Download Measurements of Photoionization Cross Sections of Positive and

Measurements of Photoionization
Cross Sections of Positive and Negative
Ions Using Synchrotron Radiation
Tommy Steindorff Jacobsen
University of Aarhus
Denmark
February 2004
ii
Preface
This thesis has been submitted to the Department of Physics and Astronomy at the University of Aarhus in order to obtain the MSc degree (Cand.
Scient). It is the result of one year of experimental work at the undulator
beamline at the ASTRID storage ring in Aarhus where beamtime has been
available during the weeks 8, (10+12 lost because of defect dipole supply at
the ring) 17, 18, 33, 36, 40 and 42 in 2003. The report will concentrate on
photodetachment measurements of negative tellurium and on photoionization of members of the beryllium, boron and carbon isoelectronic sequences,
but I have also contributed to the measurements of V− , Cr− , Co− and Ni−
as well as Ba2+ , La3+ and Ce4+
Throughout the period I have had the great privilege of working with
a number of kind and skilled people. To begin with, I am indepted to my
supervisor Henrik Kjeldsen for teaching me about experimental details and
physics in general in a patient and inspiring way and I wish him the best of
luck in his future activities within the area of AMS. Also, I wish to thank
the other group member Finn Folkmann for a very pleasant co-operation
and for taking the time to help with the proof-reading. Being a part of their
team, sharing both successes and distresses, has been an enjoyable as well as
instructive experience and in many ways it can be regarded as the highlight
of my time as a student at the University.
Of foreign collaborators I have had the pleasure of working with people
such as John West, Jean-Marc Bizau, Francis Penent, Denis Cubaynes, Nacer
Adrouche and Jørgen Hansen. Finally, I have benefitted from the expertise
iii
Preface
of members of the IFA and ISA staff - in particular Egon Jans, Kaare Iversen
and Jens Vestergaard - and their help is gratefully acknowledged.
Randers
February 2004
Tommy Steindorff Jacobsen
iv
Contents
Preface
iii
Introduction
1
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1 Theory
1.1
5
Atomic theory . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.1.1
The Schrödinger equation . . . . . . . . . . . . . . . .
5
1.1.2
Many-electron atomic systems . . . . . . . . . . . . . .
6
1.1.3
The independent particle model . . . . . . . . . . . . .
7
1.1.4
Configuration and shell structure . . . . . . . . . . . .
9
1.1.5
LS-coupling and spin-orbit splitting . . . . . . . . . . . 10
1.2
Photoionization cross section . . . . . . . . . . . . . . . . . . . 11
1.3
Interpreting experimental spectra . . . . . . . . . . . . . . . . 13
1.4
1.3.1
Positive ions . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.2
Negative ions . . . . . . . . . . . . . . . . . . . . . . . 16
“Hardcore” theoretical approaches . . . . . . . . . . . . . . . . 18
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2 The experimental setup
2.1
21
The light source . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.1.1
The undulator . . . . . . . . . . . . . . . . . . . . . . . 22
v
CONTENTS
2.1.2
2.2
2.3
CONTENTS
The Miyake monochromator . . . . . . . . . . . . . . . 25
The absolute cross section . . . . . . . . . . . . . . . . . . . . 28
2.2.1
Form factors . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2.2
Detector calibration . . . . . . . . . . . . . . . . . . . . 30
2.2.3
Photodiode calibration . . . . . . . . . . . . . . . . . . 31
Recording spectra - an overview . . . . . . . . . . . . . . . . . 33
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3 Ion production
35
3.1
General remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2
The Middleton sputter source . . . . . . . . . . . . . . . . . . 36
3.3
3.2.1
Surface ionization . . . . . . . . . . . . . . . . . . . . . 36
3.2.2
Sputtering . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2.3
Description of the Middleton source . . . . . . . . . . . 37
The ECRIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3.1
The ECRIS principle . . . . . . . . . . . . . . . . . . . 41
3.3.2
Description of the ECR source . . . . . . . . . . . . . . 43
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4 Photodetachment of Te−
47
4.1
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.2
Results and analysis . . . . . . . . . . . . . . . . . . . . . . . 48
4.2.1
The continuum . . . . . . . . . . . . . . . . . . . . . . 49
4.2.2
Resonances . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2.3
Oscillator strengths . . . . . . . . . . . . . . . . . . . . 52
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5 Photoionization results
5.1
55
B-like . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.1.1
Results and analysis . . . . . . . . . . . . . . . . . . . 56
5.1.2
Metastable fractions . . . . . . . . . . . . . . . . . . . 63
5.1.3
Comparison with theory . . . . . . . . . . . . . . . . . 63
vi
CONTENTS
5.2
Be-like . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.2.1
5.3
5.4
CONTENTS
Results and analysis . . . . . . . . . . . . . . . . . . . 66
C-like . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.3.1
Results and analysis . . . . . . . . . . . . . . . . . . . 68
5.3.2
Comparison with theory . . . . . . . . . . . . . . . . . 73
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Concluding remarks
77
A Useful material
79
A.1 Selection rules in LS-coupling . . . . . . . . . . . . . . . . . . 79
A.2 Coupling of equivalent electrons . . . . . . . . . . . . . . . . . 80
A.3 Hund’s rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
A.4 The periodic table . . . . . . . . . . . . . . . . . . . . . . . . 81
vii
viii
Introduction
Motivation
Photoionization of neutral atoms and positive ions and photodetachment of
negative ions are important examples of fundamental processes where light
and matter interact. Such processes occur in Nature all the time e.g. in stellar
and planetary atmospheres, interstellar nebulae, plasmas etc. Schematically
photoionization or detachment can be written:
Photoionization:
Aq+ + hν → A(q+1)+ + e−
(1)
A− + hν → A + e− .
(2)
Photodetachment:
In the case of inner-shell detachment the neutral atom in (2) will be in an
excited state that most likely will decay further by Auger decay, resulting in
a positively charged end-product.
As should be apparent, absolute photoionization cross sections are extremely important to know especially in astrophysics, since such data are
needed to interpret the observations made by space and Earth observatories and for modelling stellar objects. Another important issue is the study
of electron-correlation effects which are more dominant compared to the
Coulomb interaction with the nucleus in negative ions, due to the more effective screening. Despite these obvious demands of experimental results,
by far the largest amount of data has been brought by theorists. For instance, large international collaborations such as The OPAL Project (Roger
1
Motivation
Introduction
& Iglesias 1994), The OPACITY Project (1995) and The Iron Project (Nahar & Pradhan 1994, Hummer et al 1993) have put considerable effort into
the calculations of photoionization cross sections for all elements up to iron,
and substantial theoretical work has also been performed for negative ions
(Ivanov 1999). The different theoretical approaches (some will be mentioned
in the next chapter) all use state-of-the-art techniques, but it is crucial that
these models are tested experimentally.
The lack of absolute experimental contributions in this field stems mainly
from technical difficulties, like the determination of the target-atom density
for neutral atoms other than the noble gases. The same problem does not
occur with ions since the density of an ion beam can be determined very
accurately, but the fact that ions are charged particles prevents the density
from getting very high1 . The first absolute measurements for ions using the
so-called merged-beam method was obtained in the mid-eighties at Daresbury Laboratory, UK, in the pioneering work of Lyon et al (see Wuilleumier
et al 1994 for a review), who used synchrotron radiation from a bending
magnet as the source of light. The photon flux allowed only for the measurement of rather large cross sections (& 5M b = 5 · 10−18 cm2 ) and results were
obtained for Ba+ , Ca+ , Sr+ , K+ , Zn+ and Ga+ . With the advent of insertion
devices such as undulators it has been made possible to achieve intensities
that are orders of magnitude higher than from ordinary bending magnets
thus switching the main limitations to the production of stable ion beams.
Undulator-based beam lines at synchrotron radiation facilities in Denmark
(ASTRID), France (Super ACO), USA (ALS) and Japan (SPring8) are using
the merged-beam method to measure absolute photoionization cross sections,
and results have been obtained for many singly-charged positive ions. Some
multiply-charged ions, produced mainly in Electron Cyclotron Resonance
Ion Sources (ECRIS), have also been investigated thereby allowing for the
study of interesting trends and features along isonuclear and isoelectronic se1
typical densities are in the range of 105 − 106 cm−3 which for comparison corresponds
to a pressure of about 10−11 mBar, i.e. it is as low as a good UHV.
2
Introduction
Outline
quences. So far, inner-shell photodetachment measurements have only been
performed for a very small number of negative ions and the need for experimental data is therefore especially profound in this field.
Outline
In my opinion, a report of this kind should be more or less self-explanatory,
which, of course, is a rather ambitious point of view that could probably fill
several books if truly carried out. It is my aim and hope, however, that what
I have selected will give the reader a basic understanding of the experiment
and not leave too many unanswered questions. The content of this thesis is
divided into three major parts covering theoretical and experimental details
as well as selected experimental results.
PART ONE
Chapter 1: Here some of the basic theoretical concepts from quantum mechanics and atomic theory are refreshed and more sophisticated models for
treating many-electron atomic systems are mentioned briefly.
PART TWO
Chapter 2: Describes the experimental setup. To a large extent this will
follow the work done by former students (see references in chapter 2) and
will therefore not be so detailed.
Chapter 3: The subject of producing an appropriate ion beam is adressed
with emphasis on the description of the two types of ion sources used; namely
Middleton’s high-intensity sputter source and the ECRIS.
3
Outline
Introduction
PART THREE
Chapter 4: Presents the results obtained for photodetachment of 4d electrons
in Te−
Chapter 5: The photoionization cross sections of some ions of astrophysical
importance will be presented. More precisely the chapter will concentrate on
studies of N2+ , O3+ , O4+ , F3+ , F4+ and Ne4+ .
References
Hummer D. G., Berrington K. A., Eissner W., Pradhan A. K., Saraph H. E.
and Tully J. A. (1993) Astron. Astrophys. 279 298
Ivanov V. K. (1999) J. Phys. B: At. Mol. Opt. Phys. 32 R67
Nahar S. A. and Pradhan A. K. (1994) Phys. Rev. A 49 1816
Roger F. J. and Iglesias C. A. (1994) Science 263 50
The OPACITY Project (1995) (Institute of physics: Bristol, vol. 1)
Wuilleumier F. J., Bizau J.-M., Cubaynes D., Rouvelou B. and Journel L.
(1994) Nucl. Instr. Meth. B 87 190
4
Chapter 1
Theory
As one of the early motivations for performing the photoionization experiments was to provide data for testing different theoretical approaches, a little
time should be spent on this subject. On the other hand, a deep analysis
is way beyond the scope of this thesis and so only some of the basic things
of relevance for the understanding of the obtained results are treated. The
chapter is built up of sections with the first one reminding us of some key
aspects in atomic theory. Moreover the chapter will briefly deal with the
fundamental equations of photoionization (detachment), the interpretation
of cross-section spectra and “hardcore” theoretical approaches.
1.1
1.1.1
Atomic theory
The Schrödinger equation
In modern quantum mechanics a particle is not pictured as a little “glass
globe” but is described by a wave function, Ψ. If the particle is moving in a
potential, V (r, t), the wave function will satisfy the differential equation
n
o
∂
~2 2
i~ Ψ(r, t) = −
∇ + V (r, t) Ψ(r, t).
(1.1)
∂t
2m
(1.1) is the so-called non-relativistic time-dependent Schrödinger equation,
and recalling the relation p = i~∇ for the momentum operator, the term in
5
1.1 Atomic theory
Theory
the curly brackets is just an expression of the total energy of the particle and
is called the Hamiltonian operator or simply the Hamiltonian,
H=T +V =−
~2 2
∇ + V.
2m
(1.2)
If the potential is static, V (r, t) = V (r), eq. (1.1) admits stationary state
solutions of the form
Ψ(r, t) = ψE (r)e−iEt/~ ,
(1.3)
where ψE (r) is a solution to the time-independent Schrödinger equation
HψE (r) = EψE (r).
(1.4)
That is, E is an eigenvalue and ψE (r) an eigenfunction of the Hamiltonian,
H. Determining the dynamical evolution of the wave function then boils
down to solving eq. (1.4) with the set of eigenvalues making up the atom’s
energy spectrum.
1.1.2
Many-electron atomic systems
If the system is hydrogen-like, the single electron will only feel a Coulomb
attraction V (r) = − Ze
from the nucleus of charge Ze. (1.4) can then be
r
solved analytically, and the energy eigenvalues can be written
En (eV ) = −Ry ·
Z2
,
n2
(1.5)
where Ry = 13.6057 eV is the Rydberg energy and n is the main quantum
number describing the energy states of the ion. This result in the special case
of hydrogen (Z = 1) was already obtained by Bohr in his famous article from
1913 (Bohr 1913) but without using the full machinery of wavemechanics
which was developed only later. In the more general case of a many-electron
atom an electron will also feel the presence of the other electrons thereby
adding an extra term to the full Hamiltonian which now reads
H=
N X
i=1
~2 2 Ze2 X e2
−
∇ −
−
,
2m i
ri
r
i<j ij
6
(1.6)
Theory
1.1 Atomic theory
(a)
(b)
Figure 1.1: a) Niels Bohr (1885-1962). b) Bohr’s description of the atom. The
electrons can exist in special orbits and jump between orbits by absorbing or
emitting photons of energy hνa→b = Eb − Ea .
where N is the number of electrons and rij ≡ |ri − rj | is the distance between
the ith and the jth electron.
Because of the fact that changes for one electron will affect the others, i.e.
through the electron correlation, an analytical solution to (1.4) can no longer
be found. This is true even in the simplest case of helium-like systems with
only two electrons which implies the use of different approximative methods.
1.1.3
The independent particle model
Although each 1/rij term in (1.6) is usually small compared to Z/ri , the sum
over j can still be comparable in size with the Coulomb attraction between
P
electron i and the nucleus. Therefore, treating i<j 1/rij as a perturbation
is not a valid option. It is still possible, however, to use perturbation theory
by applying an independent particle model. In such a model, each electron is
most often imagined to move independently in an effective centrally symmetric potential which represents the attraction of the nucleus and the average
7
1.1 Atomic theory
Theory
effect of the repulsion between this and the other (N − 1) electrons. This
effective potential can be written
Vef f (r) = −
Ze2
+ S(r),
r
(1.7)
where S(r) is some model potential representing the screening of the nucleus
by the remaining electrons. Adding and subtracting Vef f (r) in the Hamiltonian (1.6) yields
H=
N X
i=1
|
N
N
X
~2 2
e2 X Ze2
−
∇ + Vef f (ri ) +
−
+ Vef f (ri ) .
2m i
r
ri
i<j ij
i=1
{z
} |
{z
}
Hc
(1.8)
Hper
The result of this rewriting is that the second term is now much smaller
and can be treated perturbatively. By neglecting this term at first and just
writing the Schrödinger equation for Hc , one sees that this just separates
into N one-electron Schrödinger equations:
hi φi = Eφi ,
hi ≡
−
~2 2
∇i + Vef f (ri ) .
