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
4.2 Collision strength and e�ective collision strength for the 3s
2
3p
5 2
P
o
3=2
{ 3s3p
6 2
S
1=2
transition. . . . . . . . . . . . . . . . . . . . . . . . 87
4.3 Collision strength and e�ective collision strength for the 3s
2
3p
5 2
P
o
3=2
- 3s
2
3p
4
(
3
P ) 3d
4
D
1=2
transition. . . . . . . . . . . . . . . . . . . . 88
4.4 Collision strength and e�ective collision strength for the 3s
2
3p
4
(
3
P ) 3d
4
D
1=2
- 3s
2
3p
4
(
3
P ) 3d
4
F
3=2
transition. . . . . . . . . . . . . . . . . 89
4.5 Collision strength and e�ective collision strength for the 3s
2
3p
5 2
P
o
1=2
{ 3s
2
3p
4
(
3
P )3d
4
D
5=2
transition. . . . . . . . . . . . . . . . . . . . 90
4.6 Collision strength and e�ective collision strength for the 3s
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
6.5 Identi�cation of Ni XII lines in JET pulse 31273 . . . . . . . . . . 194
6.6 Plasma conditions of JET pulse 31273 . . . . . . . . . . . . . . . 195
6.7 Magnetic �eld con�guration of JET pulse 31273 . . . . . . . . . . 196
6.8 Superimposition of lines in JET pulse 31273 . . . . . . . . . . . . 197
6.9 Integration of the lines in JET pulse 31273 . . . . . . . . . . . . . 198
6.10 Identi�cation of Ni XII lines in JET pulse 31275 . . . . . . . . . . 199
vi
LIST OF FIGURES vii
6.11 Plasma conditions of JET pulse 31275 . . . . . . . . . . . . . . . 200
6.12 Magnetic �eld con�guration of JET pulse 31275 . . . . . . . . . . 201
6.13 Superimposition of lines in JET pulse 31275 . . . . . . . . . . . . 202
6.14 Integration of the lines in JET pulse 31275 . . . . . . . . . . . . . 203
6.15 Identi�cation of Ni XII lines in JET pulse 31798 . . . . . . . . . . 204
6.16 Plasma conditions of JET pulse 31798 . . . . . . . . . . . . . . . 205
6.17 Magnetic �eld con�guration of JET pulse 31798 . . . . . . . . . . 206
6.18 Superimposition of lines in JET pulse 31798 . . . . . . . . . . . . 207
6.19 Integration of the lines in JET pulse 31798 . . . . . . . . . . . . . 208
6.20 Plot of the theoretical line ratio, R
1
, as a function of electron density.209
6.21 Plot of the theoretical line ratio, R
1
, as a function of electron tem-
perature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
6.22 Plot of the theoretical line ratio, R
2
, as a function of electron density.211
6.23 Plot of the theoretical line ratio, R
2
, as a function of electron tem-
perature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
6.24 Plot of the theoretical line ratio, R
3
, as a function of electron density.213
6.25 Plot of the theoretical line ratio, R
3
, as a function of electron tem-
perature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
6.26 Plot of the theoretical line ratio, R
4
, as a function of electron density.215
6.27 Plot of the theoretical line ratio, R
4
, as a function of electron tem-
perature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
6.28 Plot of the theoretical line ratio, R
5
, as a function of electron density.217
6.29 Plot of the theoretical line ratio, R
5
, as a function of electron tem-
perature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
6.30 Plot of the theoretical line ratio, R
6
, as a function of electron density.219
6.31 Plot of the theoretical line ratio, R
6
, as a function of electron tem-
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
1.2 Principal Methods for Electron-Impact Excitation Calculations 5
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
1.2 Principal Methods for Electron-Impact Excitation Calculations 6
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
1.2 Principal Methods for Electron-Impact Excitation Calculations 7
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
1.2 Principal Methods for Electron-Impact Excitation Calculations 8
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.
1.2 Principal Methods for Electron-Impact Excitation Calculations 9
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.
1.3 Atomic E�ects 11
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
1.3 Atomic E�ects 12
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)
1.3 Atomic E�ects 13
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
1.3 Atomic E�ects 14
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
1.3 Atomic E�ects 15
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
2.2 The Hartree-Fock method 24
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
2.2 The Hartree-Fock method 25
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
2.2 The Hartree-Fock method 26
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
2.2 The Hartree-Fock method 27
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 The Hartree-Fock method 28
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
2.3 The Con�guration Interaction method 30
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.
