1. Interpretation of Spectral
Data from Tokamaks.
A thesis submitted for the degree of
Doctor of Philosophy
by
Adrian Matthews, B.Sc., (q.u.b. 1994)
M.Sc. (q.u.b. 1995)
Faculty of Science
Department of Pure and Applied Physics
The Queen's University of Belfast
Belfast, Northern Ireland
June 1999
6. List of Tables
4.1 Orbital parameters of the radial wavefunctions. . . . . . . . . . . 100
4.2 Target state energies (in a.u.) . . . . . . . . . . . . . . . . . . . . 103
4.3 Energy points between the thresholds of Ni XII. . . . . . . . . . . 105
4.4 Oscillator strengths for optically allowed LS transitions in Ni XII. 107
4.5 E ective collision strengths for Ni XII . . . . . . . . . . . . . . . . 108
5.1 The principal JET machine parameters. The values quoted are the
maximum achieved values. . . . . . . . . . . . . . . . . . . . . . . 155
6.1 Previously measured wavelengths of Ni XII observed in the JET
tokamak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
6.2 JET pulses where laser ablation of nickel occured . . . . . . . . . 183
6.3 JET pulses checked by methods I and II . . . . . . . . . . . . . . 183
6.4 Ni XII wavelength identi cations . . . . . . . . . . . . . . . . . . 184
6.5 JET pulses where Ni XII lines were identi ed . . . . . . . . . . . 184
6.6 NiXII line ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
6.7 Derived temperatures of the plasma at an electron density = 1011cm 3189
v
7. List of Figures
2.1 Basic owchart for the CIV3 code . . . . . . . . . . . . . . . . . . 40
4.1 Collisionstrength and e ective collisionstrength for the 3s23p5 2Po
1=2
{ 3s23p5 2Po
3=2 transition. . . . . . . . . . . . . . . . . . . . . . . . 86
4.2 Collisionstrength and e ective collisionstrength for the 3s23p5 2Po
3=2
{ 3s3p6 2S1=2 transition. . . . . . . . . . . . . . . . . . . . . . . . 87
4.3 Collisionstrength and e ective collisionstrength for the 3s23p5 2Po
3=2
- 3s23p4(3P) 3d 4D1=2 transition. . . . . . . . . . . . . . . . . . . . 88
4.4 Collisionstrength and e ective collisionstrength for the 3s23p4(3P) 3d
4D1=2 - 3s23p4(3P) 3d 4F3=2 transition. . . . . . . . . . . . . . . . . 89
4.5 Collisionstrength and e ective collisionstrength for the 3s23p5 2Po
1=2
{ 3s23p4(3P)3d 4D5=2 transition. . . . . . . . . . . . . . . . . . . . 90
4.6 Collisionstrength and e ective collisionstrength for the 3s23p5 2Po
3=2
- 3s23p4(1D) 3d 2P3=2 transition. . . . . . . . . . . . . . . . . . . . 91
5.1 Tokamak geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.2 Tokamak eld con guration . . . . . . . . . . . . . . . . . . . . . 148
5.3 JET tokamak device . . . . . . . . . . . . . . . . . . . . . . . . . 152
5.4 Tokamak records . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
5.5 Multichannel detector system . . . . . . . . . . . . . . . . . . . . 159
5.6 Con guration of the KT4 multichannel spectrometer . . . . . . . 162
6.1 Basic owchart for ADAS207 . . . . . . . . . . . . . . . . . . . . 181
6.2 Identi cation of Ni XII lines in JET pulse 34938 . . . . . . . . . . 191
6.3 Plasma conditions of JET pulse 34938 . . . . . . . . . . . . . . . 192
6.4 Magnetic eld con guration of JET pulse 34938 . . . . . . . . . . 193
6.5 Identi cation of Ni XII lines in JET pulse 31273 . . . . . . . . . . 194
6.6 Plasma conditions of JET pulse 31273 . . . . . . . . . . . . . . . 195
6.7 Magnetic eld con guration of JET pulse 31273 . . . . . . . . . . 196
6.8 Superimposition of lines in JET pulse 31273 . . . . . . . . . . . . 197
6.9 Integration of the lines in JET pulse 31273 . . . . . . . . . . . . . 198
6.10 Identi cation of Ni XII lines in JET pulse 31275 . . . . . . . . . . 199
vi
8. LIST OF FIGURES vii
6.11 Plasma conditions of JET pulse 31275 . . . . . . . . . . . . . . . 200
6.12 Magnetic eld con guration of JET pulse 31275 . . . . . . . . . . 201
6.13 Superimposition of lines in JET pulse 31275 . . . . . . . . . . . . 202
6.14 Integration of the lines in JET pulse 31275 . . . . . . . . . . . . . 203
6.15 Identi cation of Ni XII lines in JET pulse 31798 . . . . . . . . . . 204
6.16 Plasma conditions of JET pulse 31798 . . . . . . . . . . . . . . . 205
6.17 Magnetic eld con guration of JET pulse 31798 . . . . . . . . . . 206
6.18 Superimposition of lines in JET pulse 31798 . . . . . . . . . . . . 207
6.19 Integration of the lines in JET pulse 31798 . . . . . . . . . . . . . 208
6.20 Plot of the theoretical line ratio, R1, as a function of electron density.209
6.21 Plot of the theoretical line ratio, R1, as a function of electron tem-
perature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
6.22 Plot of the theoretical line ratio, R2, as a function of electron density.211
6.23 Plot of the theoretical line ratio, R2, as a function of electron tem-
perature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
6.24 Plot of the theoretical line ratio, R3, as a function of electron density.213
6.25 Plot of the theoretical line ratio, R3, as a function of electron tem-
perature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
6.26 Plot of the theoretical line ratio, R4, as a function of electron density.215
6.27 Plot of the theoretical line ratio, R4, as a function of electron tem-
perature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
6.28 Plot of the theoretical line ratio, R5, as a function of electron density.217
6.29 Plot of the theoretical line ratio, R5, as a function of electron tem-
perature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
6.30 Plot of the theoretical line ratio, R6, as a function of electron density.219
6.31 Plot of the theoretical line ratio, R6, as a function of electron tem-
perature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
6.32 Identi cation of Ni XII from SOHO . . . . . . . . . . . . . . . . . 225
9. Publications
A list of publications resulting from work presented in this thesis is given below.
Matthews A., Ramsbottom C.A., Bell K.L. and Keenan F.P. :
E ective collision strengths for ne-structure transitions from the 3s(2)3p(5)
P-2 ground state of chlorine-like NiXII
Astrophys. J., 492, 415-419, (1998)
Matthews A., Ramsbottom C.A., Bell K.L. and Keenan F.P. :
E ective collision strengths for electron-impact excitation of NiXII
At. Data Nucl. Data Tables, 70, 41-61 (1998)
1
11. 1.1 Overview 3
1.1 Overview
Emission lines of highly ionized stages of the iron group elements Ti, Cr, Fe,
and Ni are used for diagnostic purposes of high temperature plasmas with central
electron temperatures up to the keV range.
The need for accurate electron-ion collision data is immense, with applications
in such diverse elds as astronomy and fusion research. Several calculations for
nickel ions have been published since the late 1960's, but these vary considerably
in sophistication and accuracy. In the intervening time period important atomic
e ects such as con guration interaction and autoionizing resonances have been
recognized and incorporated. Consequently theoreticians have been challenged to
improve their calculations to provide reliable diagnostics in high-resolution mea-
surements associated with fusion plasmas and astronomical sources in all wave-
length ranges from the infrared to hard x-ray. Unfortunately, little attention has
been paid to electron excitation rate calculations for NiX { NiXIII, with exist-
ing work having been performed in either the Distorted-Wave or Gaunt Factor
approximations, which do not consider resonance contributions (Blaha 1968 and
Krueger & Czyzak 1970). The reliability of the electron excitation rates depends
upon the accuracy of the collision strengths over the temperature range consid-
ered. In turn the reliability of the collision strengths depends most critically upon
the number of target states included in the R-matrix wavefunction expansion,
together with the con guration-interaction wavefunction representation of these
target states.
This thesis provides data for Cl-like NiXII. The knowledge of excitation en-
ergies and lifetimes of the 3s3p6 and 3p43d can be useful in the fusion and as-
trophysical applications mentioned above (Jupen et al. 1993). Theoretical data
for multiply charged ions remain relevant for astrophysical precision spectroscopy
even though several previously unidenti ed solar lines can now be assigned to
12. 1.2 Principal Methods for Electron-Impact Excitation Calculations 4
transitions of chlorine-like nickel. For fusion research, reliable data for the 3s23p5-
3s23p43d transition array are needed for reasons outlined below. The walls of
the JET are a Ni/Cr alloy | hence these elements provide impurity ions in the
plasma, with other contaminating elements, such as Fe, also contributing. NiXII
is a low ionisation stage which is unexpected within the bulk plasma of a tokamak
due to a high electron temperature (Te) of approximately 15 keV. However in the
divertor box" region and plasma edge (the scrape-o layer, or SOL) where the Te
is much lower (perhaps in the 10 | 100 eV range) this stage is expected to exist.
The derivation of plasma parameters (Te, Ne, ion concentrations) for this region
would allow the e ciency of using the divertor to extract energy and impurity
ions from the plasma to be quanti ed.
Explanation of Contents
The following sections contain a discussion of modern techniques for low-energy
electron-impact excitation calculations. Under Principal Methods is a discussion
of the theoretical methods employed in the majority of calculations, and under
Atomic E ects is a discussion of the relative importance of some of the main e ects
usually incorporated. Types of Transitions and Scaling Laws list these factors
as functions of nuclear charge and incident electron energy. The scaling laws are
sometimes useful for judging, approximately, the accuracy of the computed values.
1.2 Principal Methods for Electron-Impact Ex-
citation Calculations
The methods used in the computation of data for ions are brie y described below.
A more detailed account of the basic theory and methods for electron impact
excitation of positive ions may be found in the reviews by Seaton (1975) and
13. 1.2 Principal Methods for Electron-Impact Excitation Calculations 5
Henry (1981).
The Collisional Problem
The Schrodinger equation for the electron-ion collision problem may be expressed
in terms of the scattering electron moving in the potential of the target ion. The
radial part of the wave function of the scattering electron is written generally as
d2
dr2
i
li(li + 1)
r2
i
+ k2
i F(i;r) = 2
X
i0
fVii0 Wii0gF(i0
;r) (1.1)
where F is the radial function in a given channel (represented by i or i0
). The
summation on the right hand side is over all discrete and continuum states. Vii0
and Wii0 are direct and exchange potential operators, respectively. The Wii0 are
integral operators and therefore equation 1.1 represents an in nite set of coupled
integrodi erential equations. The following sections discuss the various approxi-
mations found in the literature for solving this equation.