2m
(1.9)
Solutions to this equation can be found and the resulting single-electron wave
functions or orbitals, φi , are described by the quantum numbers (n, l, ml ),
i.e. they are simultaneous eigenfunctions of the Hamiltonian hi , the angular
momentum operator Li and its projection onto the z-direction Lz,i . Since
electrons are fermions they have an intrinsic magnetic moment, the spin
S, which also must be included in the wave functions. Doing this, see e.g.
(Bransden & Joachain 1983), the full wave function can be expressed as a
so-called Slater determinant, i.e. an antisymmetric product of the N spinorbitals automatically fullfilling the Pauli exclusion principle stating that
two electrons in an atom cannot have the same set of quantum numbers.
8
Theory
1.1.4
1.1 Atomic theory
Configuration and shell structure
The single-electron spin-orbitals are described by the quantum numbers (n, l, ml , s, ms )
and these can take on the values
n = 1, 2, . . .
l = 0, 1, . . . , n − 1
ml = −l, . . . , l
(1.10)
s = 1/2
ms = ±1/2
where the reference to s often is omitted since this is always 1/2 for fermions.
Because of the spherical symmetry of the potential (1.7) the energy of each
electron will be independent of the magnetic quantum numbers ml and ms ,
but unlike the hydrogenic case the energies will not be degenerate in l. This
is because the screening of the nucleus due to the other electrons will be
more pronounced for electrons with large angular momentum, as these are
forced out by the centrifugal barrier. The total energy of an atom will just be
the sum of energies, Eni ,li , of each electron and therefore will be completely
determined by the configuration, by which is meant the distribution of the
electrons with respect to n and l. This arranges the electrons into shells
(electrons with same n) and subshells (electrons with same n, l) and from
(1.10) it is seen that each subshell can hold 2(2l + 1) electrons with different
ml , ms . Using the typical nomenclature where l = 0, 1, 2, 3, 4, . . . corresponds
to the letters s,p,d,f,g,. . . the configuration of e.g. Si with 14 electrons in all
can be written
1s2 2s2 2p6 3s2 3p2 ,
(1.11)
where the superscript denotes the number of electrons in the same subshell,
i.e. equivalent electrons.
9
1.1 Atomic theory
1.1.5
Theory
LS-coupling and spin-orbit splitting
In the central-field approximation, the total energy is simply given by the
configuration of the system. A more precise description must include the
term, Hper , and another important perturbation, the spin-orbit interaction,
must also be considered. This is a relativistic effect arising from the interaction of the spin of the electron with the magnetic field induced by the orbiting
nucleus (in the restframe of the electron). This interaction is described by
the Hamiltonian
HSO =
N
X
i=1
1 1 dV (ri )
Li ·Si ,
2m2 c2 ri dri
(1.12)
and will be a small correction for light atoms (ions).
The operator Hper will not commute with neither Li nor Si but it will commute with the total orbital and spin angular momenta
L=
N
X
Li
and S =
i=1
N
X
Si .
(1.13)
i=1
Including Hper therefore splits each configuration into terms, symbolized as
2S+1
L, that are (2S + 1)(2L + 1) times degenerate with respect to ML and
MS . This is the essence of the LS coupling scheme 1 and the terms may
be found by adding the spin and orbital angular momenta of the electrons
in accordance with the quantum mechanical rules. However, for equivalent
electrons not all possible terms are allowed because of the Pauli principle. In
appendix A.2 the allowed terms for coupling equivalent s, p- and d electrons
are given, as well as Hund’s rules that tell which term will be lowest in
energy. Inclusion of the spin-orbit operator will lead to a further splitting
of these terms into fine structure components, which are eigenstates of the
total angular momentum operator J = L + S, and each eigenstate will be
characterized by the symbol 2S+1 LJ . In figure 1.2 the energy splitting due to
1
This coupling scheme is actually also often used for heavier atoms although these are
more precisely described using jl coupling where Li and Si are coupled to give Ji and
then summed to give the total J. LS coupling will be used throughout this thesis.
10
Theory
1.2 Photoionization cross section
the inclusion of the perturbations is pictured schematically in the case of Si.
Figure 1.2: Effects (not to scale) of the perturbations Hper and HSO for Si.
1.2
Photoionization cross section
Consider an atom or an ion initially in the state Ψi interacting with an unpolarized electromagnetic field. If a photon of energy ~ω, exceeding the binding
energy of one of the electrons, is absorbed, the result may be that of photoionization (or detachment for negative ions) where the electron will escape
the atomic system. This will then be characterized by a final continuum wave
function, Ψf . In the dipole approximation the cross section for this process
is given by (in cgs units)
σif =
4π
ωif |Df i |2 .
3c
(1.14)
Here ωif is the energy of the absorbed photon which from energy conservation
must equal the binding plus the kinetic energy of the free electron. The
matrix element Df i is defined as
Df i ≡< Ψf |D|Ψi >,
11
(1.15)
1.2 Photoionization cross section
Theory
with D being the dipole operator
D = −e
N
X
ri .
(1.16)
i=1
Note that the sum runs independently over all electrons which means that
the dipole operator is a single-particle operator and that only one photon
can interact with one electron. This implies that only main photoprocesses,
i.e. excitation or ionization of one electron are possible in this picture. If
the wave functions in (1.15) are described by the LS coupling scheme, the
non-zero matrix elements must obey the selection rules
∆S = 0 ∆L = 0, ±1 ∆J = 0, ±1
∆π = ±1 Li + Lf ≥ 1 Ji + Jf ≥ 1.
(1.17)
As before, S, L and J are the spin, orbital and total angular momenta
respectively and π is the parity describing whether the wave function is odd
(-) or even (+) under the operation r → −r.
Another important parameter, closely related to the cross section, is the
oscillator strength, fi,f of a dipole transition between the initial and final
states Ψi and Ψf .
2mωif
|Dif |2 .
(1.18)
3e2 ~
Since both the initial and final state may be multiplet states, it is often more
fi,f =
useful to consider the mean oscillator strength defined as
1 XX
f¯i,f =
fi,f .
gi α α
i
(1.19)
f
If the states are given as terms, αi and αf will refer to the magnetic quantum
numbers of the initial and final multiplets and gi is the statistical weight of
the initial state, i.e. gi = (2li + 1). It can be shown for hydrogen-like ions
(Bransden & Joachain 1983) that both f and f¯ obey the Thomas-ReicheKuhn sum rule
X
fi,f = 1,
f
12
(1.20)
Theory
1.3 Interpreting experimental spectra
where the sum is over all possible bound or unbound final states. Finally,
the following is a useful equation for an experimental determination of the
mean oscillator strength:
f¯ = 0.00911 ·
Z
σ(E)dE,
(1.21)
where σ is in Mb and E is in eV.
1.3
Interpreting experimental spectra
Although the equations given in the last section apply to both photoionization and photodetachment, the two types of spectra differ markedly due to
the structural differences in positive and negative ions, and the description
of these spectra has been split up accordingly. Examples and the analysis of
both types are given in chapters 4 and 5 and so the following will just be an
overview of the gross details.
1.3.1
Positive ions
As shown in figure 1.3, the main features of a photoionization spectrum of
neutral atoms and positive ions will typically consist of a continuum on which
a number of autoionizing resonances are superimposed. The continuum cross
section rises abruptly at the threshold where an incoming photon has just
enough energy to send an electron directly into the continuum. The cross
section will subsequently fall off with increasing photon energy until the
threshold of a more strongly bound electron is reached. The resonances
correspond to highly excited discrete states embedded in the continuum.
Such a resonance can make a radiative transition back to a bound state,
but it is much more probable that it makes a radiationless transition into
the continuum by emission of an electron, see figure 1.4. This process is
called autoionization and is very fast compared to radiative de-excitation.
The selection rules for autoionization depend on the operator coupling the
discrete state with the continuum and here Coulomb autoionization is the
13
1.3 Interpreting experimental spectra
Theory
Figure 1.3: Absolute photoionization cross section for Cs+ (5p6 1 S) (not published). The cross section rises abruptly at the 5p−1 threshold at EIP = 23.16 eV
(McIlrath et al 1986). Superimposed on the continuum is a number of resonances
corresponding to 5p → ns,nd excitations which subsequently autoionize into the
continuum. These resonances are members of Rydberg series converging to the
Cs2+ (5p5 2 P3/2 ) series limit.
most important. However, states that are not allowed to autoionize through
the Coulomb interaction may still autoionize by means of e.g. the spinorbit interaction which will be much slower. A table of selection rules and
typical lifetimes of autoionizing states is given in appendix A.1 for the most
important interactions.
As there is no way to determine whether the final continuum state is
reached by direct ionization or by resonant ionization (i.e. excitation followed
by autoionization, see figure 1.4), the two ionization paths can interfere and
the peaks in the spectrum will have characteristic profiles called Beutler-Fano
14
Theory
1.3 Interpreting experimental spectra
Figure 1.4: Example of two possible ioniza-
Figure 1.5: Beutler-Fano profiles with differ-
tion routes of a B-like ion in the ground state.
ent values of the Fano parameter, q. The pro-
Either by direct ionization or by photoexcita-
file becomes Lorentz-like as q goes to infinity.
tion followed by autoionization.
profiles (Fano 1961). Such profiles are described by the formula
σ() = σ0
(q + )2
,
1 + 2
(1.22)
where is the reduced energy or the detuning
=
E − ER
.
Γ/2
(1.23)
σ0 denotes the direct cross section, ER the energy of the resonance and Γ is
the natural linewidth. The generally asymmetric shape of the profile is due
to interference between the two ionization routes and this is incorporated
into the Fano parameter, q. Figure 1.5 shows profiles for different values of q.
Note that the cross section is 0 whenever q = − and the case when q = = 0
gives rise to a so-called window resonance. From eq. (1.22) it is also seen
that when q is large, i.e. if one route is dominating, the profile approaches a
Lorentzian. Furthermore, when we are far from resonance, i.e. → ∞, the
cross section is simply the direct cross section, just as one would expect.
15
1.3 Interpreting experimental spectra
Theory
Quantum defects
Resonances are often members of Rydberg series, i.e. they are excited orbitals
with successively higher n-values. An example of such Rydberg members are
the 1s2 2s2 2p(2 P) → 1s2 2s2p(3 P)np(2 P) excitations that show up in the spectrum of N2+ → N3+ . These will converge to the N3+ (1s2 2s2p(3 P)) ionization
limit corresponding to n = ∞.
A very powerful tool when trying to identify the experimental peaks is to
use the simple concept of Quantum Defect Theory (QDT). The idea is that
the energy levels of a Rydberg series are given by a formula similar to (1.5)
with the difference that n is replaced by an effective quantum number, n∗ .
Z2
.
(1.24)
n ∗2
is the ionization limit of the series, Z is the charge state of the
En∗ = E∞ − Ry
Here E∞
ionized ion and n∗ ≡ (n − δnl ) where δnl is the quantum defect that takes
into account the partial screening of the nucleus. In the case of hydrogen-like
systems we have E∞ = δnl = 0 and eq. (1.24) simply reduces to (1.5). The
true power of this theory lies in the fact that the variations of δnl with n is
small meaning that δnl ≈ δl . Thus, if one knows a member of the series and
the limit to which it converges, it is possible to calculate the quantum defect
and thereby the position of all other members under the assumption that δ
is constant.
1.3.2
Negative ions
Negative ions occupy a unique place among the atomic species because of
the strong correlations of the valence electrons. Whereas correlations to
a large extent can be regarded as pertubations to the Coulomb attraction
betweem the electron and the positive nucelus in atoms and positive ions,
these correlations are essential in the binding of the extra electron in negative
ions. This fact makes them extremely difficult to handle theoretically and
the need for experimental results is therefore particularly profound in this
field (Ivanov 1999).
16
Theory
1.3 Interpreting experimental spectra
Absolute photodetachment studies have so far mainly dealt with detachment of the loosely bound outer-shell electrons utilizing lasers as the source of
photons, but in the ASTRID experiments inner-shell detachment is studied.
The creation of an inner-shell hole will almost certainly lead to the formation of a positively charged end-product which is subsequently detected (see
figure 2.1). The process of interest can be written
−
−
+
A− + ~ω → A∗ + e−
photo → A + ephoto + eAuger ,
(1.25)
where the decay of the excited neutral state is known as an Auger decay.
A cascade of Auger decays may also occur thus forming a multiply charged
end-product.
Since it is not the long-range Coulomb attraction that holds the electron
attached but instead some short-range potential induced by the polarization
of the neutral atom by the extra electron, a photodetachment spectrum of
negative ions lacks many of the features found in photoionization spectra.
Negative ions, for example, usually only exist in one bound state in contrast
to the infinity of Rydberg states found in positive ions and neutral atoms.
As a consequence, there are no resonances corresponding to different converging Rydberg series. It is sometimes possible, however, as in the case of
4d detachment in Te− (see chapter 4), that an excited state is bound with
respect to its ionization threshold. Such states give rise to relatively narrow
resonances known as Feshbach resonances below threshold in the detachment
spectrum. A more typical type of resonance is the shape resonance which is
a broad structure located above threshold and which corresponds to what is
sometimes referred to as a “nearly” bound state.
The threshold behaviour will also be different because the outgoing photoelectron will move in a potential that is very much influenced by the centrifugal barrier, Vcen (r) = l(l + 1)~2 /2mr2 , and thus show a dependence of
l. It can be shown that the cross section immediately (∼ µeV) above the
threshold energy, Eth , is given by the Wigner threshold law (Wigner 1948),
σ(E) ∝ (E − Eth )l+1/2 .
17
(1.26)
1.4 “Hardcore” theoretical approaches
1.4
Theory
“Hardcore” theoretical approaches
Returning to eq. (1.14), the calculation of the cross section is seen to depend entirely on the ability to evaluate the matrix elements (1.15) which
again means finding initial and final state wave functions as solutions to the
Schrödinger equation. One way to accomplish this, as we have seen, is to apply an independent particle picture of which the Hartree-Fock method (HF)
or Self-consistent field (SCF) is the most precise and well-known, see e.g.
(Fischer 1977). Such an approach may succeed in giving the overall shape
of the spectrum but it fails to reproduce features such as autoionizing resonances that are direct consequences of electron correlations. Therefore, one
can only hope to get quantitative agreement by using models that include
correlations. A number of such models have consequently been developed
that differ mainly in the way in which the electron-electron interaction is
included. The models most frequently encountered are the Configuration Interaction method (CI), the Random Phase Approximation (RPA), the Local
Density Approximation (LDA), Many-Body Perturbation Theory (MBPT),
Multi Configuration Dirac-Fock (MCDF) and the R-Matrix method. Time
and space allow only for a very few remarks to be given here but details as
well as some of the main achievements of these models can be found in e.g.
(Becker 1996, Chang 1993, Schmidt 1997) and references therein.
In the CI approach the correlated wave functions Ψi and Ψf are expanded
into a complete set of un-correlated basis functions with the same symmetry
properties. As an example, one would write the ground state of Li+ as
Ψcorr = a1 Ψ(1s2 1 S0e ) + a2 Ψ(1s2s 1 S0e ) + a3 Ψ(2s2 1 S0e ) + a4 Ψ(2p2 1 S0e ) + . . .
