The leading relativistic correction ()( (aZ)2 to f3 has the fonn [2.124] (2.4.3) where f3nr is the well-known non-relativistic result

f3nr = 9aJ/2Z 4

(2.4.4)

The full relativistic calculations of f3 in the range 1 :::; Z :::; 137 are presented in [2.125]. The final result for f3 is:

a3

36~4

f3 =

{

r(2Y1 +4) (YI(4YI+5) 4(4-r?)(Y2+2YI) 6(aZ)2) r(2YI + 1) 2(2YI + 3) + (4YI + 1)(Yl + Y2 + 2) - 4Yl + 1 2<1>(aZ)(2YI + 1)f2(Yl + Y2 + 2) Y2(2Y2 + 1)r(2Y2 + 1)r(2Y2 + 3)

[(Y2 - 2)(YI - 1)2(YI + Y2)(YI + Y2 - 1) + 2(2 - Y2)[3 + 5(YI + Y2) + (YI + Y2)2] +(1 + YI)(YI + Y2)(YI + Y2 - I)]} , where Yp

= Vp2 -

(aZ)

(2.4.5)

(aZ)2. (aZ) is the hypergeometrical function 3F2:

== 3F2(Y2 - YI - 1, Y2 - Yl - 1, Y2 - Yl + 1; Y2 - YI + 2, 2Y2 + 2; 1)

~ 1 + (aZ)4 144

(1

+ 3. 6! ~

n!

~ (n + 1)(n + 3)(n + 6)

) .

60

2 Atomic Structure and Spectra

Table 2.44. The ratio

f3/f3 nr in H-1ike ions in the ground state; f3nr =

9ag/2Z 4 [2.125] Z

Z

f3/f3 Dr

13 / f3nr

Z

13 /f3 nr

I I

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0.999945 0.999779 0.999503 0.999116 0.998620 0.998012 0.997295 0.996467 0.995529 0.994481 0.993323 0.992055 0.990677 0.989189 0.987592

16 17 18 19 20 25 30 35 40 45 50 55 60 65 70

0.985885 0.984068 0.982143 0.980108 0.977964 0.965616 0.950569 0.932853 0.912499 0.889547 0.864040 0.836036 0.805562 0.772706 0.737526

75 80 85 90 95/

~%

/

1 0 t 15 120 125 130 135 136 137

0.700091 0.660478 0.618769 0.575049 0.529404 0.481921 0.432680 0.381741 0.329122 0.274743 0.218278 0.158625 0.090428 0.073003 0.045000

The relativistic values for f3 are given in Table 2.44. Using Eq. (2.4.5), the leading relativistic corrections to (2.4.4) can be expressed by [2.125]: f3 =

f3nr

[1 - 28(aZ)2/27 + (2f2

+ 31)(aZ)4/432]

.

(2.4.6)

The relativistic contribution to ;the ratio f3 / f3nr as function of nuclear charge Z is shown in Fig. 2.10. At small Z the relativistic corrections to f3nr are not significant, but with increasing Z their contribution becomes quite important and reaches about 50% at Z = 100. b) He-like Ions

The Stark effect for the ground state of He-like ions is similar to that in H-like ions because the ground state is non-degenerate in the zero order perturbation theory. Taking into account the interelectronic interaction according to perturbation theory, the polarizability of the He-like ion can be presented as

f3He = f3ji>1o

+alZ-1 +a2Z-2 +a3 Z - 3 + ...)

= f3ji>1 (1

+ ~ an/Zn)

,

(2.4.7)

where f3ji>1 is the polarizability in H-like approximation, i.e. f3ji>1 = 2f3H.

(2.4.8)

The exact non-relativistic value of the coefficient al has been obtained in [2.126]: al = 207/144.

(2.4.9)

2.4

Multicharged Ions in Stationary External Fields

61

1.0 0.9 0.8 0.7 L

c ~

0.6

~

0.4

~

0.5

0.3 0.2 0.1 0.0

0

40

20

60

100

80

120

140

z Fig. 2.10. The ralativistic polarizability of H-like atoms

The numerical values for a" a2 and a3 were calculated in [2.127] by the interpolation method using the high-precision data for f3He in the range 2 ~ Z ~ 10. Finally, the non-relativistic results for the polarizabilities f3He are presented in the form: nr

f3He

=

9a6 [

Z4

I

+

1.43750 Z

+

1.43751 Z2

+

1.53899 Z3

+ O(Z

-4

J

) .

(2.4.10)

In the "screened-charge" approximation, (2.4.10) can be written in the form [2.127]: f3rIe = a6(Z - 197/500)-4[9 - 1.24489(Z - 197/500)-'

+ 1.47425(Z -

197/500)-2]

a,

(2.4.11)

in (2.4.7) has been obtained The relativistic values of the coefficient in [2.128]. The accurate variational and Hartee-Fock calculations for f3He are presented in [2.126, 127]. Extensive numerical results obtained using the relativistic random-phase method have been tabulated in [2.129]. The numerical results for f3He are listed in Tables 2.45 and 2.46. The second column in Table 2.45 gives f3He in H-like approximation according to (2.4.8), the third one gives the correction function aI/ai, the fourth are the values of f3He obtained from (2.4.7), taking into consideration the non-relativistic corrections of the order of Z-2 and Z-3 in (2.4.10), the fifth one the calculated results in the "screened-charge" approximation, i.e. ~ = 9/(Z =

+ a)4 -

28f3f/jnr(aZ)2/27

f3~~nr [I - ~ + lO(a/Z)2 + ...J - 28f3~~nr(aZ)2/27,

(2.4.12)

62

2 Atomic Structure and Spectra

Table 2.45. The polarizability (a~) of the He-like ions in the ground state (low Z) Z

fJ~;[2.125]

2 3 4 5 6 7 8 9 10

5.6250 1.1111 3.5156 1.4400 6.9444 3.7484 2.1973 1.3717 8.9999

x x x x x x x x x

10- 1 10- 1 10-2 10-2 10-3 10-3 10-3 10-3 10-4

al/a:"[2.128]

fJHe[2.25]

1.0002 1.0005 1.0009 1.0015 1.0021 1.0029 1.0038 1.0048 1.0059

1.2770 1.8837 5.1761 1.9524 8.9207 4.6346 2.6402 1.6119 1.0387

x x x x x x x x

~;[2.127]

ftHe[2.25] 10- 1 10-2 10-2 10- 3 10-3 10- 3 10-3 10-3

1.2421 1.8505 5.1201 1.9386 8.8769 4.6180 2.6330 1.6085 1.0369

x x x x x x x x

10- 1 10-2 10-2 10-3 10-3 10-3 10-3 10-3

.

1.3222 1.8947 5.1857 1.9551 8.9353 4.6449 2.6479 1.6180 1.0437

x x x x x x x x

10- 1 10- 2 10- 2 10- 3 10- 3 10- 3 10- 3 10- 3

Table 2.46. The polarizability (a~) of the He-like ions in the ground state (high Z) Z

a1lan2.128]

fJHe[2.25]

20 30 40 50 60 70 80 90 100

1.0239 1.0552 1.1019 1.1666 1.2530 1.3653 1.5088 1.6893 1.9137

5.9267 1.1113 3.3380 1.2867 5.7644 2.8429 1.4909 8.2278 4.4572

x x x x x x x x x

fJHe [2.129] 10-5 10-5 10-6 10-6 10-7 10-7 10-7 10-8 10-8

5.922 x 10-5 l.ll x 10-5 3.331 x 10-6 1.283 x 10-6

1.486 x 10-7

where the screened constant a is [2.126]: a = 197/500.

For comparison we give also the multi-configuration Dirac-Fock calculation [2.127] (the sixth column in Table 2.45) and the relativistic random-phase result [2.129] (the fourth column in Table 2.46). For small Z the interelectronic corrections to p~i are significant (60% at Z = 2 for PUe), but these corrections decrease with increasing Z (2% for Z = 10). On the contrary, the relativistic effects rapidly increase with increasing Z (90% for Z = 1(0).

2.4.2 Stark Effect. Excited States a) The States of H-Iike Ions with j
2.4 Multicharged Ions in Stationary External Fields

63

resulting from a level mixing with I = j ± 1/2. The level shift !1En has the form [2.130] ./ 2 2 nr + N aoemF !1En = ±y No - N 4N2 _ 1 - Z - , N = j + 1/2; No

=

Jn 2 - 2n r (N - AN);

YP

=

J

p2 - (aZ)2,

(2.4.13)

where nr is the radial quantum number [2.6] and m is the projection of the momentum j. The designation (±) in (2.4.13) denotes the sublevels with I = j + 1/2 and 1= j - 1/2, respectively. In the limit aZ « I, (2.4.13) can be written as !1E = !1E P n

n

(1 _

(2n + IkD(aZ)2) 2lkl(n + IkDn

(2.4.14)

where !1E! is the level shift in the Pauli approximation [2.6],

+ 1/2), (J. + 1/2)2

k = (_1)i+1/2(j

!1E P = n

±~. In2 _ 4Y

nme

j(j+1)

aoF. Z

(2.4.15)

In the generalized case the energy matrix in the intermediate range (!1 F

~

!1j) can be diagonalized only numerically. Examples of these calculations are presented in [2.131]. The states of H-like ions with Iml = n - 3/2 are an exception. The corresponding equation for !1En with the main relativistic correction ()( (aZ)2 can be written as [2.130]: 3n [ 23n3-lOn2+3n+l] !1En = nF 2Z aoeF 1 + (aZ) 3n2(n _ 1)(2n - I) ,

(2.4.16)

where nF

0, for levels with j = n - 1/2, for levels with j = n - 3/2 and I

= { ±1,

=j

T 1/2.

In the strong-field range, !1En is linear dependent on F and gives a relativistic generalization of the Stark effect in the strong field [the first term in (2.4.17)]. It should be noted that the particular case under discussion (n = j - 1/2, j - 3/2; 1m I = n - 3/2) allows a complete analytical investigation of the Stark effect for the states with n = 2, m = ±1/2 (including the Lamb splitting for 2SI/2, and 2Pl/2 levels). The non-relativistic results for !1En (in the case !1F » !1 j ) is [2.6]: !1En =

~; (nl -

n2) eaoFo -

(2~

r

[17n 2 - 3(nl - n2)2 - 9m 2 + 19]F2a6,

(2.4.17)

where n 1 and n2 are the parabolic quantum numbers which are related to the principal and magnetic quantum numbers n and m, respectively, by nl

+ n2 + Iml + 1 =

n.

64

2 Atomic Structure and Spectra

b) The States of H-Iike Ions with j=n - 1/2 The shift and the splitting of energy levels for H-like ions in exited states with j = n - 1/2 in a homogeneous electric field of strength F are determined by a quadratic law I!!Enm

= -f3nmF2/2,

(2.4.18)

where S

f3nm

T

= f3n -

f3n

j (j + 1) - 3m j(2j _ 1)

(2.4.19)

is the differential polarizability, f3~(T)is the scalar (tensor) polarizability [2.132]. The values f3~ and f3J under condition aZ « 1 can be expressed in the form [2.133]: T

f3n

For the special case Iml I!!E

=

=j

n 6 (n - 1) 2a6 6(2n - 1)Z4(aZ)2

and aZ

«

1, I!!En is given by

3 2 . - I!!E nr [ 1 aZ 2 80n + 160n + 87n + 9 n,lml=) n + ( ) 2n(n + 1)(4n + 5)(2n + 1)2

where I!!E: = -

(2~

r

2(n

+ 1)(4n + 5) F2a 6·

1 '

(2.4.20)

(2.4.21)

Equation (2.4.20) is similar to (2.4.17) at n I = n2 = O. At n = 1 (2.4.20) gives the first relativistic correction to (2.4.4). The numerical data for the 2p3/2 level in terms of parameters ai, a2 and R = I!!En,lml=j/ I!!E~r of expression I!! E 2p

= -"21

SST) al + f3nTa2m 2] F 2

[( f3 n - "2f3n

(2.4.22)

are given in Table 2.47.

c) He-like Ions In the weak-field case, where the field splitting is smaller than the level intervals between the states with opposite parity, the quadratic Stark effect takes place, because the interelectronic interaction removes the degeneration on the orbital momenta. With increasing F, the importance of the splitting effect for sublevels with opposite parity increases due to the linear Stark effect [2.6]. This phenomenon was studied in detail in [2.134] for Is2s and Is2p configurations in He-like ions. The corresponding results are given in Fig. 2.11a (for F = 0) and in Fig. 2.11b (for F = 5.15 X 106 V/cm) in notations of LS-coupling. A comparison of Figs. 2.11a and 2.11b shows that in the strong field for Z ~ 15 one has 11 sublevels instead of six for F = O.

2.4 Multicharged Ions in Stationary External Fields

65

Table 2.47. The ratio R and parameters aI, a2 (2.4.22) for the Stark shift of the 2p3/2 state in H-like ions [2.133] Z

R

al

a2

2 3 4 5 10 20 30 40 50 60 70 80 90 100 110 120 130 137

0.999984 0.999937 0.999857 0.999746 0.999603 0.998413 0.993656 0.985743 0.974694 0.960539 0.943317 0.923075 0.899872 0.873772 0.844852 0.813198 0.778909 0.742092 0.714882

0.999906 0.999626 0.999158 0.998503 0.997662 0.990656 0.962761 0.916731 0.853253 0.773283 0.678045 0.569043 0.448111 0.317578 0.180791 0.044081 0.023660 0.010047 0.000118

0.999926 0.999703 0.999331 0.998811 0.998142 0.992575 0.970404 0.933691 0.883242 0.819460 0.743348 0.566020 0.558858 0.453677 0.343253 0.233295 0.142026 0.091187 0.007127

ElZ [a.u.]

10

30

Fig. 2.11a. Energy levels of He-like ions in an electric field (F = 0)

50

66

2 Atomic Structure and Spectra

Ell [a.u.]

Fig. 2.Uh. Energy levels of He-like ions in an electric field (F = 5.2 x ]06 V/cm)

0.32 028 024 0.20 0.16 10

30

50

l.

2.4.3 Stark Effect. Hyperfine Structure The Stark effect for the hyperfine structure leads to a shift of sublevels with different momenta f = 1 ± J (I is the nuclear spin, J is the total electron momentum), and to a splitting !:l.Efm along the projection m of the moment f in the direction of the field F. Due to the smallness of hyperfine interaction, the value of !:l.Efm is several orders of magnitude smaller than !:l.En (2.4.17); !:l.Efm was measured with high accuracy by radio spectroscopic methods [2.135]. Taking into account the hyperfine interaction of the electron and the nucleus to the first order, and the interaction with the electric field F to the second order of perturbation theory, one has !:l.Efm = -f3fmF2/2.

(2.4.23)

The general expressions for the polarizability f3fm have different representations for f = 1 ± 1/2 and can be written in a form similar to the non-relativistic case [2.136]: 2 me

3 [ S

f3f=I+I/2,m = ex mp g[ao

f3 (f)

+ f3

T

(f)

3m 2

-

(I + 1/2)(1 1(2/ + 1)

+ 3/2)]

,

(2.4.24)

[/+1

2 me 3 S fJf=[-1/2,m = -ex mp g[ao -1- fJ (f)

+ fJ

T

(f)

3m 2 -(I21(2/

1/4 )] ,

+ 1)

(2.4.25) where me is the electron mass, mp is the proton mass, g[ is the g-factor of the nucleus with spin I. fJs and fJT denote the scalar and tensor parts of fJfm,

2.4 Multicharged Ions in Stationary External Fields

67

respectively, including the relativistic corrections of the order of (a Z)2 [2.137]:

fJ! [1 +

(aZ)2 (27 +

2~1 _

fJT (I + 1/2) = fJJr [1 +

(aZ)2 ('61 +

~~~ -

fJs(f) =

f) - 0.OO5;rr2) ],

31 (I +

(2.4.26)

:~(I + 3) ) ] , (2.4.27)

fJT(I _ 1/2) =

fJT [1 + nr

(aZ)2

(~+ 7;rr2 + 11

752

3(12 -1)(2/ - 1»)].

47

(2.4.28) Here

fJJr and fJ~r are the non-relativistic values [2.136] T

fJnr =

62

3

3Z3aO'

S

fJnr =

47

3

60Z 3aO '

The experimental results are given in [2.135].

2.4.4 Zeeman Effect a) H-Iike Ions In a magnetic field H the ground state lSI/2 is split into sublevels with m = ±1/2 because of the linear Zeeman effect [2.6]: Il.E(I)

= mHa(2YI +

1)/3,

Yp

= V"-p-2---(a-Z-)-2,

p

= 1,2.

(2.4.29)

The detailed analysis of the Zeeman effect including the motion and structure effects of the nucleus is given in [2.138]. With increasing H, the importance of the second-order terms in F, leading to a splitting of sublevels m = ±1/2, increases as well due to the quadratic Zeeman effect. The corresponding non-relativistic level shift is [2.6] Il.E(2) = -

Xr: H2 /2,

xI.:' is the magnetic-dipole susceptibility xI.:' = -(aZ)2a6/2Z4.

(2.4.30)

where

(2.4.31)

The exact relativistic calculation of Xis is presented in [2.139, 140]: __ (aZ)2 [ 1 4 2_ 1 r2(YI +)12 + 2) 36Z4 ( + Yl)( YI ) + 4(Y2 - yr)r(1 + 2yr)r 3 (1 + 2)12)

Xis -

x 3F2(Y2-YI +1, )I2-YI +2, )I2-YI; )I2-YI +1, 2YI +1; 1)] a6·

(2.4.32) The leading relativistic corrections to (2.4.31) can be expressed by [2.141]: Xis =

xI.:' [1 -

4(aZ)2/3 + (aZ)4(6;rr2 + 23)/432] .

(2.4.33)

68

2 Atomic Structure and Spectra

Table 2.48. The calculated diamagnetic susceptibility for the ground state in He-like ions [2.141]

z

Xis/X:,!"

I 2 3 4 5 10

0.999929 0.999716 0.999361 0.998864 0.998225 0.992905

z

Xis/Xi':

15 20 30 40 50

0.984052 0.971686 0.936539 0.887799 0.825946

z 60 70 80 90 100

XIs/X:,!"

0.751625 0.665665 0.569119 0.463318 0.349964

The relativistic numerical results for XIs in terms of the function XIs / xf: are presented in Table 2.48. For the excited states of H-like ions the situation is, in many respects, analogous to the Stark effect. In the case of the Zeeman effect where the energy shift in the magnetic field is weak as compared to the splitting of neighboring fine-structure levels (but larger than the hyperfine splitting), the unperturbed states can be obtained from the exact solution of the Dirac-Coulomb equation. The magnetic field can be considered as a first-order perturbation. This case has been considered in detail in [2.142]. The expression for the level shift t::.En was considered by the formula for the anomalous Zeeman effect with the relativistic corrections to the Lande factor [2.142]: t::.En =

NmHa [2N(n r + AN) - (-1) N-I] 4N2 - 1 No

(2.4.34)

(notations were given in Sect. 2.42). In the intermediate range (where the energy shift in the magnetic field approximately corresponds to the fine-structure splitting) the Paschen-Back effect is observed [2.6]. In strong fields both the linear and quadratic effects are observed, which can be described only on the basis of numerical methods [2.140]. The states with j = n - 1/2 and Iml = j are an exception and the corresponding level shift is given by [2.140]: X: ( (5n + 3)(aZ)2 ) t::.En,lml=j = --2- 1 - -n-'-(n-+-l-)(-'-2-n-+-'-I-) ,

(2.4.35)

where

Xn = -

(aZ)2n4(n + l)a6 4Z 4

(2.4.36)

For n = 1, (2.4.35) coincides with the first two expansion terms in (2.4.33)

b) He-like Ions in the Ground State The ground state Is2 I So shows no Zeeman splitting and the influence of the magnetic field is limited to the quadratic shift t::.E = -XHeH2/2,

(2.4.37)

2.4 Multicharged Ions in Stationary External Fields

69

Table 2.49. Calculated dipole diamagnetic susceptibility of He-like ions in the ground state Z

A,(aZ)/Ar

XHe. a~ [2.143]

XHe. a~ [2.129]

2 3 4 5 10 20 30 40 50 60 70 80 90 100

1.0004 1.0010 1.0017 1.0027 1.0107 1.0428 1.0969 1.1736 1.2741 1.3997 1.5521 1.7332 1.9453 2.1910

-0.18625 x -0.74885 x -0.39893 x -0.24668 x -0.57140 x -0.13474 x -0.57030 x -0.30240 x -0.17951 x -0.11365 x -0.73623 x -0.48174 x -0.30985 x -0.18962 x

-0.2103 -0.7901 -0.4111 -0.2515 -0.5740 -0.1348 -0.5701 -0.3021 -0.1793

where

XHe

XHe

10-4 10-5 10-5 10-5 10-6 10-6 10-7 10-7 10-7 10-7 10- 8 10- 8 10- 8 10- 8

X

x x x x x x x x

10-4 10-5 10-5 10-5 10-6 10-6 10-7 10-7 10-7

-0.4817 x 10- 8

is the diamagnetic susceptibility, which can be presented as a series (0)

= XHe (1

+ At!Z + Az/Z 2 + ...).

(2.4.38)

Here X~~ is the susceptibility in the H-lik.e approximation (0)

XHe

(2.4.39)

= 2XH.

The results of theoretical calculations for the magnetic dipole susceptibility of ions with 2 :::;; Z :::;; 100 are given in Table 2.49. These calculations are based on perturbation theory which is generalized by inclusion of relativistic effects. Table 2.49 includes the correction function At! AT"(Ayr = 0.798154 [2.126]) as well as the values of XHe XHe

(0)

= XHe [1

+ Al (aZ)/Z].

(2.4.40)

The tabulated values of XHe obtained in the relativistic random-phase approximation are in good agreement with calculated data [2.143], but the divergence up to 10% for Z = 2 is due to the failure of the perturbation theory in form of (2.4.40) in the low-Z range.

2.4.5 Multipole Electromagnetic Susceptibilities and Shielding Factors for Multicharged Ions In this section basic information on the multipole susceptibilities and shielding factors for closed shells of multicharged ions are given required for the study of the influence of non-uniform fields on the ionic spectral characteristics.

70

2 Atomic Structure and Spectra

a) H-Iike Ions

The Stark effect for the axial-symmetric electric field is linear in the quadrupole momentum of the ion [2.144] il.E(l) = !aQj(j

2

n}m

+ I) -

3m 2

j(2j+l)

,

(2.4.41)

where Q is the quadrupole momentum, j and m are the total momentum and the magnetic quantum number and a is the heterogeneity parameter (}2<1>

a

=

(}x 2

(}2<1>

=

(}y2

I (}2<1>

= -'2 (}Z2 .

(2.4.42)

Here is the scalar potential of the electric field. For the ground state 1/2 and Q = 0) no linear splitting occurs, and the influence of the non-uniform field is limited to the quadratic shift

(j

=

il.E~~~ = -/ha 2/2,

(2.4.43)

where /h is the quadrupole susceptibility. The non-relativistic value of fJz for the ground state is well known [2.104]: f3i = 15aV Z6 .

(2.4.44)

The relativistic calculations which account for the multipole electric (13K) and magnetic (XK) susceptibilities for arbitrary I are presented in [2.137]. In the case of low and intermediate values of Z, 13K is expressed with inclusion of relativistic corrections up to the order of (aZ)2 by [2.137]: 13K = f3:r[1- (aZ)2R K],

(2.4.45)

where f3:r is the non-relativistic value of 13K

f3 nr =

8(2K

K

+ I)(K + 2)(2K (2Z)2K+2

RK = 'I1(2K + 3) - '11(3) +

1)! a 2K + 1 0

4K 5 + 18K 4 + 22K3 + 7K2 - 2 2K(K + I)(K + 2)(2K + 2)2 .

(2.4.46) (2.4.47)

Here 'I1(x) is the logarithmic derivative of the gamma-function. In the generalized case, the corresponding values for XK can only be obtained using numerical methods. The non-relativistic value for X:r has the form [2.137] nr

XI = -

(aZ)2 3 2Z 4 ao'

nr

XK>I =

(aZ)2(2K)!(K + 1)(3K - I) 2K+1 (2Z)2K+2 K (K _ I) ao

(2.4.48)

The numerical results for 13K and XK are given in Table 2.50 as the functions FK(aZ) = xdx:r and GK(aZ) = f3K/f3:r (K = 2, 3), respectively. The electric (a K ) and magnetic (bK ) shielding factors are given by (2.4.49)

2.4 Multicharged Ions in Stationary External Fields

71

Table 2.50. Calculated rnultipole susceptibilities for the ground state of the H-like ions ' Z

G2(aZ)

G3(aZ)

F2(aZ)

G3(aZ)

1 2 3 4 5 10 20 30 40 50 60 70 80 90 100

0.9999 0.9996 0.9993 0.9988 0.9931 0.9922 0.9690 0.9309 0.8788 0.8138 0.7375 0.6517 0.5585 0.4605 0.3604

0.9999 0.9996 0.9992 0.9985 0.9977 0.9907 0.9633 0.9184 0.8577 0.7829 0.6966 0.6017 0.5013 0.3991 0.2982

0.9999 0.9997 0.9994 0.9990 0.9984 0.9937 0.9752 0.9444 0.9021 0.8486 0.7849 0.7119 0.6307 0.5426 0.4490

0.9999 0.9997 0.9993 0.9988 0.9981 0.9927 0.9711 0.9356 0.8870 0.8261 0.7545 0.6757 0.5854 0.4919 0.3953

Here Heff and Eeff are the effective magnetic and electric fields near the nucleus, HK and EK are the external multipole fields. In particular, bt is associated with the chemical shift for the nuclear magnetic resonance, and a2 is the quadrupole screening factor (Sternheimer coefficient). In the dipole case (I = 1), bt and at have the simple form [2.137], which is valid in the range 1 :::;; Z :::;; 100: 2(aZ)2[6 + 3Yt + 2(aZ)2(3 + 2Yt)] bt = --------------------------27ZYt(Yt + 1)(2Yt - 1)

at = I/Z,

Yt =

(2.4.50)

VI - (aZ)2.

For low and intermediate values of Z, the constants aK and bK are expressed with relativistic corrections ex (aZ)2 by the formulas [2.137]:

a K = a:r[1- (aZ)2 KK ], a~r

bK = b~r[1- (aZ)2R K],

= 2/ZK(K + 1),

(2.4.51) (2.4.52)

b¥r = a(aZ)/3,

(2.4.53)

= -a(aZ)(K + 3)/(K - 1)(2K + 1), Kt = 0, K2 = 59/150, K3 = 59/84, RJ = 97/36, R2 = 707/1350, R3 = 0.292.

(2.4.54)

b:r>t

(2.4.55) (2.4.56)

The accurate relativistic calculations of a K and bK are presented in Table 2.51 as the functions PK = aK/a: r and QK = bK/b:r . For small Z, the relativistic corrections are not significant, but with increasing Z their contribution becomes quite important.

72

2 Atomic Structure and Spectra

Table 2.S1. Calculated multipole shielding factors for the ground state of H-like ions [2.137]

Z 1 2 3 4 5 10 20 30 40 50

60 70 80 90 100

Q\(aZ)

Q2(aZ)

Q3(aZ)

P2(aZ)

P3(aZ)

1.00014 1.00057 1.00129 1.00230 1.00359 1.01446 1.05927 1.13902 1.26269 1.44624 1.71836 2.13349 2.80594 4.01698 6.67791

1.00003 1.00011 1.00025 1.00044 1.00070 1.00270 1.01134 1.02617 1.04826 1.07921 1.12163 1.17985 1.26137 1.38043 1.56782

1.00003 1.00014 1.00025 1.00046 1.00073 1.00293 1.01187 1.02734 1.05028 1.08223 1.12563 1.18445 1.26537 1.38054 1.55471

0.99998 0.99991 0.99980 0.99966 0.99947 0.99786 0.99134 0.98123 0.96361 0.94084 0.91025 0.86933 0.81376 0.73551 0.61761

0.99996 0.99983 0.99962 0.99932 0.99894 0.99575 0.98284 0.96066 0.92816 0.88363 0.82440 0.74626 0.64227 0.50018 0.29577

b) Few-Electron Ions A critical review and tabulation of experimental Stark widths and shift data for spectral lines of non-hydrogenic positive ions were carried out in [2.145]. Theoretical values of the electric and magnetic susceptibilities (fh, fh and Xl) and the shielding factors (aI, a2 and bI> calculated in the relativistic randomphase approximation are presented in [2.129] for the closed shell of He, Ne, Ar, Ni(Cu+), Kr, Pb and Xe isoelectronic sequences. Precise Hartree-Fock calculations of f3t have been performed for several isoelectronic sequences [2.146, 147]. The exchange and quantum corrections in the Thomas-Fermi calculation of diamagnetic and electric susceptibilities for multicharged ions are studied in [2.148-150]. Variational calculations for the static polarizabilities of atoms and ions are presented in [2.151]. The calculations of the Sternheimer shielding factors are reported in [2.152] for He, Ne, Ar, Kr and Xe isoelectronic sequences. The reliable semiempirical estimates of the static multipole polarizabilities 13K (quadrupole 132, octupole 133 and hexadecapole 134) are considered in [2.153]. These estimates are based on the recommended values of 131 given in [2.146]. The general correlations among the successive polarizabilities according to the relationship log f3K+1 = A log 13K

+B

(2.4.57)

are obtained. The fitting parameters are listed in Table 2.52 for He, Ne, Be, Mg, Zn, Cu+ and Ag+ isoelectronic sequences. The estimates of the dipole polarizability 131 for multicharged ions can be represented by the simple semiempirical formula [2.149, 154]

131

= a(Z - b)Ca~,

(2.4.58)

2.4 Multicharged Ions in Stationary External Fields

73

Table 2.52. Fitting parameters A and B given in (2.4.57) (in a.u.) [2.153]

K=2

K=l

K=3

Series

A

B

A

B

A

B

He Ne Ar Be Mg Zn Cu+ Ag+

1.5351 1.5329 1.6700 1.8808 1.7332 1.7561 1.6093 1.7593

0.1648 0.2232 -0.0138 -0.5256 -0.3480 -0.3768 0.2390 0.0364

1.3437 1.3850 1.3842 1.3530 1.4246 1.4254 1.3174 1.3241

0.4999 0.4069 0.3627 0.2828 0.0708 0.1139 0.2762 0.2733

1.2562 1.2931 1.3255 1.2604 1.2870 1.3126 1.2844 1.3216

0.6420 0.4220 0.1563 0.4596 0.3316 0.2576 0.3284 0.1706

Table 2.53. Parameters a, b and c in (2.4.58) for the dipole polarizabilities of multicharged ions [2.154]

Series H He Li Be B C N

0 F Ne Na Mg Ar Cr+ Cu Zn Kr Xe

Outer-shell configuration

Is Is2 2s 2s 2

2p 2p2 2p 3 2p4 2p5 2 p6 3s 3s 2 3p6 3d5 4s 4s 2 4 p6 5 p6

Total number of electrons

a

b

c

9/2 2 3 4 5 6 7 8 9 10 11 12 18 23 29 30 36 54

9 165 255 195 146 1817 1613 1201 870 951 1981 1.63 +4 235 3200 7713 2.26+5 1.7 +6

0 0.359 2.84 2.16 2.76 3.49 2.60 3.20 4.96 4.96 9.16 8.31 11.7 21.3 23.7 22.8 23.4 35.6

4 4 3 3 3 3 4 4 4 4 3 3 4 3 3 3 4 4

where Z is the nuclear charge, a, b and c are fitting parameters. The parameters a, b and c were defined from theoretical and experimental results for various electronic series and are listed in Table 2.53.

3 Transition Probabilities

The accurate values of the transition probabilities and lifetimes for highly ionized atoms are required for many applications: the interpretation of astrophysical spectra and atomic-collision studies, the development of X-ray lasers, the diagnostics of fusion plasmas, etc. In this chapter mainly the transition probabilities in H- and He-like ions are considered. The theoretical and experimental data for other sequences have been the subject of several reviews (see [3.1-3] and references therein).

3.1 Selection Rules Selection rules characterize the change of the quantum numbers of an atom or ion, making a transition under the interaction with electro-magnetic radiation. Electric (E) and magnetic (M) 2K-pole transitions are denoted as EK and MK, respectively: El = electric dipole, E2 = electric quadrupole, Ml = magnetic dipole, etc. (see also Sect. 2.1). "Exact" and "approximate" selection rules are distinguished. The exact ones follow from the conservation laws and the properties of the angular parts of the operators for electric and magnetic transitions [3.4,5]. The exact sele<;tion rules are independent of the coupling scheme of the system and formulated for "exact" quantum numbers - parity, the total momentum J and its projection M: I:l.J=O,±l, ... ,±K, I:l.M

J+J/~K

= 0, ±1, ... , ±K,

(3.1.1) (3.1.2)

and the change of parity I:l.P

={

(_1)K _(_1)K

for EK transitions, for MK transitions.

(3.1.3)

The selection rules for M components (3.1.2) are important for the determination of line polarization (Sect. 4.4). The approximate selection rules are formulated for the quantum numbers of the transition described by a certain coupling scheme. In the LS-coupling scheme one has for electric EK transitions: I:l.L=O,

±1, ... ,±K,

L+L/~K

I:l.S=O, V. G. Pal'chikov et al., Reference Data on Multicharged Ions © Spring-Verlag Berlin Heidelberg 1995

(3.1.4)

3.1 Selection Rules

75

for magnetic MK transitions: !1L=O,±I, ... ,±(K-I),

L+L'~K-I

!1S = 0, ±I, ... , ±(K - I),

S + S'

~ K-

1.

(3.1.5)

For EI transitions one has: !1L

= 0, ±I,

!1S

= 0,

(3.1.6)

and S - S transitions are ruled out. For E2 transitions: !1L=0,±I,±2,

L+L'~2,

(3.1.7)

i.e. transitions between S terms (L = L' = 0) and between S and P terms (L = 0, L' = I) are ruled out. The selection rule for spins Sand S' is the same for all EK transitions; it allows transitions between terms with equal multiplicity. A transition is called forbidden if one of the selection rules mentioned above is violated. A violation of the selection rules is connected with magnetic interactions, mostly spin-orbital ones [3.6]. The probabilities of forbidden transitions in multicharged ions rapidly increase with increasing Z. Forbidden lines can arise from any of the higher multi poles or by a multiphoton mode. Selection rules also exist for transitions between states described by other types of coupling schemes (LK, jK, jj coupling and others [3.5]). For EK and MK transitions with !1n #- 0, the order of magnitude for the transition probabilities is given by [3.6]: A(EK) ~ a(aZ)2K+2(mc2In),

(3.1.8)

A(MK) ~ (aZ)2d4(mc 2In),

(3.1.9)

i.e. the probability of an EK transition is (aZ)2 times higher than that of a MK transition. Selection rules for autoionization transitions are given in Table 3.1.

Table 3.1. The selection rules for autoionization transitions (LS coupling)* [3.7] Interaction

f1L

f1S

f1J

Order of magnitude

Coulomb Spin-orbit Spin-other-orbit Spin-spin Hyperfine

0 0, ±1 0, ±I 0, ±1, ±2 0, ±1, ±2

0 O,±I 0, ±I 0,±1,±2 0, ±I,

0 0 0 0 0, ±1

a 4 Z6 a 4 Z4 a 4 Z4 a 4 Z4 (m/M p)2

*m is the mass of the electron, Mp is the mass of the proton

76

3 Transition Probabilities

3.2 Allowed and Forbidden Transitions 3.2.1 H-like Ions Excited states in H-like ions decay spontaneously with a characteristic transition probability A and a lifetime r = A -I. The transition probability from the excited state 11 > to the ground state 10 > by emission of a photon with frequency w, angular momentum 1 and multiplicity K are given by [3.8] AIO = 2aw(2h

+ 1) L

IA(K)(1

---+ 0;

1)1

2

(3.2.1)

,

J

where A (K) (1 ---+ 0; 1) is the multipole transition amplitude. The transition is electric (EK) if 1 + II + lz is odd, and is magnetic (MK) if 1 + II + 12 is even. The corresponding oscillator strength f is written in the form: (3.2.2) where A is the wavelength (in A) and gl and go are the statistical weights of the excited and ground levels, AIO is in units of s-I and f is dimensionless. Separating angular and radial parts in (3.2.1) one has [3.8]: A(M)

= (21

+ 1)1/2 (/;2

A (E) = (_1_) 1/2 J +1

(h

1/2

//2

!1)

jo

10)

1/2

---+ 0; 1),

T(M)(1

T(E) (1

---+ 0; J),

(3.2.3) (3.2.4)

where ( jl h h ) is the Wigner 3nrsymbol, ml m2 m3 T(K) are combinations of radial integrals: T(M)(1 T(E)(1

---+ 0; 1) = (Gllh(x)lFo)

+ (FIIh(x)IGo),

x = awr,

(3.2.5)

---+ 0; 1) = (CIll~ - 1)(GIIh-IIFo)(C1;lo - 1)(FIIh-I1Go)

1

- - - ( C111 , 1 +1 0 1 + -1 -+(1C1110,

. + 1 + 1)(GIIJJ+IlFo)

+ 1 + 1)(FIIh+IIGo)

Here, h (x) is the spherical Bessel function, G i and components of the Dirac wave functions,

Fi

(3.2.6)

are the large and small

Table 3.2 presents calculated transition probabilities for H-like ions with 1 :::; Z :::; 100 (3.2.1)). Experimental data for H-like ions are given in [3.6,9].

3.2 Allowed and Forbidden Transitions

Table 3.2. Radiative one-photon transition probabilities (in s-I) in H-like ions [3.17] Z

2SI/2 - 1s1/2

I 2 3 4 5 6 7 8 9 JO II 12 13 14 15 16 l7 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

2.4946 2.5559 1.4744 2.6192 2.4406 1.5121 7.0694 2.6895 8.7423 2.5100 6.5181 1.5580 3.4739 7.3003 1.4578 2.7845 5.1152 9.0777 1.5621 2.6149 4.2695 6.8154 1.0658 1.6357 2.4673 3.6631 5.3592 7.7347 \.1023 1.5525 2.1627 2.9820 4.0721 5.5106 7.3939 9.8415 1.3000 1.7050 2.2211 2.8747 3.6980 4.7295 6.0154 7.6J07 9.5808 1.2003 1.4969 1.8587 2.2982 2.8303

x x x x x x x x x x x x X

x x x x x x x x x x x x x x x x x x x x x x x x x x X

x x x x x x x x x x

JO- 6 JO- 3 JO- I JOO JOI J02 J02 J03 J03 10" J04 J05 J05 J05 J06 J06 106 106 J07 J07 J07 J07 J08 J08 J08 J08 J08 J08 J09 J09 109 J09 J09 J09 J09 J09 JOIO JOIO JOIO JOIO JOIO JOIO JOIO JOIO JOIO JOll JOll JOll JOII JOII

2PI/2 - 1s1/2

6.2649 1.0028 5.0772 1.6048 3.9181 8.1252 1.5054 2.5684 4.1146 6.2721 9.1842 1.3009 1.7922 2.4110 3.1778 4.1145 5.2448 6.5935 8.1874 1.0054 1.2224 1.4728 1.7600 2.0872 2.4582 2.8766 3.3465 3.8719 4.4570 5. J061 5.8240 6.6153 7.4848 8.4378 9.4792 1.0615 \.1850 1.3190 1.4641 1.62JO 1. 7902 1.9723 2.1682 2.3783 2.6035 2.8443 3.J017 3.3762 3.6688 3.9800

x J08 x JOIO X JOIO X JOll x JOII x JOll x JOI2 X JOI2 x JOI2 x JOI2 X JOI2 x J013 X J013 x J013 X J013 X J013 X J013 x J013 X J013 X JOI4 X JOI4 X JOI4 X JOI4 X JOI4 x JOI4 x JOI4 x JOI4 x JOI4 x JOI4 X JOI4 X 10 14 X 10 14 X JOI4 X JOI4 x 10 14 X JOI5 x JOI5 X JOI5 X JOI5 x JOI5 x JOI5 x JOI5 x JOI5 x JOI5 x JOI5 X JOI5 x JOI5 x JOI5 x JOI5 x JOI5

2p3/2 - 1s1/2

3SI/2 - 1s1/2

6.2648 1.0027 5.0764 1.6043 3.9163 8.1198 1.5041 2.5654 4. J084 6.2604 9.1636 1.2975 1.7865 2.4022 3.1645 4.0950 5.2166 6.5538 8.1324 9.9797 1.2124 1.4596 1.7426 2.0648 2.4296 2.8405 3.3011 3.8154 4.3872 5.0206 5.7198 6.4891 7.3329 8.2559 9.2627 1.0358 \.1547 1.2834 1.4225 1.5725 1.7339 1.9073 2.0932 2.2922 2.5048 2.7317 2.9734 3.2305 3.5037 3.7935

\.1087 I. 1359 6.5524 \.1640 1.0846 6.7192 3.1411 \.1949 3.8836 1.1149 2.8949 6.9188 1.5425 3.2409 6.4706 1.2357 2.2696 4.0270 6.9282 \.1595 1.8927 3.0206 4.7224 7.2453 1.0926 1.6216 2.3717 3.4219 4.8749 6.8637 9.5579 1.3l73 1.7982 2.4324 3.2624 4.3405 5.73JO 7.5129 9.7820 1.2655 1.6270 2.0798 2.6439 3.3432 4.2062 5.2666 6.5642 8.1455 1.0065 1.2388

x J08 X JOIO X JOIO X JOII X JOll X JOll x JOI2 x JOI2 x JOI2 x J012 x JOI2 x J013 x J013 x J013 X J013 X JOI3 X JOI3 x JOI3 X J013 x J013 X JOI4 X JOI4 X JOI4 X JOI4 x JOI4 X JOI4 X JOI4 X JOI4 X JOI4 X JOI4 X 10 14 X 10 14 X JOI4 X JOI4 X 10 14 x JOI5 x JOI5 x JO 15 X JOI5 x JOI5 X JOI5 x JOI5 x JOI5 x JOI5 x JOI5 x JOI5 x JOI5 x JOI5 x JOI5 x JOI5

x JO- 6 x JO- 3 X JO- 2 x 100 X JOI X JOI x J02 x loJ x J03 x 10" x J04 x J04 x lOS x lOS x 105 x J06 x J06 x J06 x J06 x J07 X 107 X J07 x J07 x J07 x J08 X J08 x J08 x J08 x J08 x J08 X 108 X 109 x 109 x J09 X 109 x 109 X J09 x J09 X 109 x JOIO x JOIO x JOIO x JOIO x JOIO x JOIO x JOIO x JOIO x JOIO x JOII x JOll

77

78

3 Transition Probabilities

Table 3.2. Continued Z

2S I /2 -ISI/2

2PI/2 -ISI/2

2P3/2 -ISI/2

3S I /2 -ISI/2

51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

3.4722 4.2440 5.1689 6.2741 7.5908 9.1549 1.1008 1.3198 1.5779 1.8814 2.2374 2.6541 3.1409 3.7083 4.3682 5.1345 6.0225 7.0497 8.2360 9.6037 1.1178 1.2987 1.5064 1. 7443 2.0167 2.3279 2.6831 3.0881 3.5492 4.0737 4.6696 5.3459 6.1128 6.9814 7.9645 9.0760 1.0332 1.1750 1.3349 1.5151 1.7182 1.9468 2.2040 2.4932 2.8183 3.1834 3.5934 4.0537 4.5701 5.1494

4.3109 X 10 15 4.6621 X 10 15 5.0345 X 10 15 5.4291 X 10 15 5.8465 X 10 15 6.2879 X 10 15 6.7540 x 10 15 7.2458 X 10 15 7.7643 X 10 15 8.3105 X 10 15 8.8853 X 10 15 9.4897 X 10 15 1.0125 X 10 16 1.0792 X 10 16 1.1491 X 10 16 1.2225 X 10 16 1.2994 X 10 16 1.3798 x 10 16 1.4641 X 10 16 1.5521 X 10 16 1.6442 X 10 16 1.7403 X 10 16 1.8406 X 10 16 1.9453 X 10 16 2.0544 X 10 16 2.1682 X 10 16 2.2867 X 10 16 2.4100 X 10 16 2.5383 X 10 16 2.6718 X 10 16 2.8106 X 10 16 2.9547 X 10 16 3.1044 X 10 16 3.2598 X 10 16 3.4211 X 10 16 3.5884 X 10 16 3.7617 X 10 16 3.9414 X 10 16 4.1275 X 10 16 4.3202 X 10 16 4.5197 X 10 16 4.7260 X 10 16 4.9394 X 10 16 5.1599 X 10 16 5.3878 X 10 16 5.6232 X 10 16 5.8662 X 10 16 6.1170 X 10 16 6.3757 X 10 16 6.6425 X 10 16

4.1006 X 10 15 4.4256 X 10 15 4.7691 X 10 15 5.1318 X 10 15 5.5143 X 10 15 5.9172 X 10 15 6.3413 X 10 15 6.7872 X 10 15 7.2555 X 10 15 7.7469 X 10 15 8.2621 x 10 15 8.8017 X 10 15 9.3665 X 10 15 9.9570 X 10 15 1.0574 X 10 16 1.1218 X 10 16 1.1890 X 10 16 1.2590 X 10 16 1.3320 X 10 16 1.4079 X 10 16 1.4869 X 10 16 1.5690 X 10 16 1.6542 X 10 16 1.7427 X 10 16 1.8344 X 10 16 1.9296 X 10 16 2.0281 X 10 16 2.1301 X 10 16 2.2357 X 10 16 2.3448 X 10 16 2.4576 X 10 16 2.5740 X 10 16 2.6942 X 10 16 2.8181 X 10 16 2.9459 X 10 16 3.0776 X 10 16 3.2131 X 10 16 3.3526 X 10 16 3.4960 X 10 16 3.6434 X 10 16 3.7948 X 10 16 3.9502 X 10 16 4.1097 X 10 16 4.2731 X 10 16 4.4406 X 10 16 4.6121 X 10 16 4.7876 X 10 16 4.9671 X 10 16 5.1505 X 10 16 5.3377 X 10 16

1.5187 1.8550 2.2577 2.7384 3.3106 3.9898 4.7937 5.7427 6.8601 8.1728 9.7111 1.1510 1.3609 1.6052 1.8891 2.2183 2.5994 3.0397 3.5476 4.1323 4.8045 5.5759 6.4600 7.4717 8.6278 9.9471 1.1451 1.3162 1.5107 1.7316 1.9822 2.2660 2.5873 2.9505 3.3608 3.8238 4.3459 4.9340 5.5959 6.3405 7.1773 8.1171 9.1720 1.0355 1.1682 1.3168 1.4832 1.6694 1.8778 2.1109

x x x x x x x x x X

x x x X

x x x x x X

x X

x x x x x x x X

x x x x x x x x x X

x x x x x x x x x X

lOll lOll lOll lOll lOll lOll 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14

X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X

lOll lOll lOll lOll lOll lOll lOll lOll lOll lOll lOll 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 1013 1013 1013 1013 1013 10 13 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 10 14 10 14 10 14 10 14 10 14 10 14 10 14

3.2 Allowed and Forbidden Transitions

Table 3.2. Continued Z

3 PI/2 -!s1/2

3 P3/2 -

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

1.6725 x 2.6770 x 1.3553 x 4.2832 x 1.0456 x 2.1680 x 4.0161 x 6.8505 x 1.0972 x 1.6720 X 2.4475 x 3.4658 x 4.7726 x 6.4179 x 8.4555 x 1.0943 x 1.3942 x 1.7519 x 2.1741 x 2.6683 x 3.2422 x 3.9038 x 4.6617 x 5.5245 x 6.5016 x 7.6024 x 8.8370 x 1.0216 x 1.1749 x 1.3448 x 1.5324 x 1.7389 x 1.9654 x 2.2133 x 2.4838 x 2.7781 x 3.0977 x 3.4439 x 3.8180 x 4.2216 x 4.6560 x 5.1228 x 5.6235 x 6.1595 x 6.7325 x 7.3441 x 7.9958 x 8.6893 x 9.4263 x 1.0208 x

1.6725 x 2.6771 x 1.3553 X 4.2834 x 1.0457 x 2.1682 x 4.0166 x 6.8517 x 1.0974 x 1.6724 x 2.4483 x 3.4671 x 4.7748 x 6.4214 x 8.4607 x 1.0951 x 1.3953 x 1.7534 x 2.1763 x 2.6714 x 3.2463 x 3.9093 x 4.6688 x 5.5338 x 6.5135 x 7.6177 x 8.8564 x 1.0240 x 1.1779 x 1.3485 x 1.5370 x 1.7445 X 1.9723 x 2.2216 X 2.4937 x 2.7901 X 3.1119 X 3.4608 x 3.8380 x 4.2452 X 4.6837 x 5.1552 x 5.6613 x 6.2035 x 6.7836 x 7.4031 x 8.0639 x 8.7677 x 9.5162 x 1.0311 x

lOS 109 1010 1010 10 11 10 11 10 11 10 11 1012 1012 1012 10 12 1012 10 12 1012 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 10 14 1014 10 14 1014 10 14 10 14 10 14 10 14 1014 10 14 1014 1014 1014 1014 1014 1014 1014 1014 1014 1014 1014 1014 lOIS

IS I/2

108 109 1010 1010 10 11 10 11 10 11 10 11 1012 10 12 1012 1012 1012 10 12 1012 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1014 10 14 1014 10 14 1014 1014 10 14 1014 10 14 1014 1014 1014 1014 1014 1014 1014 1014 1014 1014 1014 1014 10 14 lOIS

3d3/2 -ISI/2

3ds/2 -ISI/2

5.9375 x 3.8014 x 4.3299 x 2.4326 x 9.2783 x 2.7700 x 6.9835 x 1.5557 x 3.1529 x 5.9310 x 1.0503 x 1.7697 x 2.8595 x 4.4587 x 6.7419 x 9.9250 x 1.4271 x 2.0098 x 2.7783 x 3.7770 x 5.0581 x 6.6819 x 8.7177 x 1.1245 x 1.4354 x 1.8147 x 2.2738 x 2.8256 x 3.4844 x 4.2661 x 5.1883 x 6.2702 X 7.5332 x 9.0004 X 1.0697 x 1.2652 X 1.4893 x 1.7455 x 2.0371 x 2.3679 x 2.7421 x 3.1640 x 3.6382 x 4.1698 x 4.7640 x 5.4266 x 6.1635 x 6.9812 x 7.8864 x 8.8864 x

5.9374 x 3.8010 x 4.3289 x 2.4316 x 9.2721 x 2.7673 x 6.9743 x 1.5530 x 3.1461 x 5.9151 x 1.0469 x 1.7629 x 2.8466 x 4.4353 x 6.7013 x 9.8571 x 1.4161 x 1.9924 x 2.7515 x 3.7367 x 4.9985 x 6.5955 x 8.5945 x 1.1072 x 1.4114 x 1.7819 x 2.2295 x 2.7665 x 3.4062 x 4.1636 x 5.0552 x 6.0988 X 7.3143 x 8.7229 x 1.0348 x 1.2214 x 1.4349 x 1.6782 x 1.9545 x 2.2669 x 2.6193 x 3.0153 X 3.4590 x 3.9547 x 4.5070 X 5.1207 x 5.8009 x 6.5529 X 7.3823 x 8.2951 x

102

uf UP 106 106 107 107 108 108 108 109 109 109 109 109 109 1010 1010 1010 1010 1010 1010 1010 10 11 10 11 10 11 10 11 1011 10 11 1011 1011 1011

1011 1011 1012 1012 1012 10 12 1012 10 12 1012 10 12 1012 1012 1012 1012 1012 1012 1012 1012

102

uf lOS 106 106 107 107 108 lOS 108 109 109 109 109 109 109 1010 1010 1010 1010 1010 1010 1010 10 11 10 11 10 11 1011 1011 1011 1011 1011 1011 1011 10 11 1012 1012 1012 1012 1012 1012 1012 1012 1012 1012 1012 1012 1012 10 12 1012 1012

79

80

3 Transition Probabilities

Table 3.2. Continued Z

3PI/2 -lsl/2

3P3/2 -lsl/2

51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

1.1037 1.1915 1.2842 1.3822 1.4855 1.5944 1. 7090 1. 8295 1.9560 2.0889 2.2281 2.3740 2.5266 2.6862 2.8529 3.0270 3.2085 3.3976 3.5945 3.7993 4.0123 4.2335 4.4631 4.7013 4.9481 5.2037 5.4683 5.7418 6.0244 6.3162 6.6173 6.9277 7.2475 7.5766 7.9152 8.2631 8.6204 8.9870 9.3628 9.7477 1.0142 1.0544 1.0955 1.1375 1.1803 1.2238 1.2681 1.3130 1.3586 1.4048

1.1155 1.2049 1.2995 1.3995 1.5052 1.6166 1.7341 1.8578 1.9879 2.1247 2.2683 2.4190 2.5770 2.7426 2.9158 3.0970 3.2864 3.4843 3.6908 3.9061 4.1306 4.3645 4.6080 4.8613 5.1246 5.3983 5.6826 5.9777 6.2838 6.6012 6.9301 7.2708 7.6235 7.9884 8.3658 8.7560 9.1590 9.5752 1.0005 1.0448 1.0905 1.1376 1.1861 1.2361 1.2875 1.3404 1.3947 1.4506 1.5080 1.5668

x x x x x x x x x X

x x x X

x x x x x X

x X

x x x x X

x x X

x x x x x x x x x X

x x x x x x x x x X

10 15 1015 10 15 1015 10 15 10 15 1015 1015 1015 1015 10 15 10 15 1015 1015 1015 1015 1015 10 15 1015 1015 1015 1015 1015 1015 1015 1015 1015 1015 10 15 10 15 10 15 1015 1015 1015 10 15 10 15 10 15 1015 10 15 10 15 10 16 1016 1016 10 16 1016 1016 10 16 10 16 10 16 1016

X X X X X X X X X X X X X X X X X X X X X X X X X X X X

x X

x x X

x X X

x x x X

x x x x x X

x x X

x

10 15 1015 1015 1015 10 15 10 15 10 15 10 15 1015 10 15 10 15 10 15 1015 10 15 10 15 10 15 1015 1015 10 15 10 15 1015 1015 1015 1015 10 15 1015 1015 1015 10 15 1015 10 15 1015 1015 10 15 10 15 1015 10 15 10 15 10 16 10 16 10 16 10 16 10 16 10 16 10 16 1016 10 16 1016 1016 10 16

3d3/2 -lsl/2 9.9886 1.1201 1.2532 1.3991 1.5587 1.7330 1.9229 2.1296 2.3542 2.5979 2.8618 3.1473 3.4556 3.7882 4.1465 4.5320 4.9461 5.3904 5.8667 6.3766 6.9218 7.5041 8.1254 8.7876 9.4925 1.0242 1.l039 1.1884 1.2780 1.3730 1.4734 1.5797 1.6919 1.8103 1.9351 2.0666 2.2049 2.3504 2.5032 2.6636 2.8318 3.0079 3.1923 3.3852 3.5866 3.7969 4.0162 4.2447 4.4825 4.7298

X X X X X X X X X X X X X X X X X X X X X X X X X X X X

x X X X X X X

x X

x x x X

x X X

x X X X

x x

10 12 1013 1013 10 13 1013 1013 1013 1013 1013 10 13 1013 10 13 1013 1013 1013 1013 1013 1013 10 13 1013 1013 1013 1013 1013 10 13 10 14 1014 1014 1014 10 14 10 14 10 14 10 14 10 14 10 14 1014 1014 10 14 10 14 1014 1014 10 14 1014 1014 10 14 1014 1014 1014 10 14 10 14

3d5/2 -lsl/2 9.2972 1.0395 1.1596 1.2906 1.4333 1.5885 1.7568 1.9392 2.1364 2.3494 2.5789 2.8260 3.0914 3.3763 3.6815 4.0081 4.3570 4.7292 5.1259 5.5481 5.9967 6.4730 6.9779 7.5126 8.0780 8.6753 9.3055 9.9698 1.0669 1.1404 1.2176 1.2987 1.3836 1.4725 1.5654 1.6625 1.7638 1.8693 1.9792 2.0935 2.2122 2.3354 2.4631 2.5952 2.7319 2.8730 3.0185 3.1684 3.3227 3.4811

X X X X X X X X X X X X X X X X X X X X X X X X X X

x x X X X X X

x X X X X

x X

x X X

x x X X

x X

x

10 12 10 13 1013 1013 10 13 1013 1013 1013 1013 1013 10 13 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 10 13 1013 1013 1014 10 14 10 14 10 14 10 14 10 14 1014 10 14 10 14 1014 1014 10 14 10 14 1014 1014 10 14 1014 10 14 10 14 1014 10 14 10 14

3.2 Allowed and Forbidden Transitions

Table 3.2. Continued Z

3SI/2 -2SI/2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

1.8769 1.9232 1.1 095 1.9715 1.8375 1.1388 5.3260 2.0271 6.5922 1.8937 4.9205 1.1769 2.6260 5.5226 1.1037 2.1100 3.8798 6.8921 1.1872 1.9896 3.2523 5.1980 8.1391 1.2508 1.8894 2.8091 4.1161 5.9500 8.4934 1.1983 1.6722 2.3099 3.1603 4.2851 5.7612 7.6844 1.0173 1.3371 1.7458 2.2649 2.9206 3.7447 4.7750 6.0573 7.6459 9.6057 1.2014 1.4961 1.8554 2.2920

X X X X X

x X

x x x X

x x x x x x x x X

x x x x x x x x x x x x x x x x x x x x x x x X

x x x x x x

10- 9 10-6 10-4 10- 3 10- 2 10-1 10-1 10° 10° 101 101 102 102 102 103 103 103 103 104 104 104 104 104 105 105 105 105 105 105 106 106 106 106 106 106 106 107 107 107 107 107 107 107 107 107 107 108 108 108 108

3PI/2- 2s l/2 2.2449 3.5937 1.8197 5.7528 1.4050 2.9145 5.4020 9.2206 1.4779 2.2541 3.3027 4.6814 6.4538 8.6892 1.1463 1.4855 1.8955 2.3854 2.9653 3.6458 4.4381 5.3542 6.4066 7.6086 8.9743 1.0518 1.2256 1.4204 1.6379 1.8799 2.1482 2.4448 2.7717 3.1311 3.5251 3.9562 4.4266 4.9389 5.4957 6.0997 6.7538 7.4609 8.2241 9.0465 9.9316 1.0883 1.1904 1.2998 1.4169 1.5422

X X X

x x x X X X X X

x X X

x X X

x X

x x x x x X X X X

x x x X

x x x x X X

x x x x x x x x x x x x

107 108 109 109 1010 1010 1010 1010 lOll lOll lOll lOll lOll lOll 1012 1012 1012 1012 1012 1012 1012 1012 1012 1012 1012 lOB lOB lOB lOB lOB 1013 \013 \013 1013 \013 \013 1013 lOB lOB lOB lOB lOB lOB lOB lOB 1014 1014 1014 1014 1014

3d3/2-2sl/2

2.2448 3.5930 1.8189 5.7483 1.4033 2.9094 5.3891 9.1918 1.4720 2.2430 3.2831 4.6484 6.4005 8.6058 1.1336 1.4669 1.8686 2.3475 2.9127 3.5741 4.3418 5.2265 6.2395 7.3924 8.6973 1.0167 1.1814 1.3652 1.5696 1. 7958 2.0455 2.3201 2.6212 2.9504 3.3093 3.6995 4.1227 4.5807 5.0752 5.6080 6.1809 6.7957 7.4542 8.1583 8.9098 9.7106 1.0563 1.1468 1.2428 1.3444

X X

x x x x x x x X X

x X X

x X

x x X

x x x x x X

x x x x X

x X

x x x x X

x x x x x x x x x X

x X X

107 108 109 109 1010 1010 1010 1010 lOll lOll lOll lOll lOll lOll 1012 1012 1012 1012 1012 1012 1012 1012 1012 1012 1012 lOB lOB lOB lOB 1013 lOB \013 1013 lOB 1013 1013 \013 lOB \013 lOB lOB lOB lOB lOB lOB 1013 \0 14 1014 1014 1014

3d3/2 - 2SI/2

5.1007 3.2664 3.7219 2.0921 7.9853 2.3860 6.0213 1.3429 2.7252 5.1339 9.1066 1.5371 2.4884 3.8881 5.8923 8.6951 1.2535 1.7700 2.4538 3.3461 4.4953 5.9583 7.8009 1.0099 1.2941 1.6426 2.0667 2.5794 3.1951 3.9301 4.8027 5.8333 7.0444 8.4614 1.0112 1.2028 1.4241 1.6791 1.9718 2.3068 2.6888 3.1235 3.6165 4.1745 4.8042 5.5134 6.3102 7.2037 8.2035 9.3201

X X

x X

x x x x x X X

x X X

x x x x X

x x x x x x x x x x X

x X

x X

x x X

x X

x X

x x x x x x x X X

101 103 104 105 105 106 106 107 107 107 107 108 108 108 108 108 109 109 109 109 109 109 109 1010 1010 1010 1010 1010 1010 1010 1010 \010 1010 \010 lOll lOll \011 lOll lOll lOll lOll lOll lOll lOll lOll lOll lOll lOll lOll lOll

81

82

3 Transition Probabilities

Table 3.2. Continued Z

3S1/2 - 181/2

51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