Distorted Wave (DW) Approximation
Usually for ions more than a few times charged, the DW approximation may be
employed. There are several di erent formulations of the DW method, see Henry
(1981), but the basic feature is the assumption that coupling between scattering
channels is weak and therefore the relevant matrix elements need include only the
initial (Vii0 = 0) and nal states. However, the method allows for the distortion
of the channel wave functions, from their asymptotic Coulomb form, in the target
potential. The general criterion for the validity of the DW approximation is that
the absolute value of the reactance matrix element j Kii0 j be 1, a condition
that is satis ed for highly charged ions since K (Z N) 1. For su ciently
high charge of the ion (depending on the isoelectronic sequence) the DW method
14. 1.2 Principal Methods for Electron-Impact Excitation Calculations 6
is comparable to the Close Coupling (CC) approximation (see next section) and
may be several times less expensive in terms of computing time and e ort. Dif-
ferent formulations of the DW approximation have sought to improve upon the
basic method in a number of ways, such as incorporating additional polarization
e ects, McDowell et al. (1973) and taking some account of the e ect of autoion-
izing resonances Pradhan et al. (1981). With respect to resonances, it should be
mentioned that although the DW method by its very nature does not allow for
coupling between open and closed channels (i.e., no resonances), one may intro-
duce, as in the UCL (University College London) CC formulation (IMPACT),
bound channel wave functions in the total eigenfunction expansion for the (e +
ion) system. These give rise to poles in the scattering matrix in the continuum
energy region and thus account for a limited number of resonances in the cross
sections. Most of the DW calculations found in literature neglect resonance ef-
fects. However, Badnell et al. (1991) showed how resonance structures can be
accounted for with the DW approximation and made detailed comparisons with
RMATRX calculations.
Close-Coupling (CC) Approximation
Truncating the sum on the right hand side of equation 1.1 to a nite number
of excited states of the target ion and solving the remaining coupled equations
exactly yield the NCC approximation, where N refers to the number of states
included (usually small). The CC approximation is the most accurate method for
solving the e-ion collision problem as it allows for full coupling between channels
(target ion + scattering electron), which is often strong at low energies. Pro-
vided the energy is restricted to the region below the highest term included in
the eigenfunction expansion, resonances due to the interaction between open and
closed channels are automatically included. The CC approximation is employed
for atoms and ions where one expects strong coupling between the states included
15. 1.2 Principal Methods for Electron-Impact Excitation Calculations 7
in the target expansion. This is usually the case for up to a few times charged
ions or heavy ions (like nickel), when the energy levels are close together or when
one nds transitions with strong associated multipole moments between several
levels.
Most of the existing CC calculations have been carried out using two sets of
codes, IMPACT, Crees et al.. (1978), and RMATRX, Burke and Robb (1975),
which employ di erent numerical procedures but yield results with similar accu-
racy and detail.
RMATRX refers to the R-matrix method (adopted from nuclear physics),
which incorporates the numerical procedure of matrix diagonalization of the (N +
1) electron Hamiltonian to yield the R-matrix which is related to the usual scat-
tering parameters. Another noniterative method for solving the integrodi erential
(ID) equations is NIEM (Smith & Henry 1973).
Each of the program packages" in turn consists of three main programs for
(i) calculating the target wave functions, energy levels, oscillator strengths,
etc.
(ii) computing the collision algebra" i.e. the potential operators Vii0 and
Wii0 ; and
(iii) solving the ID equations themselves, including the asymptotic region
where, due to the neglect of exchange terms, they assume the form of coupled
di erential equations.
Matching the asymptotic and the inner region" (with exchange) solutions
yields the reactance matrix, denoted as R by the IMPACT group and as K by
the RMATRX and NIEM users. For the atomic structure calculations, (i), the
RMATRX and NIEM users employ the computer program CIV3 based on the
Hartree-Fock method for computing one-electron orbitals, and IMPACT users
16. 1.2 Principal Methods for Electron-Impact Excitation Calculations 8
employ the program SUPERSTRUCTURE. Both these codes include con gura-
tion interaction.
The R-matrix method has proven to be computationally more e cient than
other methods, in particular for delineating the extensive resonance structures
in the cross sections that require calculations at a large number of energies (a
few hundred to a few thousand). The R-matrix method entails the division of
con guration space into an inner and an outer region. The inner region comprises
the target" or the core" atom or the ion and the (electron + target) system
solutions are expanded in terms of basis functions satisfying logarithmic R-matrix
boundary conditions at a radius dividing the inner and the outer regions. The
outer region solutions are obtained neglecting exchange but including long-range
multipole potentials (i.e. the terms Wii0 in equation 1.1 are omitted). Physically
relevant quantities, such as the scattering matrix, are obtained by matching the
inner and the outer solutions at the R-matrix boundary.
Coulomb-Born (CB) Approximation
For highly charged ions and for high electron energies a further approximation may
be made: neglecting the short range distortion of the Coulomb scattering waves
due to the detailed interaction between the target and the incident electron. This
Coulomb-Born approximation is unreliable for low energies (near threshold) or
for lightly charged ions. The resulting error in the cross sections may be a factor
of 3 or more; however for highly charged ions or optically allowed transitions the
error is much lower. Only the background cross sections are calculated, without
allowance for resonance e ects. In the earlier standard version of the CB method
the exchange e ect is not included and therefore probability amplitudes for spin
change transitions cannot be calculated. Most of the highly charged nickel ions
have been treated with this method.
17. 1.2 Principal Methods for Electron-Impact Excitation Calculations 9
Coulomb-Bethe (CBe) Approximation
The basis of the CBe approximation is that the collisional transition may be
treated as an induced radiative process. It is employed for optically allowed tran-
sitions where, due to the long range dipole potential involved, it is usually neces-
sary to sum over a large number of orbital angular momenta (l) of the incident
electron. The method is valid for l - waves higher than a given l0, which depends
on the ionic charge, and is used in conjunction with DW or CC approximations
for low l - waves to complete the l summation (Pradhan 1988). If one takes r
to be the mean radius of the target, the condition for the validity of the CBe
approximation is
l > (k2r2 + 2zr + 1
4
)1=2 1
2
l0 (1.2)
where z = Z N. Thus for allowed transitions the scattering calculations may be
divided according to the sets of partial waves l l0 and l0 < l < 1; the former are
treated in the DW or CC approximations that take account of the detailed close
range interaction and the latter in the CBe approximation. The partial wave sum-
mation for forbidden transitions usually converges for l l0. The CBe collision
strength is expressed in terms of the dipole oscillator strength for the transition
and radiative Coulomb integrals. Methods for the evaluation of these integrals,
their sum rules, and the collision strengths are described by Burgess, (1974) and
Burgess and Sheorey (1974). A discussion of the general forms of the Born and
Bethe approximations is given by Burgess, and Tully (1978), who also showed
that in the limit of in nite impact energies the CBe approximation overestimates
the cross sections by a factor of 2 due to the fact that the approximation is invalid
for close encounters i.e. low l - waves.
In recent years top up" procedures have been developed to complete the sum
over higher partial waves not included in the CC formulation e.g. Burke and
Seaton (1986).
18. 1.3 Atomic E ects 10
Other Approximations
Gaunt factor or the g approximation This method was used by Kato (1976)
for many ionisation stages of nickel. Analogous to the CBe approximation, the g
formula expresses the collision strength for optically allowed transitions as
ij = 8p
3
!ifij
Eij
g (1.3)
where fij is the dipole oscillator strength and g (called the e ective Gaunt factor)
is an empirically determined quantity. This expression was suggested by Burgess
(1961), Seaton (1962) and Van Regemorter (1962) and could be accurate to about
a factor of 2 or 3 (sometimes worse). The value of g may actually vary widely
depending upon the isoelectronic sequence. At high energies the collisionstrengths
for allowed transitions increase logarithmically and the proper form is given as
ij = 4!Lfij
Eij
ln 4k2
(rEij)2 (1.4)
where k2 is the incident electron energy (Rydbergs) and r is the mean ionic radius.
1.3 Atomic E ects
The accuracy of a given calculation depends not only on the principal method
employed but also on the contributing atomic e ects included. the relative im-
portance and magnitude of these e ects vary widely from ion to ion and even
within an isoelectronic sequence, large variations with Z may be found; for ex-
ample, pure LS coupling calculations become invalid for some transitions in high
Z ions. Another example is where, for the same ion, a close coupling calculation
may be less accurate than a distorted wave calculation if the target wave functions
in the latter take into account con guration interaction but those in the former
do not.
19. 1.3 Atomic E ects 11
Exchange
The total e+ion wave function should be an antisymmetrised product of the N+1
electron wave function in the system with N electrons in a bound state of the target
ion and one free electron. Nowadays nearly all scattering calculations satisfy the
antisymmetry requirement and exchange is accounted for, but there are some older
calculations in the literature where exchange is neglected. It has been shown that
apart from spin ip transitions, which proceed only through electron exchange, it
may be necessary to include exchange even for optically allowed transitions when
low l-wave contribution is signi cant
Coupling
When the coupling between the initial and the nal states is comparable to or
weaker than the coupling with other states included in the target representation,
the scattered electron ux is diverted to those other states and coupling e ects
may signi cantly a ect the cross sections. Thus the weak coupling approximations
such as the CB tends to overestimate the cross sections. As the ion charge in-
creases, the nuclear Coulomb potential dominates the electron-electron interaction
and correlation e ects (such as exchange and coupling) decrease in importance.
Optically allowed transitions are generally not a ected much.
Con guration Interaction
It is essential to obtain an accurate representation for the wave functions of the
target ion. The error in the cross sections is of the rst order with respect to the
error in the ion wave functions. Usually it is necessary to include CI between a
number of con gurations in order to obtain the proper wave functions for states
of various symmetries. The accuracy may be judged by comparing the calculated
20. 1.3 Atomic E ects 12
eigenenergies and the oscillator strengths (in the length and the velocity formula-
tion) with experimental or other theoretical data for the states of interest in the
collision.
Owing to the constraints on computer core size, it is usually impractical to
include more than the rst few con gurations in most close coupling calculations.
However, calculations that include many con gurations and involve hundreds of
scattering channels are now being carried out on supercomputers. To circumvent
the problem of core size restrictions, it is frequent practice to include pseudostates
with adjustable parameters in the total eigenfunction expansion over the target
states for additional CI. Transitions involving the pseudostates themselves are
ignored. They are used to simulate neglected con gurations. Single con gura-
tion (SC) calculations are generally less accurate than those including CI. In the
asymptotic region the coupling potentials are proportional to
pf, where f is the
corresponding oscillator strength. It is therefore particularly important that the
wave functions give accurate results for these oscillator strengths.
