iosdft

Interpretation of Spectral

Data from Tokamaks.

A thesis submitted for the degree of

Doctor of Philosophy

by

Adrian Matthews, B.Sc., (q.u.b. 1994)

M.Sc. (q.u.b. 1995)

Faculty of Science

Department of Pure and Applied Physics

The Queen's University of Belfast

Belfast, Northern Ireland

June 1999

This thesis is dedicated

to my family

Contents

Acknowledgements i

List of Tables v

List of Figures vi

Publications 1

1 Introduction 2

1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Principal Methods for Electron-Impact Excitation Calculations . . 4

1.3 Atomic E�ects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 Theoretical methods for atomic structure and the code CIV3 18

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 The Hartree-Fock method . . . . . . . . . . . . . . . . . . . . . . 23

2.2.1 Correlation energy . . . . . . . . . . . . . . . . . . . . . . 26

2.2.2 The Self-Consistent �eld method . . . . . . . . . . . . . . 28

2.3 The Con�guration Interaction method . . . . . . . . . . . . . . . 29

2.3.1 Determination of the expansion coe�cients . . . . . . . . . 31

2.3.2 Setting up the Hamiltonian matrix . . . . . . . . . . . . . 33

2.3.3 Optimization of the radial functions . . . . . . . . . . . . . 34

2.4 The Con�guration-Interaction Bound State Code - CIV3 . . . . . 36

2.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3 The R-matrix method and codes 43

3.1 The R-matrix method . . . . . . . . . . . . . . . . . . . . . . . . 44

3.1.1 Basic ideas and notation . . . . . . . . . . . . . . . . . . . 45

3.1.2 Constructing the targets . . . . . . . . . . . . . . . . . . . 47

3.1.3 The R-matrix basis . . . . . . . . . . . . . . . . . . . . . . 48

3.1.4 The internal region . . . . . . . . . . . . . . . . . . . . . . 50

ii

CONTENTS iii

3.1.5 The R-matrix . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.1.6 The Buttle correction . . . . . . . . . . . . . . . . . . . . . 55

3.1.7 The external region . . . . . . . . . . . . . . . . . . . . . . 56

3.1.8 Matching the solutions . . . . . . . . . . . . . . . . . . . . 59

3.1.9 Open Channels . . . . . . . . . . . . . . . . . . . . . . . . 59

3.1.10 Electron collision cross sections . . . . . . . . . . . . . . . 61

3.1.11 R-matrix Summary . . . . . . . . . . . . . . . . . . . . . . 63

3.2 The R-matrix Codes . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.2.1 RMATRX STG 1 . . . . . . . . . . . . . . . . . . . . . . . 64

3.2.2 RMATRX STG 2 . . . . . . . . . . . . . . . . . . . . . . . 66

3.2.3 RMATRX STG H . . . . . . . . . . . . . . . . . . . . . . 68

3.2.4 The external region codes . . . . . . . . . . . . . . . . . . 69

3.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4 Electron-impact excitation

of Ni XII 72

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.2 Calculation Details . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.2.1 Target Wave Functions . . . . . . . . . . . . . . . . . . . . 75

4.2.2 The Continuum Expansion . . . . . . . . . . . . . . . . . . 77

4.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 80

4.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.5 Explanation of Tables . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.6 Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5 Plasma source and Instrumentation 142

5.1 Tokamaks and Nuclear Fusion . . . . . . . . . . . . . . . . . . . . 143

5.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 143

5.1.2 Tokamaks and nuclear fusion . . . . . . . . . . . . . . . . . 145

5.1.3 Magnetic Con�nement . . . . . . . . . . . . . . . . . . . . 146

5.1.4 Plasma heating methods . . . . . . . . . . . . . . . . . . . 149

5.1.5 Impurities . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

5.1.6 Con�nement Modes . . . . . . . . . . . . . . . . . . . . . . 151

5.2 Plasma Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . 151

5.3 Tokamak Experiments . . . . . . . . . . . . . . . . . . . . . . . . 152

5.4 Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

5.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 156

5.4.2 The basic instrument . . . . . . . . . . . . . . . . . . . . . 157

5.4.3 Multichannel Detector Mode . . . . . . . . . . . . . . . . . 158

CONTENTS iv

5.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

6 Line Ratio Diagnostics for the JET Tokamak 166

6.1 Line Ratio Diagnostics for Tokamak Plasmas . . . . . . . . . . . . 167

6.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 167

6.1.2 Statistical equiilibrium equations . . . . . . . . . . . . . . 168

6.2 Ni XII Line Search on the JET Tokamak . . . . . . . . . . . . . . 172

6.2.1 Line search methods . . . . . . . . . . . . . . . . . . . . . 174

6.3 ADAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

6.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 167

6.3.2 Speci�c z excitation - processing of metastable and excited

populations . . . . . . . . . . . . . . . . . . . . . . . . . . 176

6.3.3 Source data . . . . . . . . . . . . . . . . . . . . . . . . . . 179

6.3.4 Metastable and excited population - processing of line emis-

sivities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

6.4 Results and Conclusions . . . . . . . . . . . . . . . . . . . . . . . 182

6.5 Thesis Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 221

6.6 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

6.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

List of Tables

4.1 Orbital parameters of the radial wavefunctions. . . . . . . . . . . 100

4.2 Target state energies (in a.u.) . . . . . . . . . . . . . . . . . . . . 103

4.3 Energy points between the thresholds of Ni XII. . . . . . . . . . . 105

4.4 Oscillator strengths for optically allowed LS transitions in Ni XII. 107

4.5 E�ective collision strengths for Ni XII . . . . . . . . . . . . . . . . 108

5.1 The principal JET machine parameters. The values quoted are the

maximum achieved values. . . . . . . . . . . . . . . . . . . . . . . 155

6.1 Previously measured wavelengths of Ni XII observed in the JET

tokamak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

6.2 JET pulses where laser ablation of nickel occured . . . . . . . . . 183

6.3 JET pulses checked by methods I and II . . . . . . . . . . . . . . 183

6.4 Ni XII wavelength identi�cations . . . . . . . . . . . . . . . . . . 184

6.5 JET pulses where Ni XII lines were identi�ed . . . . . . . . . . . 184

6.6 NiXII line ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

6.7 Derived temperatures of the plasma at an electron density = 10

11

cm

�3

189

v

List of Figures

2.1 Basic owchart for the CIV3 code . . . . . . . . . . . . . . . . . . 40

4.1 Collision strength and e�ective collision strength for the 3s

2

3p

5 2

P

o

1=2

{ 3s

2

3p

5 2

P

o

3=2

transition. . . . . . . . . . . . . . . . . . . . . . . . 86


2

3p

5 2

P

o

3=2

{ 3s3p

6 2

S

1=2

transition. . . . . . . . . . . . . . . . . . . . . . . . 87


2

3p

5 2

P

o

3=2

- 3s

2

3p

4

(

3

P ) 3d

4

D

1=2

transition. . . . . . . . . . . . . . . . . . . . 88


2

3p

4

(

3

P ) 3d

4

D

1=2

- 3s

2

3p

4

(

3

P ) 3d

4

F

3=2

transition. . . . . . . . . . . . . . . . . 89


2

3p

5 2

P

o

1=2

{ 3s

2

3p

4

(

3

P )3d

4

D

5=2

transition. . . . . . . . . . . . . . . . . . . . 90


2

3p

5 2

P

o

3=2

- 3s

2

3p

4

(

1

D) 3d

2

P

3=2

transition. . . . . . . . . . . . . . . . . . . . 91

5.1 Tokamak geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 147

5.2 Tokamak �eld con�guration . . . . . . . . . . . . . . . . . . . . . 148

5.3 JET tokamak device . . . . . . . . . . . . . . . . . . . . . . . . . 152

5.4 Tokamak records . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

5.5 Multichannel detector system . . . . . . . . . . . . . . . . . . . . 159

5.6 Con�guration of the KT4 multichannel spectrometer . . . . . . . 162

6.1 Basic owchart for ADAS207 . . . . . . . . . . . . . . . . . . . . 181

6.2 Identi�cation of Ni XII lines in JET pulse 34938 . . . . . . . . . . 191

6.3 Plasma conditions of JET pulse 34938 . . . . . . . . . . . . . . . 192

6.4 Magnetic �eld con�guration of JET pulse 34938 . . . . . . . . . . 193




6.8 Superimposition of lines in JET pulse 31273 . . . . . . . . . . . . 197

6.9 Integration of the lines in JET pulse 31273 . . . . . . . . . . . . . 198


vi

LIST OF FIGURES vii










6.20 Plot of the theoretical line ratio, R

1

, as a function of electron density.209


1

, as a function of electron tem-

perature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210


2



2


perature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212


3



3


perature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214


4



4


perature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216


5



5


perature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218


6



6


perature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

6.32 Identi�cation of Ni XII from SOHO . . . . . . . . . . . . . . . . . 225

Publications

A list of publications resulting from work presented in this thesis is given below.

� Matthews A., Ramsbottom C.A., Bell K.L. and Keenan F.P. :

E�ective collision strengths for �ne-structure transitions from the 3s(2)3p(5)

P-2 ground state of chlorine-like NiXII

Astrophys. J., 492, 415-419, (1998)

� Matthews A., Ramsbottom C.A., Bell K.L. and Keenan F.P. :

E�ective collision strengths for electron-impact excitation of NiXII

At. Data Nucl. Data Tables, 70, 41-61 (1998)

1

Chapter 1

Introduction

2

1.1 Overview 3

1.1 Overview

Emission lines of highly ionized stages of the iron group elements Ti, Cr, Fe,

and Ni are used for diagnostic purposes of high temperature plasmas with central

electron temperatures up to the keV range.

The need for accurate electron-ion collision data is immense, with applications

in such diverse �elds as astronomy and fusion research. Several calculations for

nickel ions have been published since the late 1960's, but these vary considerably

in sophistication and accuracy. In the intervening time period important atomic

e�ects such as con�guration interaction and autoionizing resonances have been

recognized and incorporated. Consequently theoreticians have been challenged to

improve their calculations to provide reliable diagnostics in high-resolution mea-

surements associated with fusion plasmas and astronomical sources in all wave-

length ranges from the infrared to hard x-ray. Unfortunately, little attention has

been paid to electron excitation rate calculations for NiX { NiXIII, with exist-

ing work having been performed in either the Distorted-Wave or Gaunt Factor

approximations, which do not consider resonance contributions (Blaha 1968 and

Krueger & Czyzak 1970). The reliability of the electron excitation rates depends

upon the accuracy of the collision strengths over the temperature range consid-

ered. In turn the reliability of the collision strengths depends most critically upon

the number of target states included in the R-matrix wavefunction expansion,

together with the con�guration-interaction wavefunction representation of these

target states.

This thesis provides data for Cl-like NiXII. The knowledge of excitation en-

ergies and lifetimes of the 3s3p

6

and 3p

4

3d can be useful in the fusion and as-

trophysical applications mentioned above (Jup�en et al. 1993). Theoretical data

for multiply charged ions remain relevant for astrophysical precision spectroscopy

even though several previously unidenti�ed solar lines can now be assigned to

1.2 Principal Methods for Electron-Impact Excitation Calculations 4

transitions of chlorine-like nickel. For fusion research, reliable data for the 3s

2

3p

5

-

3s

2

3p

4

3d transition array are needed for reasons outlined below. The walls of

the JET are a Ni/Cr alloy | hence these elements provide impurity ions in the

plasma, with other contaminating elements, such as Fe, also contributing. NiXII

is a low ionisation stage which is unexpected within the bulk plasma of a tokamak

due to a high electron temperature (T

e

) of approximately 15 keV. However in the

\divertor box" region and plasma edge (the scrape-o� layer, or SOL) where the T

e

is much lower (perhaps in the 10 | 100 eV range) this stage is expected to exist.

The derivation of plasma parameters (T

e

, N

e

, ion concentrations) for this region

would allow the e�ciency of using the divertor to extract energy and impurity

ions from the plasma to be quanti�ed.

Explanation of Contents

The following sections contain a discussion of modern techniques for low-energy

electron-impact excitation calculations. Under Principal Methods is a discussion

of the theoretical methods employed in the majority of calculations, and under

Atomic E�ects is a discussion of the relative importance of some of the main e�ects

usually incorporated. Types of Transitions and Scaling Laws list these factors

as functions of nuclear charge and incident electron energy. The scaling laws are

sometimes useful for judging, approximately, the accuracy of the computed values.

1.2 Principal Methods for Electron-Impact Ex-

citation Calculations

The methods used in the computation of data for ions are brie y described below.

A more detailed account of the basic theory and methods for electron impact

excitation of positive ions may be found in the reviews by Seaton (1975) and


Henry (1981).

The Collisional Problem

The Schr�odinger equation for the electron-ion collision problem may be expressed

in terms of the scattering electron moving in the potential of the target ion. The

radial part of the wave function of the scattering electron is written generally as

�

d

2

dr

2

i

�

l

i

(l

i

+ 1)

r

2

i

+ k

2

i

�

F (i; r) = 2

X

i

0

fV

ii

0

�W

ii

0

gF (i

0

; r) (1.1)

where F is the radial function in a given channel (represented by i or i

0

). The

summation on the right hand side is over all discrete and continuum states. V

ii

0

and W

ii

0

are direct and exchange potential operators, respectively. The W

ii

0

are

integral operators and therefore equation 1.1 represents an in�nite set of coupled

integrodi�erential equations. The following sections discuss the various approxi-

mations found in the literature for solving this equation.

Distorted Wave (DW) Approximation

Usually for ions more than a few times charged, the DW approximation may be

employed. There are several di�erent formulations of the DW method, see Henry

(1981), but the basic feature is the assumption that coupling between scattering

channels is weak and therefore the relevant matrix elements need include only the

initial (V

ii

0

= 0) and �nal states. However, the method allows for the distortion

of the channel wave functions, from their asymptotic Coulomb form, in the target

potential. The general criterion for the validity of the DW approximation is that

the absolute value of the reactance matrix element j K

ii

0

j be � 1, a condition

that is satis�ed for highly charged ions since K � (Z � N)

�1

. For su�ciently

high charge of the ion (depending on the isoelectronic sequence) the DW method


is comparable to the Close Coupling (CC) approximation (see next section) and

may be several times less expensive in terms of computing time and e�ort. Dif-

ferent formulations of the DW approximation have sought to improve upon the

basic method in a number of ways, such as incorporating additional polarization

e�ects, McDowell et al. (1973) and taking some account of the e�ect of autoion-

izing resonances Pradhan et al. (1981). With respect to resonances, it should be

mentioned that although the DW method by its very nature does not allow for

coupling between open and closed channels (i.e., no resonances), one may intro-

duce, as in the UCL (University College London) CC formulation (IMPACT),

bound channel wave functions in the total eigenfunction expansion for the (e +

ion) system. These give rise to poles in the scattering matrix in the continuum

energy region and thus account for a limited number of resonances in the cross

sections. Most of the DW calculations found in literature neglect resonance ef-

fects. However, Badnell et al. (1991) showed how resonance structures can be

accounted for with the DW approximation and made detailed comparisons with

RMATRX calculations.

Close-Coupling (CC) Approximation

Truncating the sum on the right hand side of equation 1.1 to a �nite number

of excited states of the target ion and solving the remaining coupled equations

exactly yield the NCC approximation, where N refers to the number of states

included (usually small). The CC approximation is the most accurate method for

solving the e-ion collision problem as it allows for full coupling between channels

(target ion + scattering electron), which is often strong at low energies. Pro-

vided the energy is restricted to the region below the highest term included in

the eigenfunction expansion, resonances due to the interaction between open and

closed channels are automatically included. The CC approximation is employed

for atoms and ions where one expects strong coupling between the states included


in the target expansion. This is usually the case for up to a few times charged

ions or heavy ions (like nickel), when the energy levels are close together or when

one �nds transitions with strong associated multipole moments between several

levels.

Most of the existing CC calculations have been carried out using two sets of

codes, IMPACT, Crees et al.. (1978), and RMATRX, Burke and Robb (1975),

which employ di�erent numerical procedures but yield results with similar accu-

racy and detail.

RMATRX refers to the R-matrix method (adopted from nuclear physics),

which incorporates the numerical procedure of matrix diagonalization of the (N+

1) electron Hamiltonian to yield the R-matrix which is related to the usual scat-

tering parameters. Another noniterative method for solving the integrodi�erential

(ID) equations is NIEM (Smith & Henry 1973).

Each of the program \packages" in turn consists of three main programs for

� (i) calculating the target wave functions, energy levels, oscillator strengths,

etc.

� (ii) computing the \collision algebra" i.e. the potential operators V

ii

0

and

W

ii

0

; and

� (iii) solving the ID equations themselves, including the asymptotic region

where, due to the neglect of exchange terms, they assume the form of coupled

di�erential equations.

Matching the asymptotic and the \inner region" (with exchange) solutions

yields the reactance matrix, denoted as R by the IMPACT group and as K by

the RMATRX and NIEM users. For the atomic structure calculations, (i), the

RMATRX and NIEM users employ the computer program CIV3 based on the

Hartree-Fock method for computing one-electron orbitals, and IMPACT users


employ the program SUPERSTRUCTURE. Both these codes include con�gura-

tion interaction.

The R-matrix method has proven to be computationally more e�cient than

other methods, in particular for delineating the extensive resonance structures

in the cross sections that require calculations at a large number of energies (a

few hundred to a few thousand). The R-matrix method entails the division of

con�guration space into an inner and an outer region. The inner region comprises

the \target" or the \core" atom or the ion and the (electron + target) system

solutions are expanded in terms of basis functions satisfying logarithmic R-matrix

boundary conditions at a radius dividing the inner and the outer regions. The

outer region solutions are obtained neglecting exchange but including long-range

multipole potentials (i.e. the terms W

ii

0

in equation 1.1 are omitted). Physically

relevant quantities, such as the scattering matrix, are obtained by matching the

inner and the outer solutions at the R-matrix boundary.

Coulomb-Born (CB) Approximation

For highly charged ions and for high electron energies a further approximation may

be made: neglecting the short range distortion of the Coulomb scattering waves

due to the detailed interaction between the target and the incident electron. This

Coulomb-Born approximation is unreliable for low energies (near threshold) or

for lightly charged ions. The resulting error in the cross sections may be a factor

of 3 or more; however for highly charged ions or optically allowed transitions the

error is much lower. Only the background cross sections are calculated, without

allowance for resonance e�ects. In the earlier standard version of the CB method

the exchange e�ect is not included and therefore probability amplitudes for spin

change transitions cannot be calculated. Most of the highly charged nickel ions

have been treated with this method.


Coulomb-Bethe (CBe) Approximation

The basis of the CBe approximation is that the collisional transition may be

treated as an induced radiative process. It is employed for optically allowed tran-

sitions where, due to the long range dipole potential involved, it is usually neces-

sary to sum over a large number of orbital angular momenta (l) of the incident

electron. The method is valid for l - waves higher than a given l

0

, which depends

on the ionic charge, and is used in conjunction with DW or CC approximations

for low l - waves to complete the l summation (Pradhan 1988). If one takes �r

to be the mean radius of the target, the condition for the validity of the CBe

approximation is

l > (k

2

�r

2

+ 2z�r +

1

4

)

1=2

�

1

2

� l

0

(1.2)

where z = Z�N . Thus for allowed transitions the scattering calculations may be

divided according to the sets of partial waves l � l

0

and l

0

< l <1; the former are

treated in the DW or CC approximations that take account of the detailed close

range interaction and the latter in the CBe approximation. The partial wave sum-

mation for forbidden transitions usually converges for l � l

0

. The CBe collision

strength is expressed in terms of the dipole oscillator strength for the transition

and radiative Coulomb integrals. Methods for the evaluation of these integrals,

their sum rules, and the collision strengths are described by Burgess, (1974) and

Burgess and Sheorey (1974). A discussion of the general forms of the Born and

Bethe approximations is given by Burgess, and Tully (1978), who also showed

that in the limit of in�nite impact energies the CBe approximation overestimates

the cross sections by a factor of 2 due to the fact that the approximation is invalid

for close encounters i.e. low l - waves.

In recent years \top up" procedures have been developed to complete the sum

over higher partial waves not included in the CC formulation e.g. Burke and

Seaton (1986).

1.3 Atomic E�ects 10

Other Approximations

Gaunt factor or the �g approximation This method was used by Kato (1976)

for many ionisation stages of nickel. Analogous to the CBe approximation, the �g

formula expresses the collision strength for optically allowed transitions as

ij

=

8�

p

3

!

i

f

ij

E

ij

�g (1.3)

where f

ij

is the dipole oscillator strength and �g (called the e�ective Gaunt factor)

is an empirically determined quantity. This expression was suggested by Burgess

(1961), Seaton (1962) and Van Regemorter (1962) and could be accurate to about

a factor of 2 or 3 (sometimes worse). The value of �g may actually vary widely

depending upon the isoelectronic sequence. At high energies the collision strengths

for allowed transitions increase logarithmically and the proper form is given as

ij

=

4!

L

f

ij

E

ij

ln

�

4k

2

(�rE

ij

)

2

�

(1.4)

where k

2

is the incident electron energy (Rydbergs) and �r is the mean ionic radius.

1.3 Atomic E�ects

The accuracy of a given calculation depends not only on the principal method

employed but also on the contributing atomic e�ects included. the relative im-

portance and magnitude of these e�ects vary widely from ion to ion and even

within an isoelectronic sequence, large variations with Z may be found; for ex-

ample, pure LS coupling calculations become invalid for some transitions in high

Z ions. Another example is where, for the same ion, a close coupling calculation

may be less accurate than a distorted wave calculation if the target wave functions

in the latter take into account con�guration interaction but those in the former

do not.


Exchange

The total e+ion wave function should be an antisymmetrised product of the N+1

electron wave function in the system with N electrons in a bound state of the target

ion and one free electron. Nowadays nearly all scattering calculations satisfy the

antisymmetry requirement and exchange is accounted for, but there are some older

calculations in the literature where exchange is neglected. It has been shown that

apart from spin ip transitions, which proceed only through electron exchange, it

may be necessary to include exchange even for optically allowed transitions when

low l-wave contribution is signi�cant

Coupling

When the coupling between the initial and the �nal states is comparable to or

weaker than the coupling with other states included in the target representation,

the scattered electron ux is diverted to those other states and coupling e�ects

may signi�cantly a�ect the cross sections. Thus the weak coupling approximations

such as the CB tends to overestimate the cross sections. As the ion charge in-

creases, the nuclear Coulomb potential dominates the electron-electron interaction

and correlation e�ects (such as exchange and coupling) decrease in importance.

Optically allowed transitions are generally not a�ected much.

Con�guration Interaction

It is essential to obtain an accurate representation for the wave functions of the

target ion. The error in the cross sections is of the �rst order with respect to the

error in the ion wave functions. Usually it is necessary to include CI between a

number of con�gurations in order to obtain the proper wave functions for states

of various symmetries. The accuracy may be judged by comparing the calculated


eigenenergies and the oscillator strengths (in the length and the velocity formula-

tion) with experimental or other theoretical data for the states of interest in the

collision.

Owing to the constraints on computer core size, it is usually impractical to

include more than the �rst few con�gurations in most close coupling calculations.

However, calculations that include many con�gurations and involve hundreds of

scattering channels are now being carried out on supercomputers. To circumvent

the problem of core size restrictions, it is frequent practice to include pseudostates

with adjustable parameters in the total eigenfunction expansion over the target

states for additional CI. Transitions involving the pseudostates themselves are

ignored. They are used to simulate neglected con�gurations. Single con�gura-

tion (SC) calculations are generally less accurate than those including CI. In the

asymptotic region the coupling potentials are proportional to

p

f , where f is the

corresponding oscillator strength. It is therefore particularly important that the

wave functions give accurate results for these oscillator strengths.

Relativistic or Intermediate Coupling (IC) E�ects

Relativistic e�ects become important with increasing nuclear charge and have to

be considered explicitly (Bethe and Salpeter 1980). For low Z ions (including

nickel) the cross sections for �ne structure transitions may be obtained by a pure

algebraic transformation from the LS to the IC scheme (e.g. through program

JAJOM by Saraph (1978)). In general, the ratio of the �ne structure collision

strengths to multiplet collision strengths depends on the recoupling coe�cients,

but for the case of S

i

= 0 or L

i

= 0 it can be shown that:

(S

i

L

i

J

i

; S

j

L

j

J

j

)

(S

i

L

i

; S

j

L

j

)

=

(2J

i

+ 1)

(2S

j

+ 1)(2L

j

+ 1)

(1.5)


As the relativistic e�ects become larger one may employ three di�erent approaches.

The �rst, based on the Dirac equation, is for light atoms and will not be discussed

here. The second method is to generate term coupling coe�cients < S

i

L

i

J

i

j�

i

J

i

>

which diagonalize the target Hamiltonian including relativistic terms (Breit-Pauli

Hamiltonian); �

i

J

i

is the target state representation in IC. These coe�cients are

then used together with the transformation procedure mentioned above to ac-

count for relativistic e�ects. The second method is incorporated in the program

JAJOM and is described by Eissner et al. (1974). A similar method is discussed

by Sampson et al. (1978).

The third approach is by Scott and Burke (1980), who have extended the

close coupling nonrelativistic RMATRX package to treat the entire electron-ion

scattering process in a Breit-Pauli scheme, treating intermediate coupling more

accurately. Resonances in �ne structure transitions may also be taken into account

in the relativistic RMATRX program or in an extended version of the JAJOM

program.

Resonances

For positive ions, due to the in�nite range of the Coulomb potential, there are

several in�nite series of Rydberg states converging on each bound state of the

ion. When such Rydberg states lie above the ionization limit, as is often the case

when they converge onto excited states of the target ion, they become autoionizing

(undergoing radiationless transition to the continuum) with resulting peaks and

dips in the cross section at energies that span the width of the autoionizing states.

If i and j are initial and �nal levels then there would be a series of resonances

in (i; j) belonging to excited states k > j. The magnitude of the resonance

contribution depends upon the coupling between states i; k and k; j. Neglecting

interference terms, the strength of this coupling is indicated by (i; k) and (k; j).

It follows that if the transition i ! j is weak and the coupling to higher states


is strong, then resonances might be expected to play a large role. Thus the weak

forbidden or semiforbidden transitions are particularly susceptible to resonance

enhancement. Most of the older work (pre-1970 work such as Blaha 1968) did not

take into account this resonant contribution and calculations were made either

at a single incident energy, usually near threshold, or at 2 or 3 energies above

threshold.

There are several methods for taking account of resonance e�ects. In the

RMATRX calculations, the resonance pro�les are obtained in detail by calculating

the cross section directly at a large number of energies. The RMATRX code is

capable of including resonances nearly exactly.

The e�ect of autoionization may diminish if the resonances can also decay

radiatively to a bound state, producing a recombined ion (i.e. dielectronic recom-

bination, Presnyakov & Urnov 1975 and Pradhan 1981). This would be expected

to be the case with highly charged ions where the radiative probabilities for allowed

transitions begin to approach the autoionization probabilities, approximately 10

12

-

10

14

s

�1

(the autoionization probability is nearly independent of ion charge). In

certain energy ranges the radiative decay completely dominates the autoionization

e�ect in the cross section, but the overall e�ect of dielectronic recombination is to

reduce the rate coe�cients by 10-20%. So autoionizing resonances may enhance

the excitation rates by up to several factors, with some reduction due to radiation

damping in the continuum.

Types of Transitions and Scaling Laws

Transitions may be classi�ed according to the range of the potential interaction

(V

ii

0

� W

ii

0

) in equation 1.1. Spin change transitions depend entirely on the

exchange term W

ii

0

, which is very short range since the colliding electron must

penetrate the ion for exchange to occur. Therefore, only the �rst few partial waves


are likely to contribute to the cross section, but these involve quite an elaborate

treatment (e.g. close coupling). For allowed transitions, on the other hand, a

fairly large number of partial waves contribute and similar approximations (e.g.

Coulomb-Born) often yield acceptable results. The asymptotic behaviour of the

collision strengths for allowed and forbidden transitions is as follows (x = E=E

ij

,

where E is the incident and E

ij

is the threshold energy):

(A) (i; j) � constant for forbidden transitions as x!1 �L 6= 1;�S 6= 0

(B) (i; j) � x

�2

for spin change transitions as x!1 �S 6= 0

(C) (i; j) � a ln 4x for allowed transitions as x!1 �l = 1;�S = 0

The slope a in the last equation is proportional to the dipole oscillator strength

(see equation 1.4). The above forms are valid for transitions in LS coupling. For

highly charged ions where one must allow for relativistic e�ects, through, say, an

intermediate coupling scheme, sharp deviations may occur from these asymptotic

forms, particularly for transitions of the intercombination type.

Kim and Desclaux (1988) have presented a general discussion of ther energy

dependence of electron-ion collision cross sections and have given �tting formulas

appropriate for many plasma applications.

Tests of Data Accuracy

Self-consistency checks of theoretical calculations through the analysis of quan-

tum defects (Pradhan & Saraph 1977), oscillator strengths (Doering et al. 1985),

and photoionization cross sections (Sampson et al. 1985) calculated using the same

theoretical and numerical methods as those employed to solve the scattering prob-

lem provide a reliable indicator of the accuracy of the theoretical results. It is

estimated that a detailed close coupling calculation with con�guration interaction

type target-ion wave functions and full allowance for resonance e�ects (as well

1.4 References 16

as intermediate coupling e�ects, if required, for highly charged ions) yields cross

sections with an uncertainty < 10%.

