DISS. ETH Nr. 17652
High-resolution spectroscopic
investigations of rare gas atoms and
dimers
A dissertation submitted to
ETH ZURICH
for the degree of
Doctor of Sciences
presented by
Oliver Zehnder
Dipl. Phys. ETH
born 21.05.1974
citizen of Switzerland
Accepted on the recommendation of:
Prof. Dr. F. Merkt, examiner
Prof. Dr. M. Reiher, co-examiner
2008
Abstract
The behavior of Rydberg states of rare gas atoms in static electric fields and electronically excited and ionic states of homonuclear and heteronuclear rare gas dimers have been
investigated by high-resolution resonance-enhanced multiphoton ionization (REMPI) spectroscopy and pulsed-field-ionization zero-kinetic energy (PFI-ZEKE) photoelectron spectroscopy. The atomic Rydberg spectra have been analyzed by multichannel quantum defect
theory (MQDT). The dimer spectra have been used to determine potential energy functions which include the effects of the spin-orbit, the long-range and the charge-exchange
interactions.
An experimental and theoretical investigation of the Stark effect in bound and autoionizing Rydberg states of neon is presented. The Rydberg states with principal quantum number n = 23 − 26 have been excited in a resonance-enhanced two-photon excitation sequence via either the 3s[3/2]1 or the 3s0 [1/2]1 intermediate states. Adjusting the
relative polarization of the laser beams with respect to the applied static electric fields
(F = 0 − 250 V cm−1 ) made the selective excitation of MJ = 0 and MJ = 1 Stark states
possible. Multichannel quantum defect theory (MQDT) calculations including the effects
of the external static electric fields were performed in order to identify all transitions in the
experimental spectra and to determine the eigenquantum defects µα of the np(J = 1, 2)
Rydberg series. The high resolution of the experiment in combination with the knowledge
of the total magnetic quantum number MJ of the final Stark states allowed a detailed
comparison of the experimental results and those calculated by MQDT.
The gerade autoionizing Rydberg states of Ne2 have been studied in the range 162000−
172000 cm−1 by resonance-enhanced two-photon excitation from the X 0+
g ground neutral
+
state of Ne2 via different vibrational levels of the C 0u Rydberg state of Ne2 . A rotationally
resolved part of the spectrum allowed the determination of the potential energy functions
1
5
0
of two states of 1g and 0+
g character in the vicinity of the Ne( S0 ) + Ne(2p 4p ) dissociation limit. The presence of maxima in these potential energy functions is interpreted
as originating from a repulsive interaction between the Rydberg electron and the neutral
atom.
The pulsed-field-ionization zero-kinetic-energy (PFI-ZEKE) photoelectron spectrum of
Xe2 has been measured between 97350 and 108200 cm−1 following resonant two-photon
excitation via selected vibrational levels of the C 0+
u Rydberg state of Xe2 . Transitions
to three of the six low-lying electronic states of Xe+
2 could be observed. Whereas ex+
tensive vibrational progressions were observed for the I(3/2g) ← X 0+
g and I(3/2u) ← X 0g
photoelectron transitions, only the lowest vibrational levels of the II(1/2u) state could be
i
Abstract
detected. Unambiguous assignments of the vibrational quantum numbers were derived
from the analysis of the isotopic shifts and from the modeling of the potential energy
curves. Adiabatic ionization energies, dissociation energies and vibrational constants are
reported for the I(3/2g) and the I(3/2u) states. The energies of vibrational levels, measured presently and in a previous investigation, were used to determine the potential energy
functions of the six low-lying electronic states of Xe+
2 using a global model that includes the
long-range interaction and treats, for the first time, the spin-orbit interaction as dependent
on the internuclear separation.
Photoionization and pulsed-field-ionization zero-kinetic-energy (PFI-ZEKE) photoelectron spectra of ArXe and KrXe have been recorded between 94800 and 108200 cm −1 following resonance-enhanced two-photon excitation via selected vibrational levels of the C 1
and D 0+ Rydberg states of ArXe and the C 0+ and D 1 Rydberg states of KrXe. The
PFI-ZEKE photoelectron spectra consist of three vibrational progressions corresponding
to the X 1/2 ← X 0+ , A1 3/2 ← X 0+ and A2 1/2 ← X 0+ transitions. Spectra of the
B 1/2 and C2 1/2 state of KrXe+ have been recorded in the range 111900 − 112700 cm−1
via selected vibrational levels of a Rydberg state of KrXe located in the vicinity of the
Kr∗ ([4p]5 5s[3/2]1 ) + Xe(1 S0 ) dissociation limit. From the observed progressions, adiabatic
ionization energies, dissociation energies, equilibrium internuclear distances and vibrational
constants have been derived for the X 1/2, A1 3/2 and A2 1/2 electronic states of ArXe+ and
the X 1/2, A1 3/2, A2 1/2, C1 3/2 and B 1/2 states of KrXe+ . The photoionization spectra
of ArXe and KrXe reveal long progressions of autoionizing Rydberg states converging to
the lowest vibrational levels of the A1 3/2 states of ArXe+ and KrXe+ . A potential model
has been developed that enables, for the first time, a global description of the low-lying
electronic states of the heteronuclear rare gas dimer ions. The model explicitly treats the
effects of the spin-orbit, charge-exchange and long-range interactions and was used to obtain potential energy functions for all six low-lying electronic states of ArXe+ and KrXe+
using the experimental positions derived in this work and spectroscopic data reported in
earlier studies.
A comparison of the potential energy curves and of the corresponding dissociation
energies and equilibrium internuclear distances of the six low-lying electronic states of
+
+
Xe+
2 , ArXe and KrXe enables one to quantify the contribution of the charge-exchange
interaction to the chemical bonds in these ions. The dissociation energies of the ground
electronic state increase rapidly from ArXe+ (De = 1480.3 cm−1 ), for which the charge−1
exchange is weak and nonresonant, to Xe+
2 (De = 7937.4 cm ), for which the chargeexchange can be regarded as being resonant.
ii
Zusammenfassung
In dieser Arbeit wurde das Verhalten von Rydbergzuständen von Edelgasatomen in
elektrischen Feldern sowie die Struktur und Dynamik elektronisch angeregter und ionischer Zustände von homo- und heteronuklearen Edelgasdimeren mittels hochauflösender resonanzverstärkter Mehrphotonenionisations-(REMPI)-Spektroskopie und pulsed-field-ionization zero-kinetic energy (PFI-ZEKE) Spektroskopie untersucht. Die Ionenspektren der
atomaren Rydbergzustände wurden unter Anwendung der Vielkanal-Quantendefekt-Theorie
(MQDT) analysiert. Die Photoelektronenspektren der Dimere wurden verwendet, um Potentialenergiekurven für die sechs tief liegenden elektronischen Zustände der Dimerkationen
zu bestimmen. Die Kurven beschreiben in korrekter Form die langreichweitigen Wechselwirkungen, sowie die Wechselwirkungen, welche durch die Spin-Bahn-Kopplung und den
Ladungsaustausch zwischen elektronischen Zuständen mit gleicher Quantenzahl Ω verursacht werden.
Der Stark Effekt in gebundenen und autoionisierenden Rydbergzuständen von Neon
wurde sowohl experimentell als auch theoretisch untersucht. Rydbergzustände mit effektiver Quantenzahl n = 23 − 26 wurden mittels resonanzverstärkter Zweiphotonenanregung
über den 3s[3/2]1 oder den 3s0 [1/2]1 Zwischenzustand angeregt. Durch Festlegen der relativen Polarisation der Laserstrahlen bezüglich der Richtung der statischen elektrischen
Felder (F = 0 − 250 V cm−1 ) konnten Starkzustände totaler magnetischer Quantenzahl
MJ = 0 oder MJ = 1 selektiv angeregt werden. Vielkanal-Quantendefekt-Theorie (MQDT)
Rechnungen, welche die durch das externe statische elektrische Feld verursachten Effekte
berücksichtigen, ermöglichten eine Zuordnung der im Experiment beobachteten Übergänge
und die Bestimmung der Eigenquantendefekte µα der np(J = 1, 2) Rydbergserien. Die
Aufnahme von hochaufgelösten Spektren von Rydbergzuständen mit wohldefinierten totalen magnetischen Quantenzahlen MJ erlaubte einen detaillierten Vergleich zwischen den
experimentellen Beobachtungen und den MQDT Rechnungen.
Die geraden autoionisierenden Rydbergzustände von Ne2 wurden im Bereich 162000 −
172000 cm−1 mittels resonanzverstärkter Zweiphotonenanregung vom neutralen X 0+
g Grundzustand von Ne2 über verschiedene Vibrationsniveaus des C 0+
u Rydbergzustandes von Ne2
angeregt. Die Auflösung der Rotationsstruktur einzelner Banden ermöglichte die Bestimmung der Potentialkurven zweier elektronisch angeregter Zustände mit den Symmetriebeze1
5
0
ichnungen 1g und 0+
g , welche in der Nähe der Ne( S0 ) + Ne(2p 4p ) Dissoziationsgrenze
liegen. Charakteristisch für diese Potentialkurven sind Maxima, welche energetisch höher
liegen als die zugehörige Dissoziationsgrenze und ihre Ursache in der Wechselwirkung eines
Rydbergelektrons mit dem neutralen Atom haben.
iii
Zusammenfassung
Unter Verwendung von resonanzverstärkter Zweiphotonenanregung via definierter Vibrationsniveaus des C 0+
u Rydbergzustandes von Xe2 wurden im Bereich zwischen 97350
und 108200 cm−1 Photoelektronenspektren von Xe2 aufgenommen. Übergänge zu drei
der sechs tief liegenden elektronischen Zustände von Xe+
2 konnten beobachtet werden.
+
Während ausgedehnte Vibrationsprogressionen der I(3/2g) ← X 0+
g und I(3/2u) ← X 0g
Photoelektronenübergänge gemessen worden konnten, wurden für den II(1/2u) Zustand
nur die tiefsten Vibrationsniveaus detektiert. Die Analyse der Isotopenverschiebung und
die Modellierung der Potentialkurven ermöglichte die definitive Zuordnung der Vibrationsquantenzahlen. Für die I(3/2g) und I(3/2u) Zustände von Xe+
2 wurden die adiabatischen Ionisationsenergien, Dissoziationsenergien und Vibrations- und Anharmonizitätskonstanten ermittelt. Mit den in dieser Arbeit gewonnenen experimentellen Positionen und
früher publizierten spektroskopischen Daten über die Vibrationsstruktur der elektronischen
Zustände von Xe+
2 konnten analytische Potentialkurven mit spektroskopischer Genauigkeit
von ∼ 1 meV bestimmt werden. Das zugrundeliegende globale Modell berücksichtigt
die langreichweitigen Wechselwirkungen und erstmalig die Abhängigkeit der Spin-BahnKopplung vom internuklearen Abstand R.
Photoionisations- und pulsed-field-ionization zero-kinetic-energy (PFI-ZEKE) Photoelektronspektren von ArXe und KrXe wurden zwischen 94800 und 108200 cm−1 mittels resonanzverstärkter Zweiphotonenanregung via definierter Vibrationsniveaus der C 1 and D 0 +
Rydbergzustände von ArXe und der C 0+ und D 1 Rydbergzustände von KrXe aufgenommen. Die PFI-ZEKE Photoelektronspektren beinhalten drei Vibrationsprogressionen, welche den elektronischen Übergängen X 1/2 ← X 0+ , A1 3/2 ← X 0+ und A2 1/2 ← X 0+
zugeordnet werden können. Spektren der B 1/2 und C2 1/2 Zustände von KrXe+ im Bereich 111900 − 112700 cm−1 wurden via ausgewählte Vibrationsniveaus eines Rydbergzustands von KrXe aufgenommen, der gerade unter der Kr∗ ([4p]5 5s[3/2]1 ) + Xe(1 S0 ) Dissoziationsgrenze liegt. Für die X 1/2, A1 3/2 und A2 1/2 Zustände von ArXe+ und die
X 1/2, A1 3/2, A2 1/2, C1 3/2 und B 1/2 Zustände von KrXe+ wurden die adiabatischen
Ionisationsenergien, Dissoziationsenergien, Gleichgewichtsabstände und Vibrations- und
Anharmonizitätskonstanten bestimmt. Die Photoionisationsspektren von ArXe und KrXe
zeigen lange Progressionen autoionisierender Rydbergzustände, welche zu den tiefsten Vibrationsniveaus der A1 3/2 Zustände von von ArXe+ und KrXe+ konvergieren. Ein neues
Model wurde entwickelt, welches zum ersten Mal eine globale Beschreibung der sechs tief
liegenden elektronischen Zustände der heteronuklearen Edelgasdimerkationen ermöglichte.
Das Modell behandelt explizit die Effekte der Spin-Bahn-, Ladungsaustausch- und langreichweitigen Wechselwirkungen und wurde zur Bestimmung der Potentialkurven aller
sechs tief liegenden elektronischen Zustände von ArXe+ und KrXe+ unter Verwendung der
experimentellen Positionen und früher publizierter spektroskopischer Daten benutzt.
Ein Vergleich der Potentialkurven und der zugehörigen Dissoziationsenergien und Gle+
ichgewichtsabstände der sechs tief liegenden elektronischen Zustände von Xe+
2 , ArXe und
iv
Zusammenfassung
KrXe+ erlaubt eine quantitative Diskussion des Beitrags der Ladungsaustauschwechselwirkung zur chemischen Bindung in diesen Ionen. Die Dissoziationsenergien der elektronischen Grundzustände nehmen von ArXe+ (De = 1480.3 cm−1 ), wo Ladungsaustausch
−1
schwach und nichtresonant ist, zu Xe+
2 (De = 7937.4 cm ), wo Ladungsaustausch als
resonant betrachtet werden kann, rasch zu.
v
vi
Contents
Abstract
i
1 Introduction
1
2 Background information
5
2.1
Rare gas atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.2
Rare gas dimers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.3
The electronic ground state of rare gas dimers . . . . . . . . . . . . . . . .
7
2.3.1
Homonuclear rare gas dimers
. . . . . . . . . . . . . . . . . . . . .
8
2.3.2
Heteronuclear rare gas dimers . . . . . . . . . . . . . . . . . . . . .
9
The electronically excited states of rare gas dimers . . . . . . . . . . . . . .
9
2.4
2.5
2.6
2.7
2.4.1
Homonuclear rare gas dimers
. . . . . . . . . . . . . . . . . . . . .
11
2.4.2
Heteronuclear rare gas dimers . . . . . . . . . . . . . . . . . . . . .
11
The electronic states of rare gas dimers cations . . . . . . . . . . . . . . . .
12
2.5.1
Homonuclear rare gas dimers
. . . . . . . . . . . . . . . . . . . . .
12
2.5.2
Heteronuclear rare gas dimers . . . . . . . . . . . . . . . . . . . . .
14
Potential model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
2.6.1
Long-range behavior . . . . . . . . . . . . . . . . . . . . . . . . . .
18
Vibrational assignments . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
3 Experimental setup and methods
3.1
3.2
25
Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
3.1.1
Photoionization spectroscopy . . . . . . . . . . . . . . . . . . . . .
25
3.1.2
REMPI spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . .
26
3.1.3
PFI-ZEKE photoelectron spectroscopy . . . . . . . . . . . . . . . .
27
Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
3.2.1
Laser system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
3.2.2
Spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
3.2.3
Control and detection electronics . . . . . . . . . . . . . . . . . . .
36
4 High-resolution study and MQDT analysis of the Stark effect in Rydberg
states of neon
39
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
39
Contents
4.2
Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
4.3
Multichannel quantum defect theory of the Stark effect . . . . . . . . . . .
43
4.3.1
Field-free MQDT of the rare gas atoms . . . . . . . . . . . . . . . .
43
4.3.2
MQDT including the effect of a Stark field . . . . . . . . . . . . . .
43
4.3.3
Methodology in the MQDT calculations . . . . . . . . . . . . . . .
45
Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
4.4
4.4.1
Stark spectra of the bound Rydberg states of neon excited via 3s[3/2]1 47
4.4.2
Stark spectra of bound Rydberg states of neon excited via 3s0 [1/2]1
4.4.3
4.5
4.6
49
0
Stark map of the MJ = 0 Rydberg states of neon excited via 3s [1/2]1
in the autoionizing region . . . . . . . . . . . . . . . . . . . . . . .
50
MQDT calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
4.5.1
Spectral assignments and fitting procedure . . . . . . . . . . . . . .
51
4.5.2
Characterization of the 3s[3/2]1 and 3s0 [1/2]1 intermediate states . .
53
4.5.3
Comparison of measured and calculated intensity distribution . . .
54
Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
5 Spectroscopic characterization of the potential energy functions of Ne2
Rydberg states in the vicinity of the Ne(1 S0 ) + Ne(4p0 ) dissociation limits 57
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
5.2
Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
5.3
Photoionization spectra and their assignment
. . . . . . . . . . . . . . . .
61
5.4
The potential energy functions of Ne2 Rydberg states . . . . . . . . . . . .
69
5.4.1
Potential model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
5.4.2
Fitting procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
5.4.3
Potential functions . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
5.4.4
Calculation of the tunneling predissociation linewidths . . . . . . .
76
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
5.5
6 The low-lying electronic states of Xe+
2 and their potential energy functions
79
6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
6.2
Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
6.3
Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
6.3.1
The
Rydberg state of Xe2 . . . . . . . . . . . . . . . . . . . .
81
6.3.2
The I(3/2g) and I(3/2u) states of Xe+
. . . . . . . . . . . . . . . .
2
82
6.3.3
6.3.4
6.4
C 0+
u
The II(1/2u) state of
Xe+
2
. . . . . . . . . . . . . . . . . . . . . . .
Spectroscopic constants of the I(3/2g) and I(3/2u) states of
Xe+
2
. .
84
85
Xe+
2
86
6.4.1
Potential model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
6.4.2
Fitting procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
The potential energy functions of the six low-lying electronic states of
viii
Contents
6.4.3
6.5
Potential energy functions . . . . . . . . . . . . . . . . . . . . . . .
93
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
7 The low-lying electronic states of ArXe+ and KrXe+ and their potential
energy functions
97
7.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
7.2
Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
7.3
The C and D Rydberg states of ArXe and KrXe . . . . . . . . . . . . . . . 101
7.3.1
The C 1 and D 0+ Rydberg states of ArXe located in the vicinity of
the Xe∗ (6s0 [1/2]1 ) + Ar(1 S0 ) dissociation limit . . . . . . . . . . . . 102
7.3.2
The C 0+ and D 1 Rydberg states of KrXe located in the vicinity of
the Xe∗ (6s0 [1/2]1 ) + Kr(1 S0 ) dissociation limit . . . . . . . . . . . . 104
7.4
The Rydberg state of KrXe located in the vicinity of the Kr∗ ([4p]5 5s[3/2]1 ) + Xe(1 S0 )
dissociation limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
7.5
The X 1/2 and A1 3/2 states of ArXe+ and KrXe+ . . . . . . . . . . . . . . 106
7.6
7.7
7.5.1
The X 1/2 and A1 3/2 states of ArXe+ . . . . . . . . . . . . . . . . 107
7.5.2
The X 1/2 and A1 3/2 states of KrXe+ . . . . . . . . . . . . . . . . 111
The A2 1/2 state of ArXe+ and KrXe+ . . . . . . . . . . . . . . . . . . . . 113
7.6.1
The A2 1/2 state of ArXe+ . . . . . . . . . . . . . . . . . . . . . . . 113
7.6.2
The A2 1/2 state of KrXe+ . . . . . . . . . . . . . . . . . . . . . . . 114
The B 1/2 and C1 3/2 states of ArXe+ and KrXe+ . . . . . . . . . . . . . . 116
7.7.1
The B 1/2 and C1 3/2 states of KrXe+ . . . . . . . . . . . . . . . . 116
7.8
Spectroscopic constants for the ionic states . . . . . . . . . . . . . . . . . . 118
7.9
The potential energy functions of the six low-lying electronic states of ArXe+
and KrXe+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
7.9.1
A global potential model for the six low-lying electronic states of the
heteronuclear rare gas dimers . . . . . . . . . . . . . . . . . . . . . 119
7.9.2
Fitting procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7.9.3
Potential energy functions . . . . . . . . . . . . . . . . . . . . . . . 128
7.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
8 Outlook
135
A Physical constants
139
B Fundamental data on rare gas atoms and dimers
141
C Experimental data
147
C.1 Ne2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
C.2 Xe+
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
2
C.3 ArXe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
ix
Contents
C.4 KrXe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
Currculum Vitae
198
Publications
200
Danksagung
202
x
Chapter 1
Introduction
This dissertation is primarily concerned with the properties of rare gas atoms and diatomic
molecules made of these atoms. The rare gas atoms He, Ne, Ar, Kr, Xe and Rn are
prototypes of inert gases and are chemically exceptionally inert. This property can be
explained by the fact that rare gas atoms have a fully occupied valence shell in their
ground electronic state. Although rare gas atoms are chemically inert they can form
dimers, trimers and higher clusters as a result of the weak van der Waals interactions
between ground state atoms and stronger interactions in electronically excited or charged
clusters. Rare gas dimers are encountered in a variety of environments. They are ubiquitous
in high-pressure lamps [1], plasma display panels [2], excimer and rare gas ion lasers [3, 4, 5]
and have attracted scientific interest as a potential source of continuous radiation in the
vacuum ultraviolet (VUV) region of the electromagnetic spectrum [4, 5]. They are model
systems for studying atom-atom and atom-ion interactions and represent a starting point
to describe the properties of larger rare-gas clusters [6, 7].
+
Rare gas dimers in their ground neutral state (X 1 Σ+
g or X 0g for homonuclear dimers
and X 1 Σ+ or X 0+ for heteronuclear dimers) are prototypical examples of van der Waals
molecules. Despite their fundamental importance in discussions of the nature of chemical
bonds in most chemistry textbooks [8, 9, 10, 11, 12] - to explain the repulsive nature of the
ground state potential energy function it is usually argued that the antibonding molecular
orbitals are fully occupied - very few detailed spectroscopic investigations have been carried
out and the data available are often incomplete.
Rare gas dimers in electronically excited states represent prototypes of Rydberg molecules [13] and, for a long time, He∗2 represented the only known and spectroscopically
characterized example of this class of molecules which includes now species such as other
rare gas dimers and molecules like H∗3 , H3 O∗ , NH∗4 and CH∗5 . The characteristic properties
of these Rydberg molecules are an essentially repulsive ground state and strongly bound
electronically excited states. Molecules with such properties are so-called excimer (excited
dimer) or exciplex (excited complex) molecules and find a broad range of application in
laser physics and technology [5].
1
Chapter 1. Introduction
Spectroscopic investigations of the electronically excited states of rare gas dimers are
complicated by the fact that homonuclear dimers have neither rotational nor vibrational
spectra and that electronic emission spectra lie in the VUV and often consist of structureless
emission bands. Mulliken [14, 15] has given a qualitative analysis of the electronic structure
of the diatomic rare gas neutral and positively charged molecules. In the limit of high values
of the principal quantum number n, the potential energy functions of Rydberg states
are identical to those of the corresponding singly-charged ions. In addition to families
of bound Rydberg configurations, there are families of repulsive Rydberg configurations.
At low n values, configuration interactions between bound and repulsive Rydberg states
associated with ion cores in different electronic states lead to unusually complex structural
and dynamical properties [14, 15, 16] and most electronically excited states are complex
mixtures of these configurations.
The rare gas dimer ions are prototypes of molecules built from a full shell neutral atom
and an open shell atomic ion and play a role in excimer and rare gas ion lasers as a source
of absorption losses. In chemistry textbooks [17, 10, 11, 12], they are often discussed as
prototypes of molecules of bond order 21 , but, here again, surprisingly little spectroscopic
data is available with which to characterize their structural properties.
The main motivation of this thesis was to obtain high-resolution spectroscopic data of
+
+
the Rydberg states of Ne2 and on the low-lying electronic states of Xe+
2 , ArXe and KrXe
and use this data to determine potential energy functions which satisfactorily reproduce
all the experimental observations. A detailed discussion of the electronic structure of these
systems necessitates the consideration of long-range electrostatic, spin-orbit and charge-
exchange interactions. The rare gas dimers offer ideal systems with which to quantify
the contributions of these interactions to the potential energy functions by systematically
varying the nature of the atoms involved, the degrees of electronic excitation, and the
electronic symmetry.
Chapter 2 provides background information on rare gas atoms and dimers which has
been classified according to whether the dimers are homonuclear or heteronuclear, neutral
or positively charged, or in their ground or electronically excited states. Care has been
taken to select what we feel are the most reliable data from the vast literature on these
systems and to present them in compact form as tables. The analytical potential energy
functions and models used to describe the electronic states of rare gas dimer ions and the
procedure employed to determine the vibrational assignment of electronic states are also
summarized in this chapter.
Chapter 3 describes the experimental set-up consisting of a laser system to generate
vacuum ultraviolet (VUV) radiation and a dual electron/ion time-of-flight spectrometer.
All experimental results presented in this work have been derived by applying two common spectroscopic methods, resonance-enhanced multiphoton ionization (REMPI) spectroscopy and pulsed-field-ionization zero-kinetic-energy (PFI-ZEKE) photoelectron spec2
troscopy, the principles of which are explained in this chapter.
Chapter 4 is concerned with the spectroscopy of Rydberg states of atomic neon and
how these states are affected by an externally applied electric field. The analysis of the
spectra provides several fundamental parameters such as quantum defects that are essential
to characterize the properties of Rydberg states. An atom in a homogeneous electric field
has some common features with a diatomic molecule; the overall symmetry is reduced
from spherical to cylindrical, and the electronic structure of the atom responds, in first
approximation, to the electric field as an atom would to a charged atom placed at large
distance. Although this analogy is not pursued further in Chapter 4, it is worth keeping
it in mind in view of the later chapters devoted to the electronic structure of the rare gas
dimers and the rare gas dimer ions.
Chapter 5 is devoted to a high-resolution spectroscopic study of electronically excited
states of Ne2 and represents the first detailed investigation of the low-lying (principal
quantum number n = 4 − 7) gerade Rydberg states of this molecule. This chapter reveals
the astonishing diversity of the potential energy functions of these states and discusses
a simple ion-atom scattering model with which to characterize the multiple minima and
barriers in the potential energy functions.
Chapter 6 describes the determination of the potential energy functions of the six
low-lying electronic states of Xe+
2 by combining high-resolution spectroscopic data with a
model including the results of an independent ab initio quantum chemical calculation of
the dependence of the spin-orbit coupling constant on the internuclear separation R. Until
the present work, all attempts at treating the spin-orbit interaction in the homonuclear rare
gas dimer ions had relied on the approximation of an R-independent spin-orbit coupling
constant originally introduced by Cohen and Schneider [18]. The results of Chapter 6 are of
particular significance because they enable to quantify the consequences of this widely-used
approximation for the first time.
Chapter 7 finally provides the first global survey of the low-lying electronic states of the
heteronuclear rare gas dimer cations ArXe+ and KrXe+ . From the experimentally derived
vibrational positions potential energy functions have been determined using a global model
for all six low-lying electronic states which explicitly treats the effects of the spin-orbit,
charge-exchange and long-range interactions.
This dissertation ends with a brief conclusion and outlook in Chapter 8.
3
Chapter 1. Introduction
4
Chapter 2
Background information
This chapter provides background information on rare gas atoms (Section 2.1) and homonuclear and heteronuclear rare gas dimers in their ground (Section 2.3), electronically excited
(Section 2.4) and singly-charged ionic (Section 2.5) states, respectively. The last two sections deal with two rather technical but central aspects of the present thesis, Section 2.6
with an analytical description of potential energy functions for the electronic states of the
rare gas dimer ions including the long-range interactions, and Section 2.7 with the determination of the vibrational assignments in photoelectron spectra, which are of crucial
importance in the derivation of reliable potential energy functions, dissociation energies
and adiabatic ionization energies.
2.1
Rare gas atoms
The rare gases He, Ne, Ar, Kr, Xe and Rn exhibit an extremely low chemical reactivity
because they possess a fully occupied valence shell in their electronic ground state. The
ground state electronic configuration is labeled (1s2 ) for He and (ns2 )(np6 ) for the heavier
rare gas atoms, where n describes the principal quantum number of the outermost shell
with values ranging from 2 for Ne up to 6 for Rn. The corresponding term symbol of
the electronic ground state is 1 S0 . Valence electrons are the outermost electrons of an
atom and are, in first approximation, the only electrons which participate in chemical
bonds. According to atomic theory derived from quantum mechanics and experimental
observations, atoms with full valence shells are stable and do not easily form chemical
bonds. The stability of a rare gas atom configuration can also be seen in the high ionization
energies and negligible electronegativities of these elements. He is the element with the
highest ionization energy, followed by Ne, F and Ar. Interactions between rare gas atoms
are very weak, resulting in low melting and boiling points. Under normal conditions, rare
gases exist as monoatomic gases, even those with larger atomic masses than many normally
solid elements.
The electronic ground state of a rare gas cation possesses the configuration (ns 2 )(np5 )
5
Chapter 2. Background information
(He+ : 1s1 ) with the corresponding term symbol 2 P (He+ : 2 S1/2 ). As a result of the spinorbit interaction this configuration leads to two states labeled as 2 P3/2 and 2 P1/2 with
different values of the total angular momentum quantum number J. Therefore, the first
two ionization energies of a rare gas atom are Ei (2 P3/2 ) and Ei (2 P1/2 ). According to Hund’s
third rule [19, 20], the 2 P1/2 state lies higher in energy than the 2 P3/2 state because the
valence shell is more than half occupied. Table 2.1 summarizes the values of selected
properties of the rare gas atoms.
Table 2.1:
First and second ionization energies Ei 2 P3/2 and Ei 2 P1/2 , spin-orbit splittings
A = Ei 2 P1/2 − Ei 2 P3/2 and static polarizabilities αd of the rare gas atoms He, Ne, Ar, Kr and
Xe.
Rare gas Ei
He
Ne
Ar
Kr
Xe
2
P3/2 /(hc cm−1 )
Ei
198310.6672(15) b
173929.7726(6) c
127109.842(4) f
112914.434(16) g
97833.790(11) i
2
P1/2 /(hc cm−1 )
174710.1966(11) d
128541.425(4) d
118284.728(44) h
108370.714(16) j
A/cm−1
780.4240(11) e
1431.5831(7) e
5370.294(44) h
10536.925(19) j
αd /a30
a
1.383112
2.6629
11.074(13)
16.737
27.2903
a
From Teachout and Pack
[21].
Value for Ei 2 S1/2 /(hc) for 4 He from Eikema et al.
[22].
Bergeson et al.
[23] report
198310.6712(16) cm−1 .
c
From Chang et al. [24].
d
Calculated with the values of Ei 2 P3/2 /(hc) and A.
e
From Yamada et al. [25].
f
From Velchev et al. [26].
g
Value for 84 Kr from Hollenstein et al. [27], where the values for the other isotopes are also reported.
h
Value for 84 Kr from Paul and Merkt [28], where the values for the other isotopes are also reported.
i
Value for 132 Xe from Brandi et al. [29], where the values for the other isotopes are also reported.
j
Value for 132 Xe from Wörner et al. [30].
b
All rare gases except Rn possess several stable, naturally occurring isotopes ( A Rg with
mass number A). The masses of these isotopes and their natural abundances p(A Rg) and
nuclear spins are listed in Table B.1 in Appendix B. The average mass m(Rg) of a rare
gas atom is calculated as
m(Rg) =
N
X
p
Ai
i=1
Rg · m
Ai
Rg ,
(2.1)
where N corresponds to the total number of stable naturally occurring isotopes.
2.2
Rare gas dimers
N (N + 1)/2 isotopomers of a homonuclear rare gas dimer can be formed from a monomer
with N naturally occurring isotopes. The reduced mass µ and the natural abundance p
6
2.3. The electronic ground state of rare gas dimers
0
of a homonuclear rare gas isotopomer A Rg−A Rg can be determined with Eqs. (2.2) and
(2.3), respectively
0
m A Rg · m A Rg
A
A0
µ Rg− Rg =
(2.2)
m (A Rg) + m (A0 Rg)

1 if A = A0
0
0
p A Rg−A Rg = ζ · p A Rg · p A Rg
with ζ =
(2.3)
2 if A 6= A0 .
The reduced mass µ and natural abundance p of a heteronuclear isotopomer A Rg−A Rg0
consisting of two different rare gase atoms Rg and Rg0 with masses m and m0 and natural
abundances p and p0 are calculated with Eqs. (2.4) and (2.5). The total number of different
heteronuclear rare gas dimers that can be formed amounts to N · N 0 where N and N 0
0
correspond to the number of naturally occurring isotopes of the relevant rare gas atoms
0
A
Rg and A Rg0
0
m A Rg · m0 A Rg0
A
A0
0
(2.4)
µ Rg− Rg =
m (A Rg) + m0 (A0 Rg0 )
0
0
A
A0
A
0 A0
Rg .
p Rg− Rg = p Rg · p
(2.5)
The reduced masses and natural abundances of the different homonuclear and het-
eronuclear rare gas dimers are summarized in Tables B.3 - B.9 in Appendix B. In the
following sections the characteristics of homonuclear and heteronuclear rare gas dimers in
their ground, electronically excited and ionic states are discussed.
2.3
The electronic ground state of rare gas dimers
The interactions of two rare gas atoms, which are both in the electronic 1 S0 ground state,
can be divided into an attractive and a repulsive part. At large internuclear distances,
where the atoms can be considered as separated systems [31, 32], the attractive van der
Waals interactions are dominant. They originate in the interactions of the instantaneous,
induced multipole moments of the two atoms involved [33, 34]. Because both atoms only
consist of fully occupied shells the exchange interaction does not lead to the formation of a
covalent bond [35, 36]. At short internuclear distances, the two full shell atoms repel each
other. If the molecular orbitals are described as linear combinations of atomic orbitals, the
dominant electronic configuration of the ground state of a homonuclear rare gas dimer is
(σg )2 (πu )4 (πg∗ )4 (σu∗ )2 [36] where only the valence p orbitals have been considered. In this
picture, the number of electrons occupying bonding molecular orbitals (σg , πu ) is equal to
the number of electrons in antibonding molecular orbitals (πg∗ , σu∗ ) leading to a bond order
of zero [10]. A similar argumentation applies to the heteronuclear rare gas dimers, but as a
result of the absence of the g/u symmetry, the electronic configuration of the ground state
of a heteronuclear rare gas dimer is now (σ)2 (π)4 (π ∗ )4 (σ ∗ )2 .