Relativistic or Intermediate Coupling (IC) E ects
Relativistic e ects become important with increasing nuclear charge and have to
be considered explicitly (Bethe and Salpeter 1980). For low Z ions (including
nickel) the cross sections for ne structure transitions may be obtained by a pure
algebraic transformation from the LS to the IC scheme (e.g. through program
JAJOM by Saraph (1978)). In general, the ratio of the ne structure collision
strengths to multiplet collision strengths depends on the recoupling coe cients,
but for the case of Si = 0 or Li = 0 it can be shown that:
(SiLiJi;SjLjJj)
(SiLi;SjLj)
= (2Ji + 1)
(2Sj + 1)(2Lj + 1)
(1.5)
21. 1.3 Atomic E ects 13
As the relativistice ects become larger one may employ three di erent approaches.
The rst, based on the Dirac equation, is for light atoms and will not be discussed
here. The second method is to generate term coupling coe cients < SiLiJij iJi >
which diagonalize the target Hamiltonian including relativistic terms (Breit-Pauli
Hamiltonian); iJi is the target state representation in IC. These coe cients are
then used together with the transformation procedure mentioned above to ac-
count for relativistic e ects. The second method is incorporated in the program
JAJOM and is described by Eissner et al. (1974). A similar method is discussed
by Sampson et al. (1978).
The third approach is by Scott and Burke (1980), who have extended the
close coupling nonrelativistic RMATRX package to treat the entire electron-ion
scattering process in a Breit-Pauli scheme, treating intermediate coupling more
accurately. Resonances in ne structure transitions may also be taken into account
in the relativistic RMATRX program or in an extended version of the JAJOM
program.
Resonances
For positive ions, due to the in nite range of the Coulomb potential, there are
several in nite series of Rydberg states converging on each bound state of the
ion. When such Rydberg states lie above the ionization limit, as is often the case
when they converge onto excited states of the target ion, they become autoionizing
(undergoing radiationless transition to the continuum) with resulting peaks and
dips in the cross section at energies that span the width of the autoionizing states.
If i and j are initial and nal levels then there would be a series of resonances
in (i;j) belonging to excited states k > j. The magnitude of the resonance
contribution depends upon the coupling between states i;k and k;j. Neglecting
interference terms, the strength of this coupling is indicated by (i;k) and (k;j).
It follows that if the transition i ! j is weak and the coupling to higher states
22. 1.3 Atomic E ects 14
is strong, then resonances might be expected to play a large role. Thus the weak
forbidden or semiforbidden transitions are particularly susceptible to resonance
enhancement. Most of the older work (pre-1970 work such as Blaha 1968) did not
take into account this resonant contribution and calculations were made either
at a single incident energy, usually near threshold, or at 2 or 3 energies above
threshold.
There are several methods for taking account of resonance e ects. In the
RMATRX calculations, the resonance pro les are obtained in detail by calculating
the cross section directly at a large number of energies. The RMATRX code is
capable of including resonances nearly exactly.
The e ect of autoionization may diminish if the resonances can also decay
radiatively to a bound state, producing a recombined ion (i.e. dielectronic recom-
bination, Presnyakov & Urnov 1975 and Pradhan 1981). This would be expected
to be the case with highly charged ions where the radiative probabilitiesfor allowed
transitions begin to approach the autoionizationprobabilities, approximately 1012-
1014s 1 (the autoionization probability is nearly independent of ion charge). In
certain energy ranges the radiative decay completely dominates the autoionization
e ect in the cross section, but the overall e ect of dielectronic recombination is to
reduce the rate coe cients by 10-20%. So autoionizing resonances may enhance
the excitation rates by up to several factors, with some reduction due to radiation
damping in the continuum.
Types of Transitions and Scaling Laws
Transitions may be classi ed according to the range of the potential interaction
(Vii0 Wii0) in equation 1.1. Spin change transitions depend entirely on the
exchange term Wii0, which is very short range since the colliding electron must
penetrate the ion for exchange to occur. Therefore, only the rst few partial waves
23. 1.3 Atomic E ects 15
are likely to contribute to the cross section, but these involve quite an elaborate
treatment (e.g. close coupling). For allowed transitions, on the other hand, a
fairly large number of partial waves contribute and similar approximations (e.g.
Coulomb-Born) often yield acceptable results. The asymptotic behaviour of the
collision strengths for allowed and forbidden transitions is as follows (x = E=Eij,
where E is the incident and Eij is the threshold energy):
(A) (i;j) constantforforbiddentransitions asx ! 1 L 6= 1; S 6= 0
(B) (i;j) x 2 forspinchangetransitions asx ! 1 S 6= 0
(C) (i;j) aln4xforallowedtransitions asx! 1 l = 1; S = 0
The slope a in the last equation is proportional to the dipole oscillator strength
(see equation 1.4). The above forms are valid for transitions in LS coupling. For
highly charged ions where one must allow for relativistic e ects, through, say, an
intermediate coupling scheme, sharp deviations may occur from these asymptotic
forms, particularly for transitions of the intercombination type.
Kim and Desclaux (1988) have presented a general discussion of ther energy
dependence of electron-ion collision cross sections and have given tting formulas
appropriate for many plasma applications.
Tests of Data Accuracy
Self-consistency checks of theoretical calculations through the analysis of quan-
tum defects (Pradhan & Saraph 1977), oscillator strengths (Doering et al. 1985),
and photoionization cross sections (Sampson et al. 1985) calculated using the same
theoretical and numerical methods as those employed to solve the scattering prob-
lem provide a reliable indicator of the accuracy of the theoretical results. It is
estimated that a detailed close coupling calculation with con guration interaction
type target-ion wave functions and full allowance for resonance e ects (as well
24. 1.4 References 16
as intermediate coupling e ects, if required, for highly charged ions) yields cross
sections with an uncertainty < 10%.
1.4 References
Badnell, N.R., Pindzola, M.S. and Gri n, D.C. Phys. Rev. A 43 (1991) 2250
Bethe, H. and Salpeter, E. Quantum Mechanics of One and Two Electron Atoms
(Springer-Verlag, New York/Berlin, 1980)
Blaha, M., Ann. Astrophys. 31 (1968) 311
Burgess, A. Mem. Soc. R. Sci. Liege 4 (1961) 299
Burgess, A. J. Phys. B 7 (1974) L364
Burgess, A. and Sheorey, V.B. J. Phys. B 7 (1974) 2403
Burgess, A. and Tully, J. J. Phys. B 11 (1978) 4271
Burke, P.G. and Robb, W.D. Adv. At. Mol. Phys. 11 (1975) 143
Burke, V.M. and Seaton, M.J. J. Phys. B 19 (1986) L527
Crees, M.A., Seaton, M.J. and Wilson, P.M.H. Comp. Phys. Commun. 15 (1978)
23
Doering J.P., Gulcicek, E.E. and Vaughn, J. Geophys. Res. 90 (1985) 5279
Eissner, W., Jones, M. and Nussbaumer, H. Comp. Phys. Commun. 8 (1974)
270
Henry, R.J.W., Phys. Rep. 68 (1981) 1
Jupen, C., Isler, R.C. and Trabert, E., Mon. Not. R. Astron. Soc. 264 (1993)
627
Kato, T. Astrophys. J. Suppl. 30 (1976) 397
Kim, Y.K. and Desclaux, J.P. Phys. Rev. A 38 (1988) 1805
Krueger, T.K. and Czyzak, S.J. Proc. R. Soc. London Ser. A 318 (1970) 531
McDowell, M.R.C., Morgan, L.A. and Myerscough, V.P. J.Phys. B 6 (1973) 1435
Pradhan, A.K. and Saraph, H.E. J. Phys. B 10 (1977) 3365
Pradhan, A.K. Phys. Rev. Lett. 47 (1981) 79
25. 1.4 References 17
Pradhan, A.K. At. Data Nucl. Data Tables 40 (1988) 335
Presnyakov L.P. and Urnov. M J. Phys. B 8 (1975) 1280
Sampson, D.H., Parks, A.D. and Clark, R.E.H. Phys. Rev. A 17 (1978) 1619
Sampson, J.A.R. and Pareek, P.N. Phys. Rev. A 31 (1985) 1470
Saraph, H.E. Comp. Phys. Commun. 3, 256 (1972) and 15 (1978) 247
Scott, N. and Burke, P.G. J. Phys. B 13 (1980) 4299
Seaton, M.J. in Atomic and Molecular Processes edited by D.R. Bates (Academic
Press , San Diego, 1962) 374
Seaton, M.J., Adv. At. Mol. Phys. 11 (1975) 83
Smith, E.R. and Henry, R.J.W. Phys. Rev. A 7 (1973) 1585
Van Regemorter, H. Astrophys. J. 136 (1962) 906
27. 2.1 Introduction 19
2.1 Introduction
In this chapter and the next the theory of some numerical techniques used exten-
sively throughout the course of this work to solve the coupled integro-di erential
equations which occur in low-energy electron-ion collision processes is reviewed.
A brief description is also presented of the main computer packages that are cur-
rently available and widely used in this eld. The main concern of the review is
with collisions involving complex atoms and ions where the target contains more
than two electrons. Low-energy scattering processes of this kind have certain
intrinsic e ects:
exchange between the incident and target electrons
distortion of the target by the incident electron
short range correlation e ects between the incident and target electrons
All these e ects are important and none of them should be excluded from any
general theoretical description.
The study of electron scattering by complex atoms can conveniently be divided
into two parts. Firstly, it is necessary to obtain wavefunctions which describe
the target atomic states and secondly, these wavefunctions must be incorporated
into a description of the collision problem. Numerous theoretical methods are
currently available such as the Many Body Perturbation Theory (MBPT) (Kelly
1969), Bethe-Goldstone equations (Nesbet 1968), the Born Approximation. the
Polarized Orbital Method (Tempkin 1957) and the Matrix Variational Method
(Harris and Mitchels 1971). This chapter and the next describe in detail one of
the most accurate techniques currently available to solve the collision process.
Con guration Interaction (CI) wavefunctions containing just a few well chosen
con gurations are used to describe the target states, whilst the Close-Coupling
R-matrix method of Burke (1971) gives a good description of the collision problem
28. 2.1 Introduction 20
over an extended energy range. A combination of these two methods is the basis
of the current approach.
The technique which has become the principal computational method of low-
energy electron scattering theory is found in the Close-Coupling (CC) approx-
imation. This method was implemented for practical computations by Seaton
(1953(a,b),1955). The CC approximation is based on the use of a truncated eigen-
state expansion as a representation of the total wavefunction, thereby reducing
the problem to solving a set of ordinary di erential (or integro-di erential) equa-
tions. The concept of this technique is not altogether new, the general procedure
of expansion in target eigenstates being originally proposed by Massey and Mohr
(1932). The CC method is probably best known for its accurate prediction of many
closed channel resonances which have subsequently been detected in experiments.