1.4 References

Badnell, N.R., Pindzola, M.S. and Gri�n, D.C. Phys. Rev. A 43 (1991) 2250

Bethe, H. and Salpeter, E. Quantum Mechanics of One and Two Electron Atoms

(Springer-Verlag, New York/Berlin, 1980)

Blaha, M., Ann. Astrophys. 31 (1968) 311

Burgess, A. Mem. Soc. R. Sci. Liege 4 (1961) 299

Burgess, A. J. Phys. B 7 (1974) L364

Burgess, A. and Sheorey, V.B. J. Phys. B 7 (1974) 2403

Burgess, A. and Tully, J. J. Phys. B 11 (1978) 4271

Burke, P.G. and Robb, W.D. Adv. At. Mol. Phys. 11 (1975) 143

Burke, V.M. and Seaton, M.J. J. Phys. B 19 (1986) L527

Crees, M.A., Seaton, M.J. and Wilson, P.M.H. Comp. Phys. Commun. 15 (1978)

23

Doering J.P., Gulcicek, E.E. and Vaughn, J. Geophys. Res. 90 (1985) 5279

Eissner, W., Jones, M. and Nussbaumer, H. Comp. Phys. Commun. 8 (1974)

270

Henry, R.J.W., Phys. Rep. 68 (1981) 1

Jupen, C., Isler, R.C. and Tr�abert, E., Mon. Not. R. Astron. Soc. 264 (1993)

627

Kato, T. Astrophys. J. Suppl. 30 (1976) 397

Kim, Y.K. and Desclaux, J.P. Phys. Rev. A 38 (1988) 1805

Krueger, T.K. and Czyzak, S.J. Proc. R. Soc. London Ser. A 318 (1970) 531

McDowell, M.R.C., Morgan, L.A. and Myerscough, V.P. J.Phys. B 6 (1973) 1435

Pradhan, A.K. and Saraph, H.E. J. Phys. B 10 (1977) 3365

Pradhan, A.K. Phys. Rev. Lett. 47 (1981) 79

1.4 References 17

Pradhan, A.K. At. Data Nucl. Data Tables 40 (1988) 335

Presnyakov L.P. and Urnov. M J. Phys. B 8 (1975) 1280

Sampson, D.H., Parks, A.D. and Clark, R.E.H. Phys. Rev. A 17 (1978) 1619

Sampson, J.A.R. and Pareek, P.N. Phys. Rev. A 31 (1985) 1470

Saraph, H.E. Comp. Phys. Commun. 3, 256 (1972) and 15 (1978) 247

Scott, N. and Burke, P.G. J. Phys. B 13 (1980) 4299

Seaton, M.J. in Atomic and Molecular Processes edited by D.R. Bates (Academic

Press , San Diego, 1962) 374

Seaton, M.J., Adv. At. Mol. Phys. 11 (1975) 83

Smith, E.R. and Henry, R.J.W. Phys. Rev. A 7 (1973) 1585

Van Regemorter, H. Astrophys. J. 136 (1962) 906

Chapter 2

Theoretical methods for atomic

structure and the code CIV3

18

2.1 Introduction 19

2.1 Introduction

In this chapter and the next the theory of some numerical techniques used exten-

sively throughout the course of this work to solve the coupled integro-di�erential

equations which occur in low-energy electron-ion collision processes is reviewed.

A brief description is also presented of the main computer packages that are cur-

rently available and widely used in this �eld. The main concern of the review is

with collisions involving complex atoms and ions where the target contains more

than two electrons. Low-energy scattering processes of this kind have certain

intrinsic e�ects:

� exchange between the incident and target electrons

� distortion of the target by the incident electron

� short range correlation e�ects between the incident and target electrons

All these e�ects are important and none of them should be excluded from any

general theoretical description.

The study of electron scattering by complex atoms can conveniently be divided

into two parts. Firstly, it is necessary to obtain wavefunctions which describe

the target atomic states and secondly, these wavefunctions must be incorporated

into a description of the collision problem. Numerous theoretical methods are

currently available such as the Many Body Perturbation Theory (MBPT) (Kelly

1969), Bethe-Goldstone equations (Nesbet 1968), the Born Approximation. the

Polarized Orbital Method (Tempkin 1957) and the Matrix Variational Method

(Harris and Mitchels 1971). This chapter and the next describe in detail one of

the most accurate techniques currently available to solve the collision process.

Con�guration Interaction (CI) wavefunctions containing just a few well chosen

con�gurations are used to describe the target states, whilst the Close-Coupling

R-matrix method of Burke (1971) gives a good description of the collision problem

2.1 Introduction 20

over an extended energy range. A combination of these two methods is the basis

of the current approach.

The technique which has become the principal computational method of low-

energy electron scattering theory is found in the Close-Coupling (CC) approx-

imation. This method was implemented for practical computations by Seaton

(1953(a,b),1955). The CC approximation is based on the use of a truncated eigen-

state expansion as a representation of the total wavefunction, thereby reducing

the problem to solving a set of ordinary di�erential (or integro-di�erential) equa-

tions. The concept of this technique is not altogether new, the general procedure

of expansion in target eigenstates being originally proposed by Massey and Mohr

(1932). The CC method is probably best known for its accurate prediction of many

closed channel resonances which have subsequently been detected in experiments.

These resonances are mainly coupled to just a few closed channels and hence in-

cluding them in the approximation together with the open channels will give a

reliable result for the resonance position and width. As with all approximations,

however, this method does in fact have unfortunate computational limitations. It

is obviously di�cult to increase the accuracy of calculations by taking into account

a larger number of states, as this leads to a considerable increase in computing

time whilst the contribution of each successive state gets less and less. As pointed

out by Burke (1963), an increased number of channels causes convergence of the

CC method to be very slow. It has been shown, however, that the inclusion of a

few suitably chosen pseudo-states in the expansion can considerably improve the

convergence of the results. These pseudo-states can allow for perturbing e�ects

of highly excited and continuum (ionization) channels, which cannot be included

directly in the formalism. Explicit `correlation functions' can also be included to

make the CC method fully general and allowing it to be applied, in principle, to

many calculations of arbitrary accuracy.

2.1 Introduction 21

Due to the computational limitations apparent in the use of the CC method,

equivalent, but more practical methods, such as the R-matrix method of Burke

(1971) and the algebraic reduction method of Seaton (1970) have been developed.

The theory of the R-matrix method is presented in chapter 3 along with a brief

description of the associated computer codes.

Initially the scattering of electrons by atoms and ions where relativistic e�ects

may be neglected is considered. The time independent Schr�odinger equation

(H

N

� E) = 0 (2.1)

must be solved for an N-electron target where the non-relativistic many-electron

Hamiltonian (in a.u.) takes the form

H

N

= �

1

2

N

X

i=1

(r

2

r

i

+

2Z

r

i

) +

N

X

i<j

1

r

ij

(2.2)

Z is the charge on the nucleus and the Hamiltonian is diagonal in both the total

orbital angular momentum L and the total spin S. The interelectronic distance,

r

ij

, is de�ned as r

ij

= jr

i

� r

j

j. The �rst term in equation (2.2) denotes the

one-electron contribution to the Hamliltonian while the second term denotes the

two-electron contribution.

The solution of equation (2.1) yields the wavefunctions, where =

(r

1

; r

2

; :::; r

N

). However due to the

1

r

ij

term in the Hamiltonian this equation

is not a separable one and thus cannot be solved exactly (except for hydrogenic

systems which contain only one electron).

The �rst of the methods developed to obtain approximate solutions to equation

(2.1) was the central �eld approximationmethod. This uses the basic idea that the

electrons of an atomic system move in an e�ective spherically symmetric potential

V (r) due to the nucleus and the other electrons of the system so that the total

2.1 Introduction 22

wavefunction can be expressed as a product of one-electron wavefunctions. This

is a good approximation provided that the potential V (r) of an electron does

not change signi�cantly when a second electron passes the electron in question

reasonably closely. This turns out to be the case for all but the lightest of atoms

due to the nuclear charge being an order of Z greater than the charge of an

electron. The two principal problems involved are thus the calculation of the

central �eld potential and the correct formulation of the wavefunction.

Two ways of performing these tasks were then developed. The �rst of these

was the Thomas-Fermi model of the atom which used semi-classical and statistical

methods to obtain expressions for the potential. The other was a method �rst de-

veloped by Hartree (1927(a,b), 1957) and later extended by Fock (1930) and Slater

(1930). This method used the central �eld approximation as a starting point and

combined with a variational principle, equations for the potential were produced.

This method is known as the Hartree-Fock method. Unfortunately results of cal-

culations for helium, lithium and potassium showed that this method produced

results that were not entirely satisfactory. This is due to the lack of consideration

of electron correlation e�ects i.e. the fact that V (r) does change with the passage

of another electron. The accurate computation of quantities such as transition

probabilities, electron a�nities and hyper�ne-structure constants require meth-

ods which provide solutions of a greater accuracy than the Hartree-Fock method.

That does not remove the value of this method as various modi�cations can be

made to correct this oversight such as the con�guration interaction method (which

will be used extensively here) and the random phase approximation method both

of which produce results which are highly satisfactory.

2.2 The Hartree-Fock method 23

2.2 The Hartree-Fock method

Consider an atomic system consisting of a nucleus of charge Z (atomic units) and

N -electrons. As demonstrated by the Hamiltonian, each of these electrons experi-

ences an attraction to the nucleus and a repulsion from the other (N - 1) electrons.

Suppose these interactions were represented by an e�ective potential V (r) which

can be stated to be spherically symmetric. This leads to the conclusion that each

electron in a multi-electron system can be represented by its own wavefunction,

�

i

(i = 1,...,N), which depends on the coordinates of the electron and are known

as orbitals.

In Hartree's original approach, in 1928, he assumed that the wave function

(approximate solution of equation (2.1)) was the product of these orbitals. That

is

(q

1

; q

2

; :::; q

N

) = �

1

(q

1

); �

2

(q

2

):::; �

N

(q

N

) (2.3)

where q

i

denotes the collection of spatial coordinates, r

i

, and spin coordinates of

electron i. However this wavefunction violates the Pauli-exclusion principle which

states that the wave function of a system of identical electrons must be totally

antisymmetric in the combined space and spin coordinates of the particles.

To correct this problem an alternative form of the wavefunction was introduced

by Fock and Slater in 1930 to replace that of equation (2.3). This wavefunction,

represented by a Slater determinant is given in equation (2.4) where the symbols

(� = 1; � = 2; :::; � = N) represent the set of quantum numbers (n; l) which

uniquely de�ne each of the N -electrons. Thus is the total wavefunction de-

scribing an atom in which one electron is in state �, another in state � and so

on. The electron spin orbitals,�

�

; �

�

:::�

�

, are chosen to be orthonormal over space

and spin. However orbitals with spin m

s

= +1=2 are automatically orthogonal

to those with spin m

s

= �1=2. Therefore space orbitals corresponding to the

same spin function must be orthonormal, which ensures the normalization of


i.e. hji = 1.

(q

1

; q

2

; :::; q

N

) =

1

p

N !

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

(q

1

) �

�

(q

1

) : : : �

�

(q

1

)

�

�

(q

2

) �

�

(q

2

) : : : �

�

(q

2

)

.

.

.

.

.

.

.

.

.

.

.

.

�

�

(q

N

) �

�

(q

N

) : : : �

�

(q

N

)

�

�

�

�

�

�

�

�

�

�

�

�

�

(2.4)

The orbitals are chosen subject to the condition

h�

i

j�

j

i =

Z

�

�

i

(q)�

j

(q)dq = �

ij

(2.5)

where

R

dq represents integration over all space coordinates and a summation over

all spin coordinates. It is then customary to split the orbitals �

i

into their space

and spin components as follows

�

i

(q

j

) = u

i

(r

j

)�

1

2

;m

s

i

(2.6)

where �

1

2

;m

s

i

is the spin function and u

i

(r

j

) is the spatial function which due to

its one electron nature is an eigenfunction of the one-electron Hamiltonian

h

i

= �

1

2

r

2

i

�

Z

r

i

(2.7)

which includes the kinetic energy of the electron i and its potential energy due to

interaction with the nucleus. These eigenfunctions can be shown to take the form

of a product of a radial function P

n

i

l

i

(r) and a spherical harmonic Y

m

l

i

l

i

(�; �)

u

i

(r) =

1

r

P

n

i

l

i

(r)Y

m

l

i

l

i

(�; �) (2.8)

The problem of obtaining the wavefunction thus is reduced to �nding these radial

functions which can be achieved by using a variational principle. That is if the

ground state energy of the system is denoted as E

0

and the energy of the system


when it resides in the state represented by the wavefunction (given by equation

(2.4)) by E then

E

0

< E = hjHji (2.9)

where it is assumed that the wavefunctions are normalized to unity (the source of

the factor

1

p

N !

in equation (2.4).

The problem using the variational principle then becomes one of minimizing

the energy E of equation (2.9). This problem is of considerable length and Brans-

den and Joachain [11] prove it can be resolved to give the variational equation.

�E �

X

i

�

i

�hu

i

ju

i

i = 0 (2.10)

This gives rise to the following set of coupled integro-diferential equations courtesy

of Slater (1930).

F

i

u

i

= �

i

u

i

(2.11)

where F

i

is the Fock operator given by

F

i

= h

i

+ J

i

�K

i

(2.12)

where h

i

is the one-electron Hamiltonian given by equation (2.7), J

i

is the direct

operator given by

J

i

=

N

X

j 6=i

Z

ju

j

(r

j

)j

2

r

ij

dr

j

(2.13)

and K

i

is the non-local exchange operator given by

K

i

u

i

(r

i

) =

N

X

j 6=i

u

j

(r

i

)

Z

u

�

j

(r

j

)u

i

(r

i

)

r

ij

dr

j

(2.14)

The set of equations (2.11) are known as the Hartree-Fock equations for the

wavefunction (2.4) and each operator listed here can be attributed to a certain

phenomenon within the atom. The one-electron Hamiltonian has already been


discussed above. The other two operators represent electron-electron interaction

e�ects. The �rst of these, the direct operator, can be interpreted as being the

potential associated with the electron charge density of the other electrons (i.e. the

repulsion e�ect from the other electrons). The �nal term, the exchange operator,

gives the interaction between two states obtained by interchanging two electrons.

This �nal term is what separates Hartree's original method from the Hartree-

Fock method and is a direct consequence of the antisymmetric nature of the

wavefunction. Finally the parameter �

i

may be interpreted as the energy required

to remove an electron from the orbital u

i

. This is a result of Koopman's theorem

(Cohen and Kelly 1966) and �

i

is thus referred to as the orbital energy. It should

be noted that this method will provide an in�nite number of orbitals as solutions

and not just the N number expected. Therefore the following distinction is made:

the orbitals that for a given state occur in the wavefunction are said to be occupied

while the remainder are unoccupied.

2.2.1 Correlation energy

It has been clearly pointed out that the Hartree-Fock method produces only ap-

proximate wavefunctions and thus approximate energies; denoted by

HF

and

E

HF

respectively. Comparison with exact energies E

exact

shows di�erences E

corr

between exact and Hartree-Fock energies. That is

E

corr

= E

exact

� E

HF

(2.15)

This di�erence is known as the correlation energy. It should be noted that the

Hartree-Fock wavefunction does include a certain amount of electron correlation

due to the total antisymmetry of the wavefunction and so the term correlation

e�ects, which create the correlation energy, refers to electron correlations not

present in the Hartree-Fock wavefunction. It should also be noted that E

exact

is not


the experimental energy but the exact energy of the non-relativistic Hamiltonian.

This error in the Hartree-Fock method clearly lies with the wavefunctions pro-

duced. These wavefunctions, however, do result in energies that are greater than

exact energies by less than one percent. This may be considered an acceptable per-

centage error but in the regions of con�guration space which do not play a major

role in the determination of the energy of the state in question the wavefunctions

may be in serious error and thus observables calculated from these wavefunctions

may be extremely inaccurate.

Numerous attempts have been made to understand the role of correlation ef-

fects in in uencing wavefunctions and energies of atoms. The methods commonly

used for improving on the Hartree-Fock wavefunction can be classi�ed broadly

into two categories. The �rst is that developed by Hylleraas (1930) in which the

total wavefunction is a power series expansion which includes the inter-electronic

coordinates r

ij

explicitly. This method has been applied with great success to

several states of the helium-like ions (Pekeris (1958, 1959)) but for more complex

svstems such a solution is of little value, due to the mathematical complexity of

the process and the di�culty of interpreting it physically. The second method

is that of Con�guration Interaction (CI) which involves a linear combination of

\determinantal function" each representing a particular con�guration of the elec-

trons in the atom. This method is used extensively in the present work and a more

detailed description is presented in Section 2.3. It can be said, however, that both

these methods have many common features, especially their dependence on the

Hartree-Fock approximation. Each has its advantages and disadvantages, but all

are, in principle, capable of re�nement to give a result of arbitrary accuracy.


2.2.2 The Self-Consistent �eld method

Due to the complicated nature of the Hartree-Fock equations, normal methods

are inadequate for the task of obtaining solutions to these equations. An iterative

method, based on the requirement of self-consistency, is thus required in their

solution which involves the representation of the radial function P

nl

(r) by the

following linear combination of analytical basis functions

P

nl

(r) =

k

X

j=1

c

jnl

r

I

jnl

e

��

jnl

r

(2.16)

or

P

nl

(r) =

k

X

j=1

c

0

jnl

�

jnl

(r) (2.17)

where �

jnl

is the normalized Slater-type orbital of the form

�

jnl

(r) =

�

(2�

jnl

)

2I

jnl

+1

(2I

jnl

)!

�

1

2

r

I

jnl

e

��

jnl

r

(2.18)

and the radial functions P

nl

obey the orthonormality conditions

Z

1

0

P

nl

(r)P

n

0

l

(r)dr = �

nn

0

(2.19)

The iteration method utilized in solving the Hartree-Fock equations is known

as the self consistent �eld method and consists of the following steps.

� Estimate P

nl

(r) by specifying the Clementi-type (Clementi and Roetti 1974),

c

jnl

, or Slater-type, c

0

jnl

, coe�cients, the exponents �

jnl

and the powers of r,

I

jnl

. Values are available in past literature of atoms or ions either isoelec-

tronic with the one you are considering or close to it.

� Using these values the actual orbital, u(r) = u

i

(r), is determined.

� The values of the terms K

i

u

i

and J

i

u

i

are determined using the estimated

2.3 The Con�guration Interaction method 29

value of u

i

. This results in a set of eigenvalue di�erential equations for a

new set of orbitals, u

(2)

i

say.

� These di�erential equations are solved by substituting equations (2.8) and

(2.16) to give a set of algebraic equations which include the coe�cients of the

new orbital, c

(2)

jnl

, where �

(2)

i

are treated as variational parameters in order

to minimize the energy while the powers of r, I

(2)

jnl

are �xed. The initial

estimate for the coe�cients c

jnl

are substituted in to give a set of solvable

equations for c

(2)

jnl

. The values obtained from solving these equations are then

resubstituted into the algebraic equations for the coe�cients to give further,

better results. The process is repeated until convergence is obtained for the

series of solutions, within a desired tolerance. Using the �nal set, a radial

function is found as a �rst solution of the Hartree-Fock equations

� Using the new radial function, the previous two stages are repeated until a

satisfactory degree of convergence is obtained for the radial functions.

There are various other ways of dealing with stage 4 of this method such as

solving the equations numerically. Various tabulations of orbitals obtained from

this method exist. The one referred to is by Clementi and Roetti (1974).

2.3 The Con�guration Interaction method

The lack of inclusion of electron correlation in the Hartree-Fock wavefunctions

is due to the restriction that each electron is assigned to a speci�c nl orbital

resulting in each state being represented by a single Slater determinant. The

assignment of these electrons to speci�c nl orbitals, and their couplings, are known

as con�gurations. Consider the replacement of the Hartree-Fock wavefunction

with one that represents more than just a single con�guration. This is achieved

by allowing a particular state with a certain LS symmetry to be represented by a


linear combination of Slater determinants where each determinant represents one

con�guration whose individual orbital angular momenta of the electrons couple in

one particular way to give the same total orbital angular momenta value L and

spin value S. That is, the wavefunctions can be expressed in the form,

(LS) =

M

X

i=1

a

i

�

i

(�

i

LS) (2.20)

where the �

i

(�

i

LS) are the con�guration state functions which represent a par-

ticular assignment of electrons to orbitals with speci�c n and l values. They are

eigenfunctions of L

2

and S

2

since these operators commute with the Hamiltonian,

so long as other relativistic interactions can be neglected i.e. for light atomic sys-

tems. Each of these con�guration state functions are linear combinations of Slater

determinants, the set of which is denoted by �

i

. The total wavefunction (LS)

represents the state possessing a total angular momentum L and total spin S

and the coe�cients a

i

indicate the contribution made by each con�guration state

function to this total wavefunction. The means by which the a

i

and one-electron

radial functions are obtained is called con�guration interaction. Note that the

sum should be to in�nity but in practice it is restricted to a �nite number of

con�gurations M.

The con�guration state functions introduced here represent three di�erent

types of electron correlation e�ects.

1. Internal correlation : The Hartree-Fock orbitals are those which occupy the

ground state con�guration of the system being considered. Internal correlation

corresponds to the con�gurations which are constructed solely from these orbitals

or orbitals which have the same n value i.e. those nearly degenerate with them.

2. Semi-internal correlation : These e�ects arise from con�gurations con-

structed from (N - 1) Hartree-Fock orbitals and one other orbital not included in

this set.


3. External correlation: Con�gurations that are constructed from (N-2) Hartree-

Fock orbitals and two from outside this set cause these e�ects.

As expected, of the three types of e�ects mentioned above, it is the internal

e�ects that contribute the most to expansion (2.20) (i.e. they have the largest

values of a

i

) so while the external e�ects create the most con�guration state func-

tions, in practice accurate energy levels are obtained from including all the internal

and semi-internal con�gurations but only some of the external ones (Oskuz I. and

Singanoglu O. (1969)).

2.3.1 Determination of the expansion coe�cients

The problem is now one of obtaining the expansion coe�cients a

i

and the radial

functions P

nl

(r) (and thus the con�guration state functions �

i

). One method

of calculating the CI wavefunctions is to use a con�guration basis set which in-

cludes the Hartree-Fock con�guration along with other con�gurations built from

Hartree-Fock and variationally determined orbital functions. This scheme is called

Superposition Of Con�gurations (SOC). It is employed in the CI code CIV3 which

is described in section 2.4

Another way to achieve this is by going through the same analysis as the

Hartree-Fock method to give a set of integrodi�erential equations for the radial

functions. This is known as the multi-con�gurational Hartree-Fock method. How-

ever, the radial functions derived from the SOC method are analytic whereas the

MCHF radial functions are numerical. The radial orbital functions for the Hartree-

Fock con�gurations are usually taken from the tables of Clementi and Roetti

(1974) or other Roothaan Hartree-Fock calculations. The parameters describing

these orbitals are changed when using the MCHF method and then con�gura-

tion interaction is applied. In contrast the parameters remain �xed throughout

the calculation with the SOC method so one can use the same orbital basis for


all con�gurations and states. Re-optimization of the orbitals is normally, though

not necessarily, performed with the MCHF method although both methods are

equally easy to apply.

Consider the set of con�guration state functions �

i

and the corresponding set

of coe�cients a

i

where the �

i

, and their radial functions, are �xed while the a

i

are

free to vary. That is the expansion coe�cients are the only variational parameters.

Then minimizing the energy of the state being used subject to the normalization

condition that

hji = 1 (2.21)

gives rise to the variational equation

�[hjHji � E(hji)] = 0 (2.22)

where E is a Lagrange multiplier. Substitution of the wavefunction equation

(2.20) into this expression results in

�

"

X

i

X

j

a

i

a

j

h�

i

jHj�

j

i � E

X

i

X

j

a

i

a

j

h�

i

j�

j

i

#

= 0 (2.23)

Now de�ning the Hamiltonian matrix by its general element H

ij

which is given b

H

ij

= h�jHj�i (2.24)

where H is the N-electron Hamiltonian of equation (2.2) and assuming that the

con�guration state functions are orthonormal (i.e. h�j�i = �

ij

) it follows that

X

j

a

j

(H

ij

� E�

ij

) = 0 (2.25)

where the possible values of E are in fact the corresponding eigenvalues E

j

of

the Hamiltonian matrix, H, while the a

i

are the components of the associated


eigenvectors, E

(j)

i

. Equation (2.26) may also be written as

hjHji = E�

ij

(2.26)

according to the Hylleraas-Undheim theorem (see section 2.3.3).

It follows that diagonalization of the Hamiltonian matrix will produce both

the expansion coe�cients and the energy of the state.

2.3.2 Setting up the Hamiltonian matrix

However, before energy levels are determined it is essential to form the Hamilto-

nian matrix in order to diagonalize it. First adopt the approach of writing the

matrix elements as a weighted sum of the one and two electron integrals as follows.

Split the Hamiltonian into two parts

H = H

o

+ V (2.27)

where H

o

is the one electron term and V is the two electron term which are

respectively given by

H

o

=

N

X

i=1

h

i

(2.28)

and

V =

X

i<j

1

r

ij

(2.29)

This enables us to write the Hamiltonian matrix elements as the sum of two matrix

elements associated with the operators H

o

and V . That is

H

ij

= h�

i

jH

0

j�

j

i+ h�

i

jV j�

j

i (2.30)

Each of these two new matrix elements can be expressed in the form of one and


two electron radial integrals respectively

h�

i

jH

0

j�

j

i =

X

�;�

0

x(�; �

0

)

�

P

n

�

l

�

�

�

�

�

�

1

2

d

2

dr

2

�

Z

r

+

l

�

(l

�

+ 1)

2r

2

�

�

�

�

P

n

�

0

l

�

0

�

�

l

�

l

�

0

(2.31)

and

h�

i

jV j�

j

i =

X

�;�;�

0

;�

0

;k

y(�; �; �

0

; �

0

; k)R

k

(n

�

l

�

; n

�

l

�

; n

�

0

l

�

0

; n

�

0

l

�

0

) (2.32)

where R

k

represent the two-electron radial integrals and � and � are the indices

which represent the status of the �rst and second electron respectively subject

to the restriction that the wavefunctions �

i

and �

j

must have at least (N - 2)

electrons in common for the two electron term while they must have at least (N

- 1) electrons in common for the one-electron term.

The coe�cients x and y are weighting coe�cients which Fano [47] has already

described in terms of Racah algebra. Several programs created by Hibbert (1970,

1971, 1973) exist that calculate these coe�cients by using other computer packages

which calculate recoupling coe�cients (Burke (1970)) and fractional parentage

coe�cients (Allison (1983), Chivers (1973)). All of these packages have been

incorporated into the computer package CIV3 written by Hibbert (1975) which

performs the entire task of setting up and diagonalizing the Hamiltonian matrix

to obtain the coe�cients a

i

, and energies, E

(j)

.

2.3.3 Optimization of the radial functions

The Hylleraas-Undeim theorem (Hylleraas and Undheim (1930)) states that

`The upper bound to the exact non-relativistic energies of the states of a given

symmetry obtained using a variational principle are greater than or equal to the

exact energies'


That is

E

i

� E

exact

i

(2.33)

where E

exact

i

are the exact non-relativistic energies of the state of a given symme-

try. As demonstrated this value E

i

depends on the radial functions used in the

calculation. Therefore the resulting energy of the diagonalization of the Hamilto-

nian will always be greater than the exact energy no matter what values of the

radial functions are chosen but the more accurate the radial function the closer

E

i

will be to E

exact

i

. Variation of the radial functions is thus necessary to �nd the

lowest energy possible. This process is known as the optimization of the radial

functions (or orbitals).

Using equation (2.19) (n - 1) of the linear coe�cients, c

jnl

, of equation (2.16)

are �xed where n and l are the principal and orbital angular momentum quantum

numbers of the orbital whose radial function is being varied. Since E

i

then depends

upon the remaining linear coe�cients and non-linear exponents, E

i

can be used as

a variational function in order to obtain values for the coe�cients and exponents.

The process is repeated until overall convergence for the energies is obtained.

If, for the purpose of obtaining the orbitals, only one con�guration is used

in the original expansion given by equation (2.20), then the above process will

produce an approximate solution of the Hartree-Fock equations and the orbitals

will be the Hartree-Fock orbitals. If on the other hand more than one con�guration

is included involving additional orbitals then the problem becomes non-physical

as are the orbitals thus obtained. These orbitals are known as pseudo orbitals

and they are distinguished from real orbitals by placing a bar over them. These

orbitals satisfy the same conditions as real orbitals such as orthonormality to the

other orbitals in the generated set (including real orbitals) but are important in

con�guration interaction calculations in order to accurately describe correlation

e�ects within a particular state.

2.4 The Con�guration-Interaction Bound State Code - CIV3 36

The method described here is the one used by the previously mentioned CIV3

code in order to obtain orbitals and energy levels.

2.4 The Con�guration-Interaction Bound State

Code - CIV3

A general FORTRAN program to calculate Con�guration-Interaction wavefunc-

tions and electric- dipole oscillator strengths has been formulated by Hibbert

(1975). It encompasses the entire range of calculations introduced in the previous

section including the calculation of energy levels and expansion coe�cients and

setting up the con�guration interaction wavefunctions from this data. The code

can use these wavefunctions to evaluate such observables as oscillator strengths.