(1.27)
where the absolute square of the mixing coefficients, an , is an expression of
the weights of the corresponding states. Such an expansion requires, in principle, an infinite number of basis functions which is not possible for practical
purposes, since all expansions have to be truncated into a finite number for
the calculations to be carried out. The quality of a CI calculation therefore
18
Theory
1.4 “Hardcore” theoretical approaches
depends on the number and type (usually HF one-electron wave functions)
of basis function used (Schmidt 1997). This way of incorporating electron
correlations is common for many of the different models and in chapter 5
some of the experimental results will be compared to both R-matrix results found in the OPACITY database (TOPbase, Cunto et al 1993) and to
new MCDF calculations provided by (Bizau 2003). One should note that
although these methods are both based on a CI description of initial and
final states, they are somewhat different. The R-matrix method calculates
the photoionization cross section whereas the MCDF method calculates the
photoexcitation cross section under the assumption that the excited states
decay by autoionization. Moreover, any interference between direct and resonant ionization, which would lead to asymmetric profiles of the resonances,
is neglected in the latter.
References
Becker U. and Shirley D. A. (1996) VUV and Soft X-Ray Photoionization
(Plenum Press)
Bizau J. M. (2003) - Private communication
Bohr N. (1913) Phil. Mag. 26 1
Bransden B. H. and Joachain C. J. (1983) Physics of Atoms and Molecules
(Longman Group Limited)
Chang T.-N. (1993) Many-Body Theory of Atomic Structure and Photoionization (World Scientific Publishing Co. Pte. Ltd.)
Cunto W., Mendoza C., Ochsenbein F. and Zeippen C. J. (1993) Astronomy
and Astrophysics 275 L5
Fano U. (1961) Phys. Rev. 124 1866
Fischer C. F. (1977) The Hartree-Fock method for atoms (John Wiley &
Sons, Inc.)
Gasiorowicz S. (1996) 2.ed. Quantum Physics (John Wiley & Sons, Inc.)
Ivanov V. K. (1999) J. Phys. B: At. Mol. Opt. Phys. 32 R67
19
1.4 “Hardcore” theoretical approaches
Theory
McIlrath T. J., Sugar J., Kaufman V., Cooper D., and Hill W. T. III. (1986)
J. Opt. Soc. Am. B 3 398
Sakurai J. J. (1994) Modern Quantum Mechanics (Addison-Wesley Publishing Company, Inc.)
Schmidt V. (1997) Electron Spectrometry of Atoms using Synchrotron Radiation (Cambridge University Press)
TOPbase - www.heasarc.gsfc.nasa.gov/topbase/topbase.html
Wigner E. P. (1948) Phys. Rev. 73 1002
20
Chapter 2
The experimental setup
Figure 2.1: The experimental setup: IS ion source, EL Einzel lens, M1-2 Bending
magnets, ED Electrostatic deflector, RC Reaction chamber, D1-2 Ion detectors,
FC1-2 Faraday cups (positive target beams), FC3 Faraday cup (negative target
beam), PD Photodiode, UN Undulator and MO Monochromator.
Fig. 2.1 above shows schematically the experimental setup. In short, a
beam of target ions is produced in the ion source (IS) and accelerated by
a high voltage, Vacc (typically 2 kV), to an energy E = qeVacc . The beam
is then focused (EL) and mass analyzed (M1) before it is steered into the
reaction chamber (RC) by means of an electrostatic deflector (ED), and here
the target beam is overlapped colinearly with monochromatized synchrotron
radiation from the ASTRID undulator (UN and MO). The absolute cross
section can now be determined by measuring the photon current in the pho21
2.1 The light source
The experimental setup
todiode (PD), the current of the primary beam (FC1-2 or FC3 depending on
charge), the ion-photon overlap and the number of ionized ions (D1-2). In
this chapter the primary experimental parts will be addressed, and it will be
shown exactly how measurements of the above mentioned quantities are used
to determine absolute cross sections. Since such a description has already
been written (Kristensen 2001, Andersen 2001) (both in danish), (Kjeldsen
1999) and (Schwebs 1999), the present text is not intended to be an in-depth
treatment and is mainly included for the sake of completeness. A little more
attention (next chapter) has been paid to the production of ions, as I have
spent a considerable amount of time working with the ion sources that we
have used.
2.1
The light source
As already mentioned, it is of immense importance to have an extremely
high photon flux in order to carry out absolute photoionization cross-section
measurements on ions. Therefore, high-intensity light from the undulator at
the storage ring ASTRID is used in the experiment. This section will give
a short description of both the undulator and the monochromator which
together constitutes the light source. A more thorough treatment can be
found in (Kjeldsen 1999).
2.1.1
The undulator
An undulator is basically built up by a set of permanent magnets arranged
in a periodic structure (period λ0 ), as illustrated in fig. 2.2. The magnetic
structure gives rise to a periodically changing magnetic field
B = y B0 sin(
2πz
)
λ0
(2.1)
that will exert a Lorentz force F = −ev × B on an entering electron. Since
F and v are perpendicular, the magnetic field does no work on the electron
22
The experimental setup
2.1 The light source
Figure 2.2: Sketch of an undulator with magnetic period λ0 . The strength of the
magnetic field is varied by changing the undulator gap, g.
but instead causes it to oscillate in the xz-plane and thereby to emit light as
it traverses the undulator.
Electrons circulating in the ASTRID storage ring have energies of E =
580 MeV and are therefore highly relativistic:
E
1
γ=
=p
= 1135 1,
2
m0 c
1 − β2
with β =
v
c
(2.2)
and m0 the electron mass.
In the electrons’ rest frame x0 y 0 z 0 the magnetic period will be much smaller
due to the Lorentz contraction
λ0 =
λ0
,
γ
(2.3)
and the emitted light will correspond to classical dipole radiation from an
oscillating point charge with frequency
c
cγ
ν0 = 0 = .
(2.4)
λ
λ0
In the laboratory frame the emitted ligth will be pushed dramatically in the
forward direction and will have a Doppler shifted frequency
ν=
ν0
c
=
,
γ(1 − βcosθ)
λ0 (1 − βcosθ)
23
(2.5)
2.1 The light source
The experimental setup
where θ is the angle from the z-axis. Using the approximations
γ2 =
1
1
≈
(1 − β)(1 + β)
2(1 − β)
(2.6)
and
θ2
,
(2.7)
2
which are valid when β ∼ 1 and θ ∼ 0, we can use equation (2.5) to write
cos θ ≈ 1 −
λ=
c
λ0
= 2 (1 + γ 2 θ2 ).
ν
2γ
(2.8)
In the calculations leading to equation (2.8) we have assumed that v =
vz , that is we have neglected the x-component of the electron velocity and
we have therefore overestimated the effect of Lorentz contraction and the
Doppler shift. By introducing the so-called undulator parameter
K=
eB0 λ0
,
2πm0 c
(2.9)
which is proportional to the maximum electron deflection angle, K = γα,
inside the undulator, it can be shown that (2.8) should be extended to
λ0 K2
λ= 2 1+
+ γ 2 θ2 .
(2.10)
2γ
2
By adjusting the undulator gap, B0 and thereby K can be varied between 0
and 2.3 which by insertion yields energies on the z-axis (i.e. θ = 0) between
15 and 58 eV1 . Since K does not greatly exceed 1 in an undulator the electron oscillations will be small and light emitted at different points inside the
undulator can interfere, which will lead to the presence of higher harmonics, n, of the fundamental in the undulator spectrum2 . Therefore, the final
undulator equation can be written
λn =
1
λ0 K2
2 2
1
+
+
γ
θ
2γ 2 n
2
(2.11)
This lies in the UV regime and since air is opaque to photons above 6 eV, the beamline
must be evacuated. For this reason the light is known as Vacuum Ultra Violet or VUV
radiation.
2
In the case where K 1 the insertion device is called a wiggler and here interference
effects are minimal.
24
The experimental setup
2.1 The light source
In this way one can at a given wavelenght maximize the flux through the
monochromator with the appropriate choiche of n and g (or equivalently K).
Figure 2.3: Calculated spectrum of the ASTRID undulator at three different
gaps with the numbers designating the harmonic value. Lowering the value of K
(increasing g) shifts the spectrum towards higher energies.
2.1.2
The Miyake monochromator
Since the spectrum of an undulator includes a wide range of energies (see
figure 2.3), a monochromator is a necessary supplement in order to pick a
specific energy. The Miyake monochromator was constructed at Daresbury
Laboratory in 1974 (West et al 1974). It is shown schematically in figure
(2.4) and consists of three optical components:
• A grating which can be rotated.
• A cylindrical, focusing mirror which can both be rotated and translated.
• Exit slit with adjustable width.
25
2.1 The light source
The experimental setup
When the incoming beam of radiation from the undulator enters the monochromator, the grating will select a wavelength in accordance with the grating
equation
1
(sinα − sinβ),
(2.12)
N
with k being the diffractive order (k = −1 is used) and N the number of
kλ =
lines per mm, N = 1200mm−1 . Because the beam must hit the exit slit, we
can write another equation relating the mirror angle, θ and the distance d:
h
sin(π − 2θ) = sin(2θ) = p
h2
+ (l − d)2
,
(2.13)
where h and l are constant since the grating can only be rotated.
Figure 2.4: The principle of the Miyake monochromator. The grating angle and
the mirror position and angle are controlled by stepping motors.
By considering similar expressions for the focusing of the mirror and the
grating, see (Kjeldsen 1999), it is possible to determine d, θ and α as a function of the beam energy. However, experience has shown that the monochromator should not be operated in this so-called in-focus mode, where the mirror is continuously being moved as to focus the beam on the exit slit. This is
due to mechanical instabilities in the mechanism controlling the mirror position, which are considerable especially at low energies where the mirror has
26
The experimental setup
2.1 The light source
to be moved a lot. Instead the fixed-focus mode is used, where the mirror is
locked at a specific position (d, θ) and only the grating angle, which is highly
accurate and reproducable, is varied. This, of course, means that the exit slit
will be in focus only at a specific energy and therefore the energy resolution
due to defocus will be worse when we are away from this energy.
Another important issue is the fact that a nonvanishing amount of secondorder radiation will be transmitted through the monochromator at energies
below ∼ 80 eV despite that it was designed to suppress this. This means
that a fraction of the light in the reaction chamber will be of second instead
of first order, i.e. it will be twice as energetic as expected. The measured
energy-dependent cross section will therefore not be correct, but instead be
some mean value of the cross section at the energy E, and 2E.
σmeasured (E) = (1 − X2. (E))σ(E) + X2. (E)σ(2E).
(2.14)
Usually the second order fraction, X2. (E), is small, but its contribution to
the measured cross section can still be large if σ(2E) σ(E). A way to circumvent this problem is to let light from the monochromator pass through
a foil that will absorb the higher-order radiation, but which is semitransparent at lower energy, before it enters the reaction chamber. Figure 2.5 shows
the transmission curves of 4 foils of different materials in the energy range
between 20 and 130 eV. From this it is clear that Al for instance, which has
an L2,3 absorption edge at 72 eV, is practically fully absorbant above this
energy and therefore can be used from 36 to 72 eV. It is also apparent from
the figure that the insertion of a foil will lead to a pronounced reduction in
the first order flux, which is highly undesirable. For this reason the data are
often recorded without a foil and thus with a contaminated photon beam.
It is possible, however, to correct for this contamination, and according to
eq. (2.14) it requires that one can determine the content of second order
radiation as a function of energy as well as σ(2E). The technique for doing
so is described in (Kristensen 2001).
27
2.2 The absolute cross section
The experimental setup
Figure 2.5: The figure shows the transmission of radiation of 4 absorption foils as
a function of energy. Only foils made of Mg, Al and Si are installed in front of the
reaction chamber, whereas both an Al- and a polyimide foil are used as window
foils for the ionization chamber.
2.2
The absolute cross section
As already mentioned, to measure the cross section on an absolute scale it
is necessary to know precisely the number of ions produced (i.e. the signal),
the density of target ions, the photon flux and the amount of overlap between
the two beams. The photoionization signal S is given by the Lambert-Beer
law as
S = F (1 − e−σnl ) ≈ F σ n l,
(2.15)
with F being the photon flux, n the target density, σ the cross section and l
the interaction length. The validity of the expansion is recognized by the fact
that typical values of the factors in the exponent are σ ∼ 1 Mb = 10−18 cm2 ,
n ∼ 106 cm−3 and l ∼ 50 cm.
If now we by dSxyz refer to the part of the total signal arising from the
small volume element dxdydz and let i and j indicate the ion- and photon
28
The experimental setup
2.2 The absolute cross section
current, respectively, passing through the area dxdy we get
dSxyz =
Ωi
j
σ
dz,
eη
qevdxdy
|{z} | {z }
(2.16)
n
F
where e is the elementary charge, Ω and η are the detector- and photodiodeefficiencies and v is the target ion velocity. Integration over the xy-plane
yields
Ωσ
dSz = 2
·
e ηv
RR
ij dxdy
Ωσ
IJ
· dz = 2
·
· dz,
dxdy
q e η v dxdy F (z)
where we have defined the 2-dimensional form factor F (z)
RR
RR
i dxdy · j dxdy
IJ
RR
F (z) ≡
= RR
.
ij dxdy
ij dxdy
(2.17)
(2.18)
To get the complete signal, eq. (2.17) must be integrated along the z-direction
and from this the final expression for the cross section is found to be
σ=
S · q · e2 · v · η
R
.
I · J · Ω · l dxdy1F (z) dz
(2.19)
As before, l is the length over which the photons and ions can interact, but
referring to fig. (2.1) this is not a well-defined number. To overcome this,
a bias of the order of hundreds of volts is applied to the reaction chamber thereby energetically tagging the ions produced in this region and thus
allowing for them to be separated from ions produced outside the chamber.
The parameters contained in (2.19) all need to be determined with great
care and high precission in order to keep the total error within a desired limit
of ∼ 10%. The procedure for calibrating the detectors and measurements of
form factors is described in great detail in e.g. (Kristensen 2001) and only a
few remarks will be given here.
2.2.1
Form factors
Beam profiles or form factors are measured with five sets of horizontal and
vertical beam scanners installed along the reaction chamber. Two approximations are made in the evaluation of F (z):
29
2.2 The absolute cross section
The experimental setup
1. The beam scanners move in steps of finite size which means that inteR
P
grals are replaced by sums, i.e.
→
2. The two-dimensional overlap is approximated as the product of two
one-dimensional overlaps; F (z) → Fx (z) · Fy (z). This has the advantage that the time consumption is reduced by a factor of ten or so, and
the error has experimentally been shown to be very small.
As a consequence (2.18) is replaced by
P P
P P
RR
RR
j
i dxdy · j dxdy
x
yi·
RR
P P x y
F (z) =
≈
ij dxdy
x
y ij
P P
P P
j·
i yj
xi
Px P y
≈
= Fx (z) · Fy (z).
x ij
y ij
(2.20)
In this way, five values of the form factor {F (z1 ), . . . , F (z5 )} are measured
at the points {z1 , . . . , z5 } in the reaction chamber and by fitting these with
R
a second-order polynomial, l dxdy1F (z) dz can be evaluated. The polynomial
description of the z-dependence is remarkably well and experience has shown
that it is even safe to leave out measurements at z2 and z4 which is another
timesaving factor. All in all an accurate determination of the overlap can be
done in less than ten minutes.