2.8207 x 3.4588 x 4.2265 x 5.1476 x 6.2495 x 7.5641 x 9.1284 x 1.0985 x 1.3183 x 1.5781 X 1.8842 x 2.2443 x 2.6670 x 3.1622 X 3.7413 x 4.4173 x 5.2050 x 6.1213 x 7.1855 x 8.4198 X 9.8490 x 1.1502 x 1.3411 x 1.5612 x 1.8148 x 2.1066 x 2.4419 x 2.8269 x 3.2685 x 3.7744 X 4.3536 x 5.0161 x 5.7732 x 6.6378 x 7.6244 x 8.7496 x 1.0032 x 1.1492 x 1.3155 x 1.5047 x 1.7199 x 1. 9645 x 2.2426 x 2.5585 x 2.9174 x 3.3250 x 3.7879 x 4.3135 x 4.9103 x 5.5881 X

108 108 108 108 108 108 108 109 109 109 109 109 109 109 109 109 109 109 109 109 109 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 10 11 10 11 10 11 10 11 10 11 10 11 10 11 10 11 10 11 10 11 10 11 10 11 10 11 10 11

3PI/2 - 2S1/2

3d3/2 - 2S1/2

1.6760 1.8189 1.9712 2.1334 2.3060 2.4896 2.6846 2.8916 3.1113 3.3441 3.5908 3.8520 4.1284 4.4206 4.7295 5.0557 5.4002 5.7638 6.1473 6.5517 6.9779 7.4269 7.8998 8.3977 8.9218 9.4733 1.0053 1.0663 1.1305 1.1979 1.2688 1.3433 1.4215 1.5037 1.5900 1.6807 1.7760 1.8760 1. 9810 2.0913 2.2072 2.3288 2.4566 2.5908 2.7318 2.8799 3.0356 3.1991 3.3711 3.5519

1.4520 1.5656 1.6854 1.8116 1.9444 2.0840 2.2305 2.3840 2.5448 2.7130 2.8887 3.0720 3.2631 3.4621 3.6690 3.8840 4.1071 4.3384 4.5779 4.8256 5.0815 5.3456 5.6178 5.8980 6.1862 6.4821 6.7856 7.0965 7.4146 7.7395 8.0708 8.4083 8.7514 9.0996 9.4525 9.8092 1.0169 1.0532 1.0896 1.1261 1.1625 1.1988 1.2349 1.2705 1.3056 1.3400 1.3735 1.4059 1.4371 1.4669

X X X X X X X X X X

x x X X X X

x X X X X X X

x X X X X X X X X X X X

x X X

x x X X X

x X X

x X X

x

10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 1014 10 14 10 14 10 14 10 14 10 14 10 14 10 15 10 15 10 15 10 15 10 15 10 15 10 15 10 15 10 15 10 15 10 15 10 15 10 15 10 15 10 15 10 15 10 15 10 15 10 15 10 15 10 15 10 15 10 15 10 15

X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X

x X X X

x X

x X

x x X

x X X

x

10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 1014 10 14 1014 10 14 10 14 10 14 1014 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 15 10 15 10 15 10 15 10 15 10 15 10 15 10 15 10 15 10 15 10 15 10 15 10 15 10 15

3d3/2 - 2S1/2

1.0565 X 1.1950 X 1.3489 X 1.5196 X 1.7086 X 1.9177 X 2.1485 X 2.4031 X 2.6835 X 2.9919 X 3.3308 X 3.7027 X 4.1104 X 4.5568 X 5.0452 X 5.5791 X 6.1619 X 6.7979 X 7.4910 X 8.2460 X 9.0677 x 9.9612 X 1.0932 X 1.1987 X 1.3131 X 1.4372 X 1.5718 X 1.7176 X 1.8754 X 2.0462 X 2.2309 X 2.4306 x 2.6465 X 2.8796 X 3.1314 x 3.4032 x 3.6965 X 4.0128 X 4.3540 x 4.7218 X 5.1182 x 5.5455 x 6.0057 x 6.5015 X 7.0355 x 7.6106 X 8.2298 x 8.8965 x 9.6143 X 1.0387 X

10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013. 10 13 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 10 14

3.2 Allowed and Forbidden Transitions

Table 3.2.

Continued

Z

3d5j2-2slj2

1 2 3 4 5 6 7 8 9 to 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

5.1007 3.2665 3.7222 2.0925 7.9872 2.3868 6.0241 1.3437 2.7273 5.1387 9.1168 1.5391 2.4923 3.8952 5.9046 8.7157 1.2568 1.7753 2.4620 3.3584 4.5135 5.9847 7.8386 1.0152 1.3015 1.6527 2.0804 2.5977 3.2193 3.9619 4.8441 5.8867 7.1128 8.5484 1.0222 1.2165 1.4413 1.7004 1.9980 2.3389 2.7280 3.17to 3.6740 4.2436 4.8871 5.6123 6.4279 7.3431 8.3680 9.5137

X

x x x x x x x x x X

x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x

to l to3 to4 to5 to5 to6 to6 to7 to7 to7 to7 to 8 to8 to 8 to8 to8 to9 to9 to9 to9 to9 to9 to9 to lO to lO to lO to lO to lO to lO to lO to lO to lO to lO to lO to" to" to" to" 10" to" to" to" to" to" to" to" to" to" to" to"

3Slj2-2Plj2

3Plj2- 2 Plj2

3P3j2 -2Plj2

2.1046 3.3691 1.7060 5.3931 1.3171 2.7321 5.0638 8.6431 1.3853 2.1127 3.0954 4.3874 6.0482 8.1425 1.0741 1.3919 1.7759 2.2347 2.7777 3.4148 4.1566 5.0140 5.9989 7.1236 8.4013 9.8455 1.1471 1.3292 1.5325 1.7587 2.0094 2.2865 2.5918 2.9274 3.2952 3.6974 4.1363 4.6141 5.1332 5.6962 6.3057 6.9643 7.6749 8.4404 9.2639 1.0148 1.1097 1.2114 1.3202 1.4366

4.9047 5.0257 2.8995 5.1520 4.8019 2.9760 1.3919 5.2975 1.7228 4.9490 1.2860 3.0759 6.8632 1.4434 2.8847 5.5151 1.0141 1.8015 3.to34 5.2008 8.5017 1.3588 2.1278 3.2699 4.9395 7.3445 1.0762 1.5557 2.2208 3.1334 4.3729 6.0406 8.2649 1.1207 1.5068 2.0099 2.6609 3.4977 4.5669 5.9251 7.6409 9.7971 1.2493 1.5849 2.0007 2.5136 3.1439 3.9153 4.8559 5.9989

1.1954 7.6544 8.7204 4.9006 1.8698 5.5847 1.4087 3.1399 6.3681 1.1988 2.1248 3.5833 5.7957 9.0465 1.3695 2.0185 2.9063 4.0986 5.6741 7.7258 1.0363 1.3713 1.7924 2.3163 2.9625 3.7529 4.7123 5.8688 7.2537 8.9020 1.0853 1.3149 1.5839 1.8975 2.2615 2.6823 3.1668 3.7227 4.3581 5.0822 5.9046 6.8360 7.8877 9.0721 1.0402 1.1893 1.3560 1.5418 I. 7486 1.9784

X to 6 x to7 X to 8 x to 8 x to9 x to9 x to9 x to9 x to lO X to lO x to lO x to lO x to lO x to lO x to" x to" x to" x to" x toll x to" x to" x to" x to" x to" x to" x to" x to l2 x to 12 x to 12 x to l2 x to l2 x to l2 x to 12 x to l2 X to l2 x to l2 X to 12 x to l2 x to l2 x to 12 x to l2 x to 12 x to 12 x to l2 x to 12 x to13 x to 13 x to 13 x to 13 x to 13

x x x x x x x x x x x x x x x X X

x x x x x x x x X

x x x x x x x X X

x x x x x x x x x x x x x x x

to- IO to- 7 to- 5 to- 4 to- 3 to- 2 to-I to-I toO toO to l to l to l to2 to2 to2 103 to3 to3 to 3 to3 104 104 104 104 to4 to5

lOS to5 to5 to5

lOS to5 to6 to6 to6 to6 to6 to6 to6 to6 to6 to 7 to7 to 7 to 7 to7 to 7 to7 to 7

x x x x x x x x x x X X X

x x X X

x x x x x x x x x x X

x x x X

x x x x x x x x x x x x x x x x x x

to l to2 to3 to4 to5 to5 to6 to6 to6 to7 to7 to7 to7 to7 to8 to 8 to8 to 8 to8 to 8 to9 to9 to9 to9 to9 to9 to9 to9 to9 to9 to lO to lO to lO to lO to lO to lO to lO to lO to lO to lO to lO to lO to lO to lO to" to 11 to" to" to" to"

83

84

3 Transition Probabilities

Table 3.2. Continued Z

3d5/2-2sl/2

51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

1.0792 x 1.2216 x 1.3799 x 1.5556 x 1.7503 x 1.9659 x 2.2041 x 2.4671 x 2.7569 x 3.0759 X 3.4267 x 3.8120 x 4.2346 x 4.6978 x 5.2049 x 5.7594 x 6.3654 x 7.0269 x 7.7483 x 8.5346 x 9.3907 x 1.0322 x 1.1335 x 1.2435 x 1.3630 x 1.4926 x 1.6331 x 1.7854 x 1.9503 x 2.1288 x 2.3219 x 2.5306 x 2.7562 x 2.9999 x 3.2629 x 3.5468 x 3.8531 x 4.1833 x 4.5392 x 4.9228 x 5.3359 x 5.7807 x 6.2595 x 6.7747 x 7.3290 x 7.9253 x 8.5664 x 9.2557 x 9.9966 x 1.0793 x

10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 10 14

3SI/2 -2PI/2

3PI/2 -2PI/2

1.5608 x 1.6932 x 1.8344 x 1.9847 x 2.1446 X 2.3145 x 2.4949 x 2.6863 x 2.8892 x 3.1042 X 3.3318 x 3.5727 x 3.8273 x 4.0964 x 4.3806 x 4.6805 x 4.9970 x 5.3307 x 5.6824 x 6.0529 x 6.4430 x 6.8537 x 7.2857 x 7.7402 x 8.2180 x 8.7202 x 9.2479 x 9.8023 X 1.0384 X 1.0996 x 1.1637 x 1.2310 X 1.3016 X 1.3757 X 1.4534 x 1.5349 X 1.6204 x 1.7100 x 1.8039 x 1.9023 x 2.0055 x 2.1137 x 2.2270 x 2.3458 x 2.4704 x 2.6009 x 2.7377 x 2.8812 x 3.0316 x 3.1893 X

7.3830 9.0536 1.1064 1.3476 1.6361 1.9804 2.3900 2.8763 3.4521 4.1323 4.9342 5.8774 6.9846 8.2819 9.7990 1.1570 1.3633 1.6034 1.8822 2.2056 2.5800 3.0130 3.5131 4.0898 4.7542 5.5186 6.3970 7.4056 8.5622 9.8875 1.1404 1.3139 1.5122 1. 7386 1.9969 2.2915 2.6271 3.0093 3.4444 3.9394 4.5022 5.1420 5.8689 6.6947 7.6325 8.6973 9.9061 1.1278 1.2836 1.4603

1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14

X X

x x x X

x x x x X

x X

x x x x X

x x x x x x x X

x X

x x X X

x X

x X

x x x x x x x x x x x X X X

107 107 108 108 108 108 108 108 108 108 108 108 108 108 108 109 109 109 109 109 109 109 109 109 109 109 109 109 109 109 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1011 1011 1011

3P3/2- 2 PI/2

2.2330 2.5147 2.8257 3.1686 3.5459 3.9605 4.4152 4.9132 5.4579 6.0527 6.7014 7.4080 8.1765 9.0116 9.9176 1.0900 1.1963 1.3113 1.4355 1.5696 1.7141 1.8698 2.0373 2.2174 2.4109 2.6185 2.8411 3.0796 3.3350 3.6081 3.9000 4.2119 4.5447 4.8997 5.2780 5.6810 6.1099 6.5661 7.0512 7.5664 8.1136 8.6941 9.3099 9.9626 1.0654 1.1386 1.2161 1.2981 1.3848 1.4764

X X

x x x X

x x x x X

x X

x x x x x X

x x x x x x x x X

x x X

x x x x x x x x x x x x x x x x x x x

1011 1011 1011 1011 1011 1011 1011 1011 1011 1011 1011 1011 1011 1011 1011 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 1013 1013 1013 1013 1013 1013

3.2 Allowed and Forbidden Transitions

Table 3.2. Continued Z

3d3/2 - 2PI/2

3d 5/2- 2 PI/2

3SI/2 - 2p3/2

I 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

5.3877 X 8.6244 x 4.3669 x 1.3804 x 3.3708 x 6.9914 X 1.2956 x 2.2110 X 3.5430 X 5.4024 x 7.9134 x 1.1213 x 1.5454 x 2.0798 x 2.7426 x 3.5529 x 4.5312 x 5.6997 x 7.0816 x 8.7019 X 1.0587 x 1.2764 x 1.5264 x 1.8115 x 2.1352 x 2.5007 x 2.9116 x 3.3717 x 3.8847 x 4.4547 x 5.0859 x 5.7827 x 6.5497 x 7.3914 x 8.3129 x 9.3191 x 1.0415 x 1.1607 x 1.2900 x 1.4300 X 1.5813 x 1.7446 x 1.9204 x 2.1094 X 2.3124 x 2.5300 X 2.7630 x 3.0121 x 3.2782 x 3.5619 x

5.1873 1.3288 3.4072 3.4053 2.0312 8.7418 3.0036 8.7520 2.2487 5.2320 1.1234 2.2577 4.2920 7.7820 1.3546 2.2759 3.7064 5.8720 9.0772 1.3726 2.0348 2.9626 4.2434 5.9876 8.3336 1.1453 1.5557 2.0905 2.7810 3.6652 4.7885 6.2053 7.9801 1.0189 1.2922 1.6284 2.0397 2.5405 3.1474 3.8794 4.7586 5.8104 7.0638 8.5519 1.0313 1.2389 1.4830 1.7691 2.1035 2.4932

4.2097 6.7411 3.4154 1.0805 2.6415 5.4860 1.0183 1.7409 2.7954 4.2723 6.2739 8.9151 1.2323 1.6640 2.2019 2.8631 3.6659 4.6306 5.7789 7.1343 8.7222 1.0570 1.2707 1.5165 1.7978 2.1182 2.4817 2.8923 3.3546 3.8733 4.4535 5.1006 5.8203 6.6190 7.5030 8.4794 9.5555 1.0739 1.2039 1.3464 1.5024 1.6728 1.8587 2.0613 2.2818 2.5214 2.7814 3.0634 3.3688 3.6993

107 108 109 1010 1010 1010 lO" lO" lO" lO" lO" 1012 1012 1012 10 12 10 12 10 12 1012 10 12 1012 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1014 1014 1014 1014 1014 1014 1014 1014 1014 10 14 10 14 10 14 10 14 10 14

X

10-5 10- 2 10- 1 10° 101 101

x x

loZ loZ

X

x x x x

103 x .103 x 10" x 10" x 10" x 10" x lOS x lOS x lOS x lOS x lOS x 106 x 106 x 106 x 106 x 106 x 106 x 107 x 107 X 107 X 107 x 107 x 107 X 107 x 107 x 108 X 108 X 108 X 108 X 108 X 108 X 108 x 108 x 108 x 108 x 108 x 109 X 109 x 109 x 109 x 109 x 109 X

x 106 x 107 X 108 X 109 x 109 x 109 x 1010 x 1010 x 1010 x 1010 x 1010 X 1010 x lO" x lO" X lO" X lO" X lO" x lO" x lO" X lO" X lO" x 1012 x 1012 x 1012 x 1012 x 1012 x 10 12 X 10 12 X 10 12 x 1012 x 1012 X 10 12 X 10 12 X 10 12 x 1012 x 10 12 X 10 12 X 1013 x 1013 X 1013 x 1013 x 1013 x 1013 x 1013 x 1013 X 1013 X 1013 x 1013 x 1013 X 1013

3 PI/2 - 2p3/2 2.3908 1.5308 1.7438 9.7987 3.7382 1.1163 2.8153 6.2738 1.2721 2.3940 4.2418 7.1508 1.1562 1.8039 2.7296 4.0213 5.7870 8.1566 1.1285 1.5357 2.0586 2.7222 3.5555 4.5914 5.8677 7.4271 9.3179 1.1594 1.4317 1.7553 2.1378 2.5874 3.1134 3.7256 4.4351 5.2540 6.1954 7.2734 8.5036 9.9026 1.1489 1.3281 1.5301 1.7571 2.0115 2.2959 2.6131 2.9660 3.3577 3.7917

x x x x x x x x x x x x X X X X X X X

x X

x x x x x X

x x x x X X X

x X

x X X

x x x x x X X

x x x x

101 loJ 10" 10" lOS 106 106 106 107 107 107 107 108 108 108 108 108 108 109 109 109 109 109 109 109 109 109 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 lO" lO" lO" lO" lO" lO" lO" lO" lO" lO"

85

86

3 Transition Probabilities

Table 3.2. Continued Z

3d3/2-2PI/2

51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

3.8642 x 4.1858 x 4.5278 x 4.8909 x 5.2761 x 5.6843 x 6.1166 x 6.5740 x 7.0575 x 7.5681 X 8.1070 x 8.6753 x 9.2742 x 9.9049 X 1.0568 x 1.1266 x 1.2000 x 1.2770 x 1.3579 x 1.4427 X 1.5316 x 1.6247 X 1.7223 x 1.8244 x 1.9313 x 2.0430 x 2.1597 X 2.2817 x 2.4090 x 2.5419 x 2.6805 x 2.8251 x 2.9757 x 3.1327 x 3.2961 x 3.4663 x 3.6433 x 3.8274 x 4.0188 x 4.2177 x 4.4244 x 4.6390 x 4.8617 x 5.0928 x 5.3325 x 5.5809 x 5.8384 x 6.1051 x 6.3813 x 6.6671 x

1014 10 14 1014 1014 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 10 14 1015 1015 10 15 10 15 10 15 1015 1015 10 15 10 15 10 15 10 15 10 15 1015 10 15 10 15 1015 10 15 1015 1015 1015 1015 10 15 1015 10 15 10 15 1015 1015 1015 10 15 10 15 10 15 1015 10 15 10 15 10 15 10 15

3d5/2 - 2PI/2

3SI/2 - 2p3/2

2.9461 X 3.4713 X 4.0788 X 4.7798 X 5.5870 X 6.5144 X 7.5779 X 8.7950 X 1.0185 X 1.1770 X 1.3575 X 1.5625 X 1.7951 X 2.0585 X 2.3565 X 2.6930 X 3.0726 X 3.5001 X 3.9811 X 4.5214 X 5.1277 X 5.8074 X 6.5683 X 7.4194 X 8.3703 X 9.4317 X 1.0615 X 1.1934 x 1.3402 x 1.5034 x 1.6848 X 1.8862 x 2.1097 x 2.3574 X 2.6319 x 2.9358 x 3.2720 x 3.6438 x 4.0547 x 4.5084 X 5.0093 x 5.5620 x 6.1714 x 6.8430 x 7.5830 x 8.3979 x 9.2950 x 1.0282 X 1.1368 x 1.2562 x

4.0565 4.4422 4.8584 5.3070 5.7901 6.3101 6.8693 7.4701 8.1153 8.8076 9.5499 1.0345 1.1197 1.2109 1.3085 1.4127 1.5241 1.6431 1.7701 1.9055 2.0499 2.2038 2.3677 2.5422 2.7279 2.9254 3.1354 3.3585 3.5956 3.8472 4.1143 4.3976 4.6980 5.0164 5.3536 5.7108 6.0889 6.4889 6.9119 7.3590 7.8315 8.3304 8.8571 9.4127 9.9986 1.0616 1.1267 1.1952 1.2672 1.3430

109 109 109 109 109 109 109 109 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 lOll 10 II 10 II lOll lOll lOll lOll lOll lOll lOll lOll lOll lOll lOll lOll 10II lOll lOll lOll lOll lOll 1012 10 12 10 12

X X X X X X X X X X X X X X X X X X X X X X X X X X X X X

x X

x X

x X X

x x X X

x x X

x x x x x x x

1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1014 10 14 10 14 1014 1014 1014 1014 10 14 10 14 1014 1014 1014 1014 1014 10 14 10 14 10 14 10 14 10 14 1014 1014 1014 10 14 10 14 10 14 10 14 10 14 10 14 1014 10 14 10 14 10 14 1014 10 14 10 15 10 15 1015 1015 1015

3PI/2- 2p3/2

4.2714 X 4.8006 X 5.3833 X 6.0237 X . 6.7263 X 7.4957 X 8.3370 X 9.2553 X 1.0256 X 1.1345 X 1.2529 X 1.3813 X 1.5204 X 1.6710 X 1.8336 X 2.0092 X 2.1984 X 2.4021 X 2.6211 X 2.8563 X 3.1087 X 3.3790 X 3.6683 X 3.9776 X 4.3080 X 4.6603 X 5.0359 X 5.4356 x 5.8608 X 6.3124 x 6.7918 X 7.3000 X 7.8382 X 8.4078 x 9.0098 x 9.6456 x 1.0316 x 1.1023 x 1.1767 x 1.2550 x 1.3372 x 1.4235 x 1.5140 x 1.6088 x 1. 7079 x 1.8115 x 1.9197 x 2.0324 X 2.1498 x 2.2719 x

lOll lOll lOll lOll lOll lOll lOll lOll 10 12 10 12 1012 1012 10 12 10 12 1012 10 12 10 12 1012 10 12 1012 1012 1012 10 12 1012 10 12 10 12 10 12 10 12 1012 1012 1012 10 12 10 12 10 12 1012 10 12 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013

3.2 Allowed and Forbidden Transitions

Table 3.2. Continued Z

3P3/2- 2p3/2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

1.1954 X 7.6536 x 8.7183 x 4.8985 x 1.8686 x 5.5795 x 1.4069 x 3.1347 x 6.3547 x 1.1957 X 2.1181 x 3.5699 x 5.7704 X 9:0008 x 1.3615 x 2.0052 x 2.8847 x 4.0645 x 5.6215 x 7.6465 x 1.0246 x 1.3544 x 1.7681 x 2.2823 x 2.9153 x 3.6884 x 4.6251 x 5.7522 x 7.0993 x 8.6996 x 1.0590 x 1.2810 x 1.5405 x 1.8425 X 2.1922 x 2.5955 x 3.0588 x 3.5891 x 4.1938 x 4.8811 x 5.6597 x 6.5392 x 7.5297 x 8.6421 X 9.8882 x 1.1281 x 1.2833 x 1.4558 x 1.6474 x 1.8594 x

10 1 102 103 104 lOS lOS 1
3d3/2 - 2p3/2

1.0775 X 1.7247 x 8.7317 x 2.7596 x 6.7372 x 1.3970 x 2.5880 x 4.4148 x 7.0712 x 1.0777 x 1.5777 x 2.2344 x 3.0773 X 4.1387 x 5.4535 X 7.0590 X 8.9952 X 1.1304 x 1.4032 x 1.7225 x 2.0934 x 2.5212 x 3.0113 x 3.5696 x 4.2021 x 4.9149 x 5.7147 x 6.6083 x 7.6026 x 8.7049 X 9.9227 x 1.1264 X 1.2736 x 1.4348 x 1.6108 x 1.8025 x 2.0107 X 2.2365 x 2.4807 x 2.7443 x 3.0283 x 3.3337 x 3.6616 x 4.0130 x 4.3891 x 4.7908 x 5.2194 x 5.6760 x 6.1618 x 6.6780 x

107 108 108 109 109 1010 1010 1010 1010 1011 lO" lO" 1011 lO" 1011 lO" lO" 1012 1012 1012 1012 1012 10 12 1012 10 12 1012 10 12 1012 1012 1012 10 12 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013 1013

3ds/2 - 2p3/2

6.4651 1.0348 5.2391 1.6558 4.0425 8.3823 1.5529 2.6491 4.2432 6.4670 9.4679 1.3409 1.8467 2.4838 3.2730 4.2367 5.3990 6.7854 8.4229 1.0340 1.2567 1.5136 1.8080 2.1433 2.5231 2.9514 3.4319 3.9688 4.5662 5.2287 5.9606 6.7668 7.6520 8.6212 9.6795 1.0832 1.2085 1.3443 1.4912 1.6498 1.8208 2.0046 2.2020 2.4136 2.6401 2.8821 3.1403 3.4155 3.7083 4.0195

X 107 x 109 X 109 x 1010 x 1010 x 1010 X 1011 x 1011 x 1011 x lO" X lO" X 10 12 X 10 12 x 1012 X 10 12 x 1012 x 1012 x 1012 x 1012 x 1013 x 1013 x 1013 x 1013 X 1013 x 1013 x 1013 x 1013 x 1013 x 1013 x 1013 x 1013 X 1013 x 1013 x 1013 x 1013 X 10 14 X 10 14 x 1014 x 10 14 x 1014 x 10 14 x 1014 x 1014 x 1014 x 1014 x 1014 x 1014 x 1014 x 1014 x 10 14

87

88

3 Transition Probabilities

Table 3.2. Continued Z

3 P3/2 - 2p3/2

3d3/2 -2p3/2

3d5/2-2p3/2

51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 .80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

2.0937 2.3522 2.6367 2.9493 3.2922 3.6677 4.0782 4.5264 5.0149 5.5466 6.1245 6.7518 7.4318 8.1680 8.9641 9.8239 1.0751 1.1751 1.2827 1.3984 1.5227 1.6561 1.7992 1.9524 2.1164 2.2917 2.4791 2.6791 2.8924 3.1198 3.3620 3.6197 3.8939 4.1852 4.4946 4.8230 5.1713 5.5406 5.9318 6.3459 6.7842 7.2477 7.7376 8.2551 8.8016 9.3784 9.9867 1.0628 1.1304 1.2016

7.2258 X 7.8065 X 8.4213 X 9.0714 X 9.7584 X 1.0483 X 1.1248 X 1.2053 X 1.2900 X 1.3791 X 1.4727 X 1.5709 X 1.6740 X 1. 7820 X 1.8951 X 2.0134 X 2.1371 X 2.2664 X 2.4015 X 2.5424 X 2.6893 X 2.8425 X 3.0020 X 3.1681 X 3.3409 X 3.5206 X 3.7074 X 3.9013 X 4.1027 X 4.3117 X 4.5285 X 4.7532 X 4.9860 X 5.2272 X 5.4768 X 5.7351 X 6.0023 x 6.2786 X 6.5641 X 6.8591 X 7.1636 X 7.4781 X 7.8025 X 8.1371 X 8.4822 X 8.8379 X 9.2044 X 9.5819 X 9.9705 X 1.0371 X

4.3498' X 4.6999 X 5.0708 X 5.4631 X 5.8776 X 6.3152 X 6.7767 X 7.2630 X 7.7748 X 8.3131 X 8.8788 X 9.4727 X 1.0096 X 1.0749 X 1.1433 X 1.2149 X 1.2899 X 1.3682 X 1.4500 X 1.5353 X 1.6244 X 1.7173 X 1.8141 X 1.9149 X 2.0197 X 2.1289 X 2.2423 X 2.3602 X 2.4826 X 2.6097 X 2.7416 X 2.8784 X 3.0201 X 3.1670 X 3.3192 X 3.4767 X 3.6397 X 3.8083 X 3.9826 X 4.1628 X 4.3489 X 4.5412 X 4.7397 X 4.9445 X 5.1559 X 5.3738 X 5.5985 X 5.8301 X 6.0687 X 6.3144 X

x x x x x x x x x X

x x x X

x X

x x x X

x X

x x x x X

x x X

x x x x x x x x x X

x x x x x x x x X X

1011 1011 1011 1011 1011 1011 1011 1011 1011 1011 1011 1011 1011 1011 1011 1011 10 12 10 12 10 12 1012 1012 10 12 10 12 1012 1012 1012 1012 1012 1012 1012 1012 1012 10 12 1012 1012 1012 10 12 10 12 10 12 1012 10 12 10 12 10 12 1012 1012 1012 1012 1013 1013 1013

1013 1013 1013 1013 1013 1014 10 14 10 14 1014 10 14 10 14 1014 1014 1014 1014 1014 1014 1014 1014 1014 1014 1014 1014 1014 1014 10 14 1014 1014 10 14 10 14 1014 10 14 1014 1014 10 14 10 14 1014 1014 1014 1014 10 14 1014 1014 1014 1014 1014 1014 1014 1014 10 15

1014 10 14 1014 1014 10 14 10 14 10 14 1014 10 14 10 14 10 14 1014 10 15 10 15 10 15 10 15 10 15 10 15 10 15 10 15 10 15 10 15 10 15 10 15 10 15 10 15 1015 10 15 1015 1015 1015 10 15 10 15 1015 10 15 10 15 10 15 1015 1015 10 15 10 15 10 15 10 15 10 15 1015 10 15 10 15 10 15 10 15 10 15

3.2 Allowed and Forbidden Transitions

89

3.2.2 Radiative Decays of the n = 2 States in He-like Ions Radiative transitions from n = 2 levels have been intensively studied both experimentally and theoretically. In recent years, precision measurements of forbidden lifetimes in He-like ions have been extended to higher Z = 92 [3.10]. Except for the 2 I PI state, transitions from the levels n = 2 to the ground state are very retarded because of angular momentum and parity restrictions. However, transitions between levels with n = 2 are often allowed. In this section we examine the main results for allowed and intercombination transitions between different n = 2 levels in the He-like isoelectronic sequence. The theoretical and experimental results for the 21 So-II So transition are presented in Sect. 3.2.3. The 2 3 SI state in He-like ions is expected to decay to the II So ground state via a relativistically induced magnetic-dipole Ml transition. The 2 3 SI state can also decay by two-photon emission. The transition probability for this process has been studied in [3.11] and is expected to be three to four orders of magnitude less than the magnetic-dipole transition. The first measurement of the decay of the 2 3 SI state was performed for He-like argon in [3.12] using beam-foil spectroscopy. The experimental data for the 2 3 SI-ll So interval in lJ9O+ are given in [3.10]. The experimental lifetimes have been obtained over a range extending from Z = 2 - 36 [3.13]. These values are presented in Table 3.3 together with accurate results of theoretical calculations. All three 3 PO,I,2 levels decay to the 2 3 SI state by allowed (electric-dipole) El transitions. This is the dominant decay mode for low ionization stages. Accurate calculations of this probability have been performed in [3.14]. The 2 3 P2 level also decays by magnetic-quadrupole M2 transition to the ground state. Theoretical analysis [3.14] shows that the transition probability is small for low Z systems, but, since it scales as Z8, it becomes the dominat'lBb1e 3.3. Theoretical and experimental lifetimes (in s) of the 2 3 S. state in He-like ions Z

2 16 17 18 22 23 26

36

Experiment [3.6] (0.909 ± 0.027) x 10" (0.9 ± 0.3) x 10" (0.706 ± 0.083) x 10-6 (0.354 ± 0.024) x 10-6 (0.172 ± 0.012) x 10-6 (0.202 ± 0.02) x 10-6 (0.258±0.013) x 10-7 (0.169 ± 0.07) x 10-7 (0.50 ± 0.05) x 10-8 (0.48 ± 0.06) x 10-8 (0.20 ± 0.06) x 10-9

Theory [3.17]

0.700 x 10-6 0.375 X 10-6 0.208 X 10-6 0.266 X 10-7 0.169 x 10-7 0.481 x 10-8 0.171 x 10-9

90

3 Transition Probabilities

ing depopulation mode of the 3 P2 level for Z ~ 20. This was demonstrated experimentally in [3.15]. The magnetic-quadrupole M2 transition 2 3 P2 - 2 I So is also possible but its probability remains low for all Z [3.14]. In He-like ions the 2 3 PI state is mixed with the 2 I PI and other I PI states by fine-structure interaction. The mixing is very small in helium but increases rapidly with nuclear charge. Z. The allowed decay mode for the 2 3 PI states is electric-dipole El to the 2 3 81 state. The 2 I PI level, however, can decay directly to the ground state by electric-dipole El radiation and has a very short lifetime. Finally, due to singlet-triplet mixing, the 3 PI state can also decay directly to the ground state, the decay probabilities scaling approximately as ZIO. The 3 P2,O levels do not mix with the singlet system and have lifetimes 103 times higher as compared to the 3 PI level [3.16]. The 2 3 Po level can decay to the ground level 1 I So only by two- or three-photon transitions. The calculated and measured values for transition probabilities between the states with n = 2 are presented in Tables 3.3-10 and in Figs. 3.1, 2. 1BbIe 3A. The radiative transition probabilities (in s-I) for the 2 3PI -1 I So transition in He-like ions [3.14] Z 2 3 4 5 6 7 8 9 10 11 12

13 14 15 16 17 18 19 20 21 22 23

A 1.76 x 1()2 1.79 x 10" 3.99 x lOS .4.22 x 106 2.83 X 107 1.39 x loB 5.50 x loB 1.83 X 109 5.36 X 109 1.41 X 1010 3.38 x 1010 7.52 X 1010 1.57 x 1011 3.10 x 1011 5.82 x 1011 1.05 x 1012 1.80 x 1012 2.99 x 1012 4.79 x 1012 7.43 x 1012 1.12 x 1013 1.64 x 1013

Z

A

24 25 26 27 28 29 30 32

2.34 x 1013 3.25 x 1013 4.43 x 1013 5.90 x 1013 7.71 x 1013 9.91 x 1013 1.25 X 1014 1.92 x 1014 2.80 X 1014 3.92 x 1014 7.02 X 1014 9.05 x 1014 1.28 x lOIS 2.12 X lOIS 3.57 x lOIS 4.85 x 1012 6.89 x lOIS 9.48 x lOIS 1.20 x 1016 1.67 X 1016 2.16 x 1016 2.74 X 1016

34 36

40 42 45 50 56 60 65 ·70 74 80 85 90

3.2 Allowed and Forbidden Transitions 18ble 3.5. Theoretical and experimental transItion probabilities (in s-I) for the 2 3 PI -I I So transition

z

Experiment [3.6,18]

Theory [3.19]

7 8

(1.7±0.30) X 108 (5.80 ± 0.50) x lOS (6.01 ±0.33) x lOS (6.01 ± 0.42) x lOS (6.58 ± 0.34) x 108 (5.52 ± 0.39) x lOS (5.91 ± 1.0) x 108 (6.58 ± 0.48) x 108 (6.80 ± 0.09) x 108 (6.80 ± 0.37) x lOS (1.77±0.1O) x 109 (1.77 ± 0.07) x 109 (1.88 ± 0.07) x 109 (1.86 ± 0.07) x 109 (1.87 ± 0.07) x 109 (1.57 ± 0.09) x 1011 (6.37 ± 0.73) x 1011

1.39 x lOS 5.50 x 108

9

14 16

1.57 5.82

X X

1011 1011

Table 3.6. The radiative transition probabilities (in s-I) for the 2 3 po-2 3 SI transition in He-like ions [3.14] Z

A

2 4 6 8 10 11 12 13 14 15 16 17 18 19 20 21 22

1.15 X 3.57 x 5.79 x 8.07 x 1.04 X 1.21 x 1.34 x 1.47 X 1.61 x 1.75 x 1.89 x 2.05 x 1.97 x 2.38 x 2.55 x 2.74 x 2.93 x

107 107 107 107 108 108 108 108 108 108 108 108 108 108 108 108 108

Z

A

23 24 25 26 27 28 29 30 36 42 45 50 56 65 74 80 100

3.14 x 3.36 x 3.59 x 3.82 x 4.08 x 4.35 x 4.64 x 4.94 x 7.18 x 1.04 x 1.25 x 1.71 X 2.49 x 4.40 x 7.85 x 1.16 x 4.57 x

108 108 108 108 108 108

lOS 108 108 109 109 109 109 109 109 1010 1010

91

92

3 Transition Probabilities

Table 3.7. The radiative transition probabilities (in s-I) for the 2 3 PI-2 3 S1 transition in He-like ions [3.14] Z

A

2 4 6 8 10 II 12 13 14 15 16 17 18 19 20 21 22

U5 X 3.57 X 5.79 X 8.07 X 1.05 x 1.23 X 1.37 x 1.52 X 1.68 x 1.84 x 2.02 x 2.21 x 2.33 x 2.62 x 2.84 x 3.07 x 3.31 x

107 107 107 107 lOS 108 108 108 108 108 108 108 108 lOS lOS 108 lOS

Z

A

23 24 25 26 27 28 29 30 36 42 45 50 56 65 74 80 100

3.56 x 3.80 x 4.06 x 4.31 x 4.56 x 4.82 x 5.07 x 5.32 X 6.86 x 8.50 x 9.41 x UI X 1.35 x 1.83 x 2.51 x 3.14 x 7.37 x

lOS lOS lOS 108 lOS 108 108 108 108 108 108 109 109 109 109 109 109

Table 3.S. The radiative transition probabilities (in s-I) for the 2 I PI - 2 3 SI transition in He-like ions [3.14] Z

A

2 4 6 8 10 II 12 13 14 15 16 17 18 19 20 21 22

1.29 3.65 x 7.69 X 6.37 X 3.24 x 6.49 x 1.22 x 2.19 x 3.76 x 6.22 x 9.96 x 1.55 x 2.36 x 3.50 x 5.10 X 7.28 x 1.02 x

102 103 104

UP UP 106 106 106 106 106 107 107 107 107 107 108

Z

A

23 24 25 26 27 28 29 30 36 42 45 50 56 65 74 80 100

1.41 x 1.93 x 2.60 x 3.46 x 4.56 x 5.95 x 7.70 x 9.89 x 3.93 x 1.36 x 2.43 x 6.14 x 1.75" x 7.47 x 2.84 x 6.53 X 8.53 x

108 108 108 108 lOS 108 108 lOS 109 1010 1010 1010 lOll lOll 1012 1012 1013

3.2 Allowed and Forbidden Transitions

93

1Bble 3.9. The radiative transition probabilities (in s-I) for the 21 PI-2 I Sa transition in He-like ions [3.14] Z

A

2 4 6 8 10 11 12 13 14 15 16 17 18 19 20 21 22

1.93 x 9.49 x 1.79 x 2.69 x 3.71 x 4.29 x 4.93 x 5.66 x 6.49 x 7.47 x 8.61 x 9.97 x 1.16 x 1.36 x 1.61 x 1.91 x 2.28 x

lW lW 107 107 107 107 107 107 107 107 107 107 lOS lOS 108 lOS lOS

Z

A

23 24 25 26 27 28 29 30 36 42 45 50 56 65 74 80 100

2.74 x 108 3.32 x lOS 4.05 x 108 4.96 x lOS 6.10 x 108 7.53 x 108 9.33 x lOS 1.16 x 109 4.36 x 109 1.60 x 1010 2.98 x 1010 8.03 x 1010 2.45 x lOll 1.13 x 1012 4.53 x 1012 1.07 X 1013 1.49 X 1014

1Bble 3.10. Theoretical and experimental transition probabilities (in s-I) for the 2 3 1'2 -1 I Sa transition Z

Experiment [3.6]

Theory [3.17]

16 17 18 22 23 26

(1.7 ± 0.3) (2.7 ± 0.3) (2.3 ± 1) x (1.6 ± 0.2) (2.5 ± 0.4) (7.5 ± 2) x

1.15 x 1.90 x 3.06 x 1.61 x 2.33 x 6.36 x

x lOS x lOS lOS x 1()9 x 109 109

108 108 lOS 109 109 109

50 Transition

::i ::..

30

235 1-1 15 0

L

.:

::::: .J-

10

a.

-10

I

~

I

I

I

t

T

+ Fig. 3.1. Theoretical [3.17] and experimental [3.6] values of A-I for the 2 3S,-I'So transition in He-like ions as a function of nuclear charge

-30 -50 16

17

22

18

z

23

26

36

3 Transition Probabilities

Fig. 3.2. Theoretical [3.17] and experimental [3.6] values of A-I for the 2 3 1'2-1 180 transition in He-like ions as a function of nuclear charge

0 0 0

c.. ><

l!,

j j

Transition 23P2-1'So

0

I

0

T

-3 0

-5 0

-7 "v

16

17

18

22

23

26

Z

3.3 Two-Photon Transitions A two-photon process in an ion is a transition from an initial state with energy E 1 to a final state with energy Eo, followed by the radiation of two photons. This transition is described by: initial state +n photons -+ final state +n' photons, with n +n' = 2. If n = 2 and n' = 0, the process is a two-photon absorption. It requires the energy conservation

11£01 +1iW2 = Eo - El, where 1iwl and 1iW2 are the energies of the two simultaneously absorbed photons. If n = 0 and n' = 2, the process is a two-photon emission. If n = 1 and n' = 1 the process is the scattering of radiation. In this case, the energy conservation law gives

1iwl

+ El =

1iW2 + Eo.

If the atomic state is unchanged, then WI = W2, the scattering is elastic (Rayleigh scattering). If Eo is a bound state with Eo < El the inelastic scattering (WI #= W2) is the Raman scattering (the scattering leads to the excitation), while for El > Eo the process is the anti-Stokes Raman scattering (de-excitation). If Eo is a state in the continuum, the process is called Compton scattering. In the following, we consider in detail only the emission of two photons in H- and He-like multicharged ions. 3.3.1 Two-Photon Decay of the lsl/2 State in H-Uke Ions The 2s1/2 state in H-like ions is metastable because angular-momentum and parity selection rules prevent the usual electric-dipole transitions to the ground

3.3 Two-Photon Transitions

95

Thble 3.11. Calculated two-photon decay probabilities (in s-I) for the 2s1/2 -ISI/2 transition in H-Iike ions Z

Theory [3.27]

Theory [3.28]

1 2 3 4 5 6 7 8 9 10 12 14 16 18 20 22 24 28 30 34 38 42

8.2291 5.2661 x 5.9973 x 3.3689 x 1.2847 x 3.8347 x 9.6654 x 2.1525 x 4.3612 x 8.2010 x 2.4451 x 6.1547 x 1.3686 x 2.7682 x 5.1956 x 9.1785 x 1.5423 x 3.8628 x 5.8217 x 1.2236 x

8.2291 5.2660 x 5.9973 x 3.3689 x 1.2847 x 3.8348 x 9.6657 x 2.1526 x 4.3614 x 8.2015 x 2.4453 x 6.1554 x 1.3688 x 2.7686 x 5.1965 x 9.1804 x 1.5427 x 3.8637 x 5.8231 x 1.2238 x 2.3636 x 4.2657 x 7.2824 x 1.1868 X 1.8595 x 2.8162 x 4.1409 x 5.9328 x 8.3060 X 1.1391 x 1.5331 x 2.0287 x 2.6427 x 3.3931 X 3.8251 X

102 UP 104 lOS lOS lOS IW IW IW 107 107 108 108 lOS 108 109 109 109 1010

4.2651 x 1010

46

50 54 58 62 66 70 74 78 82 86

90 92

1.1869 x 1011

8.3139 X 1011 1.1404 x 1012

3.4021 X 1012 3.8361 x 1012

102 UP 104 lOS lOS lOS IW IW IW 107 107 108 108 lOS 108 109 109 109 1010 1010 1010 1010 1011 1011 1011 1011 1011 1011 1012 1012 1012 1012 1012 1012

state. The simultaneous emission of two photons is the dominant decay mode. This follows from a second-order interaction between an ion and the electromagnetic field in the process (3.3.1) where photon frequencies satisfy the conservation law (3.3.2)

96

3 Transition Probabilities

The possibility of this process, which takes place via intennediate states, was first pointed out in [3.20]. The first measurement of the lifetime of the 2s1/2 state in H-like ions was performed using the fast Arl7+ beam from a heavy-ion accelerator [3.12]. The lifetimes have now been measured in several H-like ions: He+ [3.21-23], 07+ [3.24], p8+ [3.24], SI5+ [3.15], Arl7+ [3.15] and Ni27+ [3.9]. The development of the beam-foil technique used to measure the metastable lifetimes made it possible to compare the calculated and measured transition probabilities. The non-relativistic numerical results of the two-photon decay probability of the 2SI/2 state obtained in [3.25], A2EI = (8.2283 ± 0.0001)Z6s-l,

(3.3.3)

is believed to be the most accurate one [3.26]. Accurate calculations including all relativistic and retardation effects and combinations of photon multipoles (2El, EI-M2, 2Ml, 2E2, 2M2 and E2-Ml-types) are presented in [3.27-29]. Both contributions of high multipoles and relativistic corrections are small in the case of low-Z ions, while they rapidly increase with increasing nuclear charge Z. The calculated probabilities are given in Table 3.11. Figure 3.3 shows the contribution from different relativistic effects [3.29]. Relativistic corrections make the curve deviate from unity at high Z. The transition probability for the 2SI/2-lsl/2 decay can be expressed by a simple formula [3.27] A(2s

1/2

-Is

1/2

) = 8.22943Z6

(1 +

3.9448(aZ)2 - 2.040(aZ)4) ( -I) 1 + 4.6019(aZ)2 s (3.3.4)

The estimated accuracy of this result is ±0.05% in the range 1 ~ Z ~ 92. A comparison of the measured probabilities with the theoretical data is given in Table 3.12. The values of the two-photon transition probabilities also

1.0 0.8 ~ 0.6

«

--«

w

N

Transition 2s,/2- 1s,/2

0.4 0.2 0.0 4--,.--,-oooor-,..,..-y-....-,--r..-r-,-.........,-.-y--.--,,......,....ro--r--l o 10 20 30 40 50 60 70 80 90 100 110 120 Z

Fig. 3.3. Relativistic corrections to the transition probability of the 2SI/2 -ISI/2 transition in H-like ions [3.29]

3.3 Two-Photon Transitions

rn

Table 3.12. Theoretical and experimental data for the decay probabilities of the 181/2 state (in s-I) in H-Iike ions Ion

Theory [3.27]

Theory [3.28]

Experiment

He+

526.61

526.61

491~~~[3.21]

2.1553 4.3702 1.3966 2.8594

520 ± 21[3.22] 525 ± 5[3.23] (2.21 ± 0.22) x (4.22 ± 0.28) x (1.37 ± 0.13) x (2.82 ± 0.20) x

07+ p8+ SI5+ ArI7+

2.1552 4.3699 1.3964 2.8590

x x x x

106 106 108 108

x 106 x 106 X

108

x lOS

106[3.24] 106[3.24] IOS[3.15] IOS[3.15]

include the corrections due to the magnetic-dipole (MI) decay mode. Note that all theoretical values are based on relativistic calculations. 3.3.2 Two-Photon Decay of the 21S8 State of He-like Ions All single-photon transitions from the 21 So- state are strictly forbidden in the absence of nuclear spin, leaving two-photon emission as the dominant decay mode. Even if the nucleus has a spin, two-photon emission remains dominant up to very large values of Z. The two-photon emission in He-like ions has been observed in laboratory sources: He [3.30], Li+ [3.31], Ar16+ [3.21] and Kf34+ [3.33]. The calculation of two-photon process

Is2s I So -+ Is2 I So + hWI

+ nW2

(3.3.5)

in He-like ions is much more complicated because of the two-electron nature of the problem. The first reliable estimations for helium by explicitly summing over the discrete and continuous oscillator-strength distributions was obtained in [3.34]. Two-photon decay probabilities have been calculated by the variational method from He to Ne8+ [3.11] . .Detailed calculations of relativistic corrections to two-photon decay probabilities have been performed in [3.35,36]. Good results are given by the fit [3.35]. A(2 1SO-IISO) = A (0) (2 1 So-I ISO) (Z ; P

r

(I

+ (Z: b)2) ,

(3.3.6)

with the fitting parameters: p = 0.806389, a = 1.539, b = 2.5 and A(0)(2ISoII So) is the probability in the H-like approximation, i.e., A(0)(2 I S -IISo) = 16.548762Z6 [1- (aZ)20.6571 + 2.040(aZ)2] (s-I). o I + 4.6019(aZ)2 (3.3.7)

98

3 Transition Probabilities

-------- --------

eli 0.8

t

//.

o

'"

---- ------~~~~~--~---------

CI)

~

o
:::a

0.4

CI)

t 0

CI)

~ -0.0 <{

-0.4 0

10

20

30

40

50

60

70

80

90

100

Z

