ATOMIC DATA GENERATION AND COLLISIONAL RADIATIVE MODELING OF AR II, AR III,
AND NE I FOR LABORATORY AND ASTROPHYSICAL PLASMAS
Except where reference is made to the work of others, the work described in this dissertation is my
own or was done in collaboration with my advisory committee. This dissertation does not include
proprietary or classified information.
Jorge Manuel Mu?noz Burgos
Certificate of Approval:
Robert F. Boivin, Co-Chair
Assistant Professor
Physics
Stuart D. Loch, Co-Chair
Assistant Professor
Physics
Michael S. Pindzola
Professor
Physics
Yu Lin
Professor
Physics
George T. Flowers
Dean
Graduate School
ATOMIC DATA GENERATION AND COLLISIONAL RADIATIVE MODELING OF AR II, AR III,
AND NE I FOR LABORATORY AND ASTROPHYSICAL PLASMAS
Jorge Manuel Mu?noz Burgos
A Dissertation
Submitted to
the Graduate Faculty of
Auburn University
in Partial Fulfillment of the
Requirements for the
Degree of
Doctor of Philosophy
Auburn, Alabama
August 10, 2009
ATOMIC DATA GENERATION AND COLLISIONAL RADIATIVE MODELING OF AR II, AR III,
AND NE I FOR LABORATORY AND ASTROPHYSICAL PLASMAS
Jorge Manuel Mu?noz Burgos
Permission is granted to Auburn University to make copies of this dissertation at its
discretion, upon the request of individuals or institutions and at
their expense. The author reserves all publication rights.
Signature of Author
Date of Graduation
iii
VITA
Jorge Manuel Mu?noz Burgos, son of Jorge Armando Mu?noz Pacheco and Esther Burgos
Serrano, was born on October 21th, 1976, in the city of Guadalajara in the state of Jalisco,
M`exico. He attended Preparatoria Varonil U.A.G. in Guadalajara, and graduated June 10th, 1994.
He entered Universidad Autonoma de Guadalajara in August, 1994 and received a B. S. degree in
Electrical Engineering and Communications in June 27th, 1998. After graduation he worked from
June 1998 to August 1999 as Research and Development Engineer for Sistemas y Accesos Con-
trolados S.A. de C.V. (Now Hydra Technologies, www.hydra-technologies.com) in Guadalajara,
M`exico. In January, 2000, he entered the graduate program at Florida Institute of Technology in
Melbourne, Florida. He obtained a M. S. degree in Electrical Engineering in Electromagnetics in
July, 2001, and a M. S. degree in Physics in August, 2003. He attended Auburn University from
August, 2003 to July, 2009, where he obtained the Ph.D. degree in Physics. He has been offered a
postdoctoral position at General Atomics in San Diego, California, to conduct research in Plasma
Boundary Interface at the DIII-D TOKAMAK Nuclear Fusion Experimental Reactor.
iv
DISSERTATION ABSTRACT
ATOMIC DATA GENERATION AND COLLISIONAL RADIATIVE MODELING OF AR II, AR III,
AND NE I FOR LABORATORY AND ASTROPHYSICAL PLASMAS
Jorge Manuel Mu?noz Burgos
Doctor of Philosophy, August 10, 2009
(M.S. Physics, Florida Institute of Technology, 2003)
(M.S. EE, Florida Institute of Technology, 2001)
(B.S. EE, Universidad Autonoma de Guadalajara, 1998)
206 Typed Pages
Directed by Stuart D. Loch and Robert F. Boivin
Accurate knowledge of atomic processes plays a key role in modeling the emission in labo-
ratory as well as in astrophysical plasmas. These processes are included in a collisional-radiative
model and the results are compared with experimental measurements for Ar and Ne ions from the
ASTRAL (Auburn Steady sTate Research fAciLity) experiment. The accuracy of our model de-
pends upon the quality of the atomic data we use. Atomic data for near neutral systems present
a challenge due to the low accuracy of perturbative methods for these systems. In order to im-
prove our model we rely on non-perturbative methods such as R-Matrix and RMPS (R-Matrix
with Pseudo-States) to include correlation in the collision cross-sections. These methods are com-
putationally demanding, requiring supercomputing resources, and producing very accurate atomic
collision data. For Ar+ and Ne, R-Matrix data was already available, however for Ar2+ we had
to set up new R-Matrix calculations. To set up a new calculation we require good quality atomic
structure. A new code (LAMDA) was developed to optimize the atomic structure for different ions
in AUTOSTRUCTURE. The AUTOSTRUCTURE code was used and optimized by systematically
adjusting the orbital scale factors with the help of a Singular Value Decomposition algorithm.
v
We then tested the quality of our newly optimized atomic structure by comparing the level or term
energies, and line strengths from our optimized structure with those given by NIST.
In the case of Ar+ we compared R-Matrix electron-impact excitation data against the results
from a new RMPS calculation. The aim was to assess the effects of continuum-coupling effects on
the atomic data and the resulting spectrum. We do our spectral modeling using the ADAS suite of
codes. Our collisional-radiative formalism assumes that the excited levels are in quasi-static equi-
librium with the ground and metastable populations. In our model we allow for Ne and Te variation
along the line of sight by fitting our densities and temperature profiles with those measured within
the experiment. The best results so far have been obtained by the fitting of the experimental temper-
ature and density profiles with Gaussian and polynomial distribution functions. The line of sight
effects were found to have a significant effect on the emission modeling.
The relative emission rates were measured in the ASTRAL helicon plasma source. A spec-
trometer which features a 0.33 m Criss-Cross Scanning monochromator and a CCD camera is used
for this study. ASTRAL produces bright intense Ar and Ne plasmas with ne = 1011 to 1013 cm?3
and Te = 2 to 10 eV. A series of 7 large coils produce an axial magnetic field up to 1.3 kGauss. A
fractional helix antenna is used to introduce RF power up to 2 kWatt. Two RF compensated Lang-
muir probes are used to measureTe andNe. In a series of experiment Ar II, Ar III, and Ne transitions
are monitored as a function of Te, while Ne is kept nearly constant. Observations revealed that Te
is by far the most significant parameter affecting the emission rate coefficients, thus confirming our
predictions. The spectroscopy measurements are compared with those from our spectral modeling
which in turn help us to compare the effectiveness of the new atomic data calculations with those
from other calculations. It also shows some differences between the R-Matrix and the RMPS data
due to continuum coupling effects for Ar II, and Ne. We believe that this is the first experimental
observation of continuum-coupling effects.
vi
We performed a new R-Matrix calculation for Ar2+. Emission from Ar2+ is seen in planetary
nebulae, in H II regions, and from laboratory plasmas. Our calculation improved upon existing
electron-impact excitation data for the 3p4 configuration of Ar2+ and calculated new data for the
excited levels. Electron-impact excitation collision strengths were calculated using the R-Matrix
intermediate-coupling (IC) frame-transformation method and the R-Matrix Breit-Pauli method. Ex-
citation cross-sections are calculated between all levels of the configurations 3s2 3p4, 3s 3p5, 3p6,
3p5 3d, and 3s2 3p3 nl (3d ? nl ? 5s). Maxwellianeffective collision strengths are generated from
the collision strength data. Good agreement is found in the collision strengths calculated using the
two R-Matrix methods. The effects of the new data on line ratio diagnostics were studied. The col-
lision strengths are compared with literature values for transitions within the 3s2 3p4 configuration.
The new data has a small effect on Te values obtained from the I(?7135 ?A+?7751 ?A)/I(?5192 ?A)
line ratio, and a larger effect on the Ne values obtained from the I(?7135 ?A)/I(?9?m) line ratio.
The final effective collision strength data is archived online.
Neon as well as Argon is a species of current interest in fusion TOKAMAK studies. It is used
for radiative cooling of the divertor region and for disruption mitigation. It could also be useful
as a spectral diagnostic if better atomic data were available. We present results from modeling
emission line intensity for neutral neon by using Plane Wave Born,R-Matrix, and RMPS electron-
impact excitation calculations. We benchmark our theoretical calculations against cross-section
measurements, then against spectral measurements from ASTRAL.
vii
ACKNOWLEDGMENTS
It is written: Thefear of theLordisthebeginning of knowledge, Butfoolsdespise
wisdom and instruction (Proverbs 1:7). Or, as Psalm 19:1,2 says: The heavens declare the
glory of God; And the firmament shows His handiwork. Day unto day utters speech,
Andnightuntonight revealsknowledge. To You my God, Lord, Savior, and King Jesus
Christ, for always being with me, and for showing me that through You all things are possible. To
You alone be the kingdom, the power, and the glory forever. To my loving mother Esther, for her
faith, hard work, and prayers. Thank you for taking care of us during the hard times and for teaching
us to love the God of Abraham, Isaac, and Jacob in Whose loving and mighty hand we stand. To
my brother Alex, and sister Erika, for all their love and support. To my deceased dad Jorge whom I
hope to see one day in glory. For all the rest of my dear and supportive family and friends, thank you
for all your encouragement and prayers. To the selfless help, support, and hard working example
from my two academic advisors and friends Dr. Stuart D. Loch in the theoretical aspect, and Dr.
Robert F. Boivin in the experimental. Thank you for all your help, patience, and guidance. To Dr.
Connor P. Ballance, for all his help, time, hard work, and patience in helping us understand the
R-Matrix theory, and teaching us to use the parallel codes that played a key role in the development
of this dissertation. To all my doctoral committee members; Dr. Michael S. Pindzola, Dr. Yu Lin,
and Dr. Michael L. McKee, thank you for taking the time to review this dissertation. To Dr. Oleg
V. Batishchev from MIT, thank you for your encouragement and support. To Auburn University, a
very special place where I made such wonderful friends. To the hard working people of my native
country M?exico, to the wonderful people of TheUnitedStatesof America for kindly opening
the doors for me in your beautiful country, and to the beloved children of Israel, a testimony that
God is always faithful and truthful. May God bless you all.
viii
I want to thank also the following institutions for their financial support in the development of
these different projects. The US Department of Energy with grant DE-FG02-99ER54367. Rollins
College and US DoE with grants DE-FG05-96-ER54348 and DE-FG02-01ER54633. To Auburn
University with funding from the Oak Ridge National Lab. Computational work was carried out
at the National Energy Research Scientific Computing Center in Oakland, California, at the Na-
tional Energy Research Scientific Computing Center in Oakland, California, and at the Alabama
Supercomputer in Huntsville, Alabama.
ix
Style manual or journal used Journals of the American Physical Society (together with the style
known as ?auphd?). Bibliography follows the style used by the American Physical Society.
Computer software used The document preparation package TEX (specifically LATEX) together
with the departmental style-file auphd.sty.
x
TABLE OF CONTENTS
LIST OF FIGURES xiv
LIST OF TABLES xviii
1 INTRODUCTION 1
2 COLLISIONAL-RADIATIVE MODEL 6
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Atomic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Equations and Matrix Representation . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 Timescales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4.1 Plasma Timescales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4.2 Atomic Timescales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.5 Solution of the CR Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.6 Ionization Balance Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3 ATOMIC STRUCTURE DATA 20
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2 Atomic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3 Atomic Structure Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3.1 AUTOSTRUCTURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3.2 GASP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4 Atomic Structure Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.4.2 Linearization of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.4.3 Inverse Matrix Computation . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4 ATOMIC COLLISION DATA 40
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.2 Plane Wave Born . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2.1 Classical Scattering Theory . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2.2 Plane Wave Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2.3 The Born Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.3 Distorted Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.4 R-Matrix Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.5 Effective Collision Strengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.5.1 Collision Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
xi
4.5.2 Effective Collision Strength . . . . . . . . . . . . . . . . . . . . . . . . . 62
5 EXPERIMENTAL SETUP 65
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.3 Langmuir Probe Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.4 Spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.5 Data Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
6 AR+ MODELING AND EXPERIMENT 88
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.2 Atomic Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.3 Collisional-Radiative Modeling and Ionization Balance . . . . . . . . . . . . . . . 90
6.4 Emission Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.5 Metastable Lifetimes and Opacities . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.6 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
7 AR2+ MODELING [48] 111
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.2 Atomic Structure Calculation and Optimization . . . . . . . . . . . . . . . . . . . 114
7.2.1 R-Matrix Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
7.2.2 Collisional-Radiative Model . . . . . . . . . . . . . . . . . . . . . . . . . 116
7.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.3.1 Atomic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.3.2 Scattering Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
7.3.3 Emission Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
8 NE MODELING AND EXPERIMENT 133
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
8.2 Atomic Data Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
8.2.1 Atomic Structure Comparison . . . . . . . . . . . . . . . . . . . . . . . . 135
8.2.2 Cross-Sections Comparison . . . . . . . . . . . . . . . . . . . . . . . . . 140
8.3 Emission Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
8.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
8.4.1 Spectral Dipole Lines Connected to the Ground . . . . . . . . . . . . . . . 152
8.4.2 Spectral Dipole Lines Connected to the Metastable . . . . . . . . . . . . . 155
8.4.3 Spectral Dipole Lines Connected to the Ground and Metastable . . . . . . 158
8.4.4 Spectral Dipole Lines Not Connected to the Ground Nor the Metastable . . 160
8.4.5 Other Dipole Line Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
9 CONCLUSIONS 164
APPENDICES 167
xii
A Green?S FUNCTION SOLUTION FOR THE NON-HOMOGENEOUS Helmholtz EQUATION 168
A.1 The Helmholtz Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
A.2 Green?s Function Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
B SYSTEM RESPONSE FUNCTION FOR THE ASTRAL SPECTROMETER 173
B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
B.2 Calibration Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
B.2.1 Mounting the lamp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
B.2.2 Power Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
B.2.3 Measurement of Lamp Current . . . . . . . . . . . . . . . . . . . . . . . . 173
B.2.4 CCD Camera Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
B.2.5 Oriel Model QTH 200W Calibration Curve . . . . . . . . . . . . . . . . . 174