2.2.2
Detector calibration
The efficiency of the detectors (Johnston multipliers) changes slowly over
time and therefore needs to be calibrated once in a while. The procedure is
simply to make a stable ion beam and measure the count rate in FC1 (see
figure 2.1), which has a 100% efficiency, and then by changing the magnetic
field in M2 send it to one of the detectors D1 or D2 and measure the corresponding count rate here. Because the efficiency also depends on the mass,
charge and energy of the ions, it is essential that the ions used for the calibration have preferably the same or at least similar (M,q,E)-values as the
ones detected in the photoionization experiments. For example, in the measurement of Te− → Te+ , Te2+ a bias of 200 V was applied to the reaction
30
The experimental setup
Ions
2.2 The absolute cross section
Energy (keV) ΩD1 (%) ΩD2 (%)
Xe+
2.4
59
—–
Xe2+
2.6
68
58
2+
Ar
2.6
77
70
Ar4+
5.2
82
75
Ar6+
7.8
85
76
Table 2.1: Selected measured values of detector efficiencies.
chamber so with an initial target energy of 2 keV it can be seen that the
Te+ -ions detected in D1 had energies of E = (2 + ∆q · Vbias ) keV = 2.4 keV.
Similarly the ions detected in D2 had energies of 2.6 keV and so D1 should
in principle be calibrated with Te+ -ions at 2.4 keV and D2 with Te2+ -ions at
2.6 keV, but to avoid the difficulties in producing these ions, Xe+ at 2.4 keV
and Xe2+ at 2.6 keV were used instead.
2.2.3
Photodiode calibration
Figure 2.6: The ionization chamber used to calibrate the photodiode. The length
of the collectors and the distances to the window foils are l = 200 mm, x0 = 266
mm and x1 = 428 mm.
The photon flux is measured with a calibrated Al2 O3 photodiode but this
is unable to detect all photons (actually it detects only a few in a hundred).
For this reason, the signal measured with the diode needs to be compared
31
2.2 The absolute cross section
The experimental setup
with the true flux in order to determine the exact energy-dependent efficiency
of the diode, η(E). For this purpose a so-called double ionization chamber
sketched in figure 2.6 has been installed in front of the photodiode. The
principle is to let in a noble gas (Neon in our case) and measure the resulting
ionization current on the two collectors. Using the Lambert-Beer law the
true flux can be shown to be given as (Kristensen 2001)
J xli (J )2 /e
1
1
Ftrue =
·
,
(2.21)
J2
J1 − J2
where i ∈ {0, 1} and J1 and J2 are the currents on the two collectors. Note
that for this equation to make sense J1 6= J2 , meaning that the pressure in
the chamber needs to be large enough for the signals on the two collectors
to be different.
In (2.21) it is explicitly assumed that exactly one ion is produced for every
incoming photon, which is approximately true for the noble gases (Samson &
Haddad 1974). However, at larger energies multiple ionization starts to play
a role resulting in a larger registered current on the collectors. Secondary
ionization is another phenomenon with the same consequence, and this will
occur whenever the photon energy exceeds twice the ionization potential of
Ne
the gas (EIP
= 21.56 eV) where the emitted photoelectrons will carry enough
kinetic energy to ionize other atoms by impact. Furthermore the electrons
will gain energy from the repeller bias which therefore must not be too large
but still large enough as to prevent the electrons from reaching the collectors.
Since both multiple and secondary ionization are responsible for an increase
in the collector current (and thereby a value of Ftrue that is too high or
equivalently too low an efficiency) these effects must be taken into account.
Multiple ionization is easily handled as the relative fraction of single and
multiple ionization is known from the litterature (Bartlett et al 1992, Samson
et al 1992) but the subject of treating secondary ionization is more subtle,
since this will depend on the amount of gas in the chamber (i.e. the pressure)
and on the vertical position of the photon beam. How compensations are
made will not be mentioned here but the reader is recommended to consult
(Kristensen 2001) or (Kjeldsen 1999) for details.
32
The experimental setup
2.3
2.3 Recording spectra - an overview
Recording spectra - an overview
To summarize the chapter it can be instructive to have an overview of some of
the basic steps that must be performed when making an actual cross-section
measurement. Practically all operations on the undulator, the monochromator and the variety of power supplies can be done remotely using a PC
connected to the ISA Controle System (www.isa.au.dk/consys), with some
home-made software written in the graphical programming interface Labview.
1. The first step is to produce a target ion beam and steer it all the way
to one of the Faraday cups FC1-3.
2. The mirror in the monochromator is then moved and locked at a position corresponding to a photon energy where the highest resolution is
wanted (typically around threshold).
3. At an energy where a signal is expected (i.e. σ(E) 6= 0) the beam
profiles are measured with the beam scanners and the two beams are
aligned as to ensure maximum overlap.
4. Next, the analyzing magnet M2 is scanned to see which magnetic field
will send the primary beam to the Faraday cup and the signal to one
of the detectors (or both if doubly ionized ions are produced and can
be detected simultaneously).
5. A spectrum over a given energy range can now be recorded. At some
point before or after the scan the photodiode needs to be calibrated
covering this entire range. This must be done every time the diode
is hit differently by the photons (e.g. if the width of the exit slit is
changed between scans). The monochromator also needs to be energy
calibrated which is done by letting in different gases in the ionization
chamber and observe the position of known resonances (e.g. 3d → 5p
in Kr known to be positioned at 92.463 eV (NIST 2003)).
33
2.3 Recording spectra - an overview
The experimental setup
References
Andersen P. A. (2001) Master thesis University of Aarhus
Bartlett R. J., Walsh P. J., He Z. X., Chung Y., Lee E.-M. and Samson J.
A. R. (1992) Phys. Rev. A 46 5574
Kjeldsen H. (1999) PhD thesis University of Aarhus
Kristensen B. (2001) Master thesis University of Aarhus
NIST (2003) http://physics.nist.gov/cgi-bin/AtData/levels form
Samson J. A. R., He Z. X. and Bartlett R. J. (1992) Phys. Rev. A 46 7277
Schwebs M. (1999) Master thesis University of Aarhus
West J. B., Codling K. and Marr G. V. (1974) J. Phys. E 7 137
34
Chapter 3
Ion production
This chapter briefly deals with the problem of producing a target ion beam
and describes the two types of ion sources that I have used, namely Middleton’s high-intensity sputter source for negative ion production and the
ECRIS (Electron Cyclotron Resonance Ion Source) for the production of
multiply-charged positive ions.
3.1
General remarks
The production of ion beams has many applications in several areas of physics
and other natural sciences as well as for industrial purposes. Each application
may have its own needs (charge state, intensity, purity, divergence, etc.) and
unfortunately, no universal ion source exists that is able to comply with all
the different demands; hence numerous different types of sources are currently
in use worldwide.
In our experiments at the ASTRID undulator beamline, three points are
of main interest:
• The extracted ion beam should be stable, since a high degree of stability
is absolutely necessary for making precise measurements.
• The beam should preferentially be cold, i.e. it should not contain any
metastable components.
35
3.2 The Middleton sputter source
Ion production
• The intensity should be as high as possible, since the measured photoionization signal is directly proportional to the target ion current, see
eq. (2.19).
3.2
The Middleton sputter source
The ion source used for the production of negative ions is a Cs sputter source
which was originally designed in 1983 by Middleton (Middleton 1983). The
ions are produced after sputtering of the sample by Cs+ ions created at a
spherical ionizer by surface ionization. A schematic overview of the source is
shown in figure 3.1 and a more detailed description is given in section 3.2.3.
3.2.1
Surface ionization
For atoms that are very near to or adsorbed on a hot metal surface, it is
sometimes possible for electrons to move from the metal to the atom or
from the atom to the metal, depending on the electropositive/electronegative
character of the atom in relation to the work function of the metal. Thus,
atoms may be emitted from the surface in ionic form which is known as
surface ionization. If we consider atoms with first ionization potential Ip
striking a surface with work function φ that is heated to a temperature T ,
the probability that they leave the surface as positive ions can be written as
(Septier 1967, Alton & Mills 1996)
P
+
!
φ − Ip
= A · exp
,
kT
(3.1)
where A is a constant including the statistical weigths of the ions and neutral
atoms. From this equation it is obvious that the surface-ionization efficiency
is large only for metals with high work funcktions (e.g. tungsten (4.55 eV),
platinum (5.65 eV), rhodium (4.98 eV), tantalum (4.25 eV)) and atoms with
low ionization potentials (especially the alkali metals).
36
Ion production
3.2.2
3.2 The Middleton sputter source
Sputtering
That negative ions can be formed by sputtering a solid surface with positive
ions has been an experimental fact for several decades and different variants
of sources based on this principle have been developed (Middleton & Adams
1974, Alton 1993). No satisfactory microscopic theory has yet emerged that
agrees quantitatively with a large range of experimental results, but the
negative ion yield seems to follow a law similar to eq. (3.1).
!
E
−
φ
A
P − ∝ exp
kT
(3.2)
Here EA is the electron affinity, φ is the work function of the sputter surface
and T is its temperature which problably has only little physical significance.
From this equation, one would expect a very low ionization efficiency since
EA < φ for most elements, but a dramatic enhancement in the negative ion
formation can be achieved if the sputtered surface is covered with a layer of
an alkali metal (Cs in particular). The reason for this can be ascribed to
a decrease in the surface work function from the presence of the adsorbed
alkali element. The main theoretical difficulty is the lack of a method for
estimating such changes in the work function as well as a realistic model
for negative ion formation, so the mechanism of a sputter-type negative ion
source is far from beeing completely understood although some promising
proposals have been made (e.g. Rao et al 1992).
3.2.3
Description of the Middleton source
Figure 3.1 shows the buildup of the Middleton Cs sputter source. The
Cs reservoir is heated and the Cs vapour is led to the chamber containing the spherical ionizer and sample through the Cs pipe. The sample rod
can slide on an o-ring under vacuum and samples can be by changed sliding the rod back and closing the gate valve. This small compartment can
be pumped though the valve, thereby keeping the ionizer under vacuum
during the whole loading procedure. The ionizer is heated by sending a
37
3.2 The Middleton sputter source
Ion production
Figure 3.1: Overview of the spherical ionizer sputter ion source, see text. 1) water
cooling, 2) gate valve, 3) sample rod, 4) sample, 5) spherical ionizer, 6) Cs pipe
and 7) Cs reservoir.
Figure 3.2: Details of the region of the ionizer and sample.
large current through it, thus resulting in a surface ionization of some of
the Cs atoms. The generated Cs+ ions are accelerated towards the sample
38
Ion production
3.2 The Middleton sputter source
which is kept at a negative potential and this
leads to a sputtering of the sample, see also
figure 3.2. The sample itself is pressed into a
cathode made of Cu or Al that can be mounted
directly on the sample rod (see figure 3.3). By
cooling the sample rod, the cathode and sample are prevented from getting very hot which
Figure 3.3: cathode with sample.
allows for the Cs vapour to condense and cover
the sample, thus lowering the work function as described above. The negative ions formed are then extracted in the form of a beam through a hole in
the ionizer towards the extraction electrode (not shown in fig. 3.1). Some
typical values of the ion source parameters are listed in table 3.1.
Parameter
Value
Ionizer current
18 A
Cathode potential
-8 kV
Reservoir temp.
160 ◦ C
Extraction potential
5 kV
Pressure
5 · 10−6 mBar
Table 3.1: Some typical values of ion source parameters
Another important parameter is the cathode distance, i.e. the distance
from the sample to the ionizer, which can be varied by moving the sample rod.
This adjustment cannot be done continuously while the source is running,
since one must push or pull the rod manually and this is on high voltage.
This means that the high voltage must be switched off before it is safe to go
and change the distance. From our measurements we have experienced that
changes of as little as 1-2 mm have had a large effect on the intensity of the
extracted ion beam, but generally the same distance seems to apply well to
the different samples and only small changes should be made.
The main advantages of this source is that negative ions can be made from
39
3.3 The ECRIS
Ion production
almost every element in the periodic table (see Middleton’s ”A Negative Ion
Cookbook”). Moreover, the design makes it extremely easy to reload the
source without breaking the vacuum and the whole procedure can be done
within 5 min. A little patience is often required when inserting a new cathode
since a sputter crater usually has to be formed before the ion yield starts to
rise and one must also play around with the different source parameters to
find the right settings. This time, of course, depends on how easily the
various elements form negative ions, so while for example C and Cu starts
to rise immediately, other elements such as Al or Cr may take a few hours.
In table 3.2 some of the obtained ion currents measured in FC3 (figure 2.1)
are presented.
Ion
Current (nA) Cathode
Te−
30
CdTe
−
15
V
Cr−
40
Cr
Co−
90
Co
−
95
Ni
V
Ni
Table 3.2: Obtained ion currents in FC3.
3.3
The ECRIS
An ECR ion source is a so-called hot plasma source where the plasma is
maintained by electrons interacting resonantly with electromagnetic waves.
Many different variants of such sources are now succesfully being used to
produce beams of highly charged positive ions and research is still being done
in order to enhance their performance. Different attempts have been made
to control and optimize on important plasma parameters such as the electron
number density, electron temperature (energy) and the plasma confinement.
The following is only intended as a short explanation of how an ECRIS work.
40
Ion production
3.3 The ECRIS
For more information the reader is referred to e.g. Ciavola & Gammino
(1996), Geller (1998), Girard & Melin (1996) or Alton 1996.
3.3.1
The ECRIS principle
Figure 3.4: The principle of an ECR ion source.
The principle of an ECRIS is shown in figure 3.4 above. A plasma is
created inside a chamber surrounded by strong magnets by the injection
of radiofrequency (rf) power. The magnets are arranged in a minimum Bstructure where conventional or superconducting solenoids or permanent ring
magnets produce a strong axial field and a multipole magnet produces a
radial field in such a way that the magnetic field is at a minimum in the
middle of the chamber and increases in all directions, see figures 3.5 and 3.6.
This magnetic configuration thus constitutes a magnetic trap whichs tends to
confine the electrons in the center. The reason for this is a direct consequence
of the invariance of the magnetic moment, µ =
2
mv⊥
2B
since electrons moving
in a direction of increasing B must increase their perpendicular velocity,
which by energy conservation leads to a decrease in the parallel velocity,
41
3.3 The ECRIS
Ion production
v|| . Depending on the mirror ratio, Bmax /Bmin , the outgoing motion can
be stopped hence leading to a reflection of the electron. Furthermore, the
electrons will gyrate around the magnetic field lines and Electron Cyclotron
Resonance will occur whenever the frequency of gyration matches the rf
frequency:
ωrf = ωcyc =
Figure 3.5: Axial magnetic field.
e
B.
m
(3.3)
Figure 3.6: Radial magnetic field.
By adapting a proper combination of magnetic field strength and rf frequency, a closed surface is obtained on which the resonance condition (3.3)
is fulfilled. This defines the boundary of the ECR zone where electrons can
be accelerated to very high energies, which enables the plasma atoms/ions
to be stripped to high charge states by electron impact.