Fig. 3.4. The function A(2 1SI-II So)/ A (0) (2 1SI - IISo) [3.36]: _ _ relativistic calculation by A(2 1 SI - II So) = A(0)(2 1 SI - II So)(1 + B(aZ)/Z) with coefficient B(aZ) in relativistic approxi-

mation; - - - non-relativistic calculation; - . - . - . - calculation in "screened-charge" approximation by (3.3.6)

----------

----

15.0 'i

CI)

:;;N .......

"0

!!> 10.0 t 0

CI)

N

<"

5.0 I

t i 0.0

0

10

20

30

40

50 Z

60

70

80

90

100

Fig. 3.S. Transition probability A(2 1 SI -II So) in He-like ions [3.36] by: _ _ A(2 1 SI - II So) = A(0)(2 I SI - IISo)(1 + B(aZ)/Z), with coefficient B(aZ) in relativistic approximation; - - -

non-relativistic calculation; - . - . - . - calculation in "screened-charge" approximation by (3.3.6)

3.3 Two-Photon Transitions

99

1Bble 3.13. Scaled two-photon decay probabilities A/Z6 (in s-I) of the 21 So state [3.35] in He-like ions

Z

A(2 1So-II So)/Z6

Z

A(2ISo-I I So)/Z6

2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 19 20 21 22 23 24 25

0.7960 2.6592 4.4309 5.8895 7.0645 8.1067 8.798 9.447 9.993 10.458 10.857 11.203 11.504 11.769 12.002 12.209 12.393 12.557 12.704 12.836 12.954 13.060 13.156 13.242

26 27 28 29 30 32 34 36 38 40 42 45 50 54 56 60 65 70 74 80 82 85 90 92

13.319 13.389 13.451 13.507 13.556 13.639 13.703 13.750 13.783 13.803 13.811 13.805 13.751 13.674 13.626 13.512 13.340 13.14 12.96 12.65 12.54 12.37 12.07 11.94

A power-series non-relativistic expansion of (3.3.6) yields AM(21So-11So) = 16.458762Z6 (1 _ 4.8383

Z

+ 11.293

87.15 ... ) +--+ Z4

Z2

I

(s-) .

_ 25.63 Z3 (3.3.8)

Figures 3.4, 5 show the contribution of relativistic and correlation effects to the two-photon decay probabilities. Table 3.13 gives the relativistically corrected two-photon decay probabilities for ions in the range 2 ::;; Z ::;; 92 [3.35]. 1BbIe 3.14. Calculated and measured two-photon decay probabilities (in s-I) of 21 So state in He-like ions Ion

Theory [3.35]

Experiment

50.953 ± 0.073 1938.56 ± 0.18 (4.215 ± 0.010) x lOS (2.993 ± 0.012) x 1010

50 ± 2.5[3.30] 1988 ± 98[3.31] (4.31 ± 0.34) x IOS[3.32] (2.934 ± 0.030) x 1010 [3.33]

100

3 Transition Probabilities

For He-like ions the direct measurements of the two-photon decay probability have been made only for the ions He, Li+, Ar16+ and 1(r34+ as shown in Table 3.14.

3.4 Semiempirical and Asymptotic Formulas for Osclllator Strengths and Transition Probabilities in H- and He-like Ions 3.4.1 H-Uke Ions The simplest approximation for oscillator strengths by the Kramers formula for transitions n-n' [3.4]:

of H-like ions is given

32

n'

In =

I

M 31l' 'V 3n 5n'3[n- 2 -

(3.4.1)

(n')-2]3

Using the dipole approximation n'l' _ 2c.omax(1, I') Inl 3(21 + 1)

IRn"'1 2 nl

(3.4.2)

'

values of I for transitions nl-n'l' are tabulated in [3.4]. Equation (3.4.2) does not include the relativistic corrections needed for high-Z values. The relativistic model [3.39-40] uses a local spherically symmetric potential in the Dirac equation. 'J1le formula for the oscillator strength with relativistic .. corrections reads I::j'j' =

~6E:;?' max(1,1')(2j' + 1) (~

f, I?) IR:;?r,

(3.4.3)

where n'l'j' 6 E nlj = En'l' j' >

Enl. = J

-

Enlj ,

Ry [Z2 + a4n2Z n2

4

(

4

j

4n

+ 1/2

-

3)] + 6ECL.

(3.4.4)

Here 6E CL is the relativistic correction [3.39]. The semiempirical expressions of radial integrals for Lyman and Balmer series are [3.39]: Rnp -

ls

= 1.290/(n - 1)1.32 Z,

Rnp -

ls

= 2.17/n 3/ 2 Z,

n < 5,

(3.4.5)

n > 5,

(3.4.6)

Rnp -2s = 3.065/(n - 2)1.257 Zx,

(3.4.7)

where x

= 1 for Z < 40 and x = 1.02 for higher Z,

Rnd-2p = 4.748/(n - 2)1.43 Z,

(3.4.8)

Rns -2p = 0.938/(n - 2)1.3 Z.

(3.4.9)

3.4 Semiempirical and Asymptotic Formulas for Oscillator Strengths

101

Table 3.15. Oscillator strengths for H-like ions. QM: quantum calculations with Dirac wavefunctions [3.39]; SM: semiempirical calculations [3.40] with (3.4.3) Transition QM SM QM SM QM SM QM SM

2PI/2 -ISI/2 2P3/2 -lsl/2 3PI/2- 2s l/2 3 P3/2 - 2SI/2

Z=20

Z =40

Z =60

Z =80

0.1373 0.1393 0.2750 0.2791 0.1448 0.1461 0.2829 0.2930

0.1330 0.1313 0.2673 0.2644 0.1443 0.1388 0.2617 0.2806

0.1254 0.1231 0.2536 0.2495 0.1433 0.1346 0.2257 0.2695

0.1136 0.1149 0.2326 0.2359 0.1416 0.1252 0.1741 0.2605

For other transitions the radial integrals have the form Rn1-n'I' = R~-n'l,/(n' - n)1.3Zx,

n, n' ~ 3,

(3.4.10)

where R~-n'l' are the H-like values [3.4]. Oscillator strengths calculated for high Z are given in Table 3.15, which' shows that the semiempirical formula (3.4.3) gives quite accurate results.

3.4.2 He-like Ions The accurate calculations of the oscillator strengths in H-like ions have been performed in [3.41,42]. An alternative approximate approach has been applied using an effective charge of electrons in the initial and final states. Such a treatment, based on the assumption of single-electron H-like wave functions, has been reasonably successful in the calculations of the oscillator strengths and transition probabilities in He-like ions. For Is2 1 S -lsnp I P transitions, the square of the radial integral R was calculated in the form [3.43]: R2(ls 21 S-1snp 1 P) = 28n\n 2 - l)Z?Zi(2Zj - Zf)2<1>(Zj, Zf), (3.4.11) where (Zj, Zr) = (nZj - Zr)2n-6/(nZj

+ Zf)2n+6.

(3.4.12)

Here, Zj and Zf are the effective charge parameters (in the H-like approximation) for the initial and final states of the optical electron. In this method, the corresponding expression for the oscillator strengths f is given by [3.44]: f(1s

where a with A

21

I

S-lsnp P) .

= A(2Zr -

zi j ="32 Z?(2Z I

c = -na +b -n 5 +-, 3 n7

Zf), b 2

= aB -

Azl, c

Zf) exp(-4Zr/Zj),

(3.4.13)

= aD -

ABzl, (3.4.14)

102

3 Transition Probabilities 2

4

2

B = 6(Zr/Zj) - 3(Zr/Zj) - 1,

8

6

44

D = g(Zr/Zi) - S(Zr/Zj)

5

(3.4.15)

+ 21 (Zr/Zj)4+43(Zr/Zi) 3

- 6(Zr/Zj)2.

(3.4.16)

The quantities Zj, ZI' a, b and c for the He-like ions up to Z = 30 are given in Table 3.16. On the basis of sophisticated relativistic calculations of the transition probabilities A for He-like ions, the approximation for A was obtained in the form [3.45]: (3.4.17) where z is the spectroscopic symbol of an ion, a and b are the fitting parameters; the parameter y accounts for the dependence of A on z. The parameters a, b Table 3.16. Calculated parameters for oscillator strengths in He-like ions [3.44] Ion

Zj

Zr

a

b

c

He

1.70403 2.69814 3.69539 4.69386 5.692% 6.69246 7.69227 8.89233 9.69263 10.6903 11.6937 12.6950 13.6897 14.689 15.689 16.6893 17.6892 18.6891 19.689 20.6889 21.6889 22.6888 23.6888 24.6887 25.6887 26.6886 27.6886 28.6885 29.6885

1.183 2.35 3.22 4.08 5 6 7 8 9 10 II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

1.81766 2.55555 2.77175 2.89556 2.97417 3.02121 3.05072 3.07040 3.08419 3.09198 3.10171 3.10787 3.10850 3.11139 3.11405 3.11635 3.11804 3.11943 3.12057 3.12153 3.12237 3.12305 3.12365 3.12413 3.12458 3.12494 3.12527 3.12553 3.12579

1.62205 4.98628 5.20484 5.26529 5.52440 5.96577 6.29113 6.54005 6.73626 6.89239 7.02517 7.13486 7.22339 7.30231 7.37123 7.43178 7.48502 7.53237 7.57474 7.61289 7.64745 7.67883 7.70752 7.73378 7.75798 7.78027 7.80094 7.82010 7.83797

0.568893 6.31253 6.30342 6.12889 6.59017 7.64609 8.47433 9.13782 9.67964 10.1304 10.5089 10.8325 11.1133 11.3571 11.5712 11.7608 11.9303 12.0818 12.2188 12.3430 12.4560 12.5595 12.6544 12.7418 12.8226 12.8975 12.%72 13.0320 13.0926

Li+

Be2+ B3+ c4+

N5+ 06+

F7+ Ne8+ Na9+ Mg lO+ All 1+

Si l 2+ p13+

SI4+

0 15+ Ar 16+ KI7+ Ca18+ Sc l 9+ Ti2O+ V21+

cr22+ Mn23 + Fe24+ Co25+ Ni26+ Cu27+ Zn28+

3.5 Autoionization Probabilities

103

18b1e 3.17. Parameters a, b and c (3.4.17) for radiative transition probabilities in He-like ions [3.45] Transition

Type

Range of Z

a

b

c

Appr. error [%]

2IPI-IISo -2 1 So -23SI 2 I So-23SI _23PI -I 'So 23S,-I'So -I 'So 2 3 p,-I'So

EI EI EI MI EI 2EI MI 2EI EI

Z < 80 Z < 30 Z>4 I
9.6 x lOS 1.5 x 10" 0.66 1.8 x 10-8 10.8 16.5 1.8 x 10-6 9.2 x 10- 10 551 6.48 x lOS 1.33 X 107 1.1 x 106 3.73 X 109 5.3 x lOS 0.038 1.23 x 10-9 1.33 x 107

1.05 3.14 -0.18 -0.25 0.38 0.20 0.44 0 -1.57 -3.13 1.57 -17.25 -9.97 -4.29 0.31 -16.5 1.57

4 3 6 7 2 6 10 10 8 4 I 2 I 4 8 8 I

10 20 5 20 10 7 25 10 30 15 20 25 10 10 5 10 20

IO
Z>I 4
-2 3 S,

EI

2 3 Pz-2 3 S,

EI

-1'So -2'So 2 3 fb-2 3 S,

M2 M2

EI

Z < 50 Z ~50 15 < Z < 20 Z ~20 Z < 80 Z ~20 Z <50

and y for transitions between terms of configurations with n = 2 and n = 1 in He-like ions are given in Table 3.17 together with the approximation accuracy (last column).

3.5 Autoionization Probabilities The autoionization probabilities are used for many applications [3.46-55] mainly as a diagnostic tool to measure plasma parameters [3.54] and for astrophysical investigations [3.55]. The autoionization probabilities can be calculated in the following form:

r

=

21r1

<

"'01 L

"'I "'0

Vap

1"'1 >12 p(£),

(3.5.1)

a
where and are the initial (bound) and final (free) states, respectively, the two-electron operator VaP is the sum of the Coulomb and Breit interactions, i.e., (3.5.2) , TI2 = 1TI - T2 1, 2 p(£) is the energy density of the outgoing continuum electron states. The wave is usually normalized to the Dirac-delta function. function To construct the N-electron system wave functions, the Z-expansion (MZ) method is often used with hydrogenic wave functions [3.46,48,49]. Vap

= (l-ala2) TI2

"'0

(aIVI)(a2 V 2)TI2

104

3 Transition Probabilities

Table 3.18a. Autoionization probabilities r(l013 s-l) for He satellites ions [3.49] Transition

Z=IO

2p3d 3P2 -ls2s 3SI 2s3p 3 p-ls2s 3SI 2p3s 3PI -ls2s 3SI 2p3s I PI -ls2s I So 2s3d 3D3 -ls2p 3P2 2p3p I D2-ls2 p 3P2 2p3 p 3 P2-ls2p3 PI 2p3p3P2-ls2p3P2 2p3p 3Po-ls2p 3PI 2s3d I D2 -ls2p I PI 2p3p3D3-1s2p3P2 2s3d 3D2 -ls2p I PI 2s3p I PI-ls2s I So 2p3p I PI -ls2p I PI 2p3d 3P2 -ls3s 3SI 2p3p I So-ls3p I PI 2p3d I PI -ls2d I D2 2p3d I F3 -ls3d 3D2 2p3d I F3 -ls3d I D2 2p3d 3 Po-ls3d 3DI 2p3d 3PI-ls3d 3DI 2p3d 3PI -ls3d 3D2 2p3d 3P2 -ls3d 3D2 2s3d I D2- 1s3p I PI 2p3s I PI -ls3s ISo 2p3p 3SJ -Is3 p 3 P2 2p3s I PI -ls3d I D2 2s3d 3D2-1s3p3 PI 2s3d 3D3- 1s3p 3P2 2s3d 3D2-ls3 p 3l',. 2p3p I D2-1s3 p 3PI 2p3d 3D3-1s3d3 DJ 2s3d 3DI -ls3p 3Po 2s3d 3DI-ls3 p 3PI 2p3d 3D2-1s3d 3DI 2p3d 3D2-ls3d 3D2 2p3d 3DI -ls3d 3DI 2p3d 3DI -ls3d 3D2 2p3 p 3l',.-ls3p 3PI 2p3p3P2-ls3p3P2 2p3p 3 PI-ls3p 3 Po 2p3p 3 PI-ls3 p 3PI 2p3p 3 PI-ls3 p 3 P2 2p3p 3Po-ls3 p 3pl 2p3s 3p,. -ls3s 3SI 2s3p 3PI -ls3s 3SI 2s3 p 3 p,.-ls3s 3S1 2p3s 3PI -ls3s 3SI

1.79 x 10-4 6.74 x 10- 1 6.21 X 10- 1 9.43 2.53 X 10- 1 3.46 1.84 1.84 7.92 X 10-3 6.83 1.41 x 10-3 4.84 4.95 X 10-2 2.20 X 10-4 1.79 X 10-4 5.86 X 10- 1 5.72 X 10- 1 2.60 2.60 2.44 x 10-4 2.35 X 10-4 2.35 X 10-4 1.79 x 10-4 6.83 9.43 9.13 X 10-3 9.43 4.84 2.53 X 10- 1 4.84 3.46 4.52 x 10-3 2.52 X 10- 1 2.52 X 10- 1 1.09 x 10-3 1.09 X 10-3 3.22 X 10-2 3.22 X 10-2 1.84 1.84 2.42 x 10-3 2.42 x 10-3 2.42 x 10-3 7.92 x 10-3 1.95 X 10-2 7.92 X 10-2 6.74 x 10- 1 6.21 X 10- 1

Z= 14 6.92 X 6.07 X 9.31 2.50 X 4.09 2.95 2.95 4.51 X 7.85 4.39 X 2.02

10- 1 10- 1 10- 1

to resonance lines of H-like

Z= 18

Z =26

6.70 X 10- 1

5.95

8.78 2.36

8.83 1.20 X 10- 1 1.10 x 101

X

10- 1

7.23 x 10- 1

1.57

3.60

1.31 1359

X

10- 1

10-2 10-3

X

10- 1

3.10 7.65

X

10- 1

8.05 X 4.36 X 2.58 2.58 1.15 x 1.45 X 1.45 x 8.13 X 7.85 9.31 1.10 X 9.31 2.02 2.50 X

10- 1 10- 1

10- 1

2.36

X

10- 1

1.20 1.92

X

10- 1

2.45 2.14

10-2 10- 1

7.73

X

10-2

2.64

X

10- 1

X X

10-3 10-3 10-3 10-4

1.15 2.56 X 2.52 2.52 3.83 x 8.63 X 8.63 x 2.87 x 3.60 8.78

10- 1

10-3 10-3 10-3 10-3

2.29 3.63 X 10-2 2.23 2.23 1.56 X 10-2 2.49 X 10-2

1.31 8.83

10-2 8.78

6.52 x 10-3 6.52 x 10-3 2.39 X 10- 1 2.39 X 10- 1 2.95 2.95

4.02 X 10-2 4.51 x 10-2 2.04 X 10-5 1.31 X 10- 1 6.92 X 10- 1 6.07 X 10- 1

8.83

1.71 X 2.57 x 2.57 x 7.98 X 7.98 X 7.23 x 7.23 x 8.06 X

10- 1 10-2 10-2 10- 1 10- 1 10- 1 10- 1 10-2

8.06 X 1.26 X 2.01 x 2.57 X 6.70 X 6.09 X

10-2 10- 1 10-2 10- 1 10- 1 10- 1

3.12 x 10-2 1.57 2.25

X

10-2

2.13 X 10- 1 1.50 x 10- 1 9.30 X 10- 1 1.67

X

10- 1

3.5 Autoionization Probabilities

105

Table 3.18a. Continued

Transition

Z = 10

Z = 14

Z = 18

Z =26

2s3d 3D2-ls3plpl 2p3d I D2 -ls3d I D2 2p3p I D2- 1s3 p I PI 2p3 p 3 P2- 1s3p I PI 2p3d 3 F4 -ls3d 3 D3 2p3d 3 F3 -ls3d 3 D2 2p3d 3 F3 -ls3d 3 D3 2p3d 3 F3 -ls3d I D2 2p3d 3 F2 -ls3d 3 DI 2p3d 3 F2 -ls3d 3 D2 2p3 p 3 D3- 1s3 p 3 P2 2p3p 3 D2-ls3p 3 PI 2p3p 3 D2- 1s3p 3 P2 2p3p 3 DI -ls3p 3 Po 2p3p 3DI-ls3p 3 PI 2s3p 3 PI-ls3d 3 D2 2s3p 3 Po-ls3d 3DI 2p3p I PI-ls3p I PI 2s3s I So-ls3p I PI 2s3p I PI-ls3s I So 2s3s 3 SI-ls3p 3 PI 2s3s 3 SI-ls3p 3 P2

4.84 1.77 x 3.46 1.84 5.34 X 5.33 x 5.33 x 5.33 x 5.15 X 5.15 x 1.41 x 9.35 x 9.35 x 1.55 X 1.55 X 7.92 X 1.09 X 1.20 X 1.77 X 4.95 X 8.31 x 8.31 x

2.02 5.58 x 4.09 2.95 5.34 X 5.33 X 5.33 X 5.33 X 4.72 X 4.72 X 4.39 x 5.96 x 5.96 x 3.08 X 3.08 x 1.31 X 1.84 X 8.37 X 1.74 X 5.46 x 8.11 x 8.11 x

3.10 6.38 x 10- 2 9.31

1.92

5.34 X 5.41 X 5.41 X 5.41 X 4.44 X 4.44 X 1.09 X 2.31 X 2.31 x 4.99 X 4.99 x

10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 10-2 10- 1 10- 1 10-4 10-4

5.34 X 6.44 X 6.44 X 6.44 X 3.61 X 3.61 X

2.66

X

10- 1

1.70 X 7.15 x 7.77 x 7.77 X

101 10- 2 10-2 10-2

10- 2

10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 10- 3 10- 3 10-3 10-4 10-4 10- 2 10- 1 10-4 10- 1 10-2 10- 2 10- 2

10- 2

10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 10-3 10- 2 10-2 10- 5 10-5 10- 1 10- 1 10-4 10 1 10- 2 10-2 10-2

10- 1 10- 1 10- 1 10- 1 10- 1 10- 1

1.16 1.16 4.03 x 10- 3 9.30 X 10- 1 3.70 X 10- 1 1.95 X 10-2 1.67 X 10- 1 7.65 X 10- 1 6.78

X

10-2

Table 3.18b. Radiative (A) and autoionization (f) probabilities (lOl3 s-l) for He-satellites to resonance lines of H-like ions [3.48]

Z = 10 Transition

A

2p2 I So-ls2 p 3 PI 2s2p I PI-ls2s 3 SI 2p 21 So-1s2p I PI 2s2p I PI -ls2s I So 2p2 I D2- 1s2p 3 PI 2p2 I D2- 1s2p 3 Po 2p2 I D2- ls2p 3 P2 2s2p 3 P2 -ls2s 3 SI 2s2p 3 PI -ls2s 3 SI 2s2p 3 Po -ls2s 3 SI 2p2 3 P2- 1s2p 3 PI 2p2 3 P2- ls2p 3 Po 2p2 3 P2- ls2p 3 P2 2p 23 PI_Is2p 3PI 2p 23 PI-ls2p 3Po 2p2 3 PI -ls2p 3 P2 2p2 3 Po-Is2p 3 PI 2p21D2-ls2p1PI

1.66 x 1.23 X 8.99 X 6.02 X 0 5.05 x 1.80 x 5.81 X 5.80 x 5.80 x 0 2.92 X 8.71 x 3.88 X 2.91 x 1.16 4.85 x 1.17

Z = 14

r 10- 5 10-4 10- 1 10- 1 10-6 10- 3 10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 10- 1

1.65 2.03 X 1.65 2.03 X 3.73 X 3.73 x 3.73 x 1.36 1.36 1.36 7.61 x 7.61 x 7.61 x 0 0 5.36 x 0 3.73 x

10 1 101 10 1 10 1 10 1

10- 2 10-2 10- 2

10- 3 101

A

r

3.63 x 10-4 3.31 x 10- 3 3.61 2.35 0 4.51 X 10-5 5.10 X 10-2 2.29 2.29 2.29 0 1.16 3.39 1.53 1.14 4.56 1.91 4.55

1.92 2.03 x 10 1 1.92 2.03 x 10 1 3.68 X 10 1 3.68 X 10 1 3.68 X 10 1 1.36 1.39 1.36 5.49 x 10- 1 5.49 x 10- 1 5.49 x 10- 1 0 0 3.53 x 10-2 0 3.68 + I

106

3 Transition Probabilities

Table 3.18b. Continued

Z= 14

z= 10 Transition

A

2p2 3P2- ls2p I PI 2p2 3 PI -ls2p I PI 2p2 3Po-ls2p I PI 2s2p 3P2 -ls2s I So 2s2p 3PI-ls2s I So 2s2p 3Po -ls2s I So 2s2IS0-ls2p3PI 2s 2 ISo-ls2p I PI 2p2 I So-ls2 p 3 PI 2s2p I PI -ls2s 3SI 2p2 I So-1s2p I PI 2s2p I PI -ls2s I So 2p2 I D2-1s2p 3PI 2p21D2-ls2p3Po

1.25 x 1.71 X 3.34 X 0 1.17 X 0 8.73 x 2.45 X 3.11 x 3.75 x 1.07 X 6.47 0 1.37 x 6.22 x 6.38 6.33 6.36 0 3.35 8.95 4.26 3.14 1.25 X 5.31 1.22 X 4.63 X 5.03 X 7.42 X 0 3.64 x 0 2.45 X 2.26

2p21~-ls2p3p2

2s2p 3P2 -ls2s 3SI 2s2p 3PI -ls2s 3SI 2s2p 3Po-ls2s 3SI 2p2 3P2- ls2p 3PI 2p2 3 P2-ls2p3 Po 2p23P2-ls2p3P2 2p2 3PI -ls2p 3PI 2p2 3PI -ls2p 3Po 2p2 3PI -ls2p 3 Pz 2p2 3 Po-ls2p 3PI 2p2lD2-ls2plpl 2p23 P2 -ls2p I PI 2p2 3 PI-ls2p I PI 2p2 3Po-ls2p I PI 2s2p 3 P2 -ls2s I So 2s2p 3PI -ls2s I So 2s2 p 3 Po-ls2s I So 2s2 I So-ls2p 3PI 2s2 I So-ls2p I PI

r 10- 3 10-4 10-4 10-4 10-4 10- 1 10-3 10-2 101

10-4 10- 1

10 1 101 10- 1 10-2 10-2 10-2 10- 1

7.61 x 0 5.36 x 1.36 1.36 1.36 3.45 X 3.45 X 2.35 2.02 x 2.35 2.02 x 3.50 X 3.50 X 3.50 X 1.36 1.47 1.36 2.41 2.41 2.41 0 0 1.19 X 0 3.50 X 2.41 0 1.19 X 1.36 1.47 1.36 3.37 X 3.37 x

10-2 10-3

10 1 10 1 10 1 101 10 1 101 10 1

10- 1 101

10- 1

101 101

3.64 x 4.59 X 8.09 x 0 3.19x 0 2.35 x 9.16 X 2.52 x 1.08 5.02 x 2.79 x 0 6.27 X 1.40 X 2.87 x 2.87 x 2.74 x 0 1.80 x 2.89 x 1.91 x 1.31 5.19 X 2.37 x 4.33 X 1.03 x 1.19 8.82 X 0 1.06 0 4.91 5.90

In this method, the probability of autoionization r(aLSJ)

r

A

10-2 10-3 10-3 10-3 10- 2 10- 1 10- 2 10 1 10 1 10-2 101 101 101 101 101 101 101 101 101 101 101 10- 1

5.49 x 0 3.53 x 1.36 1.39 1.36 3.42 x 3.42 X 3.87 1.96 x 3.87 1.96 x 2.53 X 2.53 X 2.53 X 1.36 1.36 2.06 1.21 x 1.21 X 1.21 X 0 0 3.59 X 0 2.53 X 1.21 X 0 3.59 X 1.36 2.06 1.36 3.19 x 3.19 x

10- 1 10- 2

10 1 101 10 1 10 1 10 1 101 10 1

101 101 10 1

10- 1 101 10 1 10- 1

101 101

r is given by [3.48]:

= ro L r O{J(aLSJ), O{J

ao

= /colo, ro = 4.13410 x

10 16 s-I,

where r a is the partial probability for autoionization into the final state of an ion, k'5/2 and 10 are the energy and the orbital momentum of the ejected electron, respectively. ra is given by

2s4f 3F4

2s4f3 F4

2p 4d3F4

2p4f IG4

2p4f 3G 4

2p4f 3F4

2p4f 3G 5

Designation LSI

1.10 1.28 8.55

1.24 1.50 9.20

1.29 X 10- 2 1.54 X 10- 2 1.38 x 10- 2

10- 1 10- 1 10- 1

10- 2

1.62 2.10 1.12

5.43 x 6.85 X 10- 2 3.47 X 10- 2

1.91 x 10- 3 2.11 X 10- 3 2.71 x 10- 3

a

a

c

b

a

c

b

c

b

C

X

X

X

10- 2

X 10- 1 X 10- 1

X

X

10- 1 10- 1 10- 2

4.17 X 10-2 4.43 x 10-2 3.18 X 10-2

3.88 x 10-2 4.47 x 10- 2 2.65 X 10- 2

a

b

C

X

1.39 X 10- 2 4.17 x 10- 3 9.55 x 10- 3

1.94 x 10- 2 9.09 X 10- 3 1.17 X 10- 2

b

a

c

X

1.72 x 10-2 1.42 X 10-2 1.39 x 10- 2

b

a

C

b

5.24 x 10- 3 8.93 X 10- 3 4.89 x 10- 3

Z = 14

2.58 X 10- 2 2.95 x 10- 2 1.83 X 10-2

= 10

2.25 x 10- 2 2.95 x 10- 2 1.39 X 10- 2

a

Z

X

X

X

10-2 10-2 10- 2

4.37 5.04 4.30

2.26 2.62 1.80

X

X

X

X

10- 2 10- 2 10- 2

10- 1

X 10- 1 X 10- 1

1.66 x 10- 2 1.65 X 10- 2 1.50 X 10- 2

4.27 4.31 3.49

1.12 X 10- 2 2.53 x 10-3 8.54 x 10-3

2.34 X 10-2 1.71 x 10- 2 1.93 x 10- 2

2.74 X 10- 2 2.95 X 10- 2 2.09 x 10- 2

Z = 18

6.99 7.77 6.71

2.48 2.78 2.06

X

X

X

X

10- 2 10- 2 10- 2

10- 1

X 10- 1 X 10- 1

9.71 x 10-4 7.70 X 10-4 1.18 x 10- 3

4.38 x 10- 2 4.27 X 10- 2 3.69 X 10- 2

1.04 x 10- 2 1.45 X 10- 3 8.37 x 10- 3

2.67 X 10- 2 1.86 x 10- 2 2.23 X 10- 2

2.88 X 10- 2 2.95 X 10- 2 2.26 x 10-2

Z =22

X

X

X

X

X

X

10- 2 10-2 10- 2

10- 2 10- 2 10-2

8.29 9.05 7.92

2.53 2.78 2.15

X

X

X

X

10-2 10- 2 10- 2

10- 1

X 10- 1 X 10- 1

1.97 X 10- 5 6.11 X 10- 5 7.50 x 10-7

4.41 x 10- 2 4.28 x 10- 2 3.79 X 10- 2

1.05 x 10- 2 3.09 X 10-4 8.74 X 10-3

2.85 1.94 2.41

2.95 2.95 2.36

Z =26

X

X

X

10- 2 10- 2 10- 2

8.99 9.68 8.56

2.56 2.78 2.20

4.30 5.70 2.22

10-4 10-4 10-4

X

X

X

X

10- 2 10- 2 10- 2

10- 1

X 10- 1 X 10- 1

X

X

X

4.38 x 10- 2 4.29 x 10- 2 3.79 X 10- 2

1.15 x 10- 2 2.68 X 10-5 9.95 X 10- 3

2.98 1.99 2.53

3.02 x 10- 2 2.95 X 10- 2 2.42 X 10-2

Z =30

Table 3.19. Comparison of autoionization probabilities (1013 s-l) for 2141 levels, (a) AUTOLSI, (b) MZ, (c) MCDP [3.56]

X

X

X

10-2 10- 2 10- 2

9.58 1.02 9.05

2.59 2.77 2.25

X

X

X

X

10- 2 10- 1 10- 2

10- 1

X 10- 1 X 10- 1

1.06 X 10- 3 1.24 x 10- 3 6.94 X 10-4

3.98 x 10- 2 3.98 x 10- 2 3.39 X 10- 2

1.62 x 10- 2 2.58 X 10-3 1.52 X 10-2

3.09 2.03 2.64

3.09 x 10- 2 2.95 X 10- 2 2.46 x 10-2

Z = 36

9.91 1.04 9.29

2.61 2.77 2.30

X

X

X

X

10-2 10- 1 10- 2

10- 1

X 10- 1 X 10- 1

1.45 X 10- 3 1.65 x 10- 3 1.02 X 10- 3

3.16 x 10- 2 3.20 X 10-2 2.55 X 10- 2

2.50 x 10- 2 1.02 X 10- 2 2.36 X 10- 2

3.16 X 10- 2 2.05 x 10-2 2.64 X 10-2

3.14 X 10-2 2.95 X 10- 2 2.41 x 10- 2

Z =42

V.

0-..l

'"

[ g:

~0-

=

0

a.

~.

~.

> c

w

2p 4d3D3

2p4d 3i'"3

2p4j 3D3

2p4j 3G3

2p4j 3 F3

2p4j IF3

2s4d3D3

1.28 x 10- 3 4.52 x 10-4 3.52 x 10-4 8.09 x 10- 2 9.17 x 10- 2 7.40 x 10- 2

2.15 x 10- 1 2.61 x 10- 1 1.98 x 10- 1 7.60 x 10- 2 8.74 x 10-2 6.23 x 10-2

4.68 x 10- 2 1.92 x 10- 3 1.17 x 10- 2

1.03 x 10- 2 1.02 x 10- 2 2.31 x 10- 2

1.12 x 10- 2 1.09 x 10- 5 7.58 x 10- 3

1.57 x 10- 3 1.53 x 10- 3 1.38 x 10- 3

1.88 X 10- 1 2.31 x 10- 1 1.53 x 10- 1

4.68 x 10-2 6.24 x 10- 2 3.77 x 10- 2

3.36 x 10- 3 2.00 X 10- 3 4.65 X 10- 3

4.33 x 10- 3 2.49 X 10-4 4.90 x 10- 3

1.71 x 10- 2 9.91 x 10-4 9.64 x 10- 3

4.34 x 10- 3 4.33 X 10- 3 4.35 x 10- 3

4.33 x 10- 1 2.27 x 10- 1 1.22 X 10- 1

5.63 x 10- 3 1.33 X 10-2 7.79 X 10-3

a b

a b

a b

a b

a b

a b

C

C

C

C

C

C

C

2.81 x 10-4 2.54 x 10-4 1.63 x 10-4

9.01 x 10- 3 1.63 x 10-4 6.82 x 10- 3

1.78 X 10- 2 1.15 x 10- 2 1.44 x 10- 2

9.67 X 10- 2 1.16 x 10- 1 7.28 x 10- 2

1.04 x 10- 1 1.34 X 10- 1 8.94 X 10- 2

a b

9.49 x 10- 2 1.03 x 10- 1 7.25 x 10- 2

2.56 x 10- 1 3.04 X 10- 1 2.54 X 10- 1

1.52 X 10-5 8.09 x 10-6 4.82 x 10-7

3.37 x 10-3 8.27 x 10-4 6.63 x 10- 3

5.74 x 10- 2 6.36 x 10-2 5.21 x 10- 2

X

X

X

10- 1 10- 1 10- 1 1.05 x 10- 1 1.08 X 10- 1 7.08 X 10- 2

2.91 3.41 3.05

2.74 x 10-5 1.74 X 10-5 5.90 x 10- 5

8.07 x 10- 3 6.89 x 10-4 6.60 X 10- 3

4.56 x 10- 2 5.00 x 10- 2 4.15 X 10- 2

2.44 x 10- 3 2.34 x 10-3 3.42 X 10- 3

1.77 x 10- 3 1.62 X 10- 3 2.08 X 10- 3

X

X

2.11 X 10- 2 1.35 x 10-2 1.79 X 10-2

X

10- 2 10- 2 10- 2

2.01 1.26 1.65

Z =26

10- 1 10- 1 10- 1 1.12 x 10- 1 1.09 X 10- 1 6.32 X 10-2

X

X

X

1.22 1.08 4.84

X

X

X

10- 1 10- 1 10- 2

3.42 X 10- 1 3.52 x 10- 1 3.92 X 10- 1

2.60 x 10-4 1.17 x 10-4 2.58 x 10-4

1.10 x 10-4 6.41 x 10- 5 1.45 X 10-4 3.17 3.48 3.46

7.85 x 10- 3 3.50 x 10- 3 6.61 X 10- 3

3.63 x 10- 2 3.90 X 10-2 3.35 X 10-2

2.21 x 10- 3 2.02 x 10- 3 4.46 x 10- 3

2.26 X 10- 2 1.70 x 10- 2 1.99 X 10- 2

Z =36

7.96 x 10-3 3.05 x 10- 3 6.61 X 10- 3

4.01 x 10- 2 4.36 x 10-2 3.68 X 10- 2

2.51 x 10- 3 2.39 x 10- 3 4.09 X 10- 3

2.19 X 10- 2 1.51 x 10- 2 1.89 x 10-2

Z =30

1.34 1.08 3.50

X

X

X

10- 1 10- 1 10- 2

3.61 X 10- 1 3.93 x 10- 1 4.28 X 10- 1

2.30 x 10- 2 1.43 x 10-4 3.34 X 10-4

3.94 x 10-4 5.60 x 10- 3 6.46 x 10-3

7.76 x 10- 3 3.63 x 10- 2 3.18 X 10-2

3.46 x 10-2 1.47 x 10- 3 4.39 x 10-3

1.76 x 10- 3 1.90 x 10- 2 2.04 x 10- 2

Z =42

...,

Z = 14

Z = 10

Z =22

Designation LSI

Z = 18

::;

'"

...&:

~

~

:;

'"g:

~

:;l

00

Table 3.19. Continued

2s4d 3 D2

2p4p 3P2

2p4p ID2

2p4p 3 D2

2p4d 1 F3

2s4/ 1 F3

2s4/ 3 F3

LSJ

Designation

1.94 X 10- 1 2.20 x 10- 1 2.25 x 10- 1

1.89 1.90 1.82

2.51 2.64 2.94

2.56 x 10- 2 2.61 x 10- 2 4.00 X 10- 2

2.52 2.75 2.44

1.04 x 10- 1 1.40 X 10- 1 2.58 X 10- 1

1.13 x 10- 1 1.40 X 10- 1 1.11 x 10- 1

a b

a b

a b

c

C

c

c

X

X

X

10- 1 10- 1 10- 1

10- 1

3.35 x 10- 2 4.11 x 10- 2 3.77 x 10- 2

X

a b

C

10- 1

1.03 1.15 9.27

1.07 1.24 9.57

1.03 1.28 9.04

a b

C

X

8.50 x 10- 2 1.08 x 10- 1 7.69 x 10- 2

5.14 x 10- 2 6.42 x 10- 2 4.61 x 10-2

8.10 x 10-4 4.58 x 10- 2 1.37 x 10- 5

a b

C

X

X

X

10- 1 10- 1 10- 1

2.09 x 10- 3 2.00 x 10- 3 6.70 X 10- 3

1.25 1.04 1.41 2.65 x 10- 3 2.99 x 10- 3 9.31 x 10- 3

4.43 x 10- 2 5.80 x 10- 2 2.80 x 10-2

6.60 x 10- 1 6.82 x 10- 1 5.92 X 10- 1

1.65 x 10- 1 1.06 9.91 x 10- 1

3.99 x 10- 3 4.80 x 10- 3 4.95 x 10- 2

1.49 x 10- 1 1.65 x 10- 1 5.63 x 10- 2

4.95 x 10- 1 5.14 x 10- 1 4.19 X 10- 1

1.01 1.08 1.12

8.92 X 10- 1 9.57 x 10- 1 8.09 x 10- 1

9.61 x 10- 1 1.04 8.70 x 10- 1 7.88 x 10- 1 8.51 x 10- 1 8.70 x 10- 1

10- 1

1.63 x 10- 1 1.91 x 10- 1 1.53 X 10- 1

1.39 X 10- 2 7.40 x 10- 3 1.97 x 10- 3

Z =26

2.56 x 10- 1 1.50 x 10- 1 1.14 x 10- 1

7.29 x 10- 3 2.76 x 10- 3 4.67 x 10-4

Z =22

4.94 x 10- 1 3.43 x 10- 1 5.53 x 10- 1

X

2.45 x 10- 3 1.65 x 10- 3 4.80 x 10-6

7.54 x 10-4 5.49 x 10- 3 2.05 x 10-4

4.57 x 10- 2 3.76 x 10- 2 3.08 x 10- 2

a b

Z = 18

Z= 14

Z = 10

Table 3.19. Continued

X

X

X

10- 1 10- 1 10- 1

5.77 x 10- 3 7.32 x 10- 3 1.63 x 10- 1

2.47 x 10- 1 2.51 X 10- 1 1.02 x 10- 2

4.01 4.17 3.19

1.74 1.26 1.32

8.31 X 10- 1 8.80 x 10- 1 7.48 x 10- 1

2.06 x 10- 1 2.35 x 10- 1 1.96 x 10- 1

1.86 X 10- 2 1.17 x 10-2 3.45 x 10- 3

Z=30

X

X

X

10- 1 10- 1 10- 1

6.72 x 10- 3 1.02 x 10- 1 2.09 X 10- 1

3.15 x 10- 1 9.67 x 10- 3 4.52 x 10- 3

3.07 3.19 2.20

1.38 1.46 1.56

7.39 X 10- 1 7.70 x 10- 1 6.54 x 10- 1

2.81 x 10- 1 3.97 x 10- 1 2.73 x 10- 1

1.99 x 10- 2 1.35 x 10- 2 4.21 x 10- 3

Z = 36

X

X

X

10- 1 10- 1 10- 1

5.41 x 10- 3 2.93 X 10- 1 2.02 X 10- 1

3.17 x 10- 1 9.35 x 10- 3 2.90 x 10- 3

2.38 2.52 1.59

1.55 1.69 1.75

1.71 X 10- 2 6.75 x 10- 1 5.70 x 10- 1

3.53 X 10- 1 4.16 X 10- 1 3.58 x 10- 1

1.71 x 10- 2 1.17 x 10- 2 3.74 x 10- 3

Z =42

V.

~

''""

a:

~

g;

0 ::I

10

N' ::t.

::I

o·

0

> r::

....

2p4d 3 F2

2p4s 3 P2

2s4p 3 p"

2p4J ID2

2s4d ID2

2p4J 3D2

2p4J 3F2

LSJ

Designation

2.97 x 10- 2 3.13 X 10-2 3.53 x 10-2 2.32 2.48 2.47

5.63 X 10-2 8.18 x 10- 2 5.09 X 10- 2 5.54 X 10-2 5.51 x 10-2 6.07 x 10-2 2.42 x 10- 1 2.64 X 10- 1 2.55 X 10- 1

2.57 x 10- 1 1.31 x 10- 1 1.53 x 10- 1

1.36 x 10- 1 1.06 x 10- 1 1.18 X 10- 1

2.54 x 10- 1 2.82 X 10- 1 2.59 X 10- 1

2.38 X 10-2 2.74 x 10-2 1.75 x 10- 2

1.46 X 10- 1 1.75 x 10- 1 1.12 x 10- 1

1.80 x 10- 1 2.20 x 10- 1 1.48

1.11 2.06 x 10- 1 3.39 x 10- 1

2.44 x 10- 1 2.84 X 10- 1 2.31 X 10- 1

5.28 x 10-4 1.95 x 10- 3 1.44 x 10- 2

1.59 x 10- 1 2.10 x 10- 1 1.12 x 10- 1

a b

a b

a b

a b

a b

10- 1 10- 1 10- 1

3.39 4.12 2.88

C

C

C

C

C

c

1.91 X 10-2 2.08 x 10-2 1.61 x 10-2

1.04 1.91 X 10- 1 1.38 X 10- 1

4.33 4.75 3.86

6.77 x 10-3 7.80 x 10- 3 1.31 x 10-2

1.13 X 10-2 1.24 x 10- 2 1.54 x 10-2

4.55 x 10-3 5.59 x 10- 3 1.18 X 10- 2

1.76 x 10- 1 1.93 X 10- 1 1.62 X 10- 1

1.73 x 10- 1 1.91 X 10- 1 1.55 X 10- 1

2.77 x 10-3 3.89 x 10-3 1.03 x 10-2

1.81 x 10- 1 1.95 x 10- 1 1.72 x 10- 1

X

10- 1 10- 1 10- 1

1.70 x 10- 1 1.91 X 10- 1 1.48 X 10- 1

X

X

X

2.20 X 10- 1 2.32 X 10- 1 2.44 x 10- 1

2.24 X 10- 1 2.38 X 10- 1 2.43 x 10- 1

10- 1 10- 1 10- 1

X

2.17 2.27 2.52

5.60 x 10-3 6.14 X 10-3 7.34 x 10-3

1.09 x 10-2 1.14 x 10-2 1.39 x 10- 2

1.75 x 10- 2 1.89 x 10-2 2.17 x 10-2

X

4.56 x 10-3 4.77 x 10-3 4.89 x 10-3

9.11 X 10- 3 9.71 x 10-3 9.76 x 10- 3

1.49 x 10-2 1.59 x 10- 2 1.57 x 10-2

2.61 x 10-2 3.74 x 10- 2 2.64 X 10-2

1.82 1.90 1.78

Z=36

2.67 2.77 2.47

1.55 1.66 1.52

Z=30

3.05 3.16 2.82

3.49 3.63 3.20

1.22 1.34 1.19

Z=26

4.11 4.31 3.74

X

X

X

1.52 x 10- 1 3.27 4.95 x 10- 1

6.77 8.13 6.52

a b

c

1.38 X 10- 1 2.09 x 10- 1 1.19 x 10- 1

2.84 X 10-3 2.46 x 10-3 8.45 x 10-3

1.67 x 10-3 1.07 x 10-3 4.38 x 10-3

Z =22

a b

Z = 18

Z = 14

Z = 10

18ble 3.19. Continued

X

X

X

10- 1 10- 1 10- 1

2.15 X 10-3 3.46 x 10- 3 8.71 x 10-3

1.85 X 10- 1 1.98 x 10- 1 1.82 x 10- 1

2.16 2.25 2.62

2.68 x 10-3 3.20 x 10-3 3.94 X 10-3

2.24 x 10-3 2.39 X 10-3 2.49 X 10-3

2.48 2.56 2.27

1.95 2.02 1.90

Z=42

~:

=:

I

g

fa.

ia.

...,

0

-

2p4p

Ipi

2p4p 3 DI

2s4s 3 SI

2p4d 3 p2

2s4f 3F2

2p4d 3D 2

2p4d ID2

LSI

Designation

9.23 x 10-4 5.33 X 10- 2 4.22 X 10- 2

7.39 9.23 2.45

4.70 5.53 6.04

1.78 5.44 6.66

2.31 2.70 2.56

4.02 x 10-2 5.02 X 10- 2 2.48 X 10- 2

2.48 x 10- 5 4.02 X 10- 5 2.91 x 10- 3

10- 2 10- 2 10- 2

4.56 5.79 6.24

4.96 x 10- 5 7.05 X 10-6 1.78 X 10-4

3.33 x 10-4 4.01 X 10-4 6.22 x 10-4

b

c

b

a

C

b

a

c

b

a

c

b

a

C

a

C

X

X

X

9.27 1.33 1.05

4.64 x 10-4 5.81 X 10-4 1.54 X 10- 3

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

10- 3 10- 3 10- 3

10- 5 10- 5 10- 5

10- 2 10- 2 10- 2

10-4 10-4 10-4

10- 3 10- 2 10- 3

10- 2 10- 2 10- 2

b

a

C

X

X

1.22 1.33 1.01

1.63 x 10- 2 1.81 X 10- 2 1.23 X 10- 2

b

z= 14

Z = 10

a

Table 3.19. Continued

6.43 6.93 4.37

2.07 3.64 3.16

4.55 5.16 5.74

3.93 4.62 2.19

4.62 5.33 4.11

2.56 2.82 1.63

2.19 2.77 4.93

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

z= 18

10- 3 10- 3 10- 3

10-4 10-4 10-4

10- 2 10- 2 10- 2

10- 3 10- 3 10- 3

10- 3 10- 2 10- 2

10- 2 10- 2 10- 2

10-3 10- 3 10-4

9.86 9.71 4.85

8.38 1.16 1.18

4.30 4.75 5.39

7.64 8.59 5.77

8.59 5.62 4.42

2.05 2.12 1.45

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

10- 3 10- 3 10- 3

10-4 10- 3 10- 3

10- 2 10- 2 10- 2

10- 3 10- 3 10- 3

10- 3 10- 2 10- 2

10- 2 10- 2 10- 2

2.55 x 10- 2 2.73 X 10- 2 1.71 X 10- 2

Z = 22

X

X

X

X

X

X

X

X

X

X

X

X

10- 2 10- 2 10- 2

10- 2 10- 2 10- 3

10- 2 10- 2 10- 2

10- 2 10- 2 10- 2

1.26 1.15 4.89

X

X

X

10- 2 10- 2 10- 3

2.11 X 10- 3 2.60 x 10- 3 2.83 X 10- 3

4.03 4.38 5.06

1.00 1.10 8.35

1.10 5.80 4.74

3.10 3.20 2.63

2.57 x 10- 2 2.66 x 10- 2 1.68 X 10- 2

Z =26

X

X

X

X

X

X

X

X

X

X

X

X

10- 2 10- 2 10- 2

10- 2 10- 2 10- 3

10- 2 10- 2 10- 2

10- 2 10-2 10- 2

1.47 1.26 4.47

X

X

X

10- 2 10- 2 10- 3

4.03 x 10- 3 4.62 x 10- 3 5.23 X 10- 3

3.80 4.07 4.80

1.13 1.22 9.65

1.22 5.94 5.03

3.76 3.85 3.50

2.45 X 10- 2 2.46 x 10- 2 1.54 X 10- 2

Z =30 X

X

X

X

X

X

X

X

X

10- 2 10- 2 10- 2

10- 2 10- 2 10- 2

10- 2 10- 2 10-2

X

X

X

10- 2 10- 2 10- 2

1.69 1.82 3.68

X

X

X

10-2 10- 2 10- 3

7.50 x 10- 3 8.10 X 10- 3 9.74 X 10- 3

3.51 3.72 4.52

1.25 X 10- 2 1.33 x 10- 2 1.05 X 10-2

1.33 6.03 5.39

4.35 4.43 4.51

2.17 2.03 1.26

Z=36

X

X

X

X

X

X

X

X

X

10- 2 10- 2 10- 2

10- 2 10- 2 10- 2

10- 2 10- 2 10-2

1.85 1.38 3.09

1.05 1.13 1.53

3.30 3.47 4.27

X

X

X

X

X

X

X

X

X

10- 2 10- 2 10- 3

10- 2 10- 2 10-2

10- 2 10- 2 10- 2

1.40 x 10- 2 1.48 x 10- 2 1.10 X 10- 2

1.48 5.97 5.72

4.69 4.76 5.34

1.86 1.71 1.00

Z =42

'"

g:

g

~

=

~:

=

~.

> c

V.

w

2p4s 3 PI

2s4p 3 PI

2s4p I PI

2p4f 3 PI

2p4p 3 S 1

2s4d 3 DI

2p4p 3 PI

LSI

Designation

c

b

a

C

b

a

C

b

a

C

b

a

C

b

a

c

b

a

C

b

a

Table 3.19. Continued

3.64 X 10-1 4.13 X 10-1 4.37x 10- 1

10-1 10-1 10-1

2.86 X 10-1 3.24 X 10-1 3.10 x 10-1

9.22 x 10- 2 1.22 X 10-1 1.17 X 10-1

2.41 x 10- 1 2.78 X 10- 1 2.25 X 10-1

1.75 x 10- 2 3.04 X 10- 2 4.75 x 10- 2

4.23 3.03 4.73

8.12 8.85 6.96

6.68 X 10-2 7.86 X 10- 2 6.94 x 10-2

X

X

X

X

X

X

X

10-2 10- 2 10-2

10-4 10-4 10-4

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

10- 2 10-2 10- 2

10- 5 10-5 10-6

10- 2 10- 2 10- 3

10- 2 10- 2 10- 2

10- 1 10- 1 10-1

1.02 1.12 1.09

4.54 X 10-1 5.05 X 10- 1 5.57 x 10- 1

7.17 7.36 5.58

1.97 1.16 1.77

1.37 1.58 8.75

10-3 10-2 10- 3

4.17 x 10- 2 5.48 X 10- 2 5.11 x 10- 2

X X

1.04 1.15 1.14

2.76 2.51 1.16

X

X

X

X

X

1.23 x 10- 3 1.19 X 10- 3 8.89 X 10-4

X

X

X

X

X

1.08 1.21 1.10

Z =22

10-2 10- 2 10-2

10-1 10-2

3.28 x 10- 3 3.26 X 10- 3 2.87 X 10- 3

X

X

X

9.94 1.20 4.34

9.89 1.14 9.54

10- 3 10- 3 10-4

6.71 8.59 9.11

4.21 x 10- 3 6.16 X 10- 3 1.11 X 10-4

X

10- 2 10-2 2.00 2.18 2.18

7.46 7.90 8.42

1.08 x 10- 1 1.37 X 10- 1 9.99 X 10- 2

X

X

X 10-2

Z = 18

10- 2 10- 2 10-2

2.56 6.01 2.76

1.44 x 2.61 X 10- 3 7.76 X 10-4

X 10-2

Z = 14

10- 3

Z = 10

1.49 1.58 1.55

5.38 5.90 6.71

3.10 5.00 4.46

1.02 1.83 7.07

4.56 1.87 1.22

7.77 8.53 9.93

1.10 1.22 1.16

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

Z =26

6.27 6.80 7.96

10- 1 10- 1 10- 1

1.74 1.82 1.80

3.30 3.15 4.82

6.34 8.13 1.57

1.76 1.94 1.34

8.03 8.73 1.18

1.09 1.21 1.19

10-2 10- 2 10-2

10- 5 10- 5 10-5

10- 2 10- 2 10- 2

10- 3 10- 3 10- 3

10- 1 10- 1 10-1

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

10- 2 10- 2 10-2

10- 3 10- 3 10- 2

10- 1 10- 1 10- 1

10-1 10- 1 10-1

10- 2 10-2 10- 2

10-5 10-5 10-4

Z =30

1.87 1.92 1.93

7.72 8.22 1.01

2.33 2.28 9.05

1.47 1.70 2.44

1.72 1.86 1.35

9.98 1.07 1.66

1.07 1.19 1.23

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

9.14 9.57 1.20

10- 1 10- 1

1.86 1.90 1.93

1.14 3.45 1.68

2.08 2.31 2.86

1.68 1.80 1.33

4.69 1.22 2.19

1.05 1.17 1.26

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

Z =42

10-2 10-2 10-2

10-4 10-4 10-4

10- 2 10- 2 10- 2

10- 3 10- 2 10- 2

10-1 10- 1 10-1

Z = 36

10-1 10-1

10-2 10- 2 10-1

10-4 10-4 10-4

10- 2 10-2 10- 2

10- 2 10- 2 10- 2

10-1 10-1 10-1

-

'"

g:

:::.:

~

'3"

~

= f!l. r:t. 0 =

~

w

N

2p4p l So

2p4p 3 Po

2s4s ISO

2p4d l pl

2p4d 3 PI

2p4s Ipi

2p4d 3 DI

Designation LSJ

1.26 x lO-1 1.39 x lO-1 1.62 x lO-1 9.04 x lO-1 9.67 x lO-1 8.25 X lO-1

1.08 x lO-1 1.20 x lO-1 1.20 x lO-1 5.81 X lO-1 6.22 x lO-1 5.42 x lO-1

6.27 6.89 5.33

5.30 x lO-2 6.19 x lO-2 5.58 x lO-2

3.41 x lO-1 3.67 x lO-1 3.24 x lO-1

6.18 7.lO 4.90

1.11 x lO-2 1.42 x lO-2 1.22 x lO-2

1.98 x lO-1 2.lO x lO-1 2.03 x lO-1

a b c

a b

a b c

C

C

6.11 6.57 5.45

5.87 6.22 5.44

1.72 x lO-3 1.56 x lO-3 2.28 x lO-3

3.20 x lO-2 2.46 x lO-2 8.09 X lO-3

5.48 x lO-2 1.21 x lO-1 6.56 X lO-2

3.33 x lO-1 2.51 x lO-1 1.41 x lO-1

a b

C

1.89 x lO-2 1.92 x lO-2 1.38 x lO-2

3.41 x lO-2 3.04 x lO-2 2.43 x lO-2

5.48 x lO-2 4.24 x lO-2 3.89 x lO-2

2.77 x lO-2 1.93 x lO-2 2.07 x lO-2

a b

2.51 2.60 2.37

2.98 3.15 2.86

3.04 3.21 2.97

3.03 3.23 3.08

a b c

C

3.58 x lO-3 2.07 x lO-3 3.84 x lO-3

1.04 X lO-1 1.08 x lO-1 1.07 x lO-1

2.43 x lO-1 4.22 x lO-1 2.81 x lO-1

8.25 x lO-2 4.59 x lO-1 1.19 x lO-1

a b

Z =22

Z = 18

Z= 14

Z = lO

Table 3.19. Continued

1.20 1.27 1.09

1.13 X lO-1 1.28 X lO-1 1.84 x lO-1

5.64 5.92 5.43

6.34 x lO-3 7.38 x lO-3 1.42 x lO-2

1.45 X lO-2 1.57 x lO-2 1.14 X lO-2

2.00 2.06 1.89

8.62 x lO-3 1.02 x lO-2 1.13 x lO-2

Z =26

1.44 1.51 1.29

9.38 x lO-2 1.11 X lO-1 2.lO x lO-1

5.48 5.71 5.45

1.67 1.73 1.49

7.07 x lO-2 9.06 x lO-2 2.79 x lO-1

5.32 5.50 5.60

2.36 x lO-2 2.42 x lO-2 3.29 x lO-2

1.39 x lO-2 1.48 x lO-2 1.28 x lO-2

1.37 X lO-2 1.48 x lO-2 1.15 x lO-2 1.52 x lO-2 1.62 x lO-2 2.47 x lO-2

1.43 1.48 1.40

2.77 x lO-3 3.26 x lO-2 5.50 x lO-2

Z =36

1.68 1.73 1.61

2.46 x lO-2 2.57 x lO-2 3.36 x lO-2

Z =30

1.81 1.86 1.62

5.40 X lO-2 8.56 x lO-2 4.07 x lO-1

5.23 5.38 5.85

2.71 x lO-2 2.73 x lO-2 3.50 x lO-2

1.44 x lO-2 1.51 x lO-2 1.42 x lO-2

1.32 1.36 1.31

5.62 x lO-2 2.96 x lO-2 7.03 x lO-2

Z =42

Vol

-

g. '"

~

1f

=

g.