B.2.6 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
B.3 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
BIBLIOGRAPHY 184
xiii
LIST OF FIGURES
2.1 Populating processes for the ith atomic level. . . . . . . . . . . . . . . . . . . . . 7
2.2 Normalized level population dependence with respect to electron density. . . . . . 15
2.3 Ionization balance of Ar (ne = 1011 cm?3). . . . . . . . . . . . . . . . . . . . . . 19
2.4 Ionization balance of Ne (ne = 1011 cm?3). . . . . . . . . . . . . . . . . . . . . . 19
3.1 AUTOSTRUCTURE input file for a case of the Ar2+ ion. . . . . . . . . . . . . . . 28
3.2 GASP interface showing a computation for a case of the Ar2+ ion. . . . . . . . . . 29
3.3 Representation of the variation of the orbitals with respect to the scale factors ?s. . 36
3.4 LAMDA code output file for the Ar2+ ion. . . . . . . . . . . . . . . . . . . . . . . 38
4.1 Scattering of a light particle by a heavy ion. . . . . . . . . . . . . . . . . . . . . . 41
4.2 Scattering of an incident plane wave. . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.3 Partitioning of configuration space R-Matrix theory. . . . . . . . . . . . . . . . . . 56
4.4 Representation of the continuum by the introduction of pseudo-states. . . . . . . . 59
5.1 The ASTRAL helicon plasma source. . . . . . . . . . . . . . . . . . . . . . . . . 66
5.2 Ar plasma in ASTRAL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.3 Ne plasma in ASTRAL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.4 Schematic upper view of the ASTRAL helicon plasma source. . . . . . . . . . . . 68
5.5 Langmuir probe settings in ASTRAL. . . . . . . . . . . . . . . . . . . . . . . . 70
5.6 Langmuir probe inside of ASTRAL. . . . . . . . . . . . . . . . . . . . . . . . . 71
5.7 Typical I vs V plot in a Langmuir probe. . . . . . . . . . . . . . . . . . . . . . . 72
5.8 McPherson Model 218 Spectrometer. . . . . . . . . . . . . . . . . . . . . . . . . . 77
xiv
5.9 McPherson Model 789A-3 Digital Scan Control. . . . . . . . . . . . . . . . . . . 78
5.10 Monochromator optical set-up. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.11 Measured spectrum of Ne on the 650 nm region. . . . . . . . . . . . . . . . . . . . 80
5.12 Namelist input file (spectrum.in) for the program spectrum.x. . . . . . . . . . . . . 82
5.13 Input data file (in this case 875a.txt) for the program spectrum.x. . . . . . . . . . . 84
5.14 Output file (intensity.out) from the program spectrum.x. . . . . . . . . . . . . . . . 85
5.15 Namelist input file (dens.in) for the program dens.x. . . . . . . . . . . . . . . . . . 86
5.16 Output file (in this case Ne8.47E12.dat) for the program dens.x. . . . . . . . . . . 87
6.1 Fractional abundance results with new data. . . . . . . . . . . . . . . . . . . . . . 91
6.2 Norm. electron density distribution along the diameter of the vac. chamber. . . . . 93
6.3 Norm. electron temperature distributions along the diameter of the vac. chamber. . 93
6.4 Contribution to the total intensity along the line of sight as a function of central Te. 95
6.5 Traveled distance of the Ar+ ions as a function of electron temperature. . . . . . . 97
6.6 Grotrian diagram of Ar+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.7 Intensity line ratio of I(664.37)/I(668.43). . . . . . . . . . . . . . . . . . . . . . . 101
6.8 New ionization balance data gives good agreement. . . . . . . . . . . . . . . . . . 102
6.9 New ionization balance data gives good agreement. . . . . . . . . . . . . . . . . . 103
6.10 New ionization balance data gives good agreement. . . . . . . . . . . . . . . . . . 103
6.11 New ionization balance data gives good agreement. . . . . . . . . . . . . . . . . . 104
6.12 New ionization balance data gives good agreement. . . . . . . . . . . . . . . . . . 104
6.13 Intensity line ratio of I(487.99)/I(460.60). . . . . . . . . . . . . . . . . . . . . . . 105
6.14 Intensity line ratio of I(465.79)/I(484.78). . . . . . . . . . . . . . . . . . . . . . . 106
6.15 Intensity line ratio of I(473.59)/I(487.99). . . . . . . . . . . . . . . . . . . . . . . 106
6.16 Intensity line ratio of I(472.69)/I(476.49). . . . . . . . . . . . . . . . . . . . . . . 107
xv
6.17 Intensity line ratio of I(472.69)/I(480.60). . . . . . . . . . . . . . . . . . . . . . . 107
6.18 Intensity line ratio of I(484.78)/I(472.69). . . . . . . . . . . . . . . . . . . . . . . 108
6.19 Intensity line ratio of I(349.15)/I(668.43). . . . . . . . . . . . . . . . . . . . . . . 110
6.20 Intensity line ratio of I(354.55)/I(668.43). . . . . . . . . . . . . . . . . . . . . . . 110
7.1 Comparison of the ICFT and Breit-Pauli collision strengths. . . . . . . . . . . . . 121
7.2 Scatter plot showing the ratio of effective collision strengths. . . . . . . . . . . . . 123
7.3 Comparison of selected Breit-Pauli collision strengths. . . . . . . . . . . . . . . . 124
7.4 Comparison of selected Breit-Pauli effective collision strengths. . . . . . . . . . . 126
7.5 Burgess Tully plot of effective collision strength vs reduced temperature (X). . . . 128
7.6 R1 line ratio as a function of electron temperature. . . . . . . . . . . . . . . . . . . 130
7.7 R2 line ratio as a function of electron density. . . . . . . . . . . . . . . . . . . . . 131
8.1 Excitation cross-section for the 2p6 (1S0) ? 3s 2[1/2]o1 (1P1) transition. . . . . . . 140
8.2 Excitation cross-section for the 2p6 (1S0) ? 3s 2[1/2]o1 (1P1) transition. . . . . . . 141
8.3 Excitation cross-section for the 2p6 (1S0) ? 3p 2[1/2]1 (3S1) transition. . . . . . . 142
8.4 Excitation cross-section for the 2p6 (1S0) ? 3p 2[1/2]1 (3S1) transition. . . . . . . 143
8.5 Excitation cross-section for the 3s 2[3/2]o2 (3P2) ? 3p 2[5/2]3 (3D3) transition. . . . 144
8.6 Excitation cross-section for the 3s 2[3/2]o2 (3P2) ? 3p 2[5/2]2 (1D2) transition. . . . 145
8.7 Excitation cross-section for the 3s 2[3/2]o2 (3P2) ? 3p 2[3/2]2 (3P2) transition. . . . 146
8.8 Excitation cross-section for the 3s 2[3/2]o2 (3P2) ? 3p 2[3/2]2 (3D2) transition. . . . 146
8.9 Ionization balance of Ne (ne = 1011 cm?3). . . . . . . . . . . . . . . . . . . . . . 147
8.10 Excitation Grotrian diagram of Ne. . . . . . . . . . . . . . . . . . . . . . . . . . 148
8.11 Norm. electron temperature distributions along the diameter of the vac. chamber. . 149
8.12 Norm. electron density distribution along the diameter of the vac. chamber. . . . . 149
8.13 Line intensity for the 2p53p 2[3/2]1 upper level. . . . . . . . . . . . . . . . . . . . 152
xvi
8.14 Line intensity for the 2p53p 2[3/2]1 upper level. . . . . . . . . . . . . . . . . . . . 153
8.15 Intensity line ratio of I(638.30)/I(626.65). . . . . . . . . . . . . . . . . . . . . . . 153
8.16 Intensity line ratio of I(653.29)/I(621.73). . . . . . . . . . . . . . . . . . . . . . . 154
8.17 Line intensity for the 2p53p 2[5/2]3 and 2p53p 2[1/2]1 upper levels. . . . . . . . . . 155
8.18 Line intensity for the 2p53p 2[1/2]1 upper level. . . . . . . . . . . . . . . . . . . . 155
8.19 Intensity line ratio of I(602.99)/I(640.22). . . . . . . . . . . . . . . . . . . . . . . 156
8.20 Intensity line ratio of I(659.89)/I(616.36). . . . . . . . . . . . . . . . . . . . . . . 156
8.21 Intensity line ratio of I(703.24)/I(743.89). . . . . . . . . . . . . . . . . . . . . . . 157
8.22 Line intensity for the 2p53p 2[5/2]2 upper level. . . . . . . . . . . . . . . . . . . . 158
8.23 Line intensity for the 2p53p 2[1/2]0 upper level. . . . . . . . . . . . . . . . . . . . 158
8.24 Intensity line ratio of I(607.43)/I(585.25). . . . . . . . . . . . . . . . . . . . . . . 159
8.25 Line intensity for the 2p53p 2[1/2]0 upper level. . . . . . . . . . . . . . . . . . . . 160
8.26 Line intensity for the 2p53p 2[1/2]0 upper level. . . . . . . . . . . . . . . . . . . . 160
8.27 Intensity line ratio of I(609.62)/I(630.48). . . . . . . . . . . . . . . . . . . . . . . 161
8.28 Intensity line ratio of I(614.31)/I(594.48). . . . . . . . . . . . . . . . . . . . . . . 161
8.29 Intensity line ratio of I(667.83)/I(692.95). . . . . . . . . . . . . . . . . . . . . . . 162
8.30 Intensity line ratio of I(585.25)/I(594.48). . . . . . . . . . . . . . . . . . . . . . . 163
A.1 Spherical coordinates representation. . . . . . . . . . . . . . . . . . . . . . . . . . 170
A.2 Complex contour integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
B.1 Is(?) Oriels Quartz Tungsten Halogen lamp calibration curve. . . . . . . . . . . . 174
B.2 Relation between actual wavelength and instrumental wavelength. . . . . . . . . . 177
B.3 Im(?) Experimental averaged irradiance as a function of wavelength. . . . . . . . 178
B.4 R(?) System response function as a function of wavelength. . . . . . . . . . . . . 179
xvii
LIST OF TABLES
2.1 Typical collision time values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
5.1 Specifications for the McPherson scanning monochromator model 218. . . . . . . 78
5.2 Description of the namelist input file for the SPECTRUM program. . . . . . . . . 83
5.3 Description of the namelist input file for the DENS program. . . . . . . . . . . . . 86
6.1 Overview of atomic data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.2 Lifetimes of the Ar+ ion for the 3s3p6 (2S) term. . . . . . . . . . . . . . . . . . . 96
6.3 Table of some of the extracted spectral lines. . . . . . . . . . . . . . . . . . . . . . 100
7.1 Final ? values for the 1s-5s orbitals. . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.2 Energies in Rydbergs for the lowest 29 levels of Ar2+. . . . . . . . . . . . . . . . 118
7.3 Comparisons of selected radiative rates for transitions in Ar2+ . . . . . . . . . . . 119
7.4 Effective collision strengths for transitions between the 3s23p4 levels. . . . . . . . 125
8.1 Energies in Rydbergs used in O?Mullane?s Plane Wave Born calculation. . . . . . 135
8.2 Energies in Rydbergs used in Zatsarinny?s group R-Matrix calculation. . . . . . . 136
8.3 Energies in Rydbergs used in Griffin?s group R-Matrix (IC) calculation. . . . . . . 137
8.4 Energies in Rydbergs used in Griffin?s group RMPS (LS) calculation. . . . . . . . 138
8.5 Dipole transitions to the ground and metastable. . . . . . . . . . . . . . . . . . . . 139
8.6 Ne spectral transitions measured in the ASTRAL experiment. . . . . . . . . . . . . 151
B.1 Oriel Model QTH 200W Calibration Curve Parameters . . . . . . . . . . . . . . . 175
B.2 Irradiance uncertainty at different wavelengths for the QTH 200W lamp. . . . . . . 175
B.3 Selected experimental central wavelengths . . . . . . . . . . . . . . . . . . . . . . 176
B.4 Measured experimental central wavelengths . . . . . . . . . . . . . . . . . . . . . 176
B.5 Numerical values for the system response as a function of wavelength (? vs R(?)). 180
xviii
CHAPTER 1
INTRODUCTION
Spectral emission modeling has been key for many diagnostics in astrophysical and laboratory
plasmas. This modeling has particular interest in the field of astrophysics when studying emission
from planetary nebulae, solar corona, or the interstellar medium, which cannot be accessed with
probes. In laboratory plasmas the conditions are often too hostile for probe measurements. There-
fore we require accurate models for the atomic processes involved in plasma emission in order to get
a reliable interpretation of spectral observations. Several challenges arise when trying to model the
spectral emission coming from a hot plasma, and in developing plasma spectral diagnostics. These
difficulties include the need for accurate atomic data, knowledge of the temperature and density
distributions within the plasma, and an understanding of the plasma and atomic timescales.
Emission from Ar II, Ar III, and Ne I will be the main focus of the work described here. The
emission from these species in the ASTRAL Auburn helicon plasma source will be modeled with
two main purposes in mind. Firstly we seek to use the experiment to test the atomic data for Ar
and Ne. Secondly we intend to develop spectral diagnostics that can be used for other Ar and Ne
plasmas. Ar and Ne are of interest in fusion TOKAMAK plasmas to radiatively cool the divertor
and to mitigate plasma disruptions. Ar is of special interest in planetary nebulae spectroscopy, with
forbidden lines for Ar III being used as Te and Ne diagnostics. In the remainder of this chapter a
brief overview of the rest of the dissertation will be given.
1
Chapter 2 deals with the general concepts of modeling spectral emission and introduces dif-
ferent atomic processes that contribute to the population of specific atomic levels. We introduce the
collisional-radiative model [1] used in modeling the emission from plasmas. We also describe our
application of collisional-radiative theory to the calculation of excited populations and ionization
balance using the Atomic Data Analysis Structure (ADAS) suite of codes. This method encom-
passes both the low density coronal, and high density Local Thermodynamic Equilibrium (LTE)
description of the emission from the ion we intend to model. It also includes ionization and recom-
bination processes to and from metastable levels of the next ionization stage.
Chapter 3 describes calculations of atomic structure that are needed for computing electron-
impact ionization as well as electron-impact excitation cross-sections. The accuracy of the atomic
structure of the atom and/or ion is essential in the calculation of atomic collision quantities. In order
to calculate our atomic structure we make use of the AUTOSTRUCTURE [2] code. We also make
use of the graphical interface version of the AUTOSTRUCTURE code, GASP [3], to simplify our
calculation procedures. We also explore the capability of fine tuning our atomic structure by the
introduction of variational adjustment of scale factors (?s). We propose and use a new optimization
procedure base on Singular Value Decomposition (SVD) [4] to calculate the optimal scale factors
that we need in order to optimize and improve our the quality of our atomic structure.
Chapter 4 gives a general description of some of the most widely used methods to calculate
electron-impact ionization data and electron-impact excitation data. We start in section 4.2 with an
explanation of the Plane Wave Born approximation, where the incoming electron is described by a
plane wave, and the target as a static isotropic potential.
2
In section 4.3 we give a short introduction to Distorted Wave theory, where we allow the incoming
wave to be affected by the target potential and therefore be distorted. We also give a short intro-
duction to R-Matrix theory in section 4.5, and explain the use of pseudo-states, or R-Matrix with
Pseudo-States (RMPS), in order to model the interactions with the continuum which also help us
improve our atomic structure. These methods will be used throughout the dissertation for the col-
lisional atomic data that will be used in the spectral emission modeling. In section 4.5 we give an
overview of the Burgess-Tully plots [5]. These plots allow us to display electron-impact excitation
data in a dimensionless way by using suitable scaling procedures. These scaling procedures remove
the main asymptotic energy (or temperature) dependence for the given data. The energy (tempera-
ture) is also scaled so as to become a dimensionless variable which ranges from 0 at the threshold
energy (zero temperature) to 1 at infinite energy (temperature). This way we can display the whole
variation of a collision strength in a single graph, and help us to compare the whole data with previ-
ous calculations.
Chapter 5 gives a description of the Auburn Steady sTate Research fAciLity ASTRAL. This
is a helicon device where we generate intense Ar and Ne plasma columns in order to study and
measure spectral-line emission of plasmas and test them at different plasma conditions. Although
in our work we focus mainly in Ar and Ne emission, ASTRAL has also been used to study He and
CO2 plasma emission. We discuss the efficiency of the helicon source as a mean to generate dense
plasmas heated by radio waves. Helicon sources are very useful for basic plasma studies. Helicon
devices also have the advantage that the antenna is outside the plasma, which helps reducing the
introduction of contaminants into the plasma. In this chapter we also describe measurements of the
plasma densities and temperatures by using two RF compensated Langmuir probes.