2.3 The Con�guration Interaction method 31
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
2.3 The Con�guration Interaction method 32
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
2.3 The Con�guration Interaction method 33
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
2.3 The Con�guration Interaction method 34
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'
2.3 The Con�guration Interaction method 35
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.
2.4 The Con�guration-Interaction Bound State Code - CIV3 37
� 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-
2.4 The Con�guration-Interaction Bound State Code - CIV3 38
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
2.4 The Con�guration-Interaction Bound State Code - CIV3 39
� inclusion of relativistic e�ects
2.4 The Con�guration-Interaction Bound State Code - CIV3 40
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
Hibbert A. Comput. Phys. Commun. 2 (1971) 180
Hibbert A. Comput. Phys. Commun. 6 (1973) 59
Hibbert A. Comput. Phys. Commun. 9 (1975) 141
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
3.1 The R-matrix method 45
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)
3.1 The R-matrix method 46
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 The R-matrix method 47
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
3.1 The R-matrix method 48
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.
3.1 The R-matrix method 49
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.
3.1 The R-matrix method 50
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
3.1 The R-matrix method 51
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-
3.1 The R-matrix method 52
�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
3.1 The R-matrix method 53
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)
3.1 The R-matrix method 54
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)
3.1 The R-matrix method 55
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
3.1 The R-matrix method 56
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
3.1 The R-matrix method 57
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)
3.1 The R-matrix method 58
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 The R-matrix method 59
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
3.1 The R-matrix method 60
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)
3.1 The R-matrix method 61
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)
3.1 The R-matrix method 62
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 The R-matrix method 63
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
3.2 The R-matrix Codes 65
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)
3.2 The R-matrix Codes 66
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.
3.2 The R-matrix Codes 67
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
3.2 The R-matrix Codes 68
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
3.2 The R-matrix Codes 69
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
3.2 The R-matrix Codes 70
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
4.2 Calculation Details 76
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
4.2 Calculation Details 77
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)
4.2 Calculation Details 78
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
4.2 Calculation Details 79
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
4.3 Results and Discussion 81
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-
4.3 Results and Discussion 82
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
4.3 Results and Discussion 83
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
4.3 Results and Discussion 84
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
4.3 Results and Discussion 85
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.
4.3 Results and Discussion 86
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.
4.3 Results and Discussion 87
Figure 4.2: 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
3=2
{ 3s3p
6 2
S
1=2
dipole-allowed �ne-structure transition.
4.3 Results and Discussion 88
Figure 4.3: 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
3=2
- 3s
2
3p
4
(
3
P ) 3d
4
D
1=2
forbidden �ne-structure transition.
4.3 Results and Discussion 89
Figure 4.4: 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
4
(
3
P ) 3d
4
D
1=2
- 3s
2
3p
4
(
3
P ) 3d
4
F
3=2
forbidden �ne-structure transition.
4.3 Results and Discussion 90
Figure 4.5: 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
4
(
3
P )3d
4
D
5=2
forbidden �ne-structure transition.
4.3 Results and Discussion 91
Figure 4.6: 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
3=2
- 3s
2
3p
4
(
1
D) 3d
2
P
3=2
forbidden �ne-structure transition.
4.3 Results and Discussion 92
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.
4.5 Explanation of Tables 96
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
4.5 Explanation of Tables 97
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.
4.5 Explanation of Tables 98
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)
4.5 Explanation of Tables 99
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
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2
2
.
2
4
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2
2
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0
5
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2
1
.
8
7
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1
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1
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1
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5
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6
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2
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1
2
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1
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5
.
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1
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2
4
.
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0
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0
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2
3
.
9
4
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2
3
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5
9
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2
3
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2
7
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2
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9
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3
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5
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8
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2
2
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0
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0
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5
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4.6 Tables 127
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6
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5
5
.
6
5
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7
5
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8
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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
5.1 Tokamaks and Nuclear Fusion 144
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
5.1 Tokamaks and Nuclear Fusion 145
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.
5.1 Tokamaks and Nuclear Fusion 146
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
5.1 Tokamaks and Nuclear Fusion 147
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.
5.1 Tokamaks and Nuclear Fusion 148
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.
5.1 Tokamaks and Nuclear Fusion 149
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
5.1 Tokamaks and Nuclear Fusion 150
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
5.3 Tokamak Experiments 153
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.