Because chemical bonds are usually much stronger than Van der Waals bonds the
7
Chapter 2. Background information
dissociation energies De of the electronic ground states of the homo- and heteronuclear
rare gas dimers are much smaller, and the equilibrium internuclear distances are much
larger than those of other diatomic molecules exhibiting a chemical bond (e.g. Cl2 ).
2.3.1
Homonuclear rare gas dimers
The body of literature on the X 0+
g electronic ground states of the homonuclear rare gas
dimers He2 , Ne2 , Ar2 , Kr2 and Xe2 is vast and therefore only the contributions most relevant
to this thesis are cited here. A detailed review of previous work on the ground neutral
states of homonuclear rare gas dimers is given in Ref. [37] where also the determination of
the potential energy functions of the X 0+
g states of Ne2 and Xe2 from spectroscopic and
thermodynamic (virial coefficients) data is presented.
Table 2.2 lists the dissociation energies D0 and De , the equilibrium internuclear distances Re and the first adiabatic ionization energies of the homonuclear rare gas dimers.
The data presented in this table, corresponds to our critical evaluation of the literature.
Other values for D0 , De and Re can be found in Refs. [38, 39, 40, 41].
Table 2.2: Dissociation energies D0 and De , and equilibrium internuclear distances Re of the neutral
X 0+
g ground states of He2 , Ne2 , Ar2 , Kr2 and Xe2 and first adiabatic ionization energies Ei (I(1/2u)) of
these dimers.
Rg2
D0 /cm−1
De /cm−1
Re /Å
Ei (I(1/2u))/(hc cm−1 )
He2
Ne2
Ar2
Kr2
Xe2
0.00120(3) a
16.836 c
84.5(10) e
126.8(18) h
186.0 k
7.650(3) a
29.40(12) c
99.55 f
146.87 i
196.1(11) k
2.968(6) a
3.094(1) c
3.7570 f
3.9569 i
4.378(9) k
179205.45(6) b
162340(100) d
116594.2(7) g
103773.6(20) j
90147.4(10) l
a
From Jeziorska et al. [42].
From Raunhardt et al. [43].
c
From Wüest und Merkt [44].
d
Calculated by numerically solving the Schrödinger equation with the potential energy function
of
+
−1
2
the I(1/2u) state of Ne+
from
Ha
et
al.
[45],
D
(Ne
,
X
0
)
=
16.836
cm
[44]
and
E
P
/hc
=
0
2
i
3/2
g
2
173929.7726(6) cm−1 [24].
e
From Herman et al. [46].
f
From Aziz [47].
g
From Rupper and Merkt [48].
h
From Tanaka et al. [49].
i
From El-Kader [50].
j
From Signorell et al. [51].
k
From Wüest et al. [52].
l
From Rupper et al. [53].
b
The dissociation energies De of the X 0+
g electronic ground states of the homonuclear
rare gas dimers lie in the range between 7.650 cm−1 for He2 [42] and 196.1 cm−1 for Xe2 [52].
The increase of De from He2 to Xe2 is a result of the increasing number of electrons and the
8
2.4. The electronically excited states of rare gas dimers
resulting increasing magnitude of the polarizabilities of the atoms and of long-range van
der Waals interactions. [31, 54]. The equilibrium internuclear distances Re increase from
Re = 2.968 Å for He2 [42] to Re = 4.378 Å for Xe2 [52] and primarily reflect the increasing
size of the atoms involved. The reduced masses µ of the homonuclear rare gas dimers
increase
qwith the mass numbers of the atoms. Because the harmonic vibrational frequency
1
ν = 2π µk is inversely proportional to the square root of the reduced mass and De rapidly
increases from He2 to Xe2 the number of bound vibrational levels of the electronic ground
state also rapidly increases from He2 (1 vibrational level) to Xe2 (25 vibrational levels).
2.3.2
Heteronuclear rare gas dimers
The ground neutral states of the heteronuclear rare gas dimers have been investigated by
ab initio quantum chemistry [38, 55, 39, 56, 57, 41, 58], differential elastic scattering crosssection measurements [59, 60] and Fourier-transform microwave spectroscopy [61, 62, 63].
The modelling and determination of their potential energy functions has been discussed
extensively in the literature [64, 65, 66, 67, 68, 69, 70, 40].
The most recent values for the dissociation energies D0 and De , and the equilibrium
internuclear distances Re of the X 0+ neutral ground states of all heteronuclear rare gas
dimers and the first adiabatic ionization energies Ei (X 1/2) of these dimers are summarized
in Table 2.3. Other theoretical studies [64, 65, 66, 38] led to similar values for De and Re ,
but to slightly different potential energy curves.
The equilibrium internuclear distances Re of the X 0+ ground states of the heteronuclear
rare gas dimers HeRg (Rg=Ne, Ar, Kr, Xe) increase from Ne to Xe, whereas the well depths
become deeper from HeNe to HeKr. Surprisingly, the dissociation energy De of HeXe is
slightly smaller than that of HeKr.
The other heteronuclear rare gas dimers can be divided into two groups. The NeRg
(Rg=Ar, Kr, Xe) dimers show the same overall trends as the HeRg dimers: The values of
the equilibrium internuclear distances Re increase from Ar to Xe whereas the dissociation
energies De are all about the same. The other heteronuclear rare gas dimers ArKr, ArXe
and KrXe are characterized by a rapid increase of the dissociation energy with increasing
size of both rare gas partners, whereas the equilibrium internuclear distance changes less
percentage-wise from one system to the next.
2.4
The electronically excited states of rare gas dimers
In contrast to the ground state, electronically excited states can be chemically bound.
However, several electronically excited states are only weakly bound or repulsive with
a shallow long-range potential well. The electronic structure of excited states is very
complex as a result of configuration and spin-orbit interactions [15]. The variety of potential
9
Chapter 2. Background information
Table 2.3: Dissociation energies D0 and De , and equilibrium internuclear distances Re of the X 0+
neutral ground states of HeNe, HeAr, HeKr, HeXe, NeAr, NeKr, NeXe, ArKr, ArXe and KrXe and the
first adiabatic ionization energies Ei (X 1/2) of these dimers.
A
RgA Rg0
D0 /cm−1
De /cm−1
Re /Å
Ei (X 1/2)/(hc cm−1 )
HeNe
HeAr
HeKr
HeXe
NeAr
NeKr
NeXe
ArKr
ArXe
KrXe
0.5665 a
1.5433 a
7.85 c
14.6207 a
20.6842 a
20.77 c
20.41 e
45.1957 a
47.84 c
49.38 e
116.54 e
130.37 e
162.41 e
3.028 a
3.492 a
3.7042 c
3.974 e
3.493 a
3.6487 c
3.889 e
3.889 e
4.091 e
4.196 e
168328 b
126911 b
113400 d
0
7.1289 a
37.89 c
102.4
117.3
151.3
g
g
g
126508(32) f
112514(24) f
97545(32) f
108689.9(50) h
96515.6(13) i
94821.6(10) i
a
From López Cacheiro et al. [58].
From Lias et al. [71].
c
From Haley et al. [41].
d
From Ma et al. [38].
e
From Tang and Toennies [40].
f
From Pratt and Dehmer [72].
g
Calculated by numerically solving the Schrödinger equation with the potential energy function from
Tang and Toennies [40].
h
From Kleimenov et al. [73].
i
This work.
b
energy functions for different electronically excited states is large and includes strongly
bound curves, weakly bound curves, double-well potentials, potentials with humps, purely
repulsive and highly perturbed curves [14, 15, 16]. This diversity often makes an assignment
of experimental spectra very difficult. All vibrational levels located energetically close
to and above a crossing of a bound with a repulsive curve are unstable with respect to
dissociation and are predissociative.
The determination of interaction potentials of rare gas dimers in electronically excited
states is important for the elucidation of the processes involved in collisions of ground state
and excited state rare gas atoms. It is also a prerequisite for the understanding of the complex chemistry and physics taking place in molecular excimer and atomic rare gas lasers
and in plasmas, and allows the elucidation of the mechanisms of formation and relaxation
of the active species in the laser media or in plasmas. Of particular importance, e.g. in
the Xe atomic laser, are the excited states of XeAr, since the highest laser gain is observed
for Xe/Ar mixtures [74]. Exact calculations of the excited electronic state potential energy
functions of XeRg (Rg=Ne, Ar, Kr) are difficult, and consequently spectroscopic experiments are essential: They allow accurate determinations of potential energy functions
which may be used directly, or as guides in the development of semiempirical models of
10
2.4. The electronically excited states of rare gas dimers
the electronic structure of excited XeRg systems, or as references to evaluate the accuracy
of ab initio quantum chemical calculations.
The knowledge of the electronic states of the cation can help to generate potential
energy functions of the electronically excited states because the simplest description of
electronically excited states of the rare gas dimers is in terms of a dimer ion core and a
nonbonding, weakly interacting electron [75]. The theory most adequate to treat molecular
Rydberg states, multichannel quantum defect theory [76], directly relies on the potential
curves of the ionic species as essential input parameters. Unfortunately no systematic
investigations of the electronically excited states of the rare gas dimers have been conducted
so far with the exception of the triplet states of He2 [77, 78, 43].
2.4.1
Homonuclear rare gas dimers
The first experimental results on the electronically excited states of the homonuclear rare
gas dimers were already reported in 1913 when Goldstein [79] and Curtis [80] discovered the
Rydberg spectrum of He2 . Since then, the electronic structure, the decay kinetics and the
potential energy functions of the excited states of He2 , Ne2 , Ar2 , Kr2 and Xe2 have been
investigated experimentally and theoretically. Experimental techniques include, among
others, absorption [81, 82, 83, 49, 84, 85], emission [86, 87, 88, 89, 90, 91, 92, 46, 93, 84, 85]
and photoionization [94, 95, 96, 97] spectroscopy. From the many ab initio quantum
chemical studies describing the potential energy functions of the electronically excited
homonuclear rare gas dimers those of Spiegelmann and co-workers are the most extensive
[98, 99, 100, 101, 102, 103, 104, 105]. The main contributions to the investigation of the
homonuclear rare gas dimer Rydberg states were summarized in a recent review article by
Ginter and Eden [106].
2.4.2
Heteronuclear rare gas dimers
Most of the experimental studies reported on the excited states of the heteronuclear rare
gas dimers were related to the systems containing xenon [107, 108, 109, 110, 111, 112,
113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125] whereas the other systems
[83, 49, 107, 110, 96, 112, 126, 127, 128] have been less investigated. The potential energy
functions of several electronically excited states have been derived semiempirically [129,
130, 131, 132, 133, 134, 135, 16] and by ab initio quantum chemistry [136, 137, 138, 139,
140].
High quality potential energy functions for the electronically excited states of heteronuclear rare gas dimers are still rare. The main reason probably lies in the disparate nature
of experimental findings and their interpretation, e.g. the C 1 Rydberg state of KrXe (see
Section 7.3.2) or the Rydberg states of Ne2 (see Chapter 5), which illustrate how difficult
it is to observe spectra and assign the measured spectral features to vibrational progres11
Chapter 2. Background information
sions. Another reason lies in the nontrivial theoretical treatment of dense manifolds of
electronically excited states and the need to include relativistic effects in the calculations.
The simplest description of the electronically excited states of a heteronuclear rare gas
dimer is a dimer ion core interacting weakly with a Rydberg electron. However, as the
mass of the Rg atom decreases, the RgXe∗ potential energy curves become anomalously
shallow and do not resemble that of their associated RgXe+ ion cores anymore. An extreme case is found for NeXe where the potential energy curves of Rydberg states that
dissociate to Ne(1 S0 ) + Xe∗ (6p) are repulsive except for shallow minima and humps which
lie at energies higher than their asymptotic dissociation limits [120]. The binding energies
of Rydberg states of KrXe [116] and ArXe [119] associated with the Rg(1 S0 ) + Xe∗ (6p)
(Rg=Ar, Kr) dissociation limits range from 350 to 1300 cm−1 and 130 to 500 cm−1 , respectively. Several ArXe Rydberg states also exhibit potential humps the origins of which
are not readily explained by avoided crossings [119]. NeXe potential minima on the other
hand lie around 15 to 130 cm−1 above their respective asymptotic limits, and are separated
by low-lying potential barriers with maxima located typically between 15 and 200 cm −1
above the dissociation limits [120]. In order to treat cases where the simple description
mentioned above fails, Lipson and Field [16] have developed a global model that provides
a satisfactory interpretation of humps and dips in the potential energy functions. Their
model is also applicable to homonuclear rare gas dimers and represents the starting point
of the analysis of the singlet gerade Rydberg states of Ne2 presented in Chapter 5.
2.5
The electronic states of rare gas dimers cations
The rare gas dimer ions represent model systems to study ion-atom collisions. They are
a source of absorption losses in excimer and ion lasers [141, 4, 5] and are encountered in
high-pressure lamps and plasmas [1]. The potential energy curves of rare gas dimer ions
are the key to understanding their decay and fragmentation dynamics [142, 143, 144, 145,
146, 147, 148], and are required to model highly electronically excited states of the neutral
dimers by multichannel quantum defect theory. Detailed knowledge of the potential energy
functions of these ions is also important for a fundamental understanding of the chemical
binding and the fragmentation dynamics of larger rare gas cluster ions [6, 149, 150, 151,
152, 153, 154, 155, 7, 156, 157], and is a prerequisite for the calculation of charge-exchange
[158] and differential scattering cross sections [159] that are necessary to model the effects
of rare gas plasmas associated with electric space propulsion thrusters.
2.5.1
Homonuclear rare gas dimers
Whereas early investigations were primarily concerned with the qualitative features of the
electronic structure of the rare gas dimer ions [15], recent experimental and theoretical
12
2.5. The electronic states of rare gas dimers cations
studies have provided quantitative information on the potential energy curves of several
+
+
+
electronic states of Ne+
2 [160, 45], Ar2 [161, 45, 162], Kr2 [45, 163] and Xe2 [164, 53].
Experimental approaches to the determination of the potential energy curves of the rare
gas dimer ions include differential elastic-scattering measurements [165], photoionization
spectroscopy [166, 167, 97], spectroscopic studies of the microwave electronic spectrum
close to the dissociation threshold [168], the measurement of kinetic energy release following
dissociation [142, 150, 145, 147, 148] and photoelectron spectroscopy [169, 170, 171, 172,
173, 174, 175, 51, 48, 53, 163].
Table 2.4 lists the most recent values for the dissociation energies D0 and De and the
equilibrium internuclear distances Re of the ground electronic state of the homonuclear
rare gas dimer cations. The missing value for the dissociation energy D0 of the I(1/2u)
state of Ne+
2 originates from the fact that the origin of this state has not been observed
experimentally so far. Hall et al. [174] reported two different assignments of the I(1/2u)
−1
state of Ne+
2 of their photoelectron spectrum resulting to values of D0 = 9324(81) cm
and D0 = 10413(81) cm−1 , respectively. Investigations on the potential energy functions
of Ne+
2 by microwave electronic spectroscopy close to the dissociation threshold [168, 160]
and by ab initio quantum chemistry [45] suggest D0 of the I(1/2u) state of Ne+
2 to be
∼ 10900 cm−1 and ∼ 11500 cm−1 , respectively.
Table 2.4: Dissociation energies D0 and De and equilibrium internuclear distances Re of the I(1/2u)
+
+
+
+
ionic ground states of He+
2 , Ne2 , Ar2 , Kr2 and Xe2 .
Rg+
2
He+
2
Ne+
2
Ar+
2
Kr+
2
Xe+
2
D0 /cm−1
19105
a
10600.4(12) d
9267.8(28) f
7872.4(11) h
De /cm−1
19945.82 a
11907 b
10759 e
9360.1 g
7937.4 i
Re /Å
1.0806(5)
1.765 c
2.392(41)
2.697 g
3.118 i
a
d
a
From Raunhardt et al. [43].
From Ha et al. [45].
c
From Carrington et al. [160].
d
From Rupper and Merkt [48].
e
From Wüest and Merkt [162].
f
From Signorell et al. [51].
g
From Wüest and Merkt [163].
h
From Rupper et al. [53].
i
This work.
b
The homonuclear rare gas dimer ions Rg+
2 (Rg = Ne, Ar, Kr, Xe and Rn) possess
six low-lying, closely spaced electronic states. At short internuclear distances these states
2
2
2 +
can be labeled using Mulliken’s notation [15] as A 2 Σ+
1/2u , B ΠΩg , C ΠΩu and D Σ1/2g
(Ω = 1/2, 3/2) and correspond to the electronic configurations (σg )2 (πu )4 (πg∗ )4 (σu∗ )1 ,
13
Chapter 2. Background information
(σg )2 (πu )4 (πg∗ )3 (σu∗ )2 , (σg )2 (πu )3 (πg∗ )4 (σu∗ )2 and (σg )1 (πu )4 (πg∗ )4 (σu∗ )2 which are based on lin+
ear combinations of the atomic valence np orbitals (n = 2 − 6 for Ne+
2 - Rn2 ).
At large internuclear distances the spin-orbit interaction mixes Σ and Π electronic
states of the same value of Ω and same g/u symmetry. The appropriate labeling then is
I(1/2u), I(3/2g), I(3/2u), I(1/2g), II(1/2u) and II(1/2g), where the value of Ω and the
g/u symmetry are given in parentheses and the labels I and II designate states correlating
with the first Rg(1 S0 ) + Rg+ (2 P3/2 ) and second Rg(1 S0 ) + Rg+ (2 P1/2 ) dissociation limits,
respectively (see Fig. 2.1).
+
2
2
4
4
1
A S1/2u (sg) (pu) (pg) (su)
D
2
4
3
2
B PWg (sg) (pu) (pg) (su)
Pg
5p AO
2
5p AO
Pu
2
2
3
4
2
C PWu (sg) (pu) (pg) (su)
Potential energy
su
2
+
1
4
4
II (1/2u)
+ 2
B
1
Xe ( P1/2)+Xe( S0)
C
A
sg
II (1/2g)
I (1/2g)
I (3/2u) Xe+(2P3/2)+Xe(1S0)
I (3/2g)
I (1/2u)
Hunds case Hunds case (c)
(a) + (b)
2
D S1/2g (sg) (pu) (pg) (su)
Internuclear distance
Figure 2.1: Left part: Molecular orbital scheme of the homonuclear rare gas dimer cations (shown for
Xe2 ) resulting from the linear combination of the np atomic valence orbitals (AO, n = 5 for Xe). Removing
an electron from one of these six molecular orbitals (MO) (in the ground neutral state every MO is fully
occupied by two electrons) leads to the first six closely spaced electronic states of the homonuclear rare
gas dimer cations, of which the potential energy curves (for Xe+
2 ) are displayed on the right part.
2.5.2
Heteronuclear rare gas dimers
The first observation of a pair of emission band systems from binary rare gas mixtures
was made by Druyvesteyn [176] who excited helium and neon, each at a pressure of 6 torr,
with a DC discharge. Druyvesteyn’s observations were later confirmed by Oskam and
Jongerius [177] and extended to Ar/Xe, Ar/Kr and Xe/Kr mixtures using DC discharge
[178] and electron beam excitation [179, 180, 181]. Tanaka et al. [182] excited all ten binary
mixtures of He, Ne, Ar, Kr and Xe by various types of electric discharges and investigated
the emission spectra. They were the first to attribute these emission bands to charge
exchange transitions of the heteronuclear rare gas ions YZ+ (Y + Z+ ) → (Y+ + Z), where
Z represents the lighter of the two rare gases (see below).
The cationic states of the heteronuclear rare gas dimers HeNe, HeAr, HeKr and HeXe
and their potential energy curves have been studied experimentally by emission spectroscopy [182, 183, 184], infrared predissociation spectroscopy [185] and microwave spec-
troscopy [186, 187] and theoretically by ab initio quantum chemistry [188, 189, 190, 191,
14
2.5. The electronic states of rare gas dimers cations
+
2
2
4
4
1
X S1/2 (s) (p) (p) (s)
s
2
4
3
P
P
5p AO
2
2
3
C1 3/2
2
4
2
B P (s) (p) (p) (s)
Potential energy
2
A P (s) (p) (p) (s)
3p AO
C2 1/2
C
s
2
+
1
4
4
B 1/2
A2 1/2
A
1
+ 2
Xe( S0)+Ar ( P3/2)
+ 2
1
Xe ( P1/2)+Ar( S0)
A1 3/2
X 1/2
Hunds case
(a) + (b)
2
+ 2
B
X
C S1/2 (s) (p) (p) (s)
1
Xe( S0)+Ar ( P1/2)
+ 2
1
Xe ( P3/2)+Ar( S0)
Hunds case (c)
Internuclear distance
Figure 2.2: Left part: Molecular orbital scheme of the heteronuclear rare gas dimer cations (shown for
ArXe) resulting from the linear combination of the np atomic valence orbitals (AO, n = 3 for Ar, n = 5
for Xe). Removing an electron from one of these six molecular orbitals (MO) (in the ground neutral state
every MO is fully occupied by two electrons) leads to the first six electronic states of the heteronuclear
rare gas dimer cations, of which the potential energy curves (for ArXe) are displayed on the right part.
192, 193, 194].
To our knowledge no ab initio potential energy curves of the six low-lying electronic
states of the heteronuclear rare gas dimers consisting of Ne, Ar, Kr and Xe have been
reported yet, with the exception of ArKr+ [195]. Hausamann and Morgner [196] have
developed a simple semiempirical model for the calculation of the potential energy curves
of the heteronuclear rare gas dimer cations. The model relied on the then available experimental and theoretical data on the homo- and heteronuclear rare gas dimer cations
and incorporates the Pauli repulsion as well as the charge-exchange and spin-orbit interactions. Experimental approaches to the investigation of the electronic states of the heteronuclear rare gas dimer cations include emission [182, 197, 198, 199, 200], photoionization
[201, 202, 72], photoelectron [203, 111, 204], threshold photoelectron-photoion coincidence
(TPEPICO) [205, 206, 207] and pulsed-field-ionization zero-kinetic-energy (PFI-ZEKE)
photoelectron [208, 73, 209] spectroscopy.
Table 2.5 summarizes the most recent values for the dissociation energies D0 and De and
the equilibrium internuclear distances Re of the ground electronic state of the heteronuclear
rare gas dimer cations. Values for all these systems have been reported by Hausamman
and Morgner [196], but were not considered in the table as a consequence of the large
deviation of their values from ours for ArXe+ (see Table 7.11) and KrXe+ (see Table 7.12).
The heteronuclear rare gas dimer ions YZ+ (Y, Z = Ne, Ar, Kr, Xe and Rn) possess
six low-lying electronic states, of which three are associated with the Y+ (2 PJ ) + Z(1 S0 )
(J = 1/2, 3/2) and three with the Y(1 S0 ) + Z+ (2 PJ ) (J = 1/2, 3/2) dissociation limits
(see Fig. 2.2). Y represents the heavier of the two rare gases. Neglecting spin-orbit inter2
2
2 +
action, the four possible ionic states can be labeled as X 2 Σ+
1/2 , A Π, B Π and C Σ1/2
15
Chapter 2. Background information
Table 2.5: Dissociation energies D0 and De , and equilibrium internuclear distances Re of the X 1/2 ionic
ground states of HeNe+ , HeAr+ , HeKr+ , HeXe+ , NeAr+ , NeKr+ , NeXe+ , ArKr+ , ArXe+ and KrXe+ .
RgRg0+
D0 /cm−1
De /cm−1
Re /Å
HeNe+
HeAr+
HeKr+
HeXe+
NeAr+
NeKr+
NeXe+
ArKr+
ArXe+
KrXe+
4698 a
207 b
5651 a
281.6 c
208 d
1.3931 a
2.573 c
2.87 d
637(32) e
444(24) e
331(32) e
4326.9(50) f
1435.5(14) g
3163.5(10) g
4432.90(50)
1480.3 g
3220.8 g
f
2.54 f
3.154 g
3.177 g
a
From Falcetta et al.[194].
From Dabrowski and Herzberg [184].
c
From Carrington et al. [186].
d
From Carrington et al. [187].
e
From Pratt and Dehmer [72].
f
From Kleimenov et al. [73].
g
This work.
b
and correspond to the electronic configurations (σ)2 (π)4 (π ∗ )4 (σ ∗ )1 , (σ)2 (π)4 (π ∗ )3 (σ ∗ )2 ,
(σ)2 (π)3 (π ∗ )4 (σ ∗ )2 and (σ)1 (π)4 (π ∗ )4 (σ ∗ )2 which are based on linear combinations of the
atomic valence np orbitals (n = 2 − 6 for Ne - Rn). Because of the absence of the g/u
symmetry in the heteronuclear rare gas dimers the two Σ (X and C) and Π (A and B)
states are coupled to each other by charge-exchange interaction. The spin-orbit interaction further mixes Σ and Π electronic states of the same value of Ω, and the appropriate
labeling of the six low-lying electronic states of the heteronuclear rare gas dimers is X 1/2,
A1 3/2, A2 1/2, B 1/2, C1 3/2 and C2 1/2.
2.6
Potential model
The potential model to describe the electronic states of rare gas dimer cations used in this
thesis has been first applied in the analysis of the six low-lying electronic states of Ar+
2 by
Wüest and Merkt [162]. The potential energy function of an ionic Σ or Π state can be
expressed as a sum of three terms,
VΛ (R) = VΛshort (R) + VΛattr (R) + Vdiss ,
(2.6)
where VΛshort (R) and VΛattr (R) (Λ = Σ, Π) describe the short- and long-range parts of the
potentials, respectively. Vdiss is a constant used to relate the potential energies to the
energy of the ground neutral state of the rare gas dimer.
16
2.6. Potential model
The short-range part of the potential energy curves are expressed as
VΛshort (R) = AΛ e−bΛ R − BΛ e−bΛ R/βΛ .
(2.7)
The first term in Eq. (2.7) describes the Born-Mayer repulsion whereas the second term
includes the effects of charge-transfer and chemical bonding. Weakly bound states can
therefore be described by setting BΛ = 0. The form of VΛshort (R) represents a generalization of the Morse potential [210] and was first reported by Siska [211] with β = 2. The
parameters AΛ and BΛ can be related to the equilibrium internuclear distance Re,Λ and
the dissociation energy De,Λ using Eqs. (2.8) and (2.9),
dVΛ (R) = 0,
(2.8)
dR R=Re,Λ
VΛ (Re,Λ ) = Vdiss − De,Λ ,
(2.9)
as derived in the appendix of Ref. [162]. Four independent fit parameters (Re , De , β, b)
result for each potential energy curve. Weakly bound curves (BΛ = 0) can be described by
only two fit parameters (Re , b), and the parameter AΛ in Eq. (2.7) can be derived using
ebΛ Re,Λ dVΛattr (R) .
(2.10)
AΛ =
b
dR
R=Re,Λ
The long-range part of the Σ and Π potential energy curves is approximated by the
first two members of the long-range series with n = 2 and 3
VΛattr (R)
=−
∞
X
f2n (R, bΛ )
n=2
C2n,Λ
,
R2n
(2.11)
where C4,Σ = C4,Π but C6,Σ 6= C6,Π as a result of isotropic and anisotropic contributions
to VΛattr (R) (see Refs. [186, 187, 160] and Section 2.6.1). In Eq. (2.11) the Tang-Toennies
damping functions
f2n (R, bΛ ) = 1 − e
−bΛ R
2n
X
(bΛ R)k
k=0
k!
(2.12)
ensure that the contributions from the long-range interaction become insignificant at short
internuclear distances [212, 162]. Describing the long-range part of the potential up to 2n =
6 is equivalent to retaining only the charge ↔ induced dipole, charge ↔ induced quadrupole,
quadrupole ↔ induced dipole, and induced dipole ↔ induced dipole interactions also called
dispersion interactions. The formalism for the estimation of the values of the C4 and C6
long-range coefficients is discussed in Section 2.6.1.
The potential model presented above can, with some modifications, also be used for
the description of the potential energy functions of the ground neutral state [44, 52] and
electronically excited states [213] of rare gas dimers. The excited states sometimes require
17
Chapter 2. Background information
a more complex model (see Chapter 5). For the ground neutral state and weakly bound
electronically excited states, the second term in the short-range part describing the chemical
bond (see Eq. (2.7)) can be neglected which then leads to the model potential for van der
Waals molecules proposed by Tang and Toennies [212]. The members of the long-range
expansion described in Eq. (2.11) are different for neutral, electronically excited and ionic
states as a result of different interactions. The long-range part of the interaction of two
atoms in their ground state configuration (1 S0 for rare gases) contains contributions from
n = 3 (dipole-dipole dispersion) up to n = 8 (higher dispersion orders) [31, 212, 40]. The
interactions at large internuclear distances of an atom in its ground state with an atom
in an electronically excited state are described by the dipole-dipole resonance coefficient
C3 (proportional to the dipole matrix element connecting the excited atomic state to the
ground state) and the dipole-dipole dispersion coefficient C6 (proportional to α1 · α2 , where
α1 and α2 represent the polarizabilities of the two interacting atoms) [214, 31, 18].
2.6.1
Long-range behavior
The long-range coefficients for the neutral ground state of all homo- and heteronuclear rare
gas dimers are summarized in Ref. [40]. For the electronically excited states much less data
are available; in fact, only values for the Rydberg states of Ne2 formed from Ne(3s;1,3 P)
and Ne(1 S0 ) have been reported [18]. However, a procedure to calculate the values of C3
and C6 is also presented in Ref.[18].
Because the main focus in this thesis lies in the determination of the potential energy
functions of rare gas dimer cations, the formalism to derive the long-range coefficients for
cations is discussed here in more detail. The long-range behavior of a dimer cation can be
described in terms of the interaction between a neutral atom and a singly-charged ion. In
the following, the procedure for the determination of the long-range coefficients C4 and C6
for ArXe+ is outlined. The formalism can be applied to all other homo- and heteronuclear
rare gas dimer cations.
The four uncoupled potential energy curves of ArXe+ are correlated with the
Xe+ (2 P) + Ar(1 S0 ) (Σ1 , Π1 ) and Xe(1 S0 ) + Ar+ (2 P) (Σ2 , Π2 ) dissociation limits. Consequently, the long-range coefficients C4 and C6 are different for the two pairs of curves.
The long-range behavior of the Σ1 and Π1 potential energy functions is described in terms
of the interaction between a neutral Ar atom and a Xe+ ion, whereas for the Σ2 and Π2
potentials the interaction corresponds to that of an Ar+ ion and a neutral Xe atom. The
expressions listed below for the C4 and C6 coefficients are suitable for the curves correlating
with the Xe+ (2 P) + Ar(1 S0 ) dissociation limit. Exchanging Ar with Xe and Xe+ with Ar+
in Eqs. (2.13), (2.16), (2.17), (2.18), (2.19) leads to the corresponding expressions for the
curves correlating with the Xe(1 S0 ) + Ar+ (2 P) dissociation limit.
The charge ↔ induced dipole interaction is purely isotropic [186, 187, 160, 162] and can
18
2.6. Potential model
be determined to be
1
C4,Λ = C4,Σ = C4,Π = C4 = αdAr = 5.537 Eh a40
2
(2.13)
using the literature value for the static polarizability of the neutral Ar atom αdAr =
11.074 a0 3 [21]. Anisotropic contributions have to be considered in the interaction terms
proportional to R−6 [186, 187, 160, 162]: The charge ↔ induced quadrupole interaction is
purely isotropic, the quadrupole ↔ induced dipole interaction purely anisotropic and the
dispersion interaction consists of isotropic and anisotropic components. The long-range
coefficients C6 have the Λ -dependence summarized by equations (2.14) and (2.15):
2
C6,Σ = C6,0 + C6,2 ,
5
(2.14)
1
C6,Π = C6,0 − C6,2 ,
(2.15)
5
where C6,0 contains the isotropic and C6,2 the anisotropic contributions. Because in the
case of ArXe (and also of Xe2 and KrXe) no ab initio values were available in the literature
to our knowledge, the C6,Λ values had to be estimated. The charge ↔ induced quadrupole
interaction contributes purely isotropically to C6 as
1
C6,0 = αqAr = 25.105 Eh a60 ,
2
(2.16)
and was calculated using the literature value for the quadrupole polarizability of the neutral
Ar atom αqAr = 50.21 a0 5 [215]. The interaction of the charge-induced dipole of the neutral
Ar atom with the quadrupole of the Xe+ ion is purely anisotropic and can be written as
C6,2 =
15 Ar Xe+
α Θ
= 166.11 Eh a60 .
2 d
(2.17)
+
The value for the quadrupole of Xe+ , Θ Xe = 2.0 ea0 2 , was extrapolated using literature
values for the quadrupoles of Ne+ , Ar+ and Kr+ [216]. The third contribution to the
coefficient C6 originating from the dispersion interaction consists of an isotropic [54]
+
C6,0 =
ArAr
Xe
· C6,0
2 · αdAr · αdXe · C6,0
Xe
(αdAr )2 · C6,0
+
Xe+
+
+
Xe+
ArAr
+ (αdXe )2 · C6,0
(2.18)
and an anisotropic [216]
+
C6,2
∆αdXe
=
+
C6,0
3ᾱdXe
(2.19)
+
part. The values for the static polarizability αdXe = 19.1 a0 3 and the polarizability
+
anisotropy ∆αdXe = −1.8 a0 3 of Xe+ were deduced by comparing the literature values
of these quantities for the Ne, Ar, and Kr neutral atoms [21] with the values of their
cations [216] using αdXe = 27.2903 a0 3 [21] and approximating the mean polarizability
19
Chapter 2. Background information
+
+
+
+
Xe Xe
ᾱdXe by αdXe = 19.1 a0 3 . The value of the dispersion coefficient C6,0
is calculated
using the Slater-Kirkwood formula [216, 217] to be 165.08 Eh a0 6 . Using the value from
ArAr
Kumar and Meath [218] of C6,0
= 64.30 Eh a0 6 and the values derived above results
in C6,0 = 102.7485 Eha0 6 and C6,2 = −3.2277 Eh a0 6 . Summing all contributions yields
C6,0 = 127.8535 Eh a0 6 and C6,2 = 162.8823 Eh a0 6 and, thus, C6,Σ = 193.0065 Eh a0 6 and
C6,Π = 95.2771 Eh a0 6 .