= N'

S o·

:>

Vol

v.

2p4d 3 Po

2p4s 3 Po

2s4 p 3 Po

Designation LSI

2.71 4.04 6.16

7.93 9.80 2.77

1.62 x 10- 2 3.05 X 10- 2 5.50 x 10- 2

1.59 x 10-6 1.60 X 10-6 2.21 X 10- 3

a b

a b

C

c

C

2.31 2.45 2.23

2.41 x 2.56 X 10- 1 2.05 X 10- 1

X

X

X

X

X

X

X

X

10-4 10-4 10-6

10- 2 10- 2 10- 2

10- 1 10- 1

X 10- 1

Z = 14

a b

Z = 10

10- 1

Table 3.19. Continued

3.94 4.59 1.74

3.56 4.80 7.14

2.24 2.34 2.37

X

X

X

X

X

X

X

X

10- 3 10- 3 10- 3

10- 2 10- 2 10- 2

10- 1 10- 1

X 10- 1

Z = 18

8.65 9.64 5.93

4.02 5.19 7.97

2.19 2.25 2.54

X

X

X

X

X

X

X

X

10- 3 10- 3 10- 3

10- 2 10- 2 10- 2

10- 1 10- 1

X 10- 1

Z =22

1.31 1.43 1.04

4.20 5.36 8.85

2.15 2.19 2.77

X

X

X

X

X

X

X

X

10- 2 10- 2 10- 2

10- 2 10- 2 10- 2

10- 1 10- 1

X 10- 1

Z =26

X

X

10- 1 10- 1

1.67 1.72 1.41 X

X

X

10- 2 10- 2 10- 2

4.25 x 10- 2 5.44 X 10- 2 1.00 X 10- 1

2.13 2.14 3.06

X 10- 1

Z = 30

2.02 2.29 1.84

X

X

X

10- 2 10- 2 10- 2

4.21 x 10- 2 5.53 X 10- 2 1.27 X 10- 1

2.13xlO- 1 2.10 X 10- 1 3.65 X 10- 1

Z =36

X

X

X

10- 1 10- 1 10- 1

2.24 X 10- 2 2.35 X 10- 2 2.12 x 10- 2

4.10 x 10- 2 5.64 x 10- 2 1.67 X 10- 1

2.13 2.07 4.37

Z =42

'"

e: co

I

=

g:

i'"

...,

~

10 14 16 18 20 22 23 24 25 26 27 28 29 30 36 42 48 54

z

101

10 1 10 1 10 1 10 1 10 1 101 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1

1.28 x 1.44 x 1.48 x 1.51 x 1.52 x 1.51 x 1.49 x 1.47 x 1.44 x 1.40 x 1.36 x 1.32 x 1.27 x 1.22 x 9.64 7.99 7.07 6.61

10- 3 10- 2 10- 2 10- 1 10- 1 10- 1

4.11 x 4.02 x 9.91 x 2.20 x 4.45 x 8.30 x 1.10 1.43 1.82 2.25 2.74 3.27 3.81 4.37 7.25 8.93 9.84 1.05 x

1s2PI/22P3/2(5/2)

1,2PI/2 2P3/2(5/2) 10 1 10 1 10 1 10 1 10 1 10 1 101 101 10 1 10 1 10 1 10 1 10 1 101 10 1 10 1 10 1 10 1

1.27 x 1.40 x 1.42 x 1.41 x 1.38 x 1.34 x 1.31 x 1.29 x 1.27 x 1.25 x 1.24 x 1.22 x 1.21 x 1.19 x 1.16 x 1.14 x 1.14 x 1.18 x

10-4 10-3 10- 3 10- 2 10- 2 10- 2 10- 2 10- 2 10- 2 10- 2 10- 2 10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 10- 2

2.00 x 2.53 x 6.14 x 1.28 x 2.32 x 3.80 x 4.71 x 5.73 x 6.81 x 7.94 x 9.15 x 1.04 x 1.16 x 1.27 x 1.78 x 1.73 x 1.20 x 4.94 x

1,2PI/2 2P3/2(3/2)

1,2PI/22P3/2(3/2) 3.77 x 10- 2 2.94 x 10- 1 6.14 x 10- 1 1.09 1.69 2.34 2.65 2.95 2.23 3.47 3.70 3.90 4.07 4.23 4.79 4.90 4.83 4.74

1,2P3/22P3/2(3/2) 1.13 x 1.25 x 1.29 x 1.32 x 1.35 x 1.38 x 1.39 x 1.40 x 1.41 x 1.42 x 1.43 x 1.44 x 1.45 x 1.45 x 1.51 x 1.60 x 1.70 x 1.81 x

10 1 10 1 10 1 10 1 10 1 10 1 101 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1

1,2'1/22s 1/2(1/2) 10- 4 10- 3 10- 3 10- 2 10- 2 10- 2 10- 2 10- 2 10- 2 10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 10- 2 3.91 x 3.95 x 9.08 x 1.82 x 3.28 x 5.29 x 6.46 x 7.69 x 8.96 x 1.02 x 1.14 x 1.24 x 1.33 x 1.40 x 1.44 x 1.21 x 1.02 x 9.46 x

10-5 10-4 10- 4 10- 3 10- 3 10- 3 10-3 10- 2 10- 2 10- 2 10- 2 10- 2 10- 2 10- 2 10- 1 10- 1 10- 1 10- 1

3.77 x 2.27 x 5.64 x 1.33 x 3.10 x 6.67 x 9.46 x 1.33 x 1.82 x 2.47 x 3.25 x 4.22 x 5.37 x 6.66 x 1.68 x 2.72 x 3.47 x 3.88 x

1,2PI /22P3/2 (1 /2)

1,2PI/2 2PI/2(1/2) 1.77 2.07 2.18 2.29 2.38 2.47 2.52 2.55 2.59 2.64 2.68 2.72 2.76 2.80 3.02 3.24 3.38 3.38

1,2P3/22P3/2 (1 /2)

Table 3.20. Autoionization probabilities r(lol3 s -l) for transitions from Is2121' levels (designation in }}-coupling scheme) to the Is 22p3/2 level [3.57]

VI

--

'"

=:

Ig:

i:

6~.

w

v.

116

3 Transition Probabilities

(3.5.3) Here A(a, ao) is the decay amplitude for the states a, ao and C J (aLS, alLIS.) are eigenvectors obtained by diagonalization of the energy matrix. In the particular case of two- and three-electron states (21'31" and Is21'31") only one, the final state, is possible and therefore 10 = L. The amplitudes ALS(a, ao) are then given by A LS (nllln212) = N

(II0

12

0

L) 0

[Rll (11 12)

+ (-1) s R12(1211)],

(3.5.4)

where N = (1 + ~nllln212)-1/2. The radial integrals Rx(1112) were calculated in the Coulomb approximation [2.48,49]. In the Multi-Configuration Dirac-Fock (MCDF) method [3.56], the continuum states are obtained by distorted-wave Dirac calculations. In the first step, a set of single-configuration Dirac orbitals are obtained in an averaged-configuration Hartree-Fock approximation. Using these orthonormal one-electron orbitals to construct the single-configuration wave functions, the Hamiltonian matrix is calculated including the Breit corrections to the Coulomb interactions between electrons. The AUTOLSJ method [3.53] determines a set of non-relativistic wave functions by diagonalization of the Hamiltonian using orbitals calculated in the scaled Thomas-Fermi-Dirac potential. As the second step, the method diagonalizes the Hamiltonian in the Breit-Pauli approximation. Tables 3.18-20 present autonization probabilities for two- and threeelectron systems. The Z-dependence of the quantities are shown (Table 3.19) for each calculational method (MZ, MCDF and AUTOLSJ).

3.6 Branching Ratios of Inner-Shell Vacancies Ionization of the k-th inner-shell electron of an ion Xz leads to creation of an ion Xz+I in the autoionizing state, which can decay by emission of an electron (autoionization decay) with the probability Bk(Z -+ Z

+ 2) = r /(r + A).

(3.6.1)

Table 3.21. Calculated branching-ratio coefficienls (3.6.2) for ionization of inner shells in oxygen-like ions [3.58:t, z is the spectroscopic symbol

z nl B(z B(z

~ ~

z+ 1) z+2)

2

3

4

5

Is 0.0

Is 0.0

Is 0.0

Is 0.0

Is 0.0

1.0

1.0

1.0

1.0

1.0

nl B(z B(z B(z B(z

+ 1) -+ z + 2) -+ z + 3) -+ z + 4)

-+ z

Is 0.05 0.37 0.0 0.58

2s 0.0 1.0 0.0

9

-

2p 0.0 1.0

Is 0.05 0.37 0.0 0.58

2s 0.0 1.0

10

2p 0.0 1.0

Is 0.04 0.37 0.0 0.59

-

2s 0.0 1.0 0.0

11

-

2p 0.0 1.0

Is 0.03 0.37 0.0 0.60

2s 0.0 1.0

12 2p 0.0 1.0

Is 0.02 0.37 0.0 0.61

2s 0.0 1.0

13

2p 0.0 1.0

Is 0.01 0.40 0.0 0.59

-

2s 0.0 1.0 0.0

14

-

2p 0.0 1.0

Is 0.0 0.37 0.0 0.63

Table 3.22. Calculated branching-ratio coefficients for ionization of inner shells in iron-like ions [3.58]

2s 0.0 1.0

15 2p 0.0 1.0

Is 0.35 0.65 0.0 0.61

16 2s 0.0 1.0

Is 0.37 0.63

17

Is 0.35 0.65

18 Is 0.38 0.62

19 Is 0.36 0.64

20 Is 0.36 0.64

21 Is 0.55 0.45

22

Is 0

23

t.H

-..J

-

"'

iii'

~

n

~

~

~Vl

:I

.....

i"'g,

~.

::r

n

§

Ol

0-

x x x x x

101 101 101 102 102

7.51 X 2.65 X 1.04 x 3.33 X 9.10 X 2.15 3.14 4.45 6.10 8.10 1.05 x 1.31 x 1.60 x 1.90 X 3.69 X 4.89 X 5.59 x 6.09 x

10-4 x 10- 3 x 10- 2 x 10- 2 X 10- 1 x 10- 1

1.29 4.95 2.09 7.44 2.30 6.31 1.00 1.54 2.31 3.37 4.77 6.58 8.85 1.16 4.01 9.27 1.76 2.99

10 14 16 18 20 22 23 24 25 26 27 28 29 30 36 42 48 54

X

Qd

A

z

10 1 10 1 10 1 101 101 10 1 10 1 10 1

10-4 10- 2 10- 1 10- 1 10- 1

Is2pl/2 2p3/2(5/2)

1.83 X 10- 1 6.59 x 10- 1 1.05 1.52 2.05 2.59 2.86 3.11 3.36 3.59 3.81 4.01 4.19 4.36 5.08 5.47 5.68 5.80

gW

4.11 X 1.79 3.16 5.18 7.97 1.16 x 1.37 x 1.61 x 1.86 x 2.12 x 2.40 x 2.69 x 2.98 x 3.29 x 5.43 x 8.51 x 1.31 x 1.98 x

A

101 101 10 1 10 1 10 1 10 1 101 10 1 10 1 10 1 102 102

10 1

10- 1 2.39 9.55 1.56 2.32 3.14 3.94 4.29 4.61 4.87 5.07 5.22 5.31 5.34 5.34 4.91 4.38 4.02 3.84

Qd

x 10 1 x 10 1 x 10 1 X 10 1 x 10 1 X 10 1 x 10 1 X 10 1 X 10 1 X 10 1 X 10 1 X 10 1 X 10 1 X 10 1 X 10 1 X 10 1

Is2plj22p3/2(5/2)

1.87 x 10- 1 6.64 x 10- 1 1.05 1.53 2.06 2.61 2.88 3.13 3.38 3.61 3.82 4.02 4.21 4.38 5.10 5.49 5.69 5.81

gW

x x x x

x x x x x x x x

10 1 10 1 10 1 102

2.75 5.67 1.63 3.81 7.53 1.31 1.66 2.06 2.48 2.94 3.42 3.91 4.39 4.87 6.93 6.73 4.58 1.81

1.05 3.23 1.21 3.77 1.01 2.41 3.57 5.17 7.32 1.01 1.38 1.85 2.43 3.15 1.16 3.13 6.98 1.38

10-4 10-3 10- 2 10- 2 10- 1 10- 1 10- 1 10- 1 10- 1

Qd

A X

X

X

X

X

x x x x x x

X

X

X

x x x x

X

10-4 10- 3 10- 2 10- 2 10- 2 10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 10- 1

Is2pl/2 2p3/2(3/2)

1.38 2.24 2.65 2.99 3.25 3.45 3.52 3.59 3.65 3.70 3.74 3.77 3.80 3.83 3.89 3.88 3.81 3.67

gW

Thble 3.23. Atomic characteristics Qd and gW for Is212l' levels (designation in jj-coupling scheme) [3.57], radiative probabilities A(1013 s-l) to the Is 22p3/2 level

g. '"

~

~ cr'

=

f!l.

g.

..::;l=

Vol

00

16 18 20 22 23 24 25 26 27 28 29 30 36 42 48 54

x x x x

X

10 1 10 1 10 1 102 102

1.61 1.82 4.56 3.06 4.62 1.45 2.10 2.79 3.50 4.17 4.82 5.41 5.94 6.41 8.30 9.01 9.18 9.15

10- 2 x 10- 2 x 10- 2 x 10- 2 X 10- 1 x 10- 1

X

4.16 5.16 1.42 1.09 2.01 8.03 1.33 2.03 2.94 4.07 5.43 7.04 8.91 1.11 3.03 6.18 1.07 1.63

10

14

Qd

A

X

X

X

X

X

10- 1 10- 1 10- 2 10- 2 10- 1

ls2Plj2 2P3j2(3/2)

Z

Table 3.23. Continued

1.27 1.30 3.21 2.16 3.35 1.09 1.60 2.16 2.75 3.34 3.90 4.43 4.93 5.38 7.18 7.91 8.02 7.77

gW

x x x x x x x x x x x x x

X

X

X

X

X

10-2 10-2 10- 3 10-3 10- 2 10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 1.10 4.86 8.66 1.43 X 10 1 2.20 X 101 3.22 X 10 1 3.83 X 10 1 4.52 X 101 5.29 X 101 6.15 X 101 7.10 x 10 1 8.16xlOl 9.33 X 101 1.06 X 102 2.14 X 102 3.88 X 102 6.57 X 102 1.05 X 103

A

1.25 X 10- 1 9.92 x 10- 1 2.10 3.81 6.02 8.47 9.67 1.09 X 10 1 1.20 X 10 1 1.30 X 101 1.39 X 10 1 1.48 X 10 1 1.55 X 101 1.62 X 10 1 1.87 X 10 1 1.93 X 10 1 1.92 X 101 1.89 X 10 1

Qd

ls2p3j2 2P3j2(3/2)

3.32 3.38 3.43 3.49 3.55 3.62 3.65 3.68 3.71 3.74 3.76 3.79 3.81 3.83 3.91 3.95 3.97 3.98

gW

6.15 2.38 3.94 5.99 8.49 1.13 1.28 1.43 1.58 1.73 1.87 2.01 2.13 2.24 2.61 2.49 2.06 1.57

10- 2 10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 10- 1

3.09 1.21 2.02 3.11 4.48 6.10 6.99 7.91 8.87 9.83 1.08 1.18 1.27 1.36 1.81 2.03 2.05 1.93

x x x x x x x x ~ x

Qd

A

X

X

X

X

X

10- 2 10- 1 10- 1 10- 1 10- 1

ls2sl/22sl/2 (l /2)

5.46 1.91 3.05 4.52 6.28 8.23 9.24 1.03 1.23 1.22 1.31 1.40 1.47 1.54 1.73 1.56 1.21 8.71

gW

x x x x x x x x x x x x x

X

X

X

X

X

10- 3 10- 2 10- 2 10- 2 10- 2 10- 2 10- 2 10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 10- 1

~

n

,,''"

~

n

~

~

::0-

CIl

"7

::0 ::0

.....,

'"0

g.

';I:J

OQ

::0

e:

::0

t:C OJ n

'"

...,

A

0 0 0 8.54 2.11 4.20 5.44 6.64 7.60 8.10 7.97 7.11 0 0 0 6.88 1.46 2.13

z

10 14 16 18 20 22 23 24 25 26 27 28 29 30 36 42 48 54

10-4 10- 3 10- 3 10- 3 10- 3 10- 3 10- 3 10- 3 10- 3

x 10-2 x 10-2 x 10-2

x x x x x x x x x

0 0 0 5.80 1.07 1.62 1.83 1.98 2.01 1.92 1.67 1.34 0 0 0 3.37 4.42 4.08

Qd

10-4 10-4 10-4 10-4 10-4 10-4 10-4

x 10-4 x 10-4 x 10-4

X

x x x x x

X

x 10- 5 x 10-4

Is2p1/2 2P1/2(1/2)

Table 3.23. Continued

0 0 0 4.37 3.44 2.42 1.94 1.49 1.10 7.76 5.15 3.17 0 0 0 1.24 1.27 1.05

gW

10-2 10-2 10-2 10-2 10- 3 10- 3 10- 3

10- 3 x 10- 3 x 10- 3

X

X

x x x x

X

X

x 10-2 x 10-2

x x

X

X

X

X

X

x x x x

X

10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 102 102 102

2.54 2.46 5.45 1.05 1.79 2.72 3.24 3.76 4.27 4.78 5.21 5.62 5.96 6.24 6.70 6.17 5.66 5.51

10- 1

4.19 1.72 2.93 4.59 6.74 9.39 1.09 1.26 1.44 1.65 1.87 2.12 2.39 2.70 5.41 1.03 1.84 3.06 X

Qd

A

x x

X

X

x x

X

X

X

x x

X

x x x x x

X

10-4 10- 3 10- 3 10-2 10-2 10-2 10-2 10-2 10-2 10-2 10-2 10-2 10-2 10-2 10-2 10-2 10-2 10-2

Is2p1/2 2 p3/2(1/2)

6.51 6.22 6.00 5.74 5.45 5.15 5.01 4.88 4.76 4.66 4.58 4.52 4.48 4.46 4.66 5.12 5.53 5.83

gW

X

X

X

X

x x

X

x

X

x x

X

x

X

X

x x

X

10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 10- 1

2.62 X 10- 1 1.23 2.30 4.03 6.73 1.07 X 10 1 1.33 X 101 1.64 X 101 2.00 X 10 1 2.41 X 10 1 2.88 x 10 1 3.41 X 101 4.01 X 101 4.68 X 101 1.03 x 102 1.94 X 102 3.32 X 102 5.31 X 10-2

A

4.31 1.36 1.95 2.55 3.13 3.67 3.93 4.15 4.36 4.57 4.74 4.90 5.06 5.20 5.85 6.36 6.69 6.72

Qd X

10- 1

Is2p3/2 2P3/2(1/2)

2.43 X 10- 1 6.58 x 10- 1 8.93 x 10- 1 1.12 1.32 1.49 1.56 1.63 1.68 1.73 1.77 1.80 1.83 1.86 1.94 1.97 1.98 1.99

gW

g. '"

~

rj

~

o· ::s

~.

.,::s::;l

w

0

N

-

3.6 Branching Ratios of Inner-Shell Vacancies

121

Table 3.24. Theoretical lifetimes r(lO-13 s) and branching ratio B(l02) for Is21(L' S')nl' LSI levels of ArI5+ [3.59]

Initial state

r

Is2s 22 SI/2 Is2p2s 4 PI/2 Is2p2s 4 P3/2 Is2p2s 4 P5/2 Is2p(1 P)2s 2 PI/2 Is2p(1 P)2s 2 P3/2 Is2p2 4 PI/2 Is2 p2 4 P3/2 Is2pe P)2s 2PI/2 Is2p2 4 P5/2 Is2pC P)2s 2 P3/2 Is2 p2 2D3/2 Is2 p2 2D5/2 Is2p2 2 PI/2 Is3d(1 D)2s 2D5/2 Is3sC S)2s 2SI/2 Is2p3s 4 PI/2 Is2p3s 4 P3/2 Is2p3s 4 P5/2 Is2p3p4DI/2 Is2pC P)3s 2 PI/2 Is2p3p 4D3/2 Is2pC P)3s 2 P3/2 Is2p3p 4 D5/2 Is2pC P)3 p 2 P3/2 Is2pC P)3p 2 PI/2 Is2p3p4D7/2 Is3p(3 P)2s 2 PI/2 Is3pC D)3d 2 D3/2 Is2pC P)3p 2 DS/2 Is2p2d 4 Ds/2 Is2p(1 P)3s 2 P3/2 Is2p(1 P)3s 2 PI/2 Is2pC P)3d 2 F5/2 Is2p3d 4 D7/2 Is2p3d 4 P5/2 Is2p 3d4P3/2 Is2p3d 4 PI/2 Is2pC P)3 p 2SI/2 Is2pC P)3d 2 F7/2 Is2pe P)3d 2 P3/2 Is2pe P)3d 2PI/2 Is2p(1 P)3p 2 PI/2 Is2p(1 P)3 p 2D3/2 Is4p(1 P)2s 2 PI/2 Is4p2s 4 DI/2 Is4 p2s 4 D3/2 Is4p2s 4 D5/2

0.077 69.336 23.967 6120.000 0.105 0.102 28.642 19.770 0.096 3.917 0.098 0.052 0.052 0.067 1.190 0.312 17.263 6.307 100.635 2.855 0.675 2.390 0.553 6.269 0.629 0.559 19.070 0.186 2.676 0.173 1.358 0.107 0.103 0.439 1.407 1.790 2.048 2.218 0.184 0.465 0.559 0.867 0.089 0.080 0.926 4.268 4.291 4.331

B

96.6 14.7 15.7 83.7 13.6 2.5 1.8 41.8 82.5 74.8 95.0 70.0 74.6 0 42.2 71.6 10.1 11.1 0 0.2 0.4 1.0 1.0 19.3 1.7 0.2 0 66.3 0 45.8 0.1 8.5 13.1 0.3 0.7 0.2 0.6 0.6 4.1 0.2 0 0 0 26.6 18.4 0 0 0

Initial state

r

Is2p2 2 P3/2 Is2p 22 SI/2 Is3s2s 4 S3/2 Is3s(1 S)2s 2SI/2 Is3p2s 4 PI/2 Is3p2s 4 P3/2 Is3p2s 4 P5/2 Is3p(1 P)2s 2 P3/2 Is3 p(1 P)2s 2 PI/2 Is3d2s 4 DI/2 Is3d2s 4 D3/2 Is3d2s 4 D5/2 Is3d2s 4 D7/2 Is3de D)2s 2D3/2 Is3 pC D)3d 2 D5/2 Is2p3p 4S3/2 Is2p3p4pl/2 Is2p3d 4 P3/2 Is3 pC P)2s 2P3/2 Is2p3p 4 P3/2 Is2p3d 4 F5/2 Is2p3p 4 P5/2 Is2p3d 4 F7/2 Is2pC P)3p 2D3/2 Is2p3d 4 F9/2 Is3dC D)2s 2 D3/2 Is3dC D)2s 2 DS/2 Is2p2d 4 DI/2 Is2p2d 4 D3/2 Is2p(1 P)3p 2DS/2 Is2p(1 P)3p 2 P3/2 Is2p(1 P)3d 2 D3/2 Is2p(1 P)3d 2 DS/2 Is2p(1 P)3p 2SI/2 Is2p(1 P)3d 2 F7/2 Is2p(1 P)3d 2 FS/2 Is2pe P)3d 2 PI/2 Is2pe P)3d 2P3/2 Is4s2s 4 S3/2 Is4s(1 S)2s 2SI/2 Is4p2s 4 PI/2 Is4 p2s 4 P3/2 Is4p2s 4 P5/2 Is4p(1 P)2s 2P3/2 Is4sC S)2s 2 SI/2 Is2p4p 4 DI/2 Is2pC P)4s 2PI/2 Is2p4p 4D3/2

0.065 0.147 41.415 0.240 16.488 4.356 86.423 0.548 0.545 1.531 1.546 1.575 1.618 1.089 2.132 1.483 4.388 4.284 0.215 4.569 16.290 2.728 9.924 0.172 5938.965 0.546 0.506 1.103 1.072 0.078 0.085 0.093 0.092 0.116 0.095 0.097 0.098 0.100 33.369 0.687 27.978 15.620 39.161 0.933 0.774 4.921 2.609 4.974

B

7.4 30.6 0 95.1 8.7 10.3 0 8.5 7.6 0 0 0 0 42.3 0.4 2.0 0.8 49.6 83.3 9.2 6.1 34.3 9.3 18.5 0 52.9 2.2 0.5 0.3 31.7 12.4 0 0.4 13.4 12.6 12.4 0 0 0 95.8 6.7 13.5 0 18.6 66.4 1.5 45.9 11.3

122

3 Transition Probabilities

Table 3.24. Continued Initial state

r

B

Initial state

r

B

Is4p2s 4 D7/2 Is4d(1 D)2s 2 DS/2 Is4d(1 D)2s 2 D3/2 Is4f2s 4 F3/2 Is4f 2s4FS/2 Is4f2s 4 F7/2 Is4f2s 4 F9/2 Is4f(1 F)2s 2 F7/2 Is4fe F)2s 2 FS/2 Is2p4s 4 PI/2 Is2p4s 4 P3/2 Is2p 4d4F3/2 Is2p4p4D7/2 Is2pC P)4s 2 P3/2 Is2p 4d4F7/2 Is2p4p 4 P3/2 Is2p4f 4GS/2 Is2p4p 4 Ps/2 Is2p4f 4G7/2 Is2p 4d4D3/2 Is2p4d 4 DI/2 Is2pC P)4p 2 D3/2 Is2p4d 4 PS/2 Is4dC D)2s 2 D3/2 Is4dC D)2s 2 DS/2 Is2pC P)4f 2 F7/2 Is2p(3 P)4f 2G7/2 Is2PC P)4d 2 P3/2 Is2p4f 4G9/2 Is2pC P)4f 2 DS/2 Is2p4f 4F7/2 Is2p4f 4Gll/2 Is2p4 f 4D 3/2 Is2PC P)4d 2 F7/2 Is2p4 f 4DS/2 Is2PC P)4f 2G9/2 Is2p4f 4 DI/2 Is2pC P)4 f 2 D3/2 Is2pC P)4d 2 PI/2 Is2p(1 P)4s 2 PI/2 Is2p(1 P)4s 2 P3/2

4.390 4.256 4.184 10.424 10.435 10.460 10.502 10.913 10.918 13.750 4.671 2.151 12.782 0.529 3.448 2.562 4.561 5.663 9.835 3.090 2.767 0.721 2.501 2.216 2.915 4.071 9.368 2.243 10.199 6.848 8.696 10.035 7.055 2.746 6.465 11.285 7.186 5.021 2.732 0.095 0.096

0 19.2 20.3 0 0 0 0 1.4 1.5 18.5 8.1 46.4 0 91.5 5.6 38.4 29.1 13.8 0.1 0 2.5 42.6 1.0 12.6 1.9 0 0 9.1 0 8.5 0 0 0.4 11.7 5.1 0.7 0 0.6 8.5 2.7 1.4

Is2p4s 4 PS/2 Is2p4p 4 DS/2 Is2p4p 4 S3/2 Is4pC P)2s 2 P3/2 Is2p4p 4 PI/2 Is2pC P)4d 2 D3/2 Is4pC P)2s 2 PI/2 Is2pC P)4p 2 PI/2 Is2pC P)4d 2 DS/2 Is2pC P)4p 2 P3/2 Is2p4d 4 FS/2 Is2p4f 4F9/2 Is2p4f 4F3/2 Is2pC P)4f 2 FS/2 Is2p4f 4 Fs/2 Is2p4f 4D7/2 Is2p 4d4F9/2 Is2pC P)4p 2 DS/2 Is2pC P)4d 2 FS/2 Is2p4d 4 D7/2 Is4fC F)2s 2 F7/2 Is4fC F)2s 2 FS/2 Is2p4d 4 DS/2 Is2p 4d4p3/2 Is2p4d 4 PI/2 Is2pC P)4p 2 SI/2 Is2pe P)4p 2 D3/2 Is2p(1 P)4p 2 PI/2 Is2p(1 P)4p 2 DS/2 Is2p(1 P)4p 2 P3/2 Is2p(1 P)4p 2SI/2 Is2p(1 P)4d 2 D3/2 Is2pe P)4d 2 DS/2 Is2pe P)4d 2 F7/2 Is2pe P)4f 2 FS/2 Is2p(1 P)4d 2 FS/2 Is2p(1 P)4f 2 F7/2 Is2pe P)4d 2 PI/2 Is2pe P)4d 2 P3/2 Is2p(1 P)4f 2G7/2 Is2pe P)4f 2 F9/2

43.090 3.484 1.727 1.489 1.806 4.844 0.411 2.058 2.988 0.715 5.030 4.581 5.575 8.140 2.065 3.689 6.261 0.431 2.430 4.063 5.696 5.376 4.253 4.356 5.166 0.653 0.086 0.092 0.086 0.091 0.093 0.093 0.093 0.088 0.095 0.090 0.094 0.071 0.072 0.094 0.094

0 29.7 10.8 18.4 0.7 3.5 72.8 5.0 0.2 39.2 3.6 0.5 11.3 4.5 29.3 0 0 62.7 3.7 2.4 1.9 0.7 0.8 0.8 0 23.1 9.1 0 7.7 1.7 4.0 0 0.7 6.0 0 5.8 0 24.4 24.5 0.4 0.4

3.6 Branching Ratios of Inner-Shell Vacancies

123

Here 1 and A are the total probabilities of autoionization and radiative decay, respectively. The coefficients Bk are termed the branching-ratio coefficients. They are defined in a similar way for the emission of more than one electron: Bk(Z --+ Z + k), k > 2. If an ion with a vacancy ("hole") makes a transition to the stable state by emission of a photon, i.e., without changing its charge, the branching ratio is Bk(Z --+ Z + 1)

=

A/(1

+ A).

(3.6.2)

The values of the branching-ratio coefficients (3.6.1,2), calculated in the LS-coupling scheme for oxygen and iron ions, are given in Tables 3.21, 22. A vacancy in the inner shell of an ion can also be created by direct innershell excitation by electron impact. In this case the more appropriate quantities are the dielectronic satellite factor Qd and the statistically weighted linefluorescence yield Y: Qd Y

= goA(O --+

1)1/(1 + A),

= gW = goA(O --+

1)/(1 + A),

(3.6.3)

where go is the statistical weight of the excited state, A(O --+ 1) is the radiative transition probability from the excited state to a lower one. In Table 3.23 we present A, Qd and gW for doubly excited Is2/2/ 1 states of three-electron systems for 10 ~ Z ~ 54. The branching ratios and autoionization probabilities for the states of the excited electron configurations, ls2snl and Is2pnl, of Li-like argon are presented in Table 3.24. The n-values change in the range from n = 2 to n = 4 with all allowed values of I: 1 = 0 to I=n-l.

4 Radiative Characteristics

In this chapter the radiative processes, involving free electrons in the initial or final state, are considered (photoionization, dielectronic recombination, bremsstrahlung), as well as the polarization of X-ray radiation of highly charged ions, induced by electron impact.

4.1 Radiative Recombination Radiative Recombination (RR) (or photorecombination) Xz+1 (al)

+ e ~ Xz(ao} + nw

(4.1.1)

consists in electron capture and radiation of a photon; here ao and al denote sets of the quantum numbers. Together with other recombination reactions of free electrons with ions (dielectronic recombination, three-body recombination), the RR processes influence the ionization balance of high-temperature plasmas, contribute significantly to the energy loss of the plasma and are a very important cooling mechanism in electron-ion interaction. RR is the primary recombination process in very low-temperature and low-density plasmas. At high electron temperatures its importance rapidly decreases. The RR cross sections and rates are investigated mostly theoretically [4.1-7], because these values are difficult to measure directly due to their smallness. So there are only a few cases that have been studied experimentally for photorecombination [4.8-12] and photoic;mization [4.13-15].

4.1.1 General Properties. Photoionization The inverse process to RR is photoionization Xz(ao}

+ nw ~

Xz+1 (al)

+ e(e, A},

(4.1.2)

which consists in photon absorption and ejection of a bound electron into the continuum. Here e and A are the energy and orbital momentum of the photoelectron, respectively. The cross sections of photoionization (Tv and RR process (Tr are related by the detailed-balance principle (the Miln formula):

e2

W g(T gz+1 u r - 2. 1372 e z v,

eo + e = ew

= nw/ Ry

V. G. Pal'chikov et al., Reference Data on Multicharged Ions © Spring-Verlag Berlin Heidelberg 1995

(4.1.3)

4.1 Radiative Recombination

125

where BO is the binding energy of the state ao, gz+1 and gz are the statistical weights of the ions Xz+I and Xz, respectively. The total probability of photoionization is given by

J 00

W =

CN"PV(W) dw [s-I],

o where N w is the density of photons at a given frequency w, c is the speed of light. The RR rate Kr , averaged over a Maxwellian energy distribution of the incident electrons with a temperature T, is given by Kr(ao) = K B~/2 f33/2

J 00

u a(u; e-/3u du, rrao

o

= BIBo,

= BolT,

= 2,.Jiinaolm = 2.17 x

10-8 cm3 s- 1 (4.1.4) The photoionization cross section is related to the imaginary part of the dipole dynamic polarizability f3(w) by relation [4.4] Im{f3(w)} = a v (w)/4rraw, (4.1.5) where a is the fine-structure constant. u

f3

K

4.1.2 The Kramers Formulas and the Gaunt Factor For ions, being in excited hydrogenic states n, the Kramers formulas [4.1] can be used: a

Kr

(n)

v Kr

a r (n)

=

64n

3J3 . 137. Z2

(WO)3 W

= 3yM3 . 13732w5n 3 w(w .

=

WO)

0.0899z

n2B~/2(u

= 2.39 x

2

+ 1)3

[rra ],

-6

10

(4.1.6)

0 Z 1/2

eo (u

+ l)u

2

[rrao],

(4.1.7) (4.1.8) where z is the spectroscopic symbol, BO is the binding energy of the state with the principal quantum number n. A comparison of the Kramers cross sections with the numerical calculations for H-like ions with n = 1 and 2 is shown in Fig. 4.1. One can see, that the curve a~(n = 2) is in between a v (2s) and a v (2p), because the Kramers formulas are valid for I-averaged cross sections (not for their sum): a(n) = n- 2 ~)21

+ 1)anl.

I

The Kramers cross sections are connected by relation Kr

a r (n) =

BO(U

+ 1)2n 2

Kr

2.137 2 . u a v (n)

in accordance with (4.1.3).

(4.1.9)

126

4 Radiative Characteristics Fig. 4.1. Scaled photoionization cross sections from nl states (n = I, 2) for H-like ions with z = 50 as a function of photoelectron energy £T . u Kr : the Kramers formula (4.1.6); Is, 2s and 2p: corresponding to uv(nl), calculated by the code ATOM [4.16]

[H]

...... ' ....... UK'

(n = 1)

"" UK'

0.01

"",

(n =2)

",, "

to

0.1

The RR rate coefficient, corresponding to (4.1.4,7), has the form K~(n)

= 2KlztJ 3/ 2 eIl IEi(-tJ)l, z2 Ry

13 = n 2 T'

KI

=

64J7Ta5c

3.J3. 1374

= 2.60 x

1O- 14 'cm

3 s-I,

(4.1.10)

where T is the electron temperature and Ei(x) is the exponential integral. According to (4.1.10), K~(n) ~ n- I for 13 » 1 and K~(n) ~ n- 3 for 13 « 1. The function Ei(x) is well fit by 0.562 + l.4X) eXIEi(-x)1 ~ In ( 1 + x(1 + l.4x) ,

x > O.

(4.1.11)

The Kramers formulas are very useful for the estimation of the contribution from the I-averaged H-like states, especially for the calculation of the total RR rate: n-I

K;

=L

00

LKr(nl)

n=no

+ LK~(n),

(4.1.12)

n=n

I
where Kr(nl) is obtained numerically from (4.1.4); the second sum gives the contribution from all excited states n above a certain level n. Using (4.1.10) and changing summation over n by integration, one has: 00

L K~(n) = K 1ntJJ/2z[ln(1.78 tJI) + elll (1 + tJIIn)IEi( -tJdl]' n=n

z2Ry

131 = n2 T' where

KI

is given in (4.1.10).

(4.1.13)

4.1 Radiative Recombination

127

The cross sections a r and a v for an arbitrary ion in the nl state can be written in the form Qv 2 Kr av(nl) = - - n GnWv (n), 21 + 1 ar(nl) = QrGnWrKr(n),

(4.1.14)

where the Q's are the angular coefficients [4.7]. The dimensionless quantity Gnl is termed the bound-free Gaunt factor. The analytical formula of the I-averaged Gaunt factor Gnl(n) = n- 2 ~)21

+ l)G nl

I

for H-like ions was obtained in [4.17]. The I-dependence in Gnl was investigated in [4.18]. Along with the Krarnres formulas for av,r(n) other hydrogenic formulas for specific nl-states can be used [4.19-21].

4.1.3 Theory and Experiment Photoionization and photorecombination cross sections are required for many applications, i.e., for the Opacity Project [4.22,23] - an international collaboration program, the aim of which is to determine the stellar envelope opacities with high precision. The number of experimental cross sections for multicharged ions are quite limited [4.13-15], because the measurement of corresponding cross sections is a very complicated problem. Therefore, one has to rely upon sophisticated computer calculations and recommended data [4.24-28]. Calculations of a v of complex ions [4.29-31] showed the significant importance of Photo-Excitation of the Core electrons (PEC). PEC processes lead to the appearance of large resonances followed by autoionization, and can strongly modify the photoionization cross section over a wide range of frequencies (Fig. 4.2). PEC resonances are especially important for photoionization from excited states of ions with not very high ion charge [4.29,32]. log a 1mb)

1po.

I I

Fig. 4.2. Calculated photoionization cross sections from the 4J state for He II (broken curve) and C II (full curve) ions [4.29]. 3 pO and 1 pO are positions of thresholds in C III ions. PEC resonances 2s24J -2s2p4J followed by autoionization, strongly modify the C II (4f) cross section

I

~u ~"

-0.6

-0..4

-0..2

0.

Q2 log wIRy)

128

4 Radiative Characteristics

10-

12

[cm 3 5- 1 )

500.----.----.---.---.----.

o

o

1 0

0.2

0.4

0.6

0.8

Fig. 4.3. RR rates for H-like ions as a function of relative energy E. Experiment [4.8): eF8 +, 607+, OC 5+; solid curves: the Kramers formula (4.1.7) with Stobbe correction [4.2), convoluted with the experimental velocity distribution function

E reV)

The first measurements of RR rates for multicharged ions were performed in [4.8-12] using a merged-beam technique. The typical behavior of the RR rate is shown in Fig. 4.3 for recombination of H-like ions. For bare and H-like ions experimental data are well described by the Kramers hydrogenic formulas with the Stobbe correction [4.2]. For complex ions the evidence of significant incomplete screening in RR for Si6+ ions was shown [4.12]. Calculated RR rates for multicharged ions can also be found in [4.21,33,34].

4.2 Dielectronic Recombination Dielectric Recombination (DR) is the dominant electron-ion recombination process in high-temperature plasmas and, therefore, is very important in determining the ionization balance. Another very important role of DR is connected with the presence of so-called satellite lines (Sect. 2.1), arisen due to the radiative decay of autoionizing states of an ion I . Such lines contain quantitative information not only about ion structure, but also about plasma macroparameters (electron density, temperature etc.) and, therefore, play an essential role in X-ray spectroscopy and plasma diagnostics. DR is a resonant, two-step process

X;:.

(4.2.1) in which a multicharged ion captures a free electron with simultaneous excitation of the target electron and creates a doubly excited ion, which then decays by a radiative channel. Another competitive channel for the decay of an ion X;:'J is the autoionization process (Sect. 3.8).

4.2 Dielectronic Recombination

129

The capture of an electron in (4.2.1) is resonant in nature, i.e. it is possible only within a narrow interval of the free-electron energy E around the value

E

= Eres = E~ -

Eao ~ Eaao/Ry - (z - 1)2Ry/n 2,

(4.2.2)

where Eaao is the excitation energy of the ion Xz and n is the principal quantum number of the captured electron. The width of the energy interval (4.2.2) is equal to the level width ~~, which is defined by the total decay probability ~~

= h{A(~)

+ r(~)], + r(~)],

~dRy] = 4.83 x 1O-17[A(~)

(4.2.3)

where A and r (in S-I) are the radiative and autoionization decay probabilities of the state ~, respectively. Two main cases are distinguished in DR: a) inter-shell excitation transitions with ll.nc = 0, b) intra-shell excitation transitions with ll.nc # 0, where nc is the principal quantum number of the target electron. Depending on whether it is case a) or b), the free electron tends to be captured to low or high n-states.

4.2.1 DR Cross Sections and Rates The theory of DR was developed in [4.5,35-38]. In the isolated resonance approximation the DR cross section for a transition via the intermediate state ~ to all final states ~I of an ion Xz-I can be written in the form [4.16,4.34]: 21l' Ry2

2

O"d(~, E) = - ( ) -E TOq(~, ao)cp~(E)[1l'ao]' g ao res

a

-

r

a

A(~) + A(i;)'

q(i;, 0) - q(i;) (i;, 0) i(i;)

TO

= aoh/e2 = 2.419 x

10- 17 S-I,

(4.2.4)

where f(~) and A(n are the total Auger and radiative probabilities of the state g is the statistical weight, Eres and ~~ are defined in (4.2.2) and (4.2.3), respectively; cp~(E) is the Lorentzian profile, normalized to unity. ~,

E _ ~d21l' cpd ) - (E - Eres)2 + ~f/4'

J

cpdE)dE = 1.

The difference between A and A in (4.2.4) is that A contains the sum over stable final states only, i.e., those states that are below the ionization limit of the ion Xz- 1• The typical behaviour of the DR cross section, consisting of narrow peaks at resonant electron energies E res , is shown in Fig. 4.4. If one is not interested in the shape of the resonances but in the DR cross section averaged over resonances, then

130

4 Radiative Characteristics Fig. 4.4. The total DR cross section for reaction e + u9 1+ --* (tfO+)* + hw as a function of free electron energy. calculated with relativistic wavefunctions and the Breit interaction [4.39]

0.8 0.6

0.4 0.2 0.0 63.8 64D 642 68.4

_

6M np, 73.073.2

2rr

Ry2

g ao

reS O

ad(~. E) = -(-) E

E [keV]

2

~Eroq(~,ao)[rrao],

(4.2.5)

where 8E is an averaging interval. The measured DR cross sections of multicharged ions can be found in [4.8,10,40-43]. Precise high-resolution measurements of DR cross sections were performed in [4.8,10,43] using a merged-beam technique [4.44]. Usually, the rate coefficients (va) are measured, averaged over a certain electron velocity distribution. For the cooled-ion beams a two-temperature Maxwelliandistribution function is used [4.45] with a longitudinal (111) and transverse (T..L) temperature of the electron beam. Fig. 4.5 shows a resonance structure of the averaged DR cross section for Li-like Cu26+ ions as a function of the centerof-mass energy. Calculations of the DR cross sections for multicharged ions can be found in [4.39,43,46-49]. DR satellite spectra were considered in Sect. 2.1. The DR rate, averaged over a Maxwellian velocity distribution F(E), is given by

Jvad(~, 00

Kd(~) =

E)F(E)dE,

o

I

I

I I I I 11111111111

120 111L I I I 1111111. 12P1/2

90 60 30 O~~~~~--~L---~--

20

40

60

80

Ecm leV)

Fig. 4.5. Measured DR rate coefficient (vu) for Cu 26+ as a function of centerof-mass energy [4.43]

4.2 Dielectronic Recombination

131

(4.2.6) where T is the electron temperature. To obtain the total DR cross section or rate, one has to sum in (4.2.4-6) over all intermediate states ~. The DR rates Kd, averaged over a Maxwellian velocity distribution of electrons, are usually fit by (4.2.7)

Table 4.1. Parameters Ad and Xd [4.50] in (4.2.7) for DR rates of the processes: Fez (ao) + e --. Fe;~\ (a\) + flw; z is the spectroscopic symbol, 0.1 ~ f3 ~ 10 Transition

z-1 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

aO-a\

Ad

Xd

3p-4d -3d 3p-4d -3d 3p-4d -3d 3p-4d -3d 3p-4d -3d 3p-4d -3d 3p-4d -3d 2p-3d 2s-2p 2p-3d 2s-2p 2p-3d 2s-2p 2p-3d 2s-2p 2p-3d 2s-2p 2p-3d 2s-2p 2p-3d 2s-2p 2p-3d 2s-2p Is-2p Is-2p

0.86 66.5 0.82 43.6 1.32 27.3 1.57 14.2 1.33 5.54 0.50 11.0 0.59 4.46 50.5 5.1 49.3 0.84 42.4 1.3 32.4 1.4 21.6 1.3 10.5 1.1 1.82 0.78 1.30 0.21 8.1 4.0

0.054 0.041 0.087 0.034 0.061 0.026 0.062 0.021 0.055 0.Ql8 0.05 0.01 0.048 0.008 0.16 0.17 0.14 0.021 0.13 0.017 0.124 0.014 0.12 0.011 0.11 0.009 0.13 0.007 0.14 0.005 0.628 0.598

132

4 Radiative Characteristics

where T is the electron temperature; the parameters A and X are found from theoretical models or from fitting procedures. The parameters A and X for transitions in multicharged ions are given in Tables 4.1-3. Fitting parameters for Kd of light ions of astrophysical interest are given in [4.33]. For the estimation of DR rates, the serniempirical Burgess-Merts formula [4.35,36] is often used in the form (4.2.7), where A - 480 ( (z - 1)x ) dfaoa (Z _ 1)2 + 13.4

F(X)

1/2

F

(X),

= (1 + 0.105xz + 0.015z 2X 2)-1

for ll.n

= 0.5[1 +0.2Ixz +0.03z 2X2ri

for ll.n = I,

f3 = z 2Ry / T,

X

= Eaao/Z2Ry,

Xd

= 0,

= X[I + 0.015(z -

(4.2.8) 1)3/ z2rl.

Here, faoa and Eaao denote the oscillator strength and transition energy of an ion Xz-I, respectively. The Burgess-Merts formula (4.2.8) usually agrees well with experimental data and accurate theoretical calculations. For some cases, however, it has a significant error of about a factor of 5. Besides, it does not take into account the additional channels of autoionization decay. Table 4.2. Same as in Table 4.1, for oxygen ions: Oz(ao) +e --+ O;~l(al) +hw Transition

z-

I

2 3 4

5

6 7

ao -al

Ad

Xd

2p-3d 2s-2p 2p-3d 2s-2p 2p-3d 2s-2p 2s-3p 2s-2p 2s-3p 2s-2p Is-2p Is-2p

3.87 42.9 7.94 40.5 8.93 32.7 2.26 30.6 1.79 4.90 43.2 31.1

0.26 0.27 0.30 0.12 0.22 0.072 0.20 0.058 0.16 0.024 0.804 0.69

Table 4.3. Total DR rates, see Eq. (4.2.7) Initial ion Be III C IV NiXXVII Ti XXI Ti XXII

36.0 30.2 6.10 8.36 5.90

Xd

Source

0.86 0.92 0.60 0.67 0.69

Theory [4.50,51] Theory [4.51] Experiment [4.52] Experiment [4.53] Experiment + Theory [4.52]

4.2 Dielectronic Recombination

133

Table 4.4. Total DR rates for metallic ions [4.56]. E: experiment, T: theory; accuracy ratings: A = 0 - 10%, B = 10 - 25%, C = 25 - 50%, F > 100%, N is the total number of electrons in the ion Xz Cr

Ti N

E

2 3 4

T

Fe T

E

B B

B

B

B

B

E

5

8 9 10