Optimization, otherwise known as minimizing, of radial functions to give the most

accurate energies (and therefore wavefunctions) possible is also performed. This

makes it ideal for obtaining orbitals and energy levels that will be essential for

utilizing the R-matrix code that is discussed in the next chapter. This section

describes how the CIV3 code computes these values using the theory. The basic

structure of the code is presented in the schematic ow diagram, Figure 2.1.

The input required for the code can be grouped as follows:

� Initially the type of calculation to be performed must be determined. The

choice between radial function optimization, oscillator strength calculation

and others is provided although this discussion concerns the former only.

There is also the option of how much output to produce. For example the

Hamiltonian before and after diagonalizationto can be output.

� Some basic data about the ion being considered is included: e.g. the nuclear

charge Z, the maximum n and l values and the maximum powers of r among

the orbital set.


� The radial functions are input analytically in either Clementi or Slater type

form corresponding to equations (2.16) and (2.17) respectively. Distinction

between Hartree-Fock orbitals and orbitals calculated by the user must be

made. For the orbital to be optimized, an initial estimate for the radial

function is included here. The radial functions are sums of STO's, implying

the radial integrals are performed analytically.

� The con�gurations including the various coupling schemes are input. This

section includes the n and l values of each occupied orbital. In an optimiza-

tion calculation it is suggested that a minimum number of con�gurations

are needed to include the dominant contributors to the electron correlation

e�ects being introduced, with further con�guration state functions having

only a minor e�ect on the optimal radial function parameters. While for

energy level calculations the selection of all internal and semi-internal con-

�gurations with some external con�gurations is recommended.

� Data speci�c to the calculation being attempted is included. For the energy

level case this could include the option to split the Hamiltonian into separate

total symmetries and thereby increase e�ciency.

Once the correct input data has been established and checked, and the type

of calculation speci�ed, the CIV3 code proceeds to generate the radial function

parameters for the Hartree-Fock orbitals required together with any further neces-

sary pseudo-orbitals. The orbitals are generated in the order of increasing angular

momentum and principal quantum number. Each radial function to be optimized

is varied separately, by treating its parameters as the variables in the minimiza-

tion routine. When the last radial function in the list has been optimized, the

process begins again with the �rst in the list. The process terminates when the

net change in the functional is less than a preassigned amount.

The �nal Hamiltonian matrix may now be constructed and diagonalized to ob-


tain upper bounds to the exact energies, E

exact

i

,(eigenvalues) and the components

of the con�gurations in the corresponding wavefunctions, a

i

,(eigenvectors)(see

equation (2.25)). If further con�gurations are to be included and the con�gu-

ration set extended then a new Hamiltonian matrix must be constructed and re-

diagonalized. If necessary, the new partitioning of the matrix is de�ned. Finally

the SOC wavefunctions and the corresponding energies can easily be established.

Once an SOC wavefunction has been constructed it may then be used to

evaluate other atomic properties. One property of particular interest in atomic

structure is oscillator strengths (transition probabilities). Speci�cally the code

allows the calculation of absorption multiplets oscillator strengths between two

states, each of course being described by an SOC wavefunction. Length, velocity

and acceleration forms of these transition probabilities may be evaluated together

with the geometric mean. One �nal option is available to the user of CIV3, that

of subdiagonalization. It is sometimes of interest to examine the convergence

of the inclusion of more and more con�gurations, either for the energy or for

oscillator strengths. Once the Hamiltonian has been set up (after optimization)

it is possible to diagonalize sub-matrices to see the e�ect of including a limited

number of con�gurations.

There are some limitations to the complexity of any calculation performed

using CIV3. The maximum number of electrons is allowed in s, p and d subshells

but only up to 2 electrons in f or g subshells. Subshells with l > 4 may only be

included when the code has been modi�ed. The typical execution time depends

on a number of factors:

� size of the ion

� the extent of the optimization required

� the number of con�gurations involved

� the number of basis functions in each radial function


� inclusion of relativistic e�ects


Basic Data

Radial Functions

Configuration SetsCIV3

Optimize radial

functions

Pnl

(r) set up as a

sum of STO’s

Hamiltonian matrix set up

and diagonalized

Extend list of configs,

set up new Hamiltonian

and re-diagonalize

Set up SOC wavefunction and

energies

strengths

Oscillator

Output

Minimization

Figure 2.1: Basic owchart for the CIV3 code

2.5 References 41

2.5 References

Allison D.C.S. Comput. Phys. Commun. 1 (1969) 15

Bransden B.H. and Joachain C.J. Physics of Atoms and Molecules (Longman

1983)

Burke P.G. Proc. Phys. Soc. 82 (1963) 443

Burke P.G. Comput. Phys. Commun. 1 (1970) 241

Burke P.G., Hibbert A. and Robb W.D. J. Phys. B4(1971) 153

Chivers A.T. Comput. Phys. Commun. 6 (1973) 88

Clementi E. and Roetti C. At. Data Nucl. Data Tables 14 (1974)

Cohen M. and Kelly P.S. Can. J. Phys. 44 (1966) 3227

Fock V.Z. Z. Phys. 60 (1930) 126

Harris F.E. and Mitchels H.H. Methods Comp. Phys. 10 (1971) 143

Hartree D.R. Proc. Camb. Phil. Soc. 24 (1927a) 89

Hartree D.R. Proc. Camb. Phil. Soc. 24 (1927b) 111

Hartree D.R. The Calculation of Atomic Structures (Wiley 1957)

Hibbert A. Comput. Phys. Commun. 1 (1970) 359




Hylleraas E.A. and Undheim B. Z. Phys. 65 (1930) 759

Kelly H.P. Phys. Rev. 182 (1969) 84

Massey H.S.W. and Mohr C.B.O. Proc. Roy. Soc. Ser. A 136 (1932) 289

Nesbet R.K. Phys. Rev 175 (1968) 2

Oskuz I. and Sinanoglu O. Phys. Rev. 181 (1969) 42

Pekeris C.L. Phys. Rev 112 (1958) 1649

Pekeris C.L. Phys. Rev 115 (1959) 1216

Seaton M.J. Phil. Trans. R. Soc. London Ser. A 245 (1953a) 469

Seaton M.J. Proc. R. Soc. Lond. Ser. A 218 (1953b) 400

2.5 References 42

Seaton M.J. Proc. R. Soc. Lond. Ser. A 231 (1955) 37

Seaton M.J. Proceedings of the 2nd annual Conference on Computational Physics

(1970)

Slater J.C. Phys. Rev. 35 (1930) 210

Tempkin A. Phys. Rev. 107 (1957) 1004

Chapter 3

The R-matrix method and codes

43

3.1 The R-matrix method 44

3.1 The R-matrix method

TheR-matrix method is used to calculate reliable cross-sections, which are then

used to produce e�ective collision strengths (see chapter 4) for use in the ADAS

application (see chapter 6).

A cross-section, �

i!j

, is related to the probability per second that a particular

event will occur in the system considered, measured over a range of energies.

Consider a beam of electrons of known ux density impacting upon the target.

There is a probability associated with exciting the initial target to a particular

state and this is dependent on the ux density. The constant of proportionality

is the cross section �

i!j

which has units of area. It is dependent on the target

element, residual charge (for an ion) and is a complicated function of energy which

may include many resonance features. Cross-sections can be obtained for a system

from its wavefunction, but the problem is that the wavefunctions of equations

(2.4) and (2.20) should include a description of the impact electron. However, the

con�guration interaction wavefunction does not include any continuum terms -

i.e. it only considers bound states, and not continuum states like that of an (ion

+ free electron) state. `Interaction with the continuum' must be considered.

One method which deals with this interaction is called the R-matrix method.

The principle behind theR-matrix scattering method (Wigner and Eisenbud 1947,

Lane and Thomas 1958) is that con�guration space describing both the scattered

and target particles can be split into an inner region and an outer region. In

the outer region the scattered particle is outside the charge distribution of the

target so that the system is easily solvable i.e. interaction is weak and, in many

cases, is determined exactly in terms of plane or coulomb waves, modi�ed by long-

range multipole potentials. In the inner region the converse is the case so that

correlation and exchange e�ects are very strong and the collision is di�cult to

evaluate. The solution is to impose spherical boundary conditions on the surface


of the inner region centred on the target nucleus giving a complete set of states

describing all enclosed particles.

While it is not a recent development, it has only seen its fullest e�ect and use

in the previous thirty years due to the development of supercomputers. Today it

continues to be developed for di�erent and more accurate applications. Below is

a brief history of the most pertinent developments in R-matrix theory:

� 1947. R-matrix method published by Wigner and Eisenbud.

� 1971. Burke and Seaton, Burke, Hibbert and Robb. R-matrix theory applied

to electron scattering problems.

� 1975. Burke & Robb. The complete description of the R-matrix theory but

excluding relativistic e�ects.

� 1980. Scott & Burke The modi�cation of the R-matrix method to include

relativistic e�ects.

Parallel to the mathematical development of the theory has been the produc-

tion of a computer code which can perform calculations utilizing this method

where the motivation comes from the realization that manual calculations would

be unfeasable for all but the simplest atomic systems. The �rst of these codes was

written by Berrington et al. (1974, 1978) but there have since been many modi�-

cations accompanying the theoretical developments. Subsequently there are now

several versions of the code in existence but the one used herein is RMATRX1.

3.1.1 Basic ideas and notation

The equation which describes electron-impact excitation is

A + e

�

! A

�

+ e

�

(3.1)


where in general notation A is an arbitrary atom or ion target, A

�

is a �nal excited

state of A and e

�

is the continuum (or free) electron.

Consider a trivialised description of an electron colliding inelastically with an

atom or ion target. As the electron approaches the target it experiences the

target's complicated electrostatic �eld but as it gets closer there comes a point

where it is indistinguishable from the electrons around the target. It also disturbs

the electron `cloud'. At this point there are many possible outcomes permitted by

quantum mechanics - so called channels. A channel is simply a possible mode of

fragmentation of the composite system (A+e

�

) during the collision. The outcomes

are limited by the energy conservation laws. If E is the total energy and �

i

is the

energy of the target state coupled to the i-th channel then the channel energy of

the free electron, k

2

i

, is therefore

k

2

i

= 2(E � �

i

) (3.2)

If k

2

i

> 0 the channel is open while if k

2

i

< 0 it is closed. If k

2

i

= 0 then it indicates

that the system is at a threshold energy for excitation to occur.

The critical point in the collision process occurs when correlation and electron

exchange e�ects are of importance with regards to the outcome of the impact.

This point is at a distance, r

a

,known as the R-matrix radius, from the nucleus. It

encloses a sphere which is su�ciently large for the electron charge distribution of

the target to be permanently contained within said sphere while the target plus

free electron system are included in a sphere of in�nite radius which is referred

to as the (N + 1)-electron system. The (N + 1)th electron is thus considered to

be free if it occupies the region r > r

a

(known as the external region) while it is

bound if it resides in the region where r � r

a

(known as the internal region).


3.1.2 Constructing the targets

Initially an adequate description of the target must be found, from which ex-

pressions for the wavefunction in both internal and external regions may also be

found. The target may be represented by wavefunctions known as target states,

�

i

. They are de�ned by their total angular momentum and spin quantum numbers

and by the arrangement of the orbital electrons which couple in particular ways

to yield these numbers. The target states are solutions of the time independent

Schr�odinger equation.

H

N

�

i

= �

i

�

i

(3.3)

where H

N

is the non-relativistic N -electron Hamiltonian and �

i

is the energy of

the corresponding target state. It is also necessary to include a certain amount

of con�guration-interaction in the target state wavefunctions f�

i

g to describe

them accurately. This feature can be introduced by describing each of the target

states in terms of some basis con�gurations, �

k

, using the following con�guration

interaction expansion.

�

i

(x

1

;x

2

; :::;x

N

) =

M

X

k=1

a

ik

�

k

(x

1

;x

2

; :::;x

N

) (3.4)

where x

i

= (r

i

; �

i

) denotes the space (r

i

= r

i

r̂

i

) and spin (�

i

) coordinates of the

ith electron while the fa

ik

g are the con�guration mixing coe�cients which are

unique to each state.

This is the same type of wavefunction as that from equation (2.20) and the

problem of describing the target thus becomes a con�guration interaction problem

which is solved using the method of section (2.3); that is the expansion coe�cients

are determined by diagonalizing the N -electron Hamiltonian matrix while the

basis con�gurations are constructed from a bound orbital basis consisting of a

set of real orbitals, and possibly pseudo orbitals, introduced to model correlation

e�ects. This will result in expressions for the target state functions and their


energies. The con�gurations used in the expansion, naturally all have the same

total spin and orbital angular momentum values.

The R-matrix method, understandably, uses the same notation and form of

the orbitals as used in the con�guration interaction method (i.e equations (2.6),

(2.8) and (2.16)). In fact it is usual to use the CIV3 package (although other

packages such as SUPERSTRUCTURE (Eissner et al. 1974) do exist) to de-

termine the required radial functions and these can be input directly into the

R-matrix code. Note that it is important that the orbitals obtained are su�cient

for the representation of both the target and the (N + 1)-electron system.

From these radial functions it is then possible to clearly de�ne the internal and

external regions by the determination of a value for the R-matrix radius. Since

the radial functions tend to zero exponentially as r tends to in�nity, indicating

that the probability of �nding an electron signi�cantly outside the atom is quite

small, a value for r

a

can be chosen at which point it can be claimed that the

charge distribution of the states of interest are included within the sphere de�ned

by this radius. In mathematical terms if � is taken to be a suitably chosen small

number, then r

a

is chosen such that

jP

nl

(r) < �j r � r

a

(3.5)

for each of the bound orbitals. In practice the R-matrix radius is taken to be the

value of r at which the orbitals have decreased to about 10

�3

of their maximum

value.

3.1.3 The R-matrix basis

The most signi�cant problem in applying the R-matrix method to the scattering

of electrons by ions, is de�ning a suitable zero-order basis for expansion of the

(N + 1)-electron wavefunction.


This basis is constructed from three di�erent orbital types ; the real orbitals,

the pseudo orbitals and the continuum orbitals (although pseudo orbitals are

optional). The �rst two types have already been introduced while the continuum

orbitals are included to represent the motion of the free electron. For a particular

angular momentum value l

i

; the set of continuum orbitals f�

ij

g are obtained by

solving the following equation.

(

d

2

dr

2

�

l

i

(l

i

+ 1)

r

2

+ V

0

(r) + k

2

ij

)�

ij

(r) =

X

n

�

ijn

P

nl

i

(r) (3.6)

subject to the R-matrix boundary conditions:

�

ij

(0) = 0 (3.7)

�

r

a

�

ij

(r

a

)

��

d�

ij

dr

�

r=r

a

= b (3.8)

where b is an arbitrary constant known as the logarithmic derivative that is usually

chosen set to zero and k

2

ij

are the eigenvalues of the continuum electron which are

also the previously introduced channel energies. The �

ijn

are Lagrange multipliers

that ensure orthonormality of the continuum orbitals to the bound orbitals of the

same angular symmetry. That is

Z

r

a

0

�

ij

(r)P

nl

i

(r)dr = 0 (3.9)

while analogous to the bound orbital case the continuum orbitals also obey the

orthonormality conditions

Z

r

a

0

�

ij

(r)�

ij

0

(r)dr = �

jj

0

(3.10)

Finally V (r) is a zero-order potential which behaves like

�2Z

r

near the nucleus and

is usually chosen to be the static potential of the target. By default this is taken

to be the static potential but other options can be used.


Now that real and continuum orbitals have been considered, with necessary

orthogonality conditions, the �nal orbitals used for the R-matrix basis are the

pseudo-orbitals. They were omitted previously from equation (3.6) since they

would have negated the physical justi�cation for this equation, and cause the

R-matrix expansion to converge much more slowly. The process of Schmidt or-

thogonalisation is used to further orthogonalise the continuum orbitals to the

pseudo-orbitals. This does not a�ect the worth of the continuum orbitals or the

previous orthogonality conditions satis�ed, and is useful for the matrix mathe-

matics later. The �nal result is an orthonormal basis for each value of l

i

ranging

from r = 0 to r = r

a

.

3.1.4 The internal region

Within the internal region, the (N + 1)th electron is indistinguishable from the

other N electrons - i.e. it is no longer part of the `continuum'. So, the overall

wavefunction can be found by solving the time independent Schr�odinger equation

for the (N + 1)-electron Hamiltonian:

H

N+1

= E (3.11)

subject to appropriate boundary conditions where H

N+1

is the (N + 1)-electron

Hamiltonian given by equation (2.2) withN replaced by (N+1) and where E is the

total energy of the system. A con�guration expansion of the wavefunction similar

to that of equation (3.4) in the bound state problem is now appropriate. However,

interaction between the bound states and the continuum is of importance in this

region and whenever this interaction is particularly strong the inclusion of the

continuum orbitals in a con�guration interaction expansion of the wavefunction

may be insu�cient to model the continuum. Unfortunately, the inclusion of the

integral necessary to model the continuum completely in an expression for the


wavefunction is an impractical alternative. A summation is introduced to approx-

imate the continuum using special types of target states known as pseudo states.

Pseudo states satisfy the equations introduced to describe the target states but

are constructed from a combination of real and pseudo orbitals.

It should be noted that pseudo states are not a de�nite requirement of the

R-matrix method but in cases where strong continuum interaction occurs they

convert the problem from one of discrete-continuum interaction to one of discrete-

discrete interaction allowing a con�guration interaction expansion to be used to

represent the total wavefunction. That is

=

X

k

A

Ek

k

(3.12)

where the energy dependence is carried through the A

Ek

coe�cients and

k

are

states which form a basis for the total wavefunction in the inner region (r < r

a

),

are energy independent and are given by the expansion

k

(x

1

; x

2

; :::; x

N+1

) = A

X

ij

c

ijk

�

i

(x

1

; :::; x

N

; r̂

N+1

�

N+1

)

1

r

N+1

�

ij

(r

N+1

)

+

X

j

d

jk

�

j

(x

1

; x

2

; :::; x

N+1

) (3.13)

where f�

i

g are called the channel functions, obtained by coupling the target

states �

i

(including any pseudo states) with the angular and spin functions of

the continuum electron to form states of total angular momentum and parity. A

is the antisymmetrization operator which accounts for electron exchange between

the target electrons and the free electron (i.e. it imposes the requirements of the

Pauli exclusion principle).

A =

1

p

N + 1

N+1

X

n=1

(�1)

n

(3.14)

�

i

represents the quadratically integrable (L

2

) functions (or (N +1)-electron con-


�gurations) which vanish at the surface of the internal region, are formed from the

bound orbitals and are included to ensure completeness of the total wavefunction.

�

ij

are the continuum orbitals corresponding to the appropriate angular momen-

tum obtained from equation (3.6) and are the only terms in equation (3.13) that

are non-zero on the surface of the internal region. c

ijk

and d

jk

are coe�cients and

are determined by diagonalizing H

N+1

, in this �nite space.

(

k

jH

N+1

j

k

0

) = E

k

�

kk

0

(3.15)

where the round brackets here are used to indicate that the radial integrals are

calculated using the �nite range of integration from r = 0 to r = r

a

.

Given the form of the basis states

k

the determination of these coe�cients,

however, is exceedingly di�cult so the following approach is used. Let f'

�

g denote

collectively the set of basis functions (real, pseudo and continuum orbitals) and

let fV

k�

g denote collectively the set of coe�cients (fc

ijk

g and fd

ijk

g) so that

k

=

X

�

'

�

V

k�

(3.16)

The Hamiltonian matrix elements can then be rewritten as

H

��

0

= ('

�

jH

N+1

j'

�

0

) (3.17)

which are evaluated in exactly the same way as that demonstrated in section 2.3.2

where all the radial integrals involving continuum orbitals are taken over a �nite

range of r. Subsequent diagonalization of this matrix will then provide V

k�

along

with the eigenvalues E

k

. Since H

N+1

is a hermitian operator then the eigenvalues

are real.

To summarise, equation (3.13) may be described generally as follows. The

�rst expansion on the right hand side of the equation includes all target states


of interest - both the initial and �nal states in the collision process speci�ed

plus other states which are expected to be closely coupled to them during the

collision. Psuedo-states may also be included in this expansion to approximately

represent continuum states of the target. The second expansion of equation (3.13)

performs two roles. Firstly, it ensures that the total wavefunction is complete

i.e. all con�gurations have been accounted for. Secondly it represents short-

range correlation e�ects, since the functions �

j

adequately represent part of the

continuum omitted from the �rst expansion.

3.1.5 The R-matrix

It has been demonstrated that it is possible to �nd the inner region basis wave-

functions f

k

g. Therefore, to completely solve for the inner region total wave-

function , the energy-dependent coe�cients fA

E

k

g from equation (3.12) must

be found using the R-matrix. This matrix relates each continuum orbital value

at the R-matrix radius to the value of the others, and their �rst derivatives, at

the boundary. Its use will be made clear later.

Beginning with the following equation

(

k

jH

N+1

j)� (jH

N+1

j

k

) = E(

k

j)� E

k

(

k

j)

= (E � E

k

)(

k

j)

(3.18)

which is obtained from equations (3.11), (3.12) and (3.15). To simplify this, note

that only the kinetic energy operator in the Hamiltonian contributes to the left

hand side of this equation leaving

�

1

2

(N + 1)

�

(

k

jr

2

N+1

j)� (jr

2

N+1

j

k

)

�

= (E � E

k

)(

k

j) (3.19)


Now de�ne the surface amplitudes !

ik

(r) by

!

ik

(r) =

X

j

c

ijk

�

ij

(r) = r(�

i

j

k

): (3.20)

Note that in expression (3.19) the only non-zero contribution to the left hand

side occurs whenever the kinetic energy operator (r

2

N+1

)acts on the continuum

orbitals. Using this fact and equations (3.12) and (3.20):

�

1

2

X

ijk

0

A

Ek

0

[(�

i

!

ik

(r

N+1

)jr

2

N+1

j�

j

!

jk

0

(r

N+1

))

�(�

j

!

jk

0

(r

N+1

)jr

2

N+1

j�

i

!

ik

(r

N+1

))]

= (E � E

k

)(

k

j) (3.21)

as (�

i

j!

ik

) =

k

. Now de�ne the reduced radial wavefunction of the continuum

electron in channel i at energy E by

F

i

(r) =

X

k

A

Ek

!

ik

(r) = r(�

i

j) (3.22)

This is a form of the total electron wavefunction. Using the orthonormality of the

channel functions then gives

�

1

2

X

i

��

!

ik

�

�

�

�

d

2

dr

2

�

�

�

�

F

i

�

�

�

F

i

�

�

�

�

d

2

dr

2

�

�

�

�

!

ik

��

= (E � E

k

)A

E

k

(3.23)

Now apply Green's theorem and use the boundary conditions given by equations

(3.7) and (3.8) to obtain

�

1

2

X

i

!

ik

(r

a

)

�

dF

i

dr

�

b

r

a

F

i

�

r=r

a

= (E � E

k

)A

E

k

(3.24)

Rearranging gives an expression for the coe�cients and so

A

Ek

=

1

2r

a

1

(E

k

� E)

X

i

!

ik

(r

a

)

�

r

a

dF

i

dr

� bF

i

�

r=r

a

(3.25)


De�ning the R-matrix by its elements

R

ij

(E) =

1

2r

a

X

k

!

ik

(r

a

)!

jk

(r

a

)

(E

k

� E)

(3.26)

so equation (3.22) can be written in the form

F

i

(r

a

) =

X

j

R

ij

(E)

�

r

a

dF

j

dr

� bF

j

�

r=r

a

(3.27)

by multiplying equation (3.24) by !

ij

and summing over k.

The two unknowns on the right hand side of these two equations ((3.26) and

(3.27)), namely the surface amplitudes, !

ik

(r

a

), and the R-matrix poles, E

k

, can

easily be obtained from the eigenvalues and eigenvectors of the Hamiltonian ma-

trix. These two equations are in fact the basic equations from which the wave-

functions for the internal region can be obtained, thereby describing the electron

scattering problem there. The logarithmic derivative of the reduced radial wave-

function of the scattered electron on the boundary of the internal region is given

by equation (3.27) and must be matched across the boundary to the external

region.

3.1.6 The Buttle correction

The most important source of error in this method is the truncation of the ex-

pansion in equation (3.26) to a �nite number of terms. Suppose the R-matrix

expansion is truncated in such a way that the R-matrix is calculated from the

�rst N -terms in the continuum expansion. These are low lying contributions

which are obtained from the eigenvectors and eigenvalues of the Hamiltonian ma-

trix. The remaining distant, neglected contributions can play an important role

in the diagonal elements of the R-matrix where they add coherently. They can


be included in equation (3.26) by solving the zero-order equation

(

d

2

dr

2

�

l

i

(l

i

+ 1)

r

2

+ V (r) + k

2

i

)u

0

i

(r) =

X

n

�

ijk

P

k

(r) (3.28)

which is the same as equation (3.6) but is solved here at channel energies k

2

i

without applying the boundary conditions (3.7) and (3.8) at r = r

a

The correction

R

c

ii

to the diagonal elements of the R-matrix at the energy k

2

i

necessary due to

truncation is then given using the formula discussed by Buttle (1967).

R

c

ii

(N; k

2

i

) �

1

r

a

1

X

j=N+1

u

ij

(r

a

)

2

k

2

ij

� k

2

i

=

�

r

a

u

0

i

(r

a

)

�

du

0

i

dr

�

r=r

a

� b

�

�1

�

1

r

a

N

X

j=1

u

ij

(r

a

)

2

(k

2

ij

� k

2

i

)

(3.29)

where u

ij

(r) and k

ij

refer to the jth eigensolution of equation (3.6) satisfying

the boundary conditions of equations (3.7) and (3.8) and u

0

i

is the solution for

channel energy

1

2

k

2

i

in atomic units. Note that the second term of equation (3.29)

subtracts those levels which have already been included. Henceforth the Buttle

corrected R-matrix is used.

R

ij

(E) =

1

2r

a

N

X

k=1

!

ik

(r

a

)!

jk

(r

a

)

(E

k

� E)

+R

c

ii

(N; k

2

i

)�

ij

(3.30)

3.1.7 The external region

The next stage in the calculation is to solve the electron-target scattering problem

in the external region, r > r

a

which is less complex due to the lack of exchange

and correlation with the continuum electron. In this region the colliding electron

is outside the ion and can be considered distinguishable from the target electrons


i.e. antisymmetrisation can be neglected.

(x

1

; :::; x

N+1

) =

X

i

�

i

(x

1

; :::; x

N

; r̂

N+1

�

N+1

)F

i

(r

N+1

) (3.31)

where the same channel functions, �

i

, have been used which were present in

equation (3.13) with the exception that antisymmetrization is no longer required.

Substituting this expansion into the Schr�odinger equation (3.11) produces

X

i

(H

N+1

� E)�

i

F

i

(r

N+1

) = 0 (3.32)

From the de�nition of the Hamiltonian operator given in equation (3.11) the N -

electron and (N + 1)-electron Hamiltonians can be related to give the following

equation

"

X

j

(H

N

� E)�

i

�

1

2

r

2

N+1

�

Z

r

N+1

�

i

+

N

X

k=1

1

r

k;N+1

�

i

!#

F

i

(r

N+1

) = 0 (3.33)

From equations (3.2) and (3.3),

(H

N

� E)�

i

= (�

i

� E)�

i

= �

k

2

i

2

�

i

(3.34)

where k

2

i

were the channel energies and �

i

were the target energies. Combining

these two equations, premultiplying by �

i

and integrating over all coordinates

except r

N+1

it can be seen, due to the orthonormality of the channel functions

(i.e.

R

�

i

�

j

dr = �

ij

), that

X

i

Z

�

j

n

X

k=1

1

r

k;N+1

�

j

dr

!

F

j

(r

N+1

)�

�

k

2

i

2

+

1

2

r

2

N+1

+

Z

r

N+1

�

F

i

(r

N+1

) = 0

(3.35)

where n represents the number of channel functions which were used in equations

(3.13) and (3.31). The potential matrix V

ij

(r) (the long range multipole potentials)


is de�ned by

V

ij

(r) =

*

�

i

�

�

�

�

�

N

X

k=1

1

r

k;N+1

�

�

�

�

�

�

j

+

(3.36)

which upon substitution into equation (3.35) produces the following set of coupled

di�erential equations

�

d

2

dr

2

�

l

i

(l

i

+ 1)

r

2

+

2Z

r

+ k

2

i

�

F

i

(r) = 2

n

X

�=1

V

ij

(r)F

j

(r) i = 1; n(r � r

a

) (3.37)

where l

i

is the channel angular momentum. Now de�ne the long range potential

coe�cient by

a

�

ij

=

*

�

i

�

�

�

�

�

N

X

k=1

r

�

k

P

�

(cos �

k;N+1

)

�

�

�

�

�

�

j

+

(3.38)

Due to the orthonormality of the channel functions

a

0

ij

= N�

ij

(3.39)

which combined with the following expansion of

1

r

k;N+1

in terms of Legendre Poly-

nomials

N

X

k=1

1

r

k;N+1

=

1

X

�=0

1

r

�+1

N+1

N

X

k=1

r

�

k

P

�

(cos �

k;N+1

) (3.40)

reduces the di�erential equations of (3.37) to

�

d

2

dr

2

�

l

i

(l

i

+ 1)

r

2

+

2z

r

+ k

2

i

�

F

i

(r) = 2

�

max

X

�=1

n

X

j=1

a

�

ij

r

�+1

F

j

(r) (3.41)

where z = Z � N is the residual target charge. Note that �

max

is �nite and is

determined by the angular momentum algebra in equation (3.38). This type of

equation has been the subject of much discussion and computer programs are

available for its solution (Norcross 1969, Norcross and Seaton 1969 and Chivers

1973) but before this can be accomplished various boundary conditions must be

set up to ensure that the solutions obtained from this equation match the solutions

already obtained from the internal region problem at the R-matrix radius.