3
These measurements form the benchmark in which we compare our collisional-radiative model for
Ar and Ne, with the aim of developing reliable non-invasive methods for plasma diagnostic based
on spectral line ratio measurements. In our spectral analysis we compensate for wavelength due to
the response of the spectrometer to different spectral regions. We have used an absolutely calibrated
Oriel Halogen Lamp in order to measure the response of the spectrometer as a function of wave-
length. In appendix B we present a short discussion of the wavelength calibration procedures.
Chapter 6 deals with the modeling of the Ar II emission, as well as experimental measurements
of line intensities and ratios in order to develop temperature diagnostics for argon plasmas. The
accuracy of the modeling depends on the quality of the atomic data. Therefore one of the aims of this
work is to use this experimental data to determine if the Ar experimental emission from ASTRAL
can be used to test newly calculated atomic data sets in our collisional-radiative model. We use
the spectral measurements to test recent dielectronic recombination (DR) data for the low charge
states of argon. We also identify Ar II line ratios that are sensitive to continuum coupling effects in
the excitation cross-sections. These ratios could be used to benchmark new RMPS electron-impact
excitation data [6], and show where previous non-pseudo-states data [47] may be insufficient for the
modeling. This could provide the first experimental observation of continuum coupling effects.
4
Chapter 7 presents a new R-Matrix calculation for the Ar2+ ion with an optimized atomic
structure calculation. We compare our new calculation with previous R-Matrix calculations per-
formed by Johnson & Kingston [7], and Galavis et al. [8]. Johnson & Kingston calculated excita-
tions within the configuration 3s23p4 and 3s3p5 of Ar2+. Their calculation was generated in (LS)
coupling and transformed to level-resolution using the JAJOM (Saraph [9]) method. Galavis also
used the R-Matrix method to calculate level-resolved excitations within the 3s23p4 configuration.
They used a large configuration-interaction calculation to get their atomic structure. We compare
these calculations and we then discuss the applications of Ar III forbidden line spectra as electron
temperature and density diagnostics of planetary nebula [10]. We also identify some temperature
sensitive line ratios that could be observed in laboratory plasmas.
Chapter 8 employs different sets of electron-impact excitation data in the collisional-radiative
model, in order to predict intensity line emission from neutral neon plasmas. We first use Plane
Wave Born (see section 4.2) electron-impact excitation data calculated by Martin O?Mullane, and
available in the ADAS [11] database. This represents the modeling currently used in the fusion
community. We also use R-Matrix excitation data calculated by Zatsarinny and Bartschat [72].
We then compare the atomic structures, excitation cross-sections, and emission modeling with new
RMPS (LS) and R-Matrix (IC) electron-impact excitation data calculated by Griffin and Ballance
[71]. The line emission modeling predictions from each of the data sets is then compared with
different sets of experimental measurements from ASTRAL. We find that none of the data sets are
in complete agreement with the experimental measurements, likely due to physical effects missing
in each of the data sets. The comparison suggests that a level-resolved R-Matrix with Pseudo-States
calculation is required to model Ne spectral emission in low temperature, high density plasmas.
5
CHAPTER 2
COLLISIONAL-RADIATIVE MODEL
2.1 Introduction
To produce a modeled spectrum one needs to account for all the populating mechanisms in
a collisional-radiative model. Our application of collisional-radiative theory to the calculation of
excited populations is based on the Atomic Data Analysis Structure (ADAS) suite of codes to our
population and emission modeling [11]. These codes are based on the collisional-radiative theory
first developed by Bates, Kingston, and McWhirter [1] in 1962, and later generalized by Summers
and Hooper [12, 13]. Supplementary details related to the collisional-radiative formalism can be
found in Burgess and Summers [14]. The method aims to encompass both the low density coronal
and the high density Local Thermodynamic Equilibrium (LTE) description of an ion, and to track
the shifting balance between radiative and collisional processes. The ion consists of a set of levels
with radiative and collisional couplings. Ionization and recombination to and from metastables of
the next ionization stage (i.e. the plus ion stage) are included.
6
2.2 Atomic Processes
There are many kinds of processes that play a role in populating a level. In order to accurately
build the collisional-radiative model we must account for each atomic process that contributes to
the population in an individual level. These include (but are not limited to)
? Spontaneous decay (Ai?j)
? Auger rate (Aai??)
? Electronic collisional excitation/de-excitation (qei?j/ qej?i)
? Ionization (Si??)
? Recombination: radiative (?ri ), dielectronic (?di ), and three-body (?3i )
Figure 2.1: Populating processes for the ith atomic level.
Figure 2.1 illustrates some of the processes that contribute to the population for the ith atomic
level. Whereirepresents the specific level we describe, j represents any higher or lower energy level
than i, and ? denotes the ground and metastable indices of the z + 1 ion stage. Notice that other
processes such as charge exchange or proton collisions can be included in the collisional-radiative
formalism. For this work we will ignore both of these processes.
7
2.3 Equations and Matrix Representation
The ion consists of a set of levels with radiative and collisional couplings. The time dependence
of the population Ni of an arbitrary level i, in ion stage +z is given by the next set of coupled
differential equations
dNi
dt =
summationdisplay
?
neNz+1? (?ri +?di +ne?3i) +
summationdisplay
j*i
Nj(neqej?i +Aj?i)
?Ni
braceleftBigsummationdisplay
j>i
neqei?j +
summationdisplay
j**j
Ai?j +ne
summationdisplay
jnegationslash=i
qei?j +
summationdisplay
?
neSi? +
summationdisplay
?
Aai?
parenrightbigg
(2.4)
8
and a composite recombination coefficient ri? = ?ri +?di +Ne?3i . This way we can rewrite
equation (2.2) s
summationdisplay
j
CijNj = dNidt ?
summationdisplay
?
neNz+1? ri? (2.5)
or in the matrix form
?
??
??
??
??
C11 C12 ... C1N
C21 C22 ... C2N
... ... ... ...
CN1 CN2 ... CNN
?
??
??
??
??
?
?
??
??
??
??
N1
N2
...
NN
?
??
??
??
??
=
?
??
??
??
??
??
?
dN1
dt ?
summationdisplay
?
neNz+1? r1?
dN2
dt ?
summationdisplay
?
neNz+1? r2?
...
dNN
dt ?
summationdisplay
?
neNz+1? rN?
?
??
??
??
??
??
?
(2.6)
where we define C as the collisional-radiative matrix. In order to solve the system we can
simplify it considerably by taking into account the timescales of the system.
2.4 Timescales
For typical plasma conditions for TOKAMAK or Helicon devices, the excited levels have
extremely fast radiative decay rates while the ground and metastable levels have much longer life-
times. The excited levels can be assumed to be in instantaneous equilibrium with the ground and
metastable populations. This is called the quasi-static approximation and results in all but the ground
and metastable rate of change of populations being set to zero in equation (2.6). This allows the
calculation of the excited populations to be split into two parts; an ionization balance calculation to
work out the ground and metastable populations of each ion stage, and an excited level population
calculation for the levels within a given ion stage.
9
2.4.1 Plasma Timescales
It can be shown [15] that the particle self-collision time is given by
? = 0.12 1?ca2
o
parenleftbigg m
me
parenrightbigg1/2parenleftbiggkT
e
IH
parenrightbigg3/2 1
Nz4 ln? (2.7)
where ? = 12?ne?3D and ?D is the Debye length. From equation (2.7) we get the relative
collision times
?ee : ?ii : ?ie = 1 : 1z4
parenleftbiggm
i
me
parenrightbigg1/2parenleftbiggT
i
Te
parenrightbigg3/2
: 1z2
parenleftbigg?
6
parenrightbigg1/2parenleftbiggm
i
me
parenrightbigg
(2.8)
Table 2.1 shows some approximate collision time values for coronal, as well as fusion divertor
plasmas
Time (sec) Solar Corona Fusion Plasma
ne = 5 ? 108 cm?3 ne = 1 ? 1013 cm?3
Te ? 106 K Te ? 1 keV
?ee 0.18 3 ? 10?4
?ii 8 1.3 ? 10?2
?ie 200 0.5
Table 2.1: Typical collision time values.
In all of our modeling we will assume that the free electrons have a Maxwellian distribu-
tion, thus all of our electron-impact excitation effective collision strengths will be generated for
Maxwellian free electrons.
10
2.4.2 Atomic Timescales
The relaxation timescales of an excited level i, can be estimated from
?o ? 1summationdisplay
j
neqi?j +
summationdisplay
j
Ai?j
(2.9)
Taking Ai?j ? 108(z + 1)4 we can approximate equation (2.9) by
?o ? 10
?8
(z + 1)4 (2.10)
Since the ground state cannot radiatively decay, therefore its time scale is determined by ion-
ization
?g ? 10
7
ne (z + 1)
2
parenleftbiggI
H
kT
parenrightbigg1/2
exp(?/kT) (2.11)
where ? is the ionization potential. If an excited state?s radiative routes to lower levels all
have low radiative transition probability (e.g. spin changing transitions) then it is classified as a
metastable and ?m ? ?g. Bound states with energy above the ionization limit may autoionize via
interaction with the continuum. These states have extremely short lifetimes ?a ? 10?12 sec. Thus,
we have the following timescale relations
?a ??o ??m ??g (2.12)
11
2.5 Solution of the CR Matrix
Taking into consideration the relaxation time of the ground and metastables in comparison with
the relaxation time from any other excited levels, we make use of the quasi-static approximation.
Therefore, the excited levels reach equilibrium much faster than the ground/metastable due to all the
excitations, de-excitations, and all the other processes that take place to contribute to their popula-
tion. Taking into account an m number of metastables (including the ground state), we can rewrite
some of the time derivatives in equation (2.6) subject to the conditions dN?dt negationslash= 0 for 1 ? ? ? m,
and dNidt = 0 for i>m, therefore we get
?
??
??
??
??
??
??
??
??
??
C11 C12 ... C1N
C21 C22 ... C2N
... ... ... ...
Cm1 Cm2 ... CmN
Cm+11 Cm+12 ... Cm+1N
... ... ... ...
CN1 CN2 ... CNN
?
??
??
??
??
??
??
??
??
??
?
?
??
??
??
??
??
??
??
??
??
N1
N2
...
Nm
Nm+1
...
NN
?
??
??
??
??
??
??
??
??
??
=
?
??
??
??
??
??
??
??
??
??
??
??
??
dN1
dt ?
summationdisplay
?
neNz+1? r1?
dN2
dt ?
summationdisplay
?
neNz+1? r2?
...
dNm
dt ?
summationdisplay
?
neNz+1? rm?
?
summationdisplay
?
neNz+1? rm+1?
...
?
summationdisplay
?
neNz+1? rN?
?
??
??
??
??
??
??
??
??
??
??
??
??
(2.13)
Setting the time dependence of the excited levels to zero, allows the population of an ?ordinary?
level to be determined as a function of the ground and metastable populations of the Z ion stage
(N?), and of the Z = 1 ion stage (NZ+1? ). In order to achieve this goal we want to eliminate
any time dependence on our system of equations. We can do this in just few steps by eliminating
unnecessary time dependent differential equations. We start by rearranging equation (2.13).
12
We subtract every term that is multiplied by the first metastable state N1 from the LHS, and
add it to the RHS of equation (2.13). From here we get
?
??
??
??
??
??
??
??
??
??
C12 ... C1N
C22 ... C2N
... ... ...
Cm2 ... CmN
Cm+12 ... Cm+1N
... ... ...
CN2 ... CNN
?
??
??
??
??
??
??
??
??
??
?
?
??
??
??
??
??
??
??
?
N2
...
Nm
Nm+1
...
NN
?
??
??
??
??
??
??
??
?
=
?
??
??
??
??
??
??
??
??
??
??
??
??
dN1
dt ?
summationdisplay
?
neNz+1? r1? ?C11N1
dN2
dt ?
summationdisplay
?
neNz+1? r2? ?C21N1
...
dNm
dt ?
summationdisplay
?
neNz+1? rm? ?Cm1N1
?
summationdisplay
?
neNz+1? rm+1? ?Cm+11N1
...
?
summationdisplay
?
neNz+1? rN? ?CN1N1
?
??
??
??
??
??
??
??
??
??
??
??
??
(2.14)
Since we are only solving for the excited levels, we extract from equation (2.14) the differential
equation for the first metastable state N1
Nsummationdisplay
n=2
C1nNn = dN1dt ?
summationdisplay
?
neNz+1? r1? ?C11N1 (2.15)
13
therefore we obtain our equation in the reduced form
?
??
??
??
??
??
??
??
?
C22 ... C2N
... ... ...
Cm2 ... CmN
Cm+12 ... Cm+1N
... ... ...
CN2 ... CNN
?
??
??
??
??
??
??
??
?
?
?
??
??
??
??
??
??
??
?
N2
...
Nm
Nm+1
...
NN
?
??
??
??
??
??
??
??
?
=
?
??
??
??
??
??
??
??
??
??
?
dN2
dt ?
summationdisplay
?
neNz+1? r2? ?C21N1
...
dNm
dt ?
summationdisplay
?
neNz+1? rm? ?Cm1N1
?
summationdisplay
?
neNz+1? rm+1? ?Cm+11N1
...
?
summationdisplay
?
neNz+1? rN? ?CN1N1
?
??
??
??
??
??
??
??
??
??
?
(2.16)
By repeating the same procedure m?1 number of times for the rest of metastables we end up
reducing the equation to the form
?
??
??
?
Cm+12 ... Cm+1N
... ... ...
CN2 ... CNN
?
??
??
?
?
?
??
??
?
Nm+1
...
NN
?
??
??
?
=
?
??
??
??
??
?
summationdisplay
?
neNz+1? rm+1? ?
msummationdisplay
?=1
Cm+1?N?
...
?
summationdisplay
?
neNz+1? rN? ?
msummationdisplay
?=1
CN?N?
?
??
??
??
??
(2.17)
Finally, we can easily solve equation (2.17) and get the solution for the population of the jth
?ordinary? level in the form
Nzj = ?
summationdisplay
?
summationdisplay
i
C?1ji(r)Ci?Nz? ?
summationdisplay
?
summationdisplay
i
C?1ji(r)ri?Nz+1? ne (2.18)
14
Notice that the inverse matrix in equation (2.18) C?1ji(r) is not the inverse of the collisional-
radiative matrix defined by equation (2.6), but rather the inverse of the reduced collisional-radiative
matrix from equation (2.17). As equation (2.18) shows, the solution for the equilibrium population
for any ?ordinary? level depends upon the ?known? population of the ground and metastable levels.
In order to get the ground and metastable population we calculate the ionization balance (see section
2.6) for the specified ion stage. We notice that in general these ground and metastable population
could be generated from a non-equilibrium ionization balance calculation, with account taken for
plasma transport effects. For the modeling of the ASTRAL plasma presented later in this disserta-
tion, it will be shown that an equilibrium ionization balance calculation will be sufficient. Figure 2.2
shows the results of the population dependence on metastables ( NineN1 ) as a function of electron den-
sity for the Ar+ ion. We can clearly see the three different regimes (Coronal, Collisional-Radiative,
and LTE). Our ASTRAL plasma has an electron density of ? 1 ? 1012 cm?3, and is clearly in the
collisional-radiative regime.