Table 7.7 summarizes all contributions to the long-range coefficients C4 and C6 for
the two cases in which the long-range behavior of the ArXe+ cation is described by the
interaction of an Ar neutral atom with a Xe+ ion and of an Ar+ ion with a Xe neutral atom. The values for the latter case are obtained by applying equations (2.13),
(2.16), (2.17), (2.18) and (2.19) and using literature values of αdXe = 27.2903 a0 3 [21],
+
+
XeXe
= 285.9 Eh a0 6
αqXe = 212.6 a0 5 [215], ΘAr = 1.122 a0 2 [216], αdAr = 6.71 a0 3 [216], C6,0
+
Ar
[218], C6,0
Ar+
+
+
= 30.37 Eh a0 6 [216], ∆αdAr = −0.21 a0 3 [216], ᾱdAr = 6.85 a0 3 [216].
+
With the same formalism the C4 and C6 coefficients for Xe+
2 and KrXe summarized
in Tables 6.3 and 7.7 were determined using the values of the static polarizabilities αd ,
Rg−Rg
quadrupole polarizabilities αq , quadrupoles Θ, disperion coefficients C6,0
, polarizability
+
+
anisotropies ∆αd and mean polarizabilities ᾱd of Kr, Kr , Xe and Xe in Table B.2 of
Appendix B.
2.7
Vibrational assignments
A correct assignment of the vibrational quantum numbers of initial and final states observed
in a spectrum is essential for the derivation of dissociation energies and bond lengths.
In our spectroscopic investigations the vibrational level of the initial state is either the
ground (v 00 = 0) level in the case of the ground electronic state of the neutral dimers,
or a well-known vibrational level of an electronically excited state used as intermediate
state in a multiphoton excitation process. The assignment of the vibrational quantum
number of the final state cannot be derived from the measurement of a single vibrational
progression, no matter how accurately the origins of the successive vibrational bands are
determined. The only way to unambiguously derive the vibrational quantum numbers is
by measuring and analyzing isotopic shifts in the spectra of at least two isotopomers of the
same diatomic molecule. This procedure, which has been extensively used in the present
work, is described in this section and is a standard method described in many textbooks
(see e.g. [36, 219, 220]).
Fig. 2.3 schematically shows the Born-Oppenheimer potential energy functions of two
electronic states (indicated with 00 for the lower state (usually the ground electronic state)
and
+
for the upper state (electronically excited or ionic state)). Neglecting the rotational
20
2.7. Vibrational assignments
v
+
+
D0
Gv+
+
+
De
Gv+=0
v =0
Wave number
~
νatom
Te
~
ν00
~
νv+v’’
v’’
D0’’
Gv’’
v’’=0
De’’
Gv’’=0
Internuclear distance
Figure 2.3:
Schematic view of Born-Oppenheimer potential functions of two electronic states of a
diatomic molecule.
(j)
structure, the transition wave number ν̃v+ v00 of isotopomer j can be deduced from Fig. 2.3:
(j)
(j)
(j)
ν̃v+ v00 = Te + Gv+ − Gv00 ,
(j)
(j)
(2.20)
(j)
Te = ν̃00 + Gv00 =0 − Gv+ =0 .
(2.21)
Te , De00 and De+ are independent of the isotopomer in the Born-Oppenheimer approximation,
but cannot be measured directly. Inserting Eq. (2.21) in Eq. (2.20) leads to
(j)
(j)
(j)
(j)
(j)
(j)
ν̃v+ v00 = ν̃00 + Gv00 =0 − Gv+ =0 + Gv+ − Gv00
i
i h
h
(j)
(j)
(j)
(j)
(j)
= ν̃00 + Gv+ − Gv+ =0 − Gv00 − Gv00 =0 ,
(2.22)
(2.23)
a result that could also directly be obtained from Fig. 2.3. Retaining only the harmonic
(j)
and first two anharmonic corrections in the expression of the vibrational energies Gv+ (see
(j)
Ref. [36] for more details) the transition wave number ν̃v+ v00 =0 from the v 00 = 0 vibrational
level of the ground state to the vibrational level v + of an ionic state can be expressed as
(j)
(j)
(v + + 1/2)2 + ωe ye+(j) (v + + 1/2)3
ν̃v+ v00 =0 = Ei /hc + ωe+(j) (v + + 1/2) − ωe x+(j)
e
+(j)
− ωe+(j) /2 − ωe x+(j)
/4
+
ω
y
/8
(2.24)
e
e
e
(j)
where Ei
(j)
ν̃00
corresponds to the adiabatic ionization energy of isotopomer j, or to the origin
of the band system in the case of a bound-to-bound electronic transition. The harmonic
21
Chapter 2. Background information
+(j)
+(j)
+(j)
(ωe ) and anharmonic (ωe xe , ωe ye ) constants and the adiabatic ionization energy
(j)
Ei can be derived from the experimental data in a least-squares fit of Eq. (2.24) to the
+(j)
observed vibrational transitions. In addition, values for the dissociation energy D0
be calculated with the relation
+(j)
D0
(j)
(j)
00(j)
= Ei,atom /hc − Ei /hc + D0 .
can
(2.25)
(k)
Because Te is independent of the isotopomer the transition wave number ν̃v+ v00 of another
isotopomer k can be expressed as
(k)
(k)
(k)
(j)
(j)
(k)
(j)
(k)
ν̃v+ v00 = Te + Gv+ − Gv00 = ν̃00 + Gv00 =0 − Gv+ =0 + Gv+ − Gv00 ,
|
{z
}
(2.26)
Te
(k)
and the transition wave number ν̃v+ v00 can be written as
(k)
(j)
ν̃v+ v00 = Ei /hc +
|
00(j)
ωe
2
+(j)
−
ωe x00(j)
ωe y00(j)
ωe
e
e
+
−
4
8{z
2
Te
+
v + 1/2 − ωe xe
+(k)
+
v + 1/2
2
ωe xe +(j) ωe ye +(j)
(2.27)
−
4
8 }
3
+ ωe ye +(k) v + + 1/2
h
i
2
3
− ωe00(k) (v 00 + 1/2) − ωe x00(k)
(v 00 + 1/2) + ωe y00(k)
(v 00 + 1/2) .
e
e
+ωe+(k)
+
In order to derive the absolute assignment of the vibrational quantum numbers v 00 and v +
one chooses a certain assignment and determines the constants of a reference isotopomer j
in a least-squares fitting procedure based on Eq. (2.24) (if v 00 = 0 is known) or Eq. (2.27)
(for k = j if v 00 is not known). The spectroscopic constants of another isotopomer k are
determined from those of the reference isotopomer j using the relations [36]
r
µj
ρkj =
µk
(2.28)
and
ωe(k) = ρkj · ωe(j)
ωe xe(k) = ρ2kj · ωe x(j)
e
ωe y(k)
= ρ3kj · ωe y(j)
e
e ,
(2.29)
(2.30)
(2.31)
and can then be used to calculate the transition wave numbers of isotopomer k with
Eq. (2.27). This procedure is repeated with different assignments of v 00 and v + until the
differences between the observed and calculated transitions of both isotopomers j and k
are minimal.
Taking into account the rotational structure, the total energy of a rovibronic state of a
22
2.7. Vibrational assignments
dimer is expressed as
T = Te + G v + FJ .
(2.32)
In the case of a Σ state in Hund’s case (b) the rotational energy can be determined using
(j)
FJ
= Bv(j) J(J + 1) − Dv(j) J 2 (J + 1)2 + · · · ,
Bv(j) = Be(j) − αe(j) (v + 1/2) + · · · ,
Dv(j) = De(j) + βe(j) (v + 1/2) + · · ·
(2.33)
(2.34)
(2.35)
(v = v + , v 00 and J = J + , J 00 ) and
Be(k) = ρ2kj · Be(j) ,
αe(k) = ρ3kj · αe(j) ,
De(k) = ρ4kj · De(j) ,
βe(k) = ρ5kj · βe(j) ,
(2.36)
(2.37)
(2.38)
(2.39)
where Be and De represent the rotational and centrifugal distortion constants at the equilibrium internuclear distance Re , respectively. Expressions for FJ in other electronic states
and Hund’s cases are more complicated and can be found in Refs. [36, 219, 221].
23
Chapter 2. Background information
24
Chapter 3
Experimental setup and methods
3.1
Methods
All experimental results presented in this work have been derived by applying two common spectroscopic methods: resonance-enhanced multiphoton-ionization (REMPI) spectroscopy and pulsed-field-ionization zero-kinetic-energy (PFI-ZEKE) photoelectron spectroscopy. Both techniques are based on the detection of charged particles leading to two
advantages compared to other methods: First, charged particles can be detected with
a probability of almost 100 % and second, these spectroscopic techniques are so-called
”background-free” techniques, i.e., the signal vanishes far away from the resonances. In
the following, the principles of both methods are briefly explained.
3.1.1
Photoionization spectroscopy
Photoionization spectroscopy consists of measuring mass selectively the photoionization
yield of atomic or molecular species as a function of the excitation energy. In a typical
experiment, the ions generated by photoionization are extracted with a pulsed electric field
toward a suitable detector, typically a microchannel plate detector. Important contributions to the development of this method were made by Watanabe [222, 223], Berkowitz
[224, 225], Chupka et al. [226], Ruscic [227] and others. Because each energetically accessible ionization threshold contributes intensity to the photoionization signal, an idealized
photoionization spectrum is expected to display a stepwise increase at each threshold which
would, in principle, allow the reconstruction of the cationic level structure, e.g. by taking
the derivative of the ion current with respect to the excitation energy. In reality, the autoionization of Rydberg series converging on excited levels of the cation masks the direct
ionization signal [228, 224] which renders the identification of ionization thresholds difficult or impossible. The lowest threshold however, is usually observed and can be used to
determine the adiabatic ionization energy. Furthermore, a comparison of the PFI-ZEKE
photoelectron spectrum with the corresponding photoionization spectrum can be helpful in
the interpretation of the intensity distribution in the PFI-ZEKE photoelectron spectrum
25
Chapter 3. Experimental setup and methods
(see below).
At high photon energies appearance thresholds for fragment ions can also be determined by photoionization mass spectrometry. When combined with the adiabatic ionization threshold, these appearance thresholds can be used to derive dissociation energies of
molecular cations. Photoionization spectroscopy has established itself as a powerful method
to measure important thermochemical quantities [228, 224, 225, 226, 227, 229, 230].
3.1.2
REMPI spectroscopy
Multiphoton ionization excitation schemes for the investigation of resonant transitions to
excited states in atoms and molecules came up with the availability of intense tunable
laser sources and have been applied early by Johnson [231, 232], Petty et al. [233], Dalby et
al. [234], Klewer et al. [235] and Lubman and Zare [236] and subsequently improved by
many others. The resonance-enhanced multiphoton ionization (REMPI) technique typically involves the absorption of one or more photons to an electronically (or vibrationally)
excited intermediate state followed by the absorption of one or more photons which ionize
the atom or molecule. Resonant ionization is much more efficient than nonresonant ionization [237] so that the REMPI excitation scheme provides a background-free and very
sensitive method for the investigation of the transitions from the ground neutral state to
electronically excited states. The possibility of mass-selectively detect the ions leads to
spectra which allow the identification of the investigated molecule and/or the investigation
of its fragmentation dynamics. States which cannot be reached from the ground neutral
state by single-photon excitation as a result of too restrictive selection rules can be investigated by REMPI because the selection rules associated with multiphoton excitation
are different from the selection rules for a single-photon excitation. Moreover, ionic states
that cannot be reached from the ground state following single-photon excitation because of
unfavorable Franck-Condon factors can become accessible when excited via a suitable intermediate state. REMPI also leads to a simplification of photoionization spectra because
only the species corresponding to the selected mass and the resonant frequency contributes.
In particular, the selection of a single vibrational or rotational level of the intermediate
state alleviates the problem of spectral congestion and facilitates the interpretation of the
spectra. Multiphoton excitation also offers more flexibility than single-photon excitation
in that the polarization of photons at several frequencies can be freely chosen and the final
states of the excitation process modified (see Section 4.2 for a detailed discussion).
The line positions of a REMPI spectrum correlate directly with the molecular transition
frequencies. The intensities have to be treated carefully: If the frequency of the ionizing
photons accidentally correspond to a transition from the intermediate state to an autoionizing Rydberg state [36, 221] the intensity of the REMPI spectrum of the intermediate
state can be strongly altered.
26
3.1. Methods
In this thesis the advantages of a (1VUV + 10VIS/UV ) REMPI excitation scheme have been
exploited to record spectra of electronically excited states of Ne, Ne2 , Xe2 , ArXe and KrXe
and to access a wide range of quantum states of the corresponding cations.
3.1.3
PFI-ZEKE photoelectron spectroscopy
Conventional photoelectron spectroscopy (PES), developed in the early 1960’s by Vilesov et
al. [238] and Turner and Jobory [239], consists of photoionizing molecules by radiation of
fixed frequency and measuring the kinetic energy distribution of the photoelectrons. From
the knowledge of the photon energy, this information can be used to determine ionization
thresholds and thus the energetic positions of a cation with respect to those of a neutral
molecule. The He I (at 21.218 eV) and He II (at 40.814 eV) line sources have proven
particularly useful because they enable the investigation of all cationic states that can be
produced by removal of an electron out of orbitals of the outer and inner valence shells.
The resolution of He I photoelectron spectroscopy is limited to ∼ 5 meV (∼ 40 cm−1 )
by technical difficulties associated with the measurement of electron kinetic energies (see
however Refs. [240, 241]).
In threshold photoelectron spectroscopy (TPES), introduced by Peatman et al. [242],
only electrons of almost zero kinetic energy, so-called threshold electrons, are detected.
The measurement of electron kinetic energies is therefore avoided, and a higher resolution
of the order of 2 − 3 meV (∼ 20 cm−1 ) can be achieved. In contrast to conventional
PES which relies on line sources, tunable light sources are needed for TPES. The rapid
development of TPES into a powerful variant of photoelectron spectroscopy was made
possible by the development of broadly tunable intense synchrotron radiation sources in
the vacuum ultraviolet range [243, 174, 244, 205, 144].
A high-resolution version of TPES is the technique of pulsed-field-ionization zerokinetic-energy (PFI-ZEKE) photoelectron spectroscopy which has been introduced by
Müller-Dethlefs et al. [245] and Reiser et al. [246] and is described in several review articles
[247, 248, 249, 250, 251, 252, 253]. This method is based on the excitation of molecules to
very high Rydberg states (principal quantum number n >> 100) located just below the
successive ionization thresholds corresponding to the cationic energy states with tunable
radiation. The Rydberg states are then field ionized by applying a pulsed electric field of
∼ 1 V/cm, and the electron signal is measured as a function of the excitation energy. Like
a conventional photoelectron spectrum a PFI-ZEKE photoelectron spectrum shows a line
at each ionization threshold, but at a position that is shifted with respect to the position
of the ionic energy levels by the binding energy of the field-ionized Rydberg states. A
homogeneous electric field F ionizes all Rydberg states lying in an energy window of width
∆E immediately below the ionization thresholds. The relationship between ∆E and the
electric field [254, 255, 249] is described by Eq. (3.1) for diabatic pulsed field ionization
27
Chapter 3. Experimental setup and methods
and by Eq. (3.2) for adiabatic field ionization and also for ionization in a homogeneous DC
electric field:
r
F
∆E
=4
cm−1
(3.1)
hc
V cm−1
r
∆E
F
= 6.12
cm−1 .
(3.2)
hc
V cm−1
A more precise and quantitative model for the calculation of the field-ionization shift which
takes into account the field-ionization dynamics has been presented by Hollenstein et al.
[256, 257]. Narrower energy windows, and thus a higher selectivity in the field-ionization
process can be achieved by using sequences of several electric field pulses [258, 259, 260,
249, 256]. The resolution of a PFI-ZEKE photoelectron spectrum is limited to ∼ 0.2 cm−1
when a sequence of two successive electric field pulses is used. The first ”discrimination”
pulse sweeps the free electrons out of the excitation region and ionizes the highest Rydberg
states, and the second ”extraction” pulse ionizes the lower Rydberg states and extracts the
corresponding electrons [258, 260]. By applying optimal electric field pulse sequences and
using narrow-bandwidth laser sources this resolution can be improved to ∼ 0.05 cm−1 [256].
All the two-pulse sequences used in the work presented in this thesis and the corresponding
field-ionization shifts are summarized in Table 3.1.
Table 3.1:
Overview of the PFI-ZEKE electric field pulse sequences used in the present work. F
represents the amplitude and ∆t the duration of the applied electric field pulses, respectively, and t ap the
time delay between the photoexcitation laser pulse and the application of the first electric field pulse. The
field ionization shifts ∆ν̃fi were determined experimentally and theoretically.
”discrimination” pulse
F / V cm−1 ∆t/ µs tap / µs
0.01
0.15
0.15
0.15
0.02
0.01
0.05
0.03
0.10
0.14
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
”extraction” pulse
F / V cm−1 ∆t/ µs
5.0
5.0
5.0
2.0
5.0
5.0
5.0
5.0
5.0
5.0
-0.30
-0.25
-0.19
-0.20
-0.25
-0.25
-0.25
-0.25
-0.25
-0.25
0.775
0.775
0.775
0.775
0.775
0.775
0.775
0.775
0.775
0.775
∆ν̃fi / cm−1
1.36
1.69
1.60
1.62
1.18
1.17
1.34
1.24
1.51
1.65
a
b
b
b
b
b
b
b
b
b
Derived experimentally by measuring the 2 P3/2 ←1 S0 ionization threshold in Xe and comparing with
the literature value of 97833.790 cm−1 [29].
b
Derived theoretically by using the quantitative model from Hollenstein et al. [256, 257].
a
The line intensities in a PFI-ZEKE photoelectron spectrum can be influenced by interactions between ionization channels [248, 261] and therefore do not always faithfully
represent the direct ionization cross sections and the expected Franck-Condon intensity
28
3.2. Experimental setup
distributions.
In addition to the methods of photoionization and photoelectron spectroscopy described
above, ionization thresholds can also be determined by extrapolating Rydberg series to their
limits [262, 263] with a precision ultimately limited by the bandwidth of the radiation
sources [264, 265, 266]. A combination of PFI-ZEKE photoelectron spectroscopy and
Rydberg series extrapolation called Rydberg-state-resolved threshold ionization (RSRTI)
spectroscopy combines the advantages of both methods [265, 267].
3.2
Experimental setup
The experimental setup, which is discussed in detail in the following subsections, consists of
a laser system to generate VUV radiation, a dual electron/ion time-of-flight spectrometer
and a data acquisition system.
3.2.1
Laser system
The narrow-band tunable radiation required to excite ground state atoms or molecules to
Rydberg states or to ionize them typically lies in the VUV (vacuum ultraviolet) range of
the electromagnetic spectrum λ < 200 nm and is usually generated by nonlinear optical
techniques. The cut-off wavelength for nonlinear frequency up conversion using nonlinear
crystals lies at ∼ 189 nm, limited by the phase-matching properties of available crystals.
As a consequence rare gases [268] or mercury vapor [269] are used as nonlinear medium
to generate VUV radiation below 189 nm. The second-order susceptibility χ(2) , which is
used for sum- and difference-frequency mixing in nonlinear crystals, vanishes in isotropic
media and therefore the lowest-order nonlinear process in gases is four-wave mixing. The
efficiency of a four-wave mixing process is proportional to the square of the third-order
susceptibility χ(3) and to the square of the density of the nonlinear gas (see Eq. (3.3)
below).
The generation of VUV radiation below the LiF cut-off wavelength of 105 nm, a region
also called XUV (extreme ultraviolet), is further complicated by the fact that no solid-state
material transmits radiation in this range of the electromagnetic spectrum. The nonlinear
medium can therefore not be contained in a cell, and the nonlinear gas must be confined
in a pulsed gas beam in the same extended vacuum system in which the spectroscopic
experiments that use the XUV radiation are performed. By using small orifices between
the different parts of a differentially-pumped vacuum system, the background pressure in
the photoexcitation region can be kept three orders of magnitude lower than the background pressure of typically 10−4 mbar in the part of the vacuum system where the XUV
radiation is generated, enabling spectroscopic experiments to be carried out in a collision
free environment.
29
Chapter 3. Experimental setup and methods
~
n2
two-photon
resonance
~
n2
~
n1
~
n1
~
nVUV
~
nVUV
~
n1
~
n1
ground state
difference-frequency
sum-frequency
mixing
mixing
Figure 3.1: Schematic view of VUV generation using resonance-enhanced four-wave mixing.
In the four-wave-mixing process schematically illustrated in Fig. 3.1, the interaction
of intense laser beams, which overlap spatially and temporally with the nonlinear gas,
gives rise to the generation of higher frequencies when they travel through the nonlinear
medium. Because the third-order susceptibility of most gases is very small compared to the
first-order susceptibility, but large compared to susceptibilities of higher orders, input laser
beams of moderate peak intensity (∼ 108 W/cm2 ) can be used to generate VUV radiation
of sufficient intensity for spectroscopic experiments to be performed.
The intensity of the generated VUV radiation can be enhanced if the combined energy
of two of the three input photons corresponds to a two-photon resonance (2ν̃1 ) of the
nonlinear gas starting from the ground neutral state (see Fig. 3.1). The output frequencies
are then equal to the third harmonic νVUV = 3ν1 , the sum frequency νVUV = 2ν1 + ν2
and the difference frequency νVUV = 2ν1 − ν2 . The generation of all other four-wave
mixing frequencies, such as νVUV = 3ν2 for instance, is not resonantly enhanced at the
two-photon level and is therefore usually negligible. Resonance enhancement of the fourwave mixing process at the one- or the three-photon levels is unfavorable because it leads
to the absorption of the input or generated radiation, respectively [270, 271].
The total power of the generated VUV radiation PVUV in a resonance-enhanced fourwave mixing process can be estimated using Eq. (3.3) [272]
b f k 00
k04 k12 k2 2 2 2
N χ P1 P2 Fj b∆k, , , 0 ,
(3.3)
PVUV ∝ 2
kVUV k 0
L L k
where ki represent the modulus of the wave vectors corresponding to the radiation with
wave numbers ν̃1 and ν̃2 in the nonlinear medium, and k0 represents the modulus of the
wave vector of the generated VUV radiation in vacuum. N stands for the particle density
of the nonlinear gas, χ represents the nonlinear susceptibility per atom of the medium and
30
3.2. Experimental setup
Pi the powers of the involved waves. Fj describes the phase-matching integral, which is a
function of the confocal parameter b of the laser beam, the phase mismatch ∆k = kVUV −k 0 ,
the location of the focus f on the z-axis (direction of propagation of the laser beams) and
the effective length of the nonlinear medium L. k 00 is defined as k 00 = 2k1 + k2 . The
definitions of k 0 and Fj are different depending on the four-wave mixing process:
1. VUV generation in a gas jet
In this case the plane-wave approximation (b L) can be used to describe the phase
matching integral Fj in Eq. (3.3). The expression is the same for sum-frequency
mixing (k 0 = 2k1 + k2 ) and difference-frequency mixing (k 0 = 2k1 − k2 ) and is given
as
∆kL
4L2
).
Fj = 2 sinc2 (
b
2
The optimal conversion efficiency is obtained for ∆k = kVUV − k 0 = 0.
2. VUV generation in a cell
The phase-matching integral Fj in Eq. (3.3) can be estimated in the tight-focusing
limit (b L) and is different for sum- and difference-frequency mixing.
(a) Sum-frequency mixing ν̃VUV = 2ν̃1 + ν̃2
( 2
π (b∆k)2 eb∆k/2 if∆k < 0
b f k 00
FSFM b∆k, , , 0 =
L L k
0
if∆k ≥ 0
k 0 = 2k1 + k2 and ∆k = kVUV − 2k1 − k2
The result shows that for sum-frequency generation in a cell a negatively dispersive medium is necessary. The best conversion efficiency is reached with
b∆k = −2.
(b) Difference-frequency mixing ν̃VUV = 2ν̃1 − ν̃2
b f k 00
FDFM b∆k, , , 0 = π 2 e−b|∆k|
L L k
k 0 = 2k1 − k2 and ∆k = kVUV − 2k1 + k2
The expression for FDFM was derived setting k 0 = k 00 and f /L = 0.5 [272].
In contrast to sum-frequency generation, difference-frequency generation is also
possible for ∆k ≥ 0 and is best for b∆k = 0.
The optimal value for b∆k can be adjusted by mixing two gases, one with positive the
other with negative dispersion, or by regulating the pressure in the cell [272, 268, 273].
The experiments described in this thesis all made use of VUV radiation generated by
resonance-enhanced sum- and difference-frequency mixing (see Fig. 3.1) in krypton using
31
Chapter 3. Experimental setup and methods
the two-photon resonance
4p5 5p[1/2]0 ← 4p6 (1 S0 )
(3.4)
at E/hc = 94092.86 cm−1 [274, 275], for which ν̃1 = 47064.43 cm−1 . Using frequencydoubling and mixing in β-barium-borate (BBO) crystals, this wave number can be reached
by doubling the wave number of a dye laser operated at 15688.14 cm−1 and mixing the doubled output with the fundamental beam. The VUV wave number was tuned by changing
the wave number ν̃2 , which corresponded either to the fundamental or the doubled output of a second dye laser. Using other two-photon resonances in Kr, Xe and Ar coherent
tunable narrow-bandwidth laser radiation can be generated between 8 and 20 eV [276].
The third photon required for the (1VUV + 10VIS/UV ) two-photon excitation scheme corresponded to the fundamental or doubled output of a third dye laser which was introduced
in the photoexcitation region through a side port in the same plane as the VUV radiation
and the probe gas making a 45◦ angle with the latter (see Fig. 3.2). This geometrical arrangement induced a Doppler shift (∼ 0.045 cm−1 at 34500 cm−1 [48]) of the wave number
of the transitions induced in the second excitation step.
Depending on whether the wavelength of the VUV radiation to be generated lay above
or below the 105 nm cut-off wavelength, the VUV generation was carried out in a cell or in
a pulsed jet, leading to different experimental configurations that are described separately
below.
VUV generation in a jet
The laser system, displayed schematically in Fig. 3.2 consists of three dye lasers (LambdaPhysik, Scanmate 2E and Scanmate pro) pumped by the 532 nm or the 355 nm output of
a pulsed Nd:YAG laser (Quantel, YG 981E) operating at a repetition rate of 16 32 Hz. Two
of these dye lasers (dye lasers 1 and 2 in Fig. 3.2) were used to generate VUV radiation
by resonance-enhanced sum-frequency mixing in a pulsed atomic beam of krypton. The
output of dye laser 1, frequency tripled in a β-barium borate (BBO) crystal, was kept
fixed at half of the energy of the 4p5 5p[1/2]0 ← 4p6 (1 S0 ) two-photon resonance in krypton
(2ν̃1 = 94092.86 cm−1 ). The VUV wave number was tuned by scanning the wave number
ν̃2 of the doubled output of laser 2.
Both laser beams (wave numbers ν̃1 and ν̃2 ) were recombined using a dichroic mirror
and focused by a 15 cm focal length lens in the middle of the four-wave mixing chamber at the exit of a pulsed nozzle delivering short pulses (∼ 200 µs) of the nonlinear gas.
The VUV radiation of wave number ν̃VUV = 2ν̃1 + ν̃2 generated by resonance-enhanced
sum-frequency mixing was separated from the fundamental frequencies and other frequencies produced by nonlinear optical processes by a vacuum monochromator equipped with
a toroidal platinum-coated dispersion grating. The toroidal geometry of the grating ensured that the sum-frequency beam was refocused at the 1 mm diameter exit hole of the
32
3.2. Experimental setup
Probe gas
XUV/VUV
Detector
Skimmer
Magnetic
shielding
Four-wave mixing
chamber
Krypton
MCP
45°
Telescope
n~2
BBO
Pump
355 nm
Dye laser 2
Dye laser 3
BBO
~
Pump
Photoexcitation
chamber
n~3
n1
Dye laser 1
BBO
BBO
Monochromator
532 nm
Nd:YAG laser
Figure 3.2: Schematic view of the experimental setup for a (1VUV + 10UV ) two-photon excitation scheme
with VUV generation in a supersonic jet. The photons with wave number ν̃1 and ν̃2 were used to generate
VUV radiation of wave number ν̃VUV , whereas the photon with wave number ν̃3 corresponded to the
second step in the two-photon excitation scheme.
monochromator chamber. The distance from the exit hole to the photoexcitation region
was chosen so that the spot size of the VUV beam at the point where it crossed the probe
gas beam was large enough to avoid the generation of too high concentrations of charged
particles, which have a negative influence on the optimal resolution achievable in PFI-ZEKE
photoelectron spectra [277]. The VUV intensity was monitored by a solar-blind electron
multiplier (Hamamatsu). Using the two-photon resonance-enhanced sum-frequency mixing
process in krypton mentioned above, more than 108 photons/pulse were reached after the
monochromator. The bandwidth of the VUV radiation was typically around 0.3 cm−1 and
limited by the bandwidth of the dye laser radiation. By operating the dye lasers with intracavity étalons, their fundamental bandwidths could be reduced from 0.15 cm−1 to better
than 0.06 cm−1 resulting in a VUV bandwidth of ∼ 0.15 cm−1 .
The experimental configuration depicted in Fig. 3.2 is dedicated to the generation of
XUV radiation because the separation and recollimation of the beams is done without
using lenses. The disadvantages of this XUV generation setup are the high losses caused
by the grating monochromator, the small dimension of the nonlinear medium limited by
the diameter of the gas beam and the limited control of the gas density at the focal point
33
Chapter 3. Experimental setup and methods
of the laser which tend to limit the intensity of the generated XUV radiation and cause
undesired shot-to-shot XUV intensity fluctuations and thus noise in the photoionization
and photoelectron spectra.
VUV generation in a cell
VUV generation in a cell leads to a significant enhancement in the VUV intensity because
(1) the interaction path length of the laser beams with the nonlinear medium is much longer
than in a jet, (2) optimal phase-matching conditions can be found by varying the pressure
of the gas in the cell, and (3) the losses caused by the prism (∼ 30%) are considerably
less than those caused by the grating monochromator (> 90%). The relevant part of the
experimental setup for the generation of VUV radiation in a cell is displayed schematically
in Fig. 3.3. Because the MgF2 lens and prism are only transparent for wavelengths greater
than 115 nm (wave numbers < 87000 cm−1 ) VUV generation is restricted to resonanceenhanced difference-frequency mixing. Two laser beams (wave numbers ν̃1 and ν̃2 ) were
recombined with a dichroic mirror and focused by a 30 cm focal length lens in a gold-coated
cell with a constant flow of krypton such that the background pressure was ∼ 16 mbar. A
10 cm focal length MgF2 lens placed at the exit of the cell prevented the divergence of the
VUV beam. The distance between the two lenses was chosen such that the VUV beam of
the desired wave number was either refocused or propagated nearly parallel. Because of its
transmission cut-off at wavelengths below 115 nm (above 87000 cm−1 ) this lens also served
the purpose of excluding higher frequency components generated by other nonlinear optical
processes in the cell. A MgF2 prism with an apex angle of 55◦ (ν̃VUV < 79000 cm−1 ) or 45◦
(ν̃VUV > 79000 cm−1 ) [278] was employed to separate the VUV difference-frequency beam
from the fundamental laser beams of wave numbers ν̃1 and ν̃2 . The difference-frequency
beam then entered the photoexcitation/photoionization region through a small aperture.
Calibration
The calibration of the wave number of the VUV radiation was carried out by calibrating
the wave numbers of the dye lasers involved in the four-wave mixing process and adding
or subtracting them from each other. Calibration of the dye laser wave numbers was
achieved by splitting off small fractions of their fundamental output and directing them
towards either an optogalvanic (OG) cell filled with neon or argon gas or an iodine cell
where optogalvanic or laser-induced fluorescence spectra were recorded, respectively. The
fundamental frequency was then determined by comparing the lines in these spectra with
tabulated transitions of neon or argon [279, 280] or transitions listed in the I2 atlas [281, 282,
283]. In addition, a monitor étalon spectrum of the fundamental output of the scanning
laser (laser 2 or laser 3 in Fig. 3.2) was recorded with every measurement in order to detect
and correct possible nonlinearities in the laser scans. The accuracy of the calibration
34
3.2. Experimental setup
Four-wave mixing
cell
Gas
MgF2 prism
SiO2 window
MgF2 lens
SiO2 lens
~
n
2
~
n1
Pump
Apex angle
Valve
Pressure
sensor
Pump
Figure 3.3: Schematic view of the experimental setup for VUV generation in a cell. Compared to
Fig. 3.2 the four-wave mixing chamber and the grating monochromator are replaced by a cell and a prism,
respectively.
procedure is estimated to be better than 0.2 cm−1 (OG cell) or 0.05 cm−1 (iodine cell) for
the fundamental outputs of the dye lasers, leading to an estimated absolute accuracy for
the VUV radiation of better than 1 cm−1 .
Photoexcitation
chamber
Skimmer
MCP for
electrons
Magnetic
shielding
Probe gas
chamber
Pulsed valve
for probe gas
Gas
jet
MCP for
cations
Pump
Pump
Figure 3.4: Detailed schematic side-view of the spectrometer.
3.2.2
Spectrometer
The actual spectrometer depicted in Fig. 3.4, which consists of the probe gas and the
photoexcitation/photoionization chambers, is attached to the four-wave mixing and the
monochromator chambers where the VUV generation and separation takes place. The photoexcitation/photoionization chamber of the spectrometer contains a 5 cm long photoexcitation region made up of six equidistant, resistively coupled cylindrical extraction plates
35
Chapter 3. Experimental setup and methods
which are used to generate DC or pulsed homogeneous electric fields, two microchannelplate (MCP) detectors and two concentric MUMETAL cylinders for magnetic shielding.
The resistors and capacitors connecting the extraction plates are chosen such that potentials of up to 6 kV, corresponding to electric fields of 1200 V cm−1 , can be pulsed with rise
times of only a few nanoseconds. The electrons and cations are accelerated perpendicularly
to the gas and the laser beams by the electric voltage applied to the extraction plates. The
top extraction plate is always set to ground whereas the bottom plate is set to a negative
high voltage for the extraction of cations and to a low voltage for the extraction of electrons
(see Fig. 3.5 for the timing). The MCP detector for the electrons (cations) is located 10 cm
(30 cm) away from the crossing point of the laser and gas beams.