3.1.8 Matching the solutions

These boundary conditions depend upon the status of the (N +1)th electron, i.e.

whether it is bound or free. If it is free (i.e. it resides in the external region)

then the channels associated with this electron are open. If on the other hand the

(N+1)th electron is bound (i.e. it resides in the internal region) then the channels

are all closed. Suppose then there are a total of n channels where n

a

is denoted by

the number of open ones leaving n�n

a

closed channels. Then a natural boundary

condition for the reduced radial wavefunction at in�nity obtained by �tting to an

asymptotic expansion is

F

ij

(r)

�

r!1

8

>

<

>

:

1

p

k

i

(sin �

i

�

ij

+ cos �

i

K

ij

)

i=1;:::;n

a

j=1;:::;n

a

0(r

�2

)

i=(n

a

+1);:::;n

j=1;:::;n

a

9

>

=

>

;

(3.42)

where

�

i

= k

i

r �

1

2

l

i

� � �

i

ln2k

i

r + �

l

i

�

i

= �

z

k

i

�

l

i

= arg[�(l

i

+ 1 + i�

i

)]

9

>

>

>

>

=

>

>

>

>

;

(3.43)

and a second index, j, has been introduced on the reduced radial wavefunction

F

ij

(r) to label the n

a

independent solutions (the �rst index i corresponds to the

channel). This equation is used as a de�nition for the reactance matrix K whose

standard matrix element is K

ij

.

3.1.9 Open Channels

An n� n dimensional R-matrix in the internal region solution now exists and an

n

a

�n

a

dimensionalK-matrix in the external region. These two matrices must be

related in order for the solutions of each region to match at the boundary. This

is achieved by introducing a set of (n + n

a

) linearly independent solutions v

ij

of


equation (3.41) which satisfy the boundary conditions

v

ij

�

r!1

8

>

>

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

>

>

:

sin �

i

�

ij

+O(

1

r

) (

i=1;:::;n

j=1;:::;n

a

)

cos �

i

�

i(j�n

a

)

+O(

1

r

) (

i=1;:::;n

j=(n

a

+1);:::;2n

a

)

e

�

i

�

i(j�n

a

)

+O(

1

r

) (

i=1;:::;n

j=(2n

a

+1);:::;(n+n

a

)

)

9

>

>

>

>

>

>

>

>

>

>

=

>

>

>

>

>

>

>

>

>

>

;

(3.44)

f�

i

g is given by the equation

�

i

= jk

i

jr � (

z

jk

i

j

) ln[(2jk

i

jr)] (3.45)

where O(

1

r

) means `terms of the order

1

r

or smaller'.

These asymptotic solutions form a set of basis functions which can be used in

an expansion of the reduced radial wavefunction. That is

F

ij

(r) =

n+n

a

X

k=1

v

ik

(r)x

kj

(r

a

� r <1) (

i=1;:::;n

j=1;:::;n

a

) (3.46)

where the coe�cients x

kj

must be chosen so that F

ij

(r) is continuous at r

a

. To

do this, use the solutions for F

ij

in the internal region given by equation (3.27)

along with the boundary conditions (3.42) to give the following equations for the

set fx

kj

g

n+n

a

X

k=1

"

v

ik

(r

a

)�

n

X

m=1

R

im

�

r

a

dv

mk

dr

� bv

mk

�

r=r

a

#

x

kj

= 0 i = 1; :::; n (3.47)

= �

kj

k = 1; :::; n

a

which must be solved for each j = 1; :::n

a

. From the boundary conditions (3.44)

it can be deduced that a matrix element of the reactance matrix is given by

K

ij

= x

i+n

a

;j

i; j = 1; :::; n

a

(3.48)


which completes the solution of equation (3.46) for the case of open channels, and

the resulting wavefunctions are those for free states. The K-matrix is real, sym-

metric and represents the asymptotic form of the entire wavefunction containing

information from both internal and external regions.

3.1.10 Electron collision cross sections

Physically measurable parameters relating to the scattering process can be ex-

tracted from the reactance matrixK. The S-matrix is determined by taking linear

contributions of the n

a

solutions in equation (3.42) such that the new solutions

satisfy the conditions

z

ij

�

r!1

k

�1=2

i

[exp(�i�

i

)�

ij

� exp(i�

i

)S

ij

]; i = 1; n

a

j = 1; n

a

z

ij

�

r!1

O(r

�2

); i = n

a

+ 1; n j = 1; n

a

(3.49)

n

a

� n

a

dimensional S-matrix is then related to the K-matrix by the matrix

equation

S =

1 + iK

1� iK

(3.50)

and the T-matrix is de�ned as follows

T = S� 1 (3.51)

Recall that these are functions of the total angular momentum and parity, i.e. LS�

and can be labelled accordingly. The cross sections can be obtained by standard

methods from the S-matrix (e.g. Blatt and Biedenharn (1952), Lane and Thomas

(1958)). The contribution to the cross section for a transition from an atomic state

with quantum numbers �

i

L

i

S

i

(where �

i

cover all additional quantum numbers)


to an atomic state with quantum numbers �

j

L

j

S

j

is

�

�

i

L

i

S

i

!�

j

L

j

S

j

=

�

k

2

i

X

l

i

l

j

(2L + 1)(2S + 1)

(2L

i

+ 1)(2S

i

+ 1)

jS

ij

� �

ij

j

2

(3.52)

where the summation is carried out over all scattered electron angular momenta

l

i

and l

j

coupled to the initial and �nal atomic states to form the eigenstate of

L

2

, S

2

and parity under consideration.

The partial collision strength for a transition from an initial target state �

i

L

i

S

i

to a �nal target state �

j

L

j

S

j

is given by

LS�

ij

=

!

2

X

l

i

l

j

jT

ij

j

2

(3.53)

where the summation is over the channels coupled to the initial and �nal states.

! = (2L+ 1) (3.54)

for LS coupling. The total collision strength is given by

ij

=

X

LS�

LS�

ij

(3.55)

which is symmetric in i and j.

It is important to note here that the contribution to the R-matrix from the

states retained in equation (3.12) is determined at all energies by a single diag-

onalization. However the Buttle correction and the solutions �

ij

(and hence the

K-matrix) in the asymptotic region (r > a) must be recalculated at each energy

considered. For a complex ion where many channels need to be retained in the

expansion of the total wavefunction, the latter part of the calculation may become

a signi�cant part of the total computational e�ort.


3.1.11 R-matrix Summary

It has been shown that theR-matrix method provides a uniform means of treating

atomic processes, and has several advantages over some earlier approaches. Firstly

it is almost identical to the SOC procedure described in chapter 2 and hence the

same well developed analytic techniques and computer codes can be used. This

is especially true in evaluating the Hamiltonian matrix elements H

ij

, except that

the radial integrals are now evaluated over the range 0 < r < a. Secondly, one

diagonalization is su�cient to determine the R-matrix and the K-matrix over a

wide range of energies. This is particularly important when the cross section is

rapidly varying due to resonances or thresholds and must therefore be calculated

at a large number of energy points. Other advantages of the R-matrix approach

are:

� The convergence of the method is good provided a suitable zero-order Hamil-

tonian is used

� The expansion basis is complete and is included in a systematic way so that

no physics are omitted

� No spurious singularities arise and thus the converged solution satis�es cer-

tain bound principles

� Resonances can be uniquely identi�ed with R-matrix poles and thus tedious

and time consuming searches for narrow resonances are avoided

One disadvantage of the R-matrix approach is that the method is unsuitable

at high energies or for highly excited states.

3.2 The R-matrix Codes 64

3.2 The R-matrix Codes

The computer package RMATRX1 (Berrington et al. (1995))has been developed

to evaluate collision processes of electrons/photons with a variety of targets. The

package has been designed with generality in mind and can be used to calculate

scattering and photoionization cross sections or dipole polarizabilities, for any

atomic or ionic system e.g. the present work concerns electron-impact excitation

of an ion. It is typical for these general purpose codes to be organized in such a

manner that data are saved at suitable stages. Each program requires input from

a previous stage and/or producing output which is used by a subsequent stage.

The R-matrix codes which are considered are written in 4 major parts: STG1,

STG2, STGH, and what are known as the `external region codes'. Each of these

codes is written in FORTRAN 77 and uses preprocessable dimensions. The

running time and memory requirements depend upon the number of target states

included as well as the number of (N +1)-electron con�gurations. Stages 2 and H

must be linked to a further set of library routines known as STGLIB. The basic

functions of all the stages are described in the following sections.

3.2.1 RMATRX STG 1

The STG1 code performs the task of calculating of the R-matrix basis along with

the calculation of all the radial integrals that will be required for the determination

of the Hamiltonian as well as the Buttle correction for the R-matrix. It performs

these tasks through the use of three main controlling routines.

Firstly the routine STGIRD controls the reading of user supplied input data.

This data involves debug parameters, basic data about the ionic system and the

bound orbitals. The debug parameters control the level of output that is generated

by the code. Without any activation of these parameters the output created


involves mainly the eigenvalues of the continuum orbitals but this can be increased

through the use of the debug options to include the radial integrals, Schmidt

coe�cients and so forth. The basic data input includes the number of electrons

in the system, the nuclear charge, the maximum bound and continuum electron

orbital angular momentum and the nl values of real orbitals and pseudo orbitals

which are input separately. The bound orbital set is then input in the form of the

coe�cients, exponents and powers of r so that the radial functions corresponding

to equation (2.16) can be constructed. These values are usually obtained from

the CIV3 package which uses the same formulation of the radial functions. The

R-matrix radius can be input at this point but through the application of these

radial functions and equation (3.5), a value for the radius can now be calculated

by the code.

The evaluation of the bound and continuum orbitals is the second task under-

taken by this module and it is a task that is controlled by the routine BASORB.

Using the parameters that have been input, the routine EVALUE constructs the

bound orbitals using equations (2.6), (2.8) and (2.16) while the routine POTF

calculates the zero order potential V (r) for use in equation (3.6). The potential

V (r) is de�ned as the static potential of the lowest possible target con�guration

with the current radial orbitals. This potential function does not in general have

a simple analytic form, but is generated automatically in numerical form. For

each value of continuum angular momentum l a call to BASFUN (Robb 1970) is

made which provides the solutions to this equation by utilizing DeVogelaeres' in-

tegration method to solve the di�erential equations at a given set of mesh points,

while the eigenvalues k

2

ij

are found by Newton's iteration method. The use of

the Lagrange multipliers is included to ensure that the real bound orbitals are

orthogonal to the set of continuum orbitals produced but if any pseudo orbitals

are included a call to SCHMIDT will construct a basis in which these pseudo or-

bitals are orthonormal to the set of orbitals produced. Once this is completed all

the variables required to evaluate the Buttle correction given by equation (4.29)


have been found. This is accomplished by the routine BUTFIT which in turn is

controlled by the routine NEWBUT. The data resulting from this stage is stored

in a �le usually called STG1.DAT.

The �nal section of this module involves the calculation of all the multipole,

one- and two-electron integrals that are required throughout the calculation. In

each of these cases the integrals are split into the following three types - continuum-

continuum, bound-continuum and bound-bound. This results is nine di�erent

cases to be considered and a separate routine is provided for the calculation of

each integral where an overall controller known as GENINT is used to regulate the

process. Each of these routines uses symmetry properties to reduce the number of

integrals to be performed as much as possible. Following this each routine will call

a further routine to perform the actual calculation of the integral which is speci�c

to the case being considered. For the case of one-electron integrals the routine

ONEELE is called which uses Simpson's rule to evaluate the integral. For the

case of two-electron integrals the routine RS is used while the multipole integrals

are calculated using the RADINT routine. Both of these also use Simpson's rule

with an integration mesh that is de�ned within the code itself but can be altered

in the input stage if so desired. The typical running time of the code depends

approximately on the square of the number of bound orbitals, the square of the

number of continuum orbitals and on the number of continuum angular momenta.

3.2.2 RMATRX STG 2

This stage may be considered to be a three-fold problem. Firstly the target state

wavefunctions and energies need to be calculated and then used to set up the

Hamiltonian matrix. Once complete this is followed by the construction of the

dipole matrices where each of these tasks, along with routines for reading data,

are performed by a separate section of the code.


The two unconnected routines STG2RD and CHEKTP are �rst of all used

to read in and verify the user supplied input data and the data from STG1 re-

spectively. The STG1 information comprises the basic data and radial integrals.

The user supplied input consists mainly of the target and (N + 1)-electron sys-

tem data. In both cases they consist mainly of the con�gurations including the

coupling schemes and total LS�-symmetries. Debug parameters are also included

which can control whether or not the con�gurations included are output as well

as the Hamiltonian and dipole matrices.

The routine BOUND then performs a calculation to solve the target state

problem. That is, it has been written in order to �nd the expansion coe�cients

a

ij

speci�ed in equation (3.4) and thus obtain the energies �

i

and target state

wavefunctions �

i

. This is done in exactly the same manner as the C1V3 code

without any orbital optimization where the one and two-electron integrals gener-

ated in STG1 are used to determine the required Hamiltonian matrix elements.

In a non-relativistic calculation, the total number of target states will probably be

less than the total number of N-electron con�gurations. In this case the number of

target states is speci�ed in the input and the code then takes the states with the

lowest energies up to this value to be the target states. If this is not desired the

energies and expansion coe�cients could be read in manually and the BOUND

routine is thus bypassed. All data obtained from this routine is stored in a �le

usually called STG2H.DAT.

The next stage involves the setting up of the Hamiltonian matrices where one

matrix for each (N + 1)-electron symmetry speci�ed is constructed in the input

data. So the following procedure is looped over for each of these symmetries. The

routine SETUP controls the determination of the channel quantum numbers in

LS-coupling subject to the restrictions enforced by the target and (N+1)-electron

system quantum numbers as well as the maximum value of channel orbital angular

momentum stated by the user. The routine SETMX1 then controls the evaluation


and storage of the (N + 1)-electron Hamiltonian matrix elements which were

introduced in equation (3.17). This routine has three sections to it corresponding

to the determination of the following three types of matrix elements - continuum-

continuum, bound-continuum and bound-bound. In each of these three cases

the routine MATANS is called to actually evaluate the matrix elements utilizing

the radial integrals calculated in STG1 and the method described in section 2.3.2.

Finally using the channel functions generated, the routine AIJS is used to evaluate

the long range potential coe�cients. All the matrix elements, coe�cients and

channel quantum numbers calculated here are stored on disc in the same �le as

the target state data, STG2H.DAT.

3.2.3 RMATRX STG H

The primary function of the STGH module is to diagonalize the Hamiltonian

matrix and the entire task can be accomplished in the following three stages.

The �rst of these is the usual task of reading all the appropriate data which is

split into the data supplied by the user and that calculated by previous modules.

The user supplied data consists of the (N + 1)-electron symmetries and debug

information, the latter of which can provide information such as eigenvectors and

eigenvalues, resulting from diagonalization, which are not normally output. The

read of the user supplied input is performed by the routine STG3RD while TA-

PERD handles the read of these �les which, as in the RECUPD case, is performed

as it is required. The RECUPD module is used to include relativistic e�ects in

the Breit-Pauli approximation and was not used here.

The RSCT routine then controls the task of diagonalizing each of the Hamil-

tonian matrices set up by STG2. For each (N +1)-electron symmetry the routine

MDIAG performs this diagonalization in the continuum basis by applying the

Householder method (Wilkinson 1960). This produces all the eigenvalues and


eigenvectors for the symmetry in question: symmetries are treated individually.

The continuum orbitals from STG1 along with these eigenvectors are then used

to calculate the surface amplitudes using equation (3.20) and these along with

the eigenvectors are stored in a �le labeled H. Each of these eigenvectors is made

up solely of the coe�cients fC

ijk

g and fd

ik

g and thus the identi�cation of a state

with a particular energy is easily performed by examination of these coe�cients.

This is invaluable in the identi�cation of resonances.

Modi�cation of the Hamiltonian

Prior to diagonalization, the option to adjust any unsatisfactory target energies

is presented. This results in the alteration of the diagonal elements of the Hamil-

tonian matrices and has the e�ect of altering the position of excitation thresholds.

Such modi�cations ensure that resonances are in their correct positions with re-

spect to any observed energies which have been input. The physical justi�cation

for this procedure comes from the ability to split the (N+1)-electron Hamiltonian

into separate parts where one of these parts is the target Hamiltonian. This was

shown earlier in the derivation of equation (3.33) from (3.32).

3.2.4 The external region codes

The only module relevant to the current calculation is that of STGF which pro-

duces free electron data.

STGF

The necessary input required for this task consists of the H �le, produced by stage

H, and the user must supply the (N+1)-electron symmetries for which free states

are to be produced as well as an energy mesh for the continuum electron. Calcu-

lation of the R-matrix is then undertaken where it is subjected to open channel

boundary conditions for the purposes of matching on the boundary the wavefunc-

tions at each value of energy speci�ed by the input mesh. The output yields the


LS collision strengths.

These can be converted to LSj collision strengths using the JAJOM package

of Saraph (1972, 1978) which uses algebraic recoupling coe�cients to transform

the K

LS

-matrix to a pair coupling K

J

-matrix. The procedure is fully adequate as

long as the term separation in the target is large compared to the �ne structure

splitting. The energy levels of the target may be signi�cantly shifted by the inclu-

sion of relativistic e�ects, however in the transformation of K

J

to term-coupling

the approximation is made of taking the reactance matrices to be independent of

energy. The collision strength in intermediate coupling

(L

i

S

i

J

i

� L

0

i

S

0

i

J

0

i

) =

1

2

X

J�

(2J + 1)

X

ll

0

�

�

T

J

ii

0

�

�

2

(3.56)

(where capital letters with subscripts refer to angular momentum quantum num-

bers of the target, � is the parity and K = J

i

+ l) is related to the LS coupling

collision strength

(L

i

S

i

� L

0

i

S

0

i

) =

1

2

X

SL�

(2S + 1)(2L+ 1)

X

ll

0

�

�

T

SL

ii

0

�

�

2

(3.57)

by

(L

i

S

i

� L

0

i

S

0

i

) =

X

J

i

J

0

i

(L

i

S

i

J

i

� L

0

i

S

0

i

J

0

i

): (3.58)

Equation (3.58) holds only if the summations are complete with respect to the

orbital angular momentum l of the scattered electron.

3.3 References 71

3.3 References

Berrington K.A., Burke P.G., Chang J.J., Chivers J.J., Robb W.D. and Taylor

K.T. Comput. Phys. Commun.8 (1974) 149

Berrington K.A., Eissner W.B. and Norrington P.H. Comput. Phys. Commun.

92 (1995) 290

Berrington K.A., Burke P.G., Le Dourneuf M., Robb W.D., Taylor K.T. and Vo

Ky Lan Comput. Phys. Commun.14 (1978) 367

Blatt J.M. and Biedenharn L.C. Rev. Mod. Phys.24 (1952) 258

Burke P.G. and Robb W.D. Adv. At. Mol. Phys.11 (1975) 143

Burke P.G. and Seaton M.J. Methods Comp. Phys.10 (1971) 1

Burke P.G., Hibbert A. and Robb W.D. J. Phys. B4 (1971) 153

Buttle P.J.A. Phys. Rev.160 (1967) 719-729

Chivers A.T. Comput. Phys. Commun.6 (1973) 88

Eissner W.B., Jones M. and Nussbaumer H. Comput. Phys. Commun.8 (1974)

270

Lane A.M. and Thomas, R.G. Rev. Mod. Phys.30 (1958) 257

Norcross D.W. Comput. Phys. Commun.1 (1969) 88

Norcross D.W. and Seaton M.J. J. Phys. B2 (1969) 731

Robb W.D. Comput. Phys. Commun.1 (1970) 457

Saraph H.E. Comput. Phys. Commun.3 (1972) 256

Saraph H.E. Comput. Phys. Commun.15 (1978) 247

Scott N.S. and Burke P.G. J. Phys. B13 (1980) 4299

Wigner E.P. and Eisenbud, L. Phys. Rev.72 (1947) 29

Wilkinson J.H. Comput. J.3 (1960) 23

Chapter 4

Electron-impact excitation

of Ni XII

72

4.1 Introduction 73

4.1 Introduction

The walls of the Joint European Torus (JET) vessel are made from a nickel{

chromium alloy. Nickel therefore provides a main source of impurity ions in the

plasma, which are observed via their emission lines in the EUV and X-ray regions

(Rebut 1987).

The bulk of the plasma in JET has an electron temperature of up to � 2 �

10

8

K. Hence emission lines arising from transitions in highly ionized nickel ions

are primarily detected, including ionization stages from typically Ni XXI all the

way up to the hydrogenic system, Ni XXVIII (Bombarda 1988).

Recently however, a device termed the \divertor box" has been installed on

JET (Bertolini 1995). The main aim of the divertor is to remove impurities (and

hence reduce energy loss) from the tokamak, and also to control recycling. In

the longer term a divertor system will be employed to extract the fusion waste

products (i.e. the helium \ash" arising from hydrogen burning) from the tokamak.

The electron temperature in the divertor, and plasma edge (or scrape o� layer,

SOL), is much lower than in the bulk plasma (perhaps as low as �100,000K),

so that the emission lines observed are expected to be primarily those in the

EUV spectral region between �100 { 500

�

A, arising from intermediate ionization

stages of nickel, in particular Ni X { Ni XIII. They can occur if the plasma has

a su�cient electron density to radiate them intensely enough.

The derivation of plasma parameters for the divertor region of JET, including

electron temperature, density and ionic concentrations, are of extreme importance,

as they would allow the e�ciency of using the divertor to remove impurity ions

from the plasma, and hence reduce energy loss, to be quanti�ed. These quantities

will need to be reliably known before nuclear fusion in tokamaks can be realised

and commercially exploited. Electron temperatures and densities in the divertor

region may in principle be determined from diagnostic intensity ratios involving

4.1 Introduction 74

the observed nickel emission lines, while the absolute line intensities may be used

to infer the abundance of the relevant nickel ionization stage. However theoretical

estimates of both emission line ratios and absolute line intensities depend critically

on the atomic data adopted in the calculations, especially for electron excitation

rates (Mason & Fossi 1994).

Unfortunately, little attention has been paid to electron excitation rate calcu-

lations for Ni X { Ni XIII, with existing work having been performed in either the

Distorted-Wave or Gaunt Factor approximations, which do not consider resonance

contributions (see, for example, Krueger and Czyzak 1970; Kato 1976). Although

there does exist a calculation by Pelan & Berrington (1995) for the transition

between the �ne{structure levels of the ground state, there are no other existing

theoretical or experimental atomic collision data for Ni XII, required for a reli-

able calculation of emission line ratios. Therefore the current work is the �rst to

present not only results, but accurate data for the appropriate e�ective collision

strengths.

The reliability of the electron excitation rates depends upon the accuracy

of the collision strengths over the temperature range considered. In turn the

reliability of the collision strengths depends most critically upon the number of

target states included in the R-matrix wavefunction expansion, together with

the con�guration-interaction wavefunction representation of these target states.

The present detailed calculations of electron excitation rates for Ni XII, which

include the explicit delineation of the resonance structure (unlike previous work),

mean results are accurate to �10% (see, for example, Ramsbottom et al. 1996,

1997). The inclusion of resonances is an important component of this approach,

as they normally greatly increase the rate at which a process occurs, which will

in turn have a major e�ect on any derived models of plasma emission. In some

instances (see, for example, Dufton & Kingston 1987) such resonances can lead

to a signi�cant enhancement in the collision rates by up to a factor of 4.

4.2 Calculation Details 75

4.2 Calculation Details

The con�guration-interaction code CIV3, (Hibbert 1975), was used to evaluate

the wave functions and energy levels of chlorine-like Ni XII in LS coupling. The

electron impact collision strengths were produced by utilising the R-matrix com-

puter codes described by Berrington et al. (1987A,B). A full discussion of both

approximation methods appears in chapters 2 and 3.

The present calculation uses the R-matrix method (Burke & Robb 1975), and

includes the 14 lowest-lying LS target states: 3s

2

3p

5 2

P

o

; 3s3p

6 2

S; 3s

2

3p

4

(

3

P )3d

4

D,

4

F;

4

P;

2

F;

2

P;

2

D; 3s

2

3p

4

(

1

D)3d

2

P;

2

D;

2

G;

2

F;

2

S; 3s

2

3p

4

(

1

S)3d

2

D. The

wavefunction for each of these target states was determined using the code CIV3

(Hibbert 1975), and so is expressed in con�guration{interaction form. The LS

coupling reactance matrices obtained from the R-matrix calculation were trans-

formed using a unitary transformation (Saraph 1978) in order to calculate the

�ne{structure collision strengths. E�ective collision strengths were then deter-

mined by averaging the collision strengths over a Maxwellian distribution of elec-

tron velocities.

4.2.1 Target Wave Functions

The con�guration interaction code CIV3 was used to calculate the wave functions

and energy levels of Ni XII in LS coupling. Each term was represented by wave

functions of the form

(LS) =

M

X

i=1

a

i

�

i

(�

i

LS) (4.1)

where each of the con�gurational wave functions f�g is built from one{electron

functions (orbitals) whose angular momenta are coupled in a manner de�ned by

f�

i

g, to form a total L and S common to all con�gurations in equation (4.1),

which is identical to equation (2.20). The mixing coe�cients fa

i

g are determined


variationally (Hibbert 1975).

The one-electron radial functions are represented by a linear combination of

Slater-type orbitals:

P

nl

(r) =

k

X

j=1

c

jnl

r

I

jnl

exp(��

jnl

r) (4.2)

and the parameters, c

jnl

, I

jnl

, and �

jnl

are also determined variationally (Hibbert

1975). Equation (4.2) is identical to equation (2.16).

Ten orthogonal orbitals were used in the calculation, six \spectroscopic" (1s,

2s, 2p, 3s, 3p, 3d) and four pseudo-orbitals (4s, 4p, 4d, 4f), the latter being

included to allow for additional correlations. The R{matrix code requires each

target state to be represented in terms of a single orbital basis so the choice of

orbital parameters was determined as follows. The 1s, 2s, 2p, 3s and 3p orbitals

were taken to be the Hartree{Fock functions for the ground 3s

2

3p

5 2

P

o

state of

NiXII (Clementi and Roetti 1974) . The 3d spectroscopic orbital was optimised

on the energy of the 3s

2

3p

4

(

3

P )3d

4

D state using the 3s

2

3p

4

(

3

P )3d con�gura-

tion; considerable care was taken in selection of state for the energy optimisation,

and also with the parameter optimisation, so that the orbital was a true spec-

troscopic orbital and not \contaminated" with a correlation orbital component.

The correlating pseudo-orbitals (4s, 4p, 4d, 4f) were optimised as follows: the

4s orbital was optimised on the energy of the 3s3p

6 2

S state of Ni XII using

the con�gurations 3s3p

6

, 3s

2

3p

4

(

1

D)3d and 3s3p

6

4s and was included in order to

make allowance for the di�erent 3s orbitals arising in the di�erent states; the 4p

orbital was optimised on the energy of the 3s

2

3p

4

(

1

S)3d

2

D state using the con-

�gurations 3s

2

3p

4

(

3

P;

1

S)3d, 3s

2

3p

3

3d4p and allows for correction to 3p orbitals;

the 4d orbital was optimised on the energy of the 3s

2

3p

4

(

1

D)3d

2

S state using

the con�gurations 3s3p

6

, 3s

2

3p

4

(

1

D)3d and 3s

2

3p

4

(

1

D)4d and is included to ac-

count for the strong coupling between 3s3p

6

and 3s

2

3p

4

nd levels; the 4f orbital


was optimised on the energy of the 3s

2

3p

4

(

3

P )3d

2

P state using the con�gura-

tions 3s

2

3p

4

(

1

D;

3

P )3d, and 3s

2

3p

3

3d 4f . In the above description of the orbital

optimisation it has been implicit that the 1s, 2s, and 2p shells remain closed.

All 14 LS eigenstates were represented as a linear combination of all possible

con�gurations arising from one electron replacement from the above orbital set in

the two basis con�gurations: 3s

2

3p

4

3d and 3s3p

6

; the 1s, 2s and 2p shells remain-

ing closed. A total of 481 con�gurations were therefore required to represent the

target states.

4.2.2 The Continuum Expansion

The total wavefunction describing the collision is expanded in the R-matrix in-

ternal region (r < a) in terms of the following basis (Burke and Robb 1975;

Berrington et al. 1978, 1987):

k

= A

X

ij

c

ijk

�

i

(x

1

; x

2

:::; x

N

; r̂

N+1

�

N+1

)�

ij

(r

N+1

) +

X

j

d

jk

�

j

(x

1

; x

2

:::; x

N+1

)

(4.3)

A is the antisymmetrisation operator which ensures the total wavefunction satis-

�es the Pauli exclusion principle. The �

i

are channel functions formed by coupling

the target states to the angular and spin function of the scattered electron. The

u

ij

are the continuum basis orbitals representing the scattered electron and the �

j

are (N + 1){electron bound con�gurations formed from the atomic orbital basis,

and are included to ensure completeness of the total wavefunction and to allow

for short range correlation.