Coronal Collisional?Radiative LTE
Figure 2.2: Normalized level population dependence with respect to electron density.
15
2.6 Ionization Balance Calculations
In order to solve our collisional-radiative model we need to know the ground and metastable
populations of the ions of an element in equilibrium in a thermal plasma. Consider an element X of
nuclear charge zo, the populations of the ionization stages are denoted by
Nz : z = 0,...,zo (2.19)
When considering the z ion stage we include in the calculation its own ionization Sz?z+1, and
only the ionization from the lowest ion stage Sz?1?z. When talking about recombination we will
only include those from the adjacent ion stages; ?z?z?1, ?z+1?z. The time dependence of the
ionization stage populations is given by
dNz
dt =neSz?1?zNz?1 ?
parenleftBig
neSz?z+1 +ne?z?z?1
parenrightBig
Nz
+ne?z+1?zNz+1 (2.20)
Rewriting equation (2.20) into matrix form we obtain
?
??
??
??
??
??
??
?neS0?1 ne?1?0 0 0 ... 0
neS0?1 ?ne
parenleftBig
S1?2 +?1?0
parenrightBig
ne?1?0 0 ... 0
0 neS1?2 . . ... .
... ... ... ... ... ...
0 0 . . ... .
?
??
??
??
??
??
??
?
?
??
??
??
??
??
?
N0
N1
N2
...
Nzo
?
??
??
??
??
??
?
=
?
??
??
??
??
??
?
dN0
dt
dN1
dt
dN2
dt
...
dNzo
dt
?
??
??
??
??
??
?
(2.21)
16
subject to the normalization condition
NTot =
zosummationdisplay
z=0
Nz (2.22)
where NTot is the number density of ions of element X in any ionization stage. Including
equation (2.22) into equation (2.21) and adding an extra column of zeros into the matrix to keep it
square we get
?
??
??
??
??
??
??
??
?
?neS0?1 ne?1?0 0 ... 0 0
neS0?1 ?ne
parenleftBig
S1?2 +?1?0
parenrightBig
ne?1?0 ... 0 0
0 neS1?2 . ... . 0
... ... ... ... ... ...
0 0 . ... . 0
1 1 1 ... 1 0
?
??
??
??
??
??
??
??
?
?
?
??
??
??
??
??
??
??
?
N0
N1
N2
...
Nzo
0
?
??
??
??
??
??
??
??
?
=
?
??
??
??
??
??
??
??
?
dN0
dt
dN1
dt
dN2
dt
...
dNzo
dt
NTot
?
??
??
??
??
??
??
??
?
(2.23)
In equilibrium ionization balance, the time derivatives are set to zero in equation (2.23). We
write the solution of our ionization balance in the form
?
??
??
??
??
??
??
??
?
N0
N1
N2
...
Nzo
0
?
??
??
??
??
??
??
??
?
=
?
??
??
??
??
??
??
??
?
?neS0?1 ne?1?0 +nHC1?0 0 ... 0 0
neS0?1 ?ne
parenleftBig
S1?2 +?1?0
parenrightBig
ne?1?0 ... 0 0
0 neS1?2 . ... . 0
... ... ... ... ... ...
0 0 . ... . 0
1 1 1 ... 1 0
?
??
??
??
??
??
??
??
?
?1
?
?
??
??
??
??
??
??
??
?
0
0
0
...
0
NTot
?
??
??
??
??
??
??
??
?
(2.24)
17
Finally, since we don?t know the exact value for NTot, we solve for the equilibrium fractional
abundances Nz/NTot at a set of temperatures and densities.
?
??
??
??
??
??
??
??
?
N0
NTot
N1
NTot
N2
NTot
...
Nzo
NTot
0
?
??
??
??
??
??
??
??
?
=
?
??
??
??
??
??
??
??
?
?neS0?1 ne?1?0 0 ... 0 0
neS0?1 ?ne
parenleftBig
S1?2 +?1?0
parenrightBig
ne?1?0 ... 0 0
0 neS1?2 . ... . 0
... ... ... ... ... ...
0 0 . ... . 0
1 1 1 ... 1 0
?
??
??
??
??
??
??
??
?
?1
?
?
??
??
??
??
??
??
??
?
0
0
0
...
0
1
?
??
??
??
??
??
??
??
?
(2.25)
We notice that the above equations can be easily extended to include contributions from charge
exchange and proton collisions. These will be negligible processes in our plasmas and will not be
considered. Also the above equation are for the ?stage-to-stage? ionization balance and do not re-
solve metastables within an ion stage. The equation can easily be generalized to include metastables
with the introduction of ?cross coupling? coefficients [13].
18
Figures 2.3 and 2.4 show the results for the ionization balance calculations of Ar, and Ne for
the neutrals and the first four ion stages.
Figure 2.3: Ionization balance of Ar (ne = 1011 cm?3).
Figure 2.4: Ionization balance of Ne (ne = 1011 cm?3).
19
CHAPTER 3
ATOMIC STRUCTURE DATA
3.1 Introduction
Spectroscopy studies of the light emitted (or absorbed) by atoms and ions depend upon an
accurate description of the collisional processes involved. In order to compute accurate collision
cross-sections for the different atomic processes, the accuracy of the atomic structure of the atom
(or ion) is essential. As we saw in chapter 2.2, these processes depend upon plasma temperature,
density, and plasma conditions. In order to compute high quality collision data we make use of the
AUTOSTRUCTURE [2], GASP [3], and the LAMDA (see section 3.4) set of codes to compute and
optimize our atomic structure that we later use to calculate the collisional atomic data we need. In
this chapter we do not intend to cover the whole complexity of the atomic structure computation
process, there are many text books that cover this subject extensively [16, 17]. We will focus on
giving a general overview of it, and the process of optimization of the atomic structure.
20
3.2 Atomic Structure
For an atom with atomic number Z, and containing N electrons where 1 ? N ? Z, and
ignoring the spin-orbit interaction, the general Hamiltonian operator is given by
H =
summationdisplay
i
braceleftbigg
??2i ? 2(Z ?N)r
i
+
summationdisplay
i>j
2
rij
bracerightbigg
(3.1)
whereri = |vectorri|is the distance to theith electron from the nucleus, rij = |vectorri?vectorrj|is the distance
between the ith and jth electrons, and the summation over i>j is over all pairs of electrons. The
distances are given inBohr units (ao), and energies inRydbergs. Plugging this Hamiltonianinto
the time-independent Schr?odinger equation (Hi?i = Ei?i) for each individual electron we get
braceleftbigg
? d
2
dr2i +
li(li + 1)
r2i +V(ri)
bracerightbigg
?i = Ei?i (3.2)
where we define the effective potential V(ri) as
V(ri) = 2(Z ?N)r
i
+
summationdisplay
i>j
2
rij (3.3)
and ?i is the solution for each individual electron. As we see in equation (3.3), if we set
to zero the interaction potential between electrons we reduce the problem to a simple hydrogenic
model which we can solve analytically for each electron. The electron-electron interaction term
means that an analytic solution is no longer possible and an exact numerical solution is still very
difficult. Therefore we simplify the problem by the use of approximations.
21
Using the expansion of moments for the electrostatic repulsion among the electrons, we rewrite
the interaction potential as
1
rij =
1
|vectorri ? vectorrj| =
?summationdisplay
l=0
rl<
rl+1> Pl(u) (3.4)
with u being the cosine of the angle between vectorri and vectorrj, and r>, r< are the greater or the
lesser of ri and rj. Let us now consider the monopole moment, and also assuming that all the other
electrons are represented as a uniform and spherical electronic cloud of density ?(r), and radius ro,
the effective potential is given by
V(ri) = ?2(Z?N)r
i
+
integraldisplay ro
0
2
r>?(r)4?r
2dr (3.5)
This is the Thomas?Fermi potential for a test charge electron at a distance ri from the
nucleus. Let us now consider the dipole and quadrupole moments, in which case the effective
potential is given by
V(ri) = ?2(Z ?N)r
i
+
integraldisplay ro
0
?(rj)
bracketleftbigg 2
r> +C1
r<
r2> +C2
r2<
r3>
bracketrightbigg
4?r2jdrj (3.6)
where C1 and C2 contain the dipole and quadrupole angular terms that depend on the angular
positions vectorri and vectorrj of every pair of electrons of the atom. Under the assumption of uncorrelated
single-electron wave functions these C1 and C2 terms are zero. However, if one allows for corre-
lation, electrons classically tend to be at opposite sides of the nucleus due to their repulsive nature,
and the C1 and C2 terms are finite [18].
22
They are referred as electron correlation terms. Terms beyond the monopole are small and can be
treated as perturbations, thus
Vc(ri) ?V(ri) + C1r2
i
integraldisplay ri
0
?(rj)r3jdrj +C1ri
integraldisplay ro
r
?(rj)drj
+ C2r3
i
integraldisplay ri
0
?(rj)r4jdrj +C2r2i
integraldisplay ro
ri
?(rj)
rj drj (3.7)
This is the Thomas?Fermi?Dirac potential, and by taking into account the contribution
of electron exchange, the charge density includes additional terms
?(r) = 12?2
braceleftBigg
1
? +
bracketleftbigg 1
?2 +Vo ?V(r)
bracketrightbigg1/2bracerightBigg3
(3.8)
where
Vo = ?1516?2 ? 2(Z ?N)r
o
On the other hand, the Thomas?Fermi?Dirac?Amaldi (TFDA) potential introduces
variational scaling parameters (?s) to the potential of the form
VTFDA(ri;?) = VTFD(ri/?) (3.9)
23
In this way, the corrected potential Vc can be computed numerically by writing ?(r) in terms
of V(r) by means of equation (3.8) while the electron correlation coefficients are determined vari-
ationally on the eigenenergies of the system, i.e.
Vc(r) =V(r) +? 83?
bracketleftbigg 1
r2
integraldisplay r
0
?(r2)r32dr2 +
integraldisplay ro
r
?(r2)dr2
bracketrightbigg
+? 83?
bracketleftbigg 1
r2
integraldisplay r
0
?(r2)r42dr2 +r2
integraldisplay ro
r
?(r2)
r2 dr2
bracketrightbigg
(3.10)
This potential is included in the code AUTOSTRUCTURE v.18 [2], an extension of the pro-
gram SUPERSTRUCTURE [19]. In section 3.4 we employ the TFDA with scaling parameters ?nl
being determined variationally by a Singular Value Decomposition (SVD) method for each orbital.
Having defined our effective potential Vc by equation (3.10), we now proceed to solve equation
(3.2) for each individual electron by giving solutions in the form
?i(vectorri) = Pnili(ri)r
i
Ymili (?i,?i)?msi(Siz) (3.11)
where Pnili is the radial function to be computed, Ymili are the spherical harmonics, and ?msi
is the term arising from the spin coordinates. Since the effective potential Vc varies due to the
screening of the nuclear charge, we need to calculate a new potential for each different electron
when calculating its wave function. We also apply the Pauli exclusion principle by including the
exchange of two electrons with antisymmetrized product functions.
24
For a given k configuration, with an N number of electrons, we get
?k = parenleftbig?1(vectorr1),?2(vectorr2),...,?N(vectorrN)parenrightbig (3.12)
where
parenleftbig?
1(vectorr1),?2(vectorr2),...,?N( vectorrN)
parenrightbig = 1?
N!
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
?1(vectorr1) ?1(vectorr2) ... ?1( vectorrN)
?2(vectorr1) ?2(vectorr2) ... ?2( vectorrN)
... ... ... ...
?N(vectorr1) ?N(vectorr2) ... ?N( vectorrN)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
(3.13)
With the properties that if ?m = ?n then ?k = 0, and if vectorrm = vectorrn then ?k = 0. For a certain
configuration k we get
?k = 1?N!
summationdisplay
p
(?1)p?1( vectorrj1)?2( vectorrj2),...,?N( vectorrjN) (3.14)
withpbeing the number of permutations that the electrons can take in the specific configuration
k. We now take the problem of determining the quantitative form of the radial factors Pnili(ri)
appearing in equation (3.11). By plugging equation (3.11) into equation (3.2) we get
braceleftbigg
? d
2
dr2i +
li(li + 1)
r2i +V
c(ri)
bracerightbigg
Pnili(ri) = EilPnili(ri) (3.15)
25
These equations may be solved by numerical or analytic methods. In the numerical case the
radial equations are treated as a set of coupled integro-differential equations in which the radial-
wave functions Pnili(ri) are variables. Because the radial wave functions must obey the boundary
conditions Pnili(0) = 0, and limr??Pnili(r) = 0. A solution can thus be obtained numerically by
integrating to self-consistency and the boundary conditions satisfied. In the analytic case the radial
functions are expressed in an analytic form. The standard approach is to express each radial-wave
function in terms of normalized radial basis functions called Slater-type orbitals [17]
Pnili(r) =
summationdisplay
j
bij (2?ij)
lij+1/2
radicalbig(2l
ij)!
rlije??ijr (3.16)
Also in order to fine tune our structure as in the case of the TFDA potential, we could also
include scale factors ?s in order to shift the radial position of the orbitals
Pnili(r) =
summationdisplay
j
bij (2?ij)
lij+1/2
radicalbig(2l
ij)!
(?r)lije??ij(?r) (3.17)
The AUTOSTRUCTURE code also includes the option of Slater-type orbitals with scaling
parameters.
26
3.3 Atomic Structure Codes
3.3.1 AUTOSTRUCTURE
AUTOSTRUCTURE [2] is a general program for the calculation of atomic and ionic energy
levels, radiative and autoionization rates, and photoionization cross-sections in (LS), or interme-
diate coupling using non-relativistic (IC, LS) or semi-relativistic (ICM, MVD) wavefunctions. It
is in effect a superset of the code SUPERSTRUCTURE [19] on which it was initially based. The
nuclear charge and the level of accuracy desired determines whether (IC), (LS), (ICM), or (MVD)
coupling should be used. The configurations to be chosen include those for which data is wanted,
plus (optionally) additional configurations to improve the accuracy of the structure by including
the Configuration Interaction (CI). This defines a unique angular algebra problem. The CI ex-
pansion is related closely to the choice of radial functions. The better the choice of radial func-
tions, the smaller the CI expansion required to obtain a given level of accuracy, which in turn
leads to a smaller computational problem. Each nl radial function is calculated in a model poten-
tial Thomas?Fermi?Dirac?Amaldi (TFDA) or Slater-Type-Orbital (STO). Both include
optional scaling parameters to fine tune the atomic structure. These scaling parameters can be op-
timized automatically by minimizing a weighted sum of term energies chosen by the user. The
(IC), (LS), (ICM), or (MVD) Hamiltonian is diagonalized to obtain eigenenergies and eigenvec-
tors with which to construct the transition rates. AUTOSTRUCTURE has also the capability to
calculate collision strengths ?ijs by using the Plane Wave Born approximation (see chapter 4.2).