The probe gas is introduced into the photoexcitation chamber using a pulsed supersonic expansion through a solenoid valve (General Valve, nozzle orifice diameter of 0.1 mm
or 0.4 mm). The supersonic beam is further collimated by a skimmer (Beam Dynamics,
orifice diameter 1.0 mm). The use of a pulsed skimmed supersonic expansion brings several
essential advantages: First, it permits a very efficient cooling of the internal (rotational,
vibrational and electronic) and translational motion of the molecules and therefore ensures that only the lowest rotational levels of the vibronic ground state are appreciably
populated. The number of particles in the lowest quantum states is thus optimized and
spectral congestion is reduced. Second, the Doppler width is strongly reduced as a result
of the beam collimation. Finally, the short length of the gas pulses (∼ 200 µs) enables one
to greatly reduce the gas load in the experimental volume and the pumping requirements
without reducing the gas density at the time and place of the measurement.
The four-wave mixing chamber and the probe gas chamber (see Fig. 3.4), which are
exposed to a higher gas load than the other chambers, are pumped with turbomolecular
pumps of 520 ls−1 (Pfeiffer TMH 521) pumping power. The monochromator and the
photoexcitation chamber, which are exposed to a much smaller gas load, are evacuated
by 260 ls−1 (Pfeiffer TMH 261) turbomolecular pumps. The background pressure in the
photoexcitation chamber typically increased from 2 · 10−7 mbar to 5 · 10−7 mbar during
operation of the pulsed valve.
3.2.3
Control and detection electronics
The ion/electron, the calibration and the monitor étalon signals, respectively, were displayed on a digital oscilloscope (Le Croy, LC684DM) and transmitted via GPIB to a
personal computer.
The timing of the experiment was controlled by three time delay generators (Stanford,
Digital Delay Generator DG 535) triggered at a frequency of 16 32 Hz corresponding to 13
of the frequency of the in-house electricity network. Having an integral ratio between
the frequencies of the trigger and the in-house electricity network, respectively, guarantees
36
3.2. Experimental setup
FWM gas
Probe gas
YAG flashlamps YAG Q-Switch
5 ms
680 ms
125 ms
284 ms
Extraction
Time / ms
PFI-ZEKE
5 ms
Photoionization
1 ms
0.775 ms
1.7 ms
0.95 ms
Figure 3.5: Timing of the experiment. The optimal time delays between the opening of the probe gas
nozzle, the four-wave mixing (FWM) gas nozzle and the Q-Switch of the Nd:YAG pump laser depend on
the optical path length between the Nd:YAG laser and the intersection points of the laser beams with the
gas beams. Typical pulse sequences for PFI-ZEKE photoelectron and photoionization spectroscopy are
displayed.
that stray fields can be kept constant and compensated if needed [284]. The timing of
the experiment is displayed in Fig. 3.5. The optimal time delays between the opening of
the probe gas nozzle, the four-wave mixing (FWM) gas nozzle and the Q-Switch of the
Nd:YAG pump laser depend on the optical path length between the Nd:YAG laser and
the intersection points of the laser beams with the gas beams. Furthermore, the relative
timing of the opening of the nozzle delivering the probe gas is an important parameter for
optimizing the ion and electron signals of the species to be measured. The nozzle delivers
a cloud of particles (atoms, dimers, trimers and other clusters) which intersects the laser
beams in the photoexcitation chamber. Because the spatial position of the clusters in the
cloud is dependent on their total mass, the ion signal corresponding to, e.g., the dimer can
be optimized by selecting the part of the cloud with the highest density of dimers.
37
Chapter 3. Experimental setup and methods
38
Chapter 4
High-resolution study and MQDT
analysis of the Stark effect in
Rydberg states of neon
4.1
Introduction
This chapter is based on Refs. [285, 286] and provides high-resolution photoionization
spectra and a multichannel quantum defect theory (MQDT) analysis of Rydberg Stark
states of neon with principal quantum number n = 23 − 26 in both the bound and the
autoionizing regions.
In the past decades, several experimental and theoretical studies of the Stark effect in
Rydberg states have been carried out on alkali atoms [287, 288, 289, 255], rare gas atoms
[290, 291, 292, 293, 294, 295] and diatomic molecules [296, 294, 297]. The understanding
of the behavior of Rydberg atoms and molecules in external electric fields is important
for applications in PFI-ZEKE photoelectron spectroscopy [247, 249, 253], in the Stark
deceleration and trapping of Rydberg-excited systems [298, 299, 300] and for potential
future use of Rydberg atoms in quantum information processing [301, 302].
MQDT [303, 304] allows the whole spectrum of Rydberg states to be calculated with
a small number of parameters, the eigenquantum defects µα and the transition dipole
moments to the optically allowed final states. Rare gas atoms have been model systems
in the development of MQDT [305, 306, 307, 308, 309, 310, 311, 30, 312] and in the first
applications of MQDT to treat the Stark effect in Rydberg states [292, 294]. In comparison
to the alkali metal Rydberg atoms, which have a closed-shell ionic core, rare-gas Rydberg
atoms possess an open shell 2 PJ (J = 1/2, 3/2) core and therefore more complex field-free
and Stark spectra.
The present study of the Stark effect in Rydberg states of neon was motivated by recent
experiments in which the translational motion of atoms and molecules in Rydberg states
were manipulated using inhomogeneous electric fields [298, 299, 313, 314]. The success of
such experiments depends on the knowledge of the Stark effect, the ability to predict the
39
Chapter 4. High-resolution study and MQDT analysis of the Stark effect in Ne Rydberg states
positions of avoided crossings between Stark states in a Stark map [299] and to estimate
the lifetimes of Rydberg Stark states. A further motivation was the desire to test the
extension of the MQDT formalism to treat the effects of electric fields by systematically
comparing Stark spectra of well defined MJ values recorded by high-resolution spectroscopy
and calculated by MQDT.
To our knowledge, no experimental or theoretical investigation of the Stark effect in
Rydberg states of neon has been carried out to date. The current experiment was performed
using a narrow-bandwidth tunable VUV laser source (see Section 3.2.1), which allowed the
excitation of even Rydberg states in both the bound and the autoionizing energy regions
around n = 23 − 26 by resonance-enhanced two-photon excitation through the 3s[3/2] 1 and
3s0 [1/2]1 intermediate states. In the measurements, the polarizations of the laser beams
with respect to the static electric field were adjusted to access either MJ = 0 or MJ = 1
final states, which together with the high spectral resolution, allowed a more detailed
comparison of the experimental spectra with theoretical predictions than in earlier studies
of the Stark effect in rare gas atoms. In particular, the experimental information was used
to test an extension of MQDT to treat the Stark effect in Rydberg states in a collaboration
with the group of T. P. Softley at Oxford University (see also Ref. [315]).
4.2
Experiment
A (1VUV + 10UV ) resonance-enhanced two-photon excitation sequence was used to excite the
bound and autoionizing Rydberg states in the wave number regions 173740 − 173780 cm −1
and 174485 − 174525 cm−1 via either the 3s[3/2]1 (134459.2871 cm−1 ) or the 3s0 [1/2]1
(135888.7173 cm−1 ) intermediate states [316]. The VUV radiation used to drive the first
step in the excitation scheme was generated in a resonant four-wave-mixing process in a
krypton jet using the 4p5 5p[1/2]0 ← 4p6 (1 S0 ) two-photon resonance at E/hc = 94092.86 cm−1
[274, 275].
Because the control of the polarizations of the laser beams was crucial to the preparation
of Rydberg states in selected MJ levels, the experimental setup used to obtain the results
discussed in this chapter (see Fig. 4.1) differs slightly from that presented in Fig. 3.2. In
order to align the polarization vector of the VUV beam parallel to the static electric field,
the polarization of the doubled output of laser 2 was rotated by 90◦ using a half wave
(λ/2) plate. Depending on the desired relative polarization (parallel or perpendicular)
between the VUV and the UV radiation a second λ/2 plate could optionally be introduced
in the beam path of the radiation with wave number ν̃3 . The polarizer in front of the
entrance window of the photoexcitation chamber served the purpose of filtering out residual
components of the wrong polarization that result from reflections in the beam steering
prisms at angles differing from 90◦ . All spectra have been measured at a resolution of
∼ 0.05 cm−1 , which was achieved by inserting an intracavity étalon in the third dye laser.
40
4.2. Experiment
Photoionization spectra were recorded by monitoring the 20 Ne+ ions as a function of
the wave number ν̃3 of the third dye laser. To record the Stark spectra, electric fields
with electric field strengths ranging between 0 − 250 V cm−1 were generated in the photoexcitation region by applying DC voltages to a set of six resistively coupled cylindrical
extraction plates (see Fig. 3.4). The polarity of the DC voltage was chosen so as to extract
all Ne+ ions produced by photoionization above the 2 P3/2 ionization threshold. Transitions
to bound Rydberg states below the 2 P3/2 ionization threshold were detected by pulsed field
ionization using pulsed electric fields of 1 kV cm−1 delayed by 800 ns with respect to the
laser pulses.
x
z
y
~
Polarizer
~
VUV beam (2 n1 + n2 )
Neon
EVUV
Electron
detection
EUV
neon beam
Ion
detection
Four-wave-mixing
chamber
Krypton
~)
UV beam ( n
3
Skimmer
MCP
l/2-plate
Photoexcitation
chamber
~
2x
BBO
n1
Dye laser 3
532 nm
Dye
laser
1
Pump
l/2-plate
BBO
~
n2
Dye laser 2
n~3
Pump
Monochromator
l/2-plate
BBO
355 nm
Nd:YAG laser
Figure 4.1: Schematic view of the experimental setup, consisting of the laser system for the generation
of tunable VUV and UV radiation, the VUV monochromator and the photoexcitation region. Inset:
Spatial arrangement of all laser beams including their polarization vectors, the neon gas beam and the ion
extraction. The static electric field is applied parallel to the ion extraction (z) axis.
To record spectra of Rydberg states of selected MJ values (either MJ = 0 or MJ = 1),
the polarizations of the VUV and UV laser beams were carefully adjusted with respect to
the direction of the applied DC field. Spectra of MJ = 0 Rydberg states were obtained
by setting the polarizations of both laser beams parallel to the applied static electric field
and spectra of |MJ | = 1 Rydberg states by setting the polarization of the VUV radiation
41
Chapter 4. High-resolution study and MQDT analysis of the Stark effect in Ne Rydberg states
parallel, and that of the UV radiation perpendicular, to the DC electric field. The transition
moment of the two-photon excitation is proportional to two 3j symbols [317],
011
1 1 J
·
(4.1)
000
0 mγ M J
for the first and second excitation step, respectively, where mγ = 0 for a parallel or mγ =
±1 for a perpendicular arrangement of the polarizations. Consequently, in the absence
of electric fields, optical two-photon selection rules from the 1 S0 ground state restrict
photoexcitation to J = 0 and J = 2 Rydberg states for a parallel arrangement of all laser
beams, and to J = 1 and J = 2 Rydberg states for a perpendicular arrangement. These
selection rules were exploited to optimize the relative polarization of the UV and VUV
laser beams which we estimate to be 0◦ (MJ = 0) or 90◦ (MJ = 1) with a 5 % absolute
accuracy.
Table 4.1 summarizes all series that are accessible from the 1 S0 ground state following
0
two-photon excitation, both in Racah-type notation `( ) [k]J (~k = J~+ + ~`; a prime designates
series converging to the 2 P1/2 limit) and LS-coupling notation `(2S+1 LJ ). Because the
3s[3/2]1 state, unlike the 3s0 [1/2]1 state, contains a significant admixture of d character
[285], the f series are only observable in experiments carried out through this intermediate
level. Moreover, the p[5/2]2 channel is expected to be very weak when excitation is carried
out through 3s0 [1/2]1 because of the ∆k = 0, ±1 propensity rule.
Table 4.1: Optically accessible Rydberg series in the absence of a static electric field for the two different
polarization arrangements in both Racah-type notation `[k]J and LS-coupling notation `(2S+1 LJ ).
p series
relative polarization
0◦ (MJ = 0)
90◦ (MJ = 1)
J = 0 channels
p[1/2]0 , p0 [1/2]0
-
p(3 P0 ), p(1 S0 )
-
-
p[1/2]1 , p[3/2]1 , p0 [1/2]1 , p0 [3/2]1
-
p(3 D1 ), p(3 P1 ), p(1 P1 ), p(3 S1 )
p[3/2]2 , p[5/2]2 , p0 [3/2]2
p[3/2]2 , p[5/2]2 , p0 [3/2]2
p(3 D2 ), p(1 D2 ), p(3 P2 )
p(3 D2 ), p(1 D2 ), p(3 P2 )
J = 0 channels
-
-
J = 1 channels
-
f[3/2]1
-
f(3 D1 )
f[3/2]2 , f[5/2]2 , f0 [5/2]2
f[3/2]2 , f[5/2]2 , f0 [5/2]2
f(3 F2 ), f(3 D2 ), f(1 D2 )
f(3 F2 ), f(3 D2 ), f(1 D2 )
J = 1 channels
J = 2 channels
f series
J = 2 channels
42
4.3. Multichannel quantum defect theory of the Stark effect
4.3
4.3.1
Multichannel quantum defect theory of the Stark
effect
Field-free MQDT of the rare gas atoms
MQDT was first introduced by Seaton [303] and treats the ionization channels as electronion collision complexes, with suitable boundary conditions for the electron wavefunctions
at large distances from the ionic core. The space in which the Rydberg electron moves is
divided into a close-coupling (r < rc ) and an outer (r > rc ) region, depicted in Fig. 4.2
as regions I and II. The electron-ion core coupling is best described by Russell-Saunders
(LS) coupling at short range and by J + j- or J + k-coupling at long range. All combinations of ion-core and electron angular momentum quantum numbers can be regarded as
distinct ionization channels, which are designated as eigenchannels or dissociation channels
in regions I and II, respectively. The electron-core collision complex can be described by
the reactance matrix Kij , which can be related to the eigenquantum defects µα of each
Russell-Saunders-coupled channel through
Kij =
X
Qiα tan(πµα )Qαj ,
(4.2)
α
with
Qiα =
X
Qiᾱ Vᾱα
(4.3)
ᾱ
+
where Qiᾱ represents the elements of the J j − LS frame-transformation matrix
p
(4.4)
Qiᾱ = (−1)L−S−l−0.5+mJ + (2J + + 1)(L + 1)(2S + 1)(2J + 1)
X L+ S + J + L+ ` L S + s S L S J ×
mS + ms −mS
mL+ m` −mL
mL mS −MJ
mL+ mS + −mJ +
m m
L+
S+
and Vᾱα incorporates the deviations of the eigenchannels from pure Russell-Saunders coupling and can be described by generalized Euler angles θjk [307].
4.3.2
MQDT including the effect of a Stark field
The treatment of the Stark effect by MQDT used in this work follows the description
given by Softley and coworkers [292, 294], which is based on the theoretical formulations
of Sakimoto [318, 319] and Harmin [289]. To account for the Stark field, a third region
of space (region III in Fig. 4.2) is defined with r > rF = F −1/2 (r, rF , F in atomic units),
where the wavefunction of the Rydberg electron with binding energy is described by
ΦFγβm =
X
γ 0 β 0 m0
Yγ 0 Ψγ 0 β 0 m δγ 0 β 0 m0 ,γβm + Ψ0γ 0 β 0 m0 KγF0 β 0 m0 ,γβm .
43
(4.5)
Chapter 4. High-resolution study and MQDT analysis of the Stark effect in Ne Rydberg states
Figure 4.2: The three regions of space defined for the MQDT treatment of the Stark effect in Rydberg
states where rc represents the outer radius of the close-coupling region. I: Close-coupling region where
channels are described by LS coupling. II: Outer region in field-free MQDT with decoupled dissociation
channels. III: Parabolic region where the electron is influenced by the Stark field, at radii r > r F = F −1/2
(r, rF , F in atomic units).
In Eq. (4.5) Ψγβm (r) and Ψ0γβm (r) represent the regular and irregular Coulomb functions
and Yγ and γ stand for the core wavefunction and the core quantum numbers, respectively.
KγF0 β 0 m0 ,γβm is related to the reactance matrix K through
KγF0 β 0 m0 ,γβm =
X
t
Uβ 0 `0 Kγ 0 `0 m0 ,γ`m Uβ`
,
(4.6)
`0 ,`
where Uβ` represents the transformation matrix element from spherical to parabolic coordinates explicitly given in Ref. [318].
p
Below the classical ionization energy Ec /(hc cm−1 ) = −6.12 F/(V cm−1 ) [255], the
wavefunctions of the Rydberg Stark states can be regarded as quasibound and therefore
be described as a superposition of the wavefunctions of Eq. (4.5):
Ψ=
X
ΦFγβm Lγβm ,
(4.7)
γβm
where L is a vector containing the coefficients of the linear combination.
The effective potentials seen by the Rydberg electron along the parabolic coordinates
ξ and η in region III are in atomic units [320]
β
Fξ
m2
−
+
,
2
8ξ
2ξ
8
1 − β Fη
m2
−
,
V (η) = 2 −
8η
2η
8
U (ξ) =
(4.8)
where β ∈ ]0, 1[ corresponds to the separation parameter that is used to describe the bound
44
4.3. Multichannel quantum defect theory of the Stark effect
motion in the ξ coordinate and the quasibound motion in the η coordinate, as depicted for
β = 0.5 in Fig. 4.3.
0.1
0.1
m=1
m=1
0
U(ξ) / a.u.
V(η) / a.u.
0
-0.1
ξa
ηc
-0.1
ξb
ηa
ηb
m=0
m=0
-0.2
-0.2
0
5
10
ξ / a.u.
15
20
0
5
10
η / a.u.
15
20
Figure 4.3: Effective potentials U (ξ) (left panel) and V (η) (right panel) as functions of the bound ξ
and unbound η motion, respectively, for β = 0.5, F = 0.04 a.u. and both m = 0 (dashed lines) and m = 1
(solid lines).
The two classically allowed regions along the coordinate η for an electron with energy are restricted to ηa < η < ηb and η > ηc , where the ηi (i = a, b, c) correspond to the classical
turning points of the potential V (η) (see Fig. 4.3). In the tunneling region ηb < η < ηc , all
√
exponentially growing terms have to vanish for < Ec = −2 F . This boundary condition
is fulfilled if
det cot ∆(F 0 β 0 m0 ) δ0 β 0 m0 ,βm − KγF0 β 0 m0 ,γβm = 0.
(4.9)
The accumulated phase shift ∆(F βm) is calculated using a Wentzel-Kramers-Brillouin
(WKB) phase integral
Z ηb
q(η)η,
(4.10)
∆(F βm) =
ηa
in which
1
1
− 2V (η) 2
2
represents the corresponding local momentum.
q(η) =
4.3.3
(4.11)
Methodology in the MQDT calculations
The MQDT calculations relied on the field-free reactance matrix K defined by Eqs. (4.2)
- (4.4) with the eigenquantum defects µα taken either from Refs. [321, 308] if available,
45
Chapter 4. High-resolution study and MQDT analysis of the Stark effect in Ne Rydberg states
or otherwise derived from a fit to the experimental spectra, as will be described in Section 4.5.1. Because our experiments only measured relative photoionization cross sections,
no absolute, but only relative values of the transition dipole moments were derived. To
our knowledge, no explicit values for the transition dipole moments for the excitation to
the ` = 1 Rydberg states exist in the literature. Consequently, initial values had been
deduced from the experimental data and the cross sections reported in Ref. [285], which
were subsequently adjusted to the observed transition intensities. No energy dependence
of the eigenquantum defects µα was incorporated in the calculations. Channel interactions
were taken into account for J = 1 − 3 and the corresponding values of the generalized Euler
angles θjk were taken from Ref. [308].
Subsequently, K F was determined using Eq. (4.6) and the separation parameter β
was calculated using second-order perturbation theory, following the procedure described
in Ref. [294]. The calculations were performed with an energy stepsize of 0.001 cm−1 and
0.0005 cm−1 in the bound and autoionizing regions, respectively. At each energy step of the
calculations in the bound region below the 2 P3/2 threshold where the boundary condition
of Eq. (4.9) was fulfilled (indicating the presence of a bound state), the intensity Iγβm (F )
of the corresponding transition was determined using
2
Iγβm (F ) ∝ dFγβm vγβm (F ) ,
(4.12)
where vγβm (F ) represents an element of the vector which solves the equation
cot ∆(F 0 β 0 m0 ) δ0 β 0 m0 ,βm − KγF0 β 0 m0 ,γβm ~v = 0
(4.13)
and dFγβm stand for the transition dipole moments, which are related to the transition dipole
moments of field-free MQDT d`γm by
dFγβm =
X
Uβ` d`γm .
(4.14)
`
In the autoionizing region, the spectral intensity was calculated by direct evaluation of the
total photoionization cross-section
σ() ∝ X X
2
F
`
,
Aγγco00`m
d
+
d
00
00
00
00
00
γ
β
m
γ
m
β m
o
c
γo `m
(4.15)
γc00 β 00 m00
which is expressed as a sum over all partial photoionization cross-sections into the open
channels. In Eq. (4.15), dFγc00 β 00 m00 , d`γo m and Aγγoc00`m
β 00 m00 , which are explicitly given in Ref. [294],
refer to the transition dipole moments to the closed channels γc00 β 00 m00 , to the open channels
γo `m and to the admixture of the bound channel γc00 β 00 m00 into the continuum channel
wavefunction γo `m, respectively. With one set of parameters, the eigenquantum defects µα ,
the mixing angles θjk and the relative values of the transition dipole moments summarized
46
4.4. Experimental results
in Tables 4.2 and 4.3, respectively, the complete Stark maps in both the bound and the
autoionizing regions were calculated.
4.4
4.4.1
Experimental results
Stark spectra of the bound Rydberg states of neon excited
via 3s[3/2]1
The experimental spectra of the Stark manifolds around n∗ = 25 recorded from the 3s[3/2]1
intermediate level are displayed in the left panels of Fig. 4.4 for both MJ = 0 (upper
panel) and MJ = 1 (lower panel) final states. The bottom trace in the upper left panel of
Fig. 4.4 corresponds to the spectrum of MJ = 0 Rydberg states recorded at zero external
electric field. This spectrum is dominated by the strong transitions to the 26p[1/2]0 and
26p[3/2]2 states, at 173758.1 cm−1 and 173756.8 cm−1 , respectively. Two much weaker
transitions to the 26p[1/2]1 (173756.0 cm−1 ) and the 25f[3/2,5/2]2 states (173754.2 cm−1 )
are also observed, the former because of a residual component of perpendicular relative
polarization of the lasers and the latter because of the admixture of d character to the
3s[3/2]1 intermediate state (see above and also Section 4.5.2 below). Neither of these final
states have been included in the calculations. The difference between the spectra recorded
using different arrangements of the polarizations is most obvious at low electric fields:
Only a weak residual contribution of the 26p[1/2]0 state caused by a small deviation of
the relative laser polarization from 90◦ is observed in the MJ = 1 Stark map and the most
intense transition is now that to the 26p[1/2]1 state.
At F = 60 V cm−1 , more transitions to p states are observed than at zero field strength,
because the fine structure components of the p Rydberg states experience different quadratic
Stark shifts and p states that are energetically degenerate or nearly degenerate at low fields
are shifted to different energies. In both the MJ = 0 and MJ = 1 Stark maps, the 26p[5/2]2
state is observed immediately below the 26p[3/2]2 state in the experimental spectra. Additionally, in the MJ = 1 Stark spectra, the 26p[3/2]1 state becomes weakly visible at
F = 60 V cm−1 .
47
Intensity / arb. units
Chapter 4. High-resolution study and MQDT analysis of the Stark effect in Ne Rydberg states
180
180
120
120
26s[3/2]2
27s[3/2]2
26p[5/2]2
60
26p[3/2]2
25d
60
26p[1/2]0
26p[1/2]1
0
25f[3/2,5/2]2
0
Intensity / arb. units
173750 173755 173760
-1
Wave number / cm
173765
120
173750 173755 173760
-1
Wave number / cm
173765
173750 173755 173760
-1
Wave number / cm
173765
120
26p[3/2]1
27s[3/2]1
27s[3/2]2
60
60
26p[1/2]1
0
25f
26p[3/2]2
26p[1/2]0
173750 173755 173760
-1
Wave number / cm
0
173765
Figure 4.4: Stark maps of the MJ = 0 (upper panels) and MJ = 1 (lower panels) Rydberg states
of neon excited via the 3s[3/2]1 intermediate state in the region around n∗ = 25. The experimental
spectra (left panels) and the corresponding MQDT calculations (right panels) have been shifted along the
vertical axis so that the origin of the intensity scale corresponds to the electric field strength in V cm −1 .
The calculated stick spectra (dotted lines) were convoluted with the laser linewidth of 0.08 cm −1 to allow
direct comparison with the experiments.
48
4.4. Experimental results
The high-` Stark manifolds rapidly gain intensity at increasing electric field strength
and form patterns of equidistant lines that are characteristic of the linear Stark effect
[255]. The Stark manifolds fan out until manifolds of adjacent n values start overlapping
at the Inglis-Teller field FIT = 3n1 5 (in atomic units) [322] (∼ 170 V cm−1 at n = 25),
at which point the spectral structures become more complex. ` is not a good quantum
number any more in the presence of the electric field and the transitions to the 26s[3/2]2
and 27s[3/2]2 states, which are forbidden at zero electric field strength, appear as intense
lines at F = 60 V cm−1 . These states start getting overlapped by the high-` Stark manifold
at F ≈ 120 V cm−1 and F ≈ 180 V cm−1 , respectively. In addition, in the MJ = 1 Stark
spectra, the ns[3/2]1 states (n = 26, 27) become observable, which leads to more complex
spectral patterns in the region of overlap with the high-` manifold than in the MJ = 0 case
(compare the F = 120 V cm−1 spectra in both left panels of Figs. 4.4).
The 25d states, which are directly coupled to the 25f states and have a small quantum
defect, are fully integrated into the high-` linear Stark manifold already at low external
field strengths and are therefore not observed as isolated low-` resonances. In contrast to
the 25d states, the 26p states retain their low-` character up to higher external electric
fields and experience observable quadratic Stark shifts towards higher energies.
4.4.2
Stark spectra of bound Rydberg states of neon excited via
3s0 [1/2]1
The Stark map presented in the left panel of Fig. 4.5 was recorded under the same conditions as the MJ = 0 Stark map presented in the upper left panel of Fig. 4.4, except that
the 3s0 [1/2]1 state was used as intermediate state of the two-photon excitation instead of
the 3s[3/2]1 state. Both sets of MJ = 0 spectra differ significantly: The transitions to the
26p[3/2]2 , the 26p[5/2]2 and the 25f states are absent from the spectrum recorded at zero
field (bottom trace in the left panel of Fig. 4.5) and only one strong transition, that to the
26p[1/2]0 state, is observed. A very weak transition to the 26p[1/2]1 state is observed at
173756.0 cm−1 because of a slight deviation of the relative laser polarization from 0◦ . The
reason for the absence of the 26p[3/2]2 and 26p[5/2]2 states in the zero-field spectrum will
be discussed in Section 4.5.2. With increasing electric field strength, the transitions to the
26p[3/2]2 and the 27s[3/2]2 states become observable in the experimental Stark spectra,
but the intensities of the transitions to all observed Rydberg states, except the dominant
26p[1/2]0 state, remain weak compared to the intensities obtained from the 3s[3/2]1 intermediate state. This observation can be explained by the weaker overall intensity of the
transitions from the 3s0 [1/2]1 level (see Ref. [285]).
MJ = 1 Stark spectra were not intense enough to be measured from the 3s0 [1/2]1
intermediate state, presumably because the only strong series observed in the MJ = 0
spectra, the np[1/2]0 series, is forbidden for a perpendicular arrangement of the laser
49
Intensity / arb. units
Chapter 4. High-resolution study and MQDT analysis of the Stark effect in Ne Rydberg states
180
180
160
160
140
140
120
120
100
80
26p[3/2]2
100
27s[3/2]2
80
60
60
40
40
20
0
20
26p[1/2]1 26p[1/2]0
173755
0
173760
173765
-1
Wave number / cm
173755
173760
173765
-1
Wave number / cm
Figure 4.5: Experimental Stark map of the MJ = 0 Rydberg states of neon excited via the 3s0 [1/2]1
intermediate state in the region around n∗ = 25. The spectra have been shifted along the vertical axis
so that the origin of the intensity scale corresponds to the electric field strength in V cm −1 . Right panel:
MQDT calculations corresponding to the spectra shown in the left panel. The stick spectrum (dotted
lines) is convoluted with a laser linewidth of 0.08 cm−1 to allow direct comparison with the experiment.
All experimental spectra and the calculated spectra at field strengths ≤ 60 V cm −1 have been scaled so
that the most intense line fills the interval between successive traces. The calculated spectra at higher field
strengths ≥ 80 V cm−1 have been normalized so that the intensity of the high-` Stark manifold corresponds
to the experimental results.
polarizations (see Table 4.1).
4.4.3
Stark map of the MJ = 0 Rydberg states of neon excited
via 3s0 [1/2]1 in the autoionizing region
The Stark spectra in the autoionizing region between n∗ = 22 and n∗ = 24 displayed in
the left panel of Fig. 4.6 have been recorded via the upper intermediate state (3s0 [1/2]1 )
using a parallel arrangement of both laser beams with respect to the direction of the
external electric field. The Stark spectra at low fields reveal strong transitions to the
np0 [1/2]0 and weaker ones to the np0 [3/2]2 autoionizing resonances, with n = 23 − 25. All
lines in the spectra, including those corresponding to transitions to the high-` and the
ns0 [1/2]0 (n = 24, 25) autoionizing resonances, possess a characteristic blue-degraded Fano
lineshape. This property is caused by the fact that the np0 [1/2]0 resonances carry most of
50
Intensity / arb. units
4.5. MQDT calculations
250
250
250
200
200
200
150
150
150
100
100
50
50
24s’[1/2]0
100
50
24p’[3/2]2
24p’[1/2]0
0
0
0
174490 174500 174510 174520
174490 174500 174510 174520
174490 174500 174510 174520
Wave number / cm
Wave number / cm
Wave number / cm
-1
-1
-1
Figure 4.6: Experimental Stark map of the MJ = 0 autoionizing Rydberg states of neon excited via the
3s0 [1/2]1 intermediate state in the region between n∗ = 22 − 24 (left panel). MQDT calculations corresponding to the spectra shown in the left panel before (right panel) and after (middle panel) convolution
with the laser linewidth. The spectra have been shifted along the vertical axis so that the origin of the
intensity scale corresponds to the electric field strength in V cm−1 .
the intensity at zero field and that all observed resonances at nonzero fields derive both
their intensities and their autoionization dynamics from the np0 [1/2]0 resonances.
4.5
4.5.1
MQDT calculations
Spectral assignments and fitting procedure
MQDT calculations of field-free and Stark spectra were used initially to derive, or confirm tentative, spectral assignments. Subsequently, the MQDT parameters (primarily the
eigenquantum defects µα but to some extent also the relative transition dipole moments)
were iteratively optimized until the best possible simultaneous agreement between the experimental spectra displayed in Figs. 4.4 - 4.6 and the corresponding calculated spectra
could be reached. The derivation of an unambiguous set of MQDT parameters giving a
satisfactory description of all spectra, i.e., spectra in the bound and autoionizing regions,
spectra of different MJ values and spectra recorded through different intermediate levels,
turned out to be a challenging task.
In particular, the np(J = 1, 2) series could not easily be identified, because they are
51
Chapter 4. High-resolution study and MQDT analysis of the Stark effect in Ne Rydberg states
Table 4.2: Eigenquantum defects µα of
`
2S+1
`
2S+1
s
3
P0
0.3156
a
p
3
s
3
P1
0.3155
a
p
s
1
P1
0.2845
a
s
3
P2
0.3149
p
3
P0
p
1
p
LJ
µα
LJ
µα
20
Ne.
`
2S+1
S1
0.84
d
d
3
3
D2
0.85
d
d
p
1
D2
0.82
d
a
p
3
P2
0.8091
0.8091
b
p
3
D3
0.85
S0
0.6783
b
d
3
P0
0.0335
3
D1
0.85
d
3
D1
p
3
P1
0.8091
d
3
p
1
P1
0.89
d
1
c,d
d
c
LJ
µα
a
F2
0.0236
3
D2
0.00576
a
d
1
D2
0.00587
a
d
3
P2
0.0332
a
d
3
F3
0.0225
a
a
d
1
F3
0.0242
a
0.0049
a
d
3
D3
0.0056
a
P1
0.0332
a
d
3
P1
0.0217
a
c
c,d
F4
0.02
c
a
From Harth et al. [308].
From Starace [321].
c
µα assumed to be independent of J.
d
This work.
b
nearly degenerate at zero field and no eigenquantum defects µα of sufficient accuracy have
so far been published in the literature. The determination of the eigenquantum defects µα of
the p 3 D2 , p 1 D2 and p 3 P2 LS-coupled channels was achieved by fitting simultaneously the
position of the bound 26p(J = 2) states and the linewidth and position of the 24p0 [3/2]2
state, assuming µα (p 3 PJ ) to be independent of J and setting µα (p 3 P2 ) to the value of
µα (p 3 P0 ) from Ref. [321]. Moreover, J-independence was assumed for the eigenquantum
defects µα (p 3 DJ ) and µα (p 3 PJ ); µα (p 1 P1 ) and µα (p 3 S1 ) were directly fitted to the line
positions of the two observed 26p(J = 1) states at F = 0 and F = 60 V cm−1 . The
interactions between the 26p[1/2]1 , 26p[3/2]1 , 26p[3/2]2 and 26p[5/2]2 Rydberg states and
the high-` Stark manifold have a strong influence on the magnitude of the quadratic Stark
shifts of the p series: The MQDT calculations at F = 60 V cm−1 were therefore crucial to
refine and verify the determined eigenquantum defects µα . For ` = 3, a value of 0.0001
was included in the calculations for all f eigenquantum defects according to Ref. [323].