The continuum orbitals �

ij

are solutions of the radial di�erential equation:

�

d

2

dr

2

�

l

i

(l

i

+ 1)

r

2

+ V (r) + k

2

i

�

�

ij

(r) =

X

n

�

ijn

P

nl

i

(r) (4.4)


which satis�es the boundary conditions:

�

ij

(0) = 0 (4.5)

a

�

ij

d�

ij

dr

�

�

�

r=a

= b: (4.6)

In equation (4.4), l

i

is the angular momentum of the scattered electron and V (r)

is the static potential of the target in its ground state. The �

ijn

are Lagrange

multipliers which are obtained by imposing the orthogonality of the continuum

orbitals to the bound radial orbitals with the same value of l

i

.

Twenty continuum orbitals were included for each channel angular momentum

to ensure convergence in the energy range considered (0 { 121 Ryd). A zero

logarithmic derivative (b = 0) was imposed on these continuum orbitals at an

R-matrix boundary radius of a = 4.8 au.

The coe�cients c

ijk

and d

jk

in equation (4.3) were found by diagonalising

the (N + 1){electron non-relativistic Hamiltonian within the inner region. The

R{matrix is then calculated on the boundary between the inner and outer regions.

Long range coupling between channels is important in the outer region, and

the coupled radial di�erential equations for r > a are solved using a perturbation

technique. This obtains the reactance K{matrices by matching solutions in the

inner and outer regions (r = a). Collision strengths are then found.

In the current 14{state R-matrix calculation, all partial waves with L � 12

for both even and odd parities and spin multiplicities (doublets and quartets)

are considered. Whilst these are su�cient to permit convergence of the collision

strength for the forbidden transitions, it is necessary for dipole{allowed transitions

to include higher partial waves with L > 12. It is assumed that the high-L

behaviour of partial collision strengths for these transitions may be represented

by a geometrical series with a geometric scaling factor equal to the ratio of two


adjacent terms. The justi�cation for this procedure has been given in earlier work

by Ramsbottom et al. (1994, 1995, 1996).

It should be noted that the collision strengths which were determined by the

R-matrix computer packages are for LS states only. The �ne structure collision

strengths are found by transforming to a jj{coupling scheme by utilising the pro-

gram of Saraph (1978), which uses algebraic recoupling coe�cients to transform

the K

LS

-matrix to a pair coupling K

J

-matrix, neglecting term coupling. The

\top{up" from the higher partial waves is again obtained using the geometric

series procedure described previously. Care has been taken in the present work

to ensure that the use of the geometric series was appropriate and provided suf-

�ciently accurate high partial wave contributions. It is di�cult to assess the

computational errors arising due to the use of the geometric series for the high

partial wave contributions. The largest errors, however, would occur for the very

high-impact energy region once the Maxwellian averaging has been performed to

evaluate the e�ective collision strengths. The temperature of maximum abun-

dance for Ni XII ions in ionization equilibrium is log T (K) = 6:2 (Arnaud and

Rothen ug 1985) and falls o� at higher temperatures. In fact at log T (K) = 6:6

the fractional abundance has decreased to N(Ni XII)/N(Ni) < 10

�5

, and hence

atomic data at very high temperatures should normally be relatively unimportant

for this ion.

In running theR-matrix codes it is customary to adjust the target energy levels

to accurate theoretical or experimental values. This ensures that the thresholds

are in the correct places and also improves the positions of resonances. In this

work the levels were therefore adjusted to those given by Fawcett (1987). Any

such adjustment must however not alter the order of the levels, and note that the

present work �nds the 3s

2

3p

4

(

3

P )3d

4

F and 3s

2

3p

4

(

1

D)3d

2

P states to be almost

degenerate, and to be in the reverse order to that found by Fawcett (see Table

4.2). Customary procedure was therefore followed and these two states were made

4.3 Results and Discussion 80

degenerate with a common value of the energy corresponding to that found by

Fawcett for the 3s

2

3p

4

(

1

D)3d

2

P state. However a caveat to note is the reliability

of the �ne-structure transitions involving these levels is in doubt. It should be

noted that this is a considerable improvement over the FeX work of Mohan (1994)

where several states were in an incorrect order. Additionally the value found by

Fawcett for the energy of the 3s3p

6 2

S state has recently been supported by the

experimental result of 3.01414 Rydbergs obtained by Tr�abert (1993).

Finally, it is important for astrophysical and plasma applications to know the

e�ective collision strengths

if

or the excitation rate coe�cients q

if

(Eissner and

Seaton 1974). These are found by averaging the electron collision strengths (

if

)

over a Maxwellian distribution of electron velocities:

if

(T

e

) =

Z

1

0

if

(E

f

)exp(�E

f

=kT )d(E

f

=kT ) (4.7)

and

q

if

=

8:63� 10

�6

!

i

T

1=2

e

if

(T

e

)exp(��E=kT

e

) cm

3

s

�1

(4.8)

where

if

is the collision strength between �ne structure levels i and f , E

f

is

the kinetic energy of the �nal electron, T

e

is the electron temperature (K), k is

Boltzmann's constant, !

i

is the statistical weight of the lower state and �E is the

energy di�erence in Rydbergs between the upper and lower state.

4.3 Results and Discussion

The collision cross sections for all 465 independent transitions in NiXII have been

calculated for the range of impact energies 0-121 Ryd. This impact energy range

was su�cient for the Maxwellian averaging employed to derive e�ective collision

strengths at the electron temperatures of interest. A very �ne energy mesh was

used to properly resolve the detailed autoionizing resonances converging to the


target state thresholds for each transition. The number of points used between

thresholds to determine the mesh can be seen in Table 4.3. Resonances found be-

low the highest excitation threshold included (ie 3p

4

(

3

P )3d

2

D) are considered as

true resonances, whereas those above this energy level are pseudo-resonances aris-

ing because of the inclusion of pseudo{orbitals in the wave function representation

(Burke 1981). The pseudo-resonances are typically found to lie in a restricted en-

ergy range - electron energy up to 20 Ryd above the �nal threshold - and because

the higher energy region becomes more important as the temperature increases it

was necessary to average over these pseudo-resonances. The background to the

total collision strength was extracted from the \raw" data.

The parameters, c

jnl

, I

jnl

, and �

jnl

, used to describe all ten orbitals are listed

in Table 4.1. Table 4.2 shows a comparison of the energies obtained in this work

with theoretical energies (averaged over J-values) calculated by Fawcett (1987).

It is noted that Fritzsche (1995) have also computed energy data but have not

considered all levels included herein. Agreement between the present work and

the values of Fawcett is satisfactory, the greatest di�erences (� 5%) occuring

for the 3s

2

3p

5 2

P

o

{ 3s

2

3p

4

(

1

D)3d

2

S, 3s

2

3p

4

(

3

P )3d

2

P and 3s

2

3p

4

(

3

P )3d

2

D

separations, while the typical di�erence is only 2%. Note that each of the 31

�ne-structure levels is assigned an index number, which are referred to again in

Table IV when denoting a particular transition.

Table 4.3 shows the `resolution' achieved for the cross-sections by the number

of energy points listed. Note that there are no points between the 3s

2

3p

4

(

3

P)3d

4

D

and 3s

2

3p

4

(

3

P)3d

4

F thresholds as they were made degenerate. No points could

be placed between the next four thresholds due to their proximity to each other.

Table 4.4 shows the oscillator strengths, produced by the con�guration-interaction

code CIV3, for all optically allowed transitions between the fourteen LS states of

Ni XII. The accuracy of the target state wave functions can often be indicated by

the amount of conformity between the length and velocity components of the os-


cillator strengths. If an exact wave function has been used then f

L

= f

V

. Except

where the values are small (f

L

; f

V

� 0:01) the present results are in very good

agreement with di�erences between f

L

and f

V

of only 6% to 13%. Comparison

is made with three other sets of values with satisfactory agreement between the

current length values and those of the most recent data of Fawcett (1987) except

for the 3s

2

3p

4

(

1

D)3d

2

P ��3s3p

6 2

S transition, where the di�erent theories have

produced a range of results. Note that the state labels 3s

2

3p

4

(

1

D)3d

2

P and

3s

2

3p

4

(

3

P )3d

2

P appear to be switched in Huang (1983).

The present energies were in good agreement with Fawcett while reasonable

agreement between the current oscillator strengths and the other values was found,

meaning there is satisfaction with the accuracy of the present target wavefunc-

tions. The e�ective collision strengths for Ni XII are presented in Table 4.5 at

temperatures ranging from log T

e

= 5.5 { 6.5 K.

To illustrate the data from Table 4.5, �gures are presented for some transitions

of interest (Figs. 4.1 to 4.6). The collision strengths are given as a function of

incident electron energy in Rydbergs, and the e�ective collision strengths as a

function of log temperature. The transitions are:

� 3p

5 2

P

o

1=2

{ 3p

5 2

P

o

3=2

(Fig. 4.1a and 4.1b): This is an example of a forbidden

transition between the �ne{structure levels of the 3p

5 2

P

o

ground state in

Ni XII. The collision strength presented in �gure (4.1a) clearly shows the

necessity of including many target states in the wavefunction, in that a

wealth of resonance structure is found converging to these thresholds across

the entire energy region considered. The �ne mesh of energies adopted

in the present calculation has clearly ensured that these resonances have

been properly resolved. The e�ect of this structure on the e�ective collision

strength is seen in �gure (4.1b) where the higher lying resonances cause a

signi�cant peak to occur at a temperature of about log T

e

= 5.5 K. Such

enhancements of the e�ective collision strength for forbidden transitions


have been previously found by Ramsbottom et al. (1994, 1995, 1996) when

investigating electron impact excitation of NIV, NeVII and SII.

The results of Pelan & Berrington (1995) for the e�ective collision strength

for the temperature range of log T

e

= 5.0 to 7.0 K lie about a factor of 0.55

below the present values. This is due to several ways in which the present

calculation improves upon that of Pelan & Berrington, namely, proper ac-

count of the d{correlation, considerably more sophisticated con�guration{

interaction wavefunctions and adjustment of the energy thresholds to \ex-

perimental" values. Pelan's calculation also used the R-matrix method in

LS coupling, with collision strengths for the �ne-structure transitions ob-

tained using an algebraic transformation to intermediate coupling. The

energies Pelan derived from CIV3 were deemed good enough to use in the

R-matrix calculation despite his target wavefunctions lacking correlating

pseudo-orbitals. His results tabulate the e�ective collision strengths within

the ground state for chlorine-like ions from ArIV to NiXII. He performed

useful tests for CaIV comparing his 14 term model LS calculation with a 2

term model using a Breit-Pauli calculation. There were di�erences in the ef-

fective collision strengths produced by the two calculations over the desired

temperature range but they were not signi�cant. It should be noted how-

ever that a full relativistic calculation, including such e�ects as spin-orbit

interaction, would be desirable for NiXII but only as more reliable, accurate

energy levels become available.

� 3s

2

3p

5 2

P

o

3=2

{ 3s3p

6 2

S

e

1=2

(Fig. 4.2a and 4.2b) This is an example of

an allowed transition with no change in the principal quantum number i.e.

only a promotion from the 3s to the 3p orbital. Resonances located at the

low energies in this collision strength cause a slight enhancement of the

e�ective collision strength in the low temperature region. For log T

e

> 5.5

K, however the e�ect of including partial waves L > 12 causes the e�ective

collision strength to increase signi�cantly as the temperature increases. Such


a behaviour is typical for an allowed transition of this kind.

� Figure 4.3a presents the collision strength as a function of incident elec-

tron energy, relative to the ground state in rydbergs, for the spin{changing

2

3p

5 2

P

o

3=2

{ 3s

2

3p

4

(

3

P ) 3d

4

D

1=2

forbidden transition. Large autoionizing

resonances located in the low energy region have led to substantially en-

hanced e�ective collision strengths (Figure 4.3b) for low temperatures.It is

di�cult to assess the degree of enhancement due to the lack of available data

for comparison. Such an enhancement of the e�ective collision strength due

to autoionizing resonances is a common feature for forbidden transitions of

this kind. Another characteristic of forbidden transitions is the rapid fall-

o� of the e�ective collision strength for higher temperatures. It should be

noted that the absence of resonance structure in Fig. 4.3a between 4.4 and

4.7 Ry (approximately) is due to the large number of target state thresh-

olds located in this region. Five LS target state thresholds (corresponding

to 13 �ne structure levels) are positioned between these incident electron

energies, and some of them are near degenerate. It proved impossible to de-

lineate auotionizing resonances located in this energy region in the collision

cross sections.

� Figures 4.4a and 4.4b show the collision strength and e�ective collision

strength, respectively, for the forbidden 3s

2

3p

4

(

3

P ) 3d

4

D

1=2

{ 3s

2

3p

4

(

3

P ) 3d

4

F

3=2

transition. A wealth of autoionizing resonances in the low-energy region has

led to the expected enhancement of the e�ective collision strength for low

temperatures. A broad resonance structure located at approximately 5.4

Ryd (Figure 4.4a) has also been responsible for a slight increase in the e�ec-

tive collision strength for temperatures in the range log T (K) = 4.5 to log

T (K) = 5.5

� 3p

5 2

P

o

1=2

{ 3p

4

(

3

P )3d

4

D

e

5=2

(Fig. 4.5a and 4.5b) This is an example of a

typical spin{forbidden transition. At low energies the collision strength is


signi�cantly enhanced by resonances whereas the resonance structure at the

higher thresholds is relatively insigni�cant. Thus, coupling the resonance

phenomena with the almost constant collision strength background one gets

the typical behaviour of e�ective collision strength versus temperature with a

large peak at the lower temperatures and a rapid decrease as the temperature

increases.

� The collision strength for the allowed

2

3p

5 2

P

o

3=2

{ 3s

2

3p

4

(

1

D) 3d

2

P

3=2

tran-

sition is shown in Fig. 4.6a. Evidently the collision strength is dominated

by large resonances across the entire energy region, leading to an enhanced

e�ective collision strength at low temperatures. The characteristic rise of

of the e�ective collision strength at high temperatures due to the inclusion

of higher partial waves, L > 12, is not evident in Fig. 4.6b for this allowed

transition. However, the e�ective collision strength does begin to increase,

as expected, with increasing temperatures above log T (K) = 6.6.


Figure 4.1: Collision strength as a function of incident electron energy in rydbergs,

and the e�ective collision strength as a function of log temperature in Kelvin for

the 3s

2

3p

5 2

P

o

1=2

{ 3s

2

3p

5 2

P

o

3=2

forbidden �ne-structure transition.




the 3s

2

3p

5 2

P

o

3=2

{ 3s3p

6 2

S

1=2

dipole-allowed �ne-structure transition.




the 3s

2

3p

5 2

P

o

3=2

- 3s

2

3p

4

(

3

P ) 3d

4

D

1=2





the 3s

2

3p

4

(

3

P ) 3d

4

D

1=2

- 3s

2

3p

4

(

3

P ) 3d

4

F

3=2





the 3s

2

3p

5 2

P

o

1=2

{ 3s

2

3p

4

(

3

P )3d

4

D

5=2





the 3s

2

3p

5 2

P

o

3=2

- 3s

2

3p

4

(

1

D) 3d

2

P

3=2



Conclusions

All e�ective collision strengths have been calculated in the temperature range

log T

e

= 3.2 { 6.6, su�cient for astrophysical applications and diagnostics. The

accuracy of the results is di�cult to assess. Indeed, the accuracy may only be

properly assessed by comparison with experiment or a more sophisticated calcula-

tion. Such a calculation will necessitate availability of more accurate experimental

energy levels for the target states and should include relativistic e�ects \correctly"

via the Hamiltonian (noting that the present calculation neglects term coupling

in the intermediate coupling scheme.)

Alternatively, assessment is possible via the level of agreement between theo-

retical emission line ratios derived using the atomic data, and those measured for

an astrophysical or laboratory plasma as will be shown in chapter 6. However,

from past experience and noting that the present 14 state R-matrix calculation

(a) uses extensive con�guration-interaction wave functions (b) delineates the com-

plex resonance structure in the collision cross sections and (c) includes correlation

terms in the total wave function to allow for the omitted higher-lying levels it is

safe to expect the e�ective collisions strengths are accurate to approximately 10%

The results contained within this chapter have recently been published (Matthews

1998a, 1998b).

4.4 References 93

4.4 References

Arnaud, M. and Rothen ug, R. Astron. Astrophys. Suppl.60 (1985) 425

Berrington, K.A., Burke, P.G., Le Dourneuf, M., Robb, W.D., Taylor, K.T. and

Vo Ky Lan Comput. Phys. Comm.14 (1978) 367

Berrington K.A., Burke P. G., Butler K., Seaton M. J., Storey P. J., Taylor K.

T., Yu Yan, J. Phys. B20 (1987A) 6379

Berrington K.A., Eissner W.B., Saraph H.E., Seaton M. J. and Storey P. J. Com-

put. Phys. Comm.44 (1987B) 105

Bertolini, E. Fusion Engineering and Design30 (1995) 53

Bombarda F., Giannella R., Kallne E., Tallents G.J., Belyduba F., Faucher P.,

Cornille M., Dubau J., Gabriel A.H. Phys. Rev. A.37 (1988) 504

Burke, P.G., and Robb, W.D. Adv. Atom. Mol. Phys.11 (1975) 143

Burke P. G., Sukumar C. V. and Berrington K. A., J.Phys.B14 (1981) 289

Clementi, E., and Roetti, C. At. Data Nucl. Data Tables14 (1974) 397

Dufton P. L. and Kingston A. E., J. Phys. B20 (1987) 3899

Eissner, W., and Seaton, M.J. J. Phys. B7 (1974) 2533

Fawcett B. C., At. Data Nucl. Data Tables36 (1987) 151

Fritzsche, S., Finkbeiner, M., Fricke, B. and Sepp, W.D. Phys. Scr.52 (1995) 258

Gabriel A. H. and C. Jordan, Case Studies Atom. Coll. Phys. 2 (1972) 209

Hibbert A., Comput. Phys. Comm.9 (1975) 141

Huang K. N., Kim Y. K., Cheng K. T., and Desclaux J. P., At. Data Nucl. Data

Tables28 (1983) 355

Kato T. Astrophys. J. Suppl.30 (1976) 397

Krueger T. K., and S. J. Czyzak Proc. Roy. Soc. London A318 (1970) 531

Mason H.E. and Monsignori Fossi B.C. Astron. Astrophys. Rev.6 (1994) 123

Matthews A., Ramsbottom C.A., Bell, K.L. and Keenan, F.P. Astrophys. J.492

(1998a) 415

Matthews A., Ramsbottom C.A., Bell, K.L. and Keenan, F.P. At. Data Nucl.

4.4 References 94

Data Tables70 (1998b) 41

Mohan M., Hibbert A. and Kingston A.E. Astrophys. J.434 (1994) 389

Pelan, J. and Berrington, K.A. Astron. Astrophys. Suppl. Ser.110 (1995 ) 209

Ramsbottom C. A., Berrington K. A., Hibbert A. and Bell K. L. Phys. Scr.50

(1994) 246

Ramsbottom C. A., Berrington K. A. and Bell K. L., At. Data Nucl. Data Ta-

bles61 (1995) 105

Ramsbottom C. A., K. L. Bell and R. P. Sta�ord, At. Data Nucl. Data Tables63

(1996) 57

Ramsbottom C. A., Bell K. L. and Keenan F. P., Mon. Not. Roy. Astr. Soc.284

(1997) 754

Rebut, P.H. and Keen B.E. Fusion Technology11 (1987) 13

Saraph H. E., Comput. Phys. Comm.15 (1978) 247

Tr�abert E., Phys. Scr.48 (1993) 699

Vajed-Samii M., MacDonald K., At. Data Nucl. Data Tables26 (1981) 467

4.5 Explanation of Tables 95

4.5 Explanation of Tables

TABLE 4.1 Orbital Parameters of the Radial Wavefunctions

This table presents a summary of the radial orbital parameters required

in equation (4.1).

Orbital The one-electron orbital de�ned in spectroscopic notation. A bar

indicates a pseudo-orbital introduced to improve the wavefunctions.

C

jnl

I

jnl

Parameters used in equation (4.1) de�ning the radial part of the

�

jnl

orbital.


TABLE 4.2 Target State Energies (in Ry) Relative to the 3s

2

3p

5 2

P

o

State of Ni XII

Index The index number assigned to each �ne-structure target

state. These index values will be used again in Table 4.5

when depicting a particular �ne-structure transition

J Level The J value of the �ne-structure state

Ni XII State Con�guration and term of the LS target state

Present LS Energy The LS target state energy levels relative to the

3s

2

3p

5 2

P

o

ground state produced by the present

14-state R-matrix calculation

Fawcett The theoretical data of Fawcett (averaged over multiplets)

No. Con�gs. The number of con�gurations retained in the wave

function expansion for each of the target states

included in the calculation


TABLE 4.3 Energy points between the thresholds of Ni XII

Ni XII State Con�guration and term of the LS target state

No. of energy The number of points at which the �ne-structure

points cross-sections have been determined between the

thresholds in the left column.


TABLE 4.4 Oscillator Strengths for Optically Allowed Transitions

in Ni XII

Transition The transition between a lower and an upper target state

for which the oscillator strengths have been evaluated

Present

f

L

Absorption oscillator strength calculated in this work in

the length approximation

f

V

Absorption oscillator strength calculated in this work in

the velocity approximation

Fawcett The length (f

L

) oscillator strengths of Fawcett

(averaged over multiplets)

Huang The length (f

L

) oscillator strengths of Huang et al.

(averaged over multiplets)

Vajed - Samii The length (f

L

) and velocity (f

V

) oscillator strengths

& MacDonald of Vajed-Samii & MacDonald (averaged over multiplets)


TABLE 4.5 E�ective collision strengths for Ni XII

Index Transition between �ne-structure states indicated as initial{�nal

according to the assigned numbers in Table II. For example, Index

2-4 denotes the transition: 3s

2

3p

5 2

P

o

3=2

{ 3s

2

3p

4

(

3

P)3d

4

D

1=2

log T The decimal logarithm of the electron temperature in K. The e�ective

collision strengths for each transition are presented in rows for a

number of electron temperatures ranging from log T

e

(K) = 5.5

to log T

e

(K) = 6.6. A superscript indicates the power of 10 with

which the number must be multiplied; that is, a

�n

= a � 10

�n

4.6 Tables 100

4.6 Tables

Table 4.1: Orbital parameters of the radial wavefunc-

tions.

Orbital C

jnl

I

jnl

�

jnl

1s 0.95461 1 27.55320

0.02840 1 41.14190

0.00308 2 12.19810

0.02251 2 23.07540

0.00100 3 6.42851

-0.00065 3 5.49538

-0.00174 3 11.04760

2s -0.31596 1 27.55320

0.00349 1 41.14190

1.00223 2 12.19810

-0.18203 2 23.07540

0.00410 3 6.42851

0.00023 3 5.49538

0.19015 3 11.04760

2p 0.79425 2 12.14670

0.08195 2 20.03440

0.01360 3 5.97089

-0.00491 3 4.98200

4.6 Tables 101

Orbital C

jnl

I

jnl

�

jnl

0.15235 3 10.76050

3s 0.13650 1 27.55320

-0.00260 1 41.14190

-0.45848 2 12.19810

0.08180 2 23.07540

0.51522 3 6.42851

0.73946 3 5.49538

-0.31533 3 11.04760

3p -0.36670 2 12.14670

-0.03145 2 20.03440

0.73000 3 5.97089

0.41793 3 4.98200

-0.13475 3 10.76050

3d 0.47061 3 7.25259

0.57638 3 4.51270

4s 1.75395 1 1.94259

-3.97343 2 4.41675

7.48160 3 4.39218

-6.21625 4 4.37937

4.6 Tables 102

Orbital C

jnl

I

jnl

�

jnl

4p 9.62702 2 2.70946

-27.01359 3 3.54135

18.09235 4 4.32525

4d 9.14032 3 3.95911

-9.12614 4 5.31968

4f 1.00000 4 5.99002

4.6 Tables 103

Table 4.2: Target state energies (in a.u.) relative to the

3s

2

3p

5 2

P

o

1=2

ground state of Ni XII.

Index J Level LS Present Fawcett No.

State LS Energy Con�gs.

1 1/2 3s

2

3p

5 2

P

o

0.000000 0.000000 64

2 3/2

3 1/2 3s3p

6 2

S 1.455289 1.506955 28

4 1/2 3s

2

3p

4

(

3

P)3d

4

D 2.060766 2.023031 48

5 3/2

6 5/2

7 7/2

8 1/2 3s

2

3p

4

(

1

D)3d

2

P 2.238575 2.221920 59

9 3/2

10 3/2 3s

2

3p

4

(

3

P)3d

4

F 2.238967 2.193423 45

11 5/2

12 7/2

13 9/2

14 1/2 3s

2

3p

4

(

3

P)3d

4

P 2.306762 2.288668 42

15 3/2

16 5/2

17 3/2 3s

2

3p

4

(

1

D)3d

2

D 2.315078 2.290847 80

18 5/2

19 5/2 3s

2

3p

4

(

3

P)3d

2

F 2.360595 2.322125 67

20 7/2

4.6 Tables 104

Index J Level LS Present Fawcett No.

State LS Energy Con�gs.

21 7/2 3s

2

3p

4

(

1

D)3d

2

G 2.380460 2.351050 48

22 9/2

23 5/2 3s

2

3p

4

(

1

D)3d

2

F 2.547676 2.519090 67

24 7/2

25 3/2 3s

2

3p

4

(

1

S)3d

2

D 2.738262 2.705899 80

26 5/2

27 1/2 3s

2

3p

4

(

1

D)3d

2

S 2.874705 2.801870 28

28 3/2 3s

2

3p

4

(

3

P)3d

2

P 3.069623 2.933095 59

29 1/2

30 3/2 3s

2

3p

4

(

3

P)3d

2

D 3.126377 2.999628 80

31 5/2

4.6 Tables 105

Table 4.3: Energy points between the thresholds of Ni

XII.

LS State No. of Energy Points

3s

2

3p

5 2

P

o

201

3s3p

6 2

S

101

3s

2

3p

4

(

3

P)3d

4

D

101

3s

2

3p

4

(

1

D)3d

2

P

-

3s

2

3p

4

(

3

P)3d

4

F

-

3s

2

3p

4

(

3

P)3d

4

P

-

3s

2

3p

4

(

1

D)3d

2

D

-

3s

2

3p

4

(

3

P)3d

2

F

-

3s

2

3p

4

(

1

D)3d

2

G

101

3s

2

3p

4

(

1

D)3d

2

F

101

3s

2

3p

4

(

1

S)3d

2

D

101

3s

2

3p

4

(

1

D)3d

2

S

4.6 Tables 106

101

3s

2

3p

4

(

3

P)3d

2

P

101

3s

2

3p

4

(

3

P)3d

2

D

TOTAL 908

4.6 Tables 107

T

a

b

l

e

4

.

4

:

O

s

c

i

l

l

a

t

o

r

s

t

r

e

n

g

t

h

s

f

o

r

o

p

t

i

c

a

l

l

y

a

l

l

o

w

e

d

L

S

t

r

a

n

s

i

t

i

o

n

s

i

n

N

i

X

I

I

.

T

r

a

n

s

i

t

i

o

n

P

r

e

s

e

n

t

F

a

w

c

e

t

t

H

u

a

n

g

V

a

j

e

d

-

S

a

m

i

i

e

t

a

l

.

.

&

M

a

c

D

o

n

a

l

d

f

L

f

V

f

L

f

L

f

L

f

V

3

s

2

3

p

5

2

P

o

�

3

s

3

p

6

2

S

e

0

.

0

3

3

3

0

.

0

3

0

9

0

.

0

1

9

0

.

0

3

4

0

.

0

1

7

0

.

0

2

8

3

s

2

3

p

5

2

P

o

�

3

s

2

3

p

4

(

1

D

)

3

d

2

P

e

0

.

0

0

1

7

0

.

0

0

2

2

0

.

0

0

1

8

0

.

0

0

2

8

0

.

0

0

1

3

0

.

0

0

0

3

3

s

2

3

p

5

2

P

o

�

3

s

2

3

p

4

(

1

D

)

3

d

2

D

e

0

.

0

0

1

6

0

.

0

0

2

1

0

.

0

0

1

8

0

.

0

0

2

7

0

.

0

1

8

0

.

0

0

5

3

s

2

3

p

5

2

P

o

�

3

s

2

3

p

4

(

1

S

)

3

d

2

D

e

0

.

0

0

1

5

0

.

0

0

2

2

0

.

0

1

0

.

0

0

5

5

0

.

0

4

8

0

.

0

3

6

3

s

2

3

p

5

2

P

o

�

3

s

2

3

p

4

(

1

D

)

3

d

2

S

e

0

.

2

5

7

0

0

.

2

4

2

5

0

.

2

8

3

0

.

2

5

9

0

.

3

0

.

2

3

s

2

3

p

5

2

P

o

�

3

s

2

3

p

4

(

3

P

)

3

d

2

P

e

0

.