27
3.3.2 GASP
GASP (Graphical AutoStructure Package) [3] is a java front end to the atomic structure pack-
age AUTOSTRUCTURE [2], developed under the Rollins College Student-Faculty Collaborative
Scholarship Research program (Loria, Blossey, Ballance and Griffin). The purpose of this program
is to offer a graphical user interface to run the AUTOSTRUCTURE code. As an example, we have
chosen the input file for the computation of the structure of the case of the Ar2+ ion. The typical
input file (das file) for the AUTOSTRUCTURE code requires the list of the orbitals as well as each
configuration, the advantage of GASP is that it generates the configurations automatically by pro-
moting the electrons in the different specified sub-shells. The typical AUTOSTRUCTURE input
file for a case of the Ar2+ ion is shown in figure 3.1
Figure 3.1: AUTOSTRUCTURE input file for a case of the Ar2+ ion.
28
As we can see from figure 3.1, we need to have knowledge of the different input parameters
that we specify for the calculation. We show in figure 3.2 the graphical interface of GASP to com-
pute the same case shown in figure 3.2.
Figure 3.2: GASP interface showing a computation for a case of the Ar2+ ion.
29
3.4 Atomic Structure Optimization
3.4.1 Introduction
Cross-section computations as well as radiative and autoionization rates, oscillator strengths,
energy levels, photoionization cross-sections, and many other important atomic quantities depend
on the basic foundation of a good atomic structure. Accurate collisional-radiative modeling depends
on such atomic quantities and plays a key role in the development of diagnostics used in fusion as
well as astrophysical plasmas. There are many challenges arising when trying to calculate good
atomic structure, especially when it comes to near neutral systems. The collision calculations can
grow very large as we include more and more configurations in our system, this is particularly true
for R-Matrix calculations. Several codes for atomic structure have been developed over the years
[20]. We are working with the AUTOSTRUCTURE [2, 3] code to generate radial files that will be
used for collision cross-section calculations using the R-Matrix codes [21]. A lot of time and effort
is spend in the optimization task of the orbitals in AUTOSTRUCTURE by varying the scale factors
(see section 3.2). This process is quite challenging since it requires experience in atomic structure
calculation to know which orbitals are to be varied. It could be a lengthy and tedious process. To
get an idea whether our structure is good or not we compute energy levels and line strengths, and
then we compare them to those available in accurate data sources such as the ones found in the
NIST tables [22]. Our aim is to provide an automatic way to compute the optimal structure by
linearizing the dependence of the energy levels Ei, and line strengths Sij to the scale factors (?s),
and from there to solve the inverse of the normalized Jacobian matrix by using Singular Value
Decomposition (SVD) [4] to get the optimal scale factors that we need.
30
3.4.2 Linearization of the Model
Linearization methods are widely used in many fields of physics [23]. We are using the same
concepts to optimize our atomic structure. In order to monitor and compare the quality of our
atomic structure we make use of the NIST atomic database. The selected quantities we use are the
NIST energies (levels or terms) Einist, and either the line strengths Sijnist, the oscillator strengths
fijnist, or Einstein?s Ajknist coefficients. In our modeling we will use line strengths instead
of oscillator strengths due to their independence to the energy of the transitions. Since both the
energies and line strengths depend upon the scale factors (?s), we linearize both and approximate
the relation between the NIST quantities and our modeled values as Enist?Emodel(?)+ ?E????, and
Snist?Smodel(?) + ?S????. We rearrange these equations as
?E = Enist ?Emodel(?)??E???? (3.18)
?S = Snist ?Smodel(?)??S???? (3.19)
In order to be able to include both of these different quantities in our optimization, we normal-
ize both of them by their respective NIST values, therefore
?E
Enist?
1
Enist
?E
???? (3.20)
?S
Snist?
1
Snist
?S
???? (3.21)
31
This way we rewrite our complete model for any n number of energies, any m number of line
strengths, and any l number of scale factors as
?
??
??
??
??
??
??
??
??
??
??
??
?E1
Enist1
?E2
Enist2
...
?En
Enistn
?S1
Snist1
?S2
Snist2
...
?Sm
Snistm
?
??
??
??
??
??
??
??
??
??
??
??
?
?
??
??
??
??
??
??
??
??
??
??
??
1
Enist1
?E1
??1
1
Enist1
?E1
??2 ...
1
Enist1
?E1
??l
1
Enist2
?E2
??1
1
Enist2
?E2
??2 ...
1
Enist2
?E2
??l
... ... ... ...
1
Enistn
?En
??1
1
Enistn
?En
??2 ...
1
Enistn
?En
??l
1
Snist1
?S1
??1
1
Snist1
?S1
??2 ...
1
Snist1
?S1
??l
1
Snist2
?S2
??1
1
Snist2
?S2
??2 ...
1
Snist2
?S2
??l
... ... ... ...
1
Snistm
?Sm
??1
1
Snistm
?Sm
??2 ...
1
Snistm
?Sm
??l
?
??
??
??
??
??
??
??
??
??
??
??
?
?
??
??
??
??
??1
??2
...
??l
?
??
??
??
??
(3.22)
We rewrite the model in vector notation as ?P?M???. Where we have defined ?P as the
normalized vector for the difference of quantities, M as the normalized Jacobianmatrix, and??
as the correction scale factors vector. We write the solution of the vector of the correction of the
scale factors as
?
??
??
??
??
??1
??2
...
??l
?
??
??
??
??
?
?
??
??
??
??
??
??
??
??
??
??
??
1
Enist1
?E1
??1
1
Enist1
?E1
??2 ...
1
Enist1
?E1
??l
1
Enist2
?E2
??1
1
Enist2
?E2
??2 ...
1
Enist2
?E2
??l
... ... .. . ...
1
Enistn
?En
??1
1
Enistn
?En
??2 ...
1
Enistn
?En
??l
1
Snist1
?S1
??1
1
Snist1
?S1
??2 ...
1
Snist1
?S1
??l
1
Snist2
?S2
??1
1
Snist2
?S2
??2 ...
1
Snist2
?S2
??l
... ... .. . ...
1
Snistm
?Sm
??1
1
Snistm
?Sm
??2 ...
1
Snistm
?Sm
??l
?
??
??
??
??
??
??
??
??
??
??
??
?1
?
?
??
??
??
??
??
??
??
??
??
??
??
?E1
Enist1
?E2
Enist2
...
?En
Enistn
?S1
Snist1
?S2
Snist2
...
?Sm
Snistm
?
??
??
??
??
??
??
??
??
??
??
??
(3.23)
32
Or also rewritten as???M?1??P. Having the correction for the scale factors, we use them
to obtain the new scale factors?new =?old+??. With these new scale factors we recompute our
model and compare it again with the NIST quantities, and rerun the process over again. To compare
the success of our optimization process, we compare the initial and final values for the least square
?2, which is given by ?2 = ?P21 +?P22 +...+?P2N+M.
3.4.3 Inverse Matrix Computation
One of the problems we face in trying to compute the inverse of the normalized (n+m)?l
Jacobianmatrix M, is that may not be square (n+m) negationslash=l, and we also run the risk that it may
be singular. To overcome these problems we get the closest solution of the inverse by decomposing
the matrix using Singular Value Decomposition [4], therefore we can express the matrix M as
M = U?S?VT (3.24)
where U is a (n+m)?(n+m) unitary matrix, S is a (n+m)?ldiagonal matrix with
non-negative real numbers on the diagonal, and VT denotes the conjugate transpose of V which is
al?l unitary matrix. These matrices have the following properties
? The columns of V form a set of orthonormal ?input? or ?analyzing? basis vector directions
for M.
? The columns of U form a set orthonormal ?output? basis vector directions for M.
? The matrix S contains the singular values, which can be thought of as scalar ?gain controls?
by which each corresponding input is multiplied to give a corresponding output.
33
Since our aim is to minimize the least square ?2, we can do this successfully by computing
the ?inverse? or the pseudo-inverse of the matrix which is defined as
M?1?V?S?1 ?UT (3.25)
The matrix S is a diagonal matrix which contains K singular values. The number of singular
values determines the rank of the matrix, and the singular values are ordered in descendent form
S1 >S2 >...>SK
S =
?
??
??
??
??
??
??
??
??
??
S1 0 0 ... 0 0 ... 0
0 S2 0 ... 0 0 ... 0
... ... ... . .. ... ... ... ...
0 0 0 ... SK 0 ... 0
0 0 0 ... 0 0 ... 0
... ... ... . .. ... ... ... ...
0 0 0 ... 0 0 ... 0
?
??
??
??
??
??
??
??
??
??
(3.26)
TheK rank of the matrix represents the number of ?dimensions?. The difficulty is to select the
P number of singular values that we need to compute this pseudo-inverse where P ? K. Since
we need to compute the vector of the scale factor corrections ??, we choose to apply some physical
restrictions. In order not to affect the different atomic orbitals by too much while we optimize
others, we choose a range of values for the total scale factors, in our case this range is
0.8 ??new =?old +??? 1.2 (3.27)
34
With this restriction in place we select theP number of singular values to compute the singular
values inverse matrix S?1
S?1 =
?
??
??
??
??
??
??
??
??
??
1
S1 0 0 ... 0 0 ... 0
0 1S2 0 ... 0 0 ... 0
... ... ... . .. ... ... ... ...
0 0 0 ... 1S
P
0 ... 0
0 0 0 ... 0 0 ... 0
... ... ... . .. ... ... ... ...
0 0 0 ... 0 0 ... 0
?
??
??
??
??
??
??
??
??
??
(3.28)
There is not a specific method to know how manyP singular values we need to use to compute
the inverse of the singular values matrix. Therefore we use trial and error until we meet condition
(3.27). In case condition (3.27) is not met, then we have to use a different strategy, namely multiply
the Jacobian matrix by a certain factor greater than one and then compute the corrections again.
If condition (3.27) still goes unmet, then again we multiply the Jacobian matrix by a greater
factor and compute the optimization again. The reason why we multiply the Jacobian matrix by
a factor is to increase the value of its derivatives, by increasing these values we reduce the size of
the corrections for the scale factors ??, in order to meet the condition (3.27). We demonstrate this
process better in figure 3.3.
35
Figure 3.3: Representation of the variation of the orbitals with respect to the scale factors ?s.
As we see in figure 3.3, the functionf(?) may represent any of the atomic orbitals that depend
on the scale factor ?, and the ?Jacobian? value or derivative is represented by the value of the
slope of the purple line. If we compute the correction ??1 by using the value of the ?real? deriva-
tive, we would violate condition (3.27), and as shown in the figure we would end up out of the set
boundaries. Therefore, if we multiply the value of the derivative by a factor greater than one, we get
the new increased value for the slope in this example represented by the green line. By using this
new value of the derivative to compute the new correction ??2, we now satisfy condition (3.27),
and our new value for ? is now within the set boundaries.
36
3.4.4 Results
As an example of the optimization process, we have chosen the Ar2+ ion, in which we have
used eleven orbitals and ten configurations. These configurations are
? 1s22s22p63s23p4 (Ground)
? 1s22s22p63s23p33d
? 1s22s22p63s23p34s
? 1s22s22p63s23p34p
? 1s22s22p63s23p34d
? 1s22s22p63s23p34f
? 1s22s22p63s23p35s
? 1s22s22p63s3p5
? 1s22s22p63p6
? 1s22s22p63p53d1
Therefore, we have eleven scaling factors ?s, one for each of the eleven orbitals 1s, 2s, 2p,
3s, 3p, 3d, 4s, 4p, 4d, 4f, and 5s. The initial values for all the ?s is set to one, and the boundaries
are set to satisfy the condition given by equation (3.27). The program will read two input files
corresponding to the ASCII tables from the NIST energy values, and the NIST Einstein?s Ajk
coefficients. The LAMDA code has the option of transforming these coefficients into line strengths
Sjk and then run the optimization.
37
Results of the optimization process for the first iteration are presented in figure 3.4.
Figure 3.4: LAMDA code output file for the Ar2+ ion.
The program starts by running the AUTOSTRUCTURE code for all the scale factors ?s set up
to one (or to any set of initial values). AUTOSTRUCTURE generates output files that give us the
computed energy for each level (or term), and the line strengths. From there we compare our results
with those values found in the NIST files, and compute the least square ?2.
38
This value for the ?2, is the one we use as a reference point for comparison (100%), and its
value is displayed on the first row of the first iteration of the program as shown in figure 3.4. From
that point on, the program starts computing corrections by using the SVD method for each individual
orbital, and recomputing the energies and line strengths by rerunning the AUTOSTRUCTURE code.
Then the code computes corrections for all the orbitals at once and chooses the correction values that
give the minimum ?2. In the case of Ar2+, we see that the best ?2 value was given by including
all the orbitals variations at once. The code then displays the results of the energy optimizations
by displaying the ?Desired Energy? (NIST Energies), and the energy values before and after the
optimization with their respective % differences from those given by NIST. The code does the same
for the line strengths by displaying the type of transition; Electric Dipole (E1), Electric Quadrupole
(E2), Magnetic Dipole (M1), and Magnetic Quadrupole (M2). It shows also the ?Desired Value?
and its respective %error from those given by NIST. The initial and final computed values before and
after the optimization are then shown in the next rows, which also give the % differences compared
from those in NIST. Finally the code prints out the values of the optimized scale factors ?s which
we can use to compute our optimal structure.
39
CHAPTER 4
ATOMIC COLLISION DATA
4.1 Introduction
The accuracy of collision cross-section calculations play a key role in collisional-radiative
modeling. Our collisional-radiative model can only be as good as the underlying atomic data we
use. Many methods and approximations have been developed to solve the problem of scattering
of an electron by an atom or ion. In this chapter we give a general overview of some of the most
widely use methods. We begin in section 4.2 with an explanation of the Plane WaveBornapprox-
imation, where we model the incoming electron by a plane wave and the target as a potential. Plane
Wave Born atomic data will be used for some of our neon modeling in chapter 8. In section 4.3
we will outline Distorted Wave theory in which we allow the incoming wave to be affected by the
potential and therefore distorted. Distorted Wave data for electron-impact ionization cross-sections
is used for argon in chapter 6. Finally we introduce R-Matrix theory in section 4.5, and also an
extension of this method called R-Matrix with Pseudo-States (RMPS), which is used to describe
electron scattering in the intermediate energy range above the ionization limit. Section 4.5 gives a
quick overview onMaxwellianeffective collision strengths calculations.
In scattering problems there are perturbative methods such as the Plane WaveBorn and Dis-
torted Wave. These methods are good for highly charged systems, but for near neutral systems the
R-Matrix and RMPS methods in particular, result in more accurate cross-sections. R-Matrix and
RMPS data for Ar+ excitation is used in chapter 6. In chapter 7 we will present the results of a
R-Matrix calculation for Ar2+ excitation, and in chapter 8 we use R-Matrix and RMPS data for
emission of modeling from neutral neon.
40
4.2 Plane Wave Born
4.2.1 Classical Scattering Theory
Figure 4.1: Scattering of a light particle by a heavy ion.