These resonances are mainly coupled to just a few closed channels and hence in-
cluding them in the approximation together with the open channels will give a
reliable result for the resonance position and width. As with all approximations,
however, this method does in fact have unfortunate computational limitations. It
is obviously di cult to increase the accuracy of calculations by taking into account
a larger number of states, as this leads to a considerable increase in computing
time whilst the contribution of each successive state gets less and less. As pointed
out by Burke (1963), an increased number of channels causes convergence of the
CC method to be very slow. It has been shown, however, that the inclusion of a
few suitably chosen pseudo-states in the expansion can considerably improve the
convergence of the results. These pseudo-states can allow for perturbing e ects
of highly excited and continuum (ionization) channels, which cannot be included
directly in the formalism. Explicit `correlation functions' can also be included to
make the CC method fully general and allowing it to be applied, in principle, to
many calculations of arbitrary accuracy.
29. 2.1 Introduction 21
Due to the computational limitations apparent in the use of the CC method,
equivalent, but more practical methods, such as the R-matrix method of Burke
(1971) and the algebraic reduction method of Seaton (1970) have been developed.
The theory of the R-matrix method is presented in chapter 3 along with a brief
description of the associated computer codes.
Initially the scattering of electrons by atoms and ions where relativistic e ects
may be neglected is considered. The time independent Schrodinger equation
(HN E) = 0 (2.1)
must be solved for an N-electron target where the non-relativistic many-electron
Hamiltonian (in a.u.) takes the form
HN = 1
2
NX
i=1
(r2
ri + 2Z
ri
) +
NX
i<j
1
rij
(2.2)
Z is the charge on the nucleus and the Hamiltonianis diagonal in both the total
orbital angular momentum L and the total spin S. The interelectronic distance,
rij, is de ned as rij = jri rjj. The rst term in equation (2.2) denotes the
one-electron contribution to the Hamliltonian while the second term denotes the
two-electron contribution.
The solution of equation (2.1) yields the wavefunctions, where =
(r1;r2;:::;rN). However due to the 1
rij term in the Hamiltonian this equation
is not a separable one and thus cannot be solved exactly (except for hydrogenic
systems which contain only one electron).
The rst of the methods developed to obtain approximate solutions to equation
(2.1) was the central eld approximation method. This uses the basic idea that the
electrons of an atomic system move in an e ective spherically symmetric potential
V(r) due to the nucleus and the other electrons of the system so that the total
30. 2.1 Introduction 22
wavefunction can be expressed as a product of one-electron wavefunctions. This
is a good approximation provided that the potential V (r) of an electron does
not change signi cantly when a second electron passes the electron in question
reasonably closely. This turns out to be the case for all but the lightest of atoms
due to the nuclear charge being an order of Z greater than the charge of an
electron. The two principal problems involved are thus the calculation of the
central eld potential and the correct formulation of the wavefunction.
Two ways of performing these tasks were then developed. The rst of these
was the Thomas-Fermi model of the atom which used semi-classical and statistical
methods to obtain expressions for the potential. The other was a method rst de-
veloped by Hartree (1927(a,b), 1957) and later extended by Fock (1930) and Slater
(1930). This method used the central eld approximation as a starting point and
combined with a variational principle, equations for the potential were produced.
This method is known as the Hartree-Fock method. Unfortunately results of cal-
culations for helium, lithium and potassium showed that this method produced
results that were not entirely satisfactory. This is due to the lack of consideration
of electron correlation e ects i.e. the fact that V (r) does change with the passage
of another electron. The accurate computation of quantities such as transition
probabilities, electron a nities and hyper ne-structure constants require meth-
ods which provide solutions of a greater accuracy than the Hartree-Fock method.
That does not remove the value of this method as various modi cations can be
made to correct this oversight such as the con guration interaction method (which
will be used extensively here) and the random phase approximation method both
of which produce results which are highly satisfactory.
31. 2.2 The Hartree-Fock method 23
2.2 The Hartree-Fock method
Consider an atomic system consisting of a nucleus of charge Z (atomic units) and
N-electrons. As demonstrated by the Hamiltonian, each of these electrons experi-
ences an attraction to the nucleus and a repulsion from the other (N - 1) electrons.
Suppose these interactions were represented by an e ective potential V (r) which
can be stated to be spherically symmetric. This leads to the conclusion that each
electron in a multi-electron system can be represented by its own wavefunction,
i (i = 1,...,N), which depends on the coordinates of the electron and are known
as orbitals.
In Hartree's original approach, in 1928, he assumed that the wave function
(approximate solution of equation (2.1)) was the product of these orbitals. That
is
(q1;q2;:::;qN) = 1(q1); 2(q2):::; N(qN) (2.3)
where qi denotes the collection of spatial coordinates, ri, and spin coordinates of
electron i. However this wavefunction violates the Pauli-exclusion principle which
states that the wave function of a system of identical electrons must be totally
antisymmetric in the combined space and spin coordinates of the particles.
To correct this problem an alternative form of the wavefunction was introduced
by Fock and Slater in 1930 to replace that of equation (2.3). This wavefunction,
represented by a Slater determinant is given in equation (2.4) where the symbols
( = 1; = 2;:::; = N) represent the set of quantum numbers (n;l) which
uniquely de ne each of the N-electrons. Thus is the total wavefunction de-
scribing an atom in which one electron is in state , another in state and so
on. The electron spin orbitals, ; ::: , are chosen to be orthonormal over space
and spin. However orbitals with spin ms = +1=2 are automatically orthogonal
to those with spin ms = 1=2. Therefore space orbitals corresponding to the
same spin function must be orthonormal, which ensures the normalization of
32. 2.2 The Hartree-Fock method 24
i.e. h j i = 1.
(q1;q2;:::;qN) = 1p
N!
(q1) (q1) ::: (q1)
(q2) (q2) ::: (q2)
... ... ... ...
(qN) (qN) ::: (qN)
(2.4)
The orbitals are chosen subject to the condition
h ij ji =
Z
i(q) j(q)dq = ij (2.5)
where
R
dq represents integration over all space coordinates and a summation over
all spin coordinates. It is then customary to split the orbitals i into their space
and spin components as follows
i(qj) = ui(rj) 1
2;msi
(2.6)
where 1
2 ;msi
is the spin function and ui(rj) is the spatial function which due to
its one electron nature is an eigenfunction of the one-electron Hamiltonian
hi = 1
2
r2
i
Z
ri
(2.7)
which includes the kinetic energy of the electron i and its potential energy due to
interaction with the nucleus. These eigenfunctions can be shown to take the form
of a product of a radial function Pnili(r) and a spherical harmonic Y mli
li ( ; )
ui(r) = 1
rPnili(r)Ymli
li ( ; ) (2.8)
The problem of obtaining the wavefunction thus is reduced to nding these radial
functions which can be achieved by using a variational principle. That is if the
ground state energy of the system is denoted as E0 and the energy of the system
33. 2.2 The Hartree-Fock method 25
when it resides in the state represented by the wavefunction (given by equation
(2.4)) by E then
E0 < E = h jHj i (2.9)
where it is assumed that the wavefunctions are normalized to unity (the source of
the factor 1p
N!
in equation (2.4).
The problem using the variational principle then becomes one of minimizing
the energy E of equation (2.9). This problem is of considerable length and Brans-
den and Joachain 11] prove it can be resolved to give the variational equation.
E
X
i
i huijuii = 0 (2.10)
This gives rise to the following set of coupled integro-diferential equations courtesy
of Slater (1930).
Fiui = iui (2.11)
where Fi is the Fock operator given by
Fi = hi + Ji Ki (2.12)
where hi is the one-electron Hamiltonian given by equation (2.7), Ji is the direct
operator given by
Ji =
NX
j6=i
Z
juj(rj)j2
rij
drj (2.13)
and Ki is the non-local exchange operator given by
Kiui(ri) =
NX
j6=i
uj(ri)
Z
uj(rj)ui(ri)
rij
drj (2.14)
The set of equations (2.11) are known as the Hartree-Fock equations for the
wavefunction (2.4) and each operator listed here can be attributed to a certain
phenomenon within the atom. The one-electron Hamiltonian has already been
34. 2.2 The Hartree-Fock method 26
discussed above. The other two operators represent electron-electron interaction
e ects. The rst of these, the direct operator, can be interpreted as being the
potential associated with the electron charge density of the other electrons (i.e. the
repulsion e ect from the other electrons). The nal term, the exchange operator,
gives the interaction between two states obtained by interchanging two electrons.
This nal term is what separates Hartree's original method from the Hartree-
Fock method and is a direct consequence of the antisymmetric nature of the
wavefunction. Finally the parameter i may be interpreted as the energy required
to remove an electron from the orbital ui. This is a result of Koopman's theorem
(Cohen and Kelly 1966) and i is thus referred to as the orbital energy. It should
be noted that this method will provide an in nite number of orbitals as solutions
and not just the N number expected. Therefore the following distinction is made:
the orbitals that for a given state occur in the wavefunction are said to be occupied
while the remainder are unoccupied.
2.2.1 Correlation energy
It has been clearly pointed out that the Hartree-Fock method produces only ap-
proximate wavefunctions and thus approximate energies; denoted by HF and
EHF respectively. Comparison with exact energies Eexact shows di erences Ecorr
between exact and Hartree-Fock energies. That is
Ecorr = Eexact EHF (2.15)
This di erence is known as the correlation energy. It should be noted that the
Hartree-Fock wavefunction does include a certain amount of electron correlation
due to the total antisymmetry of the wavefunction and so the term correlation
e ects, which create the correlation energy, refers to electron correlations not
present in the Hartree-Fock wavefunction. It should also be noted that Eexact is not
35. 2.2 The Hartree-Fock method 27
the experimental energy but the exact energy of the non-relativistic Hamiltonian.
This error in the Hartree-Fock method clearly lies with the wavefunctions pro-
duced. These wavefunctions, however, do result in energies that are greater than
exact energies by less than one percent. This may be considered an acceptable per-
centage error but in the regions of con guration space which do not play a major
role in the determination of the energy of the state in question the wavefunctions
may be in serious error and thus observables calculated from these wavefunctions
may be extremely inaccurate.