It is important, though, to realize that the presence of energetic electrons
in the magnetic trap is not the only criterion for the production of highly
charged ions. Since step-by-step ionization is the dominant stripping process,
it is also necessary that the ions stay in the plasma for a time sufficient enough
for the desired charge state to be reached. Because of their larger mass, the
ions are not affected much by the magnetic fields so the ions are kept in the
plasma by the space charge potential of the electrons. On the other hand, the
ion confinement time must not be too high since ion losses from the plasma
is what leads to the extracted output current and therefore the ECRIS is a
compromise between these needs.
42
Ion production
3.3.2
3.3 The ECRIS
Description of the ECR source
Figure 3.7: Schematic overview of the compact 10 GHz all-permanent ECRIS
used at the undulator beamline.
The ECRIS used in the experiments at ASTRID is shown scematically in
figure 3.7. The axial confining magnetic field is provided by strong NdFeB
permanent ring magnets and the radial field by a permanent hexapole and
the magnets are surrounded by a protective shield of steel. Furthermore,
the plasma chamber walls are watercooled to prevent a de-magnetization
of the magnets at high temperatures. The field strength in the chamber
corresponds to ECR operation at ∼10 GHz, but since there is no way of
adjusting the magnetic field, it is necessary to have a rf supply which is
tunable both in power and frequency. We have used different supplies with
output frequencies ranging from 9 - 10.5 GHz and with rf power up to 250
W. The electromagnetic waves are led from the supply at ground potential
to the plasma chamber by air filled wave guides and before they enter the
ion source region they must go through a high-vacuum/high-voltage window
because the ion source is evacuated and on a high potential. The source has
two gas inlets (only one is shown in figure 3.7) and the gas flow is adjusted
by needle valves which can be controlled from outside the high-voltage region
43
3.3 The ECRIS
Ion production
by long insulated rods. The produced multiply-charged ions are extracted
towards a puller electrode that is kept on negative potential.
One of the great advantages of the ECRIS is that it is extremely easy to
operate, since the only parameters one needs to manipulate with are the gas
pressure and rf power (and to some extent also the rf frequency). Moreover,
ions of many charge states are generated immediately after adding gas and rf
power, see figure 3.8. In general, not all the ions will be in the ground state
but will instead be in some metastable excited state(s). With so few knobs to
turn there is really no way that the metastable component(s) can be reduced
and in this respect the simplicity of the ECRIS may be regarded as a drawback. The production of ions from solids is also much less straightforward
than from gaseous elements. By evaporating solids from an oven designed for
this particular ECRIS, we have succeded, though, in making stable beams
of Ba2+ , La3+ and Ce4+ , but only after days of hard work. Table 3.3 below
shows some measured output currents for different ions.
Ion
Current (nA)
Made from
N2+
150
N2 (g)
3+
O
120
O2 (g)
F4+
40
SF6 (g)
O4+
115
O2 (g)
F
55
SF6 (g)
Ne4+
45
Ne (g)
Ba2+
3+
28
Ba (s)
3+
5
C15 H15 La (s)
4+
4.5
C15 H15 Ce (s)
La
Ce
Table 3.3: Performance of the ECRIS. (g) gas and (s) solid.
An example of a measured output spectrum of the ECRIS is shown in
figure 3.8. It was obtained after optimizing on the O3+ peak by scanning the
field of magnet M1 (figure 2.1). The main peaks have been labeled and it is
44
Ion production
3.3 The ECRIS
seen that all charge states of Oq+ are present up to q = 6 whereas ionization
of the rest gas is responsible for the nitrogen and hydrogen peaks. It should
also be noted that the larger output velocities of the higher charge states
results in a decrease in beam divergence and therefore narrower peaks.
Figure 3.8: Output spectrum from the ECRIS loaded with O2 gas obtained by
scanning the magnet M1 and measuring the ion current on a shutter inserted before
the reaction chamber.
References
Alton G. D. (1993) Nucl. Instr. and Meth. in Phys. B 73 221
Alton G. D. and Mills G. D. (1996) Nucl. Instr. and Meth. in Phys. Res.
A 382 232
Alton G. D. (1996) Nucl. Instr. and Meth. in Phys. Res. A 382 276
Ciavola G. and Gammino S. (1996) Nucl. Instr. and Meth. in Phys. Res. A
45
3.3 The ECRIS
Ion production
382 267
Geller R. (1998) Rev. Sci. Instrum. 69 1302
Girard A. and Melin G. (1996) Nucl. Instr. and Meth. in Phys. Res. A 382
252
Middleton R. and Adams C. T. (1974) Nucl. Instr. and Meth. 118 329
Middleton R. (1983) Nucl. Instr. and Meth. 214 139
Middleton R. A Negative Ion Cookbook available at
http://tvdg10.phy.bnl.gov/COOKBOOK/
Rao Y., Chen H., Boiling X., Dong B. and Baihua X. (1992) Rev. Sci. Instrum. 63 2643
Septier A. (1967) Focusing of Charged Particles, ed. by A. Septier (New
York Academic Press)
46
Chapter 4
Photodetachment of Te−
In this chapter, results of the 4d photodetachment cross section of negative tellurium will be presented, revealing for the first time strongly bound
inner-shell excited states. Absolute cross sections for processes leading to
the formation of both Te+ and Te2+ are measured with an uncertainty of approximately 15 % and 50 %, respectively, where the rather large uncertainty
for the latter is because of variations in the efficiency of the second detector
(D2 in figure 2.1).
4.1
Motivation
(Kr)4d10 5s2 5p5 2 P3/2
Ground state
Fine-structure splitting (J1/2−3/2 ) 620 meV
Binding energy
1.970876(7) eV
4d threshold
40.32 eV
Table 4.1: Some facts about Te− . Values of the ground state splitting is taken
from (Thøgersen et al 1996) and the electron affinity from (Haeffler et al 1996).
The 4d detachment threshold is deduced from the ground state binding energy
and the known energy of the lowest lying 4d9 5s2 5p5 level in neutral Te (38.35 eV
above Te ground state) (Murphy et al 1999).
47
Photodetachment of Te−
4.2 Results and analysis
Table 4.1 summarizes some facts about the negative tellurium ion. The
primary motivation for choosing Te− as the target for an inner-shell detachment measurement was the possibility of finding Feshbach resonances in the
4d detachment spectrum. The reason for such excited states to be expected
is that the ground state configuration of Te− lacks one electron to completely
fill the 5p subshell. Excitations to this particular subshell will therefore result
in a filling and the ion is likely to gain extra stability.
4.2
Results and analysis
Figure 4.1: Left: Absolute photodetachment cross section for Te− → Te+ (upper)
and Te− → Te2+ (lower) in the energy range from 34-130 eV. Right: Details in
the region around the resonances.
Figure 4.1 above shows the photodetachment spectra for the processes
48
Photodetachment of Te−
4.2 Results and analysis
Te− → Te+ and Te− → Te2+ . In the following, the details of the spectra will
be discussed.
4.2.1
The continuum
In both spectra the cross section increases slowly at ∼ 40.3 eV. This is in
agreement with a value of 38.35 eV + 1.97 eV = 40.32 for the 4d detachment threshold, which can be calculated from the measured energies of the
Te(4d9 5s2 5p5 ) levels (Murphy et al 1999) and the Te− binding energy (Haeffler et al ), see also table 4.1. At higher photon energy, the cross section
eventually shows a broad structure centered at ∼90 eV which corresponds to
direct detachment of an electron followed by single or double Auger decay.
According to the dipole selection rules, the emitted photoelectron can be
either an p or an f electron ( denotes the kinetic energy of the outgoing
electron) and since double and triple detachment is very unlikely, the formation of Te+ and Te2+ probably happens in step-wise processes that may be
written schematically as
Te− (4d10 5s2 5p5 ) + hν
→ Te(4d9 5s2 5p5 ) + eph
(
Te(4d9 5s2 5p5 ) →
(4.1)
Te+ (4d10 5s2 5p3 ) + eAug
Te+ (4d10 5s5p4 ) + eAug → Te2+ (4d10 5s2 5p2 ) + 2eAug .
Note that the detachment may also lead to the production of neutral Te
and possibly also Te3+ , but neither of these were detected in the experiment.
However, we estimate that these processes contribute only little (≤ 10 % of
the total oscillator strength), see (Kjeldsen et al 2002).
The small increase in the cross section just above threshold is due solely
to 4d → p detachment as there is no strength in the 4d → f channel. At
larger energies however, this transition dominates and is responsible for the
broad so-called giant resonance which have also been observed in the photoionization of low-charged positive ions in the same region of the periodic
table, e.g. the iso-electronic Xe+ (Andersen et al 2001). This behaviour is
49
Photodetachment of Te−
4.2 Results and analysis
known as a “delayed onset” and can be explained by the large centrifugal repulsion of the f wave function, which prevents the near-threshold continuum
electron of penetrating the core (Andersen 2001).
Finally, considering the magnitudes, it is seen that the cross sections for
Te+ and Te2+ production reach 7 Mb and 15 Mb, respectively, indicating that
(sequential) double-Auger decay is more probable than single-Auger decay.
4.2.2
Resonances
In the Te+ spectrum two narrow peaks appear below the threshold at 40.32
eV. These are attributed to 4d → 5p transitions in the parent ion with
the large peak corresponding to 4d10 5s2 5p5 2 P3/2 → 4d9 5s2 5p6 2 D5/2 and the
small peak to 4d10 5s2 5p5 2 P3/2 → 4d9 5s2 5p6 2 D3/2 . Details of these resonances
are put in table 4.2 and some of the relevant levels of Te− as well as Te,
Te+ and Te2+ are shown in figure 4.2.
Transition
2
2
Note that there is no resonance
Energy (eV) Binding (eV)
P3/2 → 2 D5/2
2
P3/2 → D3/2
f
Γ (meV)
200 ± 10
37.37
2.95
0.032
38.85
1.47
0.0028 171 ± 15
Table 4.2: Details of the resonances.
corresponding to 4d10 5s2 5p5 2 P1/2 → 4d9 5s2 5p6 2 D3/2 , which implies that the
2
P1/2 level was not populated in the target beam although it lies only 620
meV above the 2 P3/2 ground state. This is in accordance with previous
experiments with a similar ion source, where a population of this level of less
than 2 % was found (Kristensen et al 1993). Similar to the decay routes
of eq. (4.2), the production of Te+ via the 4d → 5p transitions happens
through sequential double-Auger, and the absence of these resonances in
the Te2+ spectrum means that triple-Auger to the Te2+ continuum is not
energetically possible.
The two resonances are bound with 2.95 eV (40.32 eV - 37.37 eV) and
1.47 eV (40.32 eV - 38.85 eV) respectively with respect to the 4d−1 threshold,
50
Photodetachment of Te−
4.2 Results and analysis
Figure 4.2: The lowest energy levels (bottom) of Te− , Te, Te+ and Te2+ and the
4d → 5p inner-shell excited states (top) of Te− and Te (Murphy et al 1999). The
observed transitions are indicated by arrows.
and the former is therefore even more strongly bound than the ground state.
This extra gain in stability must be ascribed to the full 5p subshell in the
4d9 5s2 5p6 configuration as could be expected. Similar negative ions, with
full or half-full outer subshells but one electron, are also likely candidates
to form stable inner-shell excited states, but this is so far the only example
of such. For comparison, in the recent experiment on 1s detachment of C−
(Gibson et al 2003) it was found that the 1s → 2p transition resulted in a
shape resonance, in agreement with the fact that the 1s2s2 2p4 configuration
represents a loss of stability compared to the half-full 2p-shell in the ground
state of C− .
The natural widths are found by fitting Voigt profiles to the resonance
peaks and the corresponding values (Γ5/2 = 200±10 meV and Γ3/2 = 171±15
meV) are comparable to the values of the 4d9 5s2 5p5 levels of neutral Te that
51
Photodetachment of Te−
4.2 Results and analysis
range from 93 to 195 meV (Murphy et al 1999). Furthermore, values of
around 120 meV have been measured for the iso-electronic levels of Xe+ (Andersen et al 2001). In the article the fine-structure splitting of the 2 D3/2−5/2
levels is found to be ∼2 eV, which is somewhat higher than the corresponding
splitting in Te− (2 D3/2−5/2 = 2.95 - 1.47 = 1.48 eV). This is not surprising
as the strength of the spin-orbit operator increases as Z 4 .
4.2.3
Oscillator strengths
The oscillator strengths of both resonances as well as the total oscillator
strength has been determined by applying the formula (1.21). The oscillator
strengths of the resonances (f5/2 = 0.032 and f3/2 = 0.0028) lead to an
intensity ratio of 11:1 which is close to the statistical value of 9:1 expected
in LS coupling, see e.g. (Woodgate 1970). Integrating over the whole energy
region, the total oscillator strength has also been found as ftot = fT e+ +
fT e2+ = 3.4 ± 0.5 + 6.7 ± 3.4 = 10.1 ± 3.9. This value is very close to 10,
which is the number of 4d electrons and therefore the correct value according
to the sum rule (1.20).
References
Andersen P. A. (2001) Master thesis University of Aarhus
Andersen P., Andersen T., Folkmann F., Ivanov V. K., Kjeldsen H. and West
J. B. (2001) J. Phys. B: At. Mol. Opt. Phys. 34 2009
Gibson N. D., Walter C. W., Zatsarinny O., Gorczyca T. W., Ackerman G.
D., Bozek J. D., Martins M., McLaughlin B. M. and Berrah N. (2003) Phys.
Rev. A 67 030703 (R)
Haeffler G., Klinkmüller A. E., Rangell J., Berzinh U. and Hanstorp D. (1996)
Z. Phys. D 38 211-214
Ivanov V. K. 1999 J. Phys. B: At. Mol. Opt. Phys. 32 R67
Kjeldsen H., Andersen P., Folkmann F., Hansen J. E., Kitajima M. and Andersen T. (2002) J. Phys. B: At. Mol. Opt. Phys. 35 2845
52
Photodetachment of Te−
4.2 Results and analysis
Kristensen P., Stapelfeldt H., Balling P., Andersen T. and Haugen H. K.
(1993) Phys. Rev. Lett. 71 3435
Murphy N., Costello J. T., Kennedy E. T., McGuinness C., Mosnier J. P.,
Weinmann B. and O’Sullivan G. (1999) J. Phys. B: At. Mol. Opt. Phys.
32 3905
Thøgersen J. Steele L. D., Scheer M., Haugen H. K., Kristensen P., Balling
P. and Andersen T. (1996) Phys. Rev. A 53 3023
Woodgate G. K. (1970) Elementary atomic structure (McGraw-Hill)
53
54
Chapter 5
Photoionization results
This chapter will present results on photoionization of positive ions of astrophysical importance. More precisely, absolute cross sections of N2+ , O3+
and F4+ (B-like), O4+ (Be-like) and finally F3+ and Ne4+ (C-like) will be
provided with an uncertainty of ∼20 %. This uncertainty is somewhat larger
than would normally be obtainable and it is mainly caused by problems
with the detector at the time of the measurements. The analysis of the experimental results will not deal with details of the resonance shapes (Fano
parameters), but will instead concentrate on characterizing the different transitions involved utilizing simple QDT. Moreover the experimental spectra will
be compared to R-matrix calculations taken from the OPACITY database
(TOPbase, Cunto et al 1993) as well as results of the MCDF method. It
should be noted that in the TOPbase the continuum threshold is calculated
for each particular initial state but the cross-section calculations also include
points down to 0.01·Z 2 Ry below threshold, see (Luo & Pradhan 1989). These
points represent photoabsorbtion in the discrete region where the density of
states is high. Consequently, in each of the following graphs of OPACITY
results, I have cut away the data below the stated threshold for a better
comparison with experiment.