In terms of classical scattering theory, having a light particle incident on some heavy nucleus
with an incoming energyE, and impact parameterb, the light particle will emerge at some scattered
angle ? as shown by figure 4.1. By assuming for simplicity that the potential is azimuthally sym-
metric (no dependence in ?) therefore we constrain the problem to a single plane. The problem in
scattering theory is reduced to determining the scattering angle?, from the knowledge of the impact
parameter b, and the energy of the incident particle E. In general, having a particle incident within
an infinitesimal cross-sectional area d?, will scatter into a corresponding infinitesimal solid angle
d?. The larger d? is, the bigger d? becomes. The proportionality factor will be called D, which
represents the differential scattering cross-section D = d?d?. In terms of the impact parameter b
and the azimuthal angle ?,d? =bdbd? and d? = sin(?)d?d?,
41
therefore
D(?) = bsin(?)
vextendsinglevextendsingle
vextendsinglevextendsingledb
d?
vextendsinglevextendsingle
vextendsinglevextendsingle (4.1)
By obtaining the differential cross-section D(?), as a function of the angle?, we can compute
the total cross-section by integrating the differential cross-section with respect to the solid angle
d?.
? =
integraldisplay
D(?)d? (4.2)
42
4.2.2 Plane Wave Scattering
Figure 4.2: Scattering of an incident plane wave.
In this case we model the incident ?light? particle as a plane wave ?(z) = Aeikz, traveling
in the z direction. When the plane wave interacts with the potential described by the target, it will
produce an outgoing spherical wave as shown in figure 4.2. Therefore for values ofr far away from
the target we expect solutions to the Schr?odinger equation in the general form
?(r,?) ?A
braceleftbigg
eikz +f(?)e
ikr
r
bracerightbigg
(4.3)
The spherical wave carries a factor of 1/r because |?|2 must go like 1/r2 to conserve prob-
ability. The wave number k is defined in the usual way
k =
?2mE
planckover2pi1 (4.4)
43
It can be shown [24, 25] that the differential cross-section D(?) is related to the scattering
amplitude f(?) by
D(?) = d?d? = |f(?)|2 (4.5)
Therefore the whole problem of determining the cross-section is related to finding the scatter-
ing amplitude f(?). We can see from equation (4.5) that the differential cross-section (which is the
quantity of interest to the experimentalist) is given by the absolute square of the scattering amplitude.
4.2.3 TheBorn Approximation
The time-independent Schr?odinger equation for a given potential V is given by
? planckover2pi12mvector?2?+V? = E? (4.6)
which can be rewritten in the form
(vector?2 +k2)? = Q (4.7)
where k and Q(?) are defined by
k =
?2mE
planckover2pi1 and Q=
2m
planckover2pi12 V? (4.8)
44
As we can see, equation (4.7) has the form of the non-homogeneous Helmholtz equation.
We seek a solution by using Green?s functions to find a solution in the integral form
?(vectorr) =
integraldisplay
G(vectorr?vectorro)Q(vectorro)d3vectorro (4.9)
as shown in appendix A, the final solution of the Green?s function is given by
G(vectorr) = ?e
ikvectorr
4?r (4.10)
by inserting this solution back into equation (4.9) we get
?(vectorr) =?o(vectorr)? m2?planckover2pi12
integraldisplay eik|vectorr?vectorro|
|vectorr?vectorro|V(vectorro)?(vectorro)d
3vectorro (4.11)
where ?o satisfies the free-particle Schr?odinger equation
(vector?2 +k2)?o = 0 (4.12)
Now, for the firstBornapproximation, let us suppose thatV(vectorro) is localized atvectorro = 0, and
that the potential drops to zero outside some finite region. We are interested in calculating ?(vectorr)
at points far away from the scattering center. Then |vectorr| ? |vectorro| for all points that contribute to the
integral equation (4.11) we approximate
|vectorr?vectorro|2 =r2 +r2o ?2vectorr?vectorro ?r2
parenleftbigg
1?2vectorr?vectorror2
parenrightbigg
(4.13)
45
therefore
|vectorr?vectorro| ? r? ?r?vectorro (4.14)
Letvectork =k?r, and keeping the lowest order for the denominator we can approximate |vectorr?vectorro| ?
r, so we get
eik|vectorr?vectorro|
|vectorr?vectorro| ?
eikr
r e
?ivectork?vectorro (4.15)
For the scattering problem we approximate the incident particle to a plane wave, therefore for
an incident plane wave traveling on thez axis we have
?o(vectorr) =Aeikz (4.16)
So, by inserting in the large r approximation and the incident plane wave into equation (4.11)
we get
?(vectorr) ?Aeikz ? m2?planckover2pi12e
ikr
r
integraldisplay
e?ivectork?vectorroV(vectorro)?(vectorro)d3vectorro (4.17)
Which is in the same form of equation (4.3). From it we recognize the scattering amplitude
f(?,?) = ? m2?planckover2pi12A
integraldisplay
e?ivectork?vectorroV(vectorro)?(vectorro)d3vectorro (4.18)
46
We now apply theBornapproximation, we suppose the incoming plane wave is not modified
by the potential, therefore we approximate
?(vectorro) ??o(vectorro) = Aeikzo =Aeivectork??vectorro (4.19)
wherevectork? = k?z, this wave function would be exact if the potential V = 0 which is theBorn
approximation. Plugging this wavefunction inside the integral equation (4.18) we get
f(?,?) ? ? m2?planckover2pi12
integraldisplay
ei(vectork??vectork)?vectorroV(vectorro)d3vectorro (4.20)
wherevectorkandvectork? have the same magnitude. Notice that the firstBornapproximation is just the
Fourier transform of the potential V(vectorro). Sincevectork? = k?z for the incident wave, andvectork = k?r
for the scattered wave therefore planckover2pi1(vectork?vectork?) is the momentum transferred in the process. From this
point we can get the differential cross-section D by means of equation (4.5), and our total cross-
section from equation (4.2). In particular for low energy scattering (long wavelength) (vectork?vectork?) ? 0
therefore we get
f(?,?) ? ? m2?planckover2pi12
integraldisplay
V(vectorr)d3vectorr (4.21)
47
Let us now consider a collision of an electron with a hydrogen atom initially at ground state.
Again, the electron is being considered a light particle by assuming the nucleus to have infinite
mass thus, neglecting the motion of the proton in the collision. The wave equation for the system of
incident electron and hydrogen atom is given by
bracketleftbigg planckover2pi12
2m(?
2
1 +?
2
2)+E+
e2
r1 +
e2
r2 ?
e2
r12
bracketrightbigg
?(vectorr1,vectorr2) = 0 (4.22)
where the subscript 1 is used for the incident electron and 2 for the atomic electron. The total
classical energy of the system is given by the sum of the kinetic energy of the incident electron plus
the energy of the atomic electron in the ground state (E0)
E = 12mv2 +E0 (4.23)
We might guess a solution for equation (4.22) as
?(vectorr1,vectorr2) =?n(vectorr2)Fn(vectorr1) (4.24)
The functions ?n(vectorr2) are the solutions for the hydrogen atom, which satisfy the equation
parenleftbigg planckover2pi12
2m?
2
2 +En +
e2
r2
parenrightbigg
?n(vectorr2) = 0 (4.25)
48
Substituting equation (4.24) into equation (4.22) and using the solution given by equation
(4.25) we get
parenleftbigg planckover2pi12
2m?
2
1 +E+
e2
r1 ?
e2
r12
parenrightbigg
?n(vectorr2)Fn(vectorr1)+
parenleftbigg planckover2pi12
2m?
2
2 +
e2
r2
parenrightbigg
?n(vectorr2)Fn(vectorr1) = 0
parenleftbigg planckover2pi12
2m?
2
1 +E+
e2
r1 ?
e2
r12
parenrightbigg
?n(vectorr2)Fn(vectorr1)?En?n(vectorr2)Fn(vectorr1) = 0
parenleftbigg planckover2pi12
2m?
2
1 +E?En
parenrightbigg
?n(vectorr2)Fn(vectorr1) =
parenleftbigge2
r12 ?
e2
r1
parenrightbigg
?n(vectorr2)Fn(vectorr1)
(4.26)
The hydrogenic wave functions form an orthonormal set
integraldisplay
??n(vectorr2)?n?(vectorr2)dvectorr2 = ?nn? (4.27)
By multiplying equation (4.26) by ??n(vectorr2), and integrating with respect to dvectorr2 we get
parenleftbigg planckover2pi12
2m?
2
1 +E?En
parenrightbigg
Fn(vectorr1) =
integraldisplay parenleftbigge2
r12 ?
e2
r1
parenrightbigg
?(vectorr1,vectorr2)??n(vectorr2)dvectorr2 (4.28)
In the case of r1 ?? the right hand vanishes, and Fn satisfies the wave equation
parenleftbigg
?21 + 2mplanckover2pi12 (E?En)
parenrightbigg
Fn(vectorr1) = 0 (4.29)
49
which is the wave equation for a free particle with energy E ?En. Here we are making
the assumption that the incident electron has enough energy to excite the nth state of the atom or
E>En. Letk2n = 2m(E?En)planckover2pi12 , therefore
(?21 +k2n)Fn(vectorr1) = 0 (4.30)
and its solution must have the asymptotic form given by equation (4.3). Since we are only
interested in high energy impact, the perturbation of the incident particle by the interaction with
the atom is small. Applying the first order Born approximation to F(vectorr1) as a plane wave for
?(vectorr1,vectorr2) we get
?(vectorr1,vectorr2) =eivectork?vectorr1?(vectorr2) (4.31)
Substituting this solution into equation (4.30) we obtain
(?21 +k2n)Fn(vectorr1) = 2mplanckover2pi12
integraldisplay parenleftbigge2
r12 ?
e2
r1
parenrightbigg
eivectork?vectorr1?(vectorr2)??n(vectorr2)dvectorr2 (4.32)
which is in the same form of equation (4.7) and has an integral solution given by equation
(4.9). TheGreen?s function is given by equation (4.6), therefore we get
Fn(vectorr) = m2?planckover2pi12
integraldisplay integraldisplay eikn|vectorr?vectorr1|
|vectorr?vectorr1| e
ivectork?vectorr1
parenleftbigge2
r1 ?
e2
r12
parenrightbigg
?(vectorr2)??n(vectorr2)dvectorr1dvectorr2 (4.33)
50
Approximating by using equation (4.15) we obtain
Fn(vectorr) ? m2?planckover2pi12e
ikr
r
integraldisplay integraldisplay
ei(vectork?vectorkn)?vectorr1
parenleftbigge2
r1 ?
e2
r12
parenrightbigg
?(vectorr2)??n(vectorr2)dvectorr1dvectorr2 (4.34)
By comparing this result with equation (4.3) we recognize that the scattering amplitude fn(?)
is given by
fn(?) = m2?planckover2pi12
integraldisplay integraldisplay
ei(vectork?vectorkn)?vectorr1
parenleftbigge2
r1 ?
e2
r12
parenrightbigg
?(vectorr2)??n(vectorr2)dvectorr1dvectorr2 (4.35)
and therefore by the definition given by equation (4.5), we obtain the differential cross-section
Dn(?) = knk m
2
4?2planckover2pi14
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay integraldisplay
ei(vectork?vectorkn)?vectorr1
parenleftbigge2
r1 ?
e2
r12
parenrightbigg
?(vectorr2)??n(vectorr2)dvectorr1dvectorr2
vextendsinglevextendsingle
vextendsinglevextendsingle
2
(4.36)
We notice that the interaction of an electron with a hydrogen atom is described by the potential
V(vectorr1,vectorr2) = e2r1 ? e2r12 . The differential cross-section may be written in the more compact form
Dn(?) = knk m
2
4?2planckover2pi14|?
vectorknn|V|vectork0?|2 (4.37)
51
In the case of inelastic collisions, we consider the Coulomb collision of an electron with an
atom in which, the atom is raised from the state ni to state nf by the impact. If Eni and Enf are
the energies of the two atomic states andvectorki, andvectorkf are the initial and final momentum vectors of
the colliding electron, the conservation of energy gives
1
2m(v
2
i ?v
2
f) =Enf ?Eni (4.38)
wherevectorv = planckover2pi1vectorkm. Within the range of validity of the first Born approximation, the differential
cross-section describing the collision is given by
D(?) = kfk
i
m2
4?2planckover2pi14|?
vectorkfnf|V|vectorkini?|2 (4.39)
where theCoulomb potential is given by
V(vectorr) = e
2
|vectorr? vectorR| (4.40)
The expression given by equation (4.39), is the differential cross-section under the firstBorn
approximation, and is useful in the case of high energy incident electrons. For high incident energy,
contributions to the cross-section for a wide range of momentum become important. We will use
Plain WaveBorn electron-impact excitation for neutral neon in chapter 8.
52
4.3 Distorted Wave
It was shown in the previous section that the differential cross-section for scattering may be
obtained from
D(?) = kfk
i
m2
4?2planckover2pi14|?
vectorkfnf|V|vectorkini?|2 (4.41)
where the potential V (for a neutral system) is given by
V(vectorr,vectorrj) =
Nsummationdisplay
j=1
e2
|vectorrj ?vectorr| (4.42)
and describes the Coulomb interaction between the incident electron at position vectorr and the
target electrons at positions vectorrj. N is the number of electrons at the target atom. We express the
Coulombpotential in terms of the Fourier integral transform
V(vectorr,vectorrj) = e
2
2?2
Nsummationdisplay
j=1
integraldisplay e?ivectorq?(vectorrj?vectorr)
q2 d
3vectorq (4.43)
where the integration is over all the space and we representvectorqin spherical coordinates (q,?,?).
By letting vectorRj =vectorr?vectorrj, and orienting vectorRj along the z axis, the Fourier integral becomes
e2
2?2
integraldisplay e?ivectorq?(vectorrj?vectorr)
q2 d
3vectorq
= e
2
2?2
integraldisplay ?
0
dq
integraldisplay ?
0
d?
integraldisplay 2?
o
d?sin(?)e?iqRjcos(?)
= e
2
2?2
4?
Rj
integraldisplay ?
0
sin(qRj)
q dq (4.44)
53
and by using the identity
integraldisplay ?
0
sin(ax)
x dx=
?
2sgn(a) (4.45)
Therefore we getintegraltext?0 sin(qRj)q dq = ?2sgn(Rj), since vectorRj =vectorr?vectorrj >0, thensgn(Rj) =
1, and equation (4.44) becomes
Nsummationdisplay
j=1
e2
Rj (4.46)
which is the Coulomb potential V . Expanding the exponential factor in the Fourier inte-
gral, the differential cross-section in equation (4.41) may be written as
D(?) = kfk
i
m2
4?2planckover2pi14
vextendsinglevextendsingle
vextendsinglevextendsingle e2
2?2
integraldisplay
d3vectorq 1q2?vectorkf|e?ivectorq?vectorr|vectorki?angbracketleftbignfvextendsinglevextendsingle
Nsummationdisplay
j=1
eivectorq?vectorrjvextendsinglevextendsingleniangbracketrightbig
vextendsinglevextendsingle
vextendsinglevextendsingle
2
(4.47)
This expression divides neatly into two factors
? The first matrix element deals only with incident electron momentum parameters which are
independent of the state transition involved within the target atom.
? The second matrix element involves the atomic parameters.