Numerous attempts have been made to understand the role of correlation ef-
fects in in uencing wavefunctions and energies of atoms. The methods commonly
used for improving on the Hartree-Fock wavefunction can be classi ed broadly
into two categories. The rst is that developed by Hylleraas (1930) in which the
total wavefunction is a power series expansion which includes the inter-electronic
coordinates rij explicitly. This method has been applied with great success to
several states of the helium-like ions (Pekeris (1958, 1959)) but for more complex
svstems such a solution is of little value, due to the mathematical complexity of
the process and the di culty of interpreting it physically. The second method
is that of Con guration Interaction (CI) which involves a linear combination of
determinantal function" each representing a particular con guration of the elec-
trons in the atom. This method is used extensively in the present work and a more
detailed description is presented in Section 2.3. It can be said, however, that both
these methods have many common features, especially their dependence on the
Hartree-Fock approximation. Each has its advantages and disadvantages, but all
are, in principle, capable of re nement to give a result of arbitrary accuracy.
36. 2.2 The Hartree-Fock method 28
2.2.2 The Self-Consistent eld method
Due to the complicated nature of the Hartree-Fock equations, normal methods
are inadequate for the task of obtaining solutions to these equations. An iterative
method, based on the requirement of self-consistency, is thus required in their
solution which involves the representation of the radial function Pnl(r) by the
following linear combination of analytical basis functions
Pnl(r) =
kX
j=1
cjnlrIjnle jnlr (2.16)
or
Pnl(r) =
kX
j=1
c0
jnl jnl(r) (2.17)
where jnl is the normalized Slater-type orbital of the form
jnl(r) = (2 jnl)2Ijnl+1
(2Ijnl)!
1
2
rIjnle jnlr (2.18)
and the radial functions Pnl obey the orthonormality conditions
Z 1
0
Pnl(r)Pn0l(r)dr = nn0 (2.19)
The iteration method utilized in solving the Hartree-Fock equations is known
as the self consistent eld method and consists of the following steps.
Estimate Pnl(r) by specifying the Clementi-type (Clementi and Roetti 1974),
cjnl, or Slater-type, c0
jnl, coe cients, the exponents jnl and the powers of r,
Ijnl. Values are available in past literature of atoms or ions either isoelec-
tronic with the one you are considering or close to it.
Using these values the actual orbital, u(r) = ui(r), is determined.
The values of the terms Kiui and Jiui are determined using the estimated
37. 2.3 The Con guration Interaction method 29
value of ui. This results in a set of eigenvalue di erential equations for a
new set of orbitals, u(2)
i say.
These di erential equations are solved by substituting equations (2.8) and
(2.16) to give a set of algebraic equations which include the coe cients of the
new orbital, c(2)
jnl, where (2)
i are treated as variational parameters in order
to minimize the energy while the powers of r, I(2)
jnl are xed. The initial
estimate for the coe cients cjnl are substituted in to give a set of solvable
equations for c(2)
jnl. The values obtained from solving these equations are then
resubstituted into the algebraic equations for the coe cients to give further,
better results. The process is repeated until convergence is obtained for the
series of solutions, within a desired tolerance. Using the nal set, a radial
function is found as a rst solution of the Hartree-Fock equations
Using the new radial function, the previous two stages are repeated until a
satisfactory degree of convergence is obtained for the radial functions.
There are various other ways of dealing with stage 4 of this method such as
solving the equations numerically. Various tabulations of orbitals obtained from
this method exist. The one referred to is by Clementi and Roetti (1974).
2.3 The Con guration Interaction method
The lack of inclusion of electron correlation in the Hartree-Fock wavefunctions
is due to the restriction that each electron is assigned to a speci c nl orbital
resulting in each state being represented by a single Slater determinant. The
assignment of these electrons to speci c nl orbitals, and their couplings, are known
as con gurations. Consider the replacement of the Hartree-Fock wavefunction
with one that represents more than just a single con guration. This is achieved
by allowing a particular state with a certain LS symmetry to be represented by a
38. 2.3 The Con guration Interaction method 30
linear combination of Slater determinants where each determinant represents one
con guration whose individual orbital angular momenta of the electrons couple in
one particular way to give the same total orbital angular momenta value L and
spin value S. That is, the wavefunctions can be expressed in the form,
(LS) =
MX
i=1
ai i( iLS) (2.20)
where the i( iLS) are the con guration state functions which represent a par-
ticular assignment of electrons to orbitals with speci c n and l values. They are
eigenfunctions of L2 and S2 since these operators commute with the Hamiltonian,
so long as other relativistic interactions can be neglected i.e. for light atomic sys-
tems. Each of these con guration state functions are linear combinations of Slater
determinants, the set of which is denoted by i. The total wavefunction (LS)
represents the state possessing a total angular momentum L and total spin S
and the coe cients ai indicate the contribution made by each con guration state
function to this total wavefunction. The means by which the ai and one-electron
radial functions are obtained is called con guration interaction. Note that the
sum should be to in nity but in practice it is restricted to a nite number of
con gurations M.
The con guration state functions introduced here represent three di erent
types of electron correlation e ects.
1. Internal correlation : The Hartree-Fock orbitals are those which occupy the
ground state con guration of the system being considered. Internal correlation
corresponds to the con gurations which are constructed solely from these orbitals
or orbitals which have the same n value i.e. those nearly degenerate with them.
2. Semi-internal correlation : These e ects arise from con gurations con-
structed from (N - 1) Hartree-Fock orbitals and one other orbital not included in
this set.
39. 2.3 The Con guration Interaction method 31
3. External correlation: Con gurations that are constructed from(N-2) Hartree-
Fock orbitals and two from outside this set cause these e ects.
As expected, of the three types of e ects mentioned above, it is the internal
e ects that contribute the most to expansion (2.20) (i.e. they have the largest
values of ai) so while the external e ects create the most con guration state func-
tions, in practice accurate energy levels are obtained from including all the internal
and semi-internal con gurations but only some of the external ones (Oskuz I. and
Singanoglu O. (1969)).
2.3.1 Determination of the expansion coe cients
The problem is now one of obtaining the expansion coe cients ai and the radial
functions Pnl(r) (and thus the con guration state functions i). One method
of calculating the CI wavefunctions is to use a con guration basis set which in-
cludes the Hartree-Fock con guration along with other con gurations built from
Hartree-Fock and variationallydetermined orbital functions. This scheme is called
Superposition Of Con gurations (SOC). It is employed in the CI code CIV3 which
is described in section 2.4
Another way to achieve this is by going through the same analysis as the
Hartree-Fock method to give a set of integrodi erential equations for the radial
functions. This is known as the multi-con gurational Hartree-Fock method. How-
ever, the radial functions derived from the SOC method are analytic whereas the
MCHF radialfunctions are numerical. The radialorbitalfunctions for the Hartree-
Fock con gurations are usually taken from the tables of Clementi and Roetti
(1974) or other Roothaan Hartree-Fock calculations. The parameters describing
these orbitals are changed when using the MCHF method and then con gura-
tion interaction is applied. In contrast the parameters remain xed throughout
the calculation with the SOC method so one can use the same orbital basis for
40. 2.3 The Con guration Interaction method 32
all con gurations and states. Re-optimization of the orbitals is normally, though
not necessarily, performed with the MCHF method although both methods are
equally easy to apply.
Consider the set of con guration state functions i and the corresponding set
of coe cients ai where the i, and their radial functions, are xed while the ai are
free to vary. That is the expansion coe cients are the only variational parameters.
Then minimizing the energy of the state being used subject to the normalization
condition that
h j i = 1 (2.21)
gives rise to the variational equation
h jHj i E(h j i)] = 0 (2.22)
where E is a Lagrange multiplier. Substitution of the wavefunction equation
(2.20) into this expression results in
"
X
i
X
j
aiajh ijHj ji E
X
i
X
j
aiajh ij ji
#
= 0 (2.23)
Now de ning the Hamiltonian matrix by its general element Hij which is given b
Hij = h jHj i (2.24)
where H is the N-electron Hamiltonian of equation (2.2) and assuming that the
con guration state functions are orthonormal (i.e. h j i = ij) it follows that
X
j
aj(Hij E ij) = 0 (2.25)
where the possible values of E are in fact the corresponding eigenvalues Ej of
the Hamiltonian matrix, H, while the ai are the components of the associated
41. 2.3 The Con guration Interaction method 33
eigenvectors, E(j)
i . Equation (2.26) may also be written as
h jHj i = E ij (2.26)
according to the Hylleraas-Undheim theorem (see section 2.3.3).
It follows that diagonalization of the Hamiltonian matrix will produce both
the expansion coe cients and the energy of the state.
2.3.2 Setting up the Hamiltonian matrix
However, before energy levels are determined it is essential to form the Hamilto-
nian matrix in order to diagonalize it. First adopt the approach of writing the
matrix elements as a weighted sum of the one and two electron integrals as follows.
Split the Hamiltonian into two parts
H = Ho + V (2.27)
where Ho is the one electron term and V is the two electron term which are
respectively given by
Ho =
NX
i=1
hi (2.28)
and
V =
X
i<j
1
rij
(2.29)
This enables us to write the Hamiltonianmatrix elements as the sum of two matrix
elements associated with the operators Ho and V. That is
Hij = h ijH0j ji+ h ijVj ji (2.30)
Each of these two new matrix elements can be expressed in the form of one and
42. 2.3 The Con guration Interaction method 34
two electron radial integrals respectively
h ijH0j ji =
X
; 0
x( ; 0) Pn l
1
2
d2
dr2
Z
r + l (l + 1)
2r2 Pn 0l 0 l l 0 (2.31)
and
h ijVj ji =
X
; ; 0; 0;k
y( ; ; 0; 0;k)Rk(n l ;n l ;n 0l 0;n 0l 0) (2.32)
where Rk represent the two-electron radial integrals and and are the indices
which represent the status of the rst and second electron respectively subject
to the restriction that the wavefunctions i and j must have at least (N - 2)
electrons in common for the two electron term while they must have at least (N
- 1) electrons in common for the one-electron term.
The coe cients x and y are weighting coe cients which Fano 47] has already
described in terms of Racah algebra. Several programs created by Hibbert (1970,
1971, 1973) exist that calculate these coe cients by using other computer packages
which calculate recoupling coe cients (Burke (1970)) and fractional parentage
coe cients (Allison (1983), Chivers (1973)). All of these packages have been
incorporated into the computer package CIV3 written by Hibbert (1975) which
performs the entire task of setting up and diagonalizing the Hamiltonian matrix
to obtain the coe cients ai, and energies, E(j).
2.3.3 Optimization of the radial functions
The Hylleraas-Undeim theorem (Hylleraas and Undheim (1930)) states that
`The upper bound to the exact non-relativistic energies of the states of a given
symmetry obtained using a variational principle are greater than or equal to the
exact energies'
43. 2.3 The Con guration Interaction method 35
That is
Ei Eexact
i (2.33)
where Eexact
i are the exact non-relativistic energies of the state of a given symme-
try. As demonstrated this value Ei depends on the radial functions used in the
calculation. Therefore the resulting energy of the diagonalization of the Hamilto-
nian will always be greater than the exact energy no matter what values of the
radial functions are chosen but the more accurate the radial function the closer
Ei will be to Eexact
i . Variation of the radial functions is thus necessary to nd the
lowest energy possible. This process is known as the optimization of the radial
functions (or orbitals).