The final MQDT results are displayed to the right of the corresponding experimental
spectra in Figs. 4.4 - 4.6 for direct comparison. The classification in Racah-type notation
presented in Figs. 4.4 - 4.6 corresponds to the usual nomenclature used for lower n states
(see compilation in [280]). The eigenquantum defects µα for ` = 0 − 2 channels and
the relative transition dipole moments on which the MQDT calculations are based are
summarized in Tables 4.2 and 4.3, respectively. The overall agreement between the MQDT
calculations and the experimental spectra is good, though not quantitative. The main
discrepancies, which are discussed below, concern the relative intensities and the quadratic
Stark shifts of several s and p series.
52
4.5. MQDT calculations
4.5.2
Characterization of the 3s[3/2]1 and 3s0 [1/2]1 intermediate
states
The comparison of the Stark spectra excited via the two different intermediate states
3s[3/2]1 and 3s0 [1/2]1 allows several conclusions to be drawn on the character of these low-n
Rydberg states. In the present MQDT calculations, the relative transition dipole moments
for the ` = 1 eigenchannels summarized in Table 4.3 have been used to model the intensities
of the Stark spectra. The 3s0 [1/2]1 intermediate state has dominant singlet character, i.e.
1
P1 . Consequently, nonzero relative transition dipole moments to the p 1 S0 , p 1 P1 and
p 1 D2 channels in the second excitation step were assumed to be sufficient to reproduce the
measured intensity distributions, an assumption that was confirmed by the calculations.
This assumption is less satisfactory in the case of excitation through the 3s[3/2]1 state,
which contains significant triplet character and a weak but noticeable d character [285]. In
this case, the inclusion of a nonzero contribution from the p 3 D2 transition dipole moment
is necessary to qualitatively reproduce the experimental line intensities. In the calculations
of the MJ = 0 spectra from the 3s[3/2]1 intermediate state, almost identical distributions
were obtained with relative transition dipole moments of 0.0 and 1.0 as well as 1.0 and 0.0
for the 3 P0 and 1 S0 channels, respectively. Moreover, in the case of the MJ = 1 spectra,
other combinations of transition dipole moments to the J = 1 channels than listed in the
second column of Table 4.3 are possible, in particular for transitions to triplet channels.
Table 4.3:
calculations.
Relative transition dipole moments of all ` = 1 eigenchannels µα used in the MQDT
ionization threshold
intermediate state
relative polarization
3
P3/2
2
P3/2
2
3s[3/2]1
0
3s[3/2]1
3s [1/2]1
0
3s [1/2]1
90◦ (MJ = 1)
0◦ (MJ = 0)
0◦ (MJ = 0)
0◦ (MJ = 0)
P1/2
0.0 (1.0)
a
0.0
0.0
S0
0.0
1.0 (0.0)
a
1.0
1.0
D1
0.0
0.0
0.0
0.0
3
P1
0.0
0.0
0.0
0.0
1
P1
1.0
0.0
0.0
0.0
3
S1
0.0
0.0
0.0
0.0
3
D2
1.0
1.0
0.0
0.0
1
D2
1.0
1.0
0.0
0.09
0.0
0.0
0.0
0.0
3
a
2
P3/2
0.0
P0
1
3
2
P2
Both variants are compatible with the experiments.
A recent publication on the two-photon excitation of autoionizing even-parity Rydberg states of neon in the region around n∗ = 12 − 14 [285] based on data recorded at
53
Chapter 4. High-resolution study and MQDT analysis of the Stark effect in Ne Rydberg states
zero field using the same experimental arrangement as the present work explained the
very low intensity of the transition to the 13p0 [3/2]2 Rydberg state by the presence of a
j =1/2,J=2
Cooper minimum in the partial cross section σ3sc 0
. Following the argumentation of
Ref. [285], this Cooper minimum considerably reduces the intensity of the transitions to
all np, p0 [3/2]2 states from the 3s0 [1/2]1 intermediate state. The triplet character of the
3s[3/2]1 intermediate state opens an additional pathway for the transition to the 26p[3/2]2
state. The expectation that the transition to the 26p[5/2]2 Rydberg state should be weak
in spectra recorded through the 3s0 [1/2]1 intermediate level because of the propensity rule
∆k = 0, ±1, is borne out by the experiments (see Fig. 4.5).
4.5.3
Comparison of measured and calculated intensity distribution
As mentioned above, the MQDT calculations in the bound region (right panels in Figs. 4.4
and 4.6) reproduce all experimental Stark maps well, though not quantitatively. The
positions of all states at zero field and of the high-` manifold are exactly reproduced. The
main spectral features in Figs. 4.4 and 4.5 are reproduced by the calculations, including the
intensity distribution of transitions to the components of the high-` manifold. However,
two main quantitative discrepancies remain between the calculated and the experimental
spectra: The most obvious deviation concerns the relative intensities of transitions to the p
states compared to the high-` manifold in the bound part of the spectrum, and the second
important discrepancy is the systematic overestimation of the quadratic Stark shifts of the
low-` states, which will be discussed in Section 4.6.
The relative intensities of transitions to p states compared to those to the high-` manifold systematically appear larger in the calculated than in the experimental spectra. This
observation is illustrated in Fig. 4.7, which compares the integrated intensities of the experimental (full line) and calculated (dashed line) spectra for a representative measurement
(F = 60 V cm−1 , via 3s[3/2]1 , MJ = 0). The relative intensity ratio of the p states to the
components of the high-` manifold is ∼ 3.5 times larger in the calculations than in the
experiment. We believe that, in the case of the spectra of bound Rydberg states (Figs. 4.4
and 4.5), this discrepancy is at least partly explicable by the fact that the population of
initially prepared Stark states partially decays by fluorescence before the application of the
pulsed electric field (i.e. during the first 800 ns). Indeed, the p states have shorter lifetimes
than the high-` manifold of Stark states and consequently are subject to a faster decay.
In the case of the autoionizing Rydberg Stark states, the decay by fluorescence becomes
negligible and the calculated and experimental line intensities and shapes are in excellent
agreement. The right panel of Fig. 4.6 displays the results of MQDT calculations in which
the intensities have been calculated directly using Eq. (4.15) and the middle panel of
Fig. 4.6 shows the same intensity distribution but convoluted with a Gaussian line profile
54
4.6. Discussion and Conclusion
Intensity / arb. units
200
100
0
173752
173754
173756
173758
-1
Wave number / cm
173760
173762
Figure 4.7: Integrated intensity of the experimental (full line) and calculated (dashed line) spectrum
recorded at F = 60 V cm−1 , via 3s[3/2]1 and with MJ = 0. The scaling was chosen so that the integral of
the high-` manifold is equal for both the experiment and the calculation, which allows a direct estimation
of the difference in the intensity ratio of the p states to the high-` states, which amounts to ∼ 3.5.
of 0.08 cm−1 . Calculated and experimental spectra are in almost quantitative agreement
in the autoionization region.
4.6
Discussion and Conclusion
The only real discrepancy between experimental and calculated spectra is the overestimation of the quadratic Stark shifts of the low-` states in the calculations. A similar
discrepancy was also observed in a recent study [324] of the MJ = 0 Stark states of argon
located around n∗ = 22 below the 2 P3/2 ionization limit. The use of a completely different source code in Ref. [324] and the perfect agreement between calculations of the Ar
Stark spectra carried out with both codes exclude the possibility that the deviations in the
magnitude of the quadratic Stark shift have a numerical origin. Consequently, the approximations made in the treatment of the Stark effect by MQDT may well be responsible for
these deviations.
The calculation of the separation constant β using a perturbational approach and a
WKB wavefunction are approximate in nature, but we believe them to be very accurate
in the energy ranges of interest here. A more significant potential deficiency of the current
MQDT approach lies in the neglect of the influence of the external electric field in region
II where rc < r < rF = F −1/2 (see Section 4.3). In future, it would be of interest to
55
Chapter 4. High-resolution study and MQDT analysis of the Stark effect in Ne Rydberg states
incorporate the effects of the electric field in this region into the MQDT formalism.
Compared to the more standard treatment of the Stark effect in Rydberg states by
perturbation theory [287, 288, 255], which involves matrix diagonalization, MQDT offers
distinct advantages: The cost of the calculations does not depend on n and therefore
MQDT becomes much less expensive than matrix diagonalization at high n; the MQDT
parameters can be interpreted as collisional phase shifts, and many parameters can be
determined independently from experimental or theoretical studies of Rydberg states at
zero field. Finally, MQDT provides a unified treatment of Rydberg states in the bound
and autoionizing regions of the spectrum.
In earlier studies of the Stark effect in atomic [292, 294] and molecular [294, 297]
Rydberg states, uncertainties in a too large number of MQDT parameters and difficulties
associated with the assignment of Stark states of different MJ values prevented a sufficiently
detailed test of the MQDT formalism to treat the Stark effect in Rydberg states. The
present study of Rydberg states of neon relied on a two-photon excitation scheme via two
different intermediate states, on a good control of the relative polarizations of the lasers
and the static electric field at a resolution sufficient to resolve the fine structure of most
levels. Moreover, the large body of experimental data on the odd and even Rydberg states
of neon at zero field available in the literature [325, 308, 285, 280] greatly reduced the
uncertainties of many MQDT parameters.
The following conclusions can be drawn from comparison of experimental Stark spectra of neon and spectra calculated by MQDT: The theory successfully describes the line
shapes, intensities and the positions of the components of the high-` Stark manifolds. Its
main deficiency remains at present the inability to accurately reproduce the experimentally observed quadratic Stark shifts of the low-` states. While we cannot entirely rule out
that another parameter set than the one determined here could give perfect agreement, we
suspect that the approximations made in the MQDT treatment, primarily the neglect of
field effects in region II, represent a potential source of discrepancies.
The ability of calculating Stark spectra demonstrated in this chapter represents an
important step towards a systematic exploitation of Rydberg Stark states in atom optics
experiments. Moreover, the quantum defect parameters that could be derived for neon
from the MQDT analysis are relevant for studies of Ne2 which can be described as neon
atoms in Rydberg states perturbed by a ground state Ne atom, as is explained in the next
chapter.
56
Chapter 5
Spectroscopic characterization of the
potential energy functions of Ne2
Rydberg states in the vicinity of the
Ne(1S0) + Ne(4p0) dissociation limits
5.1
Introduction
This chapter is based on Ref. [326] and describes the measurement and analysis of highresolution photoionization spectra of Rydberg states of Ne2 with principal quantum number
n = 4 − 7 in the vicinity of the Ne(1 S0 ) + Ne(4p0 ) dissociation limits and the determination
of their potential energy functions.
Several experimental and theoretical studies have been devoted to the ground and first
excited singlet states of Ne2 [83, 18, 44, 213, 106] and to its triplet Rydberg states [327, 328,
329, 330, 331, 332, 333, 334, 335, 336, 106], which were studied from the metastable a 3 Σ+
u
+
state. The six low-lying electronic states of the Ne2 ion have been studied by microwave
electronic spectroscopy [168, 160] and ab initio quantum chemistry [45]. The experimental
data obtained on higher singlet Rydberg states of Ne2 are scarce [18], and the purpose of this
chapter is to describe an investigation of the autoionizing singlet gerade Rydberg states of
Ne2 located in the wave number region between 162000 and 172000 cm−1 above the neutral
ground state by (1VUV + 10UV ) resonance-enhanced multiphoton excitation via selected
levels of the C 0+
u state of Ne2 . This wave number range corresponds to the excitation
of Rydberg levels located in the vicinity of the Ne(1 S0 ) + Ne(nl(0) [K]J ) dissociation limits
with n = 4 − 7.
Rydberg states of Ne2 in the vicinity of their equilibrium internuclear distances Re
and at longer distances are usually described in Hund’s angular momentum coupling
0
case (c) nomenclature Ωpp , where Ω is the quantum number corresponding to the projection of the total electronic angular momentum on the internuclear axis, p = g/u and
p0 = +/− denote the symmetry of the electronic wavefunction with respect to inversion
57
Chapter 5. Spectroscopic characterization of potential energy functions of Ne 2 Rydberg states
through the center of symmetry and reflection through any plane passing through both
nuclei, respectively [36]. The dissociation limits of the molecular states are denoted as
Ne(2p6 1 S0 ) + Ne(2p5 nl(0) [K]J ) (or shorter: Ne(1 S0 ) + Ne(nl(0) [K]J )), where n and l are the
principal and the orbital angular momentum quantum numbers of the Rydberg electron,
respectively, and J is the total angular momentum quantum number. K results from the
vectorial addition of the total angular momentum J~+ of the ion core and the orbital angular momentum ~l of the Rydberg electron. The nl and nl 0 notations are used to designate
Rydberg series converging to the first (2 P3/2 ) and second (2 P1/2 ) ionization limits of Ne,
respectively.
The molecular states can be correlated with their dissociation limit according to the
rules summarized by Mulliken in Ref. [337]. For the fragment atoms in Rydberg states
associated with the 2 P1/2 spin-orbit excited ionization threshold relevant to the present
study, the correlations can be expressed as
+
Ne(1 S0 ) + Ne(np(0) [K]J=0 ) ↔ 0+
g , 0u
−
Ne(1 S0 ) + Ne(np(0) [K]J=1 ) ↔ 0−
g , 0u , 1g , 1u
(5.1)
+
Ne(1 S0 ) + Ne(np(0) [K]J=2 ) ↔ 0+
g , 0u , 1g , 1u , 2g , 2u .
+
The ground X 0+
g and the excited C 0u states of Ne2 have been studied by absorption [83]
and by high-resolution REMPI [44, 213] spectroscopy. The potential energy functions of
these states were derived from vibrationally and rotationally resolved spectra [44, 213]. In
contrast to the potential curves of the ground neutral states (0+
g ) of the rare-gas dimers,
which are characterized by a single shallow potential well [338, 44], the potential curves
of the low-lying Rydberg states of the rare-gas dimers can possess one or more potential
barriers (”humps”), and several potential wells [213, 15]. Such a phenomenon was discussed
by Mulliken [15] in relation to the He2 and Xe2 molecules and was later globally and
causally interpreted by Lipson and Field [16] for mixed rare-gas dimers in a model that
forms the basis of the analysis presented below. The potential functions of the six lowest
electronic states of Ne+
2 were derived by combining the results of microwave spectroscopic
experiments and ab initio quantum chemical calculations by Carrington et al. [160]. These
potential curves are very accurate close to the dissociation limits but less so in the vicinity
of the equilibrium internuclear distance because the microwave data involve transitions
between highly excited vibrational levels of the different g and u electronic states close to
dissociation. Theoretical calculations of the same states were performed by Ha et al. [45].
A survey of earlier calculations of the low-lying electronic states of Ne+
2 can be found in
Table 1 of Ref. [160].
+
Fig. 5.1 depicts the potential energy curves of the X 0+
g and C 0u electronic states of
Ne2 taken from Refs. [44, 213], of the lowest ionic states of Ne+
2 taken from Ref. [45],
and of the two autoionizing states derived in the present work. The figure also serves
58
5.2. Experiment
the purpose of introducing the resonant two-photon excitation scheme used to study the
autoionizing Rydberg states experimentally. The C 0+
u state of Ne2 has two potential wells.
0
The v = 1 vibrational wavefunction of this state is localized in the outer well, whereas the
v 0 = 2 wavefunction is distributed over both wells and has a very small amplitude at the
internuclear distance where the v 0 = 1 wavefunction is maximal. Therefore, autoionizing
states with wavefunctions located in different internuclear distance regions can be accessed
from the v 0 = 1 and 2 levels of the C 0+
u state, which can be exploited to make spectral
assignments and study the potential energy functions over a wide range of internuclear
distances. The potential wells of the autoionizing states studied here are located at much
larger internuclear distances than that of the ground I(1/2u) electronic state of the ion,
and the vibrational wave functions of the autoionizing levels do not overlap with those of
the energetically accessible low-lying vibrational levels of the ground ionic state. It was
therefore not certain, at the outset of the present study, whether autoionization would be
observed at all.
5.2
Experiment
The excitation sequence
UV
VUV
−
∗∗
0
0
00
00
Ne+
−− Ne2 (C 0+
−−− Ne2 (X 0+
2 + e ← Ne2 ←
u, v ,J ) ←
g , v , J ),
(5.2)
also depicted in Fig. 5.1, was used to access the gerade autoionizing Rydberg states of
Ne2 in the spectral region beyond the adiabatic ionization energy of Ne2 which is located
162340(100) cm−1 above the ground neutral state according to our calculations for 20 Ne2
based on Refs. [24, 45, 44].
The experimental setup has been described in Section 3.2 and only aspects specific to
the study of Ne2 are summarized here. Neon dimers were generated in a pulsed supersonic
expansion of neat neon through a nozzle with orifice diameter 0.1 mm (nozzle stagnation
pressure 10 − 15 bar). The skimmed supersonic jet was intersected by a beam of coherent
VUV radiation at right angles and by an ultraviolet (UV) laser beam at an angle of 45 ◦ .
The VUV radiation of wavenumber ν̃VUV = 2ν̃1 + ν̃2 around 135800 cm−1 used to drive
the transition to the C 0+
u state was generated by two-photon resonance-enhanced sumfrequency mixing in a krypton jet.
+
The energies of the vibrational and rotational levels of the X 0+
g ground and the C 0u
intermediate states of Ne2 are known from Refs. [44, 213]. The C 0+
u state is known to be
predissociative [213], which reduces the Ne+
2 signal. The optical selection rules ∆Ω = 0, ±1,
0
0
u ↔ g, (p = +) ↔ (p = +) allow transitions from the C 0+
u intermediate state to final
states of 1g and 0+
g electronic symmetry.
59
Chapter 5. Spectroscopic characterization of potential energy functions of Ne 2 Rydberg states
-1
E / (hc cm )
175000
Ne (2p
+
5 2
+
5 2
Ne (2p
165000
P3/2)
+
Ne2 I(1/2u)
measurement region
170000
P1/2)
5
Ne(2p 5p)
5
Ne(2p 4p)
autoionization
5
UV
135840
Ne(2p 3s)
v’ = 2
v’ = 1
135800
Ne2 C 0u
135760
+
VUV
135720
0
6 1
Ne(2p
-15
-30
Ne2 X 0g
1
2
S0)
v’’ = 0
+
3
4
Internuclar distance R / Å
5
6
Figure 5.1: Two-photon excitation sequence used to study the autoionizing gerade Rydberg states of
Ne2 and potential energy functions of the relevant states of Ne2 and Ne+
2 . The potential energy function
+
of the Ne2 (X 0+
)
ground
state
was
taken
from
Ref.
[44],
that
of
the
Ne
2 (C 0u ) state from Ref. [213], and
g
those of Ne+
2 from Ref. [45]. The potential functions of the two Rydberg states were derived in this work.
The figure also shows the positions (dotted horizontal lines) and wave functions of the relevant vibrational
+
levels of the X 0+
g and the C 0u states. The right-hand side of the figure indicates the positions of the
dissociation limits.
The autoionization of Ne∗∗
2
+
+
+
−
Ne∗∗
2 (Ωg , v, J) → Ne2 (I(1/2u), v , J ) + e
(5.3)
is expected to be slow because of the poor overlap of the vibrational wavefunctions of the
60
5.3. Photoionization spectra and their assignment
Rydberg and ionic states. Consequently, predissociation
∗∗
Ne∗∗
2 (Ωg , v, J) → Ne + Ne
(5.4)
is an important competing decay mechanism. Predissociation can be detected by photoionization of the excited atomic fragments, as explained in Ref. [213].
The Ne+
2 isotopomers produced by autoionization were separated in a time-of-flight
(TOF) mass spectrometer and the corresponding spectra were recorded separately. Because
of the low natural abundances of 21 Ne20 Ne and 22 Ne2 no spectra of sufficient quality could
be recorded for these isotopomers. The trimers and larger clusters were not observed in
the TOF spectra because of the selectivity of the resonant two-photon excitation. The
present study is therefore restricted to the spectra of 20 Ne2 and 22 Ne20 Ne.
5.3
Photoionization spectra and their assignment
In the spectra and tables presented in this chapter, the absolute positions of the autoionizing states are given relative to the dissociation limit of the X 0+
g ground state of Ne2 :
E/hc = ν̃VUV + ν̃UV − D000 ,
(5.5)
where ν̃VUV and D000 are taken from Refs. [44, 213]. This choice facilitates the comparison
of the spectra of 20 Ne2 and 22 Ne20 Ne and enables one to directly indicate the positions of
the Ne(np0 [K]J ) dissociation limits in the photoionization spectra because they coincide
with the positions of the corresponding atomic levels.
Survey spectra of the autoionizing states recorded via the v 0 = 1 and 2 vibrational levels
of the C 0+
u state of Ne2 are displayed in Figs. 5.2(a) and 5.2(b), respectively. Vibrational
progressions associated with the Ne(1 S0 ) + Ne(np) and Ne(1 S0 ) + Ne(np0 ) dissociation limits can be recognized in the range n = 4 − 7. The ionization signal decreases rapidly with
n and is barely noticeable at n = 7.
The n = 4 progressions, which are represented on an enlarged scale in Fig. 5.3, are the
primary focus of this chapter. They are intense and well separated from the progressions
associated with n = 5 and higher dissociation limits. Excitation via different vibrational
levels v 0 of the C 0+
u state allows vibrational levels with wavefunctions localized in different
ranges of internuclear distance to be accessed (see Fig. 5.1). Consequently, the intensity
distributions of the vibrational progressions associated with the various 4p[K]J or 4p0 [K]J
atomic limits differ in Figs. 5.3(a) and 5.3(b). The vibrational progressions excited via
v 0 = 1 (Fig. 5.3(a)) have their intensity maxima at intermediate v, whereas the progressions
excited via v 0 = 2 (Fig. 5.3(b)) have two maxima at low and high v and a minimum at
intermediate v. This contrast in intensity behavior qualitatively agrees with the FranckCondon factors expected from the properties of the v 0 = 1 and 2 wavefunctions discussed
61
(b)
via v’ = 2
(a)
via v’ = 1
+
Ne2 signal / arb. units
+
Ne2 signal / arb. units
Chapter 5. Spectroscopic characterization of potential energy functions of Ne 2 Rydberg states
5
Ne 2s 4p
4p’
5p
5p’
6p
6p’ 7p
164000
166000
168000
170000
1
1
-1
Energy relative to Ne( S0)+Ne( S0) / (hc cm )
Figure 5.2: Survey spectrum of the autoionizing Ne2 Rydberg states recorded following two-photon
excitation via the v 0 = 1 (a) and v 0 = 2 (b) vibrational levels of the C 0+
u intermediate state of Ne2 . The
positions of the dissociation limits are indicated below each spectrum.
above. However, the intensity distributions are not only determined by the Franck-Condon
+
factors of the Ne∗∗
2 ← Ne2 (C 0u ) transitions, but also by the autoionization and predis-
sociation dynamics of the final states, and by perturbations caused by interactions with
Rydberg states associated with vibrationally excited levels of the I(1/2u) ionic state.
The vibrational progression in the wave number range 164070 − 164300 cm −1 (pro-
0
gression I in Fig. 5.3) displayed in Fig. 5.4 was excited via the Ne2 (C 0+
u , v = 1) intermediate state. The progression is associated with the 4p0 [1/2]0 dissociation limit at
E/hc = 164258.88 cm−1 and, therefore, the final state must be of 0+
g symmetry. The large
widths (≈ 10 cm−1 ) and the irregularity of the progressions of both isotopomers prevent the
observation of the rotational structure and complicate the assignment of the absolute value
of the vibrational quantum number. The large isotopic shifts indicate that the observed
levels have high vibrational quantum numbers. This progression will not be discussed
further.
Figs. 5.5 and 5.6 depict the progressions labeled II in Fig. 5.3 recorded via the v 0 = 1
20
22
20
and v 0 = 2 levels of the C 0+
u intermediate state, respectively, for Ne2 and Ne Ne. These
progressions have lines with full widths at half maximum (FWHM) of ∼ 1 cm−1 , narrow
enough for the rotational structure to be resolved. The rotational structure was studied by
0
recording spectra via different rotational levels of the Ne2 (C 0+
u , v = 1) intermediate state.
The key to the assignment of the final states is the difference between (Ω = 1) ← (Ω 0 = 0)
62
(b)
via v’ = 2
I
III
II
(a)
via v’ = 1
II
I
III
+
Ne2 signal / arb. units
+
Ne2 signal / arb. units
5.3. Photoionization spectra and their assignment
5
Ne 2s 4p
162800
4p’
163200
163600
164000
1
1
-1
Energy relative to Ne( S0)+Ne( S0) / (hc cm )
164400
Figure 5.3:
Enlarged view of the vibrational progressions located in the vicinity of the
Ne(1 S0 ) + Ne(4p[K]J ) and Ne(1 S0 ) + Ne(4p0 [K]J ) dissociation limits recorded following two-photon excitation via the v 0 = 1 (a) and v 0 = 2 (b) vibrational levels of the C 0+
u intermediate state of Ne2 . The
spectral regions marked as I, II and III are discussed in the text.
and (Ω = 0) ← (Ω 0 = 0) transitions. Symmetry selection rules [36] allow only even J
0
20
+
Ne2 , whereas both odd and
to be observed in the (0+
g , J) ← (C 0u , J ) transitions of
0
even J levels can be observed in the (1g , J > 0)←(C 0+
u , J ) transitions. The rotationally
resolved spectra allowed us to unambiguously distinguish between 1g and 0+
g final states
and to determine the rotational constants B and the band origins. For example, Fig. 5.7
shows the rotational assignments in the region where transitions to the 0+
g (v = 0) and
1g (v = 5) states of 20 Ne2 overlap. A survey of the relative line positions with assignments
of the rotational quantum numbers and of all relevant rotational constants are presented
in Tables C.1 and C.2 in Appendix C. The reason why the deviations between measured
and calculated positions are not distributed symmetrically in these tables comes from the
fact that the intensity distributions in the spectra were fitted pointwise and that some lines
were either weak or highly overlapped by other features and therefore effectively weighted
much less in the fit. These lines are those with the largest deviations in Tables C.1 and
C.2.
The assignment of the absolute vibrational quantum numbers of these vibrational progressions was performed on the basis of the observed isotopic shifts of the lines, and was
supported by the calculations of line positions and rotational constants based on the model
potentials presented in Section 5.4. The experimental line positions Eobs , listed in Ta63
+
1
Ne2 signal / arb. units
Ne( S0) + Ne(4p’[1/2]0)
Chapter 5. Spectroscopic characterization of potential energy functions of Ne 2 Rydberg states
20
Ne- Ne
20
20
Ne- Ne
22
164100
164150
164200
164250
1
1
-1
Energy relative to Ne( S0)+Ne( S0) / (hc cm )
1
0
Figure 5.4: Vibrational progression of the 0+
g state associated with the Ne( S0 ) + Ne(4p [1/2]0 ) disso0
ciation limit. The spectrum was recorded following resonant two-photon excitation via the C 0 +
u (v = 1)
intermediate level.
bles C.3 and C.4 in Appendix C, were fitted with the function:
E/hc = E0 /hc + ωe (v + 1/2) − ωe xe (v + 1/2)2 ,
(5.6)
with the constraints:
ωe (20 Ne −22 Ne) = ρ ωe (20 Ne −20 Ne)
ωe xe (20 Ne −22 Ne) = ρ2 ωe xe (20 Ne −20 Ne),
s
µ(20 Ne −20 Ne)
and µ is the reduced mass (see also Section 2.7). The fits were
µ(20 Ne −22 Ne)
performed for different assignments of v0 , the vibrational quantum number of the lowest
where ρ =
observed vibrational level. The assignments presented in Figs. 5.5 and 5.6 and Tables C.3
and C.4 correspond to the absolute minima of the sums of squared deviations χ2 (v0 ).
The dependence of the rotational constants B on v is displayed in Fig. 5.8. Such a
dependence, if it is linear, can be used to estimate the equilibrium internuclear distance
Re [36]. A strong deviation from a linear dependence is observed for low vibrational
quantum numbers of the 0+
g state and around v = 5 of the 1g state. These deviations and
the differences between the isotopomers suggest a local perturbation caused by another
Rydberg state.
The three dissociation limits in the vicinity of these molecular states are indicated by
64
5.3. Photoionization spectra and their assignment
v=0
5
7
1g
+
0g
5
+
0g
20
20
Ne - Ne
20
22
Ne - Ne
+
Ne2 signal / arb. units
v=0
1
Ne( S0)+Ne(4p’[3/2]2)
Ne( S0)+Ne(4p’[1/2]1)
1
Ne( S0)+Ne(4p’[3/2]1)
1
1
Ne( S0)+Ne(4p[1/2]0)
20
20
Ne - Ne
20
22
Ne - Ne
163400
163500
163600
163700
163800
1
1
-1
Energy relative to Ne( S0)+Ne( S0) / (hc cm )
1
0
Figure 5.5: Spectra of the 0+
g state associated with the Ne( S0 ) + Ne(4p [3/2]2 ) dissociation limit and
1
0
of the 1g state associated with the Ne( S0 ) + Ne(4p [K]J ; [K]J = [3/2]1,[1/2]1 ) dissociation limits of 20 Ne2
(upper traces) and 20 Ne22 Ne (lower inverted traces). The spectra were recorded following two-photon
excitation via the v 0 = 1 vibrational level of the C 0+
u intermediate state. The calculated Franck-Condon
factors for excitation of the Rydberg states are shown below the spectra.
dotted lines in Figs. 5.5 and 5.6: 4p0 [3/2]1 at E/(hc) = 163657.2726(10) cm−1 , 4p0 [1/2]1
at 163707.7261(10) cm−1 and 4p0 [3/2]2 at 163708.6029(10) cm−1 [339]. Only the 4p0 [3/2]2
65
Chapter 5. Spectroscopic characterization of potential energy functions of Ne 2 Rydberg states
v=0
5
1g
0g
v=0
+
5
+
Ne2 signal / arb. units
0g
7
20
+
20
20
Ne - Ne
22
Ne - Ne
1
1
Ne( S0)+Ne(4p’[1/2]1)
1
Ne( S0)+Ne(4p[1/2]0)
20
20
Ne( S0)+Ne(4p’[3/2]2)
1
Ne( S0)+Ne(4p’[3/2]1)
20
Ne - Ne
22
Ne - Ne
163400
163500
163600
163700
163800
1
1
-1
Energy relative to Ne( S0)+Ne( S0) / (hc cm )
1
0
Figure 5.6: Spectra of the 0+
g state associated with the Ne( S0 ) + Ne(4p [3/2]2 ) dissociation limit and
of the 1g state associated with the Ne(1 S0 ) + Ne(4p0 [K]J ; [K]J = [3/2]1,[1/2]1 ) dissociation limits of 20 Ne2
(upper traces) and 20 Ne22 Ne (lower inverted traces). The spectra were recorded following two-photon
excitation via the v 0 = 2 vibrational level of the C 0+
u intermediate state. The calculated Franck-Condon
factors for excitation of the Rydberg states are shown below the spectra.
limit can form a 0+
g molecular state because of the parity rules (see Eq. (5.1)), while each
of these three limits could be the dissociation limit of a 1g state. The 0+
g state with band
66
5.3. Photoionization spectra and their assignment
J=1
2
6
4
+
0g (v = 0)
Ne2 signal / arb. units
J’ = 5
+
J’ = 3
J’ = 1
J=1
0
2
3
5
4
2
6
1g (v = 5)
4
6
-1
Relative energy / (hc cm )
8
Figure 5.7: Rotationally resolved spectra of the transitions to the 0+
g (v = 0) and 1g (v = 5) states of
20
0
+ 0
Ne2 recorded via selected J rotational levels of the C 0u (v = 1) state. The assignments of the rotational
levels of the final states are indicated along the horizontal bars.
0.15
20
22
22
Ne- Ne
Rotational constant B / cm
-1
0.14
20
Ne- Ne
0.13
+
0g
1g
0.12
0.11
0.1
0.09
0
1
2
3
4
5
Vibrational quantum number v
6
7
Figure 5.8: Rotational constants for the 0+
g and 1g states determined from the experimental data
assuming a rigid rotor behavior.
origin at 163640.4 cm−1 is thus assigned to a level correlating to the 4p0 [3/2]2 dissociation
0
limit. Several vibrational levels of this 0+
g state lie above the 4p [3/2]2 dissociation limit.
67
Chapter 5. Spectroscopic characterization of potential energy functions of Ne 2 Rydberg states
+
1
Ne2 signal / arb. units
Ne( S0)+Ne(4p[1/2]0)
This observation can only be accounted for by assuming the existence of a potential barrier
in the potential curve of this state (see Section 5.4).
163250
20
Ne- Ne
20
22
20
Ne- Ne
163300
163350
163400
1
1
-1
Energy relative to Ne( S0)+Ne( S0) / (hc cm )
1
Figure 5.9:
Spectra of the transitions to the 0+
g state associated with the Ne( S0 ) + Ne(4p[1/2]0 )
dissociation limit recorded following two-photon excitation via the v 0 = 1 vibrational level of the C 0+
u
intermediate state. The upper trace corresponds to the spectrum of 20 Ne2 , the inverted lower trace to
that of 20 Ne22 Ne. Solid line: Experimental spectra. Dotted line: Envelope the experimental spectra, for
the calculation of which Gaussian line shapes with full widths at half maximum varying between 5 and
10 cm−1 were assumed.
Fig. 5.9 displays the vibrational progression in the wave number range 163235−163445 cm −1
0
(progression III in Fig. 5.3), excited via the C 0+
u v = 1 level. The progression consists of
seven broad features (highlighted by a dotted line) with full widths at half maximum in
the range 5 − 10 cm−1 , which consist each of several narrower lines. We can rule out that
the substructures of these broad features originate from the rotational structure. Instead,
we attribute the spectral structures to complex resonances originating from the interaction of the successive vibrational levels of a state of the n = 4 complex with Rydberg
states belonging to series converging on vibrationally excited levels (v + = 2 or 3 according
to Ref. [45]) of the I(1/2u) electronic ground state of Ne+
2 . Similar structures have been
analyzed in the spectrum of H2 by Jungen and Raoult [340] and N2 by Giusti-Suzor and
Lefebvre-Brion [341]. Unfortunately, the current uncertainty in the absolute position of
the I(1/2u) state of Ne+
2 does not allow us to assign the interfering Rydberg states. The
dissociation limit of the vibrational progression corresponds to the 4p[1/2]0 limit located
at 163401.3 cm−1 , which implies that the main vibrational progression has a 0+
g character. Significant shifts in the line positions of the
68
22
Ne20 Ne spectrum relative to those of
5.4. The potential energy functions of Ne 2 Rydberg states
the 20 Ne2 spectrum further suggest that the vibrational quantum number v0 of the lowest
observed line in this progression is nonzero.