55
5.1 B-like
5.1
Photoionization results
B-like
Because of its special role in astrophysics, C+ was one of the first ions to be
studied in Aarhus. The cross section showed a rich structure with many peaks
corresponding to different Rydberg and doubly-excited states and a comparison with OPACITY results showed a generally good agreement (Kjeldsen
1999, Kjeldsen et al 1999). The next three members of the boron sequence
are also important to astrophysics because of their abundance in e.g. stellar
and planetary atmospheres.
The ions were produced with an ECRIS (see chapter 3) with the majority
in the 1s2 2s2 2p 2 P ground state and smaller fractions in the 1s2 2s2p2 4 P
metastable state (the 1s2 2s2p2 2 S,2 P,2 D terms are not metastable since they
are allowed to decay back to the ground state by fluorescence). Hence, the
measured cross sections will contain contributions both to the continuum as
well as additional resonances arising from these metastable states.
5.1.1
Results and analysis
Figure 5.1 shows the results obtained for the isoelectronic ions N2+ , O3+
and F4+ . Also shown are theoretical spectra which have been folded with
100, 250 and 500 meV Gaussians (FWHM), respectively, which represent the
experimental resolutions. MCDF calculations for N2+ and F4+ are provided
by (Bizau 2003) whereas O3+ calculations are from (Champeaux et al 2003).
The R-matrix results by Fernley et al (2004) are available at (TOPbase)
and finally the red lines are more recent R-matrix calculations performed by
Nahar (Nahar 2004).
Referring to figure 5.1 the experimental spectra show the expected behaviour with a continuum on which different Rydberg states are superimposed. The thresholds can be read off from the figure and are located at
47.43 eV, 77.34 eV and 114.27 eV respectively, which is close to the values
reported at NIST (NIST 2003), see table 5.1.
56
Figure 5.1: Caption located at next page.
5.1 B-like
Photoionization results
Caption to figure 5.1:
Experimental and theoretical photoionization spectra of N2+ , O3+ and F4+ with
energies (horizontal axes) in eV and cross sections (vertical axes) in Mb. Rmatrix results are from Fernley et al (2004) and are found at (Topbase). The red
lines are R-matrix data obtained by Nahar (2004) and the MCDF calculations
were performed as part of the present work (N2+ and F4+ ) (Bizau 2003) or by
(Champeaux et al 2003) (O3+ ). The theoretical spectra have been convoluted
with Gaussians of 100 meV, 250 meV and 500 meV (FWHM), respectively, and
are calculated under the assumption of a (2 P,4 P) relative fraction of (0.90,0.10)
(N2+ and F4+ ) and (0.84,0.16) (O3+ ).
Energy (eV)
State
N2+
O3+
F4+
1s2 2s2 2p 2 P
0
0
0
7.10
8.88
10.69
1s2 2s2p(3 P)3p 2 D
39.80
59.86
83.84
EIP (1s2 2s2 1 S)
47.45
77.41
114.24
2
1s 2s2p
2 4
P
Table 5.1: Some relevant energy levels. From (NIST 2003).
The peaks correspond mainly to 2s → np transitions from the ground
state and 2p → nd transitions from the metastable state and most of them
can be explained from QDT. An analysis of the dipole allowed transitions in
the parent ions will be given below.
Photoexcitation of an outer 2p electron will lead to the following
1s2 2s2 2p 2 P + hν → 1s2 2s2 (1 S)ns, n0 d
(5.1)
but such Rydberg series will converge to the 1s2 2s2 1 S ground state of the
ionized ions and will therefore not be present in the photoionization signal
as they lie below the continuum limit.
If the photons are energetic enough to excite an inner 2s electron, the result58
Photoionization results
5.1 B-like
ing allowed transitions will be
1s2 2s2 2p 2 P + hν → 1s2 2s2p
1
P
3
P
!
np
2
S, 2 P, 2 D
2
S, 2 P, 2 D
!
(5.2)
Considering the selection rules for Coulomb autoionization, it is easy to
see that the 1s2 2s2p(1,3 P)np 2 P terms cannot autoionize into the 1s2 2s2 1 S
+ p continuum without breaking the parity selection rule. However, the
study of C+ showed that these states were indeed responsible for peaks in
the photoionization spectrum. This indicates that the 2 P terms may autoionize either by some mechanism other than the Coulomb interaction (e.g.
spin-orbit which is not much slower than fluorescence, see table A.2 in appendix A.1) or by a radiative decay followed by Coulomb autoionization (e.g.
1s2 2s2p(1,3 P)np 2 P → 1s2 2s2p(1,3 P)n0 s 2 P + hν → 1s2 2s2 1 S + p). All in
all we can expect the presence of 6 Rydberg series converging to the limits
of the 1 P and 3 P atomic cores. As described in chapter 1, the location of
these Rydberg states have been calculated by means of QDT using eq. (1.24)
and the known energy of the 1 P and 3 P ionization thresholds as well as one
member of each series (e.g. the 1s2 2s2p(3 P)3p 2 D) (NIST 2003).
In a similar way it is possible to analyze resonances originating from
transitions from the metastable component.
1s2 2s2p2 4 P + hν → 1s2 2s2p(3 P)
n0 s
nd
!
4
4
P
P, 4 D
!
(5.3)
These 3 Rydberg series all converge to the energy limit at E(1s2 2s2p 3 P) E(1s2 2s2p2 4 P) but they are not allowed to decay through Coulomb autoionization because of the spin selection rule. Still, from the calculated Rydberg
series shown in the experimental spectra it is evident that these transitions
are responsible for the resonances below threshold and so the states most
likely autoionize via the spin-orbit interaction which is able to flip the spin.
In figure 5.2, the experimental spectrum of N2+ is shown in greater detail.
Since this is taken with the best resolution, it reveals more structure and is
therefore best suited for an illustration of the different Rydberg series. Also
59
5.1 B-like
Photoionization results
Figure 5.2: Detailed view of the experimental photoionization spectrum of N2+ .
Peak numbers refer to table 5.2. Furthermore, the location of Rydberg states
calculated from QDT are indicated with the numbers designating the state with
the lowest value of n.
shown is the location of the different Rydberg states calculated from QDT
and this rather simple analysis is sufficient enough to characterize all major
resonances in the spectrum and these are given in table 5.2.
A few comments ought to be made about the assignments summarized
in table 5.2. The peaks below threshold clearly corresponds to transitions
from the metastable component although the limited resolution does not
allow us to resolve the 2s2p(3 P)nd (4 D,4 P) excited states (transitions to the
4
D term are statistically favoured since the statistical weight of the terms
60
Photoionization results
5.1 B-like
Peak number Energy (eV)
Transition
2
2
4
1s 2s2p ( P) → 1s2 2s2p(3 P)5s (4 P)
1
42.62
2
43.64
5d (4 D,4 P)
3
44.64
6s (4 P)
4
45.25
6d (4 D,4 P)
5
46.15
7d (4 D,4 P)
6
46.75
8d (4 D,4 P)
7
47.19
9d (4 D,4 P)
8
47.49
10d (4 D,4 P)
9
49.93
1s2 2s2 2p (2 P) → 1s2 2s2p(3 P)5p (2 P)
10
50.37
5p (2 D)
11
50.61
5p (2 S)
12
51.61
6p (2 P)
13
52.19
6p (2 D,2 S)
14
53.14
7p (2 P,2 D,2 S)
15
53.77
8p (2 P,2 D,2 S)
16
54.18
9p (2 P,2 D,2 S)
17
54.49
18
54.65
19
54.86
20
58.13
1s2 2s2 2p (2 P) → 1s2 2s2p(1 P)5p (2 P,2 D)
Table 5.2: Location and designation of the resonances in the photoionization spectrum
of N2+ . The peak numbers refer to figure 5.2.
are (4 D,4 P)=(20,12)). Similarly, in the case of 2s → np transitions from
the ground state, it has only been possible to resolve the 2 P, 2 D and 2 S
excited states for n = 5 and partially for n = 6. Note that in these cases
the 2 P state lies a little lower than the QDT value, but the assignment is
supported by the fact that the peaks are not present in the (nonrelativistic)
R-matrix calculation but do occur in the MCDF spectrum, see figure 5.1. The
assignment of the peaks labelled 17-19 is not straightforward since there will
61
5.1 B-like
Photoionization results
be some degree of configuration interaction between the 2s2p(3 P)(n ≥ 10)p
(2 D,2 P,2 S) and the corresponding 2s2p(1 P)4p terms so the peaks will be a
mixture of these. Moreover, additional weak peaks between number 19 and
20, and also higher in energy, does not seem to belong to any of these Rydberg
series and are most likely just experimental artefacts or the result of twoelectron excitations, which were also observed in the spectrum of C+ .
A comparison of the experimental spectra of the three isoelectronic ions
clearly shows that many of the same transitions are responsible for the resonances in the spectra, although the decreasing resolution tends to smear them
out. It is also evident that succesive inner-shell resonances move downward
in energy relative to the threshold when Z increases. As an example of this,
in the case of N2+ the 2s2p(3 P)5p 2 D resonance is located approximately
3 eV above the 2s2p 3 P threshold, whereas the same resonance is located
just at threshold in the O3+ spectrum. For F4+ it has plunged down below
the ionization limit and into the discrete region and is therefore not visible.
This behaviour can be explained by the fact that as Z → ∞, the ions become more and more hydrogenlike and the 2s2p and 2s2 states consequently
become degenerate. A downward movement of inner-shell resonances with
respect to outer-shell thresholds can therefore be expected for all sequences
where the outermost subshell is different from l = 0.
A general decrease of the direct photoionization cross section is also observed when moving up the sequence. This behaviour is in accordance with
the work of Msezane et al (1977) who predicted that the direct photoionization cross sections for members of isoelectronic sequences decrease approximately parallel with photon energy. At a given energy their will be only a
small increase in the cross section, due to the increase in nuclear attraction
which leads to a contraction of the orbitals and therefore a larger overlap between the wave functions. Consequently, the decrease of the cross section can
be ascribed to the increasing value of the ionization threshold with increasing
nuclear charge.
62
Photoionization results
5.1.2
5.1 B-like
Metastable fractions
The presence of the Rydberg states below threshold can be used to estimate
the fraction of metastable components in the target ion beams by comparing
the oscillator strengths of the resonances with the theoretical values. Experimental and MCDF oscillator strengths for the 2s2p(3 P)nd 4 D, 4 P states
have been obtained by fitting the peaks with Gaussians and using eq. (1.21).
This procedure is obviously not applicable for the R-matrix results since
no peaks are present in the spectra, but instead the corresponding oscillator strengths for photoexcitation, which can also be found at the OPACITY
database (TOPbase), may be used under the assumption that the excited
states autoionize with 100 % probability.
The results are presented in table 5.3 and it is estimated that the metastable
fractions are 10, 16 and 10 % respectively for the three ions, which is in agreement with previous results obtained for N2+ (8 % metastable) (Bizau et al
2003) and O3+ (16 % metastable) (Champeaux et al 2003) produced with
the same ECR source.
5.1.3
Comparison with theory
In figure 5.1 the experimental data are shown together with theoretical results of the R-matrix approach and the MCDF method. To account for the
instrumental resolution, the spectra have been convoluted with Gaussians of
100 meV, 250 meV and 500 meV (FWHM) respectively, for the three ions.
Moreover, in each case the contribution from the metastable component is
incorporated into the spectra by applying the formula
σtotal = (1 − X4 P ) · σ(2 P) + X4 P · σ(4 P)
(5.4)
except for the Nahar data (red lines) which are purely ground state. It seems
that both theoretical methods are able to calculate the direct cross section
fairly well, whereas the OPACITY results in particular fails to reproduce the
spectral structure with regards to both the number and relative intensity in
the O3+ and F4+ cases. For N2+ , on the other hand, the result is quite good.
63
5.1 B-like
Photoionization results
Table 5.3: The measured and calculated oscillator strength (f -values) of the 2s2p2
4P
→ 2s2pnd 4 P, 4 D transitions for N2+ , O3+ , and F4+ .
Transition
Oscillator strength
Ratio
Ion
n
Exp.
R-matrix
MCDF
Exp.
R−matrix
N2+
5
0.004126
0.0740
0.08717
0.0558
0.0473
6
0.003079
0.0238
0.05555
0.1294
0.0554
7
0.002656
0.0205
0.02146
0.1296
0.1238
8
0.001437
0.0138
0.01947
0.1041
0.0738
9
0.001076
0.0096
0.00925
0.1121
0.1163
5
0.002695
0.0704
0.06270
0.0383
0.0430
6
0.002388
0.0374
0.03037
0.0638
0.0786
7
0.002501
0.0223
0.01630
0.1121
0.1534
8
0.002241
0.0145
0.00955
0.1545
0.2346
9
0.001998
0.0100
0.00571
0.1998
0.3499
10
0.001206
0.0070
0.00566
0.1723
0.2130
6
0.004945
0.0724
0.04480
0.0683
0.1104
7
0.002717
0.0379
0.02566
0.0717
0.1059
8
0.002055
0.0227
0.01642
0.0905
0.1252
9
0.002247
0.0151
0.03253
0.1488
0.0691
O3+
F4+
Exp.
MCDF
Exp.: Present experimental data.
R-matrix: R-matrix calculation by Fernley et al (2004)
MCDF: Multi-Configuration Dirac-Fock calculations performed as part of the
present work (N2+ and F4+ ) or by (O3+ ) (Champeaux et al 2003).
In the OPACITY data, the 2s2p(3 P)(n=5,6)p 2 P states are not present as
the calculation does not include relativistic effects, and these are not present
64
Photoionization results
5.2 Be-like
in the calculation by Nahar (red line) either. In fact, this calculation is not
really an improvement of the OPACITY result in contrast to the O3+ case,
where better (although not convincing) agreement is found in the Nahar
result (see also Champeaux et al (2003) for the most recent relativistic Rmatrix result). The MCDF description of the cross sections is generally good.
The primary difference when comparing the experimental spectrum of N2+
with the MCDF result is that most theoretical resonances are shifted towards
lower energies (approximately -0.7 eV for the peaks below threshold and -0.8
eV for the peaks above). Two significant peaks at 47.15 eV and 55.5 eV
respectively are not found in the experiment and these are attributed to the
n = 3, 4 members of the 2s2p(1 P)np Rydberg series. The appearance of the
n = 3 member is because the calculation incorrectly places this state above
threshold and therefore allows it to autoionize into the continuum, whereas
the n = 4 member is indeed present but its strength is largely overestimated
in the calculation. The calculation for F4+ also looks very reasonable except
for a little shift in energy of the resonances above threshold. The appearance
of a theoretical double-peak at around 115 eV may be due to a too low
threshold value and/or a calculated oscillator strength for the 2s2p(3 P)5p
2
P term that is too large. The most discrepancies occur in the case of O3+
with regards to both the number and relative intensities of resonances. The
energy region from around 77-85 eV is rather difficult to handle theoretically
because of CI between 2s2p(3 P,1 P)np states and is further complicated by the
presence of a 1s2 2p2 4d 2 D doubly-excited state predicted in the calculation
to be located just above 81 eV.