Using equation (2.19) (n - 1) of the linear coe cients, cjnl, of equation (2.16)
are xed where n and l are the principal and orbital angular momentum quantum
numbers of the orbitalwhose radial function is being varied. Since Ei then depends
upon the remaining linear coe cients and non-linear exponents, Ei can be used as
a variational function in order to obtain values for the coe cients and exponents.
The process is repeated until overall convergence for the energies is obtained.
If, for the purpose of obtaining the orbitals, only one con guration is used
in the original expansion given by equation (2.20), then the above process will
produce an approximate solution of the Hartree-Fock equations and the orbitals
willbe the Hartree-Fock orbitals. If on the other hand more than one con guration
is included involving additional orbitals then the problem becomes non-physical
as are the orbitals thus obtained. These orbitals are known as pseudo orbitals
and they are distinguished from real orbitals by placing a bar over them. These
orbitals satisfy the same conditions as real orbitals such as orthonormality to the
other orbitals in the generated set (including real orbitals) but are important in
con guration interaction calculations in order to accurately describe correlation
e ects within a particular state.
44. 2.4 The Con guration-Interaction Bound State Code - CIV3 36
The method described here is the one used by the previously mentioned CIV3
code in order to obtain orbitals and energy levels.
2.4 The Con guration-Interaction Bound State
Code - CIV3
A general FORTRAN program to calculate Con guration-Interaction wavefunc-
tions and electric- dipole oscillator strengths has been formulated by Hibbert
(1975). It encompasses the entire range of calculations introduced in the previous
section including the calculation of energy levels and expansion coe cients and
setting up the con guration interaction wavefunctions from this data. The code
can use these wavefunctions to evaluate such observables as oscillator strengths.
Optimization, otherwise known as minimizing, of radial functions to give the most
accurate energies (and therefore wavefunctions) possible is also performed. This
makes it ideal for obtaining orbitals and energy levels that will be essential for
utilizing the R-matrix code that is discussed in the next chapter. This section
describes how the CIV3 code computes these values using the theory. The basic
structure of the code is presented in the schematic ow diagram, Figure 2.1.
The input required for the code can be grouped as follows:
Initially the type of calculation to be performed must be determined. The
choice between radial function optimization, oscillator strength calculation
and others is provided although this discussion concerns the former only.
There is also the option of how much output to produce. For example the
Hamiltonian before and after diagonalizationto can be output.
Some basic data about the ion being considered is included: e.g. the nuclear
charge Z, the maximum n and l values and the maximum powers of r among
the orbital set.
45. 2.4 The Con guration-Interaction Bound State Code - CIV3 37
The radial functions are input analytically in either Clementi or Slater type
form corresponding to equations (2.16) and (2.17) respectively. Distinction
between Hartree-Fock orbitals and orbitals calculated by the user must be
made. For the orbital to be optimized, an initial estimate for the radial
function is included here. The radial functions are sums of STO's, implying
the radial integrals are performed analytically.
The con gurations including the various coupling schemes are input. This
section includes the n and l values of each occupied orbital. In an optimiza-
tion calculation it is suggested that a minimum number of con gurations
are needed to include the dominant contributors to the electron correlation
e ects being introduced, with further con guration state functions having
only a minor e ect on the optimal radial function parameters. While for
energy level calculations the selection of all internal and semi-internal con-
gurations with some external con gurations is recommended.
Data speci c to the calculation being attempted is included. For the energy
level case this could include the option to split the Hamiltonian into separate
total symmetries and thereby increase e ciency.
Once the correct input data has been established and checked, and the type
of calculation speci ed, the CIV3 code proceeds to generate the radial function
parameters for the Hartree-Fock orbitals required together with any further neces-
sary pseudo-orbitals. The orbitals are generated in the order of increasing angular
momentum and principal quantum number. Each radial function to be optimized
is varied separately, by treating its parameters as the variables in the minimiza-
tion routine. When the last radial function in the list has been optimized, the
process begins again with the rst in the list. The process terminates when the
net change in the functional is less than a preassigned amount.
The nal Hamiltonian matrix may now be constructed and diagonalized to ob-
46. 2.4 The Con guration-Interaction Bound State Code - CIV3 38
tain upper bounds to the exact energies, Eexact
i ,(eigenvalues) and the components
of the con gurations in the corresponding wavefunctions, ai,(eigenvectors)(see
equation (2.25)). If further con gurations are to be included and the con gu-
ration set extended then a new Hamiltonian matrix must be constructed and re-
diagonalized. If necessary, the new partitioning of the matrix is de ned. Finally
the SOC wavefunctions and the corresponding energies can easily be established.
Once an SOC wavefunction has been constructed it may then be used to
evaluate other atomic properties. One property of particular interest in atomic
structure is oscillator strengths (transition probabilities). Speci cally the code
allows the calculation of absorption multiplets oscillator strengths between two
states, each of course being described by an SOC wavefunction. Length, velocity
and acceleration forms of these transition probabilities may be evaluated together
with the geometric mean. One nal option is available to the user of CIV3, that
of subdiagonalization. It is sometimes of interest to examine the convergence
of the inclusion of more and more con gurations, either for the energy or for
oscillator strengths. Once the Hamiltonian has been set up (after optimization)
it is possible to diagonalize sub-matrices to see the e ect of including a limited
number of con gurations.
There are some limitations to the complexity of any calculation performed
using CIV3. The maximum number of electrons is allowed in s, p and d subshells
but only up to 2 electrons in f or g subshells. Subshells with l > 4 may only be
included when the code has been modi ed. The typical execution time depends
on a number of factors:
size of the ion
the extent of the optimization required
the number of con gurations involved
the number of basis functions in each radial function
47. 2.4 The Con guration-Interaction Bound State Code - CIV3 39
inclusion of relativistic e ects
48. 2.4 The Con guration-Interaction Bound State Code - CIV3 40
Basic Data
Radial Functions
Configuration Sets
CIV3
Optimize radial
functions
P
nl
(r) set up as a
sum of STO’s
Hamiltonian matrix set up
and diagonalized
Extend list of configs,
set up new Hamiltonian
and re-diagonalize
Set up SOC
wavefunction and
energies
strengths
Oscillator
Output
Minimization
Figure 2.1: Basic owchart for the CIV3 code
49. 2.5 References 41
2.5 References
Allison D.C.S. Comput. Phys. Commun. 1 (1969) 15
Bransden B.H. and Joachain C.J. Physics of Atoms and Molecules (Longman
1983)
Burke P.G. Proc. Phys. Soc. 82 (1963) 443
Burke P.G. Comput. Phys. Commun. 1 (1970) 241
Burke P.G., Hibbert A. and Robb W.D. J. Phys. B4(1971) 153
Chivers A.T. Comput. Phys. Commun. 6 (1973) 88
Clementi E. and Roetti C. At. Data Nucl. Data Tables 14 (1974)
Cohen M. and Kelly P.S. Can. J. Phys. 44 (1966) 3227
Fock V.Z. Z. Phys. 60 (1930) 126
Harris F.E. and Mitchels H.H. Methods Comp. Phys. 10 (1971) 143
Hartree D.R. Proc. Camb. Phil. Soc. 24 (1927a) 89
Hartree D.R. Proc. Camb. Phil. Soc. 24 (1927b) 111
Hartree D.R. The Calculation of Atomic Structures (Wiley 1957)
Hibbert A. Comput. Phys. Commun. 1 (1970) 359
Hibbert A. Comput. Phys. Commun. 2 (1971) 180
Hibbert A. Comput. Phys. Commun. 6 (1973) 59
Hibbert A. Comput. Phys. Commun. 9 (1975) 141
Hylleraas E.A. and Undheim B. Z. Phys. 65 (1930) 759
Kelly H.P. Phys. Rev. 182 (1969) 84
Massey H.S.W. and Mohr C.B.O. Proc. Roy. Soc. Ser. A 136 (1932) 289
Nesbet R.K. Phys. Rev 175 (1968) 2
Oskuz I. and Sinanoglu O. Phys. Rev. 181 (1969) 42
Pekeris C.L. Phys. Rev 112 (1958) 1649
Pekeris C.L. Phys. Rev 115 (1959) 1216
Seaton M.J. Phil. Trans. R. Soc. London Ser. A 245 (1953a) 469
Seaton M.J. Proc. R. Soc. Lond. Ser. A 218 (1953b) 400
50. 2.5 References 42
Seaton M.J. Proc. R. Soc. Lond. Ser. A 231 (1955) 37
Seaton M.J. Proceedings of the 2nd annual Conference on Computational Physics
(1970)
Slater J.C. Phys. Rev. 35 (1930) 210
Tempkin A. Phys. Rev. 107 (1957) 1004
52. 3.1 The R-matrix method 44
3.1 The R-matrix method
The R-matrix method is used to calculate reliable cross-sections, which are then
used to produce e ective collision strengths (see chapter 4) for use in the ADAS
application (see chapter 6).
A cross-section, i!j, is related to the probability per second that a particular
event will occur in the system considered, measured over a range of energies.
Consider a beam of electrons of known ux density impacting upon the target.
There is a probability associated with exciting the initial target to a particular
state and this is dependent on the ux density. The constant of proportionality
is the cross section i!j which has units of area. It is dependent on the target
element, residual charge (for an ion) and is a complicated function of energy which
may include many resonance features. Cross-sections can be obtained for a system
from its wavefunction, but the problem is that the wavefunctions of equations
(2.4) and (2.20) should include a description of the impact electron. However, the
con guration interaction wavefunction does not include any continuum terms -
i.e. it only considers bound states, and not continuum states like that of an (ion
+ free electron) state. `Interaction with the continuum' must be considered.
One method which deals with this interaction is called the R-matrix method.
The principle behind the R-matrixscattering method (Wigner and Eisenbud 1947,
Lane and Thomas 1958) is that con guration space describing both the scattered
and target particles can be split into an inner region and an outer region. In
the outer region the scattered particle is outside the charge distribution of the
target so that the system is easily solvable i.e. interaction is weak and, in many
cases, is determined exactly in terms of plane or coulomb waves, modi ed by long-
range multipole potentials. In the inner region the converse is the case so that
correlation and exchange e ects are very strong and the collision is di cult to
evaluate. The solution is to impose spherical boundary conditions on the surface
53. 3.1 The R-matrix method 45
of the inner region centred on the target nucleus giving a complete set of states
describing all enclosed particles.