5.4
5.4.1
The potential energy functions of Ne2 Rydberg
states
Potential model
In order to extract potential energy functions for the 0+
g and 1g states for which a rovibrational analysis was possible, we have adapted the potential model for rare gas dimers
in low-n Rydberg states developed by Lipson and Field [16]. This model describes the
excited dimer as consisting of three interacting particles: a Ne atom in its 1 S0 ground
state, the Rydberg electron of the excited atom (here Ne(4p0 [K]J )) and the ionic core of
the excited atom. An analytical expression of the potential function V (R) can be derived
from an analysis of the relevant interactions. In our calculations, we have neglected any
perturbation of the Rydberg atom by the ground-state atom, i.e., the wavefunction of the
Rydberg electron was assumed to be that of the unperturbed atom. The potential energy
is expressed as a sum of four terms
V (R) = Vdiss + VBM (R) + Vrep (R) + Vattr (R),
(5.7)
where Vdiss is the absolute position of the dissociation limit relative to the position of
the dissociation limit of the neutral ground state, VBM (R) = A · e−bR is the Born-Mayer
repulsion term, Vrep (R) is the energy of Pauli repulsion between the Rydberg electron
and the electrons of the neutral ground-state atom, and Vattr (R) is an attractive potential
resulting from the charge↔induced-dipole interaction between the ground-state atom and
the ionic core. Vrep (R) can be derived from Mulliken’s formula [342]
0
Vrep (R) = Crep
p
∗
INe · INe
·
p
SNe,Ne∗ (R)
0
≈
C
INe · INe∗ · SNe,Ne∗ (R),
rep
2
1 − SNe,Ne
∗ (R)
(5.8)
where INe and INe∗ are the ionization energies of the corresponding atomic states, and
SNe,Ne∗ represents the overlap of the Rydberg electron wavefunction with the wavefunctions
of the neon atom. In first approximation, the wavefunctions of all 10 electrons of the neon
atom are taken to be constant within a sphere of radius rNe = 1 Å (values between 0.7 and
1.6 Å were reported in the literature for the radius of Ne [343] on the basis of solid-state
experiments) centered at the position of the Ne atom and zero outside this sphere. In this
approximation SNe,Ne∗ can be calculated as the integral over the volume of the Ne atom
sphere defined above. SNe,Ne∗ depends on both the orientation of the p Rydberg orbital
with respect to the internuclear axis and on the distance of the Ne atom from the Ne∗ core.
Absorbing the orientation dependence in a new coefficient Crep , Vrep (R) can be expressed
69
Chapter 5. Spectroscopic characterization of potential energy functions of Ne 2 Rydberg states
as
2 Z R+rNe
p
rNe
|ψR (R)|2
Vrep (R) = Crep INe INe∗
dR ,
4 R−rNe
R2
(5.9)
where ψR (R) is the radial part of the Rydberg electron wavefunction. Equation (5.9) is
reminiscent of the Fermi-pseudopotential expression used to describe the interaction of a
low-energy electron and a neutral atom in terms of the energy-dependent scattering length
A(k(R)) of the electron-neutral atom collision [344, 345, 346]
∆ = 2πA(k(R))|ψ(R)|2 ,
(5.10)
where R is the internuclear distance, ψ(R) the wavefunction of the Rydberg electron, and
k(R) = p(R)/~. For the energies (∼ 1 eV) and internuclear distances (R = 1 − 7 Å)
relevant here, the approximation of pure s-wave scattering is only a very coarse one. At
these energies, the scattering length for the collision of an electron and a ground state atom
is known to be strongly positive [347] and translates into a repulsive interaction. In this
analogy, the positive value of the scattering length can be linked to the Pauli repulsion
between the Rydberg electron and the electrons of the 1 S0 ground state Ne atom (see also
discussion in Section 5.4.3).
Vattr originates from the interaction of the ionic core with the dipole of the neutral atom
induced by that core
Z(R)e2 α
f4 (R),
(5.11)
Vattr (R) = −Cattr
R4
−41 2 2
C m /J [348] is the static polarizability of the
where α = 2.376 e2 a0 2 E−1
h = 3.9175 · 10
ground-state neon atom,
Z
R
Z(R) = 1 −
0
|ψR (R)|2 dR
(5.12)
is the effective charge of the ionic core (the charge of the core partially screened by the
Rydberg electron) and f4 (R) is a Tang-Toennies damping function [212]. The coefficients
Crep and Cattr in Eqs. (5.9) and (5.11) depend on the quantum numbers n, l and λ of the
Rydberg electron. The radial part of the Rydberg electron wavefunction was calculated
according to quantum defect theory [76] as
ψR (R) = K · Wκ,l+1/2
2R
κ
,
(5.13)
−1/2
where K = [κ2 Γ(κ + l + 1)Γ(κ − l)]
, Wκ,µ (z) is Whittaker’s function [349], and κ ≡
∗
n = n − δ, with δ = 0.84 the quantum defect.
The potential model described here is expected to be adequate in the vicinity of the
equilibrium internuclear distance Re . For internuclear distances larger than the mean
radius of the Rydberg electron orbit (7.4, 13.1 and 15.2 Å for n∗ = 3.16, 4.16 and 5.16,
respectively), the leading term in the long-range interaction series is no longer the ion70
5.4. The potential energy functions of Ne 2 Rydberg states
induced-dipole interaction and higher terms in the long-range series should be included.
These terms are neglected here.
5.4.2
Fitting procedure
The parameters A, b, Crep and Cattr describing the potential energy functions of the 1g and
0+
g states were determined in a nonlinear least-squares fit to the experimental data. The
fitting procedure was analogous to that used in Refs. [44, 213]. The radial Schrödinger
equation for the nuclear motion
2 2
~ d
+ hcV (R) χv (R) = Ev χv (R)
(5.14)
−
2µ dR2
was solved numerically using a discrete variable representation of the wavefunction [350]
on an equidistant grid with 301 points between 1.7 and 6.6 Å. In Eq. (5.14), µ is the
effective mass, χv (R) is the vibrational wavefunction of the molecular Rydberg state, and
Tv = Ev /(hc) is the vibrational term value. The energies Ev and rotational constants
Bv =
χ2
~2
2µhc
< χv | R12 |χv > were fitted by minimizing the sum of normalized square deviations
χ2 =
X E obs − E calc 2
i
i
i
σ(Eiobs )
+
X
j
Bjobs − Bjcalc
σ(Bjobs )
!2
,
(5.15)
where the superscripts ”obs” and ”calc” designate the observed and calculated quantities,
respectively, and σ represents the experimental standard deviations.
The root-mean-square (rms) deviation
rms =
s
χ2
,
ndf
(5.16)
where ndf is the number of degrees of freedom (the number of experimental points minus
the number of fitted parameters), served as an indicator of the quality of the fit. The rms
value of a reliable fit should be close to unity [351].
As discussed in Section 5.3, the 0+
g (v = 0, 1) and 1g (v = 5) states of the
20
Ne2 iso-
topomer are perturbed. Therefore, the rotational constants of these two levels were not
0
included in the fit. The 0+
g state has the dissociation limit 4p [3/2]2 , whereas the 1g state
can have three possible dissociation limits: 4p0 [3/2]1 , 4p0 [1/2]1 and 4p0 [3/2]2 . Choosing
4p0 [3/2]1 as the dissociation limit in the fit gives much larger deviations of the calculated
energies and rotational constants from the experimental values than using the other two
limits. This observation suggests that the dissociation limit of the 1g state is either 4p0 [1/2]1
or 4p0 [3/2]2 . These two limits are nearly degenerate (∆E/hc = 0.88 cm−1 ), and the choice
between them has only a small influence on the differences between calculated and measured level positions compared to the experimental error. We have somewhat arbitrarily
71
Chapter 5. Spectroscopic characterization of potential energy functions of Ne 2 Rydberg states
chosen the 4p0 [3/2]2 limit as the dissociation limit for the calculations.
Table 5.1: Parameters describing the potential energy functions of the 0+
g and 1g states located close
to the Ne(1 S0 ) + Ne(4p0 [K]J ) dissociation limits determined in a least-squares-fit procedure. Values in
parentheses represent the statistical experimental uncertainty (one standard deviation).
0+
g
1g
A/cm−1
b/Å−1
Vdiss /cm−1
Crep
Cattr
803013(1.4·106)
2.758(608)
163708.60
1.023(54)
0.830(128)
311998573(2.0·108)
5.206(251)
163708.60
0.207(69)
0.964(45)
rms
0.972
3.556
Re /Å
De /cm−1
Rhump /Å
4Ehump /(hc cm−1 )
3.69
83
6.56
112
2.79
397
7.38
15
Correlation matrices
0+
g
A
b
Crep
Cattr
A
b
Crep
1
0.9997
1
-0.9158 -0.9166
1
-0.9919 -0.9942 0.9400
Cattr
1
1g
b
A
Crep
1
0.9780
1
0.0979 0.0556
1
-0.0187 -0.1943 0.6779
Cattr
1
The potential parameters derived in the least-squares-fitting procedure are presented in
the upper part of Table 5.1. The calculated and experimental positions of the vibrational
levels and of their rotational constants B are compared in Tables C.1 and C.2 for the
0+
g and 1g states, respectively. The deviations of the calculated level positions from the
experimental values amount to less than 5 cm−1 . The calculated rotational constants B are
also in overall agreement with the experiment, except for the 20 Ne2 0+
g (v = 0) level, which
is perturbed as discussed above. Although the fit is well converged, several parameters are
strongly correlated, which causes large uncertainties in the parameter values (see Table 5.1).
The assignment of the vibrational quantum numbers of the 1g state presented in Section 5.3 was confirmed by comparing the quality of the fits of the potential energy curves to
the experimental data (Table C.4) assuming different values for v0 (the quantum number
of the lowest observed vibrational level). The assignments presented in Figs. 5.5 and 5.6
and in Table C.4 correspond to the minimum of χ2 (v0 ) (sum of squared deviations). Fits
of the 0+
g potential parameters to the experimental data (Table C.3) for different assumed
values of v0 give values of χ2 (v0 ) smaller than 1 for v0 = 0 − 2 with a minimum of χ2 (v0 )
72
5.4. The potential energy functions of Ne 2 Rydberg states
for v0 = 2 (rms = 0.87). This rms value is only slightly less than the rms value of 0.97
obtained for the assignment made on the basis of the analysis of the isotopic shifts (see
Table C.3). Although we cannot exclude the assignments v0 = 1 and v0 = 2 for the 0+
g
state, we believe that the assignment of v (v0 = 0) derived from the isotopic shifts (see
Figs. 5.5 and 5.6 and Table C.3) is correct, but that the potential model may not be flexible
enough to perfectly describe the 0+
g potential function near its minimum.
5.4.3
Potential functions
The potential energy functions of the 0+
g and 1g states derived from the fits are displayed
in Figs. 5.10 and 5.11, respectively. The figures also show the different contributions to
the potential energy, VBM (R) as dotted lines, Vattr (R) as dash-dotted lines and Vrep (R) as
dashed lines. The humps in these potentials originate from Vrep (R) and can be interpreted
as being caused by the repulsive interaction between the Rydberg electron and the 1 S0
rare gas atom. This repulsive interaction comes from the Pauli repulsion, as explained
by Lipson and Field [16], and translates into a positive scattering length (see also the
discussion following Eq. (5.9)). Because of these humps, the potentials sustain several
metastable levels. The effects of predissociation caused by tunneling through the potential
barriers on the positions of the metastable levels were calculated to be less than 0.001 cm −1
(see Section 5.4.4 and Table 5.2), and tunneling has a negligible influence on the linewidths
and line positions at the resolution of the present measurements, as can be seen from the
results presented in Table 5.2. However, the predissociation caused by the interaction with
neighboring repulsive electronic states may be significant.
The potential energy functions of the 1g and 0+
g states differ in that the former has
−1
−1
a deeper well than the latter (De (1g ) = 397 cm compared to De (0+
g ) = 83 cm ), a
shorter internuclear equilibrium separation (Re (1g ) = 2.79 Å compared to Re (0+
g ) = 3.69 Å)
−1
and a smaller barrier (∆Ehump (1g ) = 15 cm as measured from the dissociation limit
−1
compared to ∆Ehump (0+
g ) = 112 cm ) and thus supports fewer quasibound levels (one
instead of four). These differences can be accounted for qualitatively by Eqs. (5.9) and
(5.10) because the pπ Rydberg wavefunction, unlike the pσ wavefunction, has a node in the
plane containing the Ne(1 S0 ) atom and therefore does not result in a significant repulsive
contribution.
The potential energy functions of the next members of the 0+
g Rydberg series located
close to the Ne(1 S0 ) + Ne(np0 [3/2]2 ) limits with n = 5 and 6 displayed in Fig. 5.12 have
been predicted on the basis of the calculated potential parameters for the n = 4 state
(see Table 5.1) and Eq. (5.10). At increasing n values, the potential barrier shifts to
longer internuclear distances and its height decreases rapidly reflecting the smaller values
of |ψ(R)|2 at the position of the maxima, which scale as 1/R2 .
The potential humps observed at the extrema of the Rydberg electron wavefunction
73
V(R)
VBM(R)
-1
Energy relative to Ne( S0)+Ne( S0) / (hc cm )
Chapter 5. Spectroscopic characterization of potential energy functions of Ne 2 Rydberg states
Vrep(R)
163800
1
Vattr(R)
1
163750
163700
1
Ne( S0)+Ne(4p’[3/2]2)
163650
2
4
6
8
10
Internuclear distance R / Å
12
14
-1
163600
1
Energy relative to Ne( S0)+Ne( S0) / (hc cm )
163700
1
1
0
Figure 5.10: Potential energy function of the 0+
g state of Ne2 associated with the Ne( S0 ) + Ne(4p [3/2]2 )
dissociation limit. The full, dotted, dashed and dash-dotted line represent the various contributions to
the potential energy function as indicated in the top right-hand side corner (see text for details). The
positions of the vibrational levels and their wave functions are drawn as dotted and full lines, respectively.
1
Ne( S0)+Ne(4p’[3/2]2)
163500
V(R)
VBM(R)
163400
Vrep(R)
Vattr(R)
163300
2
4
6
8
10
Internuclear distance R / Å
12
14
Figure 5.11:
Potential energy function of the 1g state of Ne2 associated with either the
Ne(1 S0 ) + Ne(4p0 [K]J ; [K]J = [3/2]1,[1/2]1 ) dissociation limits. The full, dotted, dashed and dash-dotted
line represent the various contributions to the potential energy function as indicated in the bottom
right-hand side corner (see text for details). The positions of the vibrational levels and their wave functions
are drawn as dotted and full lines, respectively.
74
5.4. The potential energy functions of Ne 2 Rydberg states
100
+
0g
-1
Relative energy / (hc cm )
50
n=4
0
1
Ne( S0)+Ne(np’[3/2]2)
n=5
n=6
-50
-100
-150
4
6
8
10
Internuclear distance R / Å
12
14
Figure 5.12:
Comparison of the potential energy function of the 0+
g state of Ne2 associated with
1
0
the Ne( S0 ) + Ne(4p [3/2]2 ) dissociation limit with predictions of the potential energy functions of the
1
0
corresponding 0+
g (Ne( S0 ) + Ne(np [3/2]2 ) states for n = 5 and 6. See text for details.
ψR (R) are reminiscent of the recent discussion of multiminima potential functions of Rb2
and Cs2 Rydberg states [345, 346] (see also discussion following Eq. (5.8)). The main difference between Ne2 and the alkali-metal dimers is that potential maxima of Ne2 are located
close to the maxima of |ψR (R)|2 whereas potential minima are located at these positions
in Rb2 and Cs2 because of the opposite signs of the scattering length of the electron-atom
interaction. In both cases, a complex long-range physics results. The Rydberg levels of
Ne2 studied here may thus be seen as being the low-n precursors of the low-` ”trilobites”
discussed in Refs. [345, 346], with the difference that the minima of the potential functions
are located at energies above the dissociation limit in Ne2 rather than below, as is the case
for the alkali-metal dimers. However, the scattering length of the electron-neon collision
changes sign at low energies [347], and the high-n states of Ne2 may be expected to behave
similarly to those of the alkali-metal dimers.
The vibrational wavefunctions are also depicted in Figs. 5.10 and 5.11 and were used
+ 0
to calculate Franck-Condon factors for the 1g (v), 0+
g (v) ← C 0u (v ) transitions using the
v 0 = 1 and 2 wavefunctions of the C 0+
u state derived in Ref. [213]. These Franck-Condon
factors are depicted in Figs. 5.5 and 5.6 and are in qualitative agreement with the measured intensities. A quantitative agreement cannot be expected because the intensities of
the lines observed in the photoionization spectrum are influenced by the autoionization
and predissociation dynamics, and by perturbations with Rydberg states converging on
vibrationally excited levels of the I(1/2u) electronic ground state of Ne+
2 and also on the
75
Chapter 5. Spectroscopic characterization of potential energy functions of Ne 2 Rydberg states
vibrational levels of the other five low-lying electronic states of Ne+
2.
5.4.4
Calculation of the tunneling predissociation linewidths
A semiclassical treatment of the tunneling predissociation of a quasibound level was well
described by Huang and Le Roy in [352]. The tunneling predissociation lifetime τtp is
described as a period of vibration tvib of the mass µ between the turning points r1 and r2
divided by a tunneling probability κ
τtp = tvib /κ,
(5.17)
and the line broadening resulting from the tunneling predissociation is
Γtp = }/τtp = } κ/tvib .
(5.18)
In a first-order quasi-classical approximation [353]
tvib = 2
Z
r2
r1
2(E − V (r))
µ
−1/2
dr .
(5.19)
and the tunneling probability:
κ = e−
Z
2 r3 p
=
2µ (U (r) − E)dr
} r2
(5.20)
The results of the numerical calculations for the quasibound levels derived in this work are
listed in Table 5.2.
5.5
Conclusion
Comparing the potential energy functions of the Rydberg states determined here with
those of Ne+
2 from Ref. [45] (see Fig. 5.1) leads to the conclusions that (1) the Rydberg
states cannot have significant contributions from channels associated with the I(1/2u),
I(3/2g) and I(1/2g) electronic states of the ion, and (2) the vibrational wavefunctions of
+
= 0 − 2)
the 0+
g and 1g states do not overlap with those of the ionic levels (I(1/2u) v
associated with the open channels. The observation of autoionization therefore implies
that this process is mediated by complex channel interactions, and that Rydberg levels
associated with electronically excited levels of the ion must be involved to bridge the gap
of internuclear distances between the 1g and 0+
g levels and those of the low vibrational levels
of the I(1/2u) state. A multichannel quantum defect theory analysis would be desired to
quantify these interactions.
76
5.5. Conclusion
Table 5.2: Term values (with respect to the dissociation limit of the neutral ground state of Ne 2 )
and widths Γtp of the metastable vibrational levels of the 0+
g and 1g states of Ne2 located close to the
Ne(1 S0 ) + Ne(4p0 [K]J ) dissociation limits.
v
E/(hc cm−1 )
Γtp /cm−1
0+
g
20
Ne2
3
4
5
6
163731.2
163756.9
163780.3
163800.9
4.29 · 10−24
1.63 · 10−15
3.83 · 10−09
3.01 · 10−04
0+
g
20
Ne-22 Ne
3
4
5
6
163729.1
163754.4
163777.5
163798.0
1.39 · 10−25
1.18 · 10−16
4.05 · 10−10
5.11 · 10−05
1g
Ne2
9
163717.1
4.40 · 10−04
1g
Ne-22 Ne
9
163714.1
1.68 · 10−06
20
20
77
Chapter 5. Spectroscopic characterization of potential energy functions of Ne 2 Rydberg states
78
Chapter 6
The low-lying electronic states of Xe+
2
and their potential energy functions
6.1
Introduction
Most experimental data on the low-lying electronic states of Xe+
2 have been obtained by
He I [170, 354], threshold [169, 172, 174, 173] and PFI-ZEKE [171, 53] photoelectron
spectroscopy, and electron impact ionization [148]. Previous experimental studies of the
+
photoelectron spectrum of Xe2 have provided data on the I(1/2u) ← X 0+
g , I(3/2g) ← X 0g
and II(1/2u) ← X 0+
g photoionizing transitions [172, 174, 173, 171, 53]. The spectroscopic
+
+
data available on the I(3/2u) ← X 0+
g , I(1/2g) ← X 0g and II(1/2g) ← X 0g transitions are
much less extensive. The vibrational assignment of the I(1/2u), I(3/2g) and I(3/2u) states
are unambiguous. Lu et al. suggested the origin of the II(1/2u) state to be located at
107109 cm−1 above the neutral ground state while Rupper et al. found its origin to be
located at 107157.7 cm−1 . Tonkyn and White [171] and Rupper et al. [53] have determined
three vibrational levels (v + = 0, 1, 2) of the I(3/2u) state by PFI-ZEKE photoelectron
spectroscopy using single-photon excitation from the X 0+
g ground neutral state of Xe2 .
The I(1/2g) state has not been observed experimentally so far and is reported to be either
repulsive [355, 356] or weakly bound [149, 357, 164, 53]. Lu et al. attributed a line at
∼ 107715 cm−1 in their threshold photoelectron spectrum to the origin of the II(1/2g)
state. Rupper et al. also observed an isolated line at 108132.3 cm−1 in their spectrum
which they tentatively attributed to the II(1/2g) (v + = 0) ← X 0+
g transition.
When combined with results of semiempirical [15] and ab initio quantum chemical
[355, 356, 358, 357, 164] calculations, these experimental results provide a qualitative understanding of the relative role of short-range, long-range and spin-orbit interactions and
a qualitative description of the strengths and lengths of the bonds in all six low-lying
electronic states of Xe+
2 . However, despite the considerable experimental and theoretical efforts that have been invested to derive accurate potential energy functions for these
states, several assignments remain controversial. Moreover, all theoretical and experimental determinations of the potential energy functions of the six lowest electronic states of
79
Chapter 6. The low-lying electronic states of Xe +
2 and their potential energy functions
Xe+
2 have been based so far on the assumption that the spin-orbit coupling constant is
independent of the internuclear separation. This assumption was first made and justified
by Cohen and Schneider [18], but has never been rigorously tested for the rare gas dimers.
The main objectives of the present work were (1) to obtain a complete set of experimental data on the vibronic energy level structure of Xe+
2 , (2) to remove all persisting
ambiguities in the spectral assignments, and (3) to identify, and if possible quantify, the
effects of a potential dependence of the spin-orbit coupling constant on the internuclear
distance by combining the results of high-resolution photoelectron spectroscopy with those
of ab initio quantum chemistry.
To this end a new investigation of the I(3/2g), I(3/2u) and II(1/2u) states of Xe +
2
+
by resonance-enhanced two-photon ionization via selected vibrational levels of the C 0 u
intermediate state of Xe2 was carried out. Xe+
2 was selected because the very large spinorbit interaction seemed ideally suited for the detection of a potential R-dependence of
the spin-orbit coupling constant. The analysis of the experimental data was guided by
independent determination of a(R) by ab initio quantum chemistry carried out in the
group of Prof. M. Reiher at ETH.
A (1VUV + 10VIS/UV ) two-photon excitation scheme via selected vibrational levels of
∗
5 0
1
the C 0+
u Rydberg state of Xe2 located in the vicinity of the Xe ([5p] 6s [1/2]1 ) + Xe( S0 )
dissociation limit has been used to access the ionic levels. This species-selective excitation
scheme enables one to avoid undesirable contributions to the spectra originating from
the ionization of free xenon atoms or of large clusters which completely obscure several
regions of the single-photon PFI-ZEKE photoelectron spectrum [53]. Transitions to the
vibrational levels up to v + = 52 of the I(3/2g) state and up to v + = 22 of the I(3/2u) state
have been observed and unambiguously assigned. The assignment of the II(1/2u) state
proposed by Rupper et al. was confirmed in an analysis of the isotopic shifts. In the range
between 97800 and 97900 cm−1 several features could be observed in the spectra which
may be members of the I(1/2g) ← X 0+
g progression. However, an unambiguous assignment
was not possible. No transitions to the II(1/2g) state were observed following resonant
two-photon excitation via the C 0+
u Rydberg state.
6.2
Experiment
The investigation of the low-lying electronic states of Xe+
2 was performed by PFI-ZEKE
photoelectron spectroscopy using a (1VUV + 10VIS/UV ) two-photon excitation scheme via
selected vibrational levels of the C 0+
u Rydberg state of Xe2 . The experimental setup has
been described in Chapter 3 and is illustrated in Figs. 3.2 and 3.3. In this section only the
aspects relevant to the investigation of Xe2 are summarized.
+
The VUV radiation required to drive the C 0+
u ← X 0g transition of Xe2 was produced in
the four-wave mixing cell (see Fig. 3.3) by resonance-enhanced difference-frequency mixing
80
6.3. Experimental results
using the 4p5 5p[1/2]0 ← 4p6 (1 S0 ) two-photon resonance in krypton at 2ν̃1 = 94092.86 cm−1 .
Xe2 rare gas dimers were generated in a pulsed supersonic expansion of pure xenon (stagnation pressure of ∼ 3.6 bar) through a pulsed solenoid valve (diameter of 0.4 mm). Before
entering the photoexcitation/photoionization region the supersonic beam passed through
a skimmer (orifice diameter 1.0 mm).
+
REMPI spectra of the C 0+
u ← X 0g transition were recorded by scanning the wave
number ν̃VUV = 2ν̃1 − ν̃2 keeping the wave number ν̃3 of a third dye laser fixed so that
the wave number of the sum ν̃VUV + ν̃3 lay above the first adiabatic ionization threshold
of Xe2 . The generated ions were extracted by applying an electric field pulse of amplitude
−496 V cm−1 and duration 1.55 µs delayed by 1.7 µs with respect to the VUV laser pulse
and detected mass selectively at the MCP detector placed at the end of the TOF tube.
The PFI-ZEKE photoelectron spectra of Xe2 were measured by detecting the electrons
produced by delayed pulsed field ionization of very high Rydberg states located below the
vibronic states of Xe+
2 using a sequence of two successive electric field pulses as a function
of the wave number ν̃3 of a tunable dye laser. A (1VUV + 10VIS/UV ) excitation scheme was
used for which the wave number ν̃VUV of the VUV radiation was kept fixed at a position
corresponding to a transition from the X 0+
g ground neutral state of Xe2 to a selected
vibrational level of the C 0+
u Rydberg state. The isotopic shift of the transitions to the
0
v = 20, 21 and 22 vibrational levels allowed the selection of specific isotopomers in the
PFI-ZEKE photoelectron spectra (see Fig. 6.1).
6.3
6.3.1
Experimental results
The C 0+
u Rydberg state of Xe2
∗
5 0
1
The C 0+
u Rydberg state of Xe2 located just below the Xe ([5p] 6s [1/2]1 ) + Xe( S0 ) dissociation limit was chosen as intermediate level in the resonance-enhanced two-photon excita-
tion scheme because it is long-lived and has been well characterized [89, 52]. Vibrationally
and isotopically resolved fluorescence excitation spectra of this state were reported by Lipson et al. [89]. Wüest et al. [52] presented rotationally resolved REMPI spectra of the C 0 +
u
state recorded for several isotopomers of Xe2 at a resolution of 0.008 cm−1 . Fig. 6.1 displays
0
+ 00
the REMPI spectrum of the C 0+
u (v = 20 − 22) ← X 0g (v = 0) transitions recorded in this
work. The time gate placed for integration in the TOF trace was set such that the TOF
positions of all isotopomers were included, and thus all isotopomers contributed to the
REMPI spectrum. The resolution of this experiment (∼ 0.15 − 0.3 cm−1 ) was insufficient
to observe the rotational structure. The isotopic shifts exceeded the spectral width of the
rotational envelope of the isotopomers which allowed the selection of specific isotopomers
in the subsequent investigation by PFI-ZEKE photoelectron spectroscopy. The line positions of all observed vibrational positions of the C 0+
u Rydberg state of
81
129
Xe132 Xe and
Chapter 6. The low-lying electronic states of Xe +
2 and their potential energy functions
131
Xe132 Xe are listed in Table C.5 in Appendix C.
H J
H
J
K
K
Ion signal / arb. units
H
G
F
C
I
E
J
FG I
M
G
E
L
F
LM
C
D
D
C
B
A
0
76700
N
A
v’ = 20
A
v’ = 21
76720
E
M
D
N
B
K
I
76740
-1
Wave number / cm
B
N
v’ = 22
76760
76780
Figure 6.1: REMPI spectrum of the vibrational levels v 0 = 20 − 22 of the C 0+
u Rydberg state of Xe2 .
The capital letters correspond to the following dominant isotopomers: A: 136 Xe136 Xe, B: 134 Xe136 Xe, C:
132
Xe136 Xe, D: 131 Xe136 Xe, E: 132 Xe134 Xe, F: 129 Xe136 Xe, G: 132 Xe132 Xe, H: 131 Xe132 Xe, I: 131 Xe131 Xe,
J: 129 Xe132 Xe, K: 129 Xe131 Xe, L: 129 Xe130 Xe, M: 129 Xe129 Xe, N: 128 Xe129 Xe.
6.3.2
The I(3/2g) and I(3/2u) states of Xe+
2
PFI-ZEKE photoelectron spectra following single-photon excitation from the neutral ground
state provided only information on the low vibrational levels of the I(3/2g) and I(3/2u)
+
+
1
+ 2
states of Xe+
2 up to v = 21 and v = 2, respectively [53]. Close to the Xe( S0 ) + Xe ( P3/2 )
dissociation limit the interpretation of the single-photon PFI-ZEKE photoelectron spectrum was complicated by atomic lines belonging to the Rydberg series converging to the
2
P3/2 dissociation limit of atomic xenon [53]. These undesirable atomic contributions could
be avoided in the present work using a resonant - and thus species-selective - two-photon
excitation sequence. Moreover, different regions of the internuclear distance could be accessed by selecting different vibrational levels of the intermediate state.
The spectrum of the I(3/2g) and I(3/2u) states of 131 Xe132 Xe+ recorded via the v 0 = 21
vibrational level of the C 0+
u state is presented in Fig. 6.2. Long vibrational progressions
were observed and assigned to levels as high as v + = 52 for the I(3/2g) state and v + = 22 for
the I(3/2u) state. The unambiguous vibrational assignments were derived from an analysis
of the isotopic shifts using the procedure described in Section 2.7. In the wave number
region beyond 97910 cm−1 , an unambiguous assignment of the spectral features could not
82
6.3. Experimental results
+
30
35
40
45
50
I(3/2g)
Electron signal / arb. units
v = 25
0
I(3/2u) v+ = 0
97400
5
10
15
97600
97800
-1
Wave number / cm
20
98000
Figure 6.2:
PFI-ZEKE photoelectron spectrum of the I(3/2g) and I(3/2u) states of
recorded via the v 0 = 21 vibrational level of the C 0+
u state.
131
Xe132 Xe+
be derived because of (1) the spectral congestion, (2) the weakness of most spectral features
and (3) the irregularity of the intensity distribution partially caused by channel interactions
with autoionizing Rydberg states [248]. These autoionizing states were independently
observed in photoionization spectra recorded via the same intermediate vibrational levels
of the C 0+
u state. Spectral lines in the photoelectron spectra observed in this region could
correspond to vibrational levels of any of the four ionic states of Xe2 associated with
the Xe+ (2 P3/2 ) + Xe(1 S0 ) dissociation limit. The positions of the vibrational levels for
the I(3/2g) and I(3/2u) states of
Appendix C.
131
Xe132 Xe+ are summarized in Tables C.7 and C.8 in
The PFI-ZEKE photoelectron spectra of the I(3/2g) and I(3/2u) states of 129 Xe132 Xe+
and 131 Xe132 Xe+ between 97820 and 98000 cm−1 are compared in Fig. 6.3. The intensity
distributions strongly depend on the selected vibrational level of the intermediate C 0+
u state
because of different Franck-Condon factors and channel interactions [248] (see also above
and Section 7.5). The top trace in Fig. 6.3 corresponds to the spectrum of the 129 Xe132 Xe
0
isotopomer recorded via the C 0+
u (v = 20) intermediate level whereas the bottom two
0
traces correspond to spectra of 131 Xe132 Xe recorded through the C 0+
u (v = 21 and 22)
levels, as indicated. The line marked by an asterisk in the lower two traces of Fig. 6.3
could not be assigned to either the I(3/2g) or the I(3/2u) states of Xe+
2 . This line could
correspond to a low vibrational level of the I(1/2g) state or to a high vibrational level
(v + > 110) of the I(1/2u) state.
83
Chapter 6. The low-lying electronic states of Xe +
2 and their potential energy functions
I(3/2u)
+
v = 14
20
Electron signal / arb. units
m = 261, via v’ = 20
+
50
v = 45
I(3/2g)
+
v = 45
50
*
m = 263, via v’ = 22
*
+
v = 14
97820
20
I(3/2u)
97840
97860
97880
-1
Wave number / cm
m = 263, via v’ = 21
97900
Figure 6.3:
PFI-ZEKE photoelectron spectra of the I(3/2g) and I(3/2u) states just below the
Xe+ (2 P3/2 ) + Xe(1 S0 ) dissociation limit of 129 Xe132 Xe+ recorded via the v 0 = 20 vibrational level of
131
the C 0+
Xe132 Xe+ recorded via the v 0 = 21 and 22 vibrational levels of the
u state (upper trace) and of
C 0+
u state (lower traces).
6.3.3
The II(1/2u) state of Xe+
2
In our previous study of the lowest electronic states of Xe+
2 following single-photon excita+
tion from the X 0g ground state of Xe2 , only transitions to the lowest twelve vibrational
levels of the II(1/2u) state could be observed [53]. An unambiguous assignment of the
absolute value of the vibrational quantum number was, however, hindered by the fact that
the isotopic shifts were not large enough.