6

5

1

0

0

.

6

2

3

1

0

.

8

3

0

.

6

8

0

.

8

4

0

.

1

3

3

s

2

3

p

5

2

P

o

�

3

s

2

3

p

4

(

3

P

)

3

d

2

D

e

1

.

2

6

7

5

1

.

1

9

5

9

1

.

4

4

1

.

2

8

1

.

1

6

0

.

6

7

4.6 Tables 108

T

a

b

l

e

4

.

5

:

E

�

e

c

t

i

v

e

c

o

l

l

i

s

i

o

n

s

t

r

e

n

g

t

h

s

f

o

r

N

i

X

I

I

l

o

g

T

I

n

d

e

x

5

.

5

5

.

6

5

.

7

5

.

8

5

.

9

6

.

0

6

.

1

6

.

2

6

.

3

6

.

4

6

.

5

1

-

2

2

.

2

7

2

.

1

6

2

.

0

2

1

.

8

5

1

.

6

7

1

.

4

7

1

.

2

9

1

.

1

1

9

.

5

5

�

1

8

.

1

5

�

1

6

.

9

4

�

1

1

-

3

2

.

8

5

�

1

2

.

8

4

�

1

2

.

8

4

�

1

2

.

8

5

�

1

2

.

8

7

�

1

2

.

9

2

�

1

2

.

9

8

�

1

3

.

0

6

�

1

3

.

1

7

�

1

3

.

3

0

�

1

3

.

4

5

�

1

1

-

4

2

.

8

5

�

2

2

.

6

4

�

2

2

.

4

3

�

2

2

.

2

4

�

2

2

.

0

5

�

2

1

.

8

7

�

2

1

.

7

1

�

2

1

.

5

5

�

2

1

.

4

0

�

2

1

.

2

6

�

2

1

.

1

2

�

2

1

-

5

5

.

1

1

�

2

4

.

7

0

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2

4

.

3

0

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2

3

.

9

4

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2

3

.

5

9

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3

.

2

7

�

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2

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9

7

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2

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6

9

�

2

2

.

4

3

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2

2

.

1

8

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2

1

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4

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1

-

6

6

.

1

9

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5

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6

5

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5

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1

5

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2

4

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8

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4

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5

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2

3

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4

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3

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3

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2

2

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0

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.

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0

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2

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.

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2

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5

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5

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9

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9

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9

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9

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3

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2

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1

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6

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5

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6

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4

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8

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1

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3

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5

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3

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4

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9

2

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9

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9

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1

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0

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2

1

.

5

2

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2

1

.

3

5

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2

1

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1

3

2

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8

3

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2

2

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6

3

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2

2

.

4

3

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2

2

.

2

3

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2

2

.

0

5

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2

1

.

8

7

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2

1

.

7

0

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2

1

.

5

4

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2

1

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9

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4

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2

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1

0

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2

1

-

1

4

8

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6

7

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3

8

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1

5

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3

7

.

6

3

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3

7

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1

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6

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6

1

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1

3

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3

5

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6

6

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3

5

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2

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3

4

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7

6

�

3

4

.

3

2

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3

3

.

9

0

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3

4.6 Tables 109

l

o

g

T

I

n

d

e

x

5

.

5

5

.

6

5

.

7

5

.

8

5

.

9

6

.

0

6

.

1

6

.

2

6

.

3

6

.

4

6

.

5

1

-

1

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1

.

5

9

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2

1

.

4

9

�

2

1

.

4

0

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1

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3

0

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2

1

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1

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1

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1

2

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3

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2

9

.

4

6

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3

8

.

6

2

�

3

7

.

8

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3

7

.

0

1

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1

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1

6

2

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7

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Chapter 5

Plasma source and

Instrumentation

142

5.1 Tokamaks and Nuclear Fusion 143

5.1 Tokamaks and Nuclear Fusion

5.1.1 Introduction

For comparison with theoretical calculations, extreme ultraviolet emission line

spectra from tokamak plasmas were obtained using a high resolution, time-resolving

duo-multichannel soft x-ray spectrometer instrument. The following chapter gives

a brief overview of tokamak devices, with reference to the JET tokamak where

results were taken.

The role and importance of spectroscopic diagnostics of hot plasmas produced

in controlled thermonuclear fusion research have been extensively discussed in

recent years (Hinnov 1980, De Michelis & Mattioli 1981). In particular, the be-

haviour of impurities in magnetically con�ned tokamak plasmas have been widely

studied, mainly because of the role impurities play in power balance estimates

(Roberts 1981) and their use in particle transport studies (Isler 1984). Moreover

local values of major plasma parameters, such as particle density and temper-

ature, can be measured using the line emissions of various ionization states of

impurity atoms (Suckewer & Hinnov 1977, Suckewer 1981). As fusion plasmas

become hotter by using auxiliary heating (electron temperatures of the order of 5

keV and ion temperatures of 8 keV and, recently, up to about 20 keV have been

measured in tokamaks (Eubank et al. 1978, Strachaen et al. 1987 )), the domain

of interest for spectroscopic diagnostics shifts towards the XUV range, as highly

ionized atoms of heavy impurities emit their brightest lines in the 1-400

�

A range.

The ideal fusion plasma would consist only of fuel ions, electrons and fusion

reaction products. However, in reality the presence of impurities is unavoidable.

Particles are ejected from the wall materials due to plasma surface interactions and

enter the plasma (Bohdansky 1984). Through power loss by radiation, increased

resistivity and ion transport, impurities a�ect the energy balance and stability of


the plasma.

In a plasma, ions do not emit in isolation; a measured spectrum is in uenced by

the plasma conditions in the vicinity of the emitting ion. As a rule, in a tokamak

the radial electron density and temperature pro�les peak at the plasma centre and

decrease towards the edge. This gives rise to a series of spatially discrete regions

for each ion charge state, which vary and overlap with local plasma conditions.

Radiation from each ionisation stage is emitted forming a component of the total

power spectrum. Spectroscopy, as a passive or non-perturbing measurement, o�ers

a method of measuring plasma parameters such as electron and ion temperature,

electron density, Z

e�

(e�ective charge of the plasma) and the ionisation balance,

because of the in uence of the plasma on emission spectra. The EUV region

of the electromagnetic spectrum, in particular, contains a wealth of diagnostic

information.

The plasma conditions in tokamaks are the closest of any laboratory plasmas

to those encountered in many hot astrophysical EUV-ray sources. Tokamaks have

typical peak electron temperatures up to several keV , 12 - 15keV in JET; electron

densities range from 10

18

to 10

20

m

�3

. Their parameters are therefore comparable

with conditions found in solar ares and coronal plasmas of other active stars.

Those plasma parameters relevant to atomic physics (electron temperature T

e

,

and density n

e

, etc.) are independently measured by various non-spectroscopic

techniques. Therefore, tokamaks are well suited to the testing of atomic physics

models in a controlled environment.

The data presented later in this thesis were measured from the Joint European

Torus tokamak. This machine has been built as part of the body of intense research

activity investigating an economically viable, safe and long term energy source by

thermonuclear fusion. In this chapter a brief outline is presented of the main

aspects of fusion research and of magnetic con�nement techniques, in particular,

of which the tokamak is the most successful. The main features of the machine is


also described.

5.1.2 Tokamaks and nuclear fusion

Controlled nuclear fusion o�ers the prospect of an abundant long term energy

source. In a nuclear fusion reaction isotopes of hydrogen (deuterium (D) and

tritium (T)) are combined to create a new nucleus with a lower nuclear binding

energy. The excess energy is divided amongst the fusion products as kinetic energy

and has the potential to be harnessed for use, eventually, in commercial reactors.

The most promising fusion reaction, in terms of energy produced, high reaction

rate and low temperature threshold is that between D and T:

D + T !

4

He (3:56MeV ) + n (14:03MeV ) (5.1)

In practice tokamaks at present tend to operate with deuterium plasmas. The

non-tritium reactions have smaller cross-sections than the D-T reaction except at

large energies and they are as follows:

D +D !

3

He (0:817MeV ) + n (2:45MeV )

D +D! T (1:008MeV ) +H (3:024MeV ) (5.2)

where the reduced neutron energy and ux result in minimal vessel activation

compared with the D-T case, allowing increased access to the device for modi�ca-

tions and maintainance. The energies given are the kinetic energies of the reaction

products.

The goal of nuclear fusion research is to con�ne a plasma of these reactants at a

high enough temperature and density, for a long enough time that an appreciable

number of fusion reactions take place and yield more power than that used to

sustain the plasma.


The fusion plasma �gure of merit, the triple product n

i

T

i

�

e

, �rst derived by

Lawson (1957), summarises the trade o� between fusion heating and power loss.

n

i

and T

i

are the fuel ion density and temperature respectively and �

e

, is the

energy con�nement time, which is a ratio of total kinetic energy to input power.

A triple product of approximately 10

21

m

�3

keV s corresponds to the break-even

point, where the fusion � particle energy absorbed by the plasma is equal to the

plasma energy loss. The ratio of fusion reaction power produced to heating power

supplied is the factor Q

DT

which is equal to one in this case. Beyond this point

ignition is achieved yielding a net energy gain.

Note that it is impossible to ignite the plasma using a very low tempera-

ture, very high density plasma because the bremsstrahlung losses will always

exceed reaction gains. Also, at very high temperatures and very low densities,

the equipartition time becomes long enough for the electrons and ions to have

di�erent temperatures unless they are heated equally and have equal energy loss

time. A reactor could be operated here, but the ions would always require exter-

nal heating as the fusion �-particles slow down on the electrons rather than the

ions due to their higher velocities.

5.1.3 Magnetic Con�nement

There are two main practical approaches to the con�nement of fusion plasmas:

inertial and magnetic con�nement. In inertial con�nement a pellet of fuel is

compression heated to high temperatures (> 10keV ) by the spherically symmetric

application of high power lasers but for very short con�nement times (� few

nanoseconds) due to the plasma expansion under internal pressure. In magnetic

con�nement similar or higher temperatures are obtained with much lower fuel

densities but correspondingly higher con�nement times.

The simplest geometry is toroidal in which fuel ions are constrained to move


Figure 5.1: The tokamak geometry

around closed magnetic �eld lines. The most successful, and most widely studied,

device for such con�nement, which overcomes the Coulomb repulsion of positive

ions, is the tokamak (from the Russian for toroidal chamber-magnetic chamber).

The tokamak geometry is shown in Fig. 5.1.


Figure 5.2: Schematic of the tokamak �eld con�guration

Fig. 5.2 shows a schematic outline of the tokamak magnetic �eld con�guration.

A toroidal magnetic �eld is created by a set of coils around the toroidal vacuum

vessel. This �eld alone does not allow con�nement of the plasma. In order to

have an equilibrium in which the plasma pressure is balanced by the magnetic

forces it is necessary also to have a poloidal �eld. This is created by transformer

action, where the toroidal plasma current acts as the single secondary winding of a

transformer and gives rise to the poloidal self magnetic �eld of the plasma current.

Auxiliary poloidal �eld coils are used to shape and position the plasma. The

combination of the toroidal and poloidal magnetic �elds gives rise to a resultant

closed helical magnetic �eld which con�nes the charge particles in a ring shaped

plasma. The magnetic �eld lines lie on a series of nested ux surfaces, which are

also surfaces of constant pressure. A particle describes a helical path along a ux

surface. Collisions with other plasma constituents are necessary for the particles

to cross from one surface to another.


Since the transformer action necessary to create the poloidal �eld compo-

nent requires a time varying current in the primary winding, most tokamaks can

only operate in pulse mode. The pulse duration is set by the ux swing of the

transformer. Steady state operation is only possible with the application of non-

inductive current drive methods. However, in some machines the pulse duration

can be extended by A.C. operation, where the current is reversed within the ux

swing.

5.1.4 Plasma heating methods

The tokamak has intrinsic ohmic heating as a result of the plasma current de-

scribed above. As the temperature increases the e�ectiveness of this is reduced

since plasma resistivity scales as T

�3=2

e

(Spitzer 1962). This has the advantage

that a low resistivity means that only a small loop voltage is required to drive the

plasma current. However additional means of heating will therefore be necessary.

Additional heating methods can be categorised into Radio Frequency (RF)

heating or Neutral Beam Injection (NBI) heating. For the former of these radio

frequency waves are coupled into the plasma at the gyration frequency of the

electrons or ions as they move in a helical trajectory along the magnetic �eld

lines. For the ions the frequency is in the range 20-100MHz and this method is

known as ion cyclotron resonance heating (ICRH), while for electrons (ECRH)

the frequencies are in the 100 - 200 GHz range with subsequent heating of the

ion by collisions. Additionally, a lower hybrid current drive (LHCD) causes reso-

nance heating of the plasma by utilizing electromagnetic power in the frequency

range 1- 8 GHz, between the ion and electron gyrofrequencies. This wave has an

electric �eld parallel to the magnetic �eld and di�erentially accelerates the ions

and electrons along the magnetic �eld lines, giving current drive.

For NBI heating ions are accelerated and then neutralised so that the atoms


can penetrate across the magnetic �eld. In practice high energy hydrogen or

deuterium beams (> 10keV ) are used, and the beam energy is deposited into the

plasma by re-ionisation or charge exchange processes. Such beams also fuel the

plasma in addition to heating it and their usage is marked by a rise in plasma

density.

5.1.5 Impurities

In tokamak plasmas, emission lines originate from impurity ions present, with

small concentrations, in the main fuel gas. The con�nement of the plasma con-

stituent particles is not perfect and some interaction with the walls of the vessel

inevitably takes place. This interaction gives rise to impurity ions in the bulk

plasma. These impurities degrade the plasma by diluting the fuel and by increas-

ing power loss by radiation (De Michelis & Mattioli 1984, Isler 1984). However,

many diagnostic techniques rely on the presence of intrinsic or injected impurities.

Impurity ions are either intrinsic to the machine from erosion of material limiters

or the vessel walls or added deliberately to the plasma for a speci�c experiment.

A number of wall conditioning techniques have been developed which give

a degree of control over the material surface in contact with the plasma. Light

elements are used for wall coating materials since the power loss through radiation

is proportional to atomic number Z

2

.

Another approach for the control of impurities is the use of a magnetic limiter.

This de�nes the last closed ux surface and limits the plasma radius. The magnetic

�elds can be con�gured to provide a magnetic limiter or separatrix whereby the

plasma outside the last closed ux surface is diverted to a remote part of the

vessel from which the impurity particles can be exhausted. This is known as the

divertor con�guration or X-point plasma.

5.2 Plasma Diagnostics 151

5.1.6 Con�nement Modes

It was discovered that there are a number of di�erent con�nement modes in toka-

mak plasmas dependent on di�erent magnetic con�gurations and heating meth-

ods. As the total power into the non-divertor plasma, P

in

, is increased, the

plasma con�nement degrades, approximately as 1=

p

P

in

leading to a low con-

�nement mode of operation (L-mode). However, in tokamaks operating in the

divertor con�guration, a transition can occur above a certain threshold level of

P

in

, which leads to a high con�nement regime (H-mode) with energy con�nement

times enhanced by a factor of 2 or more relative to those in L-mode. Since their

initial discovery, H-modes have also been reported for other magnetic con�gu-

rations (Stambaugh et al. 1990) and for high density, ohmically heated plasmas

(Carolan et al. 1994). As a factor in the triple product, high con�nement regimes

are important from the point of view of the fusion reactor. However, the particle

con�nement time is also enhanced and this has severe implications for impurity

accumulation and subsequent power loss.

5.2 Plasma Diagnostics

Among the principal diagnostics required for the characterisation of a plasma are:

� Radial pro�les of density and temperature of both ions and electrons.

� Energy balance diagnostics to compare radiated and particle power losses

with power input.

� Magnetics for plasma position and size control and for mode structure mea-

surements.

� Impurity monitors.

� Fusion product monitors.

5.3 Tokamak Experiments 152

Figure 5.3: Cutaway diagram of the JET tokamak showing the main components

of this device

The diagnosis of plasma parameters involves making measurements across the

range of the electromagnetic spectrum, of neutral and charged particles and of

magnetic and electrostatic �elds. A broad range of plasma diagnostics has been

reviewed by Hutchinson (1987). The most important diagnostics for the purpose

of this work were the electron density and temperature measurements.

5.3 Tokamak Experiments

Results are presented in this thesis from the JET (Joint European Torus) tokamak.

The main machine speci�cations for the tokamak is given in Table. 5.1 and a

cutaway diagram of the JET device can be seen in �gure 5.3.

JET is currently the largest tokamak experiment in the world. The project

was designed with the objectives of obtaining and studying plasmas in conditions


and dimensions approaching those needed in a fusion reactor (Rebut 1987). The

machine, illustrated in Fig. 5.3 has overall dimensions of about 15m in diameter

and 12m in height. The D-shaped vacuum vessel is of major radius R

0

= 2.96m

with minor radii of a = 1.25m (horizontal) and b = 2.10m (vertical). The toroidal

component of the magnetic �eld is generated by 32 D-shaped coils equally spaced

around the torus and enclosing the vacuum vessel. The resultant magnetic �eld

at the plasma centre is a maximum of 3.45 T . A plasma current of up to 7

MA is produced by transformer action using an eight limbed magnetic circuit.

A set of coils around the centre limb of the magnetic circuit acts as the primary

winding, with the plasma acting as the secondary. Poloidal coils situated around

the outside of the vacuum vessel are used to shape and position the plasma.

Normally the duration of a plasma pulse in JET is 20 - 30s with the plasma

current sustainable at peak values for several seconds. The plasma duration can

be extended to 60s by the use of a non-inductive current drive system (LHCD).

Two additional heating systems are used: NBI and ICRH, with a total maximum

power availability of 25-30MW . In 1992/93 an axisymmetric pumped divertor

was installed inside the vacuum vessel in order to assist impurity control studies.

JET is equipped with over sixty di�erent diagnostic systems for the monitoring

and study of plasma parameters. Experiments have been carried out mainly using

hydrogen or deuterium plasmas, although

3

He and

4

He have also been used. In

1991, a preliminary experiment using 10% tritium in deuterium was performed.

A further successful 50% tritium phase was performed in 1997 with a peak ratio

of fusion power of 0.9 : see �gure 5.4.


Figure 5.4: Current and previous world records for power output by tokamak

devices


Table 5.1: The principal JET machine parameters. The values quoted are the

maximum achieved values.

Parameter JET

Major Radius R

0

2.96m

Minor radius horizontal a 1.25m

Minor radius vertical b 2.10m

Aspect ratio R

0

/a 2.37

Plasma elongation b=a 1.68

Toroidal magnetic �eld 3.45 T

Plasma current 7.0 MA

Flat top pulse length 60s

Additional Heating Power (Total) 32 MW

Neutral Beam Injection 21 MW

Ion Cyclotron Resonance Heating 20 MW

Lower Hybrid Current Drive 6.3 MW

5.4 Instrumentation 156

5.4 Instrumentation

5.4.1 Introduction

In recent years a new type of detector has come into use in tokamak spectroscopy.

A microchannel plate image intensi�er (Wiza 1979) coupled to a photodiode array

(Talmi 1980) (or to other types of multichannel photoelectric devices (Timothy

& Simpson 1983)) allows the recording of large spectral ranges with simultaneous

time resolution.

During the last few years tokamak plasmas have been used as laboratory

sources for basic research in atomic spectroscopy. Spectra of highly ionized heavy

elements (Z = 30-80) have been obtained from these hot plasmas and a large body

of line classi�cation work has been performed (see, for instance, the review paper

by Fawcett (1984)). Also, because electron temperatures and densities are accu-

rately measured in tokamaks, independently of spectroscopic observation, models

used in astrophysics to predict these parameters can be tested by measuring line

intensity ratios.(Feldman et al. 1982, Stratton et al. 1984, Yu et al. 1986) This

work relies heavily on accurate line brightness measurements, and therefore re-

quires high spectral resolution in order to reduce line blending.

The above arguments led to the development of the current KT4 instrument

which is a high-resolution, duo-multichannel, time-resolving EUV spectrometer,

(Schwob et al.:1983a, 1983b) working at grazing incidence. Most of the data

observed were from the KT4/2 instrument.

Due to an interferometric adjustment of both the grating and the microchan-

nel plate detector on the Rowland circle, this novel instrument achieves a very

high resolution over the whole spectral range covered. Pre-adjusted interchange-

able grating mountings allow changes, with no new adjustment, in the overall

spectral coverage from 10-85

�

A at high resolution (typically 0.05

�

A , using the


2400-g/mm grating) to 10-340

�

A (with a 600-g/mm grating and a resolution of

0.2

�

A ). Moreover, this instrument can easily be switched from a spectrograph

mode using photographic plates to the duochromator or the multichannel mode.

On the JET instrument two similar multichannel detectors (MCD) simultaneously

covering two di�erent spectral ranges are used. A relatively large Rowland circle

diameter (2m) with a very small grazing incidence angle (1

�

- 1.5

�

) and an aux-

iliary slit to eliminate UV stray light provide a high signal-to-background ratio,

even at the short wavelength side (10-30

�

A ) of the spectral domain covered.

The multichannel detector system was �rst developed and installed on a Schwob-

Fraenkel soft x-ray spectrometer at the Princeton Plasma Physics Laboratory and

operated on the PLT (Princeton Large Torus) and on the TFTR (Tokamak Fusion

Test Reactor) tokamaks. Since then, similar multichannel detector systems have

become operational on other tokamaks: TFR (Tokamak Fontenay-aux-Roses) in

France and JET (Joint European Torus) at Culham U.K.

5.4.2 The basic instrument

The basic instrument is a high-resolution 2-m grazing-incidence Schwob-Fraenkel

spectrometer built at the Hebrew University of Jerusalem (Filler et al. 1977),

operating in the following modes: spectrograph, duochromator or multichannel.

The instruments installed on the JET tokamak use the multichannel mode.

The main body of the instrument consists of a mono-bloc duraluminium piece

with an accurately machined cylindrical surface (with a precision better than

10�m), which materializes the Rowland circle. The grazing incidence angle can

be varied from less than 1

�

to 2.5

�

by moving the carriage supporting the main

entrance slit along the cylindrical surface. In the present work an angle of 1.5

�

was chosen. An accurate mounting accepting preadjusted grating holders enables

grating interchange without any further optical adjustment: this allows a quick


change of wavelength range and spectral resolution. When equipped with a 2400-

groove/mm concave grating, the instrument covers an overall spectral range of 5-

90

�

A, and with a 600-groove/mm grating, the overall wavelength coverage extends

from 10 to 360

�

A.

5.4.3 Multichannel Detector Mode

The photoelectric system employs two carriages, each of which carries an exit slit

coupled to a multichannel electron photomultiplier detector.

The microchannel plate detector The head of the detector is composed of a

at rectangular 50-mm-long microchannel plate (MCP), manufactured by Galileo

Electro-Optics Corp. This is coupled to a phosphor (P-20) screen image intensi-

�er, as shown in Fig. 5.5.

The incident XUV photons produce photoelectrons at the MCP face which

are subsequently multiplied inside the microchannels due to a cascading e�ect. A

negative voltage of up to -1 kV is applied to the MCP input face, leading to a gain

of up to 10

4

. The exiting electrons are then accelerated and proximity focused onto

the P-20 phosphor screen which converts the electron signal to visible photons.

Focusing is achieved by applying a voltage of +3 to +5 kV across the 1{mm gap

between the MCP output face and the phosphor layer. This voltage also enables

an e�cient conversion in the phosphor.

The MCP used here has 25-�m-diam channels, with a 32-�m center-to-center

spacing. It has been selected to have a relatively, high strip current allowing for

a large dynamic range of operation, and its front surface is MgF

2

, coated in order

to enhance the quantum e�ciency in the soft x-ray region (Milchberg et al. 1984).

The MCP input face is funneled to enlarge the open area from 55 % to 70% of

the total MCP input surface. Moreover the funneling leads to an enhancement


Figure 5.5: Schematic of the multichannel detector system

of the detector quantum e�ciency for grazing incidence angles (which otherwise

is very low). In the present instrument, the MCP operates at extreme grazing

incidence, from 3

�

to 12

�

. A cut is machined in the MCP holding frame, so as

to avoid the shadow cast on the MCP input face at extreme grazing incidences.

In order to increase the e�ciency, the MCP has also been oriented in such a way

as to reduce the angle between the direction of the microchannels (bias angle 8

�

)

and the incident beam.

The phosphor screen which is deposited on a �ber-optic faceplate is optically

coupled to a �ber-optic taper. This is attached to a coherent exible �ber-optic


conduit which transfers the visible photons produced by the phosphor to a Reti-

con photodiode array (PDA), as shown schematically in Fig. 5.5 The use of an

optical reducer permits the actual MCP length to be matched with the 25.6-mm-

long PDA, allowing for a larger simultaneous spectral coverage with only a small

reduction in the spectral resolution, as discussed later. The exible �ber-optic

bundle and the PDA are optically coupled by removable coupling mounts through

a �ber-optic window, which constitutes the vacuum seal interface. Thus, the elec-

tronics controlling the PDA are entirely located outside the vacuum, enabling the

use of conventional commercially available components.

The MCP detector housed in an adjustable cradle located inside a carriage is

interferometrically adjusted to be tangent at the center of its input face to the

Rowland circle. This MCD (multi-channel detector) carriage is attached to the

leadscrew (in place of an SCD (single channel detector) carriage) and can be ac-

curately moved along the machined Rowland cylinder. As in the duochromator

mode, the detector carriage is pressed by means of the guide arc against the cylin-

der surface, thus enabling either horizontal mounting of the spectrometer (with

horizontal or vertical entrance slit), or vertical mounting of the entire instrument.

The coupling of two MCP detectors moving on the Rowland circle to the PDA

through exible coherent optical conduits constitutes a unique feature speci�c to

this instrument. This allows both a simultaneous wide wavelength coverage com-

posed of two di�erent portions of the spectrum and a high spectral resolution to

be obtained, while still using a fairly conventional means of data acquisition and

detector control.

Optical multichannel analyser The visible light signals at the �ber-optic con-

duit output are analyzed by a Reticon (1024 SF) linear 1024-pixel self-scanned

silicon photodiode array which is controlled and read out by a commercially avail-

able optical mulichannel analyzer (OMA) system produced by EG&G Princeton


Applied Research Corp. This includes a PARC 1412 F detector tube which incor-

porates the Reticon PDA, and a PARC 1218 controller unit (replaced recently by

an upgraded PARC 1461 module). The only modi�cation to the standard PARC

1412 detector tube is the use of a window-less detector using a �ber-optic-faced

PDA.

The performance of the OMA system has been extensively studied and de-

scribed by Talmi and Simpson (1980) and for a similar plasma spectroscopy ap-

plication by Fonck et al. (15). The spectral resolution of the system depends on

the distance between two adjacent pixels of the Reticon PDA; each pixel here is

2.5 mm long by 25�m wide. This gives an aspect ratio of 100:1, requiring a good

alignment to make the spectral lines parallel to the pixels. As will be shown later,

the main limitation of spectral resolution of the entire MCP + OMA system lies

in the proximity focusing at the phosphor and in the various optical couplings.

The data-acquisition and data display systems are very similar to that de-

scribed by Fonck et al. (1982). The design of their interface electronics and the

necessary software were adapted to this system. The time resolution of the sys-

tem is limited by the serial scan of the PDA through the PARC package. In the

(multichannel) spectrograph mode, i.e., the readout of the entire PDA array, the

fastest scan is made in 11 ms.

One interesting feature of this detection system is the very large signal-to-

noise ratio. The dark current noise of the PDA can be reduced to 3 count/s rms

(1 count corresponds to about 1000 electrons) when cooled to -20

�

C by the Peltier

cooler. The intrinsic �xed pattern noise can be eliminated by subtracting a dark

scan from each data scan. Owing to the wide linear range of the PDA output,

the �nal dynamic range can reach 10

4

, which allows the recording of intense and

weak lines in the same spectrum.

Absolute intensity calibration of this kind of detector is discussed by Hodge

et al. (1984) and comparison with this and other instruments on PLT indicates


Figure 5.6: Schematic of the con�guration in use in the KT4 multichannel spec-

trometer

that even lines of less than 10

12

photon/cm

2

sr time-integrated intensity should

still be detectable.

The JET detector con�guration The duo-multichannel con�guration is rep-

resented in Fig. 5.6. Each MCD is coupled to its own OMA system. This mode

(selected in the instrument installed on the JET machine) permits observation of

two extended wavelength ranges with a high spectral resolution. It is suitable,

for instance, for monitoring in the same discharge the short wavelength domain

(18-41

�

A ) containing the H-like and He-like transitions of the light elements O

and C, and simultaneously a longer wavelength region where the lines of highly

ionized metallic impurities are emitted.

Although the MCD carriages may be positioned at any point along the Row-

land circle, a series of preselected 20- mm-spaced positons, y, are generally used for

convenience (or to enable �ne corrections in the wavelength calibration). It seems

that for most tokamak diagnostic applications the 600-g/mm grating (blazed at

1

�

31') gives the best compromise between wavelength coverage and spectral reso-

lution. Practically, and according to the Rayleigh criterion, the resolution reaches

values of 0.3

�

Ato 0.4

�

Ain the wavelength region currently investigated. Higher

resolution can be achieved in the 10-100

�

A range by using higher orders, which


are rather intense with this grating, or by employing a 1200- or 2400-g/mm grat-

ing, especially for dense regions in the spectra of injected impurities where the

emission in higher orders may interfere with lines in �rst order. In the 120-340

�

A

range a 600-g/mm grating blazed at 3

�

31' is more e�cient, (Dav�e et al. 1987) but

lines below 80

�

A are practically undetected (even in high orders) in the spectra

thus obtained.