Ni T

B B B B B

11

F B B C

12

C

21

B

B

E

T

B

B B B

B

B

The DR rates were mainly measured in Tokamak: [4.15,52-54] and laser [4.55] plasmas from DR satellite-line-intensity measurements. Table 4.4 shows the available data of the total DR rates for metallic ions, which are of primary interest in fusion problems. A typical behavior of the DR rate as a function of electron temperature is shown in Fig. 4.6. A comparison of Kd and RR rate at low electron temperatures is given in Fig. 4.7. Calculations of DR rates for multicharged ions can be found in [4.34,50-58].

4.2.2 Electric-Field (EF) and Electron-Density (ED) Effects The majority of calculations were made for DR cross sections and rates in the field-free zero-density limit. However, the strong influence of these effects are now generally accepted. A comparison of experiment and theory is complicated because of the presence of EF in the experiments, which produce large effects on the measured cross sections for transitions with !l.nc = O. This results both from the Stark mixing of levels with different angular quantum number I, and from the ionization of recombined ion states with large n. The EF and ED effects are strongly interrelated and accurate calculations for specific plasma conditions are extremely complicated.

134

4 Radiative Characteristics Fig. 4.6. DR rate coefficients for Ti XXII ions as a function of electron temperature. Cross: experiment [4.53]; dashed curve: the z-expansion method [4.52]; solid curve: BurgessMerts formula (4.2.8)

0.4

Ti XXII

0.2 0.10~7---L.--2L..--4L--..L.6-8'--'10

T[keV]

re[cm 3 s- l ]

109'.---r--r--.--.--..--....--.

1~0 i'-.... . .C£---.:d:-_--j . -:11 \ (Er 10 0 2 4 6 8 10 12 T[eV]

Fig. 4.7. DR and RR rates for t.n = 0 transition in 04+ ions as a function of electron temperature. Kd: intermediate-coupling calculations of DR rate [4.57]; K r : RR rate [4.33]

For I::!.n #- 0 transitions the effects are predicted to be small since the cross sections for these transitions falloff rapidly with n [4.34]. The exception, of course, will be for H- and He-like ions where I::!.n = 0 transitions are the only ones possible. For I::!.n = 0 transitions the DR rates are dominated by the contribution from the Rydberg states [4.34,43]. External EF strongly influence the DR process. Such fields can ionize electrons in high Rydberg states and thereby decrease the DR rate. Ions are rapidly ionized for all n > n max , where n max is given by the semiempirical formula [4.59] n max = (6.2 x 108z3/ F)1/4. (4.2.9) Here, z is the charge of an ion before recombination and F is the strength of electric field, in Vfcm.

4.2 Dielectronic Recombination

135

Fig. 4.8. DR rate for reaction C 3+(2s) + e --+ [C2+(2p, nl)]" --+ [C2+(2s, nl)]' + hw as a function of relative electron energy E. Circles: experiment [4.40]; curves: calculations [4.62] for different electron fields F (in V/cm) ( ... F = 0, - - - F = 5, -. - F = 25, - .. - F = 125, __ F =625)

0

0.40

5

10

15

20 E[eV]

Table 4.5. Fitting parameters A and 0.1 ~

f3

x,

see Eq. (4.2.7), for rates Fez (ao) + e --+ Fe;:' I (al) = 1020 - 1022 cm- 3 [4.50] (Table 4.1)

+ hw,

~ 10, as a function of electron density Ne

Transition

z - I ao 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

-al

Ad(Ne

3p-4d 3p-3d 3p-4d 3p-3d 3p-4d 3p-3d 3p-4d 3p-3d 3p-4d 3p-3d 3s-4p 3s-4p 2p-3d 2p-3d 2p-3d 2p-3d 2p-3d 2p-3d 2s-3p 2s-3p Is-2p Is-2p

0.29 0.17 0.2 0.1 0.2 0.08 0.2 0.05 0.14 0.01 0.017 0.0 30 31.5 25.2 16.8 12.2 6.0 0.8 0.7 7.6 3.8

= 1020 cm- 3)

Ad(Ne

0.11 0.03 0.09 0.02 0.03 0.01 0.08 0.0 0.05 0.0 0.0 0.0 16 20 15 10 8.0 4.0 0.5 0.4 6.8 3.0

= loll

cm- 3)

Ad(Ne

0.02 0.0 0.1 0.0 0.01 0.0 0.01 0.0 0.01 0.0 0.0 0.0 7 9 7.2 5.4 3.4 1.7 0.1 0.1 5.4 2.6

= 1022 cm- 3 )

Xd

0.06 0.018 0.083 0.017 0.034 0.013 0.023 0.009 0.025 0.008 0.068 0.0 0.16 0.14 0.13 0.12 0.12 0.11 0.14 0.14 0.628 0.6

136

4 Radiative Characteristics

The field enhancement of DR by EF has been verified by direct experimental measurements [4.40,60,61]. Figure 4.8 shows a comparison of experimental DR rates for t::..n = transitions in C IV ions with theoretical values calculated for different electric fields E. ED effects can also be quite significant. Like EF, collisional redistribution of states and collisional ionization can affect the DR rate. According to the equilibration model [4.63], the electrons in the Rydberg states with quantum numbers n larger than a certain quantity nt return to the plasma continuum and so do not contribute to DR. The quantity nt is called the "thermal limit" and is defined by n = 165z12/17TI/17 N- 2/ 17 (4.2.10)

°

t e e

'

where Te is in Rydberg and Ne in cm- 3 . Estimates for the ED effects on DR rates are given in [4.21,48-50,64-68]. The fitting parameters for DR rates of iron ions, calculated with account for ED effects, are given in Table 4.5 (cf. Table 4.1). In general, the DR rates can be severely affected by the presence of plasma microfields. EF and ED effects can change DR rates, obtained in the zerodensity and zero-field limits, by as much as a factor of 10 (see Table 4.5). These effects are serious and require further detailed investigations, especially for high Rydberg states which are particularly sensitive to external perturbations.

4.3 Bremsstrahlung Bremsstrahlung (BS) (or deceleration radiation) is the process in which one photon is radiated in the scattering of an electron from an atom or ion whose internal state remains unchanged XZ+

+ e(po) ~

XZ+

+ e(PI) + Tuv,

(4.3.1)

where PO,I are the electron momenta before and after collision. In the non-relativistic approximation, the photon and electron energies are related by Tuv

= (1/2m)(p5 -

pr)

= Ttkc,

(4.3.2)

where k is the momentum of the radiated photon. BS is closely related to elastic electron scattering, radiative recombination (Sect. 4.1) and pair production. BS is also a fundamental process for a variety of applications in astrophysics, biology, fusion-plasma problems, transport and energy loss in charged particle beams, etc. In general, BS is a quite complicated theoretical problem because it requires the bound conditions and the use of the continuum Coulomb wave functions. The problem is solved only in the non-relativistic limit and for some special cases where numerical calculations have been performed. Experiments are also limited in number, especially for positive ions. The general theory

4.3 Bremsstrahlung

137

of BS was considered in [4.69-77]. The BS spectra for electron energies E = 100 eV -10 MeV are tabulated in [4.78] and the angular distributions in [4.79].

4.3.1 Basic Formulas There are several analytical expressions for the BS cross section in the pure Coulomb case. The BS cross section in the frequency interval w, w + dw is usually written in the form da = g(1]O, 1]dda Kr , 2 dw

16

(4.3.3) 2

da Kr - - - a 31] -[1Ta ] - 3J3 0 w 0 '

(4.3.4)

Ze 2 1]0 = - , Two

(4.3.5)

where da Kr is the Kramers BS cross section [4.1], g is the dimensionless Gaunt factor, Z is the nuclear charge. So, in general, the BS cross section is a function of the two variables 1]0 and 1]1. Non-relativistic quantum mechanics in the dipole approximation gives the Sommerfeld formula [4.71]: g(1]o, 1]})

= (e 2

11"'10

1TJ3 d . -1)(1- e- 211"'1t) xodxo ItF2(i1]0, 11]1,1; xO)!2

Xo = -41]01]1/(1]0 - 1]1)2,

(4.3.6) (4.3.7)

where F is a hypergeometric function. In the quasiclassicallimit 1]1 > 1]0» 1, (4.3.6) gives [4.1,72]:

g

1TJ3. (1). (1)' . = -4-lVHiV (lV)Hiv (IV),

V

= Ze 2 w/mvo,3

(4.3.8)

where H(l) is the Hankel function of complex argument and index and Hi~l)' (iv) is its derivative with respect to the argument. The function (4.3.8) is monotonic; its limiting expressions are g =

I, v»l, { (J3/1T) In(2/yv),

v« 1,

(439) ..

where y = 1.781 is the Euler constant. The BS spectrum for a point Coulomb potential has a logarithmic divergence in the soft photon region (v « 1) and becomes flat for harder photons (v » 1) (Fig. 4.9). In the case of large initial electron velocity (1]0 « 1) one obtains from (4.3.6) the Bom-Elwert approximation: g = J3

1T

IE In 1]1 + 1]0, 1]1 - 1]0

1]0« 1,

(4.3.10)

138

4 Radiative Characteristics

IE is the Elwert factor 11l 1 - exp( -21r710) IE=. 710 1 - exp( -21r71d

where

(4.3.11)

If the final electron energy is also large, 21r711 non-relativistic Born approximation

g = (J3/1r)In

(11l + 710), IE = 711 - 710

1,

«

1, (4.3.10) gives the

21r710 < 21r711 «1. (4.3.12)

The general (approximate) formula for the Gaunt factor g can be obtained with the semiclassical method [4.74]: g=

1r~iV (1 + y~J Hi~l) [iV (1 + y~J] Hi~I)1 [iV ( 1 + Y~O)] , (4.3.13)

which gives both limiting cases (the Born approximation and the classical limit). For 71o/V » 1, (4.3.13) becomes (4.3.8). For the limiting case w -+ O(v « 1) and arbitrary 710 one has from (4.1.13) g = (J3/1r) In[2/v(y

+ 1/710)),

(4.3.14),

which gives the classical limit (4.3.9). In the Born approximation and for w -+ 0, one can obtain from (4.3.12) or (4.3.14) g = (J3/1r) In(2710/v),

711 ~ 710.

(4.3.15)

In general, the semiclassical result (4.3.13) is a good approximation to the Sommerfeld formula (4.3.6) for all parameters, except the so-called shortwavelength limit

v -+ 0,

710 -+ 0,

V/710 -+ 1.

This case corresponds to large electron velocities for which the Sommerfeld formula (4.3.6) is not valid, and one must take into account the relativistic and retardation effects. The relativistic theory of BS is considered in [4.70,80]. The formulas for the total intensity of BS, integrated over the whole radiation spectrum, are given in [4.7]. 4.3.2 Screening Effects The Sommerfeld formula and its modifications, considered in Sect. 4.3.1, were obtained for BS on bare nuclei. These results can be used for multicharged ions with Z » 1, if the initial electron energy is

mv5l2 < h,

(4.3.16)

4.3 Bremsstrahlung

139

where h is the first ionization potential of the ion. Corrections due to screening effects at higher energies are discussed in [4.3,72,74,80]. The general expression for BS cross sections in the one-electron approximation are given in [4.7]. The inclusion of screening effects leads to a reduction in cross section, corresponding to a reduction of the effective charge seen by the free electron [4.80]. The reduction is large near the long-wavelength region and is least in the short-wavelength region. In the Born approximation, screening leads to the change of Z2 in (4.3.4) to IZ - F(q)1 2, where F(q) is the atomic form factor: F(q) =

J

p(r)eiq.rdr,

J

p(r)dr = N,

q = Po - PI·

(4.3.17)

Here, q is the momentum transfer to the ion with a nuclear charge Z and a total number of electrons N; p(r) is the charge electron density. For large q (small distances), F ::::::: 0 and there is no screening, while for small q (large distances), F ::::::: N and shielding is complete. A comparison of the calculated BS energy spectra for neutral Mo and its ions at different kinetic energies of the incident electron is given in Fig. 4.9. The Gaunt factors for BS in the field of light positive ions were calculated in [4.81] using the scaled Thomas-Fermi approximation.

Eg1keV Z =42 _·_·-Z=32 1.5 - .. - Z=24 \ - ... - Z= 0

2.0

1.0 \ ..'"'05 '-.._ .. _ .. _ ..

2.0

05

Fig. 4.9. The calculated Gaunt factor for BS from Moz+ ions as a function of the incident electron energy Eo g

o

L.....-...J......----'----'_~~

0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

nc.lEo

140

4 Radiative Characteristics

4.4 Polarization of X-Ray Lines X-ray emission of multicharged ions is an important source of information about processes taking place in hot plasmas. The polarization of X-ray lines and bremsstrahlung is quite sensitive to the presence of electron beams in plasmas, or more generally, to the deviation of the electron-velocity distribution function from a Maxwellian one [4.82-85]. Electron beams, in turn, strongly influence the heat conductivity, oscillations and radiation from plasmas. The measurements of the polarization of X-ray lines in multicharged ions were performed in the laboratory [4.86-88] and the Solar [4.89,90] spectra. The most important process, leading to line polarization, is the excitation of ions by anisotropic electrons. This process takes place in crossed-beam experiments [4.87,88], RF current-driven Tokamak plasmas [4.91], and in the Solar plasma [4.92-94]. The theory of the polarization of emission lines, first developed for neutral atoms [4.95-97], is extended to multicharged ions [4.82-85]. The theoretical description of the subject is based on the photon polarization density matrix [4.98]: PU' = -I ( 1+'Y/2. 2 -'Y/3 -l'Y/l

-'Y/3 +i'Y/1) 1- 'Y/2 '

(4.4.1)

where I, 'Y/\. 'Y/2 and 'Y/3 are the four Stokes parameters [4.99]: I being the total photon intensity, 'Y/1 and 'Y/3 the two linear polarizations and 'Y/2 the circular polarization. The polarization degree of the radiation is P

=

('Y/T + 17~ + 17~)1/2.

(4.4.2)

For the interesting cases of X-ray lines in multicharged ions, the polarization degrees 'Y/1 = 'Y/2 = 0 [4.83,84], 'Y/3 remains and is defined by III-h 'Y/3 = III + h'

(4.4.3)

where III (h) is the intensity of the emitted photons, polarized with a polarization vector parallel (perpendicular) to the meridian plane, i.e., the plane formed by the crossing directions of the incident electron beam ke and of the detection of the radiation k. The general quantum-mechanical expression for the linear polarization of photons in the transition (1010-(11 h is given by [4.83]: 'Y/3(k) =

±L

L

N(Mo)(_1)I- M o-iJ(2J + 1)1/2 Yj2 (k)

Mo j even j~2

x

X

(j1 1j

1) (1 -2 0

[L L

10

-Mo

10 ) {1

Mo

h

10 j

10} j

N(Mo)(-1)- MO-Jl(21 + 1)1/2Yjo(k)

Mo j even j~O

4.4 Polarization of X-Ray Lines

~) (~

141

Jo) { J Jo (4.4.4) Mo JI j where j is the photon angular momentum, J is the total momentum of the system electron-target-ion, Jo. 1 are the total momenta of the ion which makes a transition, Mo.1 are the magnetic quantum numbers, Yl m is the spherical function and N (Mo) is the population of sublevel Mo. The plus sign corresponds to the electric (EK) transitions and the minus sign to the magnetic (MK) ones. From the general properties of (4.4.4), it follows that YJ3 is maximal when the final state has zero angular momentum (JI = 0). Therefore, the line corresponding to the transition 3P2 _I SO in He-like ions is expected to be more polarized than that for the 3P2-3S1 transition. Equation (4.4.4) is too complicated for general use. However, usually, the basic mechanism of level population is electron impact. In this case, the population of sublevels N (M) is expressed in terms of excitation-rate coefficients (vaex ) which are calculated in the Coulomb-Born approximation with exchange. Calculations of the polarization degree of X-ray lines (including dielectronic satellites) in H-, He- and Li-like ions were performed in [4.82-85]. In the case of He-like ions, the polarization of the following lines was investigated in detail [4.82-85]: j -I

Jo

-Mo

21 PI-II So

= the resonance

2 3PI -1 I So

= the intercombination line (y),

2 3P2 -1 I So 2 3SI-II So

= the magnetic quadrupole line

line (w), (x),

= the forbidden line (z).

The notations in parentheses correspond to those given in Sect. 2.1. The degree of polarization for wand y lines due to electric dipole transitions can be written in the form [4.85]: (No - NI) sin2 () YJ3«() = .2 ' (4.4.5) No sm () + NI (1 + cos 2 () and for the x-line: (NI - No) sin 2 () «() = (4.4.6) YJ3 , (NI + No) + 3(No - N I ) cos 2 () where () is the angle between the incident electron beam and the direction of observation of the emitted radiation. The abbreviation Ni = N «(Xi Ji Mi ) with the subscript i means the value of 1Mi I. For electric dipole transitions the following formula is valid: () _

900

YJ3( )-YJ3(

sin2 () )1-YJ3(900)cos2()'

(4.4.7)

and for the magnetic quadrupole transitions: () _ YJ3( ) - YJ3(

900

. 2 ()

sm ) 1 + YJ3(900)cos2()'

(4.4.8)

142

4 Radiative Characteristics Fig. 4.10. Calculated parameter 1/3 (6 = 90°) for x, y and w lines in Fe XXV ions. dashed curves: [4.83]; solid curves: [4.85]

60 Fe XXV

40

20

0 500

-40

-----------~--~~

...-:

-60

The dependence of the polarization for x, y and w lines for Fe24+, Ca 18+ and Mg lO+ He-like ions on the energy of the monoenergetic electron beam is shown in Figs. 4.10-12. These results were obtained in [4.85] under the assumption that the population of magnetic sublevels is proportional to the corresponding excitation cross sections. The forbidden line z in this assumption is unpolarized. The polarization of x, y and w lines as a function of energy (in threshold units) has a weak dependence on nuclear charge, while the polarization of the y line reveals a strong dependence on nuclear charge. (The relativistic coupling between 2 I PI and 2 3 PI levels which rapidly falls off with decreasing nuclear charge, strongly influences the excitation cross sections of the 2 3 PI level). The polarization of dielectronic satellites Is2nl' -ls2pnl' of He-like ions is given in Table 4.6. The emission from the level with J = 1/2 is unpolarized. Here, J and J' are the total momenta of the upper and lower states of the resulting ion. For He-like Fe24+ ions the q- and u-satellite lines, corresponding to transitions Is 22s 2 SI/2 -ls2p(I P)2s 2 P3/2 and Is 22s 2 SI/2 -ls2s2p 4 P3/2, respectively, are produced almost entirely by inner-shell excitation of Li-like ions in the ground state [4.83]. The degree of linear polarization is shown in Fig. 4.13. The polarization of w and x lines for Fe24+ ions were calculated in the model in which both thermal and non-thermal (anisotropic) electrons were taken

4.4 Polarization of X-Ray Lines

143

Fig. 4.11. The same as in Fig. 4.10, for Ca XIX ions

60

40

20

-20

-40

---

-60

60

w

Mg XI

Fig. 4.12. The same as in Fig. 4.10, for Mg XI ions

144

4 Radiative Characteristics

Table 4.6. Calculated polarization 713 [4.83] of electric dipole lines J - Jf due to [He] -+ [Li] dielectronic recombination in direction perpendicular to the incident electron beam Jf \ J

3/2

5/2

712

3/5

112 3/2 5/2

10

o

-3/4

1/2

1/7

-8/9

5111

u

700

1200 2000 E [Ry]

Fig. 4.13. Degree of linear polarization 713(%) for q and u satellites in Fe XXIV ions as a function of electron energy at () = 90° [4.83]

into account [4.84]. The lines show a measurable amount of linear polarization when the part of the anisotropic electron density is about 1% or more. For the line Is22p 2 P3/2- 2DS/2 and Is22p 2 PI/2- 2D3/2 satellites, similar calculations were carried out in [4.83].

4.5 Photon Polarization in Radiative Recombination The radiation of photons emitted in radiative recombination of multicharged ions with free electrons Xz+I

+ e(8) ---+

Xz(nl)

+ Tuv,

(4.5.1)

nw = 8 + Enl, Enl > 0, (see Sect. 4.1) is generally polarized [4.2,101,102]. Photon polarization from Radiative Recombination (RR) is very important in experiments using crystal spectrometers to perform high-resolution spectroscopy. Besides, the measurement of photon polarization can be used as a diagnostic method for an electron beam in the cooler of a storage ring. The asymptotic behavior of photorecombination a r (and photoionization) cross sections at 8 » Enl (8 and Enl being the energies of the optical electron in the continuum and in the bound state nl, respectively) is considered in [4.16]. In the opposite case of the low-energy limit 8

«

Enl,

(4.5.2)

4.5 Photon Polarization in Radiative Recombination

145

the expressions for a r and corresponding rates can be obtained in a closed analytical form, valid for nl states with n ~ Z [4.100]. Numerical calculations of the angular distributions and polarization of photons were performed in [4.100-102]. The differential RR cross section is written in the form [4.6]:

dar(nl) = ar(nl) {I dQ 4Jr

+ f3nl(e) [3(e . U 2

)2 _

I]}

p

,

(4.5.3)

where e and up are the unit vectors along the photon electric vector and electron momentum p, respectively; f3nl(e) is an anisotropy parameter describing the angular distribution of photoelectrons. In the low-energy limit (4.5.2), the parameter f3nl is energy dependent and expressed by the reduced dipole-matrix elements [4.100]. The polarization degree of radiation from recombination into nl states is written in the form [4.102]: III ((n

- h (e) III (e) + h (e)

Pnl(e) =

=

3f3nl sin 2 e

2(2 - f3nl)

+ 3f3nl sin2 e '

(4.5.4)

where e is the angle between the photon wave vector k and the electron momentum p, 111 ..1 are photon fluxes at angle e with the electric vector in and perpendicular to the (p,k)-plane, respectively; 111 ..1 ex dar(nl)ldQ. The dependence of Pnl(e) on e and quantum numbers nl is given in Fig. 4.14. We note, that for ns states f3ns = 2 and the photon radiation is completely linearly polarized with an electric vector in the (p,k)-plane. The polarization degree has its maximum at e = 90°: (4.5.5) In Fig. 4.15 the I-dependence of pn,?ax is shown for n = 30. Pn,?ax decreases from unity at I = 0 to the lowest value 1/3 at large I: f3nl ~ (1

+ 2)/(21 + 1),

Pn,?ax

~ 1/3,

I»

1.

(4.5.6)

Fig. 4.14. The calculated polarization degree Pn /«(}) of photons from RR into nl states (n ~ 3) as a function of () [4.102]

30

60

146

4 Radiative Characteristics p::\a.

Fig. 4.15. Calculated I dependence of P::;ax for n = 30 [4.102]

1.2r---~--~--....,

02

0.00':---~10---2""'0--3-'O

I

The polarization rates, i.e., the photon polarizations averaged over a Maxwellian two-temperature electron-velocity distribution f(v) =

1 (mv2 mv2) (2rrm)3/2 T.L(111)1/2 exp - 2T~ - 211:1 ,111:::; T.L

(4.5.7)

can be also obtained in a closed analytical form [4.102], where 111 and T.L are the effective longitudinal and transverse electron-beam temperatures, respectively [4.45]. For a symmetrical distribution 111 = T.L, the polarization rate is averaged out (i.e., is equal to zero), while, for a flattened distribution 111 < T.L, which is typical for electron coolers, the strongest polarization is expected at () = 90° (in the moving frame). In the laboratory system the highest degree of polarization moves towards forward angles with increasing ion velocity. A compilation of atomic processes, responsible for polarization of radiation in plasmas is given in [4.103].

5 Electron-Ion Collisions

Excitation and ionization processes in collisions of highly charged ions with electrons, including resonant and multiple ionization processes, are discussed in this chapter.

5.1 Excitation The excitation of ions by electrons

Xz +e -+ X;

+e

(5.1.1)

is governed by three main processes: (i) excitation of outer-shell electrons X z (~nlq)

+ e -+

X;(~nlq-ln'l')

+ e,

(5.1.2)

(ii) excitation of inner-shell electrons

X z (~nlq~)

+ e -+

X; (~nlq-l~n'l')

+ e,

(5.1.3)

+ e.

(5.1.4)

(iii) resonant excitation Xz(~o)

+ e -+

X;:' I «(nO -+ X;(~l)

Here, ~ and ~ denote a set of qu"ntum numbers and q is the number of equivalent electrons of the nl shell.

5.1.1 Excitation of Outer-Shell Electrons Because of the long-range attractive Coulomb force the excitation cross sections for processes (i) and (ii) are finite at threshold and, as a rule, have their maximum there. For dipole (optically allowed) transitions (tl.l = ±1, tl.S = 0) the excitation cross section at threshold can be estimated from the Van Regemorter formula d·

(J"thiP

~ 2.90J/(tl.EjRy)2[na5],

(5.1.5)

where tl. E is the transition energy and f oscillator strength. For high electron energies E » tl.E, the dipole cross section falls off according to the Bethe formula: dip A InE (J" ex: -E + BE' E ~ tl.E, (5.1.6) fi V. G. Pal'chikov et al., Reference Data on Multicharged Ions © Spring-Verlag Berlin Heidelberg 1995

148

5 Electron-Ion Collisions

where A and B are constants. B is related to the oscillator strength (B ex f) and A is obtained from numerical calculations. For other transitions and high energies one has

u ex {E-~ for transitions with I~ll =F 1, ~S = 0, (5.1.7) E- for intercombination transitions, ~S = 1. At present, absolute experimental excitation cross sections u are known only for a few ions with charges larger h'tan one: AI2+, C3+, N4+, Hg2+ [5.1] and Si3+ [5.2]. These data were obtained using the crossed-beam technique and fluorescence detection. Because of the low detection efficiencies (~ 10-4 ), these measurements become much more difficult as the ion charge increases. The invention of the Electron-Beam Ion Source (EBIS) [5.3] and the ElectronBeam Ion Trap (EBIT) [5.4] made it possible to obtain relative excitation cross sections for highly charged He-like argon and Ne-like barium and gold ions [5.5,6]. Much of the theoretical data on u can be obtained by using simple twostate theories such as Distorted-Wave (DWE) [5.7] or Coulomb-Born (CBE) [5.8] approximations with Exchange. The typical behavior of the electron-ion excitation cross section is shown in Fig. 5.1 (curve 1). In some cases the shape of the cross section can be distorted by electron exchange effects (Fig. 5.1, curve 2). A comparison of experimental excitation cross sections with theoretical calculations for C IV and Si IV ions is given in Figs. 5.2 and 5.3. The basic methods have to include close-coupling, relativistic [5.13] and even QED effects [5.14] for ions X z with z ~ 50. QED effects lead to an enormous increase of the excitation cross sections from threshold up to the high-energy region due to electromagnetic and retardation interactions in highly charged ions (Fig. 5.4). A new crossed-beam apparatus has been developed for measuring differential cross sections d u / d Q for excitation of multicharged ions by electron

A/X1I1s21S-1S2p lp

1

Fig. 5.1. Excitation cross section for transition Is2 1S - 2s2p 1 P in He-like Al XII ions as a function of scaled electron energy u E/ll.E - 1. Curve 1: Coulomb-Born approximation; curve 2: same with electron exchange included (the ATOM code [5.9])

=

o

5

10

5.1 Excitation

6 C IV (28 - 2p)

149

Fig. 5.2. Excitation cross section for transition 2s - 2p in Li-like C IV ions. Dashed curve: two-state close-coupling calculations [5.10]; solid curve: a convolution of experimental data [5.11] with calculations [5.10]

4 2

" 1"

I:

I

~ "'1-'

o~~--------------~

E[eV]

12

8 4

Si IV (38 - 3p)

11.5 12 E[eV]

Fig. 5.3. Excitation cross section for the transition 3s-3p in Na-like Si IV ions dots: experiment [5.2]; curve: close-coupling calculations [5.12] convoluted with a 0.20 eV (FWHM) electron energy distribution

impact. First results for the Ar VIII (3s-3p) transition [5.15] are shown in Fig. 5.5. Calculated do/dO. values for transitions in multicharged He-like ions are given in [5.16].

5.1.2 Excitation of Inner-Shell Electrons The excitation of inner-shell electrons (5.1.3) shows properties similar to the excitation of outer-shell electrons and is very important for ionization due to Excitation-Autoionization (EA) processes. EA involves direct excitation of an inner-shell electron and the production of an ionized ion (5.1.8)

150

5 Electron-Ion Collisions Fig. 5.4. Excitation cross sections for transition Is - 2s in H-like tJ91+ ions. QED: QED theory; D: Dirac theory [5.14]; CBE: CBE calculations with the code ATOM [5.9]

9i+

2.0

U 1s - 2s

1.0 /

QED

leSE

o

2

1

3

4u

du [cm 2 s r-'] (in

Ar VIII (3s-3p)

1015 1016 1017

1018

, 0 10

30

50e[deg]

Fig. 5.5. Differential excitation cross section for the transition 3s - 3p in Na-like Ar VIII ions as a function of angle at an electron energy E = 100 eV. Dots: experiment [5.15]; curve: Born approximation

Table 5.1. Sum of inner-shell excitatIon cross sections at threshold + (Tth(ls22s-ls2s2p) for Li-Iike ions (in units of 10- 19 cm2 )

(Tth(ls22s-ls2s 2 )

Ion

Close coupling of six states [5.17]

CBE [5.18]

Experiment [5.19]

Be II BIll C IV NV OVI

9.3 4.1 2.24 1.27 0.74

1l.5 6.96 3.77 1.98 1.07

20.0±8.0 4.0 ± 1.0 2.3 ±0.7 1.6 ± 0.4 0.8±0.3

5.1 Excitation

151

Because EA cross sections are non-zero at threshold they give distant steps in the ionization cross sections (Sect. 5.2.2). EA cross sections can be obtained from the total experimental ionization cross sections and theoretical directionization ones (Table 5.1). Excitation cross sections a of ions are often presented in the form 0'01

[1l'a~]

(5.1.9)

= Q/go(E/Ry),

where go is the statistical weight of the initial state and E is the electron energy. Q is termed the collision strength. Q-values are symmetrical on direct (0-1) and inverse (1-0) transitions: Q(O-1) = Q(1-0). Excitation cross sections (E

and

0'01

+ tlE)goaOl(E + tlE) =

0'10

are related by

EglalO(E), 0'10

--+

00,

goaol ~ gJO'IO,

E --+ O.

E» tlE,

(5.1.10)

The rate coefficient (va) is defined by (va) =

J

va(v)f(v)d 3v [cm3 S-I] ,

(5.1.11)

where a is the effective cross section, v is the relative velocity of colliding particles and f(v) is the velocity-distribution function. In the case of isotropic distribution

J J 00

(va)

=

J 00

va(v)f(v)dv

=

va(E)f(E) dE

I'lE

Vrnin

00

a -EfT EdE Ry l/2T3/2' 1l'a 2e

= K

I'lE

0

K = 2.J]ra~vo = 2.17 ... X 10-8 cm3 s-I, E = J.Lv 2/2,

Vrnin

= (2tlE/J.L)I/2,

(5.1.12)

where J.L is the reduced mass of colliding particles, tlE is the threshold energy for the process and T is the electron temperature. Particles in plasmas are usually described by a Maxwellian (isotropic) distribution function (5.1.13) or (5.1.14)

152

5 Electron-Ion Collisions

where T is the plasma temperature. The functions f(v) and f(E) are normalized to unity:

J

J

o

0

00

00

f(v)dv =

f(E)dE = 1.

The Maxwellian rate coefficients for cross sections (5.1.9) are often presented in the form (va}=K

x

(-RY) T

1/2

= Ej f:1E,

G = fJe P

go fJ

J 00

Ge-p

--,

e-PxQ(x)dx,

(5.1.15)

I

= f:1EjT,

where the constant K is defined in (5.1.12). The quantity G is termed the effective collision strength. The main types of non-Maxwellian distribution functions are shortly discussed in [5.20]. According to the detailed balance principle, excitation and de-excitation (quenching) rates are related by (5.1.16) Recommended data on electron-impact excitation cross sections and rates for multicharged ions are given in several papers [5.21-25]. A review on experimental (va) values obtained from the time-dependent plasma method [5.26] is given in [5.27]. The typical dependence of excitation rates on electron temperature is shown in Fig. 5.6. In Table 5.2 the fitting parameters for transitions nolo-nlll' nl ~ 4 in Hlike ions are given. The parameters were determined from a least-squares fit of CBE cross sections and rates in the non-relativistic approximation [5.28]; the accuracy of the fit is also given. Excitation cross sections and rates were fit by aOI

=

E )3/2 C Z2 ( _I - - [1ra~] ,

Eo u + q> 8 3 _ 10- cm s-I (EI )3/2 AfJI/2(fJ + 1) (va) Z3 Eo fJ + X ' u

= (E -

f:1E)jz 2 Ry,

fJ

(5.1.17)

= z2 RyjT,

where z is the spectroscopic symbol, EO,I are the energies of the lower and upper levels counted from the ionization limit and T is the electron temperature in Ry units. The parameters from Table 5.2 can be used for transitions in ions with z ~ 25. For higher z it is necessary to include relativistic and QED effects.

5.1 Excitation

153

Fig. 5.6. Calculated excitationrate coefficients as a function of electron temperature for transitions in He-like Ti XXI ions. Solid curves: CBE calculations; dashed curve: contribution of resonant excitation in transition Is2 1 S - Is2s 3 S (with the code ATOM [5.9])

T,XXI

T reV)

5.1.3 Resonant Excitation

Because of the long-range attractive Coulomb force, a multicharged ion X z can capture a free electron and create an excited ion Xz(~o)

+ e --+

X;~l (~),

~ = ~nlLSJ,

(5.1.18)

which, at least, is doubly-excited: an inner electron is excited to a state ~ and a free electron is captured to the nl state. The excited ion X;~ I is unstable and can decay by two competitive channels: autoionization (5.1.19) or radiative decay X;~l (~) --+ X;_I (~l)

+ nw.

(5.1.20)

The two-step process (5.1.18,20) is termed dielectronic recombination (Sect. 4.2). The autoionization (5.1.18,19) leads to excitation ~o --+ ~I (# ~o) of the ion X z via an intermediate state ~. Such an additional excitation channel with respect to the usual ones, (5.1.1) or (5.1.2), is termed resonant inelastic excitation. If ~I = ~o the process is called resonant elastic scattering. The

154

5 Electron-Ion Collisions

Table S.2. Fitting parameters C,

({J,

A and X (5.1.17) for transitions in H-like ions [5.28]; 0.02 ,:;; = 0.02

U ,:;; 36,0.5 ,:;; f3 ,:;; 16. The ao values correspond to Uo

Approx. error [%] Transition

C

({J

A

X

z4ao[1l"a~]

a

(va)

Is-2s -2p -3s -3p -3d -4s -4p -4d -4f 2s-3s -3p -3d -4s -4p -4d -4f 2p-3s -3p '-3d -4s -4p -4d -4f 3s-4s -4p -4d -4f 3p-4s -4p -4d -4f 3d-4s -4p -4d -4f

3.47 71.6 2.29 38.9 1.46 2.01 32.0 1.61 0.025 17.7 178 54.8 8.19 75.7 15.9 8.35 4.3 18.2 286 1.74 8.83 95.3 1l.5 52.9 427 109 54.7 26.6 62.7 517 121 1.20 9.60 50.4

0.92 4.96 1.02 4.54 1.06 1.04 4.37 1.01 0.20 0.17 1.91 0.28 0.22 1.93 0.27 0.23 0.84 0.14 0.77 0.54 0.15 0.71 0.37 0.055 1.17 0.16 0.082 0.98 0.058 0.54 O.ll 0.067 0.16 0.045 0.31

5.67 23.2 4.02 16.0 2.47 3.62 14.4 3.02 0.22 30.8 44.5 61.8 14.2 22.0 23.1 13.2 1.86 38 146 1.27 20.0 62.7 12.2 99.9 105 106 78.9 6.97 ll4 208 141 1.80 7.41 103 520

0.72 0.15 0.79 0.26 1.04 0.81 0.31 1.16 3.52 0.71 -0.14 0.36 0.70 -0.074 0.60 0.62 0.026 0.87 0.083 0.33 1.03 0.22 0.34 0.82 -0.23 0.13 0.47 -0.24 0.76 -O.ll 0.27 0.52 0.14 0.91 0.029