The isotope selectivity of the resonant two-photon excitation sequence used in the
present investigation allowed the observation of the II(1/2u)(v + ) ← X 0+
g progression for
different isotopomers of Xe2 . The PFI-ZEKE photoelectron spectrum of the II(1/2u) state
of 131 Xe132 Xe+ recorded via the v 0 = 21 vibrational level of the C 0+
u state of Xe2 is displayed
in Fig. 6.4 and the corresponding line positions are listed in Table C.10 in Appendix C. The
decreasing background signal in the photoelectron signal between 107100 and 107350 cm −1
was caused by a strong electron signal close to the time gate of the background window in
the TOF trace. These prompt electrons are mostly generated by processes where the laser
beams hit metallic surfaces. The line marked by an asterisk corresponds to an impurity
line. From the analysis of the well-resolved isotopic shifts (see Fig. 6.5 for v + = 5) the
vibrational assignment of the II(1/2u) state tentatively proposed by Rupper et al. [53]
could be confirmed.
84
Electron signal / arb. units
6.3. Experimental results
*
0
+
v =0
5
107200
107300
107400
-1
Wave number / cm
9
107500
Figure 6.4: PFI-ZEKE photoelectron spectrum of the II(1/2u) state of 131 Xe132 Xe+ recorded via the
v 0 = 21 vibrational level of the C 0+
u state. The line marked by an asterisk corresponds to an impurity
line.
Unfortunately, the II(1/2g) state could not be observed from any of the selected vibrational levels of the intermediate C 0+
u state and we are not able to confirm or refute the
assignment of the single line observed in our previous study [53].
6.3.4
Spectroscopic constants of the I(3/2g) and I(3/2u) states
of Xe+
2
Analyzing the vibrational progressions associated with the I(3/2g) and I(3/2u) states of
Xe132 Xe+
2 in terms of the standard expansion formula (Eq. (2.24)) led to the set of
+
+
adiabatic ionization energies Ei , harmonic (ωe+ ) and anharmonic (ωe x+
e , ωe ye , ωe ze ) vibrational constants summarized in Table 6.1. Spectroscopic constants for the I(3/2g) state
131
were derived using the transition wave numbers determined from the spectra shown in
Figs. 6.2 and 6.3 and the positions of the lower vibrational levels reported by Rupper et
al. [53]. The constants of the I(3/2u) state were derived solely from the experimental data
recorded in this work. In addition, values for the dissociation energies D0+ were determined
from Eq. (2.25) using the atomic ionization energy of 97833.783 cm−1 corresponding to the
formation of the 2 P3/2 state of Xe+ [29] and the dissociation energy D0 = 186 cm−1 of the
X 0+
g ground state of Xe2 [52]. The spectroscopic constants summarized in Table 6.1 are
consistent with earlier results of Tonkyn et al. [171], Lu et al. [173] and Rupper et al. [53],
but represent a more complete set.
85
Electron signal / arb. units
Electron signal / arb. units
Chapter 6. The low-lying electronic states of Xe +
2 and their potential energy functions
(a)
(b)
(c)
(d)
107385
107395
-1
Wave number cm
107385
107395
-1
Wave number / cm
Figure 6.5: Upper traces: PFI-ZEKE photoelectron spectra of the II(1/2u) (v + = 5) state of different
0
+
isotopomers of Xe+
2 recorded via the v = 21 vibrational level of the C 0u state. Lower inverted traces:
Simulated spectra assuming that the ionic state has vibrational quantum number v + = 5 (full line) and
v + = 6 (dashed line). (a): 129 Xe129 Xe, (b): 129 Xe132 Xe, (c): 131 Xe132 Xe, (d): 132 Xe136 Xe.
6.4
6.4.1
The potential energy functions of the six low-lying
electronic states of Xe+
2
Potential model
The semiempirical potential model first presented in Ref. [162] for the determination of the
+
+
six lowest electronic potential functions of Ar+
2 and later used for Kr2 [163] and Xe2 [53]
was based, as all previous ab initio studies [355, 356, 358, 357, 164], on the approximation of an R-independent spin-orbit coupling constant introduced by Cohen and Schneider
[18]. The consequences of this approximation have never been quantified so far. Relativistic ab initio quantum chemical calculations carried out by R. Mastalerz in the group
of Prof. M. Reiher at ETH (see Ref. [361] for a discussion on the computational details)
allowed the investigation of the R-dependence of the spin-orbit coupling constant a(R)
by comparing the potential energy functions of the Πu and Πg states derived in nonrelativistic calculations with those for the I(3/2u) and I(3/2g) states determined in relativistic
calculations. According to the spin-orbit interaction matrix presented in Table 6.2 the
potential energy functions of the 2 Π state and the 2 Π3/2 state differ by a(R)/2. Thus a(R)
can be determined at each internuclear separation for which both calculations have been
performed by simple subtraction.
86
6.4. The potential energy functions of the six low-lying electronic states of Xe +
2
Table 6.1: Adiabatic ionization energies Ei , dissociation energies D0+ and vibrational constants ωe+ ,
+
131
ω e x+
Xe132 Xe+ .
e , ωe ye of the I(3/2g) and I(3/2u) states of
State
Ei /(hc cm−1 )
D0+ /cm−1
ωe+ /cm−1
−1
ω e x+
e /cm
I(3/2g)
96220.7(15)
96220.2(10)
96226(4)
96359(24)
1799.1(15)
1799.6(15)
1793(5)
1670(24)
≥ 1220(5) f
1694(40)
1492(121)
58.17(35)
58.61(27)
58.36(34)
0.456(26)
0.506(13)
0.484(15)
1815(40)
1565(40)
1876
1178
55.5(28)
55(4)
58.74
49.7
97577.6(15)
97576.6(10)
97582(4)
97214(24)
97576(2)
97617(40)
442.2(15)
443.2(15)
437(5)
807(24)
442(2)
403(40)
< 403
23.19(54)
21.89(40)
0.419(54)
23.1
0.55
97512(40)
645(40)
613(40)
582
306
33.0(17)
33(4)
26.53
58.5
96327(40)
96545(121)
96343(40)
96143
I(3/2u)
97437
≥ 45
a
ωe ye+ /cm−1
-0.00198(73)
f
a
This work b,c,d
[53] b,c,d
[173] d,e
[354] d,e
[171] d,e
[169] d,e
[170] d,e
[359]
[360]
[164]
[355]
0.508
0.3
0.425
0.3
Reference
0.0027(15)
87
e,g
e,h
e,h
This work b,c,d
[53] b,c,d,e
[173] d,e
[354] d,e
[171] d,e
[169] d,e
[170] d,e
[359]
[360]
[164]
[355]
ωe ze+ = −0.0000266(68) cm−1 .
The uncertainties in Ei and D0+ include the full width at half maximum of the observed transitions
and potential errors in the wave number calibration and in the determination of the field-induced shifts of
the ionization thresholds.
c
The uncertainty in the vibrational constants represent one standard deviation in the fit.
d
Experimental values.
e
Values correspond to a mixture of all isotopomers which is best represented by a fictive average
isotopomer of Xe2 with reduced mass µ = 65.646 u.
f
Tonkyn and White [171] were not able to derive an absolute vibrational assignment of the I(3/2g)
state.
g
Semiempirical values using known spectroscopic data.
h
Theoretical values.
b
e,g
e,g
e,g
e,h
e,h
Chapter 6. The low-lying electronic states of Xe +
2 and their potential energy functions
8000
7500
a / cm
-1
g states
u states
7000
6500
2
3
4
5
Internuclear distance / Å
6
7
Figure 6.6: R-dependent spin-orbit coupling constant a(R) for the u (circles) and the g (squares) states
derived from the comparison of the relativistic with the nonrelativistic potential energy curves derived by
ab initio calculations (upper pair of curves). The lower pair of curves are scaled such that the asymptotic
value exactly matches the spin-orbit splitting of the Xe+ 2 P ground state. The full lines represent fits
based on the expressions in Eqs. (6.1) and (6.2).
The upper sets of open squares and circles in Fig. 6.6 correspond to the values of a(R)
determined directly from the ab initio quantum chemical calculations. At large distances,
the calculated spin-orbit coupling constants tend asymptotically to a value larger then the
atomic value, both for the g and the u states. To correct for this artefact of the calculations,
the curves were scaled by a factor of 0.933 for the u states and 0.928 for the g states so
that the asymptotic value of a(R) exactly corresponds to the spin-orbit splitting of the
Xe+ 2 P ground state. The scaled ag/u (R) values are also displayed in Fig. 6.6 (lower sets
of open squares and circles). The spin-orbit coupling constant was found to vary by about
10% over the range of internuclear distance relevant for the present study. Consequently,
the R-dependence of the spin-orbit coupling constant must be included in a quantitative
description of the potential energy functions of the six low-lying electronic states of Xe +
2.
We found that the R-dependence of the scaled spin-orbit coupling constant a(R) of
the u and g states could be described almost perfectly by two rather simple analytical
expressions,
au (R) = aatom − au,0 · 1 − 1 − e−au,1 (R−au,2 )
ag,2
where aatom = aXe = 32 AXe
2 ,
(6.1)
ag (R) = aatom + ag,0 · e−ag,1 ·R ,
(6.2)
= 7024.617 cm−1 (AXe = E(Xe+ ,2 P1/2 ) − E(Xe+ ,2 P3/2 ) /hc)
[30]. Eq. (6.1) is a Morse-type function and Eq. (6.2) falls exponentially at increasing R.
88
6.4. The potential energy functions of the six low-lying electronic states of Xe +
2
The optimal parameters au,i and ag,i (i = 0, 1, 2) are listed in the lower part of Table 6.4
and the corresponding au (R) and ag (R) curves are displayed as full lines in Fig. 6.6.
The effects of the spin-orbit interaction are treated using the same coupling matrix
as proposed by Cohen and Schneider [18], improved by the R-dependence of the spinorbit coupling constant. The procedure consists of diagonalizing, at each value of the
internuclear distance R, the spin-orbit interaction matrix given in Table 6.2. Because
the spin-orbit operator only couples states of the same g/u symmetry, two distinct 3 × 3
matrices must be set up, one for the g states, the other for the u states. The spin-orbit
interaction matrix is expressed in the basis set adequate for the short-range part of the
potentials in which the quantum numbers Λ and Σ corresponding to the projection of
the total electronic orbital and spin angular momenta onto the internuclear axis are good
2
quantum numbers. Consequently, the electronic potential energies of the 2 Σ+
1/2 , Π1/2 and
2
Π3/2 states appear as diagonal elements. The spin-orbit operator is not only described by
diagonal contributions of ±a(R)/2 which induce a splitting of the 2 Π state into the two
√
components 2 Π1/2 and 2 Π3/2 , but also by off-diagonal elements −a(R)/ 2 which couple
states of the same value of Ω (the projection of the total angular momentum onto the
2
internuclear axis), namely the 2 Σ+
1/2 and Π1/2 states.
Table 6.2: Spin-orbit interaction matrix in Hund’s case (a) basis describing the coupling between the
states of 2 Σ+ and 2 Π symmetry in the homonuclear rare gas dimer ions.
2
Σ+
1/2
2
Σ+
1/2
VΣ (R)
2
Π1/2
2
√
− a(R)
2
Π3/2
2
2
Π1/2
√
− a(R)
2
VΠ (R) +
Π3/2
a(R)
2
VΠ (R) −
a(R)
2
The potential energy functions of the 2 Σ+ and 2 Π states under neglect of the spin-orbit
interaction are expressed as
VΛ (R) = AΛ e
−bΛ R
− BΛ e
−bΛ R/βΛ
−
3
X
n=2
f2n (R, bΛ )
C2n,Λ
+ Vdiss
R2n
(6.3)
with Λ = Σ, Π (see Section 2.6 for a detailed discussion of this potential energy function).
Because the 2 Σ+
g state is only weakly bound, the second term in Eq. (6.3), which describes
the chemical bond, is neglected (BΣg = 0). Vdiss is a constant used to relate the potential
00
energies to the energy of the X 0+
g (v = 0) ground neutral state and is defined as
2
1
Vdiss = D0 (Xe2 , X 0+
g ) + Ei (Xe, P3/2 ← S0 )/hc +
aatom
2
(6.4)
with aatom = aXe . The value of Vdiss = 101532.1 cm−1 is obtained from the literature
89
Chapter 6. The low-lying electronic states of Xe +
2 and their potential energy functions
values aatom = 23 AXe = 7024.617 cm−1 [30], Ei (Xe, 2 P3/2 )/hc = 97833.783 cm−1 [29] and
−1
D0 (Xe2 , X 0+
[52].
g ) = 186 cm
Because the 2 Σ+ and 2 Π potential energy curves are correlated with the Xe(1 S0 ) + Xe+ (2 P)
dissociation limit, the long-range behavior of these states is described in terms of the interaction between a neutral Xe atom and a Xe+ ion. The long-range coefficients are equal
for the u and the g states. Values for the C4 and C6 coefficients of Xe+
2 determined with
the formalism described in Section 2.6.1 are summarized in Table 6.3.
Table 6.3: All contributions to the C4 and C6 coefficients describing the long-range behavior of the Xe+
2
ion in its low-lying electronic states. See Section 2.6.1 for details.
Xe+ + Xe
interaction
charge-induced dipole
C4 / (Eh a0 4 )
charge-induced quadrupole
quadrupole-induced dipole
dispersion (isotropic part)
dispersion (anisotropic part)
total isotropic part of C6
total anisotropic part of C6
C6,0
C6,2
C6,0
C6,2
C6,0
C6,2
(Eh a0 6 )
(Eh a0 6 )
(Eh a0 6 )
(Eh a0 6 )
(Eh a0 6 )
(Eh a0 6 )
106.3
409.355
216.515
-6.80
322.815
402.555
C6 for Σ states
C6 for Π states
C6,Σ / (Eh a0 6 )
C6,Π / (Eh a0 6 )
483.837
242.304
/
/
/
/
/
/
13.6452
The potential energy functions Vj (R) of the three u and the three g states were then
calculated from the VΣu (R), VΠu (R), VΣg (R) and VΠg (R) potentials in a straightforward
manner using the interaction matrix presented in Table 6.2. The calculated transition wave
numbers were determined by numerically solving the radial Schrödinger equation
~2 d2
−
+ hc Vj (R) ψijv (R) = Eijv ψijv (R)
(6.5)
2µi dR2
corresponding to the vibrational motion of the isotopomer with reduced mass µi in the
potential Vj (R). In Eq. (6.5) the indices i, j and v designate the isotopomer, the electronic
state and the vibrational quantum number of the ion, respectively.
6.4.2
Fitting procedure
The parameters of the model potential energy functions (Eq. (6.3)) were derived from the
experimental data in a nonlinear least-squares fitting procedure. The numerical procedure
relied on a discrete variable representation (based on the Gauss-Chebyshev quadrature of
the 1st kind [350, 362, 363, 364]) on a grid with 801 equidistant grid points in the range of
internuclear distances R = 2.0 − 10.0 Å. In the fitting procedure, all parameters describing
90
6.4. The potential energy functions of the six low-lying electronic states of Xe +
2
the long-range interaction were kept fixed at the values listed in Table 6.3.
Table 6.4: Parameters of the interaction potentials of the lowest electronic states of Xe +
2 . The uncertainties represent 95 % confidence intervals. Parameters given without uncertainties were held fixed.
u states
2
Re,Λ /Å
De,Λ /cm−1
βΛ
bΛ /Å−1
C4,Λ /(cm−1 Å4 )
C6,Λ /(cm−1 Å6 )
AΛ /cm−1 c
BΛ /cm−1 c
a0 /cm−1
a1 /Å−1
a2
a,b
a,b
Σ+
u
g states
2
3.07789 ± 0.00030
10634.87 ± 0.62
1.29029 ± 0.00047
2.15150 ± 0.00041
234837
2331794
2.458 · 107
6.764 · 106
Πu
2
Σ+
g
2
4.60705 ± 0.00051
7.5 a
470.967 ± 0.088
4.0204 ± 0.0024
2.84295 ± 0.00053 1.1157 ± 0.0010
234837
234837
1167755
2331794
8
1.094 · 10
148617
1471
Πg
3.6960 ± 0.0022
1772.25 ± 0.82
1.5162 ± 0.0024
2.6963 ± 0.0019
234837
1167755
4.858 · 107
1.743 · 106
au (R)
ag (R)
597.2 a
1.566 a
2.359 Å a
1084.5 a
0.086 a
2.500 a
a
This parameter was kept constant during the fit.
Determined from Table 6.3 using 1 Eh = 219474.6 cm−1 , 1 a0 = 0.529177211 Å.
Λ (R) c
= 0 and VΛ (Re,Λ ) = Vdiss − De,Λ . See Appendix A in
Determined from the conditions dVdR
b
R=Re,Λ
Ref. [162].
u states
The u states are experimentally well characterized and the observed and unambiguously
assigned vibrational bands (59 bands for the I(1/2u) state, 21 bands for the I(3/2u) state,
and 12 bands for the II(1/2u) state) form a large data set with which the potential energy
functions of the three electronic states of u symmetry can be determined. A total of 8
parameters (Re,Λ , De,Λ , βΛ , bΛ ; Λ = Σ, Π) were refined in the fitting procedure. The
coefficients of au (R) were kept constant at the values derived from fitting the expression of
Eq. (6.1) to the scaled spin-orbit coupling constant (see Fig. 6.6). We have also performed
2
fits with 11 parameters (8 for the 2 Σ+
u and Πu states, and 3 for au (R)). The optimized
values for au,0 , au,1 and au,2 differed by less than 1 % from the scaled ab initio values
and no noticeable improvement of the quality of the fit resulted. The inital value for
the equilibrium internuclear distance of the 2 Σ+
u state was set to the ab initio value of
Re,Σu = 3.075 Å [361]. This value was supported by an independent measurement of the
91
Chapter 6. The low-lying electronic states of Xe +
2 and their potential energy functions
rotationally resolved photoelectron spectrum of the I(1/2u) state of Xe+
2 [365] and only
changed slightly in the fit (see Table 6.4).
The calculated positions of the vibrational transitions of the I(1/2u), I(3/2u) and
II(1/2u) states using the optimal parameter set summarized in Table 6.4 are listed in
Tables C.6, C.8 and C.10 in Appendix C where they are also compared with the experimental values. The root-mean-square (rms) deviation (see Eq. (5.16)) of the best fit
amounted to 1.04. This number implies that the quality of the fit is perfect and that all
of the information contained in the experimental data could be extracted [351]. The differences between observed and calculated vibrational positions were smaller than 5 cm −1
with the exception of the v + = 11 level of the II(1/2u) state for which the difference
amounts to 5.7 cm−1 . Looking at the behavior of the discrepancies for the II(1/2u) state
(see Table C.10 in Appendix C) one recognizes that these are much smaller for the vibrational transitions observed in this work (v + = 0 − 9) than for those taken from Ref. [53]
(v + = 10 − 11). A reanalysis of the data of Ref. [53] showed that the values for the vibrational levels v + = 8 − 11 of the II(1/2u) state do not correspond to 131 Xe132 Xe but to
another isotopomer of Xe+
2 , a conclusion which could only be drawn with the knowledge
of the spectroscopic information derived in the present study.
g states
Of the three low-lying gerade electronic states only the I(3/2g) state (48 vibrational transitions) is well characterized experimentally. Because of the weakly bound nature of the
2
Σ+
g state this state is only described by two parameters (R e,Σg and bΣg ). The most recent
ab initio quantum chemical calculations suggest that the 2 Σ+
g state is repulsive in the internuclear distances probed by this experiment [361]. Therefore, and somewhat arbitrarily,
we set the value of the equilibrium internuclear distance of this state to the large value of
Re,Σg = 7.5 Å and kept it fixed during the fitting procedure. Because of the lack of data
on the I(1/2g) and II(1/2g) states the coefficients ag,i (R) (i = 0, 1, 2) were held constant
at the values derived from fitting the expression of Eq. (6.2) to the scaled ab initio a g (R)
2
curve (see Fig. 6.6). A total of 5 parameters (1 for the 2 Σ+
g and 4 for the Πg states) were
refined for the g states.
The optimal parameter set for the description of the potential energy functions of the
three g states is summarized in Table 6.4. The rms deviation of the best fit amounted to
0.41. The low value of this deviation must be interpreted with some caution. Indeed, the
experimental data set does not contain enough information on the I(1/2g) and II(1/2g)
states. The adjustable parameter of the Σ curve suffices to place the position of the only
observed level of the II(1/2g) state at its observed position. The agreement between the
calculated and observed positions of this level can thus not be considered as a proof of the
assignment. The calculated positions of the vibrational levels of the I(3/2g), I(1/2g) and
92
6.4. The potential energy functions of the six low-lying electronic states of Xe +
2
II(1/2g) states are listed in Tables C.7, C.9 and C.11 in Appendix C. A comparison of the
calculated positions of the I(3/2g) state with those from the experiment shows that the
maximum absolute deviation amounted to −1.6 cm−1 for the v + = 11 level.
6.4.3
Potential energy functions
The potential energy functions of the three u and the three g states of Xe+
2 derived from the
2 + 2
2 +
2
Σu , Πu , Σg and Πu potential energy curves and the R-dependent spin-orbit coupling
matrix displayed in Table 6.2 are presented in Figs 6.7 and 6.8. The dissociation energies
and equilibrium internuclear distances of all six states are summarized and compared with
values from other studies in Table 6.5.
Table 6.5: Dissociation energies De and equilibrium internuclear distances Re of the lowest electronic
states of Xe+
2 determined from the potential energy curves displayed in Figs. 6.7 and 6.8.
I(1/2u)
I(3/2g)
I(3/2u)
I(1/2g)
II(1/2u)
II(1/2g)
Reference
De /cm−1
7937.4
7932.7
7904
7178
8691
8646
6372
1829.5
1828.8
1905
1532
1163
1202
968
453.9
455.7
595
403
527
339
242
72.9
147.8
162
78
44
repulsive
repulsive
1419.8
1430.2
1602
1371
1690
1686
968
436.0
436.1
368
242
307
repulsive
61
This work
[53]
[164]
[357]
[149]
[355]
[356]
Re /Å
3.118
3.034
3.114
3.17
3.196
3.18
3.27
3.683
3.674
3.695
3.19
3.979
3.91
3.97
4.632
4.390
4.395
4.76
4.382
4.00
4.76
6.096
5.437
5.774
6.35
5.260
repulsive
repulsive
4.130
3.985
3.983
4.13
4.059
3.97
4.23
4.199
4.448
4.773
5.29
4.620
repulsive
5.29
This work
[53]
[164]
[357]
[149]
[355]
[356]
Potential energy functions of the u states
The potential energy functions of the u states derived from the global fit are displayed
as full lines in Fig. 6.7. Compared to the results of our previous study [53] (dashed lines
in Fig. 6.7) the dissociation energies of the three u states are almost equal whereas the
equilibrium internuclear distances have shifted to larger values. The increased bond lengths
are a result of a new rotationally resolved measurement of the low-lying electronic states of
Xe+
2 [365]. The potential energy function of the I(1/2u) state is in excellent agreement with
93
Chapter 6. The low-lying electronic states of Xe +
2 and their potential energy functions
that reported by Paidarová and Gadéa [164]. At internuclear distances smaller than ∼ 4 Å
the potential energy curves of the I(3/2u) and II(1/2u) states derived here are located
higher in energy than those reported earlier [164, 53] whereas the long-range behavior
is almost identical. This discrepancy is likely to originate from the larger equilibrium
2
internuclear distances of the 2 Σ+
u and Πu states and from the R-dependent spin-orbit
coupling constant. The effect of the R-dependence of a is further discussed in Section 6.5.
20000
-1
Potential energy / (hc cm )
15000
II(1/2u)
10000
5000
I(3/2u)
0
I(1/2u)
-5000
3
4
5
6
Internuclear distance / Å
7
8
Figure 6.7: Comparison of the potential energy functions of the I(1/2u), I(3/2u) and II(1/2u) states
of Xe+
2 derived in this work (full line), in Ref. [53] (dashed line) and in Ref. [164] (circles). The dashed
horizontal lines indicate for each state the positions of the highest observed vibrational level.
Potential energy functions of the g states
The potential energy functions of the g states derived from the global fit are displayed as
full lines in Fig. 6.8. The most remarkable feature in this figure is related to the potential
energy function of the I(1/2g) state which is only very weakly bound and gets its character
mainly from the repulsive 2 Σ+
g state. The low dissociation energy and the large internuclear
distance could explain why this state has not been observed in spectroscopic experiments
so far. Whereas the dissociation energy and the equilibrium internuclear distance of the
I(3/2g) state are comparable to those determined in our previous study the equilibrium
internuclear distance of the II(1/2g) state has shifted to lower values. Because of the
scarce experimental data on the g states, these potential energy functions strongly rely on
the value of the equilibrium internuclear distance of the 2 Σ+
g state suggested by ab initio
quantum chemistry.
94
6.5. Conclusion
-1
Potential energy / (hc cm )
15000
II(1/2g)
10000
5000
I(1/2g)
0
I(3/2g)
3
4
5
6
Internuclear distance / Å
7
8
Figure 6.8: Comparison of the potential energy functions of the I(1/2u), I(3/2u) and II(1/2u) states
of Xe+
2 derived in this work (full line), in Ref. [53] (dashed line) and in Ref. [164] (circles). The dashed
horizontal line indicates for the I(3/2g) state the position of the highest observed vibrational level.
6.5
Conclusion
The potential energy functions of the six low-lying electronic states of Xe+
2 presented in
this chapter were derived, for the first time, by explicitly considering the R-dependence
of the spin-orbit coupling constant as predicted by recent ab initio quantum chemistry
[361]. The effect of the R-dependence is best manifested by comparing the potential energy functions of the I(1/2u), I(3/2u) and II(1/2u) states calculated with an R-dependent
spin-orbit coupling constant au (R) with those calculated with an R-independent spin-orbit
coupling constant a = aatom using the same Σ and Π potential energy curves (corresponding to the parameter set for the u states in Table 6.4). These potential energy curves
and the corresponding energy differences are displayed in Fig. 6.9. The influence of the
R-dependence gets significant at internuclear distances smaller than ∼ 5 Å, where the deviation of the value of the spin-orbit-coupling constant from the atomic value becomes visible
(see Fig. 6.6). Compared to the potential energy function of the I(3/2u) state calculated
with a = aatom , that calculated with au (R) is shifted towards higher energies by ∼ 50 cm−1
in the internuclear distance probed by the experiment, whereas that for the II(1/2u) state
is shifted to lower energies by 60 − 110 cm−1 . The fact that the R-dependence of a has
a larger effect on the I(3/2u) and II(1/2u) potential energy curves than on the I(1/2u)
function can be explained by the strongly bound nature of the 2 Σ+
u curve and the fact
that, below 4.5 Å, this curve is separated from the Π curve by more than 3000 cm−1 .
The rms value of 1.04 for the best fit of the u states of Xe+
2 implies that all the in95
-400
(c)
0
(f)
-100
-800
-200
-1200
-300
0
(b)
(e)
200
-200
100
-400
0
-2000
(a)
0
80
(d)
60
-4000
40
-6000
20
-8000
3
4
5
Internuclear distance / Å
6
3
4
5
Internuclear distance / Å
6
-1
0
Energy difference / (hc cm )
-1
Potential energy relative to the dissociation limit / (hc cm )
Chapter 6. The low-lying electronic states of Xe +
2 and their potential energy functions
0
Figure 6.9: Left panels: Potential energy functions of the I(1/2u) (a), I(3/2u) (b) and II(1/2u) (c)
states of Xe+
2 calculated with R-dependent au (R) (full lines) and R-independent a = aatom (dashed lines)
using the same Σ and Π potential energy curves (parameter set for the u states in Table 6.4). The dashed
horizontal lines indicate for each state the positions of the highest observed vibrational level. Right panels:
Energy difference of the two curves displayed in the left panels for the I(1/2u) (d), I(3/2u) (e) and II(1/2u)
(f) states of Xe+
2.
formation on the potential energy functions contained in the experimental data could be
extracted in the fitting procedure. The experimental data on the g states are much less extensive and the potential model appears to contain too many adjustable parameters leading
to a rms value of 0.41 (see discussion in Section 6.4.2). The quality of the potential energy
functions of the six low–lying electronic states of Xe+
2 could be improved by rotationally
resolved spectroscopic investigations of these states which would also provide independent
information on the internuclear separations. In addition, measurements of the two (1/2g)
states would be dersirable. At present the experimental data on the g states are too sparse
for the determination of reliable potential energy functions.
The potential energy functions derived in this chapter reproduce the positions of all
experimentally observed vibrational levels within their statistical uncertainties. This ex+
cellent agreement, which could not be reached in Ar+
2 [162] and Kr2 [163], is attributed
to the inclusion, in the model, of the R-dependence of the spin-orbit coupling constant.
The calculated R-dependence of the g and u functions can be well described by an exponentially decaying function (Eq. (6.2)) and a Morse-type function (Eq. (6.1)), respectively.
+
Preliminary results on Ar+
2 and Kr2 suggest that these functions also describe a(R) in
these systems and thus have a certain universality.
96
Chapter 7
The low-lying electronic states of
ArXe+ and KrXe+ and their
potential energy functions
7.1
Introduction
Considerable effort has been invested in the studies of the excited Rydberg states of ArXe
and KrXe [107, 108, 129, 111, 112, 133, 113, 114, 134, 136, 115, 116, 118, 117, 119, 121,
16, 122, 123, 124, 139, 140, 125] because these states play an important role in excimer
lasers and rare gas ion lasers [3, 4, 5]. The simplest description of the excited electronic
states of RgXe (Rg=Ar, Kr) is in terms of an RgXe+ dimer ion core and a nonbonding,
weakly interacting electron [75]. In this description, the potential energy curves of each
excited state of the neutral resembles that of its associated ion core. The knowledge of
the potential energy curves of the six low-lying electronic states of RgXe+ represents the
starting point for a description of the potential energy curves of the Rydberg states of
RgXe.
While many experimental and theoretical studies describing the properties of the lowlying electronic states of the homonuclear rare gas dimer cations have been carried out (see
Sections 2.5.1 and 6.1) little is known about the corresponding states of the heteronuclear
rare gas dimer cations (see Section 2.5.2). Emission spectra of mixtures of Ar and Xe
excited in a discharge tube [178] or using electron beam (∼ 50 keV) bombardment [180, 181]
showed two bands in the intervals 5120 − 5507 Å and 4960 − 5081 Å which belong neither to
the Ar nor to the Xe emission spectrum. With the same technique, mixtures of Kr and Xe
were investigated which led to the observation of a new band in the interval 4600 − 4950 Å
[179, 181]. Tanaka et al. [182] excited all ten binary mixtures of He, Ne, Ar, Kr and
Xe in various types of electric discharges and investigated the resulting emission spectra.
They could detect emission from all mixtures, except from Kr/Xe, and were the first to
attribute the emission bands observed earlier [178, 179, 180, 181], to a charge exchange
transition of the heteronuclear rare gas ions YZ+ (Y + Z+ ) → (Y+ + Z), where Z represents
97
Chapter 7. The low-lying electronic states of ArXe + and KrXe+
the lighter of the two rare gases. In addition to the two known bands in ArXe, they
observed three new bands, labeled the five bands as A, B, C, D, E in order of increasing
wavelength and assigned them to the following transitions of ArXe+ : C2 1/2 → X 1/2
(A), B 1/2 → X 1/2 (B), C1 3/2 → A1 3/2 (C), C2 1/2 → A2 1/2 (D), B 1/2 → A2 1/2
(E). These five emission bands and fragments of a sixth, labeled as A0 and assigned to the
transition C2 1/2 → A1 3/2, were also observed in the emission spectrum of ArXe+ in a DC
discharge by Huber and Lipson [197]. These authors reported the vibrational analysis of
the six bands and extracted values for the dissociation energies of all six low-lying electronic
states of ArXe+ . Tsuji et al. [199] observed three emission bands (C2 1/2 → X 1/2 (A),
B 1/2 → X 1/2 (B), C1 3/2 → A1 3/2 (C)) of KrXe+ produced from Kr afterglow reactions
with Xe. On the basis of a vibrational analysis they also extracted the molecular constants
of the A1 3/2 and C1 3/2 states of KrXe+ .
Ng et al. [201] and Dehmer and Pratt [202] recorded photoionization spectra of ArXe
and KrXe in the energy region from the first adiabatic ionization potentials at ∼ 11.97 eV
for ArXe and ∼ 11.76 eV for KrXe to the Xe(1 S0 ) + Rg+ (2 P1/2 ) dissociation limits at
∼ 15.9 eV for Rg=Ar and ∼ 14.7 eV for Rg=Kr. They also reported values for the adiabatic
ionization energies of ArXe and KrXe and the dissociation energies of the X 1/2 and A 2 1/2
states of ArXe+ and the X 1/2, A2 1/2 and C2 1/2 states of KrXe+ . In the spectral region
between the two Xe+ (2 PJ , J = 3/2, 1/2) + Ar(1 S0 ) Xe(1 S0 ) + Kr+ (2 PJ , J = 3/2, 1/2) ,
Rydberg series converging to the A2 1/2 C2 1/2 state of the ion were also observed [202].
The first photoelectron spectra of the X 1/2, A1 3/2 and A2 1/2 states of ArXe+ and the
A1 3/2 state of KrXe+ were recorded by Pratt et al. [203, 204] using REMPI. Because the
resolution of their electron spectrometer was ∼ 40 meV, no vibrational structure could be
resolved. Vibrationally resolved threshold photoelectron-photoion coincidence (TPEPICO)
spectra of all six low-lying electronic states of ArXe+ and KrXe+ were later measured
by Yoshii et al. [206, 207] at a resolution of 0.05 − 0.10 Å (∼ 1 meV) using synchrotron
radiation. In the case of the B 1/2 state of ArXe+ , their vibrational assignment differed
from that proposed by Huber and Lipson [197]. Indeed, Yoshii et al. [207] detected two
additional peaks in the B 1/2 ← X 0+ transition, assigned the two new peaks to transitions
to the v + = 0 and v + = 1 levels of the B 1/2 state and therefore increased the vibrational
numbers proposed in Ref. [197] by 2.
Hausamann and Morgner [196] have developed a semiempirical model for the calculation of the potential curves of the heteronuclear rare gas dimer cations. The model relies on
experimental and theoretical data on the homo- and heteronuclear rare gas dimer cations
and explicitly considers the Pauli repulsion, the charge-exchange, and the spin-orbit interactions. However, the simultaneous treatment of these interactions was based on an
incorrect angular momentum transformation (see Section 7.9.1).