5.5 References 164

5.5 References

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Carolan P.G. et al., Plasma Phys. Contr. Fusion 36 (1994) A111

Dav�e J.H., Feldman U., Seely J.F., Wouters A., Suckewer S., Hinnov E. and

Schwob J.L. J. Opt. Soc. Am. B 4 (1987) 635

De Michelis C. and Mattioli M., Nucl. Fusion 21 (1981) 617

De Michelis C. and Mattioli M., Rep. Prog. Phys. 47 (1984) 1233

Eubank H. et al., Phys. Rev. Lett. 43 (1978) 270

Fawcett B.C., J. Opt. Soc. Am. B 1 (1984) 195

Feldman U., Doschek G. A. and Bbatia A. K., J. Appl. Phys. 53 (1982) 8554

Filler A., Schwob J.L. and Fraenkel B.S., Proceedings of the 5th International Con-

ference on Vacuum Ultraviolet Radiation Physics, Montpellier Vol. III (1977) 86

Fonck R., Ramsey A. and Yelle R. Appl. Opt. 21 (1982) 2115

Hinnov E. Atomic and Molecular Processes in Controlled Thermonuclear Fusion

(Plenum, New York) (1980) 449

Hodge W.L., Stratton B.C. and Moos H.W. Rev. Sci. Instrum. 55 (1984) 16

Hutchinson I.H. Principles of Plasma Diagnostics (Cambridge University Press)

(1987)

Isler R., Nucl. Fusion 24 (1984) 1599

Lawson J.D., Proc. Phys. Soc. B 70 (1957) 6

Milchberg H., Schwob J.L., Skinner C.H., Suckewer S. and Voorhees D. , Laser

Techniques in the Extreme UV (1984) Conf. Proc. No. 119 (American Institute

of Physics, New York) 379

Rebut P.H., Fus. Tech. 11 (1987) 11

Roberts E., Nucl. Fusion 21 (1981) 215

Schwob J.L., Finkenthal M. and Suckewer S., Proceedings of the 7th International

Conference on VUV Radiation Physics (1983a); Ann. Israel Phys. Soc. 6 (1983a)

54

5.5 References 165

Schwob J.L., Wouters A., Suckewer S. and Finkenthal M., Bull. Am. Phys. Soc.

28 (1983b) 1252

Spitzer L., Physics of Ionized Gases (Interscience Publications New York) (1962)

Stambaugh R.D. et al., Phys. Fluids B 12 2941

Strachaen J D. et al., Phys. Rev. Lett. 58 (1987) 1004

Stratton B.C., Moos M. W. and Finkenthal M. , Astrophys. J. 279 (1984) L31

Suckewer S. and E. Hinnov, Nucl. Fusion 17 (1977) 945

Suckewer S., Phys. Scr. 23 (1981) 72

Talmi Y. and Simpson R. W., Appl. Opt. 19 (1980) 1401

Timothy J.G., Publ. Astron. Soc. Pac. 95 (1983) 810

Wiza J. L., Nucl. Instrum. Methods 162, (1979) 587

Yu T.L., Finkenthal M. and Moos H.W., Astrophys. J. 305 (1986) 890

Chapter 6

Line Ratio Diagnostics for the

JET Tokamak

166

6.1 Line Ratio Diagnostics for Tokamak Plasmas 167

6.1 Line Ratio Diagnostics for Tokamak Plasmas

6.1.1 Introduction

Atomic reaction models provide the link by which quantitative diagnostic com-

ments on plasma behaviour and parameters may be made from spectral observa-

tions of emission by impurity ions in the plasma. In this chapter the conditions

under which emission line intensity ratios (which are usually called diagnostic line

ratios) are sensitive to variations in the physical conditions of a plasma, such as

electron temperature (T

e

) and density (N

e

), are discussed.

The problem of impurities must be solved for a fusion reactor, and in partic-

ular for the International Thermonuclear Experimental Reactor (ITER) which is

currently being designed and expected to succeed JET. The problem of impurities

and the power exhaust has been fully recognised in the design of ITER for which

a divertor has been incorporated for this purpose. The JET programme is now

studying divertor plasmas and in particular high power, deuterium-tritium plas-

mas. This required the installation of a pumped divertor inside the Torus. The

construction of the pumped divertor was a major undertaking for the project,

and took nearly two years to complete. Subsequently, following successful ex-

periments the design of the divertor is being progressively optimised by further

modi�cations. Recently a new divertor structure has been installed during a fur-

ther 10 month shutdown. It allows remote handling installation of various divertor

\target" designs.

Essentially the divertor consists of four large coils in the bottom of the Torus on

which the carbon-tiled (or beryllium) target plates are assembled. Alongside the

outer coil is a cryopump. Currents in the divertor coils modify the main tokamak

magnetic �eld to create a null point of the poloidal magnetic �eld above the target

plate. The bulk plasma is bounded by the last closed �eld line whilst the edge


plasma, called the scrape-o� layer (SOL), ows along the outer �eld lines until

intersecting with the divertor target plate. The impurity atoms resulting from

the plasma interaction with the divertor target plates are forced back towards the

divertor and thereafter are \pumped" from the system by the cryopump.

6.1.2 Statistical equilibrium equations

Consider a set of n levels for a given ion in a plasma where the principal popula-

tion and de-population mechanisms are collisions with electrons and spontaneous

radiative de-excitation. The change in population dN

i

=dt of a level i is then given

by

dN

i

dt

= N

e

n

X

j=1

N

j

C

ji

�N

e

N

i

n

X

j=1

C

ij

+

n

X

j=i

N

j

A

ji

�N

i

i

X

j=1

A

ij

(6.1)

where the �rst and second terms are the collisional rates in and out of level i,

respectively, the third and fourth terms are the radiative rates in and out of the

level, C

ij

is the electron collisional rate from level i ! j and unit N

e

, and A

ij

is

the spontaneous radiative de-excitation rate from i! j. For a stationary plasma,

dN

i

=dt = 0 and hence

N

i

=

N

e

n

X

j=1

N

j

C

ji

+

n

X

j=i

N

j

A

ji

N

e

n

X

j=1

C

ij

+

i

X

j=1

A

ij

(6.2)

where i = 1, . . . , n, 1 denoting the ground state. The level populations are related

to the total volume density of the ionization stage N

ion

by

N

ion

=

n

X

i=1

N

i

(6.3)

Consider low values of N

e

. If level i has an allowed transition to the ground state

(i.e. A

i1

is large), then the �rst term in the denominator of (6.2) (N

e

P

n

j=1

C

ij

)


is negligible. Also for low N

e

the level populations of the excited levels will be

very small compared with the ground state, and hence the second term in the

numerator (

P

n

j=i

N

j

A

ji

) becomes negligible. Hence the coronal approximation is

found (Elwert 1952)

N

i

=

N

e

N

1

C

1i

A

i1

(6.4)

The line intensity is therefore

I

i1

= E

i

N

i

A

i1

= E

i

N

e

N

1

C

1i

(6.5)

where E

i

is the energy of level i relative to the ground state, and is directly

proportional to the collisional excitation rate, but is independent of the A-value.

However at high values of N

e

the radiative terms in (6.2) become negligible

and

N

i

=

n

X

j=1

N

j

C

ji

n

X

j=1

C

ij

(6.6)

The relation between inverse collisional rates then gives the thermodynamic equi-

librium population distribution

N

j

N

i

=

g

j

g

i

exp(�E

ji

=kT

e

) (6.7)

where E

ji

is the energy di�erence of the levels and g is the level degeneracy. The

line intensity is therefore

I

i1

= E

i

N

i

A

i1

= E

i

N

1

g

i

g

1

A

i1

exp(�E

i

=kT

e

) (6.8)

Hence the line intensity is directly proportional to the A-value, and is independent

of the collision rate.


T

e

-diagnostics

Gabriel & Jordan (1972) originally derived T

e

and N

e

diagnostics, details of

which are given below. Consider two levels i and j for which the principal rates

are spontaneous radiative de-excitation and electron impact excitation from the

ground state (i.e. we have no metastable levels). Then the coronal approximation

gives for the emission line ratio R

R =

I

j1

I

i1

=

E

j

E

i

C

1j

C

1i

(6.9)

C

1j

may be written as

C

1j

=

8:63� 10

�6

g

1

p

T

e

�

1j

exp(�E

j

=kT

e

) (6.10)

where �

1j

is the e�ective collision strength, which is a slowly varying function of

T

e

. Hence

R =

�

1j

�

1i

E

j

E

i

exp(�(E

j

� E

i

)=kT

e

) (6.11)

so that from the observed value ofR we may derive T

e

. However note that (E

j

�E

i

)

needs to be large for R to be sensitive to variations in T

e

, so that the relevant

emission lines are often well separated in wavelength.

N

e

-diagnostics

In this instance two lines are needed, 1 { i and 1 { k, where i has a small

radiative decay rate (i.e. is a metastable level), and can be depopulated by electron

collisions to another level m with collisional loss rate C

im

. Hence the population

of level i is given by

N

i

(A

i1

+N

e

C

im

) = N

e

N

1

C

1i

(6.12)

as the N

m

A

mi

term can be neglected since N

m

is small. The line intensity ratio


is therefore given by

R =

I

k1

I

i1

=

E

k

E

i

N

e

N

1

C

1k

N

i

A

i1

=

E

k

E

i

N

e

N

1

C

1k

N

e

N

1

C

1i

A

i1

(A

i1

+N

e

C

im

)

=

E

k

E

i

C

1k

C

1i

(1 +

N

e

C

im

A

i1

) (6.13)

If N

e

C

im

� A

i1

then R is independent of N

e

(coronal approximation), but if

N

e

C

im

>

� A

i1

then R is sensitive to variations in N

e

. The presence of the C

1k

=C

1i

term in (6.13) implies that R will also be T

e

{sensitive, particularly when (E

k

�E

i

)

is large.

6.2 NiXII Line Search on the JET Tokamak 172

6.2 NiXII Line Search on the JET Tokamak

The search for NiXII lines was peformed using a variety of packages on the IBM

mainframe at JET. The current project was de�ned after previous detection of

emission lines from low ionisation stages of nickel within the tokamak (Co�ey

1997). All lines noted in this chapter have been accurately recorded in earlier

journals. Observations for the lines due to the 3s

2

3p

5

{ 3s

2

3p

4

3d transitions in the

range 152 - 155

�

A were reported by Gabriel et al. (1966), Behring et al. (1972),

Fawcett & Hayes (1972) and Malinovsky & Heroux (1973). A more accurate

measurement in the region 147 - 161

�

A was performed by Goldsmith & Fraenkel

(1970) who identi�ed the 3s

2

3p

5 2

P

o

{ 3s

2

3p

4

(

3

P )3d

2

D and

2

P arrays and

the 3s

2

3p

5 2

P

o

3=2

{ 3s

2

3p

4

(

1

D)3d

2

S

1=2

line. Ryabtsev (1979) remeasured the

region 138 - 166

�

A and obtained wavelength values for the identi�ed lines in good

agreement with the earlier values. Fawcett (1987) calculated wavelengths in the

region 69 - 317

�

A . His Slater parameters were optimised on the basis of minimising

the discrepancies between observed and computed wavelengths. Fawcett & Hatter

(1980) observed the 3s

2

3p

5

{ 3s3p

6

transitions at 295.321 and 317.475

�

A with an

accuracy of �0:008

�

A. They are identi�ed in the current work within a tokamak

for the �rst time. A table of wavelengths for the transitions observed at JET is

shown below. The A values were computed using Huang et al.'s (1983) oscillator

strengths.


�

=

� A

T

r

a

n

s

i

t

i

o

n

E

n

e

r

g

y

l

e

v

e

l

s

(

c

m

�

1

)

A

(

1

0

9

s

�

1

)

R

e

f

e

r

e

n

c

e

2

9

5

.

3

2

1

3

s

2

3

p

5

2

P

o

3

=

2

{

3

s

3

p

6

2

S

1

=

2

0

3

3

8

6

1

4

5

.

2

5

F

a

w

c

e

t

t

&

H

a

t

t

e

r

1

9

8

0

3

1

7

.

4

7

5

3

s

2

3

p

5

2

P

o

1

=

2

{

3

s

3

p

6

2

S

1

=

2

2

3

6

2

7

3

3

8

6

1

4

2

.

1

7

F

a

w

c

e

t

t

&

H

a

t

t

e

r

1

9

8

0

1

6

0

.

5

5

4

3

s

2

3

p

5

2

P

o

3

=

2

{

3

s

2

3

p

4

(

1

D

)

3

d

2

S

1

=

2

0

6

2

2

8

1

5

1

4

9

.

4

G

o

l

d

s

m

i

t

h

&

F

r

a

e

n

k

e

l

1

9

7

0

1

5

4

.

1

7

5

3

s

2

3

p

5

2

P

o

3

=

2

{

3

s

2

3

p

4

(

3

P

)

3

d

2

P

3

=

2

0

6

4

8

6

4

5

2

2

8

.

9

G

o

l

d

s

m

i

t

h

&

F

r

a

e

n

k

e

l

1

9

7

0

1

5

2

.

9

5

3

s

2

3

p

5

2

P

o

1

=

2

{

3

s

2

3

p

4

(

3

P

)

3

d

2

D

3

=

2

2

3

6

2

7

6

7

7

4

3

5

2

1

4

.

6

G

a

b

r

i

e

l

e

t

a

l

.

1

9

6

6

1

5

2

.

1

5

2

3

s

2

3

p

5

2

P

o

3

=

2

{

3

s

2

3

p

4

(

3

P

)

3

d

2

P

1

=

2

0

6

5

7

2

9

0

5

1

.

5

G

o

l

d

s

m

i

t

h

&

F

r

a

e

n

k

e

l

1

9

7

0

1

5

2

.

1

5

3

3

s

2

3

p

5

2

P

o

3

=

2

{

3

s

2

3

p

4

(

3

P

)

3

d

2

D

5

=

2

0

6

5

7

2

3

0

2

2

3

.

0

G

o

l

d

s

m

i

t

h

&

F

r

a

e

n

k

e

l

1

9

7

0

T

a

b

l

e

6

.

1

:

P

r

e

v

i

o

u

s

l

y

m

e

a

s

u

r

e

d

w

a

v

e

l

e

n

g

t

h

s

o

f

N

i

X

I

I

o

b

s

e

r

v

e

d

i

n

t

h

e

J

E

T

t

o

k

a

m

a

k


6.2.1 Line Search Methods

Nickel lines, in particular those of NiXII, were previously observed in two JET

pulses, 10355 and 31231 (Co�ey 1997). A search has been conducted to discover

how prevalent NiXII is within the plasma and to use the measured line ratios

as validation of the atomic data by comparison with theoretical lines produced

by ADAS (section 6.3). The electron temperature of maximum NiXII fractional

abundance in ionisation equilibrium within a plasma is logT (K) = 6:2 (137eV )

(Arnaud and Rothen ug 1985, Mazzota et al. 1998). In the JET tokamak device

where central temperatures can reach several keV, this temperature only occurs

in the cooler outer layers of the plasma. The detection of lines in the two JET

pulses mentioned above occured when the spectrometer KT4 was set at an angle

of 28.2

�

, meaning that NiXII was most likely to be present above the \divertor

box" (Bertolini et al. 1995). The main aim of the divertor is to remove impurities

(and hence reduce energy loss) from the tokamak, and also to control recycling.

The �rst prerequisite for �nding NiXII lines was to ensure that the data

recorded by KT4 was measured when it was placed at a relatively steep angle,

� 19:2

�

, to ensure the line of sight is through the cooler edge of the plasma

and not the bulk region. Secondly, the detector must have been at a position

where it covered the desired wavelength range. With the detector placed at

y = 280mm, a spectral range of � 136 to 187

�

A is observed, this range in-

cluding the 3s

2

3p

5 2

P

o

3=2

{ 3s

2

3p

4

(

3

P )3d

2

D

5=2

, 3s

2

3p

5 2

P

o

1=2

{ 3s

2

3p

4

(

3

P )3d

2

D

3=2

,

3s

2

3p

5 2

P

o

3=2

{ 3s

2

3p

4

(

3

P )3d

2

P

3=2

and 3s

2

3p

5 2

P

o

3=2

{ 3s

2

3p

4

(

1

D)3d

2

S

1=2

transi-

tions whose wavelengths are listed in table 6.1. Unfortunately, because the detec-

tor only views approximately 50 - 60

�

A at any one position, it proved impossible

to obtain spectra of all the lines together. At a detector position of y = 385mm,

a spectral range of 267 to 335

�

A is seen, the range including the 3s

2

3p

5 2

P

o

1=2

{

3s3p

6 2

S

1=2

and 3s

2

3p

5 2

P

o

3=2

{ 3s3p

6 2

S

1=2

transitions along with second order

wavelengths of the four previously mentioned transitions. Thirdly, to improve


signi�cantly the possibility of �nding NiXII lines, only pulses where laser abla-

tion of nickel into the plasma had taken place were considered. The spectra were

observed at the time of the ablation and several scans beyond that, where one

scan takes 11ms. LISP was the display program used to observe the spectroscopic

data. It was primarily written to produce graphs of the signals from the suite of

XUV spectrometers.

The JET pulses where ablation occured yielded no detection of the 295.321

�

A

and 317.475

�

A lines due to poor data. A search was conducted for these two lines

which proved to be very time intensive due to slow data retrieval from the IBM

mainframe and laborious spectral analysis of several hundred JET pulses. The

�rst search method employed (method I) involved recalling one hundred pulses at

a time using the JETPLOT package. The pulses believed to contain NiXII were

chosen by using the NI17 (NiXVII line at 249.18

�

A ) DDA (diagnostic data area)

as an indicator of the presence of nickel. Any nickel seen in this way has been

in uxed due to sputtering or evaporation from the wall of the tokamak device.

No reliable identi�cations were made of the two transitions using this technique

of detecting the presence of nickel.

A second search method was devised (method II) which produced more reliable

results in a faster time by utilising processed pulse �les (PPF's). Each batch of

pulses during plasma operations typically has all lines at particular pixel numbers

and this only changes if there is a shift in the spectrometers position; usually

during maintenance work. Two lines were used as calibration markers: a NiXVIII

line at 292.0

�

A and a CIV line at 312.44

�

A (both DDA's). The relative positions

of the two NiXII lines to be found were then calculated and the pixel values

used to generate PPF's. Therefore the search was conduted with a view towards

�nding directly the desired lines and not merely the presence of nickel. Care

was taken to ensure the calibration lines were placed exactly for each batch of

pulses. Data recollection was swifter and the pixel intensities were viewed using

6.3 ADAS 176

the JETDISPLAY package. Suspected �nds were viewed as spectra in LISP.

6.3 ADAS

6.3.1 Introduction

The Atomic Data and Analysis Structure (ADAS) is an interconnected set of

computer codes and data collections for modelling the radiating properties of ions

and atoms in plasmas and assisting in the analysis and interpretation of spec-

tral measurements. The current work made use of the ADAS2 series which is

concerned with evaluating excited populations of speci�c ions, here NiXII, in a

plasma environment and then their radiation emission. It relied on the availabil-

ity of reaction rate data from chapter 4. The primary excited state population

calculation (ADAS205) provided extensive tabulations and graphs of the popula-

tions of NiXII in a thermal plasma and prepared a passing data set for use by the

diagostic display routine, ADAS207. Both ADAS205 and ADAS207 are discussed

below.

6.3.2 Speci�c z excitation - processing of metastable and

excited populations

The program calculates excited state and metastable state populations of a se-

lected ion in a plasma of speci�ed temperatures and densities by drawing on

fundamental energy level and rate coe�cient data from a speci�c ion �le. The �le

for NiXII was constructed from data shown in the previous chapter.

Consider ions X

+z

of the element X. The adjacent ionisation stages are X

+z+1

and X

+z�1

. Let the levels of the ion X

+z

be separated into the metastable levels

X

+z

�

, indexed by Greek indices, and excited levels X

+z

i

,indexed by Roman indices.

6.3 ADAS 177

The collective name metastable states as used here includes the ground state. The

driving mechanism considered for populating the excited levels X

+z

i

is excitation

from the metastable levels X

+z

�

. The dominant population densities of the ions

in the plasma are those of the levels X

+z

�

and X

+z+1

1

denoted by N

�

and N

+

1

re-

spectively. They, or at least their ratios are assumed known from a dynamical

ionisation balance. In the case of excitation, the other dominant population den-

sity in the plasma is the electron density N

e

. The excited populations, denoted by

N

i

, are assumed to be in a quasistatic equilibrium with respect to the dominant

populations. The program evaluates the dependence of the excited populations

on the dominant populations with this assumption.

Let M denote the number of metastable levels and O denote the number of

excited levels, hereafter called ordinary levels. The statistical balance equations

take the form

O

X

j=1

C

ij

N

j

= �

M

X

�=1

C

i�

N

�

i = 1; 2; ::: (6.14)

The C

ij

and C

i�

are elements of the collisional-radiative matrix. The element C

ij

of the collisional-radiative matrix is composed as

C

ij

= �A

j!i

�N

e

q

(e)

j!i

i 6= j (6.15)

where A

j!i

and q

(e)

j!i

are the rate coe�cients for spontaneous transition and elec-

tron induced collisional transition respectively.

C

ii

=

X

j<i

A

i!j

+N

e

X

j 6=i

q

(e)

i!j

(6.16)

is the total loss rate from level i. The solution for the ordinary populations is

N

j

= �

P

O

i=1

C

�1

ji

P

M

�=1

C

i�

N

�

+

�

P

M

�=1

F

(exc)

j�

N

e

N

�

(6.17)

6.3 ADAS 178

where the F

(exc)

j�

is the e�ective contribution to the excited populations from ex-

citation from the metastables. This coe�cient depends on density as well as

temperature. The actual population density of an ordinary level may be obtained

from it when the dominant population densities are known.

The full statistical equilibrium of all the level populations of the ion X

+z

, that

is of metastables as well as ordinary levels relative to metastables, may also be

obtained from the equations

M

X

�=1

C

p�

N

�

= �

O

X

j=1

C

�j

N

j

(6.18)

Substitution of the quasi-equilibrium solution for the ordinary levels, eqn. 6.17,

gives

M

X

�=1

(C

��

�

O

X

j=1

C

�j

O

X

i=1

C

�1

ji

C

i�

)N

�

= 0 (6.19)

Solution of these equations gives an expression for the metastable populations N

�

of the form

N

�

� F

(exc)

�

N

1

(6.20)

The e�ective contributions to the metastable population densities (excluding the

ground level) are expressed relative to the ground population density. Note also

that a full equilibrium with respect to the adjacent X

+z+1

ion population density

is not established. The metastable to ground fractions in equilibrium when only

excitation is included are the F

(exc)

�

. Substitution of eqn. 6.20 in eqn. 6.17 gives

the statistical equilibrium population densities for the ordinary levels in terms of

the ground population density.

N

j

=

M

X

�=1

F

(exc)

j�

F

(exc)

�

N

e

N

+

1

(6.21)

6.3 ADAS 179

6.3.3 Source data

The program operates on collections of fundamental rate coe�cient data called

speci�c ion �les. The scope of operation of ADAS205 is determined by the content

of the speci�c ion �le processed. The mininum content is the ion identi�cation, ion,

e�ective ion and nuclear charges, ionisation potential, an indexed energy level and

level assignment list, a set of temperatures and a set of level to level spontaneous

transition probabilities and electron impact Maxwell averaged rate parameters at

the speci�ed temperatures, as was included in the current case. Data for upper to

lower level only is required. Electron impact rate coe�cients for both excitation

and de-excitation are evaluated by interpolation at user selected values from the

tabulated rate parameters in the speci�c ion �le. Transition rate data is not

required for all possible upper/lower level pairs, but the code checks that there

are no `untied' levels, that is without populating or depopulating processes. The

temperature range, in reduced units, for inclusion in the ion �le is limited to

500 < T (K)=(z + 1)

2

< 2� 10

5

. The temperatures chosen for the current work,

after attempting many di�erent values, are: 8:12 � 10

4

K (7 eV), 1:16 � 10

5

K

(10 eV), 2:32 � 10

5

K (20 eV), 6:58 � 10

5

K (56.7 eV), 9:32 � 10

5

K (80.3 eV),

1:10�10

6

K (95 eV), 1:16�10

6

K (100 eV), 1:74�10

6

K (150 eV), 2:32�10

6

K (200

eV), 2:90� 10

6

K (250 eV), 3:48� 10

6

K (300 eV) and 3:95� 10

6

K (340 eV). The

e�ective collision strengths were taken from table 4.5. Strict energy ordering is

not required in the speci�c ion �le, the code reorders as necessary. Proton induced

rates, free electron recombination rates and charge exchange recombination rates

may only be activated in the code if such data are present in the speci�c ion �le

but for the current work they were neither included nor necessary. A centrally

supported, speci�c ion data collection was archived in a partitioned data set.

6.3 ADAS 180

6.3.4 Metastable and excited population - processing of

line emissivities

The program evaluates and displays line emissivities and their ratios for an ion.

It uses a passing �le of excited population data from the code ADAS205.

Consider emissivities of spectrum lines arising from a single ionisation stage.

Ratios of such lines are frequently used as temperature, density or transient state

diagnostics in plasmas. The primary advantage of seeking such ratios of lines

from a single ionisation stage is that they are independent of the stage to stage

ionisation balance (often uncertain). In general it is matter of some investigation

to identify the most diagnostically useful ratios.

A necessary preliminary to evaluating line emissivities is a calculation of pop-

ulations of excited states of the ion as a function of plasma parameters. This

is provided by ADAS205 which must be executed before ADAS207. In practice,

problems of line blending and the spectral resolution of spectrometers mean that

it is useful to work with line groups rather than just individual lines. A line group

is a set of lines conveniently or necessarily measured together. ADAS207 deals

with two line groups which are built up by the user in the data entry section of

the code.

From equation 6.17, the solution for the ordinary populations is

N

j

=

M

X

�=1

F

(exc)

j�

N

e

N

�

(6.22)

where the F

(exc)

j�

is the e�ective contribution to the excited populations from ex-

citation from the metastables.

Consider a set of individual lines, or line group, G with upper levels I

G

and

lower levels J

G

. Let A

i!j

be the spontaneous emission coe�cient for the line

6.3 ADAS 181

i! j. Then the composite emissivity for the line group is

"

G

=

X

j�J

G

;i�I

G

"

j!i

=

X

j�J

G

;i�I

G

A

j!i

N

j

=

X

j�J

G

;i�I

G

A

j!i

M

X

�=1

F

(exc)

j�

N

e

N

�

= N

e

N

1

X

j�J

G

;i�I

G

A

j!i

M

X

�=1

F

(exc)

j�

N

�

N

1

(6.23)

expressed in terms of the ratio N

�

=N

1

. The photon emissivity coe�cient for the

line group is "

G

=N

e

N

1

. The coe�cient depends on electron density and tem-

perature in general. Ratios of line group emissivities cancel the leading N

e

N

1

dependence. The code prepares and operates primarily with a ratio "

G

1

="

G

2

The

program step are summarised in the �gure 6.1.

output tables

and graphs

select contour

pass file from

ADAS205

read and verify

contour pass file

read and verify

associated

specific ion file

enter user data

including graph

type

display emissivity

graphs

emissivities

END REPEAT

assemble line

BEGIN

REPEAT

Figure 6.1: Basic owchart for the processing of line emissivities using the ADAS

codes

6.4 Results and Conclusions 182

6.4 Results and Conclusions

The JET pulses examined for the presence of NiXII are listed below in tables

6.2 and 6.3. The last 17 pulses in the ablation list could have contained the

295.321

�

A and 317.475

�

A lines, because of the angle of the spectrometer and the

detector position, but the data were too poor to make any identi�cations. Table

6.4 lists the identi�cations of nickel lines in the pulses from the ablation list.

The only positive identi�cation of the 295.321

�

A and 317.475

�

A lines occured

in pulse 34938 after a search through over 3600 pulses. This �nd represents the

�rst identi�cation of these lines within a tokamak device. Previously they have

been seen in theta-pinch spectra (Fawcett and Hatter 1980). There were some

more tenuous identi�cations of the lines but the spectra were generally too poor

to con�rm them. The spectrum of pulse 34938 can be seen in Fig. 6.2 with not

only the 295.321

�

A and 317.475

�

A lines but also the second order lines of the

transitions 3s

2

3p

5 2

P

o

3=2

{ 3s

2

3p

4

(

3

P )3d

2

D

5=2

, 3s

2

3p

5 2

P

o

1=2

{ 3s

2

3p

4

(

3

P )3d

2

D

3=2

,

3s

2

3p

5 2

P

o

3=2

{ 3s

2

3p

4

(

3

P )3d

2

P

3=2

and 3s

2

3p

5 2

P

o

3=2

{ 3s

2

3p

4

(

1

D)3d

2

S

1=2

. The

bulk plasma conditions of the pulse are shown in Fig. 6.3. A plot of the magnetic

�eld con�guration in Fig. 6.4 shows that the X point has been reached in the

plasma. All the lines currently identi�ed are listed in table 6.4. There is generally

good agreement with previously measured values except for the 160.492

�

A and

152.903

�

A lines. The disagreement with calculated values can be attributed to

the close �tting of the two 3s

2

3p

5

levels in the optimization procedures used by

Fawcett.