5.2
Be-like
This section will present the absolute cross section for O4+ which belongs to
the Be isoelectronic sequence. Previously, relative data on O4+ have been obtained with a better resolution (Champeaux et al 2003) and absolute data for
N3+ were measured at ASTRID this spring by Bizau, Folkmann and cowork65
5.2 Be-like
Photoionization results
ers, but will not be discussed here. Furthermore, high-resolution absolute
measurements of C2+ (Müller et al 2002) and B+ (Schippers et al 2003) have
recently been reported and compared to relativistic R-matrix calculations.
5.2.1
Results and analysis
The experimental data are shown in figure 5.3 along with results from the
OPACITY project (Tully et al 1990) and the MCDF method (Champeaux
et al 2003).
The experimental spectrum clearly shows two onsets at 113.63 eV and
103.50 eV which correspond to ionization into the O5+ 1s2 2s 2 S continuum
from the 1s2 2s2 1 S ground state and the 1s2 2s2p 3 P metastable state of O4+
respectively. According to NIST (NIST 2003) these should be present at
113.90 and 103.74 eV. By comparing the continuum contribution from the
metastable component with theory, it was estimated that 50 % of the parent
ion beam was produced in the metastable state, which was also observed by
Champeaux et al (2003) with an identical ion source.
Two series of resonances are present and QDT calculated members of
these series are indicated in the upper graph. It is seen that 1s2 2s2p 3 P →
1s2 2p(2 P)np 3 D, 3 S, 3 P transitions from the metastable state are responsible
for the peaks labeled 1-6, whereas numbers 8-11 can be ascribed to 1s2 2s2 1 S
→ 1s2 2p(2 P)nd 1 P double-excitations from the ground state. Peak no. 7 was
not observed in the experiment by Champeaux et al and it is not predicted
in any of the theoretical approaches which suggests that this is not a physical
resonance but a result of statistical fluctuations.
A further comparison between experiment and theory shows an overall
agreement in both cases. The OPACITY data correctly describes the continuum and also approximately the positions of the resonances but not their
relative intensities. The general spectral structure is reproduced much better
in the MCDF calculation although the strength of the transitions from the
metastable 3 P term seems to be too small.
Moreover, the magnitude of the continuum above the threshold at ∼114
66
Photoionization results
5.2 Be-like
Figure 5.3: Top: Experimental cross section of O4+ . Middle: R-matrix results
from the TOPbase (Tully et al 1990). Bottom: MCDF results from (Champeaux
et al 2003). The theoretical spectra have been convolved with Gaussians of widths
(FWHM) 650 meV for the 2s2 1 S ground state and 450 meV for the 2s2p 3 P
metastable component. For easier comparison between experiment and theory the
experimental values at 107 eV and 124 eV are shown in the theoretical spectra.
67
5.3 C-like
Photoionization results
eV is somewhat smaller than the experiment by as much as 35 % at 124
eV. For further information of the MCDF calculation as well as the most
recent (relativistic) R-matrix result, the reader is referred to (Champeaux et
al 2003).
5.3
C-like
The results for F3+ and Ne4+ are shown in figure 5.4 and recent reports
of photoionization measurements of members of the carbon isoelectronic sequence also includes absolute cross sections of N+ (Kjeldsen et al 2002) and
O2+ (Champeaux et al 2003). An analysis of the data is given below but this
is complicated by the population of several metastable states in the target
ion beam.
5.3.1
Results and analysis
The ground state configuration of a C-like ion is 1s2 2s2 2p2 which gives rise
to a 3 P ground state term and the 1 D and 1 S metastable terms, see table
A.3 in appendix A.2. An additional small fraction of the ion beam was also
produced in the 1s2 2s2p3 5 S excited state and all these consequently lead to
a wealth of possible Rydberg series in the spectra.
The following dipole allowed 2s → np transitions from the 3 P ground
state and 1 D,1 S metastable states and 2p → nd,n0 s excitations from the 5 S
term may show up in the spectra:
1s2 2s2 2p2 3 P + hν → 1s2 2s2p2

4
P
2
P
2
P
!
np


3
S, 3 P, 3 D
3
S, 3 P, 3 D
1
P, 1 D
!
(5.5)


 

2
  1 1 1 
1s2 2s2 2p2 1 D + hν → 1s2 2s2p2 
 D  np  P, D, F 
2
1
S
P
68
(5.6)
Photoionization results
5.3 C-like

2
P


1
P


 

2
 np  1 P 
1s2 2s2 2p2 1 S + hν → 1s2 2s2p2 
D
 


2
1
S
P
1s2 2s2p3 5 S + hν → 1s2 2s2p2 (4 P)nd, n0 s (5 P)
(5.7)
(5.8)
The 3 S terms in eq. (5.5) cannot Coulomb autoionize into the 2s2 2p 2 P
continuum because of the parity selection rule and the excited states in (5.8)
are spin-forbidden; however, as in the case of the B-like ions, such resonances
may still be present.
It has not been possible to use QDT to calculate the location of all these
Rydberg states since for some of the series no members are reported at NIST,
but a few QDT results are shown in the upper panels of figure 5.4. For this
reason, a complete assignment of the resonances is not feasible but it still
seems reasonable to make educated guesses regarding the spectral structure,
especially in the case of F3+ .
In the experimental spectrum of F3+ , which is shown in greater detail in
figure 5.5, two peaks (1 and 2) corresponding to 2p → (n = 6, 7)d transitions
from the 2s2p3 5 S metastable term are clearly visible at 79.58 eV and 80.38
eV, respectively. By comparing the strength of these with the MCDF calculation, a target beam contamination of 2.5 % of the 5 S term was estimated.
Direct ionization from the 2p2 1 D excited state is responsible for the sharp
onset at 84.07 eV (NIST 84.01 eV, see table 5.4) where the cross section rises
to a small plateau with a magnitude of ∼ 0.9 Mb. This plateau has been
used to estimate the 1 D population of the ion beam to 30 %.
Note that there is no clear evidence of an onset at 80.50 eV because of
direct ionization from the 1 S initial excited state. Furthermore, although
it has only been possible to calculate one Rydberg series from QDT (figure
5.4), there does not seem to be any trace of transitions belonging to this. For
these two reasons it was estimated that the 2p2 1 S state was not populated.
The exact experimental location of the ionization threshold for the 3 P ground
state is hard to see because of the presence of three prominent peaks in this
69
5.3 C-like
Photoionization results
Figure 5.4: Top: Experimental photoionization cross sections for F3+ and Ne4+ .
Some possible Rydberg series are also indicated. Middle: R-matrix results (Luo
& Pradhan 1989) from the OPACITY database (TOPbase). The red line is a newer
calculation (100 % 3 P ground state) by Nahar (Nahar 2004). Bottom: MCDF
calculations performed as part of this work (Bizau 2003). The theoretical data have
been convoluted with Gaussians of 300 meV (F3+ ) and 550 meV (Ne4+ ) (FWHM)
to account for the average instrumental resolution. For easier comparison some
experimental values indicated by open circles are also shown.
70
Photoionization results
5.3 C-like
Figure 5.5: Detailed view of the experimental spectrum of F3+ where the resonances have been labeled. Also shown are relevant Rydberg series calculated
from QDT. The dotted series is calculated under the assumption that peak no. 14
corresponds to the 3 P → (2 P)5p 3 L transitions with L = S,P,D.
energy region. The NIST threshold is indicated by a thick vertical line and it
seems plausible that this marks the onset for ionization from the ground state.
It is likely that the (1 D) → (2 D)(n = 4 − 9)p transitions are responsible for
the unresolved peaks 4,7,10-13 although 7 and 11 are probably overlapping
with other states. Similarly, (3 P) → (4 P)(n = 5 − 9)p can be assigned to the
peaks 5,6,7(overlapping),8,9 with some certainty. Number 3 does not fit with
any of the QDT levels but as it lies below the ground state threshold, it is
undoubtedly due to a (2s → (2 P or 2 S)np) transition from the 1 D metastable
component. Moreover, if one carefully assumes that peak no. 14 is the
member of the 2 P → (2 P)np 3 L (L=S,P,D) series with n = 5, then peak no.
71
5.3 C-like
Photoionization results
11 (overlapping) and 15 also belongs to this series.
F3+
Ne4+
State
Energy (eV)
Population (%)
Energy (eV)
Population (%)
2s2 2p2 3 P0
0
67.5
0
56.5
2s2 2p2 1 D
3.13
30
3.76
36
S
6.64
0
7.92
6
2s2p3 5 S
9.20
2.5
10.96
1.5
EIP (2s2 2p 2 P)
87.14
—
126.22
—
2
2 1
2s 2p
Table 5.4: Estimated population of target beam states as well as their energies
according to NIST (NIST 2003).
A similar ”guessing procedure” is not possible in the case of Ne4+ since
even fewer QDT levels can be calculated and the resolution is worse. The
fractions of metastable components in the parent ion beam was estimated
in much the same way as for F3+ . Even though it is far from being obvious
whether or not there are one or two small peaks corresponding to transitions
from the 5 S metastable state it was estimated that this state constituted 1.5
% of the target beam by comparing with the MCDF data. From the small
plateau starting at 122.49 eV the 1 D contamination was determined to be
36 %. On top of this plateau a clear resonance is seen which is reproduced
by both theoretical approaches. By considering the theoretical cross sections
from the 1 S initial state (not shown), it is clear that a transition from this
state is responsible for this peak and by comparison the 1 S term was set to
6 %. The estimated metastable fractions and their energy positions are put
in table 5.4 for both C-like ions.
Finally, as in the case of the B-like ions, a small decrease of the magnitude of the continuum is seen when moving from F3+ to Ne
expected downward movement of Rydberg states.
72
4+
and also the
Photoionization results
5.3.2
5.4 Conclusion
Comparison with theory
The theoretical spectra in figure 5.4 are obtained by applying the formula
σtot = X(3 P )σ(3 P ) + X(1 D)σ(1 D) + X(1 S)σ(1 S) + X(5 S)σ(5 S)
(5.9)
where X(2S+1 L) are the fractions of each target beam state, see table 5.4.
The best agreement is found in the case of F3+ . The OPACITY data naturally fails to reproduce the peaks from the 5 S state since these occur as the
result of relativistic effects, but they are present in the MCDF result. Considering the overall spectral structure, the OPACITY result is generally better
both with regards to positions and relative intensities of the resonances. The
calculated magnitude of the cross section is within the experimental uncertainty; however, the continuum falls off too rapidly in the MCDF case.
For Ne4+ none of the theoretical approaches succesfully reproduces the
experimental spectrum. The agreement is only good below the ground state
threshold where the 1 D plateau with the prominent peak is reproduced in
both the OPACITY and the MCDF data. Above threshold, several discrepancies are seen with respect to both the observed resonances and also the
magnitude of the continuum, which seems to be too low in both cases but
worst in the MCDF calculation
5.4
Conclusion
Absolute cross sections for photoionization of important members of the
beryllium, boron and carbon isoelectronic sequences have been presented.
In most cases it has been possible to characterize the spectral features from
a simple QDT approach although the population of metastable components
of the parent ion beams makes the analysis more complicated. A comparison with theoretical results of the R-matrix and MCDF method has also
been performed and both of these are generally able to reproduce the experimental data, with some differences, though, especially regarding the relative
intensities of the resonances.
73
5.4 Conclusion
Photoionization results
References
Bizau J.-M. (2003) - Private communication
Bizau J.-M., Champeaux J.-P., Cubaynes D., Blancard C., Hitz D., Bruneau
J., Lemaire J.-L, Compant La Fontaine A., Girard A. and Wuilleumier F.
J. (2003) Proceedings of the XXIII’th International Conference on Photonic
Electronic and Atomic Collisions (ICPEAC)
Champeaux J.-P., Bizau J.-M., Cubaynes D., Blancard C., Nahar S. N., Hitz
D., Bruneau J. and Wuilleumier F. J. (2003) Astrophysical Journal Supplement Series, ApJS 148 583
Cunto W., Mendoza C., Ochsenbein F. and Zeippen C. J. (1993) Astronomy
and Astrophysics 275 L5
Fernley J. A., Hibbert A., Kingston A. E. and Seaton M. J. to be published
Kjeldsen H. (1999) PhD thesis University of Aarhus
Kjeldsen H., Folkmann F., Hansen J. E., Knudsen H., Rasmussen M. S.,
West J. B. and Andersen T. (1999) The Astrophysical Journal 524 L143
Kjeldsen H., Kristensen B., Brooks R. L., Folkmann F., Knudsen H. and
Andersen T. (2002) Astrophysical Journal Supplement Series, ApJS 138 219
Luo D. and Pradhan A. K. (1989) J. Phys. B: At. Mol. Opt. Phys. 22 3377
Msezane A., Reilman R. F., Manson S. T., Swanson J. R. and Armstrong Jr.
L. (1977) Phys. Rev. A 13 668
Müller A., Phaneuf R. A., Aguilar A., Gharaibeh M. F., Schlachter A. S.,
Alvarez I., Cisneros C., Hinojosa G. and McLaughlin B. M. (2002) J. Phys.
B: At. Mol. Opt. Phys. 35 L137
Nahar (2004) - data for N2+ , O3+ and F3+ downloaded from ftp.astronomy.ohiostate.edu
NIST (2003) - http://physics.nist.gov/cgi-bin/AtData/levels form
Schippers S., Müller A., McLaughlin B. M., Aguilar A., Cisneros C. Emmons
E. D., Gharaibeh M. F. and Phaneuf R. A. (2003) J. Phys. B: At. Mol. Opt.
Phys. 36 3371
TOPbase - www.heasarc.gsfc.nasa.gov/topbase/topbase.html
Tully J. A., Seaton M. J. and Berrington K. A. (1990) J. Phys. B: At. Mol.
74
Photoionization results
5.4 Conclusion
Opt. Phys. 23 3811
75
76
Concluding remarks
Photoionization and photodetachment are physical processes that are important in areas such as astrophysics and plasma physics and a substantial
amount of theoretical work has been done over the years. By now, absolute
photoionization cross-section measurements of a wide range of singly-charged
ions have been performed whereas results for multiply-charged and especially
negative ions are fewer.
In this thesis, results for photoionization of N2+ , O3+ , O4+ , F3+ , F4+ and
Ne4+ have been presented and compared to theoretical calculations within
the R-matrix approach and the MCDF method. Also presented is the photodetachment cross section of Te− which reveals strongly bound inner-shell
excited states.
The experimental work has been carried out at the undulator beamline
at the storage ring ASTRID in Aarhus using the so-called merged-beam
method. In this technique, an ion beam is generated and overlapped over a
known distance with an intense photon beam from an undulator. By counting
the number of ionized ions in a detector the absolute cross section can be
determined if we also know the target ion density, the number of photons
and the ion-photon overlap.
In the future, studies of trends along isoelectronic and isonuclear sequences will probably be of main interest and the production of multiplycharged ion beams in ECR ion sources makes such experiments feasible. Finally, much more work on inner-shell detachment of negative ions is needed
in order to understand effects of correlated electron motion.