While it is not a recent development, it has only seen its fullest e ect and use
in the previous thirty years due to the development of supercomputers. Today it
continues to be developed for di erent and more accurate applications. Below is
a brief history of the most pertinent developments in R-matrix theory:
1947. R-matrix method published by Wigner and Eisenbud.
1971. Burke and Seaton, Burke, Hibbert and Robb. R-matrix theory applied
to electron scattering problems.
1975. Burke & Robb. The complete description of the R-matrix theory but
excluding relativistic e ects.
1980. Scott & Burke The modi cation of the R-matrix method to include
relativistic e ects.
Parallel to the mathematical development of the theory has been the produc-
tion of a computer code which can perform calculations utilizing this method
where the motivation comes from the realization that manual calculations would
be unfeasable for all but the simplest atomic systems. The rst of these codes was
written by Berrington et al. (1974, 1978) but there have since been many modi -
cations accompanying the theoretical developments. Subsequently there are now
several versions of the code in existence but the one used herein is RMATRX1.
3.1.1 Basic ideas and notation
The equation which describes electron-impact excitation is
A + e ! A + e (3.1)
54. 3.1 The R-matrix method 46
where in general notation A is an arbitrary atom or ion target, A is a nal excited
state of A and e is the continuum (or free) electron.
Consider a trivialised description of an electron colliding inelastically with an
atom or ion target. As the electron approaches the target it experiences the
target's complicated electrostatic eld but as it gets closer there comes a point
where it is indistinguishable from the electrons around the target. It also disturbs
the electron `cloud'. At this point there are many possible outcomes permitted by
quantum mechanics - so called channels. A channel is simply a possible mode of
fragmentationof the composite system (A+e ) during the collision. The outcomes
are limited by the energy conservation laws. If E is the total energy and i is the
energy of the target state coupled to the i-th channel then the channel energy of
the free electron, k2
i , is therefore
k2
i = 2(E i) (3.2)
If k2
i > 0 the channel is open while if k2
i < 0 it is closed. If k2
i = 0 then it indicates
that the system is at a threshold energy for excitation to occur.
The critical point in the collision process occurs when correlation and electron
exchange e ects are of importance with regards to the outcome of the impact.
This point is at a distance, ra,known as the R-matrix radius, from the nucleus. It
encloses a sphere which is su ciently large for the electron charge distribution of
the target to be permanently contained within said sphere while the target plus
free electron system are included in a sphere of in nite radius which is referred
to as the (N + 1)-electron system. The (N + 1)th electron is thus considered to
be free if it occupies the region r > ra (known as the external region) while it is
bound if it resides in the region where r ra (known as the internal region).
55. 3.1 The R-matrix method 47
3.1.2 Constructing the targets
Initially an adequate description of the target must be found, from which ex-
pressions for the wavefunction in both internal and external regions may also be
found. The target may be represented by wavefunctions known as target states,
i. They are de ned by their total angular momentum and spin quantum numbers
and by the arrangement of the orbital electrons which couple in particular ways
to yield these numbers. The target states are solutions of the time independent
Schrodinger equation.
HN
i = i i (3.3)
where HN is the non-relativistic N-electron Hamiltonian and i is the energy of
the corresponding target state. It is also necessary to include a certain amount
of con guration-interaction in the target state wavefunctions f ig to describe
them accurately. This feature can be introduced by describing each of the target
states in terms of some basis con gurations, k, using the following con guration
interaction expansion.
i(x1;x2;:::;xN) =
MX
k=1
aik k(x1;x2;:::;xN) (3.4)
where xi = (ri; i) denotes the space (ri = ri^ri) and spin ( i) coordinates of the
ith electron while the faikg are the con guration mixing coe cients which are
unique to each state.
This is the same type of wavefunction as that from equation (2.20) and the
problem of describing the target thus becomes a con guration interaction problem
which is solved using the method of section (2.3); that is the expansion coe cients
are determined by diagonalizing the N-electron Hamiltonian matrix while the
basis con gurations are constructed from a bound orbital basis consisting of a
set of real orbitals, and possibly pseudo orbitals, introduced to model correlation
e ects. This will result in expressions for the target state functions and their
56. 3.1 The R-matrix method 48
energies. The con gurations used in the expansion, naturally all have the same
total spin and orbital angular momentum values.
The R-matrix method, understandably, uses the same notation and form of
the orbitals as used in the con guration interaction method (i.e equations (2.6),
(2.8) and (2.16)). In fact it is usual to use the CIV3 package (although other
packages such as SUPERSTRUCTURE (Eissner et al. 1974) do exist) to de-
termine the required radial functions and these can be input directly into the
R-matrix code. Note that it is important that the orbitals obtained are su cient
for the representation of both the target and the (N + 1)-electron system.
From these radial functions it is then possible to clearly de ne the internal and
external regions by the determination of a value for the R-matrix radius. Since
the radial functions tend to zero exponentially as r tends to in nity, indicating
that the probability of nding an electron signi cantly outside the atom is quite
small, a value for ra can be chosen at which point it can be claimed that the
charge distribution of the states of interest are included within the sphere de ned
by this radius. In mathematical terms if is taken to be a suitably chosen small
number, then ra is chosen such that
jPnl(r) < j r ra (3.5)
for each of the bound orbitals. In practice the R-matrix radius is taken to be the
value of r at which the orbitals have decreased to about 10 3 of their maximum
value.
3.1.3 The R-matrix basis
The most signi cant problem in applying the R-matrix method to the scattering
of electrons by ions, is de ning a suitable zero-order basis for expansion of the
(N + 1)-electron wavefunction.
57. 3.1 The R-matrix method 49
This basis is constructed from three di erent orbital types ; the real orbitals,
the pseudo orbitals and the continuum orbitals (although pseudo orbitals are
optional). The rst two types have already been introduced while the continuum
orbitals are included to represent the motion of the free electron. For a particular
angular momentum value li; the set of continuum orbitals f ijg are obtained by
solving the following equation.
( d2
dr2
li(li + 1)
r2 + V0(r) + k2
ij) ij(r) =
X
n
ijnPnli(r) (3.6)
subject to the R-matrix boundary conditions:
ij(0) = 0 (3.7)
ra
ij(ra)
d ij
dr r=ra
= b (3.8)
where b is an arbitrary constant known as the logarithmic derivative that is usually
chosen set to zero and k2
ij are the eigenvalues of the continuum electron which are
also the previously introduced channel energies. The ijn are Lagrange multipliers
that ensure orthonormality of the continuum orbitals to the bound orbitals of the
same angular symmetry. That is
Z ra
0
ij(r)Pnli(r)dr = 0 (3.9)
while analogous to the bound orbital case the continuum orbitals also obey the
orthonormality conditions
Z ra
0
ij(r) ij0(r)dr = jj0 (3.10)
Finally V (r) is a zero-order potential which behaves like 2Z
r near the nucleus and
is usually chosen to be the static potential of the target. By default this is taken
to be the static potential but other options can be used.
58. 3.1 The R-matrix method 50
Now that real and continuum orbitals have been considered, with necessary
orthogonality conditions, the nal orbitals used for the R-matrix basis are the
pseudo-orbitals. They were omitted previously from equation (3.6) since they
would have negated the physical justi cation for this equation, and cause the
R-matrix expansion to converge much more slowly. The process of Schmidt or-
thogonalisation is used to further orthogonalise the continuum orbitals to the
pseudo-orbitals. This does not a ect the worth of the continuum orbitals or the
previous orthogonality conditions satis ed, and is useful for the matrix mathe-
matics later. The nal result is an orthonormal basis for each value of li ranging
from r = 0 to r = ra.
3.1.4 The internal region
Within the internal region, the (N + 1)th electron is indistinguishable from the
other N electrons - i.e. it is no longer part of the `continuum'. So, the overall
wavefunction can be found by solving the time independent Schrodinger equation
for the (N + 1)-electron Hamiltonian:
HN+1
= E (3.11)
subject to appropriate boundary conditions where HN+1 is the (N + 1)-electron
Hamiltoniangiven by equation (2.2) with N replaced by (N+1) and where E is the
total energy of the system. A con guration expansion of the wavefunction similar
to that of equation (3.4) in the bound state problem is now appropriate. However,
interaction between the bound states and the continuum is of importance in this
region and whenever this interaction is particularly strong the inclusion of the
continuum orbitals in a con guration interaction expansion of the wavefunction
may be insu cient to model the continuum. Unfortunately, the inclusion of the
integral necessary to model the continuum completely in an expression for the
59. 3.1 The R-matrix method 51
wavefunction is an impractical alternative. A summation is introduced to approx-
imate the continuum using special types of target states known as pseudo states.
Pseudo states satisfy the equations introduced to describe the target states but
are constructed from a combination of real and pseudo orbitals.
It should be noted that pseudo states are not a de nite requirement of the
R-matrix method but in cases where strong continuum interaction occurs they
convert the problem from one of discrete-continuum interaction to one of discrete-
discrete interaction allowing a con guration interaction expansion to be used to
represent the total wavefunction. That is
=
X
k
AEk k (3.12)
where the energy dependence is carried through the AEk coe cients and k are
states which form a basis for the total wavefunction in the inner region (r < ra),
are energy independent and are given by the expansion
k(x1;x2;:::;xN+1) = A
X
ij
cijk i(x1;:::;xN;^rN+1 N+1) 1
rN+1
ij(rN+1)
+
X
j
djk j(x1;x2;:::;xN+1) (3.13)
where f ig are called the channel functions, obtained by coupling the target
states i (including any pseudo states) with the angular and spin functions of
the continuum electron to form states of total angular momentum and parity. A
is the antisymmetrization operator which accounts for electron exchange between
the target electrons and the free electron (i.e. it imposes the requirements of the
Pauli exclusion principle).
A = 1p
N + 1
N+1X
n=1
( 1)n (3.14)
i represents the quadratically integrable (L2) functions (or (N +1)-electron con-
60. 3.1 The R-matrix method 52
gurations) which vanish at the surface of the internal region, are formed from the
bound orbitals and are included to ensure completeness of the total wavefunction.
ij are the continuum orbitals corresponding to the appropriate angular momen-
tum obtained from equation (3.6) and are the only terms in equation (3.13) that
are non-zero on the surface of the internal region. cijk and djk are coe cients and
are determined by diagonalizing HN+1, in this nite space.
( kjHN+1j k0) = Ek kk0 (3.15)
where the round brackets here are used to indicate that the radial integrals are
calculated using the nite range of integration from r = 0 to r = ra.