Figures 6.5, 6.10 and 6.15 represent the best spectra showing the identi�cations

of the 152.153

�

A , 152.95

�

A , 154.175

�

A and 160.554

�

A lines from the transitions

mentioned above. The dominant NiXII line at 152.153

�

A is blended with the

152.152

�

A line from the 3s

2

3p

5 2

P

o

3=2

{ 3s

2

3p

4

(

3

P )3d

2

P

1=2

transition. The spec-

trometer is unable to resolve them separately (in this wavelength region it has a

resolution of approximately 0.3

�

A (FWHM)). Bulk plasma conditions and


JET pulse numbers

30092, 30095, 30196, 30197,

30776, 30778, 30779, 30780,

30783 - 30787,

31272, 31273, 31275, 31324,

31325, 31327,

31329 - 31331,

31338 - 31346,

31372 - 31374,

31376 - 31778,

31380, 31419, 31420, 31424,

31426, 31717, 31719, 31727,

31768 - 31770,

31773 - 31775,

31798 - 31800, 31881,

32408 - 32410, 32656,

32658 - 32660,

33408 - 33411,

33951 - 33956,

34292, 34308, 34309,

34416 - 34419,

34475, 34476,

34479 - 34481,

34491, 34508,

35133 - 35135

TOTAL = 89

Table 6.2: JET pulses where laser ablation of nickel occured.

JET pulse numbers Method

23500 - 25200 I

34056 - 34317 II

34320 - 35243 II

35000 - 35700 I

39428 - 39453 II

TOTAL = 3605

Table 6.3: JET pulses checked by methods I and II.


Transition Present �/

�

A Previous �/

�

A Theoretical �/

�

A

(Refs. Table 6.1) Fawcett (1987)

3s

2

3p

5 2

P

o

1=2

{ 3s3p

6 2

S

1=2

317.50 � 0.04 317.475 � 0.008 317.473

3s

2

3p

5 2

P

o

3=2

{ 3s3p

6 2

S

1=2

295.33 � 0.04 295.321 � 0.008 295.322

3s

2

3p

5 2

P

o

3=2

{ 3s

2

3p

4

3d

2

S

1=2

160.49 � 0.04 160.554 � 0.005 160.561

3s

2

3p

5 2

P

o

3=2

{ 3s

2

3p

4

3d

2

P

3=2

154.17 � 0.04 154.175 � 0.005 154.168

154.20

a

� 0.06

3s

2

3p

5 2

P

o

1=2

{ 3s

2

3p

4

3d

2

D

3=2

152.90 � 0.04 152.95 � 0.05 152.724

152.84

a

� 0.06

3s

2

3p

5 2

P

o

3=2

{ 3s

2

3p

4

3d

2

D

5=2

152.13 � 0.04 152.153 � 0.005 151.954

Table 6.4: NiXII wavelength identi�cations

JET NiXII wavelength

pulse 152.153

�

A 152.95

�

A 154.175

�

A 160.554

�

A

30776 � � � -

30778 � � � -

30779 � � � -

30780 � � � -

30784 � � � -

30785 � - � -

30787 � � � -

31272 � � � �

31273 � � � �

31275 � � � �

31324 � � � -

31327 � � � �

31330 � � � -

31373 � � � �

31798 � � � �

Table 6.5: JET pulses where NiXII lines were identi�ed.

a

Malinovsky & Heroux (1973).

magnetic �eld con�gurations are shown for each pulse in Figs. 6.6, 6.7, 6.11,

6.16 and 6.17. Power was measured with a bolometer while the bulk density was

recorded by a Michelson interferometer.

Theoretical line ratios were calculated for all possible combinations of lines

identi�ed in table 6.5 using the ADAS codes. The temperatures chosen to display


density line ratios were 8:12� 10

4

K (7 eV), 1:16� 10

5

K (10 eV), 2:32� 10

5

K(20

eV), 6:58�10

5

K (56.7 eV), 9:32�10

5

K (80.3 eV), 1:10�10

6

K (95 eV), 1:16�10

6

K

(100 eV), 1:74�10

6

K (150 eV), 2:32�10

6

K (200 eV) and 2:90�10

6

K (250 eV). The

densities chosen for temperature line ratios were 1:0� 10

10

cm

�3

, 5:0� 10

10

cm

�3

,

7:5� 10

10

cm

�3

, 1:0� 10

11

cm

�3

, 5:0� 10

11

cm

�3

, 1:0� 10

12

cm

�3

, 1:0� 10

13

cm

�3

and 1:0 � 10

14

cm

�3

. The 3s

2

3p

5 2

P

3=2

{ 3s

2

3p

4

(

3

P )3d

2

P

1=2

and 3s

2

3p

5 2

P

3=2

{ 3s

2

3p

4

(

3

P )3d

2

D

5=2

transitions were considered to be a single line group for

the calculations. NiXII line ratios are density sensitive below values of ap-

proximately 5 � 10

11

cm

�3

particularly for any ratio including the 3s

2

3p

5 2

P

1=2

{ 3s

2

3p

4

(

3

P )3d

2

D

3=2

transition. This is possibly due to two competing populat-

ing mechanisms { one between the lower

2

P

1=2

level and the excited state and one

within the ground state between the

2

P

1=2

and

2

P

3=2

levels. Temperature sensi-

tivity can be seen in Figs. 6.21, 6.23, 6.25, 6.27, 6.29 and 6.31. Unfortunately the

sensitivity is lowered signi�cantly beyond temperatures of approximately 7�10

5

K

(60 eV).


Ratio JET pulse Measured Theoretical

values values

R

1

30776 3.2 � 0.1 1.88 � 0.01

30778 2.19 � 0.05

30779 1.35 � 0.2

30780 1.6 � 0.1

30784 2.4 � 0.1

30785 3.3 � 0.2

30787 2.23 � 0.05

31272 1.6 � 0.5

31273 1.85 � 0.05

31275 1.85 � 0.05

31324 1.8 � 0.5

31327 1.45 � 0.05

31330 1.66 � 0.05

31373 2.3 � 0.1

31798 1.86 � 0.05

R

2

30776 5.4 � 0.2 2.07 � 0.02

30778 2.8 � 0.2

30779 1.9 � 0.2

30784 1.4 � 0.2

30787 2.41 � 0.3

31272 1.5 � 0.5

31273 2.8 � 0.1

31275 2.83 � 0.05

31324 5.6 � 0.5



values values

31327 2.4 � 0.1

31330 1.8 � 0.2

31373 2.5 � 0.2

31798 2.47 � 0.05

R

3

30776 1.7 � 0.2 1.10 � 0.02

30778 1.2 � 0.2

30779 1.4 � 0.2

30784 0.6 � 0.2

30787 0.93 � 0.05

31272 1.0 � 0.5

31273 1.7 � 0.1

31275 1.53 � 0.05

31324 3.2 � 0.5

31327 1.63 � 0.05

31330 1.3 � 0.2

31373 1.1 � 0.1

31798 1.33 � 0.05

R

4

31272 2.1 � 0.5 2.00 � 0.01

31273 2.6 � 0.3

31275 3.3 � 0.3

31327 3.5 � 0.3

31373 1.8 � 0.3

31798 4.1 � 0.3



values values

R

5

31272 0.5 � 0.5 0.55 � 0.02

31273 0.7 � 0.3

31275 0.5 � 0.3

31327 0.5 � 0.3

31373 0.6 � 0.3

31798 0.3 � 0.3

R

6

31272 3.3 � 0.5 3.77 � 0.02

31273 4.9 � 0.3

31275 6.0 � 0.3

31327 5.0 � 0.3

31373 4.1 � 0.3

31798 7.7 � 0.3

Table 6.6: NiXII line ratios

The ratios, R

x

, are de�ned as follows:

R

1

=

3p

5 2

P

o

3=2

� 3p

4

(

3

P )3d

2

D

5=2

+ 3p

5 2

P

o

3=2

� 3p

4

(

3

P )3d

2

P

1=2

3p

5 2

P

o

3=2

� 3p

4

(

3

P )3d

2

P

3=2

)

=

I(152:15

�

A)

I(154:17)

�

A

R

2

=

3p

5 2

P

o

3=2

� 3p

4

(

3

P )3d

2

D

5=2

+ 3p

5 2

P

3=2

� 3p

4

(

3

P )3d

2

P

1=2

3p

5 2

P

o

1=2

� 3p

4

(

3

P )3d

2

D

3=2

)

=

I(152:15)

�

A

I(152:95)

�

A

R

3

=

3p

5 2

P

o

3=2

� 3p

4

(

3

P )3d

2

P

3=2

3p

5 2

P

o

1=2

� 3p

4

(

3

P )3d

2

D

3=2

)

=

I(154:17)

�

A

I(152:95)

�

A

R

4

=

3p

5 2

P

o

3=2

� 3p

4

(

3

P )3d

2

P

3=2

3p

5 2

P

o

3=2

� 3p

4

(

1

D)3d

2

S

1=2

=

I(154:17)

�

A

I(160:55)

�

A

R

5

=

3p

5 2

P

o

3=2

� 3p

4

(

1

D)3d

2

S

1=2

3p

5 2

P

o

1=2

� 3p

4

(

3

P )3d

2

D

3=2

)

=

I(160:55)

�

A

I(152:95)

�

A


Ratio Jet pulse Derived

temperature /eV

R

1

31273 73 +100/-25

R

1

31275 73 +100/-25

R

1

31798 95 +100/-40

Table 6.7: Derived temperatures of the plasma at an electron density = 10

11

cm

�3

R

6

=

3p

5 2

P

o

3=2

� 3p

4

(

3

P )3d

2

D

5=2

+ 3p

5 2

P

o

3=2

� 3p

4

(

3

P )3d

2

P

1=2

3p

5 2

P

o

3=2

� 3p

4

(

1

D)3d

2

S

1=2

=

I(152:15)

�

A

I(160:55)

�

A

The theoretical values are named as such because they are derived from the ADAS

plots which in turn were calculated from theoretical data. They were taken at a

density of 10

11

cm

�3

after which point all the ratios become density insensitive.

Line intensities were measured in the actual spectra by considering the peaks

of the lines only. A central pixel was selected and integration of the line was

performed after choosing a � pixel range, typically varying between � 1 to �5

pixels. Values were chosen so as to avoid the background. Ratios of lines had to be

of those lines integrated over the same pixel range. Many ratios were made and the

values cross-referenced on the theoretically produced graphs to �nd the resultant

temperature and density of the plasma. The majority of ratios measured did not

�t to the ADAS plots while most of the remainder gave spurious values. Table 6.6

lists the measured ratios. Errors were based on the integration range used for each

set of lines and, particularly, the error inherent in the reading of the 160.492

�

A

line due to its very low intensity. After eliminating the possibilities that extreme

results were not due to plasma conditions e.g. a con�guration change, heating

mechanisms (which are perturbative at the plasma edge in the case of LHCD

heating) or core density, temperature changes, it was concluded that unidenti�ed

line blends and weak lines were the cause of the poor values.

Most NiXII lines were of very short duration, typically one or two detector

scans, implying they must be in a very low density region, such as the scrape-

o� layer outside the last closed ux surface. Ions in this region are subject to a


transport e�ect known as `spiralling' which occurs very rapidly and explains the

short detection time. The lines are of low intensity because the SOL is at a much

lower density than that within the bulk plasma.

The quality of the lines shown in Figs. 6.8, 6.13 and 6.18 is therefore rare

and these 3 ratios, from a total of 89 ablation shots, are the indicators for the

temperature and density within the SOL. Figures 6.9, 6.14 and 6.19 show the

integration of the lines. From the ratios the temperature of the SOL where the

lines occured can be derived, with the results listed in table 6.7. The temperature

of maximum abundance for NiXII is 137 eV so the derived temperatures may have

signi�cantly worse error margins than indicated - possibly due to the worsening

insensitivity of the ratios at higher temperatures. Alternatively, and possibly

additionally, the density could be lower than expected which can a�ect the derived

temperatures. The lines are appearing in a plasma with an electron density of

approximately 1� 10

11

cm

�3

.

Ratios of the second order lines in pulse 34938 also yielded no derived tempera-

tures due to the weakness of the lines. The measured ratio of I(317:475

�

A )=I(295:321

�

A )

was 2.24 compared to the theoretical result of 2.42; within the margin of instru-

mental error. The ratio is a branching ratio so it's value is always that of the

respective A values. It is also further proof of a positive identi�cation for the

lines.


Figure 6.2: Identi�cation of NiXII lines in JET pulse 34938 at t = 54.8 s


Figure 6.3: Various plasma conditions of JET pulse 34938


Figure 6.4: Magnetic �eld con�guration of JET pulse 34938


Figure 6.8: Superimposition of the 3p

5 2

P

o

3=2

{ 3p

4

(

3

P )3d

2

D

5=2

+ 3p

5 2

P

o

3=2

{

3p

4

(

3

P )3d

2

P

1=2

lines and the 3p

5 2

P

o

3=2

{ 3p

4

(

3

P )3d

2

P

3=2

line for JET pulse

31273


Figure 6.9: Integration of the 3p

5 2

P

o

3=2

{ 3p

4

(

3

P )3d

2

D

5=2

+ 3p

5 2

P

o

3=2

{

3p

4

(

3

P )3d

2

P

1=2

lines and the 3p

5 2

P

o

3=2

{ 3p

4

(

3

P )3d

2

P

3=2

line for JET pulse

31273 over � 5 pixels



5 2

P

o

3=2

{ 3p

4

(

3

P )3d

2

D

5=2

+ 3p

5 2

P

o

3=2

{

3p

4

(

3

P )3d

2

P

1=2

lines and the 3p

5 2

P

o

3=2

{ 3p

4

(

3

P )3d

2

P

3=2

line for JET pulse 31275


Figure 6.14: Integration of the 3p

5 2

P

o

3=2

{ 3p

4

(

3

P )3d

2

D

5=2

+ 3p

5 2

P

o

3=2

{

3p

4

(

3

P )3d

2

P

1=2

lines and the 3p

5 2

P

o

3=2

{ 3p

4

(

3

P )3d

2

P

3=2

line for JET pulse

31275 over � 5 pixels



5 2

P

o

3=2

{ 3p

4

(

3

P )3d

2

D

5=2

+ 3p

5 2

P

o

3=2

{

3p

4

(

3

P )3d

2

P

1=2

lines and the 3p

5 2

P

o

3=2

{ 3p

4

(

3

P )3d

2

P

3=2

line for JET pulse 31798


Figure 6.19: Integration of the 3s

2

3p

5 2

P

o

3=2

{ 3s

2

3p

4

(

3

P )3d

2

D

5=2

+ 3s

2

3p

5 2

P

o

3=2

{ 3s

2

3p

4

(

3

P )3d

2

P

1=2

lines and the 3s

2

3p

5 2

P

o

3=2

{ 3s

2

3p

4

(

3

P )3d

2

P

3=2

line for JET

pulse 31798 over � 5 pixels


Figure 6.20: Plot of the theoretical line ratio, R

1

, (3p

5 2

P

o

3=2

{ 3p

4

(

3

P )3d

2

D

5=2

+ 3p

5 2

P

o

3=2

{ 3p

4

(

3

P )3d

2

P

1=2

/ 3p

5 2

P

o

3=2

{ 3p

4

(

3

P )3d

2

P

3=2

) as a function of

electron density. Results are plotted at electron temperatures T

e

= 8.12�10

4

K

(1), 1.16�10

5

K (2), 2.32�10

5

K (3), 6.58�10

5

K (4), 9.32�10

5

K (5), 1.10�10

6

K

(6), 1.16�10

6

K (7), 1.74�10

6

K (8), 2.32�10

6

K (9), 2.90�10

6

K (10).



1

, (3p

5 2

P

o

3=2

{ 3p

4

(

3

P )3d

2

D

5=2

+

3p

5 2

P

o

3=2

{ 3p

4

(

3

P )3d

2

P

1=2

/ 3p

5 2

P

o

3=2

{ 3p

4

(

3

P )3d

2

P

3=2

) as a function of electron

temperature. Results are plotted at electron densities n

e

= 1.0�10

10

=; cm

�3

(1),

5.0�10

10

=; cm

�3

(2), 7.5�10

10

=; cm

�3

(3), 1.0�10

11

=; cm

�3

(4), 5.0�10

11

=; cm

�3

(5), 1.0�10

12

=; cm

�3

(6), 1.0�10

13

=; cm

�3

(7), 1.0�10

14

=; cm

�3

(8).



2

, (3p

5 2

P

o

3=2

{ 3p

4

(

3

P )3d

2

D

5=2

+ 3p

5 2

P

o

3=2

{ 3p

4

(

3

P )3d

2

P

1=2

/ 3p

5 2

P

o

1=2

{ 3p

4

(

3

P )3d

2

D

3=2

) as a function of


e

= 8.12�10

4

K

(1), 1.16�10

5

K (2), 2.32�10

5

K (3), 6.58�10

5

K (4), 9.32�10

5

K (5), 1.10�10

6

K

(6), 1.16�10

6

K (7), 1.74�10

6

K (8), 2.32�10

6

K (9), 2.90�10

6

K (10).



2

, (3p

5 2

P

o

3=2

{ 3p

4

(

3

P )3d

2

D

5=2

+

3p

5 2

P

o

3=2

{ 3p

4

(

3

P )3d

2

P

1=2

/ 3p

5 2

P

o

1=2

{ 3p

4

(

3

P )3d

2

D

3=2



e

= 1.0�10

10

=; cm

�3

(1),

5.0�10

10

=; cm

�3

(2), 7.5�10

10

=; cm

�3

(3), 1.0�10

11

=; cm

�3

(4), 5.0�10

11

=; cm

�3

(5), 1.0�10

12

=; cm

�3

(6), 1.0�10

13

=; cm

�3

(7), 1.0�10

14

=; cm

�3

(8).



3

, (3p

5 2

P

o

3=2

{ 3p

4

(

3

P )3d

2

P

3=2

/ 3p

5 2

P

o

1=2

{ 3p

4

(

3

P )3d

2

D

3=2

) as a function of density. Results are plotted

at electron temperatures T

e

= 8.12�10

4

K (1), 1.16�10

5

K (2), 2.32�10

5

K (3),

6.58�10

5

K (4), 9.32�10

5

K (5), 1.10�10

6

K (6), 1.16�10

6

K (7), 1.74�10

6

K (8),

2.32�10

6

K (9), 2.90�10

6

K (10).



3

, (3p

5 2

P

o

3=2

{ 3p

4

(

3

P )3d

2

P

3=2

/ 3p

5 2

P

o

1=2

{ 3p

4

(

3

P )3d

2

D

3=2

) as a function of electron temperature. Results

are plotted at electron densities n

e

= 1.0�10

10

=; cm

�3

(1), 5.0�10

10

=; cm

�3

(2),

7.5�10

10

=; cm

�3

(3), 1.0�10

11

=; cm

�3

(4), 5.0�10

11

=; cm

�3

(5), 1.0�10

12

=; cm

�3

(6), 1.0�10

13

=; cm

�3

(7), 1.0�10

14

=; cm

�3

(8).



4

, (3p

5 2

P

o

3=2

{ 3p

4

(

3

P )3d

2

P

3=2

/ 3p

5 2

P

o

3=2

{ 3p

4

(

1

D)3d

2

S

1=2



e

= 8.12�10

4

K (1), 1.16�10

5

K (2), 2.32�10

5

K (3),

6.58�10

5

K (4), 9.32�10

5

K (5), 1.10�10

6

K (6), 1.16�10

6

K (7), 1.74�10

6

K (8),

2.32�10

6

K (9), 2.90�10

6

K (10).



4

, (3p

5 2

P

o

3=2

{ 3p

4

(

3

P )3d

2

P

3=2

/ 3p

5 2

P

o

3=2

{ 3p

4

(

1

D)3d

2

S

1=2



e

= 1.0�10

10

=; cm

�3

(1), 5.0�10

10

=; cm

�3

(2),

7.5�10

10

=; cm

�3

(3), 1.0�10

11

=; cm

�3

(4), 5.0�10

11

=; cm

�3

(5), 1.0�10

12

=; cm

�3

(6), 1.0�10

13

=; cm

�3

(7), 1.0�10

14

=; cm

�3

(8).



5

, (3p

5 2

P

o

3=2

{ 3p

4

(

1

D)3d

2

S

1=2

/ 3p

5 2

P

o

1=2

{ 3p

4

(

3

P )3d

2

D

3=2



e

= 8.12�10

4

K (1), 1.16�10

5

K (2), 2.32�10

5

K (3),

6.58�10

5

K (4), 9.32�10

5

K (5), 1.10�10

6

K (6), 1.16�10

6

K (7), 1.74�10

6

K (8),

2.32�10

6

K (9), 2.90�10

6

K (10).



5

, (3p

5 2

P

o

3=2

{ 3p

4

(

1

D)3d

2

S

1=2

/ 3p

5 2

P

o

1=2

{ 3p

4

(

3

P )3d

2

D

3=2



e

= 1.0�10

10

=; cm

�3

(1), 5.0�10

10

=; cm

�3

(2),

7.5�10

10

=; cm

�3

(3), 1.0�10

11

=; cm

�3

(4), 5.0�10

11

=; cm

�3

(5), 1.0�10

12

=; cm

�3

(6), 1.0�10

13

=; cm

�3

(7), 1.0�10

14

=; cm

�3

(8).



6

, (3p

5 2

P

o

3=2

{ 3p

4

(

3

P )3d

2

D

5=2

+ 3p

5 2

P

o

3=2

{ 3p

4

(

3

P )3d

2

P

1=2

/ 3p

5 2

P

o

3=2

{ 3p

4

(

1

D)3d

2

S

1=2

) as a function of


e

= 8.12�10

4

K

(1), 1.16�10

5

K (2), 2.32�10

5

K (3), 6.58�10

5

K (4), 9.32�10

5

K (5), 1.10�10

6

K

(6), 1.16�10

6

K (7), 1.74�10

6

K (8), 2.32�10

6

K (9), 2.90�10

6

K (10).



6

, (3p

5 2

P

o

3=2

{ 3p

4

(

3

P )3d

2

D

5=2

+

3p

5 2

P

o

3=2

{ 3p

4

(

3

P )3d

2

P

1=2

/ 3p

5 2

P

o

3=2

{ 3p

4

(

1

D)3d

2

S

1=2



e

= 1.0�10

10

=; cm

�3

(1),

5.0�10

10

=; cm

�3

(2), 7.5�10

10

=; cm

�3

(3), 1.0�10

11

=; cm

�3

(4), 5.0�10

11

=; cm

�3

(5), 1.0�10

12

=; cm

�3

(6), 1.0�10

13

=; cm

�3

(7), 1.0�10

14

=; cm

�3

(8).

6.5 Thesis Conclusions 221

6.5 Thesis Conclusions

E�ective collision strengths computed by the R-matrix method are presented for

the electron-impact excitation of Cl-like Ni XII for the �rst time. The total

wave function used in the expansion includes the lowest 14 eigenstates of Ni XII

which arise from the 3s3p

6

and 3s

2

3p

4

3d con�gurations. The 14 LS target states

correspond to 31 �ne-structure levels, giving 465 possible transitions. All the

e�ective collision strengths for these transitions are tabulated within in the range

log T

e

=5.5 to log T

e

=6.6. Additionally the orbital parameters, energy level values

and oscillator strengths for allowed transitions are also tabulated. The e�ective

collision strengths were calculated by averaging the electron collision strengths

over a Maxwellian distribution of velocities.

Wavelengths for emission lines arising from 3s

2

3p

5

{3s3p

6

and 3s

2

3p

5

{3s

2

3p

4

3d

transitions in Ni XII have been measured in extreme ultraviolet spectra of the

Joint European Torus (JET) tokamak. The 3s

2

3p

5 2

P

1=2

{3s

2

3p

4

(

3

P)3d

2

D

3=2

line

is found to lie at 152.90�0.04

�

A, compared to the previous experimental determi-

nation of 152.95�0.5

�

A. This new wavelength is in good agreement with a solar

identi�cation at 152.84�0.06

�

A, con�rming the presence of this line in the solar

spectrum. Previously unidenti�ed emission lines of the 3s

2

3p

5 2

P

3=2

{3s3p

6 2

S

1=2

and 3s

2

3p

5 2

P

1=2

{3s3p

6 2

S

1=2

transitions within laboratory spectra have bbe found

to have wavelengths of 295.33

�

A and 317.50

�

A, respectively.

The R-matrix calculations of electron impact excitation rates in NiXII have

been used to derive several emission line ratios for this ion. The ratios are found

to be insensitive to changes in the adopted electron density (N

e

) when N

e

�

10

11

cm

�3

, typical of laboratory plasmas. However they do vary with electron

temperature (T

e

), with for example R

1

and R

3

changing by factors of 1.3 and

1.8, respectively, between T

e

= 10

5

and 10

6

K. A comparison of the theoretical

line ratios with measurements from the Joint European Torus (JET) tokamak has

6.5 Thesis Conclusions 222

revealed generally good agreement between theory and observation. This pro-

vides some experimental support for the accuracy of the diagnostic calculations,

and hence for the atomic data adopted in their derivation. However in several

instances the temperatures deduced (� 80eV ) from the R

1

ratio are much lower

than expected on the basis of ionization equilibrium, indicating that in some cases

the NiXII ions must di�use into cooler regions of the JET plasma.

6.6 Future Work 223

6.6 Future Work

The di�culty in detecting NiXII within the plasma, unless it has been laser ab-

lated, is demonstrated by the search for the 295.321

�

A and 317.475

�

A lines. How-

ever the same search method (method II) may possibly detect, for example, the

152.153

�

A line much more easily because it is of inherently greater intensity. In

fact, this is the only line to be detected in a spectrum from the Solar Heliospheric

Observatory as shown in �gure 6.32. Indeed, the usefulness of NiXII as a tem-

perature or density diagnostic in the solar corona is questionable. The resolution

of the SOHO coronal diagnostic spectrometer is not as good as that of KT4 on

the JET tokamak making line detection di�cult. Since SOHO spectra are of the

quiet sun and the 295.321

�

A and 317.475

�

A lines observed by Dere (1978) were

observed in a are, plus the fact that the lines have not been detected thus far

suggests SOHO data are unfortunately of no signi�cant use. The fact that NiXII

line ratios are sensitive to only 1 � 10

6

K or 86 eV (see size of error margins in

table 6.7) implies there would also be a signi�cant margin of error in any values

obtained in such a high temperature region of the Sun. However although the R

1

,

R

2

and R

6

ratios are density insensitive for N

e

� 10

11

cm

�3

, typical of laboratory

plasmas, they do vary with N

e

at lower densities. In particular, the solar tran-

sition region has N

e

' 10

9

{10

11

cm

�3

at log T

e

' 6.2 (Keenan et al 1991), and

hence the Ni XII line ratios may provide useful N

e

{diagnostics for the Sun.

Investigation of how NiXII is transported through the JET plasma is of im-

portance meaning a further series of nickel ablations into the SOL would be nec-

essary. More line ratios could be performed also and compared with the cur-

rent results, hopefully con�rming them. The current resolution of approximately

0.3

�

A(FWHM) in KT4 is inadequate to resolve line blending. If this value was,

perhaps, twice as good, by using a 1200g/mm grating, better results could be

achieved although at the cost of narrowing the wavelength range. It would also

be better to measure the NiXII lines in a high density, low temperature plasma,

6.6 Future Work 224

suggesting that a smaller, and therefore cooler, tokamak be used. Line of sight

covering the region containing the lines, no matter which plasma in which they

are observed, is crucial.

6.6 Future Work 225

Figure 6.32: Identi�cation of NiXII 152.153

�

A line in a quiet Sun spectrum

6.7 References 226

6.7 References

Arnaud M. and Rothen ug R. Astron. Astrophys60 (1985) 425

Behring W.E., Cohen L. and Feldman U. Astrophys. J.175 (1972) 493

Co�ey, I.H., private communication

Bertolini, E. Fusion Engineering and Design30 (1995) 53

Dere, K.P. Astrophys. J.221 (1978) 1062

Elwert G. Z. Naturforsch7A (1952) 432

Fawcett B.C. At. Data Nucl. Data Tables36 (1987) 151

Fawcett B.C. and Hatter A.T. Astron. Astrophys.84 (1980) 78

Fawcett B.C. and Hayes R.W. J. Phys. B5 (1972) 366

Gabriel A.H. and Jordan C. Case Studies Atom. Coll. Phys.2 (1972) 209

Gabriel A.H., Fawcett B.C. and Jordan C. Proc. Phys. Soc.87 (1966) 825

Goldsmith S. and Fraenkel B.S. Astrophys. J.161 (1970) 317

Huang K.N., Kim Y. K., Cheng K. T., and Desclaux J. P., At. Data Nucl. Data

Tables28 (1983) 355

Keenan F.P., Dufton P.L, Boylan M.B., Kingston A.E. and Widing K.G. Astro-

phys. J. 373 (1991) 695

Malinovsky M. and Heroux L. Astrophys. J.181 (1973) 1009

Mazzotta P. Mazzitelli G., Colafrancesco S. and Vittorio N. Astron. Astrophys

Supp. Ser133 (1998), 403

Ryabtsev A.N. Sov. Astron.23 (1979) 732

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