54
The expansion of the matrix element in equation (4.41), into two matrix element factors in
equation (4.47) is due to the fact that the solution of the Schr?odinger equation describing the
system is a product of a function of only incident electron coordinates and a function of atomic
electrons? coordinates as described at the end of section 4.2.3. We use the distorted Coulomb
wavefunction to calculate the first matrix element
?vectorkf|e?ivectorq?vectorr|vectorki? (4.48)
and the second matrix element is in the form factor which is taken between the target states
[26]
Fif(vectorq) = angbracketleftbignfvextendsinglevextendsingle
Nsummationdisplay
j=1
eivectorq?vectorrjvextendsinglevextendsingleniangbracketrightbig (4.49)
We call this method Distorted Wave Approximation (DWA). We make use of Distorted Wave
electron-impact ionization data for argon in chapter 6.
55
4.4 R-Matrix Theory
Figure 4.3: Partitioning of configuration space R-Matrix theory.
R-Matrix theory was first introduced in nuclear physics by Wigner and Eisenbud [27] in 1947
in a study of resonance reactions. In the case of electron-atom collisions, the non-perturbative R-
Matrix theory partitions the configuration space into two regions named the internal region, and the
external region as shown in figure 4.3. The internal region r ?a, where r is the coordinate of the
scattered electron relative to the target nucleus, and a is chosen to encompass the charge cloud of
the atom/ion. In this region, the wavefuctions of the scattered electron and the atom or ion overlap.
Therefore electron exchange and correlation between the scattered electron and the N-electron tar-
get atom or ion are important, and the (N + 1)-electron collision complex behaves in a similar
way to a bound state.
In this chapter we do not intend to cover the whole development ofR-Matrix theory, but simply
to emphasize the advantages that this non-perturbative method has when calculating electron-impact
excitation data, and its applications in emission modeling in plasmas. For a more comprehensive
description on the R-Matrix theory refer to Burke et al. [28], or Burke and Berrington [29].
56
For the internal region r ? a, the wave function for the electron-atom collision process is
defined by
?k(x1,...,xN+1) =A
summationdisplay
ij
cijk??i(x1,...,xN,?rN+1?N+1)uij(vectorrN+1)
+
summationdisplay
j
djk?j(x1,...,xN+1) (4.50)
where the functions ??i are formed by coupling the multi-configurational functions ?i. The
uij are the basis orbitals for the scattered electron. The operator A antisymmetrizes the scattered
electron coordinate with theN atomic electron coordinates. The functions?j are (N+1)-electron
configurations formed from the atomic orbitalsPnl(r), and are analogous to theN-electron config-
urations. Finally,cijk anddjk are expansion coefficients determined by diagonalizing the (N+1)-
electron Hamilonian. The coefficients cijk and djk in equation (4.50) are determined by diag-
onalizing
(?k|HN+1|?k?) =EN+1k ?kk? (4.51)
whereHN+1 is the (N +1)-electron Hamiltonianoperator which is projected onto the
space functions ?k. TheR-Matrix is given in the form
Rij = 12a
summationdisplay
k
wik(a)wjk(a)
EN+1k ?E (4.52)
where wik are defined as
wik(r) =
summationdisplay
j
cijkuij(r) (4.53)
57
The surface amplitudes wik(a) and the poles EN+1k of the R-Matrix are obtained directly
from the eigenvectors and eigenvalues of the Hamiltonian matrix defined by equation (4.51).
The most important source of error in this method is the truncation of the expansion in equation
(4.52) to a finite number of terms.
TheR-Matrix acts as an interface between the inner region and the outside region.
In the external region, r>a, electron exchange between the scattered electron and the target
can be neglected. The scattered electron then moves in the long-range multipole coulomb potential
of the target. Outside the R-Matrix box, the total wavefunction for a given (LS) symmetry is
expanded in basis states given by
?N+1k =
summationdisplay
i
?N+1i vi(rN+1)r
N+1
(4.54)
wherevi(r) are radial continuum functions obtained by solution of radial asymptotic coupled
differential equations. The inner and outer solutions are matched at the edge of the R-Matrix box
to extract collision strengths.
58
The R-Matrix method also is able to calculate electron-impact ionization data by introducing
pseudo-states in our atomic structure to represent the highRydberg and continuum states as shown
by figure 4.4.
Figure 4.4: Representation of the continuum by the introduction of pseudo-states.
As we show in chapter 6 and later in chapter 8, these pseudo-states make a significant differ-
ence in the electron-impact excitation cross-sections [6] between standardR-Matrix, andR-Matrix
with Pseudo-States (RMPS). These differences are due to the continuum coupling effects and also
significantly improvement of our atomic structure. The advantage of R-Matrix is that it is a non-
perturbative method. Its limitations are mainly computational. The accuracy of the method depends
on the quality of our atomic structure and the calculation can grow significantly with the addition
of configurations. This limitation is overcome by the paralellization of the method. Therefore the
use of small parallel computer clusters and massively parallel supercomputers when running large
calculations is essential.
59
We make use ofR-Matrix and RMPS data for Ar+ in chapter 6. In chapter 7 we present a new
R-Matrix calculation for electron-impact excitation of Ar2+, and in chapter 8 we useR-Matrix and
RMPS excitation data for neutral neon modeling.
4.5 Effective Collision Strengths
As shown in the previous section, the calculation of accurate collision cross-section calcula-
tions play a very important role in our emission modeling in plasmas. The format in which this data
is made available is also important since it has to be presented to the user in a simple and compact
form. It should also be consistent with no significant error in accuracy. Electron-impact excitation
cross-sections as well as electron-impact ionization cross-sections have a strong dependence with
respect to the energy of the incident electron. By using suitable scaling procedures it is possible
to remove the main asymptotic energy (or temperature) dependence for the given data. The energy
(temperature) is also scaled so as to become a dimensionless variable which ranges from 0 at thresh-
old energy (zero temperature) to 1 at infinite energy (temperature). In this way the whole variation
of a collision strength can be exhibited in a single graph since the energy is mapped onto the interval
(0,1). An introduction of the Burgess-Tully plots is also given by Burgess et al. [5].
60
4.5.1 Collision Strength
In the previous sections we have discussed several methods to calculate electron-impact exci-
tation and ionization cross-sections. For example let us consider the reaction
X+zi (Ei)+e(?i) ?X+zj (Ej)+e(?j) (4.55)
with
?i +Ei = ?j +Ej (4.56)
WhereEi is the energy of the initial level of the ionX+Z, andEj is the energy of the excited
level. The energy of the incident (scattered) electron is given by?i (?i). The reaction is described as
a cross-section as a function of the incident electron energy by ?i?j(?i). By energy concepts the
electron-impact excitation can only occur if the incident electron energy is?i ? ?Eij =Ej?Ei.
It is convenient to introduce the threshold parameter X = ?i/?Eij, with X ? [1,?]. The
cross-section can therefore be expressed in terms of the incident electron energy ?i, the scattered
electron energy?j, or the threshold parameterX. In the literature, in preference to the cross-section
?i?j(?i) it is usual to give the collision strength ?ij since is a dimension-less quantity, and is also
symmetrical between the initial and final statesi,j. It is also a slowly varying quantity with respect
to the incident electron energy. The excitation cross-section ?i?j(?i), de-excitation cross-section
?j?i(?i), and collision strength ?ij, are connected by the following relations
?ij = ?i(?i?Ei)I
H
?i?j(?i?Ei)
?a2o = ?j
(?i?Ej)
IH
?j?i(?i?Ej)
?a2o = ?ji (4.57)
61
The atomic unit of cross-section is ?a2o = 8.7972 ? 10?17 cm2. ?i and ?j are the statistical
weights of the initial and final levels, and IH = 13.6058 eV.
It is shown by Burgess et al. [5] that the collision strengths behave like
Type-1 Electric Dipole (ED) ?ij ? const. ln(?i)
Type-2 Non-ED, No Spin Change ?ij ? const.
Type-3 Spin Change ?ij ? const. ??2i
4.5.2 Effective Collision Strength
As seen in section 2.2, for our collisional-radiative model we make use of the electronic colli-
sional excitation/de-excitation (qei?j/qej?i) rate coefficients, which by assuming aMaxwellian
electron distribution are given by
qei?j(Te) = 4?
parenleftbigg m
e
2?kTe
parenrightbigg3/2integraldisplay ?
0
viexp
parenleftBig
? ?ikT
e
parenrightBig
?i?j(?i)v2idvi (4.58)
The relation between the excitation/de-excitation rate coefficients is given by
qei?j =qej?i
parenleftBig?j
?i
parenrightBig
exp
parenleftBig
? ?EijkT
e
parenrightBig
(4.59)
We now transform fromvi to?j, where?j is the colliding electron kinetic energy after excita-
tion has occurred, and rewrite equation (4.58) as follows
qei?j(Te) = 2?? aoplanckover2pi1m
e?i
parenleftBigIH
kTe
parenrightBig1/2
exp
parenleftBig
? ?EijkT
e
parenrightBig
?ij (4.60)
62
where ?ij is the effective collision strengths which is defined as
?ij =
integraldisplay ?
0
?ij exp
parenleftBig
? ?jkT
e
parenrightBig
d
parenleftBig ?j
kTe
parenrightBig
(4.61)
The advantage of using effective collision strengths over rate coefficients is that they vary very
slowly with respect to electron temperature. This simplifies our need of having a lot of data stored
since we only need a few points and the rest can be interpolated.
Type-1 ?ij ? const. ln(Te)
Type-2 ?ij ? const.
Type-3 ?ij ? const. T?1e
ADAS [11] computes effective collision strengths from our R-Matrix collision strength data
via convolution with aMaxwellianelectron distribution. The data is stored in a default temper-
ature grid in an adf04 file. In order to show the effective collision strengths from threshold to the
infinite energy point on a single plot, we make use of the Burgess-Tully plots [5]. The transforma-
tions introduce an adjustable parameter C. Again the x is defined to be zero when Te = 0, and
unity whenTe = ?.
63
These transformations are defined as
For Type 1
x= 1? ln(C)
ln
parenleftBig
kTe
?Eij +C
parenrightBig
y(x) = ?ij
ln
parenleftBig
kTe
?Eij +C
parenrightBig (4.62)
For Type 2
x=
parenleftBig
kTe
?Eij
parenrightBig
parenleftBig
kTe
?Eij +C
parenrightBig
y(x) = ?ij (4.63)
For Type 3
x=
parenleftBig
kTe
?Eij
parenrightBig
parenleftBig
kTe
?Eij +C
parenrightBig
y(x) =
parenleftBig kTe
?Eij +1
parenrightBig
?ij (4.64)
64
CHAPTER 5
EXPERIMENTAL SETUP
5.1 Introduction
The Auburn Steady sTate Research fAciLity ASTRAL is a helicon device that can generate
intense Ar and Ne plasma columns. It has also been used with He and CO2. ASTRAL Ar and Ne
typical plasma parameters are
? Ne = 1011 - 1013 cm?3
? Te = 2 - 15 eV
? Bfield = 200 - 1300 Gauss
? RFpower lessorsimilar 2 kWatt
The helicon plasma source is a very efficient method for generating high density plasmas using
radio waves. Helicon sources can be very useful for basic plasma studies because there is no large
electric current running through the plasma that can disturb the phenomenon we are trying to study,
and also the antenna is outside the plasma, thus avoiding further contamination to the plasma and
damage to the antenna by sputtering.
65
Figure 5.1: The ASTRAL helicon plasma source.
Figure 5.1 shows the ASTRAL laboratory. In the ASTRAL experiment at Auburn University
we carry on experiments with a main focus on measurements of spectral-line emission of plasmas
at different conditions. We measure plasma densities and temperatures with the use of two RF com-
pensated Langmuirprobes. These measurements form the benchmark in which we compare our
collisional-radiative model in order to generate reliable non-invasive methods for plasma diagnostic
based on spectral emission.
66
ASTRAL generates intense Ar (see figure 5.2), and Ne (see figure 5.3) plasmas.
Figure 5.2: Ar plasma in ASTRAL (Blue core with purple edge).
Figure 5.3: Ne plasma in ASTRAL (Yellow core with red/orange edge).
67
5.2 Experimental Setup
Figure 5.4: Schematic upper view of the ASTRAL helicon plasma source (not to scale).
Figure 5.4 shows the ASTRAL experimental setup, where 1. End viewport, 2. Gas Inlet, 3.
Glass section (Vacuum Chamber), 4. Plasma column, 5. Fractional helix antenna, 6. Magnetic
field coils, 7. SS section (Vacuum Chamber), 8. Light collection optics (LIF and spectrometer), 9.
Spectrometer (Monochromator and CCD camera), 10. Diode Laser LIF system, 11. Top viewport,
12. Retractable RF compensated LangmuirProbe, 13. Large viewport, 14. Toward the pumping
station.
68
ASTRAL is a 2.5 m long helicon plasma source divided by a 0.6 m long glass section, which
has the water cooled helix copper antenna wrapped around to couple RF radiation into the plasma,
and a metal section with ports distributed along its length for diagnostics. The vacuum system
consists of a turbo-molecular drag pump with a pumping speed of 400 l/s. The base pressure
in the system is 5.0 ? 10?8 Torr. We introduce gas into the vacuum chamber by means of two
flow controllers mounted in a flange at one of the ends of the device. These flow controllers also
allow us to experiment with controlled gas mixtures. The operating gas pressure ranges from 0.5
to 50.0 mTorr. The steady-state axial magnetic field ranges from 0 to 1300 Gauss, and is generated
by seven magnetic coils. The power amplifier can supply up to 2 kW of RF power to the plasma
and is coupled to the antenna through a capacitance matching circuit. The RF signal is provided
by a function generator with a 3 to 30 MHz frequency range. We have chosen to tune up the
RF frequency to 11.5 MHz since the RF power amplifier obtains a good performance around that
frequency. The plasma parameters of density and temperature can be changed by varying the RF
power, gas pressure, magnetic field intensity, and RF frequency. ASTRAL counts with a number of
diagnostics. These computerized diagnostics include: Optical Emission Spectroscopy, two radially
scanning RF compensated Langmuir Probes, and a Laser Induced Fluorescence diagnostic. In
this chapter we will focus in the spectrometer system and the Langmuir probes.
69
5.3 Langmuir Probe Settings
Figure 5.5: Langmuir probe settings in ASTRAL.
Figure 5.5 shows the schematic diagram of the two RF compensated Langmuir probes as
setup in ASTRAL. Each of the probes consists on two tips. A tip that is exposed to the plasma to
perform the measurements, and a dummy tip that is used for RF compensation inside of the ceramic
cover. Each one of them also includes a manual linear motion system to adjust the position of
the probes. This motion capability enable us to make measurements of temperatures and densities
at different locations along the inner diameter of the vacuum chamber. This help us to map the
temperature and density profiles which, as we will show in chapter 6, are necessary for a successful
spectral emission modeling.
70
Figure 5.6 shows a picture of the Langmuir probe inside the vacuum chamber (small tip at
3:00 o?clock) during an Ar plasma run.
Figure 5.6: Langmuir probe inside of ASTRAL (Ar plasma run).
Of all the different plasma diagnostics, theLangmuirprobe is probably the simplest, since it
consists of sticking a wire into the plasma and measuring the current to it at various applied voltages.
