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

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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

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Plasma Physics 2nd Edition Basic Theory with Fusion Applications By K. Nishikawa and M. Wakatani

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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

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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


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 \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


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

N

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


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(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


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


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
!!> 10.0 t 0

CI)

N

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(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

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


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


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


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


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

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)


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 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


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


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


4.4 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


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


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


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


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


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, 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


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


-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


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


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


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


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


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


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


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 + ...

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.

References

Chapter 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19

1.20 1.21 1.22 1.23 1.24 1.25 1.26 1.27

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Chapter 1 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 2.20 2.21 2.22 2.23 2.24 2.25 2.26 2.27 2.28 2.29 2.30 2.31 2.32 2.33 2.34 2.35 2.36 2.37 2.38

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2.66 2.67 2.68 2.69 2.70 2.71 2.72 2.73 2.74 2.75 2.76 2.77 2.78 2.79 2.80 2.81 2.82

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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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Data Mining on Multimedia Data

Data Mining on Multimedia Data

Data Compression The Complete Reference

Data Compression: The Complete Reference

Data Compression: The Complete Reference

Data Compression: The Complete Reference

CIBSE Guide C: Reference Data

Reference Data for Radio Engineers

Electro-diffusion of ions

Electro-diffusion of ions

Ions in Polymers

Biosorbents for Metal Ions

On Reference and Sense

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Essays on Self-Reference

On Self-Reference

Essays on Self-Reference

Engineering Data on Mixing

Stata Longitudinal-Data Panel-Data Reference Manual: Release 11

Statistical Data Editing: Impact on Data Quality

Bioelectrochemistry.. Ions, Surfaces, Membranes

The.Ethical.EquOI^ions.(1945)

Metal Ions in Fungi

Data Compression: The Complete Reference, 3E

Data Compression: The Complete Reference, Fourth Edition

Reference Data for Engineers Radio, Electronics, Computer & Communications (Reference Data for Engineers)

Physics of Atoms and Ions

Stata Data-Management Reference Manual: Release 11

Reference data on multicharged ions