0.47 1.92 0.084 0.35 0.058 0.030 0.12 0.028 1.85 x 10- 3 28.5 29.4 58.5 4.27 5.25 7.57 4.78 1.96 38 126 0.47 6.77 19.1 4.3 314 182 316 270 15.1 370 500 460 7.2 30.4 560 1450

2 10 4 14 30 3 15 25 32 1 20 15 2 20 15 20 50 4 30 50 4 32 25 4 23 28 15 34 3 38 20 20 50 2 40

5 5 4 3 4 4 3 4 3 7 20 6 6 18 3

900

I

4 4 10 4 2 7 2 6 33 20 10 35 6 20 12 5 7 4 14

resonant capture of an electron (5.1.18) is only possible if the free electron energy E is equal to (Fig. 5.7) Eres

= E~ -

E~o ~ EHo/Ry - (z - 1)2Ry/n 2,

~

= ~nl.

(5.1.21)

Experimental data on RE cross sections are rather scarce. Therefore, most of the results are obtained by theoretical calculations. There are three main theoretical methods which lead to equivalent results: the R-matrix method [5.29], the generalized quantum-defect method [5.30] and the asymptotic-expansion method [5.31].

5.1 Excitation

Fig. 5.7. Energy-level scheme for electron resonance capture (5,1,1821)

(11111111 "'" 1111

--.----------

155

" " I I I ,"l111l I

-----~nf

.; I

--+--1---- - (""" (II It

';0

_....J..._....J..._ _ -

'I l l l l l / " I " " I I / I I I I I / ! / ' ' ' ' ' ' 1

Xz

---------------~

Xz -

0,03

I

n

0,02

0,01

r- ~ 3p~

Fe XXIII

o 43

4,5

(,SB- 3 P?)

49

4,7

51

E [Ryl

Fig. 5.8. Collisional strength n for transition I So - 3 PI in Be-like Fe XXIII ions: Dirac R -matrix calculations above 3 pf threshold [5.32]

RE leads to the appearance of resonance structures on top of the "usual" potential cross sections, as shown in Fig. 5.8. The resonances are rather small for optically allowed transitions (e.g., I 2S-2 2 P in H-like or II S _21 P in Helike ions) and other strong transitions such as monopole p - p or d -d transitions and transitions between excited states. The resonances can be very important for weak transitions (e.g., intercombination transitions with l1S = 0, especially at low temperatures (see below). The contribution of RE for the transition ~o - ~I in plasmas with a Maxwellian velocity distribution is given by (va

) res

l1E

=

g(~)4Jl'3/2a6r(~, ~o)l(~, ~l) fJ3/2e-Eres/T

g(~o)(l1E/Ry)2[A(~)

= E~o -

E~I '

fJ

+ r(~)]

= l1E / T,

,

(5.1.22)

156

5 Electron-Ion Collisions

Table 5.3. Fitting parameters A and X (5.1.23) for transitions in He-like ions [5.33]: 0.1 ~ fJ ~ 10 MgXI

Fe XXV

Transition

A

X

A

X

Is2 I So-ls2s 3 SI -ls2s ISO -ls2p 3Po -ls2 p 3 PI -ls2p 3P2 -ls2p I PI

1.82 1.82 0.94 2.82 4.71 3.53

0.85 0.85 0.85 0.85 0.85 0.85

0.16 0.16 0.10 0.29 0.48 0.15

0.85 0.85 0.85 0.85 0.85 0.85

where T is the plasma temperature and g is the statistical weight; A and r are radiative and autoionization probabilities (see Chap. 3). The typical behavior of the rates for strong and weak transitions and resonant transitions is shown in Fig. 5.6. At low temperatures the contribution from RE is quite substantial. The resonant excitation rate is fit by (5.1.23) where A and X are fitting parameters. For He-like ions, parameters A and X are given in Table 5.3. Numerical calculations of RE (Resonant Excitation) cross sections and rates for multicharged ions are given in [5.24,32-36].

5.2 Ionization In the ionization process of ions by electron impact:

X z + e -+ X z+ 1 + 2e, three main processes are distinguished: (i) Direct Ionization (DI), (ii) Excitation-Autoionization (EA), (iii) resonant processes (REDA, READI).

5.2.1 Direct Ionization (01) The cross section (}' for DI of an ion (5.2.1)

5.2 Ionization

157

is a smooth function of the incident electron energy E with asymptotes u--+O, u --+ 0,

U 3/ 2 ,

a ex

{

u, AInu

- + B -, U

a max ex/- 2

ex

u --+

z=l, Z > 1,

(5.2.2)

00,

U

Z-4,

U max

~ 1.5,

where u = E I I - 1 is the scaled incident electron energy, I is the binding energy of the outer or inner shell of the target nl q ; and q is the number of equivalent electrons. Ionization cross sections a and corresponding rates (va) for DI are usually estimated by the semiempirical Lotz fonnulas [5.37]:

1n(u+l) 2 [:rrao], u = Ell-I, u+l (va) = 6qV'fi(RYII)3/2e- fJ f(fJ) x 10-8 cm3 s-I, fJ = liT, a

= 2.76q(Ryll) 2

(S.2.3) (5.2.4)

f(fJ) = efJIEi(-fJ)I,

where T is the electron temperature, Ei(x) is the integral exponent. The function f(x) is fit within 3% by [S.9]: f(x)

= eXIEi(-x)1 ~ In ( 1 + 0.S62 + 1.4X) , x(1

+ l.4x)

x> O.

(S.2.S)

The Lotz fonnulas (S.2.3,4) were obtained on the basis of numerical calculations of ionization cross sections for H-like ions in the CBE and, therefore, produce errors for multielectron systems. However, the fonnulas are convenient and useful for estimating a and (va) with an accuracy of about a factor of 2. Classical and quantum theories give the following scaling laws for a and (va) F(u)

= 12 a,

u = EII- 1,

cf)(fJ) = /3/2 efJ (va)lq,

fJ = liT.

(5.2.6) (S.2.7)

The functions F(u) and cf)(fJ) have to be universal for a fixed initial state in ions along a given isoelectronic sequence (Figs. S.9,1O). Recent measurements [5.42] and calculations [S.43] of a for multicharged ions showed that the function F(u) in (S.2.6) is not universal for ions with z > SO and it is necessary to take into account relativistic effects both for bound and incident electrons. A similar situation occurs for the ionization of inner-shell electrons in neutral atoms. For heavy atoms with nuclear charge Zn > 30 relativistic effects are very important [S.44]. For the estimation of a values at relativistic electron energies one can use the Gryzinski fonnula [S.4S] or its modification [S.46].

158

5 Electron-Ion Collisions Fig. 5.9. Scaled cross section F(u) = Z4 a , 10- 16 cm2 (5.2.6) for ionization of H-like ions from the ground state. Experiment [5.38]: (0) CH, (e) N 6 +, (~) 07+, (a) Ne9+, (0) ArI7+. Solid curve: CBE calculations [5.39] for z = 128

10

o 06

¢ (J3)

6.---~---------------r--------------------'--'

5

3 2

OL-----------------~-L--------------------~~

0.1

10 Til

1.0

Fig. 5.10. Scaled ionization rate 4>(fJ) = (l/Ry)3/2 expf3 (va)/q, 1O- 8 cm3 s- 1 (5.2.7) for ionization of H- and He-like ions from the ground state. Experiment: H-like ions: 0 B V, 0 C IV; He-like ions: ~ B IV, e C V, • N VI, x~ C V, e B IV-data from O-pinch [5.40,41]. Theory: dashed curve: Lotz formula (5.2.4); solid curve: CBE results (the ATOM code [5.9])

For ions with z < 50, ionization cross section cr and rates (vcr) can be estimated using the fitting parameters given in Table 5.4 obtained by the leastsquares method from CBE calculations [5.28]. cr and (vcr) values are fit by: cr

= q(RY//) 2

Cu (u

+ qI)(u + 1)

2

[nao],

u

= Ell -

1,

5.2 Ionization

159

Table 5.4. Fitting parameters C,
nl q

C

A

X

Isq 2sq 2pq 3s q 3pq 3d q 4s q 4pq 4d q

7.96 6.69 6.93 6.00 6.24 6.57 5.77 6.00 6.23 7.06 5.66 5.88 6.08 6.26 6.47 5.60 5.82 6.00 6.24 6.33 6.44

2.70 2.03 1.47 1.59 1.31 1.08 1.43 1.26 1.11 1.00 1.36 1.23 1.12 1.08 1.04 1.32 1.22 1.13 1.07 1.04 1.01

5.65 6.23 9.05 7.37 9.11 11.7 7.76 9.11 10.8 13.5 7.96 9.13 10.4 11.1 11.9 8.07 9.13 10.2 11.2 11.7 12.2

0.40 0.52 0.73 0.70 0.82 1.00 0.76 0.86 0.97 1.07 0.79 0.87 0.96 1.00 1.03 0.80 0.88 0.95 1.00 1.03 1.06

4j9

5s q 5pq 5dq 5j9

5g q 6s q 6pq 6d q 6j9

6g q 6h q

(va)

=

Af31/2 10- 8 cm3 s-lq(Ry j 1)3/2 f3 + X e- I / T ,

(5.2.9)

f3=ljT.

5.2.2 Excitation-Autoionization For a target ion having more than one electron shell, the total cross section can contain a structure on top of the DI cross section which is due to indirect (multistep) ionization mechanisms. One of such a mechanism is the excitation of the inner-shell electrons into autoionizing states followed by autoionization decay (Sect. 5.1): Xz

+e ~

X;*

+e ~

Xz+l

+ 2e,

(5.2.10)

which is termed Excitation Autoionization (EA). For example, for Li-like ions, EA processes are X z (ls22s )

+e ~

[X z (ls2snl)]**

+e ~

X z+l (ls2)

+ 2e,

n ~ 2.

Figure 5.11 shows the contribution of EA to the total a in the case of ionization of Ni XV ions. The sum of DI and EA cross sections is usually presented in the form a

= am + L j

BjaE(j),

(5.2.11)

160

5 Electron-Ion Collisions Fig. 5.11. Electron-impact ionization of Si-Iike Ni XV ions. Dashed curve: direct ionization (DI); solid curve: DI+EA [5.47]; points: experiment [5.48]

0.56

NiXV

0.42 0.28 0.14 0.00

780 1020 1260

E leV]

(5.2.12) where Bj is the branching-ratio coefficient of the autoionizing j state of the target, A and r are the radiative and autoionization probabilities (Chap. 3), respectively, aE is the excitation cross section of inner-shell electrons. Equation (5.2.11) is valid if DI and EA processes are treated as independent; the interference of these processes is also possible [5.49]. The autoionization probability r ~ 1013 _10 14 S-1 and weakly depends on z. The probability A r « A a for ions with z :( 10 and, therefore, for such ions B ~ 1. In the case of multicharged ions the quantities r and A differ significantly for different states and it is necessary to use the general formula (5.2.U) that strongly complicates the calculations [5.50]. EA cross sections and rates can be estimated by semiempirical formulas [5.51] or using fitting approximations [5.52,53].

5.2.3 Resonant Ionization Besides DI and EA there are other important indirect processes leading to ionization but showing resonant characteristics, i.e., they occur only at a definite (resonance) energy of the incident electron (5.1.2). Such resonant processes are also multistep processes and lead to the appearance of narrow resonances in the total ionization cross section. The first step in resonant ionization (RI) is the capture of a free electron by the target ion, i.e., the same step as in the DR process (Sect. 4.2), and the creation of an ion in the doubly (triply etc.) excited (autoionizing) state. Dielectronic capture usually involves the excitation of an inner-shell electron. An ion in an autoionizing state can decay by sequential or simultaneous emission of two electrons; the net result is a single ionization of the target ion. Thus, for

5.2 Ionization

1.8

1]5 1.7

1.65

t

~

1s2sP

~:\

NV

1s2s3!

Fig. 5.12. Resonant structure of ionization cross section of Li-like N V ions [5.54]. Arrows show the positions of threshold for ion excitation to particular autoionizing states

1s2s4f n

11

~f."'SJ2p~·i~

l

l n=~

1s2sISJ2pPO 1s2s(S)2 p"!J

161

4 561-~ LUl.:.::l

n=4 567110

~_-L_ _-'--_ _.l...-_---'_ _---U

420

440

460

480

500E[eV]

3

2 1

700

BOO

900 E[eV]

Fig. 5.13. Ionization cross section for Na-like Fe XXVI ions e: experiment [5.60]; dashed curve: direct ionization, lower solid curve: DI+EA; upper solid curve: DI+EA+REDA [5.50]

Li-like ions one has

X z(ls22s)

+ e -+

X z(ls22s)

+e -+ [Xz_l(ls2s3p3l)]**

[Xz-l (ls2s2p21)] ** -+ Xz+l (ls2)

-+ Xz+l (ls2)

+ 2e

-+

+ 2e,

[X z(ls2s21')]*

(5.2.l3)

+e (5.2.14)

Obviously, the excitation of the Is-electron into· other excited states can occur. The process (5.2.l3) is termed resonant-excitation-auto-doubleionization (READI); the process (5.2.14) is termed resonant-excitation-doubleautoionization (REDA). The contribution of REDA to the ionization of N V ions is shown in Fig. 5.12. The RI processes were predicted theoretically [5.55,56] and confirmed experimentally [5.57]. To perform the experimental measurements was possible only by using advanced crossed-beam techniques [5.58], which, in particular, permit to study narrow features like resonances and excitation thresholds

162

5 Electron-Ion Collisions

with extremely high precision to obtain spectroscopic information on multiply excited states [5.59]. The contribution of RI depends on ion charge and electronic configuration of the target ion. The largest contribution from REDA was obtained for ionization of Fe XVI ions (Fig. 5.13), where REDA contributes up to 30% to the total ionization cross section.

5.3 Multiple Ionization Multiple Ionization (MI) of ions by electron impact X z +e --+ Xz+k + (k+ 1)e,

k

~

2

(5.3.1)

is investigated mostly experimentally using the crossed-beam technique [5.61-69]. The same processes as for single ionization (Sect. 5.2) are distinguished in MI processes: direct ionization of outer- and inner-shell electrons, indirect processes, including excitation of inner shells with subsequent autoionization, and resonant processes (RETA, REQA), where T and Q stand for triple and quadruple, respectively. At present M I cross sections O"z,z+k are measured for ions X z with z < 10 for processes with k ~ 4. The O"z,z+k values are extremely large and, therefore, M I processes should be taken into account along with the single ionization processes in the investigation of the charge-state balance in hot plasmas and calibration of ion-beam probes used for plasma diagnostics in thermonuclear research reactors [5.70], etc. The measured thresholds for k-electron ionization cross sections correspond to the minimum ionization energy z+k-l Iz,z+k =

L

Ij,j+l,

(5.3.2)

j=z

where Ij,j+l is the one-electron binding energy. Figure 5.14 shows the experimental result 0"2,5 for triple ionization of Kf2+ ions [5.67]; h5 = 36.95 + 52.5 + 64.7 = 154.15 eV corresponds to the experimental threshold energy. The data on O"z,z+k provide also detailed information about the relative strengths of multiple- to single-electron interactions

R~z) = O"z,z+k/O"z,z+l'

(5.3.3)

In Fig. 5.15 the relative strength Riz) for double ionization of Xe z+ ions is given as a function of electron energy E; for Xe2+ the ratio (5.3.3) is about 0.7 at E = 700 eV. The sum Et=2 R~z) for Xe z+ as a function of z is shown in Fig. 5.16; a contribution of MI to single ionization is maximum for Xe2+ ions (80%) and is larger than for neutral Xe atoms. The comparatively high O"z,z+k cross sections can be attributed to the contribution of indirect processes, such as inner-shell excitation autoionization; the

5.3 Multiple Ionization

163

Fig. 5.14. Triple ionization cross section U2.5 of J(r2+ ions. Points: experiment; solid curve: estimation using a Lotz-type fonnula for ionization of innershell electrons 15.67]

16

12

8 4

o 100

200

500

1000 E leV]

06

04 0.2

so

200

1000

Fig. 5.15. Relative strength Riz) for double ionization of (z = 1-4) as a function of electron energy [5.64]

Xe z ions

E leV]

o

0

2

4 z

Fig. 5.16. Relative strength R(z) = L::~ Riz) for Xe z+ ions as a function of z(z = 0-4) at an energy of E = 700 eV [5.64]

164

5 Electron-Ion Collisions

relative probabilities of these processes increase with increasing z and with increasing number of electrons k, removed in a single electron impact [5.65]. There are virtually no theoretical quantum calculations of MI cross sections with the exception of ionization of neutral He [5.71,72]. It was shown [5.72] that the direct ionization cross section a;~+k falls off more rapidly than the single ionization cross section in the Born region: (d) )2k, k > I, v» 1 , (5.3.4) az,z+k ex (zp/ v where zp is the charge of the incident particle and v is the relative velocity. The a;~1+2 values for double ionization cross sections are usually estimated by the binary-encounter approximation of Gryzinski [5.45]: (6.56 x 10- 14 eV2 cm2)2 q5/3(q - 1) z,z+2 Ii4rr g

a(d)

_

Ra

If

(_E_) + /z' II

(5.3.5)

where II and 12 are the ionization potentials of the ion X z and X z+ I, respectively, q is the number of equivalent electrons in the outer shell of the X z ion, Ro is the gas-kinetic radius of the shell [5.73] and g is the universal function of electron energy E. The Gryzinski formula (5.3.5) may reproduce the shape of a direct double-ionization cross section reasonably well, but can greatly overestimate the size of z+2 (see [5.65,67]). The resonant contributions to double and triple ionization of heavy-metal ions were observed using a fast energy-scanning technique [5.69]. The resonances arise due to multi-step processes

a:'

X z + e ---+ X;~I ---+ Xz+k

+ (k + l)e,

(5.3.6)

i.e., the same way as in single ionization (Sects. 5.1.3,5.2.3). The first step is a dielectronic capture of a free electron, having a definite (resonance) energy (5.1.21), and creation of an highly excited ion X;~I; the second step is Auger decay followed by emission of k additional electrons. Emission of one electron (k = 0) corresponds to resonant electron scattering, two-electron emission

1.7 1.5

1.3 Fig. 5.17. Cross section for doubleelectron ionization of La2+ ions [5.69]. The absolute magnitudes of the resonances are comparable with the total cross section

1.1

90

94

98

E[eV]

5.3 Multiple Ionization

165

(k = 1) corresponds to REDA processes [5.55]. Resonant ionization processes with emission of a total of three (k 2) and four (k 3) electrons were named

=

=

resonant-excitation-triple/quadruple-autoionization (RETAlREQA [5.69]). The contribution of RETA or REQA processes may be also very high and exceed the total measured multiple ionization cross section by up to 30% (Fig. 5.17). In general, the MI processes of highly charged ions are much richer and of greater variety than single-electron ionization, while the inner-shell electrons play a dominant role and, in fact, determine the magnitude of effective cross sections. These processes must be taken into account in calculating the kinetics of a high-temperature plasma and its ionization stages. An adequate theory is required for such applications.

6 Ion-Atom Collisions

Collisions of multicharged ions with neutral species are currently a subject of fundamental interest processes since such as ionization, electron capture and excitation occur in many areas, e.g., confined plasma research, accelerator design and beam transport, fusion and the production of slow highly charged recoil ions for subsequent collision studies.

6.1 Electron Capture Experimental and theoretical data on Electron Capture (EC) (charge exchange) in collisions of multicharged ions with neutrals Xz+

+A

~ X(z-l)+

+ A+

(6.1.1)

are required in many fields of thermonuclear fusion research. The subject of charge exchange in ion-atom collisions has been reviewed in [6.1-3] from the experimental point of view and in theoretical works [6.4-7]. Experimental data on EC cross sections U c for collisions of protons with neutral targets are given in [6.8,9]. Numerical calculations of differential cross sections due/dO are presented in [6.10,11] and references on EC cross sections can be found in [6.12]. Reaction (6.1.1) is a rearrangement process with different particles before and after collision, so the study of such reactions is a more complicated problem as compared to electron-ion collisions. The value f:l.E = Eo - El is termed the energy defect, where Eo and El (Eo, El > 0) are the binding energies of the electron being captured in the target atom A and the resulting ion X(z-l)+, respectively. For f:l.E < 0 one has an endothermal reaction, f:l.E > 0 corresponds to an exothermal reaction; the cases with f:l.E = 0 and f:l.E ~ 0 are termed resonance and quasiresonance EC, respectively. The behavior of the electron-capture cross sections U c are usually described in two energy regions: adiabatic (v ~ ve ) and diabatic (v ~ ve ), where v is the relative velocity, Ve is the bound-electron velocity. In the first region, the bound target electrons adjust to the changing field of the projectile, and quasi-molecular or close-coupling approaches can be used. In the diabatic (fast collision) region, perturbation theory and its modifications are usually used. V. G. Pal'chikov et al., Reference Data on Multicharged Ions © Spring-Verlag Berlin Heidelberg 1995

6.1 Electron Capture

167

Some general features of U c for reaction (6.1.1) are outlined in [6.13]. For a given target atom in the low-velocity region (v ~ ve): 1) for multicharged, partially stripped ions the uc-values are large and nearly independent of v, and U c ex z. U c is constant at low energies because a lot of excited states of the resulting ion are involved. U c may have a maximum for particular states (partial cross section), but the total cross section is a sum over the partial cross sections with maxima at different ion velocities and hence, the total cross section is nearly constant; 2) for almost fully stripped ions and ions in low charge states, the capture is highly selective, uc-values do not scale with z and are dependent on v; 3) the capture into particular n-state of the X(z-l)+ ions with n ~ Z3/4 has the largest probability. In the high energy region (v ~ ve): 1) U c decreases rapidly with v increasing for one-shell targets (H or He), and decreases much slower for multielectron targets because of the contribution from the capture of inner-shell target electrons; 2) the total U c values can contain the characteristic "breaks" in the velocity dependence uc(v), where the capture of electrons from neighbor shells gives a nearly equal contribution (Fig. 6.1), 3) for partially stripped ions U c depends only on z and v, and is almost independent on ion species; 4) U c values have a scaling law U c ex za(v), where 1 ~ a(v) ~ 5 (see Fig. 6.2), 5) the distribution over n of the resulting ion X(z-l)+ becomes broader with increasing v, and its maximum moves towards low-n states.

o:.-_o_......o~

-24

10

K

Fig. 6.1. EC cross section in H+ +Ar collisions. Dashed and dashed-dotted curves: contribution from different subshells of Ar; solid curve: total cross section (modified BrinkmanKramers approximation); circles and crosses: experimental data (see [6.14] for details)

168

6 Ion-Atom Collisions

a(v)

4 3 2

10

0.1

100 v[a.u.]

Fig. 6.2. The Z-dependence of the EC cross section in XZ+-Ar collisions for different relative velocities v: U c ex za(v) (Solid curve: calculated on the basis of different models; squares: experiment [6.15])

6.1.1 Collisions with H and He The available experimental data for one-electron capture cross sections a c in collisions with simple targets H and He, X z+ + H,

He --+

X(z-I)+

+ H+, He+,

are given in [6.2,3,16,17]. At low energies the simple empirical estimation for H targets gives ac(H)

= Z1O- 15 cm2 .

(6.1.2)

Theoretical results predict ac(H)

= Az In(B/ EI/2),

(6.1.3)

where A and B are constants (see also Fig. 6.3). For fully stripped ions with z ~ 10, theory gives [6.22]: A

= 2.55.10- 16 cm2 ,

B

= 75,

where E is the energy of projectile in units of keV/u. Some other treatment [6.21] predicts considerable oscillations of the cross sections at low energies. In the high-energy region both experimental and theoretical data are described by the scaling law [6.13] G(u) = a~(E)/z,

u = E/-vIz[keV/u].

(6.1.4)

The general fitting formula for ac(E) in the case of Xz+ + H, He collisions can be written in the form [6.13]:

6.1 Electron Capture

169

Fig. 6.3. The Z-dependence of the EC cross section in Fez+ +H collisions at low energies (Solid curve: absorbing sphere model [6.18]; long-dashed curve: tunneling model [6.19]; short-dashed curve: empirical scaling (6.1.2); solid curve: experiment [6.21]

(T

c

(u) - __A_I-:n(:-B...:./_u.:....)-:-:: cm2 , - 1 + C u 2 + D u 4 .5

u=

E/v'z,

(6.1.5)

where E is in keV/u; A, B, C and D are fitting parameters; given in Table 6.1. The scaled (1c values for H and He targets are shown in Figs. 6.4 and 6.5. Measurements of two-electron capture cross sections in He (and H2) have shown that they are typically 10% or less than those of one-electron ones. However, the relative importance of the two-electron capture processes increases proportional to Z2 with increasing z [6.24,25]. A review on multiple-electron Table 6.1. Fitting parameters for capture cross sections (6.1.5) [6.13] Target

A

B

C

D

H H2 He

5.967 x 10- 17 5.707 X 10- 17 1.818 x 10- 17

5.870 x lOS 5.283 x 10" 1.856 x 106

1.913 X 10-3 7.800 x 10-4 2.753 x 10-4

1.383 2.721 1.370

Approx. error [%] X X X

10-7 10-8 10-9

13 15 70

Fig. 6.4. Scaled EC cross section for XZ+ - H collisions; (6.1.5) and Table 6.1.1

102u [keV/amu]

170

6 Ion-Atom Collisions Fig. 6.S. Same as in Fig. 6.4, for Xz+ collisions

-15 G(u) [cm 2]

10

-2

+

He

102U [keV/amu]

10

capture of highly charged ions on atoms from theoretical and experimental points of view is given in [6.26]. For one-electron state-selective electron capture (SSEC) X z+

+ H, He -+

X(z-I)+(nl)

+ H+, He+

(6.1.6)

is the relevant reaction, the knowledge of which is required for studying radiation losses from multicharged ions due to EC recombination and for plasma diagnostics based on EC. The available theoretical and experimental data on SSEC cross sections are given in [6.4]. Most of the general results for SSEC were obtained theoretically for collisions of bare ions with H atoms. The scaling law for collisions with H(ls) atoms is given by (6.1.4) [6.27]: 8z 3 :rra 2 7 0, z> 1, E/z> 10 keV/u. v For the collisions with excited H atoms,

a[H(ls)] ~

Xz+

+ H* (no) -+

X(z-l)+ (n l)

+ H+ ,

the following asymptotic behavior is predicted: a(no - nd ~

8z 3 :rra 2 7 3

~,

v nOnl

no > 1,

nl

>

Z,

v > 1 a.u.

In general, the distribution over nl-states of the ion X(z-l)+, given by theoretical models and confirmed by experiments with bare and closed-shell ions, is as follows [6.13]: 1) the principal quantum number of the dominantly populated ionic state is n = nm ~ z3/4; 2) in the very-low-velocity limit (v « ve ) and for ions with z » 1 the most populated sub-state for a given n is the p state if the initial electron state

6.1 Electron Capture

171

of H or He is nos. This results from the weak mixing of the sublevels in this velocity limit; 3) with increasing v up to Vo = 1, the most populated level 1m increases and tends to the maximum value 1m = n - 1. Because of the strong mixing of states in this collision regime, the relative distribution tends to its statistical limit undun :::::: (21 + 1)/n 2 ; 4) in the intermediate energy region E = 10-200 keVlu one has: 1m :::::: n-l for n ~ nm and 1m ex nm for n > n m; 5) with further increase of E, the maximum 1m - value decreases rapidly and tends to the s state, i.e., 1m = O. The behavior of SSEC cross sections is shown in Fig. 6.6 in the case of Fe8+ + H collisions. The total electron-removal cross section U e (electron capture plus ionization) is scaled by [6.29,30]: G(u) = ue(E)/z,

(6.1.7)

u = E/z[keV/u] ,

The universal curve for U e in collisions XZ+ + H obtained by the Classical Trajectory Monte Carlo (CTMC) method [6.29] is given in Fig. 6.7 together with experimental data and theoretical results. The solid curve is fit by the formula for E/z > 1 keV/u [6.31]: G(u) = A (

A = 7.57,

1

+ C In(1 + FU»)

1 +Bu B = 0.089,

D+u C = 2.65,

x 10- 16 cm2 , D = 58.98,

u = E/z,

(6.1.8)

F = 1.65.

6.1.2 Collisions with Multielectron Atoms In general, the properties of EC in collisions with complex atoms are similar to those in collisions with H and He (Sect. 6.1.1) with one important exception: at high energies the capture of inner-shell target electrons is dominant and the capture of outer electrons does not play any role. The effective capture of

................~~ .... 0.5

.............. Fig. 6.6. The I-distribution of EC cross sections for the Fe8+ +H .... Fe7+(6/)+H+ reaction: Calculations by semiclassical closed-coupliug method with atomic- orbital basis sets [6.28]

0.0 1

10

E[keV]

172

1015

6 Ion-Atom Collisions

o'!oss [cm2]/z

1016 I

I

I

/

/

/

z+

,.--

X +H

Fig. 6.7. Scaled removal cross section (6.1.7) in XZ+ + H collisions (Z ~ 50) (Solid curve: CTMC calculation [6.29]; dashed curve: Born approximation for ionization only; symbols: experiment [6.29]

1017

1018

1

10

100

1000 E[keV]/z

inner-shell target electrons by incident ion is the most striking property of EC reactions at large relative velocities of colliding particles. The low energy region corresponds to the projectile velocity, which is much less than the orbital velocity of the electron being captured (usually from the outer shell of the target). In this velocity region, the main features of the total a c for collisions between multicharged ions and many-electron atoms are [6.32]: 1) a c values for capture of one or several electrons are almost independent of the impact energy E; 2) a c decreases with the number of electrons transferred in a single ion-atom collision; 3) a c increases with projectile charge Z; 4) a c decreases with increasing target ionization potential I A • In the case of k-electron capture (k = 1,2,3,4) the large amount of experimental data [6.24,33,34] for collisions XZ+ + A(X = Ne, Ar, Kr, Xe, Z :0:::; 8, A = He, Ne, Ar, Kr, Xe, H2, N2, 02, C~, C02) allowed to obtain the scaling law for az,z-k- values at velocities v < 1 a.u.: (6.1.9) where IA is the first ionization potential of the target. The fitting parameters A, b and i are listed in Table 6.2. Cross sections for the capture of k electrons in collisions X e Z+ + K r are given in Fig. 6.8. For one-electron capture (k = 1) one has az,z-I

= 1.43 x lO-12 cm2z I. 17 (lA/eV)-2.76,

az,z-I

ex 1A-28.•

i.e.

Another dependence on the ionization potential of the target I A was found in [6.35], where az,z-I was measured in collisions of Xe IO+ ions with metallic

6.1 Electron Capture

173

Table '.2. The fitting parameters for k-electron capture cross sections (6.1.9) [6.24]

k

Number of cross sections

I

107

2 3 4

77

A

b

1.43 ±0.76 LOB ± 0.95 (5.50 ± 5.B) x 10- 2 (3.57 ± B.9) x 10-4

50 34

1.17 ±0.09 0.71 ±0.14 2.10 ± 0.24 4.20 ± 0.79

2.76 ±0.19 2.BO ± 0.32 2.B9 ±0.39 3.03 ±0.B6

Fig. 6.S. The k-electron capture cross sections in Xez+ + Kr collisions (Solid curve: (6.1.9) and Table 6.2; symbols: experiment [6.24])

targets A = Li, Na, Mg, K, Cd, Cs, Uz.z-I

= 2.02

U z•z-l

ex:

X

10- 12 cm2UA/eV)-1.94,

v = 3.8

X

107 cms- 1,

i.e. -2 ]A .

One-electron capture cross sections at high velocities were measured by many others [6.12]. The typical behavior of U c in collisions of X4+ ions with Ar as a target is shown in Fig. 6.9. An empirical scaling law was found [6.39] for one-electron capture by fast multicharged ions in gas targets A = H2, He, N2, Ne, Ar, Kr, Xe in the form: G(u)

= u c (E)Zf· 8/z0. 5,

U

= E/Zf·25Z0.7,

(6.1.10)

where E is the energy of the ion projectile in keVlu, ZT is the nuclear charge of the atomic target. About 70% of the experimental data lie within of a factor of 2 of the curve described by the function (Fig. 6.10) [6.39]: G(u) =

1.1 x 10-8 cm2

u4 .8

[1 - exp( -0.037u 2 .2 )]

x [1 - exp( -2.44 x 1O-5 u2 .6 )]. For u

»

1, U c according to (6.1.11) asymptotically approaches

uc(E) ~ 1.1

x 10-8 cm2z3.9zf·2 E -4.8(keVlu)

(6.1.11)

174

6 Ion-Atom Collisions

-14 10 ------... •• -

Fig. 6.9. Behavior of the Ee cross section in XZ+ + Ar collisions. Experiment: V'Xe4+ +Ar; ...Kr"+ +Ar; eAr4+ +Ar [6.32]; ON4+ + Ar [6.36]; "-N4+ + ArT6.37]. Theory: - - - [6.19]; [6.38]; ... Ar4+ + Ar collisions [6.14]

;"0\ "1!

-16 10

'..,

.,.\

.~

~

.....

.... '

-18

10

.\

0.01

Q1

1

10 v[a.u.]

G(u)

-:17 10

1021!------'-----'------'-~

1

1000 u

Fig. 6.10. Reduced EC cross section in collisions of multicharged ions with rare-gas targets (Solid curve (6.1.1); circles: experiment [6.39])

Equation (6.1.11) can be used for the prediction of a c in the reduced energy range 10 < u < 1000 and ion charges z > 3.

6.2 Ionization The measured ionization cross sections with H and He XZ+

+ H, He --+

XZ+

+ H+,

He+

aj

in collisions of multicharged ions

+ e,

at energies E > 200 keVlu are given in [6.2,40] for a wide range of ion species. According to these data ai ex za(v) for z > 5, where a(v) slowly varies with the relative velocity v. At energies below the maximum of ai, experimental data are very scarce [6.41].

6.2 Ionization

175

For ionization of H and He at energies E I z > 10 ke Vlu the scaling formula based on the Bethe approximation [6.42] can be used: O"j

= z2 exp[-O.76z 2(al,8)2]O"B(,B),

,8 = vic,

(6.2.1)

where a is the fine-structure constant, c is the velocity of light and O"B is the Bethe ionization cross section for proton impact. Another scaling law for collisions with H atoms was introduced in [6.43] on the basis of CTMC calculations:

= O"ilz,

O"(u)

u

= Elz(keV lu).

For many-electron targets, G(u)

O"j

values are suggested to scale by [6.44]:

= Ui [Z2~qj(RY/lj)2l-1,

U

= E/(M1o),

(6.2.2)

where Ij is the binding energy of j-th target shell with qj equivalent electrons; and M are the projectile charge, energy and mass, expressed in units of electron mass and 10 is the ionization potential of the target. Equation (6.2.2) does not take into account the contribution of excitationautoionization processes which involve inner-shell electrons (see Sect. 7.2). The typical behavior of the ionization cross section, as compared to electron capture (EC), is shown in Fig. 6.11. One can see that at low energies EC dominates as mechanism of electron loss from H atoms. The importance of EC extends to higher collisions energies as the charge of projectile increases. At still higher energies ionization dominates the electron loss. The total removal cross section O"e (ionization plus electron capture) in collisions with excited H atoms,

z, E

XZ+

+ H*(n),

Fig. 6.11. Comparison of electron capture (left branch of curves) and ionization (right branch) cross sections in crZ+ + H collisions (CTMC calculations [6.45])

176

6 Ion-Atom Collisions

has the scaling law [6.46]: G(u)

= u e (E)/(zn 4 ),

u

= En2/z[keV/u] .

(6.2.3)