However, it is an intrusive method which could affect our plasma conditions to a certain extent. The
probe tip must be carefully designed to not interfere much with the plasma, nor to be destroyed by
it. For this reason we use tungsten in order to withstand the heat and to reduce sputtering created
by the ion collisions against the material. The interpretation of the current-voltage curves could
be difficult. In this section we give a basic overview of the Langmuir probe theory applied to
plasma diagnostics.
71
The Langmuir probe is inserted into the plasma and biased with a voltage V , and the cur-
rentI is then measured as a function of the biased voltage. When the measured current in the probe
goes to zero at a certain value of the potential, we name it the floating potential Vf. Typically the
floating potential has a negative value (see figure 5.7) caused by the differences between the mobil-
ity of electrons and ions. Let the plasma potential be Vp, when V >Vp, an electron current Ie is
collected, and the probe current is negative. When V ?Vp we then reach the electron saturation
current value Ies. WhenV 1000 nm). The spectral region between 275
and 300 nm is subject to the greatest uncertainty because of the low irradiance of the lamp at these
wavelengths. The system response function cannot be described by a simple expression. However,
we can reproduce the data accurately by combining a set of fit equations for different wavelength
regions. Table B.5 shows the values obtained using the fit for the different wavelengths from 266
to 1075 nm. Having the values of the system response function, the acquisition time, and the
experimental measured value we now rearrange equation (B.2) to solve for the ?real? experimental
value
Iexp(?) = Im(?)T
acqR(?)
(B.5)
179
Table B.5: Numerical values for the system response as a function of wavelength (? vsR(?)).
265 1867.73 306 1916.73 347 2462.82 388 4759.08 429 10213.65
266 1860.26 307 1920.45 348 2493.15 389 4845.65 430 10418.97
267 1853.96 308 1924.33 349 2524.60 390 4933.92 431 10629.57
268 1848.75 309 1928.40 350 2557.17 391 5023.89 432 10845.64
269 1844.52 310 1932.68 351 2590.89 392 5115.61 433 11067.37
270 1841.19 311 1937.19 352 2625.78 393 5209.11 434 11294.98
271 1838.67 312 1941.95 353 2661.83 394 5304.40 435 11528.66
272 1836.88 313 1946.98 354 2699.08 395 5401.54 436 11768.64
273 1835.76 314 1952.32 355 2737.53 396 5500.54 437 12015.15
274 1835.22 315 1957.98 356 2777.20 397 5601.45 438 12268.42
275 1835.22 316 1963.99 357 2818.10 398 5704.32 439 12528.70
276 1835.68 317 1970.38 358 2860.24 399 5809.17 440 12796.24
277 1836.55 318 1977.17 359 2903.64 400 5916.05 441 13071.30
278 1837.78 319 1984.38 360 2948.30 401 6025.02 442 13392.51
279 1839.33 320 1992.05 361 2994.24 402 6136.12 443 13498.20
280 1841.14 321 2000.20 362 3041.47 403 6249.40 444 13552.14
281 1843.19 322 2008.85 363 3090.01 404 6364.91 445 13593.25
282 1845.43 323 2018.03 364 3139.85 405 6482.72 446 13622.09
283 1847.83 324 2027.78 365 3191.02 406 6602.89 447 13639.18
284 1850.37 325 2038.10 366 3243.52 407 6725.47 448 13645.06
285 1853.01 326 2049.04 367 3297.36 408 6850.55 449 13640.25
286 1855.74 327 2060.61 368 3352.56 409 6978.18 450 13625.24
287 1858.54 328 2072.84 369 3409.13 410 7108.44 451 13600.55
288 1861.38 329 2085.76 370 3467.07 411 7241.41 452 13566.66
289 1864.26 330 2099.39 371 3526.40 412 7377.18 453 13524.06
290 1867.17 331 2113.76 372 3587.12 413 7515.83 454 13473.21
291 1870.09 332 2128.89 373 3649.25 414 7657.44 455 13414.57
292 1873.03 333 2144.81 374 3712.81 415 7802.12 456 13348.61
293 1875.97 334 2161.54 375 3777.79 416 7949.96 457 13275.76
294 1878.91 335 2179.10 376 3844.22 417 8101.07 458 13196.46
295 1881.86 336 2197.52 377 3912.10 418 8255.55 459 13111.14
296 1884.82 337 2216.81 378 3981.45 419 8413.52 460 13020.21
297 1887.79 338 2237.01 379 4052.28 420 8575.10 461 12924.07
298 1890.78 339 2258.14 380 4124.61 421 8740.40 462 12823.12
299 1893.80 340 2280.20 381 4198.45 422 8909.56 463 12717.76
300 1896.86 341 2303.24 382 4273.82 423 9082.71 464 12608.36
301 1899.97 342 2327.26 383 4350.73 424 9260.00 465 12495.29
302 1903.13 343 2352.28 384 4429.20 425 9441.55 466 12378.91
303 1906.38 344 2378.33 385 4509.25 426 9627.54 467 12259.57
304 1909.72 345 2405.42 386 4590.90 427 9818.10 468 12137.62
305 1913.16 346 2433.58 387 4674.17 428 10013.42 469 12013.38
180
470 11887.17 510 7791.51 550 7087.07 590 5780.47 630 4793.19
471 11759.32 511 7748.28 551 7057.98 591 5750.76 631 4772.96
472 11630.13 512 7708.27 552 7028.13 592 5721.34 632 4752.88
473 11499.89 513 7671.39 553 6997.61 593 5692.19 633 4732.96
474 11368.88 514 7637.57 554 6966.47 594 5663.33 634 4713.18
475 11237.39 515 7606.69 555 6934.78 595 5634.75 635 4693.54
476 11105.68 516 7578.67 556 6902.60 596 5606.45 636 4674.03
477 10974.01 517 7553.38 557 6869.99 597 5578.43 637 4654.65
478 10842.62 518 7530.71 558 6837.01 598 5550.68 638 4635.38
479 10711.77 519 7510.52 559 6803.69 599 5523.21 639 4616.22
480 10581.67 520 7492.66 560 6770.10 600 5496.02 640 4597.15
481 10452.54 521 7476.99 561 6736.27 601 5469.10 641 4578.17
482 10324.61 522 7463.35 562 6702.26 602 5442.44 642 4559.26
483 10198.07 523 7451.57 563 6668.09 603 5416.05 643 4540.41
484 10073.11 524 7441.47 564 6633.81 604 5389.93 644 4521.61
485 9949.92 525 7432.86 565 6599.45 605 5364.07 645 4502.84
486 9828.68 526 7425.54 566 6565.04 606 5338.47 646 4484.09
487 9709.54 527 7419.31 567 6530.62 607 5313.12 647 4465.34
488 9592.66 528 7413.95 568 6496.22 608 5288.03 648 4446.58
489 9478.19 529 7409.24 569 6461.86 609 5263.18 649 4427.78
490 9366.27 530 7404.94 570 6427.57 610 5238.58 650 4408.93
491 9257.03 531 7400.81 571 6393.36 611 5214.23 651 4390.01
492 9150.57 532 7396.59 572 6359.27 612 5190.11 652 4370.98
493 9047.02 533 7392.02 573 6325.31 613 5166.24 653 4351.84
494 8946.47 534 7386.83 574 6291.49 614 5142.59 654 4332.55
495 8849.01 535 7380.74 575 6257.84 615 5119.18 655 4313.08
496 8754.73 536 7373.46 576 6224.38 616 5095.99 656 4293.41
497 8663.70 537 7364.69 577 6191.10 617 5073.03 657 4273.50
498 8575.97 538 7354.11 578 6158.04 618 5050.29 658 4253.33
499 8491.60 539 7341.42 579 6125.19 619 5027.76 659 4232.86
500 8410.65 540 7326.29 580 6092.57 620 5005.45 660 4212.05
501 8333.13 541 7308.36 581 6060.19 621 4983.35 661 4190.86
502 8259.08 542 7285.24 582 6028.05 622 4961.45 662 4169.25
503 8188.51 543 7263.05 583 5996.17 623 4939.75 663 4147.18
504 8121.44 544 7241.50 584 5964.55 624 4918.26 664 4124.60
505 8057.85 545 7218.59 585 5933.18 625 4896.95 665 4101.47
506 7997.74 546 7194.43 586 5902.09 626 4875.84 666 4077.73
507 7941.09 547 7169.11 587 5871.27 627 4854.91 667 4053.33
508 7887.86 548 7142.72 588 5840.73 628 4834.17 668 4028.21
509 7838.02 549 7115.35 589 5810.46 629 4813.60 669 4002.32
181
670 3975.59 710 2939.75 750 2934.04 790 2449.03 830 1782.71
671 3947.95 711 2944.93 751 2927.25 791 2432.83 831 1767.22
672 3919.34 712 2949.95 752 2920.13 792 2416.52 832 1751.86
673 3889.69 713 2954.77 753 2912.69 793 2400.11 833 1736.63
674 3858.91 714 2959.40 754 2904.93 794 2383.61 834 1721.53
675 3826.94 715 2963.81 755 2896.84 795 2367.02 835 1706.58
676 3793.68 716 2968.01 756 2888.44 796 2350.35 836 1691.77
677 3759.05 717 2971.97 757 2879.72 797 2333.60 837 1677.10
678 3722.96 718 2975.70 758 2870.70 798 2316.79 838 1662.58
679 3685.32 719 2979.18 759 2861.37 799 2299.93 839 1648.21
680 3646.02 720 2982.39 760 2851.74 800 2283.01 840 1633.99
681 3604.96 721 2985.35 761 2841.81 801 2266.04 841 1619.93
682 3562.04 722 2988.03 762 2831.59 802 2249.03 842 1606.03
683 3517.14 723 2990.43 763 2821.08 803 2232.00 843 1592.29
684 3470.14 724 2992.54 764 2810.29 804 2214.94 844 1578.72
685 3420.93 725 2994.36 765 2799.22 805 2197.85 845 1565.30
686 3369.37 726 2995.88 766 2787.88 806 2180.76 846 1552.06
687 3315.34 727 2997.10 767 2776.28 807 2163.66 847 1538.98
688 3258.69 728 2998.01 768 2764.40 808 2146.57 848 1526.07
689 3199.29 729 2998.60 769 2752.27 809 2129.47 849 1513.33
690 3136.98 730 2998.87 770 2739.90 810 2112.40 850 1500.76
691 3071.62 731 2998.83 771 2727.27 811 2095.34 851 1488.36
692 3003.03 732 2998.45 772 2714.40 812 2078.31 852 1476.13
693 2931.07 733 2997.75 773 2701.30 813 2061.31 853 1464.07
694 2850.14 734 2996.71 774 2687.98 814 2044.35 854 1452.19
695 2850.65 735 2995.34 775 2674.43 815 2027.44 855 1440.48
696 2856.66 736 2993.64 776 2660.66 816 2010.57 856 1428.94
697 2862.74 737 2991.59 777 2646.69 817 1993.76 857 1417.57
698 2868.88 738 2989.21 778 2632.52 818 1977.01 858 1406.37
699 2875.04 739 2986.49 779 2618.15 819 1960.33 859 1395.34
700 2881.22 740 2983.42 780 2603.59 820 1943.72 860 1384.48
701 2887.38 741 2980.02 781 2588.84 821 1927.18 861 1373.78
702 2893.53 742 2976.27 782 2573.92 822 1910.74 862 1363.26
703 2899.62 743 2972.18 783 2558.83 823 1894.37 863 1352.89
704 2905.66 744 2967.74 784 2543.58 824 1878.11 864 1342.69
705 2911.62 745 2962.97 785 2528.17 825 1861.93 865 1332.64
706 2917.49 746 2957.86 786 2512.61 826 1845.87 866 1322.76
707 2923.25 747 2952.41 787 2496.91 827 1829.91 867 1313.02
708 2928.89 748 2946.62 788 2481.08 828 1814.06 868 1303.44
709 2934.39 749 2940.49 789 2465.11 829 1798.32 869 1294.01
182
870 1284.72 912 948.33 954 454.57 996 493.90 1038 265.00
871 1275.58 913 938.93 955 465.05 997 489.21 1039 259.91
872 1266.57 914 929.31 956 474.81 998 484.43 1040 254.88
873 1257.70 915 919.44 957 483.87 999 479.54 1041 249.91
874 1248.95 916 909.32 958 492.26 1000 474.56 1042 245.02
875 1240.33 917 898.93 959 500.00 1001 469.49 1043 240.20
876 1231.84 918 888.25 960 507.11 1002 464.34 1044 235.46
877 1223.46 919 877.28 961 513.62 1003 459.12 1045 230.78
878 1215.19 920 866.00 962 519.54 1004 453.83 1046 226.20
879 1207.02 921 854.39 963 524.90 1005 448.48 1047 221.70
880 1198.96 922 842.43 964 529.72 1006 443.07 1048 217.30
881 1190.99 923 830.12 965 534.01 1007 437.61 1049 212.98
882 1183.10 924 817.44 966 537.80 1008 432.10 1050 208.76
883 1175.30 925 804.36 967 541.10 1009 426.55 1051 204.64
884 1167.58 926 790.88 968 543.93 1010 420.97 1052 200.63
885 1159.92 927 776.98 969 546.32 1011 415.34 1053 196.73
886 1152.33 928 762.64 970 548.27 1012 409.70 1054 192.94
887 1144.79 929 747.84 971 549.81 1013 404.03 1055 189.28
888 1137.30 930 732.57 972 550.95 1014 398.34 1056 185.73
889 1129.84 931 716.82 973 551.70 1015 392.64 1057 182.31
890 1122.42 932 700.55 974 552.08 1016 386.92 1058 179.04
891 1115.03 933 683.76 975 552.11 1017 381.20 1059 175.89
892 1107.64 934 666.43 976 551.81 1018 375.47 1060 172.90
893 1100.27 935 648.55 977 551.18 1019 369.74 1061 170.05
894 1092.89 936 630.08 978 550.24 1020 364.02 1062 167.36
895 1085.51 937 611.02 979 549.01 1021 358.30 1063 164.84
896 1078.10 938 591.34 980 547.49 1022 352.59 1064 162.48
897 1070.66 939 571.03 981 545.70 1023 346.89 1065 160.30
898 1063.18 940 550.07 982 543.65 1024 341.21 1066 158.30
899 1055.65 941 528.44 983 541.36 1025 335.55 1067 156.49
900 1048.07 942 506.12 984 538.83 1026 329.91 1068 154.88
901 1040.41 943 483.09 985 536.07 1027 324.29 1069 153.48
902 1032.67 944 459.33 986 533.11 1028 318.71 1070 152.28
903 1024.84 945 434.82 987 529.94 1029 313.15 1071 151.31
904 1016.91 946 409.54 988 526.58 1030 307.63 1072 150.57
905 1008.86 947 371.22 989 523.04 1031 302.14 1073 150.07
906 1000.69 948 375.19 990 519.32 1032 296.70 1074 149.81
907 992.38 949 390.51 991 515.44 1033 291.29 1075 149.81
908 983.92 950 404.97 992 511.41 1034 285.93
909 975.30 951 418.57 993 507.23 1035 280.63
910 966.50 952 431.36 994 502.91 1036 275.35
911 957.52 953 443.35 995 498.47 1037 270.15
183
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