77
5.4 Conclusion
Photoionization results
Figure 5.6 shows absolute results for 3p photodetacment of the V− , Cr− ,
Co− and Ni− negative ions obtained at ASTRID in the autumn 2003. The
results are preliminary and have not yet been analyzed.
Figure 5.6: Preliminary results for detachment of V− , Cr− , Co− and Ni−
78
Appendix A
Useful material
A.1
Selection rules in LS-coupling
Interaction
Dipole (E1)
∆S
∆L
0
∆J
0,±1
∆π
0,±1
Comment
τ (s)
−10
±1
La +Lb >1 10
- 10−8
Ja +Jb >1
Table A.1: Selection rules for dipole transitions
Interaction
∆S
∆L
∆J
∆π
Relative
τ (s)
rate
Coulomb
0
0
0
0
1
10−15 - 10−13
Spin-(other)-orbit
0,±1
0,±1
0
0
α4
10−7 - 10−5
Spin-spin
0,±1,±2 0,±1,±2 0
0
α4
10−7 - 10−5
Hyperfine
0,±1,±2 0,±1
0
me
)
α4 ( M
p
0,±1
Table A.2: Selection rules for autoionization
79
A.2 Coupling of equivalent electrons
A.2
Useful material
Coupling of equivalent electrons
configuration
2S
ns
ns2
1S
np
np5
np2
np4
2P
1 S, 1 D
np3
3P
2 P, 2 D
np6
4S
1S
nd
nd9
nd2
nd8
nd3
nd7
nd4
nd6
2D
1 S, 1 D, 1 G
3 P, 3 F
2 P, 2 D, 2 F, 2 G, 2 H
1 S, 1 D, 1 F, 1 G, 1 I
nd5
3 P, 3 D, 3 F, 3 G, 3 H
2 S, 2 P, 2 D, 2 F, 2 G, 2 H, 2 I
nd10
4 P, 4 F
5D
4 P, 4 D, 4 F, 4 G
1S
Table A.3: Allowed terms for electron configuration (nl)k , with l = 0,1,2
A.3
Hund’s rules
1. For a given configuration, the term with the largest value of S has the
lowest energy.
2. For a given value of S, the term with the maximum possible L has the
lowest energy.
3. For a given value of L and S, the term with J = L + S has the lowest
energy if the subshell is more than half-filled. If the subshell is less
than half-filled, the term with J = |L − S| has the lowest energy.
80
6S
81
2
S1/2
2
S1/2
2
S1/2
2
S1/2
[Rn] 7s
4.0727
(223)
1
S0
1
S0
1
S0
1
S0
(226)
2
[Rn] 7s
5.2784
Radium
Ra
88
137.327
2
[Xe] 6s
5.2117
Barium
Ba
56
87.62
2
[Kr] 5s
5.6949
Strontium
Sr
38
40.078
2
[Ar] 4s
6.1132
Calcium
Ca
20
24.3050
2
[Ne] 3s
7.6462
Magnesium
1
G°
4
T A B L E
Sc
Ni
Cu
Solids
Liquids
Gases
Artificially
Prepared
Zn
P°
1/2
B
Aluminum
Al
2
P1/2
°
10.811
2
2
1s 2s 2p
8.2980
Boron
13
5
2
13
IIIA
P0
Carbon
C
Silicon
Si
14
P0
3
12.0107
2
2
2
1s 2s 2p
11.2603
6
3
14
IVA
physics.nist.gov
Physics
Laboratory
°
S3/2
Nitrogen
N
°
S3/2
P
Phosphorus
15
4
14.0067
2
2
3
1s 2s 2p
14.5341
7
4
15
VA
P2
Oxygen
O
S
Sulfur
16
P2
3
15.9994
2
2
4
1s 2s 2p
13.6181
8
3
16
VIA
P3/2
°
Fluorine
F
2
17
VIIA
Chlorine
°
P3/2
Cl
17
2
18.9984032
2
2
5
1s 2s 2p
17.4228
9
www.nist.gov/srd
Standard Reference
Data Group
Helium
He
S0
1
18
VIIIA
S0
Argon
S0
1
Ar
18
20.1797
2
2
6
1s 2s 2p
21.5645
Neon
1
Ne
10
4.002602
2
1s
24.5874
2
Y
2
D3/2
88.90585
2
[Kr]4d 5s
6.2173
Yttrium
39
44.955910
2
[Ar]3d 4s
6.5615
Scandium
F2
F2
V
6
D1/2
F3/2
Tantalum
4
Ta
73
92.90638
4
[Kr]4d 5s
6.7589
Niobium
Nb
41
50.9415
3
2
[Ar]3d 4s
6.7462
Vanadium
Cr
S3
D0
Tungsten
5
W
74
95.94
5
[Kr]4d 5s
7.0924
Molybdenum
7
Mo
42
51.9961
5
[Ar]3d 4s
6.7665
Chromium
Mn
6
2
S5/2
S5/2
Rhenium
6
Re
75
(98)
5
2
[Kr]4d 5s
7.28
Technetium
Tc
43
[Ar]3d 4s
7.4340
5
54.938049
Manganese
F5
D4
Osmium
5
Os
76
[Kr]4d 5s
7.3605
7
101.07
Ruthenium
5
Ru
44
55.845
6
2
[Ar]3d 4s
7.9024
Iron
Fe
Cobalt
Co
4
F9/2
4
F9/2
Ir
Iridium
77
[Kr]4d 5s
7.4589
8
102.90550
Rhodium
Rh
45
58.933200
7
2
[Ar]3d 4s
7.8810
Nickel
S0
D3
Platinum
3
Pt
78
106.42
10
[Kr]4d
8.3369
Palladium
1
Pd
46
58.6934
8
2
[Ar]3d 4s
7.6398
F2 ?
2
D3/2
2
D3/2
(227)
2
[Rn] 6d7s
5.17
Actinium
Ac
89
138.9055
2
[Xe]5d 6s
5.5769
Lanthanum
La
57
(261)
14
2
2
[Rn]5f 6d 7s ?
6.0 ?
Rutherfordium
3
Rf
104
G°
4
F2
232.0381
2
2
[Rn]6d 7s
6.3067
Thorium
3
Th
90
140.116
2
[Xe]4f5d 6s
5.5387
Cerium
1
Ce
58
(262)
Dubnium
Db
105
°
I9/2
5
Nd
60
(264)
Bohrium
Bh
107
I4
K11/2
231.03588
2
2
[Rn]5f 6d7s
5.89
Protactinium
4
Pa
91
140.90765
3
2
[Xe]4f 6s
5.473
5
L°
6
H°
5/2
L11/2
(237)
4
2
[Rn]5f 6d7s
6.2657
Neptunium
6
Np
93
(145)
5
2
[Xe]4f 6s
5.582
Promethium
6
Pm
61
(277)
Hassium
Hs
108
F0
F0
(244)
6
2
[Rn]5f 7s
6.0260
Plutonium
7
Pu
94
150.36
6
2
[Xe]4f 6s
5.6437
Samarium
7
Sm
62
(268)
Meitnerium
Mt
109
S1/2
Cadmium
Mercury
S0
1
Hg
80
112.411
10
2
[Kr]4d 5s
8.9938
111
112
196.96655
200.59
14
10
2
14
10
[Xe]4f 5d 6s [Xe]4f 5d 6s
9.2255
10.4375
Gold
2
Au
79
107.8682
10
[Kr]4d 5s
7.5762
S0
1
Cd
48
Zinc
65.409
10
2
[Ar]3d 4s
9.3942
S°
7/2
°
S7/2
(243)
7
2
[Rn]5f 7s
5.9738
Americium
8
D°
2
96
D°2
(247)
7
2
[Rn]5f 6d7s
5.9914
Curium
9
157.25
7
2
[Xe]4f 5d6s
6.1498
Gadolinium
9
Gd
64
(272)
Am Cm
95
151.964
7
2
[Xe]4f 6s
5.6704
Europium
8
Eu
63
(281)
H°15/2
H°15/2
(247)
9
2
[Rn]5f 7s
6.1979
Berkelium
6
Bk
97
158.92534
9
2
[Xe]4f 6s
5.8638
Terbium
6
Tb
65
(285)
Ununbium
Uun Uuu Uub
Ununnilium
Unununium
110
S1/2
Silver
2
Ag
47
63.546
10
[Ar]3d 4s
7.7264
Copper
Ge
Arsenic
As
Se
Selenium
Br
Bromine
Kr
Krypton
I8
5
I8
(251)
10
2
[Rn]5f 7s
6.2817
Californium
Cf
98
162.500
10
2
[Xe]4f 6s
5.9389
Dysprosium
5
Dy
66
204.3833
[Hg] 6p
6.1082
Thallium
P1/2
°
2
Tl
81
Lead
°
I15/2
°
I15/2
(252)
11
2
[Rn]5f 7s
6.42
Einsteinium
4
Es
99
164.93032
11
2
[Xe]4f 6s
6.0215
Holmium
4
Ho
67
(289)
Ununquadium
Uuq
114
207.2
2
[Hg]6p
7.4167
P0
3
Pb
82
Tin
P0
3
Sn
50
Antimony
°
S3/2
4
Sb
51
Tellurium
P2
3
Te
52
°
P3/2
2
I
Iodine
53
Xenon
S0
1
Xe
54
72.64
74.92160
78.96
79.904
83.798
10
2
2
10
2
3
10
2
4
10
2
5
10
2
6
[Ar]3d 4s 4p [Ar]3d 4s 4p [Ar]3d 4s 4p [Ar]3d 4s 4p [Ar]3d 4s 4p
7.8994
9.7886
9.7524
11.8138
13.9996
Germanium
H6
H6
F°
7/2
F°
7/2
(258)
13
2
[Rn]5f 7s
6.58
Mendelevium
2
Md
101
168.93421
13
2
[Xe]4f 6s
6.1843
Thulium
2
Tm
69
(292)
Ununhexium
Uuh
116
(209)
4
[Hg] 6p
8.414
Polonium
P2
3
Po
84
S0
S0
(259)
14
2
[Rn]5f 7s
6.65
Nobelium
1
No
102
173.04
14
2
[Xe]4f 6s
6.2542
Ytterbium
1
Yb
70
(210)
5
[Hg] 6p
Astatine
°
P3/2
2
At
85
Radon
D3/2
P°
1/2?
(262)
14
2
[Rn]5f 7s 7p?
4.9 ?
Lawrencium
2
Lr
103
174.967
14
2
[Xe]4f 5d6s
5.4259
Lutetium
2
Lu
71
(222)
6
[Hg] 6p
10.7485
S0
1
Rn
86
NIST SP 966 (September 2003)
(257)
12
2
[Rn]5f 7s
6.50
Fermium
3
Fm
100
167.259
12
2
[Xe]4f 6s
6.1077
Erbium
3
Er
68
208.98038
3
[Hg]6p
7.2855
Bismuth
°
S3/2
4
Bi
83
114.818
118.710
121.760
127.60
126.90447
131.293
10
2
6
10
2
4
10
2
5
10
2
3
10
2
10
2
2
[Kr]4d 5s 5p [Kr]4d 5s 5p [Kr]4d 5s 5p [Kr]4d 5s 5p [Kr]4d 5s 5p [Kr]4d 5s 5p
5.7864
7.3439
8.6084
9.0096
10.4513
12.1298
Indium
P1/2
°
2
In
49
69.723
10
2
[Ar]3d 4s 4p
5.9993
Gallium
Ga
For a description of the data, visit physics.nist.gov/data
238.02891
3
2
[Rn]5f 6d7s
6.1941
Uranium
U
92
144.24
4
2
[Xe]4f 6s
5.5250
Praseodymium Neodymium
4
Pr
59
(266)
Seaborgium
Sg
106
178.49
180.9479
183.84
186.207
190.23
192.217
195.078
14
2
2
14
6
2
14
5
2
14
3
2
14
7
2
14
9
14
4
2
[Xe]4f 5d 6s [Xe]4f 5d 6s [Xe]4f 5d 6s [Xe]4f 5d 6s [Xe]4f 5d 6s [Xe]4f 5d 6s [Xe]4f 5d 6s
6.8251
7.5496
7.8640
7.8335
8.4382
8.9670
8.9588
Hafnium
3
Hf
72
91.224
2
2
[Kr]4d 5s
6.6339
Zirconium
3
Zr
40
47.867
2
2
[Ar]3d 4s
6.8281
Titanium
Ti
26.981538
28.0855
30.973761
32.065
35.453
39.948
3
4
5
6
7
8
9
10
11
12
2
5
2
6
2
4
2
2
2
2
3
[Ne]3s 3p
[Ne]3s 3p
[Ne]3s 3p
[Ne]3s 3p
[Ne]3s 3p
[Ne]3s 3p
IIIB
IVB
VB
VIB
VIIB
VIII
IB
IIB
5.9858
8.1517
10.4867
10.3600
12.9676
15.7596
1
3
6
5
2
3
4
7
4
1
3
3
S0 31 2P1/2
F4 29 2S1/2 30
° 32
° 34
° 36
S5/2 26
D4 27
P3/2
P0 33 4S3/2
F3/2 24
S3 25
F9/2 28
S0
P2 35
F2 23
21 2D3/2 22
For the most accurate values of these and other constants, visit physics.nist.gov/constants
1 second = 9 192 631 770 periods of radiation corresponding to the transition
between the two hyperfine levels of the ground state of 133Cs
-1
speed of light in vacuum
c
299 792 458 m s
(exact)
-34
Planck constant
h
6.6261 × 10 J s
(
/2 )
-19
elementary charge
e
1.6022 × 10 C
-31
electron mass
me
9.1094 × 10 kg
2
0.5110 MeV
me c
-27
proton mass
mp
1.6726 × 10 kg
fine-structure constant
1/137.036
-1
Rydberg constant
R
10 973 732 m
15
R c
3.289 842 × 10 Hz
R hc
13.6057 eV
-23
-1
Boltzmann constant
k
1.3807 × 10 J K
Frequently used fundamental physical constants
C. () indicates the mass number of the most stable isotope.
Ionization
Energy (eV)
140.116
2
[Xe]4f5d6s
5.5387
Cerium
Ce
58
12
S0
Mg
12
1
9.012182
2
2
1s 2s
9.3227
P E R I O D I C
Atomic Properties of the Elements
S0
Be
1
Beryllium
4
2
IIA
Ground-state
Level
Francium
Fr
87
[Xe] 6s
3.8939
132.90545
Cesium
Cs
55
85.4678
[Kr] 5s
4.1771
Rubidium
Rb
37
39.0983
[Ar] 4s
4.3407
Potassium
K
19
22.989770
[Ne] 3s
5.1391
Sodium
Based upon
†
S1/2
Na
2
6.941
2
1s 2s
5.3917
Lithium
Ground-state
Configuration
Atomic
†
Weight
Name
S1/2
Li
11
3
2
1.00794
1s
13.5984
Hydrogen
H
Atomic
Number
7
6
5
4
3
2
1
Symbol
Period
Group
1
IA
2
S1/2
1
Lanthanides
A.4
Actinides
Useful material
A.4 The periodic table
The periodic table