A general scaling law for cross section of electron removal from a neutral target is given by theoretical predictions (Sect. 6.1.1) [6.29,30]: G(u) = uj(E)/z, u = E/z[keV/u]. (6.2.4) Multiple Ionization (MI) in collisions with neutrals is caused by the high charge of the projectile when many electrons of the target are removed simultaneously at large collision distances. Theoretical approaches, used for describing MI, are based on the Independent Particle Model (IPM), where the target electrons are treated independently from each other, shell structure effects are neglected and the average binding energies for each shell are used. MI cross sections were measured in [6.47 - 52] for collisions of multicharged ions with rare-gas targets. Usually, the cross sections Uj for producing a recoil ion in charge state j are obtained by normalizing measured charge-state fractions to measured net-ionization cross section U+: U+

= LjUj.

(6.2.5)

j

CTMC calculations have shown [6.46] that the calculated U+ values for a given rare-gas target scale as G(u) = u+/z, u = E/z[keV /u] , (6.2.6) where z is the charge of the incident ion (Fig. 6.12).

5

(b)

(a)

2

2 -15 10

1015

5

5

2

2

1016

1016

5

5

~ +5C

+6 C.

+ C r-_+4

2

fa7L-:-L:---:-~-+'--;;':~~--;!, 0.02 0.05 0.1 0.2 0.5 1D

ffFL--=+-:---::--!-:--;&---;!;.----fr:"~ 0.02 0.05 0.1 0.2 as 1D

E/Z [MeV/amu] Fig. 6.12. Scaled net-ionization cross sections in collisions of ions with rare-gas targets (Symbols: experiment [6.48]; solid curves: CTMC calculations [6.47])

6.3 Excitation

177

A short review on classical, semiclassical and quantum-mechanical methods used in the theory of multiple ionization, excitation and capture in energetic ion-atom collisions is given in [6.53].

6.3 Excitation Coulomb excitation of neutral atoms is an important elementary process in neutral-beam injection for plasma heating and beam-driven currents. Especially the excitation of injected atoms by impurity ions has to decrease the mean free path of injected atoms considerably. Excitation processes in ion-atom collisions, XZ+

+ A(nolo) ---+

XZ+

+ A*(nI11),

(6.3.1)

are investigated mainly for one-shell targets (A = H, He). The database for experimental cross sections (J' in XZ+ + H collisions with z > 1 at energies E ~ 10 keY x u- l Z-l are evaluated in [6.54], for XZ+ + He collisions in [6.55-57] for 3 ~ z ~ 45. (J' values for collisions with protons, (6.3.2) H+ + H(1s) ---+ H+ + H*(nd, nl ~ 6, were measured in [6.58,59].

H(n=2) ---> H(n=3)

Fig. 6.13. Calculated excitation cross sections (6.3.3) for reaction XZ+ + H(n = 2) ~ XZ+ + H(n = 3) (longdashed curve: DACC theory [6.60], short-dashed curve: [6.61]; solid curve: CTMC calculations [6.62], dash-dotted curve: classical formula of Vriens and Smeets [6.66]

178

6 Ion-Atom Collisions Fig. 6.14. The z-dependence of excitation cross sections for Siz+. Cuz+ + He(11 S) --. He(n 1L) transitions at an energy of E = 1.4 MeV /u (Symbols: experiment; solid curves: z2-scaling [6.57])

D 31P X 41S

o+ 10

',p0

+ 10

20

30

40

Z

6+ •• D.+:SI6+ o,~:(u

~9L-~__~__~~~~

100

200

400

1000 E[keV]

Fig. 6.1S. Energy dependence of excitation cross sections for Si6+. Cu6+ + He(ll S) --. He· (n 1L) transitions (Symbols: experiment [6.571. solid line: guide to the eye)

For H and He targets, cross sections scale as [6.60]: G(u)

= u(E)/z,

u

= E/z[keV/u].

(6.3.3)

For H targets, u values are well described by the semiempirical Lodge formula [6.61] in coordinates (6.3.3) for u ~ 100.

6.3 Excitation a

5

179

Fig. 6.16. Scaled excitation cross sections (6.3.3) for the dipole transition He (lIS-3 1p). (0. Experiment; [6.55,56))

[10- 18 Cm 2]/Z D:X~+6~Z .;45 .: H+

4 3

• ••

2

••

0

•

••

••

••••

10

100

E/Z [keV/u]

There are several theoretical approaches beyond the first Born approximation used for calculations of 0": the Dipole-Close-Coupling (DACC Approximation [6.60]), the Classical-Trajectory Monte-Carlo approximation (CTMC [6.62]), the Atomic-Orbital approach (AO) [6.63], the Unitarized Distorted-Wave-Approximation (UDWA [6.64)] and the close-coupling method (CC) [6.65]. In general, different approaches give quite different results (Fig. 6.13). In order to test theory, more experimental data on excitation cross sections are required. Experimental data of 0" values for transitions in He targets are given in Figs. 6.14-16.

7 Ion-Ion Collisions

Theoretical and experimental data on ion-ion collisions are quite limited because this problem is a relatively young field of physics stimulated by recent research on thermonuclear fusion with inertial and magnetic confinement. Only since 1977, the experimental data on charge-exchange and ionization, involving singly or doubly charged ions, obtained by the intersecting-beam technique, have become available. Such data for collisions between two multicharged ions A ZI + and B Z2+ with ZI, Z2 > 2 are not available. The problems arising in the investigation of ion-ion collisions are described in reviews and monographs [7.1-4], the general features of the intersecting - beam techniques in [7.5]. A theoretical description of ionization, electron capture, excitation and other processes in heavy-ion collisions at relativistic velocities is given in [7.6]. Usually, the effective cross sections in ion-ion collisions are given as a function of the relative ion velocity or the center-of-mass energy Ecm: Ecm:::;

MIM2 MI + M2

[.§.. + MI

E2 _ 2 ( EIE2 ) 1/2 coso] , M2 MIM2

where E and M denote the laboratory energies and masses of colliding ions, respectively, 0 is the angle at which two beams intersect.

7.1 Electron Capture The Electron Capture (BC),

H+ + x+ --+ H + X 2+,

(7.1.1)

of protons in singly charged ions is mainly investigated. The crossed-beam technique provides to measure capture cross sections O'c and cross sections 0'(X2+) for the production of doubly-ionized atoms in the final channel. The reaction

H+ +He+ --+ H+He++

(7.1.2)

represents the simplest system to study, involving only one electron. At present all known measurements of O'c are in excellent agreement at the center-of-mass energies Ecm = 10-150 keY (Fig. 7.1). Among different theoretical calculations of BC (7.1.2), the best description of O'c is given by the coupled-state approach [7.7]. V. G. Pal'chikov et al., Reference Data on Multicharged Ions © Spring-Verlag Berlin Heidelberg 1995

7.1 Electron Capture

181

Fig. 7.1. Cross section (Tc for H+ + He+ reaction ( _ : combined experimental data from [7.4]; e: Atomic Orbital (AO) calculations [7.7])

30 20 10

5

11:..LL.------,~~~

10

50 100Ecm [keV]

EC in collisions between protons and H -like ions, z=3,4,5,6,

H++Hz ,

were studied theoretically in [7.8] using a coupled Sturmian-pseudostate approximation, and a scaling law for U c was obtained: G(u) = Z7uc ,

G max ~ 3.2

U = (v/Z)2,

X 10- 15

cm2 ,

(7.1.3) U max

~ 0.5,

where v is the relative velocity of colliding particles. Experimental data on H-like ions with z > 2 are not available. Measured capture cross sections in collisions of protons with ions of the group-IIIB elements Al+, Ga+, In+ and Tl+ [7.9-11] are shown to be in poor agreement with available theoretical predictions. In the case of the simplest system of homonuclear ions, (7.1.4) the measured U c values are also in good agreement with sophisticated calculations (Fig. 7.2). In collisions of identical heavy-ion pairs X+

+ X+

~ X

+ X2+,

X = Li, Na, Ar, K, Rb, Xe, Cs,

(7.1.5)

u(X2+) values were measured [7.4].

For heavy-ion-fusion applications it is necessary to know U c and Uj separately for obtaining the total beam-loss cross section UL, which, for identical ion pairs, is equal to UL

= 2(2uc + Uj) = 2[uc + u(X 2+)].

(7.1.6)

In (7.1.6) the factor 2 appears because two particles are lost per interaction in the electron-capture reaction and both ions simultaneously act as projectile and target particles.

182

7 Ion-Ion Collisions Fig. 7.2. Cross section (Tc for He+ + He+ reaction ( _ : combined experimental data from [7.4]; AO calculation: ... [7.12], ••• [7.13])

20

•

5

2

5

20

1.0

100 Ecm [ke V]

10 Ecm [ke

V]

Fig. 7.3. Calculated (Tc values [7.16] for quasi-resonant

reactions (7.1.7)

First separate measurements of CTi and CTC for systems Xe+ + Xe+ and Bi+ + Bi+ have been described in [7.14]. Theoretical calculations for collisions of heavy systems are not available. Up to now, the only experimental data for ion-ion collisions involving multicharged ions exist for electron-loss cross sections CTL = CTc + CTi. These data have been obtained in [7.15] using a folded-beam ion-ion collider for pairs Af3+ + Af3+ and 1(r3+ + 1(r3+ at a fixed energy Ecrn = 60 keV:

+ .ru-J+) = CTd~+ + ~+) =

CTd.ru-J+

(6.1 ± 1.7) x 10- 16 cm2, (2.9 ± 0.8) x 10- 16 cm2 •

Theoretical calculations of CTc for multicharged ions were performed for quasiresonant electron capture [7.16] in collisions N 3+ + C2+ ~ N 2+ + C3+ - 0.44 eV,

F5+ + 04+ ~ F4+ + 05+ + 0.34 eV, showing the resonant structure of CTc (Fig. 7.3).

(7.1.7)

7.2 Ionization

183

"'

A bibliography on electron-transfer processes in ion-ion collisions can be found in [7.17].

7.2 Ionization The ionization processes in ion-ion collisions are mainly investigated for collisions between singly charged ions. Experimentally the one-electron ionization cross section O"i is usually obtained from the difference (7.2.1) where O"e is the capture cross section, 0"(X2+) is the total cross section for production of doubly-ionized atom in the final channel. For simple colliding partners H+ + He+ and He+ + He+ the measured O"i values obtained from (7.2.1) are shown in Figs. 7.4,5. Even for these one- and two-electron systems there is no agreement between experimental results and theoretical calculations [7.4]. In general, ionization of these systems is much less understood as compared to electron capture (Sect. 7.1). Ionization in collisions between protons and H-like ions with z = 3,4,5,6 was studied in [7.8] using a coupled-Sturmian-pseudostate approach and a scaling law was obtained, (see (7.1.3), G(u)

= Z4O"e,-

G max ~ 1.6

U

X

= (viz) 2 ,

10- 16 cm2 ,

umax ~ 1.1.

(7.2.2)

The proton-impact ionization of many-electron singly charged ions H+

+ X+,

X = AI, Ga, In, n, Ba, Sr

was also studied experimentally [7.9,10]. The corresponding sented in Fig. 7.6 in a scaled form, suggested in [7.9,21]:

O"i

values are pre-

Fig. 7.4. Cross sections for H+ + He+ collisions [7.18]. O"c-capture cross section I1j was obtained using (7.2.1), 0"2+ = O"C + Gj

184

7 Ion-Ion Collisions Fig. 7.5. Cross sections for He+ + He+ collisions: • [7.19], 0 [7.20]; O"j was obtained using (7.2.1): 0 [7.18], • [7.20]; !T2+ =!Tc +!Tj

20 10 5

2 1 0.5 10

5

20

50

100 Ecm [ke V]

G(u)

1&6

0 00 0 0 0 ...

o

0

00

_ ..

,"

~"i~ a.a 0

-17

10

5

G(u)

10

15

• .. •• • • •• 20

25

30 u

Fig. 7.6. Scaled ionization cross section (7.2.3) for collisions of protons with singly charged ions (0, 0, 0, '1/: AI+, Ga+, In+, n+ [7.9]; f).,.: Ba+, Sr+ [7.10]; T, +: Ba+, Sr+ [7.10] with allowance for autoionization)

~ [22 ~ qJCRyJ[j)2r', u ~ EJ(M 0;

[0),

(7.2.3)

where Ij is the binding energy of j-th target shell with qj equivalent electrons; z, E and M are the projectile charge, energy and mass, expressed in units of the electron mass, and 10 is the ionization potential of the target. As can be seen from Fig. 7.6, the accuracy of the scaling formula (7.2.3) is a factor of 2 even at high energies, because it does not take into account the contribution from excitation-autoionization processes, that was clearly indicated in [7.10].

7.3 Excitation Excitation of ions in ion-ion collisions was investigated mainly theoretically for proton impact H+ + X z -+ H+ + X;.

7.3 Excitation

185

The important characteristic of interaction in ion-ion collisions is the Coulomb repulsion which is described by the parameter 1'/

=

Zp(Z - 1)e2 !:l.E

f.Lv

2

hv

(7.3.1)

'

where zp is the charge of the projectile, Z is the spectroscopic symbol of the target ion, f.L is the reduced mass, v is the relative velocity and !:l.E is the transition energy in Ry. If v » ,JIi, the Born approximation can be used. For small and medium relative velocities, the close-coupling method gives the best results [7.22-27,35]. In the Born approximation, the excitation cross section of an ion by a charged structureless particle can be written in a closed analytical form, if the model potential for 2K-pole transitions is used [7.28]: . M

VK (R)

= -Zp(Olr

K 11)

RK (R

2

2 1/2' + RO)K+

K

#- 0

(7.3.2)

Here Ro is the effective (cut-off) radius, and VKM (R) ex {

RR~,K-l,

R ---+ 00, R ---+ O.

Usually Ro

= nont!z,

(7.3.3)

where nO,1 are the principal quantum numbers of the initial and final states of the target. The use of other types of Ro is also possible [7.28-30]. The typical behavior of the model dipole (K = 1) and quadrupole (K = 2) potentials (7.3.3) is shown in Fig. 7.7. Using first -order perturbation theory in the impact parameter representation and (7.3.2) the excitation probability can be written in the form [7.31]:

=

W[(p, v)

=

J

2

VKM(R)exp(i!:l.Et/h)dt

-=

2 K+l

= 2~~K+l (!:l.E)2Kq; La;vY';v(X, Xo, 1'/), em

X = p!:l.E/2v,

v

Xo

= Ro!:l.E/2v,

2 QK(Loso]o, L 1s1]1) (lollcKll/d qK = 2K + 1 2/0 + 1

(7.3.4)

r

(7.3.5)

[1

Po(r) PI (r )r'dr

(7.3.6) where p is the impact parameter, P(r) are the radial wave functions of the target, liCK II is the reduced matrix element, QK is an angular coefficient depending on quantum numbers LSI [7.32]; Eem and !:l.E are in Ry units. Here aKV are the

186

7 lon-Ion Collisions

RV (R)

ae=1

0.5

a!=2

2p-3p

+0.1

2s-2p

0

03

VR/aO

-0.1 0.1 0

2

4

6

VR/aO

Fig. 7.7. Dipole (K = 1) (left) and quadrupole (K = 2) (right) potentials for transitions in H-like ions. Solid curves: exact potentials; dashed curves: model potentials (7.3.2,3)

numeral coefficients: aKIJ = (_I)(K+IJ)/2 [(K - V)!(K

+ v)!]1/2 [(K -

V)!!(K

+ v)!!r l .

The functions YKIJ were obtained by integration over hyperbolic trajectories, corresponding to the Coulomb repulsion [7.33,34]. For v -+ 0, YKIJ <X exp(-Jrl1) and the excitation cross section u, <X exp( - 2Jr l1) = exp [ -

RoAEfJ}/2 ( Z (z EI/2 Jr P RoE

1»)]

.

(7.3.7)

For v -+ 00, YKIJ are expressed in terms of the McDonald functions Kn (x) and dipole and quadrupole excitation cross sections have the form:

J 00

u{ (v) = 2Jr

W; (p, v)pdp,

o I

u 1 (v)

3Z~JL

2

2

8fZ~JL

2

= TqlqJl(xo)[Jrao] = EAE qJl(XO)[Jrao],

(7.3.8) (7.3.9)

7.3 Excitation

187

4

3

+

+ \

+ \

2

~~

10

+

Na

2

VI (2 3 PO -2 3 P2 )

10 Fig.7.8. Proton-induced quadrupole transitions 2 3 PI-2 3 P2 and 2 3 Po-2 3 P2 in C-like Ne V and Na VI ions (Crosses: normalized cross sections (7.3.4,12); solid curves: close-coupling (low energies) and the Born calculations (high energies) [7.27]; dashed curves are for the interpolation of solid curves)

(7.3.10) where f is the dipole oscillator strength. Cross sections (7.3.8-10) for v » 1 exactly coincide with those obtained in the quantum-mechanical Born approxi91ation [7.29] with the model potential (7.3.2). The functions (7.3.10) are fit within a few percent by:

Q~1

f/11(x)~exp(-2x)ln { 2.193+x -[l+ln(1+0.8JX)]

f/12(X) ~ exp( -2x)[2 + x(31l" /2)1/2].

} ,

(7.3.11)

188

7 Ion-Ion Collisions

4

2

o~~~----~------------------~--~

103

102

E

IRy)

Fig. 7.9. Proton-induced quadrupole transitions 2 3 P3/2 - 22 PI/2 in F-like Fe XVIII ions (Solid curve: quasiclassical close-coupling [7.22]; dashed curve: normalized cross sections (7.3.4,12»

7 \

\

\ \

5

\

\ \

\ \ \

3

\ (8)

1

0.01

0.1

1.0

E IRy)

Fig. 7.10. Proton-induced excitation cross sections in H-like ions (dash-dotted curve: first-order perturbation theory (7.3.9); dashed curves: normalized cross sections (7.3.4,12); solid curves: closecoupling calculations [7.35] (8) He II; (b) Ar XVIII; (c) C VI)

Normalization of the Born cross sections can be performed by simple procedure:

J 00

O';'(v)

= 21l'

pdp sin2(W:)1/2

o

(7.3.12)

7.3 Excitation

189

1.0

05 (c)

--- -----------

o t)

100

1000

E [Ry]

10

100 E [Ry]

Fig. 7.10. Continued

Table 7.1. Proton-impact excitation cross sections (in units of a5) for the n = 21evels in ions ArIH, SI5+, Mgll+, C5+ and He+ as a function of the scaled energy E/z2 [7.351; z is the spectroscopic symbol of the ion

E[eV1/z 2

a(2plj2- 2s lj2) [a51

0.084 0.21 0.42 0.63 0.84 1.68 3.02 6.30 8.40 10.50 12.60 14.70 16.80 20.16 25.19 33.59

0.028 16.4 44.2 46.6 43.4 29.5 19.5 11.2 9.01 7.57 6.51 5.79 5.20 4.48 3.74 2.94

1.0 X 10-4 5.8 x 10- 3 0.46 1.86 3.50 3.65 3.66 3.64 3.45 3.31 3.10 2.78 2.54

6.5 X 10- 5 5.1 X 10- 3 0.437 0.946 0.871 0.687 0.582 0.495 0.440 0.392 0.329 0.257 0.181

2.17 72.7 84.0 64.5

2.81 X 10-4 0.292

1.89 X 10-4 0.326

0.106 0.319 0.531 1.063

190

7 Ion-Ion Collisions

Table 7.1. Continued E[eVl/z2

a(2pI/2- 2s l/2)

2.126 5.314 8.503 12.75 17.01 21.26 25.51 38.26

40.6 20.8 14.4 10.5 8.30 6.93 5.98 4.27

0.094 0.189 0.472 0.945 1.890 3.780 6.803 10.58 18.90 23.62 24.56 27.21

30.6 2.39 3.22 2.28 1.44 86.6 55.4 39.2 24.6 20.5 19.8 18.3

[a51

a(2s l/r2p3/2)

2.61 6.51 6.80 6.41 5.79 5.33 4.82 3.81

[a51

a(2pI/2- 2p3/2)

1.78 1.78 1.20 0.851 0.661 0.521 0.433 0.265

Mgll+

x x x x

102 102 102 102

x x x x x x x x x x x x x

102

0.06 3.60 15.6 27.1 28.7 26.0 20.2 17.7 17.4 16.4

0.06 4.32 9.69 8.02 5.29 3.66 2.07 1.60 1.54 1.38

C5+ 0.0378 0.0756 0.189 0.378 0.756 1.512 3.023 4.535 6.047 11.34 15.12 24.19 34.01

5.17 5.93 9.17 7.01 4.64 2.91 1.76 1.29 1.03 6.24 4.93 3.24 2.51

103 103 103 103 103 103 103 103 102 102 102 102

7.40 1.53 5.18 8.99 9.61 8.75 7.83 5.71 4.79 3.51 2.79

x 102 X 102 x 102 X 102 X 102 x 102 x 102 X 102 X 102 x 102

6.26 2.07 4.10 3.40 2.04 1.40 1.07 5.73 4.30 2.57 1.86

102 103 104 lOS lOS 105 105 104 104 104 104 104

15.2 2.91 5.41 1.09 6.55 2.82 1.41 7.20 4.82 3.64 2.03 1.42

x x

102 102 102 102 102 102 10 1 10 1 102 102

x x x x x x x x x x x

103 104 lOS 104 104 104 103 103 103 103 103

x x x x X

x x X

He+

0.0068 0.0204 0.034 0.068 0.340 0.680 1.701 3.40 6.80 10.20 13.61 23.81 34.01

4.67 1.38 1.80 1.70 7.30 4.58 2.34 1.36 7.76 5.56 4.34 2.72 1.98

x x x x x x x x x x x x x

104 106 106 106 105 105 lOS 105 104 104 104 104 104

0.12 1.02 6.95 5.59 2.28 2.55 1.97 1.37 8.77 6.59 5.32 3.50 2.66

x x x x x x X X

x x x x

[a51

7.3 Excitation

Table 7.2. Proton-impact excitation rate coefficients (va) for the n = 2 levels in the ions ArI7+, SI5+, MgIl+, C5+ and He+ [7.35]. The Z3 -scaled rates are expressed in units of 10- 7 cm3 sec -I, as a function of the scaled temperature kT /z2 (a.u.) z3(va) kT/Z2

z3(va)

z3(va)

2SI/2- 2 p3/2

2PI/2 -2p3/2

2PI/2 -

2SI/2

0.025 0.05 0.10 0.20 0.40 0.80

1.27 1.38 1.31 1.17 9.90 8.16

x x x x x x

103 103 103 103 102 102

0.25 0.05 0.10 0.20 0.40 0.80

1.38 1.45 1.35 1.19 1.01 8.28

x x x x x x

103 103 103 103 103 102

2.85 1.04 2.36 3.86 5.00 5.53 Mg II +

x x x x x x

0.025 0.05 0.10 0.20 0.40 0.80

1.55 1.55 1.42 1.23 1.04 8.51

x x x x x x

103 103 103 103 103 102

6.62 1.85 3.53 5.12 6.12 6.39 C5+

x x x x x x

0.025 0.05 0.10 0.20 0.40 0.80

1.95 1.81 1.59 1.35 1.12 9.08

x x x x x x

103 103 103 103 103 103

3.20 5.57 7.64 8.85 9.12 8.67

0.025 0.05 0.10 0.20 0.40 0.80

2.54 2.21 1.86 1.52 1.22 9.62

x x x x x x

103 103 103 103 103 102

ArI7+ 1.70 x Wi 7.35 x Wi 1.87 x 102 3.28 x 102 4.48 x 102 5.19x102 SI5+

1.08 3.21 5.54 6.68 6.39 5.25

x Wi X Wi x Wi x Wi x Wi x Wi

Wi 102 102 102 102 102

1.80 4.59 7.08 7.90 7.22 5.79

x x x x x

Wi 102 102 102 102 102

4.06 7.59 9.76 9.90 8.58 6.72

x x x x x

X

X

Wi Wi Wi Wi Wi Wi Wi Wi Wi Wi Wi Wi

x 102 x 102 X 102 x 102 x 102 x 102

1.46 x 102 1.71 x 102 1.62 X 102 1.35 x 102 1.04 x 102 7.67 x Wi

x x x x x x

3.16 2.65 2.06 1.55 1.14 8.20

He+ 1.35 1.57 1.62 1.54 1.37 1.19

103 103 103 103 103 103

x 102 X 102 x 102 x 102 x 102 x Wi

191

192

7 Ion-Ion Collisions

or

J 00

u:(v) = 2rr

pdp {I - exp(-wf)}.

(7.3.13)

o The use of (7.3.12) or (7.3.13) leads, as a rule, to identical results with an accuracy of 20-30%. Cross sections of excitation of multicharged ions by proton impact, calculated by close-coupling methods and using (7.3.4) and (7.3.12) are shown in Figs. 7.8-10. It is seen that calculations with the model potential give a quite good description of the excitation cross sections, except for the cases where the close-coupling effects are very important (Figs. 7.lOb,c). Recommended cross sections and rate coefficients for the excitation of transitions in H-like ions, calculated with the close-coupling method [7.35], are given in Tables 7.1,2.

7.4 Collisions Involving H- Ions The growth of interest to the physics of collisions of H- ions with positive and negative ions is stimulated by both fundamental and practical importance of H- ions, especially with respect to conversion efficiencies of H- to neutral If> ion-plasma neutralizers proposed for efficient neutral-beam heating of future fusion devices [7.36]. A review on experimental and theoretical data of singleand multiple-electron detachment ion collisions of negative ions with neutrals is given in [7.37]. In recent years, the use of advanced crossed-beam or mergedbeam techniques made it possible to obtain, for the first time, results covering collisions of H- ions with protons, multicharged ions and H- ions.

7.4.1 H+ + H- Collisions Three fundamental reactions in H+ + H- collisions were investigated (i) Transfer Ionization (TI): H+ + H- -+ H + H+ + e, (7.4.1) (ii) Electron detachment (ED): (7.4.2) H+ + H- -+ H+ + H + e (iii) Mutual Neutralization (MN): (7.4.3) H+ + H- -+ H + H. TI is a two-electron process in which one of the target electrons is captured by the projectile and the other one is ejected. The cross section UTI, measured with the use of merged-beam [7.38] techniques is given in Fig. 7.11. Theoretical treatment of TI is quite complicated because the strong coupling of many discrete states and a whole continuum has to be taken into account. The low-energy behavior of UTI (Fig. 7.11, dashed curve) is well described by the approach given in [7.39], when IT is considered as mutual neutralization (7.4.3) followed by autoionization.

7.4 Collisions Involving H- Ions

a [10- 16 cm 2] ~Dr-~-----'----~~--~

..-ft. 1"'

_1..J1h~. ~.T

1D

193

Fig. 7.11. Experimental TI cross section in collision H+ + H- _ H + H+ + e (Symbols: experiment; dashed curve: model calculations [7.38]

•

...

Ol

•tt t

+

001~~~~--~--~~~

Q1

1D

10D EI:Ill [ke V]

e

0.001

0,01

0,1

10 Ecm [keVj

Fig. 7.12. MN cross sections for reaction H+ + H- _ H + H. Experiment: f1 [7.47]. x [7.48] •• [7.49]. -e- [7.40.47,45-461. Theory: ... MO calculations [7.50]; - - - MO calculations [7.52]. _ _ close-coupling calculations [7.441

The ED processes were investigated in [7.40-44]. The MN process was investigated extensively in experiments [7.40,4S-49] and theoretically [7.44,SO-S2]. First measurements of MN cross sections are presented in Fig. 7.12 together with theoretical calculations. It has been pointed out that for energies Eern < 2 keY the contribution of H*(n = 2,3) + H(1s) channels to the MN process is very important and for energies Eern < S keV the capture of the active Is' electron in H- plays the main role and gives the largest contribution (80%) to the total cross section. The puzzling structures of O"MN observed in (7.44] still remain unclear.

7.4.2 H-+ H- Collisions Mutual Ionization (MI) in H- + H- collisions were investigated experimentally [7.S3] by means of the crossed-beam technique for two reactions double

194

7 Ion-Ion Collisions Fig. 7.13. Double u(2) (7.4.4) and triple u(3) (7.4.5) ionization cross sections. Symbols: experiment [7.53]. C1MC calculations: _ _ with potential (7.4.80), - - - with (7.4.6)

-15 10

-16

10

~(3)

___

/- t ftt01 -17 10

10

100

Ecm [keV] ionization: H-

+ H- -+ If + HO +2e

(7.4.4)

and triple ionization: H-

+ H-

-+

HO

+ If + 3e.

(7.4.5)

The measured double a(2) and triple a(3) ionization cross sections are presented in Fig. 7.13. The results are compared with calculations based on the three-body Classical-Trajectory Monte-Carlo method (CTMC) [7.54] and the independent particle model [7.55]. Three forms of model potentials were used to describe the interaction between the outer Is' electron and the If core (i.e., nucleus and inner Is electron):

VI = -air, V2 = -(1

a = 0.2354,

+ 1/r)e-2r -

0.405 V3 = ---[(1 r

2.25r-4e-2.55/r2

(7.4.6) (7.4.7)

+ ar)e- 2ar + (1 + r)e- 2r + 0.94(3 + a) + e-(a+l)r]. (7.4.8)

The use of potentials VI and V3 gives the best results (Fig. 7.13). The discrepancy between theory and experiment is explained by the fact that the CTMC method does not describe accurately the interchange of target and projectile electrons, which becomes increasingly important for decreasing impact energies.

7.4.3 Collisions of H- with Multicharged Ions Reactions, arising in collisions of H- -ions with singly and doubly charged ions H- + XZ+, Z = 1, 2, are investigated experimentally [7.56,57] and theoretically

7.4 Collisions Involving H- Ions (1

1~ 1

10

195

Fig. 7.14. Single u(1) and double u(2) electron-removal cross sections (7.4.9,10). Experiment: u(l): 0 [7.60], x [7.64]; u(2): 0 [7.60]. Theory: ~ CTMC calculations [7.60], - - - Keldysh theory [7.62]

[10- 15 cm 2 ]

~1)

~ 'I"

i
-1 10 0 2 4 6 8 10 Z

Table 7.3. Calculated single-electron removal cross sections in X2+-H- collisions (X = Ne, Ar and Xe) in units of 10- 14 cm2 as a function of Ecm [7.62]

Z

2

3

4

5

6

7

8

Ecm [keY] NeZ+-H-

10 15 20 30 40 50 60 80 100 120 140 160 200 250 300 400

0.769 0.675 0.614 0.521 0.462 0.412 0.373 0.316 0.276 0.247 0.224 0.206 0.178 0.155 0.137 0.112

1.867 1.674 1.537 1.350 1.227 1.124 1.042 0.923 0.824 0.745 0.684 0.631 0.552 0.482 0.429 0.356

3.088 2.800 2.595 2.306 2.109 1.961 1.841 1.643 1.495 1.385 1.281 1.192 1.055 0.924 0.828 0.694

4.399 5.770 3.976 5.287 3.734 4.901 3.348 4.437 3.075 4.101 2.867 3.844 2.700 3.626 2.455 3.308 2.247 3.069 2.085 2.852 1.953 2.682 1.846 2.538 1.649 2.308 1.457 2.061 1.312 1.863 1.104 1.578 AfZ+-H-

7.260 6.599 6.176 5.601 5.190 4.854 4.612 4.217 3.921 3.682 3.463 3.286 2.991 2.717 2.473 2.110

8.750 7.910 7.487 6.767 6.301 5.943 5.607 5.155 4.809 4.534 4.296 4.077 3.720 3.391 3.127 2.681

10.240 9.400 8.798 7.952 7.468 7.032 6.696 6.150 5.735 5.400 5.147 4.910 4.495 4.102 3.799 3.297

10 15 20 30 40 50 60 80 100 120

0.772 0.679 0.617 0.524 0.465 0.416 0.376 0.318 0.279 0.249

1.874 1.682 1.545 1.358 1.234 1.130 1.049 0.930 0.831 0.752

3.101 2.811 2.606 2.317 2.119 1.970 1.852 1.654 1.506 1.395

4.413 3.996 3.749 3.364 3.089 2.881 2.716 2.469 2.261 2.098

7.287 6.619 6.203 5.623 5.211 4.881 .4.634 4.239 3.941 3.703

8.777 7.950 7.514 6.790 6.330 5.970 5.639 5.184 4.835 4.557

10.267 9.440 8.825 7.993 7.497 7.059 6.728 6.179 5.761 5.432

5.797 5.308 4.915 4.456 4.122 3.862 3.642 3.324 3.086 2.870

196

7 lon-Ion Collisions

Table 7.3. Continued 2

Z

3

4

5

6

7

8

3.488 3.308 3.013 2.738 2.494 2.129

4.320 4.105 3.747 3.412 3.152 2.705

5.174 4.937 4.522 4.129 3.824 3.325

7.328 6.650 6.244 5.657 5.244 4.922 4.667 4.271 3.971 3.736 3.525 3.341 3.046 2.770 2.527 2.159

8.818 8.012 7.555 6.824 6.376 6.011 5.688 5.228 4.876 4.593 4.359 4.148 3.789 3.445 3.190 2.741

10.308 9.502 8.867 8.055 7.543 7.100 6.777 6.223 5.802 5.481 5.216 4.982 4.564 4.170 3.863 3.369

Ecm [keY] M+-H140 160 200 250 300 400

0.226 0.208 0.180 0.156 0.138 0.113

0.690 0.637 0.557 0.487 0.434 0.360

1.292 1.203 1.065 0.933 0.836 0.701

1.965 1.860 1.663 1.469 1.325 1.115

2.700 2.555 2.325 2.078 1.880 1.592

Xez+-H10 15 20 30 40 50 60 80 100 120 140 160 200 250 300 400

0.778 0.685 0.623 0.530 0.470 0.421 0.381 0.323 0.283 0.253 0.230 0.212 0.183 0.159 0.140 0.115

1.886 1.694 1.556 1.370 1.245 1.141 1.059 0.940 0.842 0.763 0.700 0.646 0.566 0.494 0.441 0.367

3.122 2.829 2.622 2.334 2.136 1.985 1.868 1.670 1.523 1.410 1.310 1.221 1.079 0.947 0.848 0.712

4.433 4.027 3.771 3.389 3.111 2.901 2.740 2.491 2.282 2.118 1.987 1.881 1.685 1.490 1.345 1.131

5.838 5.339 4.938 4.490 4.155 3.889 3.672 3.348 3.114 2.903 2.730 2.582 2.351 2.106 1.907 1.615

[7.58,59]. Very recently [7.59-61], the absolute cross sections were measured and calculated for singleH-

+ XZ+

~

If> + ...

and double-electron H- + XZ+ ~ H+

+ ...

(7.4.9) (7.4.10)

removal in collisions of H- with multicharged ions of inert gases XZ+, X = Ne, Ar, Xe, Z < 10. Experimental single- and double-electron removal cross sections «1(1) and (1(2») for H- -AfZ+ (z :::;; 8) collisions at Ecm = 50 keY are given in Fig. 7.14 together with theoretical calculations. In the case of singleelectron removal the best prediction is given by the Keldysh approximation [7.63]. The ratio of measured cross sections for Ar ions is (1(2) /(1(1) = (4.7 ± 1.2)%. Each of the (1(1) and (1(2) cross sections consists of two components: ionization and transfer ionization. For high energies both single- and doubleelectron removal is due to impact ionization rather than electron capture [7.60]. The Z2/ v 2 dependence for (1(1) and Z4/ v 4 for (1(2) (v is the relative velocity), predicted by the Born approximation, are invalid especially for high z. The measured (1(1) values are scaled by a z1.3 dependence. However, theory [7.63]

7.4 Collisions

Invo1vi~g

H- Ions

197

predicts another scaling law: G(u) =

u(v)/z,

u =

v2 /z.

(7.4.11)

The calculated cross sections u(l) for Nez+, M+ and Xe z+ ions (Table 7.3) follow the dependence (7.4.11). Both measurements and calculations show that the efficiency of H- neutralization mainly depends on the ion charge z, but not on the structure of the incident ion.

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Chapter 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17

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Subject Index

Amplitude 76 Analytical fitting approximations for cross sections 152, 157, 169, 171 rate coefficients 152, 156, 158 Anomalous magnetic moment 7, 20, 32 Asymptotic formulas 100 Atomic data banks 2 Autoionization 5, 75, 103, 116, 129 Bethe logarithm 7, 31, 129 Binding energy 48, 49 effect 8 Born approximation 185, 188 Branching ratio 116, 160 Breit interaction 27, 28, 58, 103 Bremsstrahlung 139 Burgess-Merts formula 132 Charge exchange 166, 171, 180 Charge nucleus 3, 7 Classic trajectory Monte Carlo method (CTMC) 176 Classification of spectral lines 3 Correction finite-nuclear mass 16 finite-nuclear size 16 mass polarization 30 quantum electrodynamical 2, 6, 31, 44, 150 relativistic I, 7, 59, 67, 70, 96 Coulomb interaction 1, 28 potential 10, 137, 185 screened potential 10 Coupling schemes 3 Decay 76, 89, 94, 97 Degeneration 7, 62, 64 Delta function 7, 9, 12, 16 Dirac equation 27, 68, 100 Effective charge 57 Electric polarizability 58, 60, 66 Electron anomalous magnetic moment capture 166, 171, 180 density 139

7, 32

mass 16, 66 total moment 4, 7, 16, 66 spin 4, 7 Electron-impact excitation 147 ionization 156, 162 Elwert factor 138 Energy 41, 48 conservation 94 degeneration 7, 64 level 6 shift 7, 11, 17, 63 Equation Dirac 27, 68, 100 Schroedinger 27 Excitation-autoionization 149, 159 Excitation of atoms 177 of ions 147, 184, 189 Excited states 32, 62, 76, 177 External field 12, 58, 71 Field stationary 58 monochromatic 58 radiation 74 strong 63 Fine-structure intervals 42 Fitting parameters 73, 131, 135, 152, 158, 169 Gaunt factor 125 Green's function 8 Hartree-Fock 61, 72 Heterogeneity parameters 70 Hyperfine structure 66, 68 Hyperpolarizability 58 Interaction radiation 74 interelectronic 62, 64 with external field 58, 71 Intermediate states 96 Ion H-like 3, 6, 11, 17, 22, 25, 58, 62, 67, 70,76 few-electron 43, 58, 72 He-like 3, 5, 20, 27, 60, 64, 68, 89 Li-like 5, 43, 123 Ion-atom collisions 166

214

Subject Index

Ion-ion collisions 180 Ionization by electrons 156 by heavy particles 174, 183 potential 23 Kramers formulas

Radiative corrections 7, 31, 44 transitions 58, 76, 89 Rate coefficients 126, 130, 151, 155 Recombination dielectronic 5, 128, 134 radiative 124, 144 Resonant excitation 153 ionization 160

125, 137

Lamb shift 7, 18 H-like ions 8, 15, 17, 21 He-like ions 26 Lifetime 76, 89 Li-like ions 5, 43 123, 150 Linear Stark effect 62, 70 Lotz formulas 157 Magnetic-dipole transition 4, 89 Magnetic moment of electron 7, 20, 32 Mixing singlet-triplet 28, 90 Momentum 74 Monochromatic field 58 Multielectron processes 162, 173, 176 Multipole polarizabilities 69 Neutral atoms I, 8, 30, 48, 58 Nonrelativistic limit 59, 66 Normalization of cross section 188 Nuclear mass 16 size correction 16 Order of magnitude 75 Oscillator strengths 23, 76, 100, 101, 147 Parity 89, 94 Particle velocity distributions Pauli approximation 28, 63 Photoionization and photorecombination 144 Photon angular momentum 76 Polarizability 58, 72 Polarization 140, 145 Potential 100, 185 Probabilities 74, 96, 103 Quantum field theory

7

124,

Satellites 15, 141 Scaling laws 157, 168, 175, 181, 197 Scattering of radiation 94 Schroedinger equation 27 Screening effects 138 Selection rules 74 Self-energy 7, 48 Semiempirical approximation 100 expression 72, 100, 101 Shielding factor 69 Sommerfeld formula 137 Spectroscopic symbol 3 Spherical Bessel function 76 Spin 4, 7, 66, 97 Susceptibilities 69 Stark effect 58, 62, 66, 70, 72 linear effect 62, 64, 70 quadratic effect 64 shift 59, 72 Stokes parameters 140 Strong field 63 Transition autoionization 75, 103, 105 electric 4, 74 magnetic 4, 74 radiative 76, 90, 105 Two-photon decay 94 Vacuum polarization Wavelengths

7, II, 33

20, 25, 34, 40, 46

X-ray spectra

I, 2, 140

Zeeman effect

67

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The paper in this book is made from low- or no-chlorine pulp and is acid free, in conformance with international standards for paper permanency.

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