5.3 Tokamak Experiments 154
Figure 5.4: Current and previous world records for power output by tokamak
devices
5.3 Tokamak Experiments 155
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
5.4 Instrumentation 157
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
5.4 Instrumentation 158
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
5.4 Instrumentation 159
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
5.4 Instrumentation 160
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
5.4 Instrumentation 161
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
5.4 Instrumentation 162
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
5.4 Instrumentation 163
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
Bohdansky J., Nucl. Fusion (1984) Special Issue 61
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
6.1 Line Ratio Diagnostics for Tokamak Plasmas 168
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
)
6.1 Line Ratio Diagnostics for Tokamak Plasmas 169
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.
6.1 Line Ratio Diagnostics for Tokamak Plasmas 170
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
6.1 Line Ratio Diagnostics for Tokamak Plasmas 171
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.
6.2 NiXII Line Search on the JET Tokamak 173
�
=
� 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 NiXII Line Search on the JET Tokamak 174
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
6.2 NiXII Line Search on the JET Tokamak 175
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
6.4 Results and Conclusions 183
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.
6.4 Results and Conclusions 184
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
6.4 Results and Conclusions 185
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).
6.4 Results and Conclusions 186
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
6.4 Results and Conclusions 187
Ratio JET pulse Measured Theoretical
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
6.4 Results and Conclusions 188
Ratio JET pulse Measured Theoretical
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
6.4 Results and Conclusions 189
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
6.4 Results and Conclusions 190
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.
6.4 Results and Conclusions 191
Figure 6.2: Identi�cation of NiXII lines in JET pulse 34938 at t = 54.8 s
6.4 Results and Conclusions 192
Figure 6.3: Various plasma conditions of JET pulse 34938
6.4 Results and Conclusions 193
Figure 6.4: Magnetic �eld con�guration of JET pulse 34938
6.4 Results and Conclusions 194
Figure 6.5: Identi�cation of NiXII lines in JET pulse 31273 at t = 61.0 s
6.4 Results and Conclusions 195
Figure 6.6: Various plasma conditions of JET pulse 31273
6.4 Results and Conclusions 196
Figure 6.7: Magnetic �eld con�guration of JET pulse 31273
6.4 Results and Conclusions 197
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
6.4 Results and Conclusions 198
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
6.4 Results and Conclusions 199
Figure 6.10: Identi�cation of NiXII lines in JET pulse 31275 at t = 57.7 s
6.4 Results and Conclusions 200
Figure 6.11: Various plasma conditions of JET pulse 31275
6.4 Results and Conclusions 201
Figure 6.12: Magnetic �eld con�guration of JET pulse 31275
6.4 Results and Conclusions 202
Figure 6.13: 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 31275
6.4 Results and Conclusions 203
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
6.4 Results and Conclusions 204
Figure 6.15: Identi�cation of NiXII lines in JET pulse 31798 at t = 51.0 s
6.4 Results and Conclusions 205
Figure 6.16: Various plasma conditions of JET pulse 31798
6.4 Results and Conclusions 206
Figure 6.17: Magnetic �eld con�guration of JET pulse 31798
6.4 Results and Conclusions 207
Figure 6.18: 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 31798
6.4 Results and Conclusions 208
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
6.4 Results and Conclusions 209
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).
6.4 Results and Conclusions 210
Figure 6.21: 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
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).
6.4 Results and Conclusions 211
Figure 6.22: Plot of the theoretical line ratio, R
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
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).
6.4 Results and Conclusions 212
Figure 6.23: Plot of the theoretical line ratio, R
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 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).
6.4 Results and Conclusions 213
Figure 6.24: Plot of the theoretical line ratio, 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
) 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).
6.4 Results and Conclusions 214
Figure 6.25: Plot of the theoretical line ratio, 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
) 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).
6.4 Results and Conclusions 215
Figure 6.26: Plot of the theoretical line ratio, 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
) 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).
6.4 Results and Conclusions 216
Figure 6.27: Plot of the theoretical line ratio, 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
) 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).
6.4 Results and Conclusions 217
Figure 6.28: Plot of the theoretical line ratio, 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
) 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).
6.4 Results and Conclusions 218
Figure 6.29: Plot of the theoretical line ratio, 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
) 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).
6.4 Results and Conclusions 219
Figure 6.30: Plot of the theoretical line ratio, 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
) 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).
6.4 Results and Conclusions 220
Figure 6.31: Plot of the theoretical line ratio, 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
) 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).
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
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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
Recommended