Given the form of the basis states k the determination of these coe cients,
however, is exceedingly di cult so the followingapproach is used. Let f' gdenote
collectively the set of basis functions (real, pseudo and continuum orbitals) and
let fVk g denote collectively the set of coe cients (fcijkg and fdijkg) so that
k =
X
' Vk (3.16)
The Hamiltonian matrix elements can then be rewritten as
H 0 = (' jHN+1j' 0) (3.17)
which are evaluated in exactly the same way as that demonstrated in section 2.3.2
where all the radial integrals involving continuum orbitals are taken over a nite
range of r. Subsequent diagonalization of this matrix will then provide Vk along
with the eigenvalues Ek. Since HN+1 is a hermitian operator then the eigenvalues
are real.
To summarise, equation (3.13) may be described generally as follows. The
rst expansion on the right hand side of the equation includes all target states
61. 3.1 The R-matrix method 53
of interest - both the initial and nal states in the collision process speci ed
plus other states which are expected to be closely coupled to them during the
collision. Psuedo-states may also be included in this expansion to approximately
represent continuum states of the target. The second expansion of equation (3.13)
performs two roles. Firstly, it ensures that the total wavefunction is complete
i.e. all con gurations have been accounted for. Secondly it represents short-
range correlation e ects, since the functions j adequately represent part of the
continuum omitted from the rst expansion.
3.1.5 The R-matrix
It has been demonstrated that it is possible to nd the inner region basis wave-
functions f kg. Therefore, to completely solve for the inner region total wave-
function , the energy-dependent coe cients fAEkg from equation (3.12) must
be found using the R-matrix. This matrix relates each continuum orbital value
at the R-matrix radius to the value of the others, and their rst derivatives, at
the boundary. Its use will be made clear later.
Beginning with the following equation
( kjHN+1j ) ( jHN+1j k) = E( kj ) Ek( kj )
= (E Ek)( kj )
(3.18)
which is obtained from equations (3.11), (3.12) and (3.15). To simplify this, note
that only the kinetic energy operator in the Hamiltonian contributes to the left
hand side of this equation leaving
1
2(N + 1) ( kjr2
N+1j ) ( jr2
N+1j k) = (E Ek)( kj ) (3.19)
62. 3.1 The R-matrix method 54
Now de ne the surface amplitudes !ik(r) by
!ik(r) =
X
j
cijk ij(r) = r( ij k): (3.20)
Note that in expression (3.19) the only non-zero contribution to the left hand
side occurs whenever the kinetic energy operator (r2
N+1)acts on the continuum
orbitals. Using this fact and equations (3.12) and (3.20):
1
2
X
ijk0
AEk0 ( i!ik(rN+1)jr2
N+1j j!jk0(rN+1))
( j!jk0(rN+1)jr2
N+1j i!ik(rN+1))]
= (E Ek)( kj ) (3.21)
as ( ij!ik) = k. Now de ne the reduced radial wavefunction of the continuum
electron in channel i at energy E by
Fi(r) =
X
k
AEk!ik(r) = r( ij ) (3.22)
This is a form of the total electron wavefunction. Using the orthonormality of the
channel functions then gives
1
2
X
i
!ik
d2
dr2 Fi Fi
d2
dr2 !ik = (E Ek)AEk (3.23)
Now apply Green's theorem and use the boundary conditions given by equations
(3.7) and (3.8) to obtain
1
2
X
i
!ik(ra) dFi
dr
b
ra
Fi
r=ra
= (E Ek)AEk (3.24)
Rearranging gives an expression for the coe cients and so
AEk = 1
2ra
1
(Ek E)
X
i
!ik(ra) ra
dFi
dr bFi
r=ra
(3.25)
63. 3.1 The R-matrix method 55
De ning the R-matrix by its elements
Rij(E) = 1
2ra
X
k
!ik(ra)!jk(ra)
(Ek E)
(3.26)
so equation (3.22) can be written in the form
Fi(ra) =
X
j
Rij(E) ra
dFj
dr bFj
r=ra
(3.27)
by multiplying equation (3.24) by !ij and summing over k.
The two unknowns on the right hand side of these two equations ((3.26) and
(3.27)), namely the surface amplitudes, !ik(ra), and the R-matrix poles, Ek, can
easily be obtained from the eigenvalues and eigenvectors of the Hamiltonian ma-
trix. These two equations are in fact the basic equations from which the wave-
functions for the internal region can be obtained, thereby describing the electron
scattering problem there. The logarithmic derivative of the reduced radial wave-
function of the scattered electron on the boundary of the internal region is given
by equation (3.27) and must be matched across the boundary to the external
region.
3.1.6 The Buttle correction
The most important source of error in this method is the truncation of the ex-
pansion in equation (3.26) to a nite number of terms. Suppose the R-matrix
expansion is truncated in such a way that the R-matrix is calculated from the
rst N-terms in the continuum expansion. These are low lying contributions
which are obtained from the eigenvectors and eigenvalues of the Hamiltonian ma-
trix. The remaining distant, neglected contributions can play an important role
in the diagonal elements of the R-matrix where they add coherently. They can
64. 3.1 The R-matrix method 56
be included in equation (3.26) by solving the zero-order equation
( d2
dr2
li(li + 1)
r2 + V(r) + k2
i )u0
i(r) =
X
n
ijkPk(r) (3.28)
which is the same as equation (3.6) but is solved here at channel energies k2
i
without applying the boundary conditions (3.7) and (3.8) at r = ra The correction
Rc
ii to the diagonal elements of the R-matrix at the energy k2
i necessary due to
truncation is then given using the formula discussed by Buttle (1967).
Rc
ii(N;k2
i ) 1
ra
1X
j=N+1
uij(ra)2
k2
ij k2
i
= ra
u0
i(ra)
du0
i
dr r=ra
b
1
1
ra
NX
j=1
uij(ra)2
(k2
ij k2
i ) (3.29)
where uij(r) and kij refer to the jth eigensolution of equation (3.6) satisfying
the boundary conditions of equations (3.7) and (3.8) and u0
i is the solution for
channel energy 1
2k2
i in atomic units. Note that the second term of equation (3.29)
subtracts those levels which have already been included. Henceforth the Buttle
corrected R-matrix is used.
Rij(E) = 1
2ra
NX
k=1
!ik(ra)!jk(ra)
(Ek E) + Rc
ii(N;k2
i ) ij (3.30)
3.1.7 The external region
The next stage in the calculation is to solve the electron-target scattering problem
in the external region, r > ra which is less complex due to the lack of exchange
and correlation with the continuum electron. In this region the colliding electron
is outside the ion and can be considered distinguishable from the target electrons
65. 3.1 The R-matrix method 57
i.e. antisymmetrisation can be neglected.
(x1;:::;xN+1) =
X
i
i(x1;:::;xN;^rN+1 N+1)Fi(rN+1) (3.31)
where the same channel functions, i, have been used which were present in
equation (3.13) with the exception that antisymmetrization is no longer required.
Substituting this expansion into the Schrodinger equation (3.11) produces
X
i
(HN+1 E) iFi(rN+1) = 0 (3.32)
From the de nition of the Hamiltonian operator given in equation (3.11) the N-
electron and (N + 1)-electron Hamiltonians can be related to give the following
equation
"
X
j
(HN E) i
1
2
r2
N+1
Z
rN+1
i +
NX
k=1
1
rk;N+1
i
!#
Fi(rN+1) = 0 (3.33)
From equations (3.2) and (3.3),
(HN E) i = ( i E) i = k2
i
2 i (3.34)
where k2
i were the channel energies and i were the target energies. Combining
these two equations, premultiplying by i and integrating over all coordinates
except rN+1 it can be seen, due to the orthonormality of the channel functions
(i.e.
R
i jdr = ij), that
X
i
Z
j
nX
k=1
1
rk;N+1
jdr
!
Fj(rN+1) k2
i
2
+ 1
2
r2
N+1 + Z
rN+1
Fi(rN+1) = 0
(3.35)
where n represents the number of channel functions which were used in equations
(3.13) and (3.31). The potential matrix Vij(r) (the long range multipole potentials)
66. 3.1 The R-matrix method 58
is de ned by
Vij(r) =
*
i
NX
k=1
1
rk;N+1
j
+
(3.36)
which upon substitution into equation (3.35) produces the following set of coupled
di erential equations
d2
dr2
li(li + 1)
r2 + 2Z
r + k2
i Fi(r) = 2
nX
=1
Vij(r)Fj(r) i = 1;n(r ra) (3.37)
where li is the channel angular momentum. Now de ne the long range potential
coe cient by
aij =
*
i
NX
k=1
rkP (cos k;N+1) j
+
(3.38)
Due to the orthonormality of the channel functions
a0
ij = N ij (3.39)
which combined with the following expansion of 1
rk;N+1
in terms of Legendre Poly-
nomials
NX
k=1
1
rk;N+1
=
1X
=0
1
r +1
N+1
NX
k=1
rkP (cos k;N+1) (3.40)
reduces the di erential equations of (3.37) to
d2
dr2
li(li + 1)
r2 + 2z
r + k2
i Fi(r) = 2
maxX
=1
nX
j=1
aij
r +1 Fj(r) (3.41)
where z = Z N is the residual target charge. Note that max is nite and is
determined by the angular momentum algebra in equation (3.38). This type of
equation has been the subject of much discussion and computer programs are
available for its solution (Norcross 1969, Norcross and Seaton 1969 and Chivers
1973) but before this can be accomplished various boundary conditions must be
set up to ensure that the solutions obtained from this equation match the solutions
already obtained from the internal region problem at the R-matrix radius.
67. 3.1 The R-matrix method 59
3.1.8 Matching the solutions
These boundary conditions depend upon the status of the (N +1)th electron, i.e.
whether it is bound or free. If it is free (i.e. it resides in the external region)
then the channels associated with this electron are open. If on the other hand the
(N+1)th electron is bound (i.e. it resides in the internal region) then the channels
are all closed. Suppose then there are a total of n channels where na is denoted by
the number of open ones leaving n na closed channels. Then a natural boundary
condition for the reduced radial wavefunction at in nity obtained by tting to an
asymptotic expansion is
Fij(r) r!1
8
><
>:
1pki
(sin i ij + cos iKij) i=1;:::;na
j=1;:::;na
0(r 2) i=(na+1);:::;n
j=1;:::;na
9
>=
>;
(3.42)
where
i = kir 1
2li iln2kir + li
i = z
ki
li = arg (li + 1 + i i)]
9
>>>>=
>>>>;
(3.43)
and a second index, j, has been introduced on the reduced radial wavefunction
Fij(r) to label the na independent solutions (the rst index i corresponds to the
channel). This equation is used as a de nition for the reactance matrix K whose
standard matrix element is Kij.
3.1.9 Open Channels
An n n dimensional R-matrix in the internal region solution now exists and an
na na dimensional K-matrix in the external region. These two matrices must be
related in order for the solutions of each region to match at the boundary. This
is achieved by introducing a set of (n + na) linearly independent solutions vij of