We present here a new determination of the potential energy functions of the six lowlying electronic states of ArXe+ and KrXe+ by combining an improved global model
98
7.2. Experiment
for the potential energy functions of these states with new informations contained in
high-resolution PFI-ZEKE photoelectron spectra and with experimental data reported
previously in the literature [197, 206, 207]. A (1VUV + 10VIS/UV ) two-photon excitation
scheme via selected intermediate vibrational levels of the C 1 and D 0+ Rydberg states
of ArXe and the C 0+ and D 1 Rydberg states of KrXe located in the vicinity of the
Xe∗ ([5p]5 6s0 [1/2]1 ) + Rg(1 S0 ) (Rg=Ar, Kr) dissociation limit has been used to access the
three lowest ionic levels. Transitions to the higher lying B 1/2 and C1 3/2 states of KrXe+
were only observable via selected vibrational levels of a Rydberg state located in the vicinity of the Kr∗ ([4p]5 5s[3/2]1 ) + Xe(1 S0 ) dissociation limit. With this resonance-enhanced
two-photon excitation scheme, which is species selective, undesirable contributions to the
spectra originating from the ionization of the corresponding free atoms and homonuclear
rare gas dimers could be avoided.
7.2
Experiment
The Rydberg states of ArXe and KrXe associated with the Xe∗ ([5p]5 6s0 [1/2]1 ) + Rg(1 S0 )
(Rg=Ar, Kr) and Kr∗ ([4p]5 5s[3/2]1 ) + Xe(1 S0 ) dissociation limits were investigated by
(1VUV + 10VIS/UV ) REMPI spectroscopy. The characterization of the ionic states was performed by PFI-ZEKE photoelectron spectroscopy using a resonant two-photon excitation
sequence via selected vibrational levels of these Rydberg states. The experimental setup
has been described in Chapter 3 and illustrated in Figs. 3.2 and 3.3. In this section only
the aspects relevant to the investigation of ArXe and KrXe are summarized in detail.
The VUV radiation required to drive transitions from the X 0+ ground neutral state of
ArXe and KrXe to their low-lying Rydberg states was produced in the four-wave mixing cell
(see Fig. 3.3) by resonance-enhanced difference-frequency mixing using the 4p5 5p[1/2]0 ←
4p6 (1 S0 ) two-photon resonance in krypton at 2ν̃1 = 94092.86 cm−1 . RgXe rare gas dimers
were generated in a pulsed supersonic expansion of a 20:1 (10:1) mixture of Ar (Kr) and
Xe held at a stagnation pressure of ∼ 2.5 bar through a solenoid valve with a diameter of
0.4 mm.
REMPI spectra were recorded by scanning the wave number ν̃VUV = 2ν̃1 − ν̃2 keeping
the wave number ν̃3 of a third dye laser fixed so that the wave number of the sum ν̃VUV + ν̃3
lay above the first adiabatic ionization threshold of RgXe (Rg=Ar, Kr). The generated ions
were extracted by applying an electric field pulse of amplitude −520 V cm−1 and duration
950 ns delayed by 500 ns with respect to the VUV laser pulse and detected mass selectively
at the MCP detector placed at the end of the time-of-flight (TOF) tube. TOF mass
spectra illustrating the distribution of the different isotopomers of ArXe+ and KrXe+ are
displayed in Figs. 7.1 and 7.2, respectively. The intensity distribution corresponds closely
to the relative abundances of the isotopomers occurring in a natural mixture of Ar and Xe
and Kr and Xe, respectively. The sensitivity of the experiment was sufficient to observe
99
Xe (26.78%)
Xe (21.10%)
Ar
132
131
Xe (26.33%)
Ar
134
136
Ar
40
Ar
40
40
Ar
130
Xe (1.91%)
128
Ar
40
13.2
Time / µs
Xe (4.06%)
(x 85)
Xe (8.83%)
Xe (10.40%)
40
40
Ar
40
129
Xe (0.10%)
126
Ar
40
40
Ar
Ion signal / arb. units
124
Xe (0.10%)
Chapter 7. The low-lying electronic states of ArXe + and KrXe+
0
164
166
13.2
168
13.4
169
170
171
Time / µs
172
13.6
176
174
13.8
Figure 7.1: Time-of-flight (TOF) mass spectrum of ArXe+ recorded following (1VUV + 10VIS ) resonance-enhanced two-photon ionization via the D 0+ (v 0 = 1) state of ArXe with ν̃3 = 20001.4 cm−1 . The
mass resolution of the spectrometer was sufficient to distinguish between isotopomers differing in mass by
1 u. Different isotopomers with the same mass number could not be resolved. For each mass the most
abundant isotopomer is indicated above the corresponding TOF peak. The inset shows the ion signal
originating from the isotopomers with masses 164 u and 166 u on an enhanced scale (× 85) and illustrates
the sensitivity of the present experiments.
even minor isotopomers such as 40 Ar124 Xe with a natural abundance of only ∼ 0.1 % (see
inset of Fig. 7.1). REMPI spectra of selected isotopomers were recorded by setting time
gates at the desired TOF positions and monitoring the corresponding ion currents as a
function of the VUV excitation wave number.
The PFI-ZEKE photoelectron spectra of ArXe and KrXe were measured by detecting
the electrons produced by delayed pulsed field ionization (using a sequence of two successive
electric field pulses) of very high Rydberg states located below the vibronic states of RgXe +
(Rg=Ar, Kr) as a function of the wave number ν̃3 of a tunable dye laser. A (1VUV +10VIS/UV )
excitation scheme was used in which the wave number ν̃VUV of the VUV radiation was
kept fixed at a position corresponding to a transition from the X 0+ ground neutral state
of RgXe to a selected vibrational level of the intermediate Rydberg states. The spectral
resolution was insufficient to observe the rotational structure of the REMPI and PFI-ZEKE
photoelectron spectra. Single isotopomers of ArXe and KrXe could be selected whenever
the isotopic shifts of the vibrational bands of the C 1, D 0+ ← X 0+ transitions of ArXe and
the C 0+ , D 1 ← X 0+ transitions of KrXe exceeded the widths of the rotational envelopes
of the vibrational bands.
100
210
15.8
211
Xe (5.05%)
136
136
136
Kr
Kr
213
214
215
216
16.2
Time / µs
217
218
16.4
219
86
83
212
16
Xe (1.53%)
Kr
Kr
Xe (1.02%)
131
86
84
134
Kr
84
Xe (3.67%)
Xe (5.95%)
Xe (15.33%)
84
Kr
132
Xe (3.12%)
Kr
132
82
Xe (3.04%)
129
Kr
83
Xe (3.06%)
129
Kr
128
Kr
82
0
16
Time / µs
Xe (0.22%)
15.8
82
Ion signal / arb. units
84
Kr
84
131
Kr
129
(x 14)
Xe (12.10%)
Xe (15.04%)
7.3. The C and D Rydberg states of ArXe and KrXe
220
222
16.6
Figure 7.2: Time-of-flight (TOF) mass spectrum of KrXe+ recorded following (1VUV + 10VIS ) resonance-enhanced two-photon ionization via the D 1 (v 0 = m + 1) state of KrXe with ν̃3 = 19609.2 cm−1 .
The mass resolution of the spectrometer was sufficient to distinguish between isotopomers differing in mass
by 1 u. Different isotopomers with the same mass number could not be resolved. For each mass the most
abundant isotopomer is indicated above the corresponding TOF peak. The inset shows the ion signal
originating from the isotopomers with masses 210 u and 211 u on an enhanced scale (× 14).
7.3
The C and D Rydberg states of ArXe and KrXe
The C and D Rydberg states of ArXe and KrXe located in the vicinity of the
Xe∗ ([5p]5 6s0 [1/2]1 ) + Rg(1 S0 ) (Rg=Ar, Kr) dissociation limits were chosen as intermediate states in the (1VUV + 10VIS/UV ) two-photon excitation scheme because of their long
lifetimes and because spectroscopic information was already available in the literature
[107, 111, 114, 118]. Freeman et al. [107] were the first to obtain VUV absorption spectra of ArXe and KrXe. They identified several band systems, among others transitions
to the C and D Rydberg states. The two vibrational progressions of the C ← X 0+ and
D ← X 0+ transitions were further studied by laser induced fluorescence (LIF) and REMPI
spectroscopy by Tsuchizawa et al. [114] who determined the symmetries (C 1, D 0+ ) of the
states which were subsequently confirmed for ArXe but reversed to C 0+ and D 1 for KrXe
by Mao et al. [118] in two-photon-excitation experiments using linearly and circularly polarized laser radiation. The latter symmetry assignment was further supported by ab initio
calculations [139]. Irregularities in the vibrational level spacings of the C 1 state of ArXe
were attributed to a perturbation of this state by a repulsive electronic state of Ω = 1
symmetry associated with the Xe∗ ([5p]5 6s[3/2]1 ) + Ar(1 S0 ) dissociation limit [114, 121] or
101
Chapter 7. The low-lying electronic states of ArXe + and KrXe+
to interactions with several electronic states associated with the 6s and 6p dissociation
channels [136]. Hickman et al. [136] also suggested that the D 0+ state of ArXe has a
double well potential.
7.3.1
The C 1 and D 0+ Rydberg states of ArXe located in the
vicinity of the Xe∗ (6s0 [1/2]1 ) + Ar(1 S0) dissociation limit
The REMPI spectrum of the C 1, D 0+ ← X 0+ transitions of 40 Ar129 Xe recorded in this
work is presented in Fig. 7.3. Line positions (see Table C.12 in Appendix C) and intensities
match very well with the REMPI spectrum of Tsuchizawa et al. [114]. Because of the
perturbations it was not possible to unambiguously assign the vibrational levels of the C 1
state on the basis of the isotopic shifts. We have adopted the vibrational assignment first
proposed by Tsuchizawa et al. [114] and later also used by Mao et al. [118] and Liu et al.
[121]. The assignment of the origin of the D 0+ progression to the line at ∼ 77200 cm−1
proposed by Tsuchizawa et al. [114] was confirmed by the analysis of the isotopic shifts and
the intensity distribution (see below). The vibrational numbering of the two progressions
are indicated along the assignment bars in Fig. 7.3.
C1
2
3
4
5
6
7
Ion signal / arb. units
v’ = 1
0
v’ = 0
77100
77150
-1
Wave number / cm
77200
1
D0
+
2
3
77250
Figure 7.3: REMPI spectrum of the C 1 and D 0+ Rydberg states of 40 Ar129 Xe located in the vicinity
of the Xe∗ ([5p]5 6s0 [1/2]1 ) + Ar(1 S0 ) dissociation limit at 77302.34 cm−1 .
The Franck-Condon factors displayed in Fig. 7.4 for the D 0+ ← X 0+ transition were
calculated using the neutral ground state potential energy function derived by Tang and
102
7.3. The C and D Rydberg states of ArXe and KrXe
Toennies [40] and a model potential energy function
VD (R) = A e−bR − f3 (R, b)
C3
R3
(7.1)
for the D 0+ state which we assume is valid in the range of energy and internuclear distance
probed by our experiment. The equilibrium internuclear separation of the D 0+ state was
varied until good agreement with the measured intensity distribution was reached. The
first and second term of Eq. (7.1) describe the short-range part and the leading term of
the long-range expansion, respectively, and f3 (R, b) represents a Tang-Toennies damping
function [212]. The parameters of the model potential were derived from the experimental
line positions in a nonlinear least-squares fitting procedure, in which the transition wave
numbers were determined by numerically solving the radial Schrödinger equation (see Section 7.9). Comparison of calculated Franck-Condon factors with the measured intensity
distribution suggests that the equilibrium internuclear distance of the D 0+ state lies in
the range between 3.73 and 3.77 Å which is in good agreement with the value of 3.77 Å
derived by Tsuchizawa et al. [114] who used a similar method, but a different potential
energy function for the ground neutral state. The potential parameters extracted for the
D 0+ state assuming an equilibrium internuclear distance Re = 3.75 Å are summarized in
Table 7.1.
Table 7.1: Parameters of the potential energy function of the D 0+ state of ArXe. The uncertainties
represent 95 % confidence intervals. Parameters given without uncertainties were held fixed.
Re /Å
a
3.75
a
b
c
b/Å−1
C3 /(cm−1 Å3 )
8.67 ± 1.04
6662 ± 30
A/cm−1
b
1.574·1015
De /cm−1
c
114.7
This parameter was kept constant during
the fit.
dVD (R) = 0. See Appendix A in Ref. [162].
Determined from the condition dR R=Re
Determined from the condition De = −VD (Re ).
The vibrational levels C 1 (v 0 = 6), D 0+ (v 0 = 0) and D 0+ (v 0 = 1) were selected as
intermediate states in the (1VUV + 10VIS/UV ) two-photon excitation sequence because they
enabled us to access distinct regions of the ionic potential curves as a result of different
Franck-Condon factors. The spectral positions of the different isotopomers could not be
fully resolved for any of these three intermediate levels so that all PFI-ZEKE photoelectron
spectra presented in Sections 7.5.1 and 7.6.1 contain contributions from more than one
isotopomer of ArXe.
103
Chapter 7. The low-lying electronic states of ArXe + and KrXe+
Re = 3.70 Å
Re = 3.73 Å
Re = 3.75 Å
Re = 3.77 Å
Ion signal / arb. units
Re = 3.80 Å
0
v’ = 0
77200
1
77210
77220
77230
-1
Wave number / cm
2
77240
3
77250
Figure 7.4: The intensity distribution of the REMPI spectrum is compared with Franck-Condon factors
for the D 0+ ← X 0+ transition calculated for different equilibrium internuclear distances.
7.3.2
The C 0+ and D 1 Rydberg states of KrXe located in the
vicinity of the Xe∗ (6s0 [1/2]1 ) + Kr(1 S0) dissociation limit
The REMPI spectrum of the C 0+ , D 1 ← X 0+ transitions of 84 Kr132 Xe recorded in this
work is presented in Fig. 7.5. The line positions (see Table C.19 in Appendix C) are in
excellent agreement with those determined by Tsuchizawa et al. [114] and Mao et al. [118],
but the vibrational assignment of the D 1 state probably needs to be revised: Analyzing the
isotopic shifts of the D 1 ← X 0+ transition leads to the conclusion that the lowest observed
level (v 0 = m) of the D 1 state in Fig. 7.5 is unlikely to be v 0 = 0 [114] or v 0 = 1 [118] (see
inset of Fig. 7.5). Unfortunately, an unambiguous assignment of the vibrational levels of
the D 1 state on the basis of the isotopic shifts was not possible because of the irregular
vibrational level spacings. The lines marked by asterisks in Fig. 7.5 could originate from
vibrational levels of a Rydberg state associated with another dissociation limit or from
other vibrational levels of the D 1 state if this state has a double well potential function.
The analysis of the isotopic shifts of the C 0+ state confirmed the vibrational assignment
first proposed by Tsuchizawa et al. [114] and later also used by Mao et al. [118].
The v 0 = 19 vibrational level of the C 0+ state was selected as intermediate state in
the (1VUV + 10VIS/UV ) two-photon excitation sequence because the isotopic shifts exceeded
the spectral widths of the rotational envelopes of the vibrational bands which permitted the recording of isotopomer-selective PFI-ZEKE photoelectron spectra. Using the
D 1 (v 0 = m + 1) vibrational level as intermediate state made a broader region of the
104
7.4. The Rydberg state of KrXe located in the vicinity of the Kr ∗ ([4p]5 5s[3/2]1 ) + Xe(1 S0 )
dissociation limit
Ion signal / arb.units
Ion signal / arb.units
D 1 (v’ = m)
C0
+
84
129
84
131
84
132
Kr
Xe
Kr
Xe
Kr
Xe
D1
0
77288
77292
Wave number / cm
-1
77296
*
0
v’ = 16
77000
21
77050
77100
77150
77200
77250
-1
Wave number / cm
* *
*
v’ = m
77300
m+3
77350
Figure 7.5:
REMPI spectrum of the C 0+ and D 1 Rydberg states of 84 Kr132 Xe located in the
vicinity of the Xe∗ ([5p]5 6s0 [1/2]1) + Kr(1 S0 ) dissociation limit at 77336.34 cm−1 . The inset shows the
D 1(v 0 = m) ← X 0+ transition for 84 Kr129 Xe, 84 Kr131 Xe and 84 Kr132 Xe on an enhanced scale.
ionic potential curves accessible as a result of different Franck-Condon factors. The spectra recorded through this intermediate level contain contributions from more than one
isotopomer because the vibrational bands of the different isotopomers overlapped in the
REMPI spectra.
7.4
The Rydberg state of KrXe located in the vicinity of the Kr∗([4p]55s[3/2]1) + Xe(1S0) dissociation
limit
The three electronic states of KrXe+ associated with the Xe(1 S0 ) + Kr+ (2 PJ , J = 3/2, 1/2)
dissociation limits could not be observed from the C 0+ and D 1 intermediate levels of KrXe.
To study these states a Rydberg state located in the vicinity of the Kr∗ ([4p]5 5s[3/2]1 ) +
Xe(1 S0 ) dissociation limit was chosen as intermediate level in the resonant two-photon
excitation sequence. This state was first observed by Freeman et al. [107] in vacuum
ultraviolet absorption spectra. A vibrationally resolved fluorescence spectrum was reported
by Balakrishnan et al. [113].
The REMPI spectrum of the transitions of KrXe from the ground neutral state to this
Rydberg state recorded in the present work is displayed in Fig. 7.6. The time gate placed
for integration in the TOF trace was set such that the TOF positions of all isotopomers
105
Chapter 7. The low-lying electronic states of ArXe + and KrXe+
were covered, and thus all isotopomers contributed to the observed spectrum. The line
positions (see Table C.20 in Appendix C) correspond closely to those derived from the fluorescence spectrum [113]. Although we measured mass selective spectra, an unambiguous
assignment of the vibrational levels of this Rydberg state by analyzing the isotopic shifts
was not possible. Moreover, the symmetry of this state remains unknown. Possible labels
derived for a transition from the X 0+ ground state of KrXe to states associated with the
Kr∗ ([4p]5 5s[3/2]1 ) + Xe(1 S0 ) dissociation limit are 0+ and 1.
The v 0 = n + 6 vibrational level was selected as intermediate state in the (1VUV + 10UV )
two-photon excitation sequence state because it led to the strongest ionization signal when
integrating over all contributing isotopomers. The spectral positions of the vibrational levels of the different isotopomers could not be fully resolved so that more than one isotopomer
Ion signal / arb.units
contributed to the PFI-ZEKE photoelectron spectrum.
0
n+5
v’ = n
81010
81020
n+10
81030
81040
81050
-1
Wave number / cm
n+15
81060
81070
Figure 7.6:
REMPI spectrum of the Rydberg state of KrXe located in the vicinity of the
∗
5
Kr ([4p] 5s[3/2]1) + Xe(1 S0 ) dissociation limit at 81068.07 cm−1 .
7.5
The X 1/2 and A1 3/2 states of ArXe+ and KrXe+
The PFI-ZEKE photoelectron spectra of the transitions to the two lowest electronic states
X 1/2 and A1 3/2 of ArXe+ recorded from the D 0+ (v 0 = 0), D 0+ (v 0 = 1) and
C 1 (v 0 = 6) intermediate levels are presented in Fig. 7.7 and those of KrXe+ recorded
from the C 0+ (v 0 = 19) and D 1 (v 0 = m + 1) intermediate levels are displayed in Fig. 7.11.
The intensity distributions strongly depend on the selected intermediate levels because of
106
7.5. The X 1/2 and A1 3/2 states of ArXe+ and KrXe+
different Franck-Condon factors and of the effects of channel interactions (see below).
The assignment of the vibrational levels of the X 1/2 state was derived from the isotopic
shifts following the procedure described in Section 2.7. The irregular intensity distribution
in these spectra is characteristic of situations in which the photoionization spectrum is
highly structured (see Figs. 7.8, 7.9, 7.12 and 7.13). In these cases the intensities of
the lines observed in the PFI-ZEKE photoelectron spectrum are enhanced by channel
interactions with low-lying autoionizing Rydberg levels that appear as strong sharp lines
in the corresponding photoionization spectra [248]. The intensity perturbations apparent
in the spectra of the X 1/2 state presented in Figs. 7.7 and 7.11 hindered an analysis of the
intensity distributions in terms of Franck-Condon factors. Several lines in the PFI-ZEKE
spectra of ArXe and KrXe presented in this section are marked by an asterisk. These lines
appeared regardless of the exact position of the VUV laser and therefore do not belong
to the spectra of ArXe and KrXe. The positions of the vibrational levels of all measured
electronic states of ArXe+ and KrXe+ are summarized in Appendix C.
Photoionization spectra were also recorded from the same intermediate states, in order
to obtain more information on the photoionization of ArXe and KrXe. These spectra are
displayed in the upper panels of Figs. 7.8, 7.9, 7.12 and 7.13. In contrast to the PFI-ZEKE
photoelectron spectra, which sometimes have contributions from several isotopomers of
ArXe and KrXe, the photoionization spectra were always recorded mass selectively.
The spectra of the X 1/2 and A1 3/2 states of ArXe+ and KrXe+ are discussed separately
in Subsections 7.5.1 and 7.5.2 below.
7.5.1
The X 1/2 and A1 3/2 states of ArXe+
The strong line at 97430 cm−1 in the PFI-ZEKE photoelectron spectrum of ArXe displayed
in Fig. 7.7(c) could be unambiguously assigned to the origin of the A1 3/2 state for three
reasons: First, neither the position of this line nor its width fits in the vibrational progression of the X 1/2 state. Second, its position corresponds to the limit of the strongest
Rydberg series observed in the photoionization spectrum recorded via the D 0+ (v + = 0)
level (see Fig. 7.9). Third, the assignment is supported by the analysis of Franck-Condon
factors (see below). The assignments presented in Fig. 7.7 are in accordance with those
reported by Huber and Lipson [197] and Yoshii et al. [207].
The PFI-ZEKE photoelectron spectrum of ArXe and the photoionization spectra of
40
Ar129 Xe recorded via the C 1 (v 0 = 6) intermediate level are compared in Fig. 7.8. The
photoionization spectrum shows two distinct steps at the positions of the v + = 0 and
v + = 1 thresholds and a weaker one at the v + = 2 threshold of the X 1/2 state. No
ion signal could be detected below the lowest of these three steps, an observation which
supports the assignment of the origin of the X 1/2 vibrational progression derived from the
PFI-ZEKE photoelectron spectra.
107
Chapter 7. The low-lying electronic states of ArXe + and KrXe+
Electron signal / arb. units
(a)
+
v =0
5
X 1/2
10
15
0
+
5
v =1
(b)
+
v =0
X 1/2
5
10
15
0
+
(c)
0
96400
v =0
+
v =0
5
X 1/2
*
*
10
A1 3/2
*
+
96800
5
15
v =0
96600
A1 3/2
97000
97200
-1
Wave number / cm
97400
4 A1 3/2
97600
Figure 7.7: PFI ZEKE photoelectron spectra of the X 1/2 and A1 3/2 states of ArXe+ recorded via the
C 1 (v 0 = 6) (a), D 0+ (v 0 = 1) (b) and D 0+ (v 0 = 0) (c) intermediate levels. The lines marked by an
asterisk correspond to impurity lines.
The observation of steps in the photoionization spectra of diatomic molecules at the
positions of the successive ionization thresholds is rather exceptional because of the dominance of autoionization processes [226]. These steps indicate that the continua associated
with the lowest vibrational levels of the X 1/2 ionic state are directly accessible from the
v 0 = 6 level of the C 1 state. At higher energies strong autoionizing lines obscure the steplike structure. These lines form regular Rydberg progressions that converge to the v + = 3
vibrational level of the A1 3/2 state (see Fig. 7.8 and columns four and five in Table 7.2).
The same spectral region was also recorded via the v 0 = 0 level of the D 0+ state (see
Fig. 7.9). In this case, the photoionization spectrum is entirely dominated by the contributions of autoionizing levels converging to the lowest vibrational levels of the A 1 3/2
state. The positions of the autoionizing levels could be well reproduced by using Rydberg’s
formula. The analysis of these Rydberg series enabled the determination of the ionization
limits and quantum defects listed in the first three columns of Table 7.2. All extrapolated series limits correspond, within the experimental uncertainty, to the positions of the
transitions to the v + = 0−3 vibrational levels of the A1 3/2 state determined from the PFIZEKE photoelectron spectra. The small values of the quantum defects and the sharpness
of the corresponding autoionizing lines suggest that these series are largely nonpenetrating
and correspond to l = 3 or l = 4 Rydberg series. One of the two series converging to the
108
Ion signal / arb. units
7.5. The X 1/2 and A1 3/2 states of ArXe+ and KrXe+
40
Xe
+
0
15
n* = 12
n* = 12
Electron signal / arb. units
129
Ar
+
v =0
15
X1/2
5
A1 3/2 (v = 3)
20
+
A1 3/2 (v = 3)
20
10
15
0
+
5
v =1
96400
96600
96800
97000
97200
-1
Wave number / cm
97400
A1 3/2
97600
Figure 7.8:
Comparison of the PFI-ZEKE photoelectron spectrum (lower panel) of the X 1/2 and
A1 3/2 vibrational progressions of ArXe+ with the photoionization spectrum (upper panel) of 40 Ar129 Xe+
Both spectra were recorded via the C 1 (v 0 = 6) intermediate level. In the photoionization spectrum two
progressions belonging to Rydberg series converging to the v + = 3 vibrational level of the A1 3/2 state of
ArXe+ have been observed and assigned.
v + = 3 level has a larger quantum defect and must correspond to a low l series (s, p or d).
Table 7.2: Series limits and quantum defects of the Rydberg series of
(columns 1 to 3) and 7.8 (columns 4 and 5).
A1 3/2
(v + = 0)
Series limit / cm−1
Quantum defect
A1 3/2
(v + = 1)
A1 3/2
(v + = 2)
40
Ar129 Xe displayed in Figs. 7.9
A1 3/2
(v + = 3)
A1 3/2
(v + = 3)
97432.9(3) 97479.5(15) 97522.9(22) 97562.8(7) 97563.1(7)
0.063(22)
0.067(87)
0.055(118) 0.0082(500)
0.5(1)
In contrast to the vibrational progressions associated with the X 1/2 ionic state which
are strongly perturbed (see above) the vibrational progressions associated with the A 1 3/2
state are regular and do not appear to be strongly perturbed by channel interactions. A
Franck-Condon analysis of the intensity distribution was possible in this case. With the
value of the equilibrium internuclear distance Re = 3.75 ± 0.02 Å and the model potential
energy function derived for the D 0+ state (see Section 7.3.1), Franck-Condon factors for
the A1 3/2 ← D 0+ (v 0 = 0) transitions were calculated assuming different equilibrium
109
Chapter 7. The low-lying electronic states of ArXe + and KrXe+
Ion signal / arb. units
40
0
129
Xe
+
n* = 12
A1 3/2 (v = 0)
20
15
n* = 11
+
A1 3/2 (v = 1)
20
15
n* = 14
Electron signal / arb. units
Ar
+
A1 3/2 (v = 2)
20
15
+
v =0
*
0
*
+
v =0
96400
5
96600
96800
A1 3/2
5
*
10
97000
97200
-1
Wave number / cm
15
97400
X 1/2
97600
Figure 7.9: Comparison of the PFI-ZEKE photoelectron spectrum (lower panel) of the X 1/2 and A 1 3/2
vibrational progressions of 40 Ar129 Xe+ with the photoionization spectrum (upper panel) of ArXe. Both
spectra were recorded via the D 0+ (v 0 = 0) intermediate level. In the photoionization spectrum three
progressions belonging to Rydberg series converging to the v + = 0, 1, 2 vibrational levels of the A1 3/2
state of ArXe+ have been observed and assigned. The lines in the PFI-ZEKE photoelectron spectrum
marked by an asterisk correspond to impurity lines.
internuclear distances for the A1 3/2 state and the model potential
VA1 (R) = A e−bR − B e−bR/β − f4 (R, b)
C4
.
R4
(7.2)
The short-range part of the potential is described by the first two terms of Eq. (7.2), and
the leading term in the long-range expansion series (charge ↔ induced dipole interaction)
by the third term in which f4 (R, b) represents a Tang-Toennies damping function [212].
The potential parameters were extracted from the experimental line positions using the
same procedure as outlined in Section 7.3.1 for the D 0+ state and are summarized in the
first column of Table 7.3. A comparison of the calculated Franck-Condon factors with
the intensity distribution in the PFI-ZEKE photoelectron spectrum displayed in Fig. 7.10
suggests that the equilibrium internuclear distance of the A1 3/2 state lies in the range
between 3.70 and 3.77 Å. This value is an important information for the determination of
the potential energy curves of the six low-lying electronic states of ArXe+ as discussed in
Section 7.9.
110
7.5. The X 1/2 and A1 3/2 states of ArXe+ and KrXe+
Table 7.3: Parameters of the A1 3/2 and A2 1/2 potential energy functions of ArXe+ . The uncertainties
represent 95 % confidence intervals. Parameters given without uncertainties were held fixed.
A1 3/2
Re /Å a
De /cm−1
β
b/Å−1
C4 /(cm−1 Å4 )
A/cm−1 c
B/cm−1 c
A2 1/2
3.735
3.45
542.615 ± 0.060 884.54 ± 0.20
2.318 ± 0.078 2.688 ± 0.052
3.472 ± 0.044 4.006 ± 0.035
95294
95294
8
1.301·10
4.365·108
96354
110774
a,b
a
This parameter was kept constant during the fit.
Value derived from Ref. [21] (C4 = 12 αAr
d ).
dV (R) c
= 0 and V (Re ) = −De . See Appendix A in Ref. [162].
Determined from the condition dR b
R=Re
Re = 3.65 Å
Re = 3.70 Å
Re = 3.72 Å
Electron signal / arb.units
Re = 3.75 Å
Re = 3.77 Å
Re = 3.80 Å
Re = 3.85 Å
0
+
v =0
2
1
97450
97500
-1
Wave number / cm
3
97550
4
97600
Figure 7.10: Intensity distribution of the PFI-ZEKE photoelectron spectrum and comparison with
Franck-Condon factors for the A1 3/2 ← D 0+ (v 0 = 0) transition of ArXe+ calculated for different equilibrium internuclear distances.
7.5.2
The X 1/2 and A1 3/2 states of KrXe+
The unknown vibrational assignment of the D 1 intermediate state and the strong intensity
perturbations apparent in the spectra of the X 1/2 and A1 3/2 states presented in Fig. 7.11
precluded an analysis of the Franck-Condon factors. The assignment of the vibrational
111
Chapter 7. The low-lying electronic states of ArXe + and KrXe+
levels of the A1 3/2 state is therefore based on the observation that the width of the lines
assigned to this progression are much narrower and do not show any isotopic substructure
when excited from the D 1 (v 0 = m + 1) intermediate state. The assignments presented in
Electron signal / arb.units
Electron signal / arb. units
Fig. 7.11 are in accordance with those reported by Yoshii et al. [206].
*
(a)
84
*
+
v =0
5
X 1/2
10
15
20
25
132
Kr
Xe
30
*
*
*
*
*
0
A1 3/2 v+ = 0
(b)
+
5
10
A13/2
v =0
5
30
X1/2
10 15
*
*
0
*
*
*
*
*
+
v =0
5
95000
95500
10
15
20
96000
96500
-1
Wave number / cm
25
97000
97500
Figure 7.11: PFI ZEKE photoelectron spectra of the X 1/2 and A1 3/2 states of KrXe+ recorded via the
C 0+ (v 0 = 19) (a) and the D 1 (v 0 = m + 1) (b) intermediate levels. The spectrum in panel (a) corresponds
to 84 Kr132 Xe+ . The lines marked by an asterisk correspond to impurity lines.
The PFI-ZEKE photoelectron spectrum and the photoionization spectrum of 84 Kr132 Xe
recorded via the C 0+ (v 0 = 19) intermediate level are compared in Fig. 7.12. The photoionization spectrum is dominated by autoionizing lines which are members of a Rydberg series
converging to the v + = 4 vibrational level of the A1 3/2 state of KrXe+ . Extrapolating the
series with Rydberg’s formula gives a series limit of 97433.7(5) cm−1 and a quantum defect
of 0.991(51). This series limit corresponds, within the experimental uncertainty, to the position of the transition to the v + = 4 vibrational level of the A1 3/2 state determined from
the PFI-ZEKE photoelectron spectra. The close-to-integer value of the quantum defect
suggests that this series corresponds to a largely nonpenetrating Rydberg series (compare
with the analysis of Rydberg series of ArXe summarized in Table 7.2).
The same spectral region was also recorded via the v 0 = m + 1 level of the D 1 state
(see Fig. 7.13). In this case, the photoionization spectrum shows three distinct steps at
the positions of the v + = 0 − 2 thresholds of the X 1/2 state of KrXe+ . No ion signal
could be detected below the lowest of these three steps, an observation which supports the
assignment of the origin of the X 1/2 vibrational progression derived from the PFI-ZEKE
112
7.6. The A2 1/2 state of ArXe+ and KrXe+
Electron signal / arb. units
Ion signal / arb.units
84
0
Kr
132
Xe
+
n* = 8
10
A1 3/2 (v = 4)
15
*
84
*
+
v =0
5
X 1/2
10
20
15
20
Kr
132
Xe
25
*
*
*
0
*
*
A1 3/2 v+ = 0
95000
95500
96000
96500
-1
Wave number / cm
97000
5
10
97500
Figure 7.12:
Comparison of the PFI-ZEKE photoelectron spectrum (lower panel) of the X 1/2
and A1 3/2 vibrational progressions of 84 Kr132 Xe+ with the photoionization spectrum (upper panel) of
84
Kr132 Xe. Both spectra were recorded via the C 0+ (v 0 = 19) intermediate level and detected mass
selectively. The lines marked by an asterisk correspond to impurity lines.
photoelectron spectra.
7.6
7.6.1
The A2 1/2 state of ArXe+ and KrXe+
The A2 1/2 state of ArXe+
The PFI-ZEKE photoelectron spectra of the A2 1/2 state of ArXe+ measured via the
D 0+ (v 0 = 0), D 0+ (v 0 = 1) and C 1 (v 0 = 6) intermediate levels are presented in Fig. 7.14.
The intensity distributions in the spectra recorded from the D 0+ levels were much more
regular than in the case of the transitions to the X 1/2 state and correspond closely to
distributions of Franck-Condon factors expected for transitions out of v 0 = 0 (no node
in the vibrational wave function) and v 0 = 1 (one node in the vibrational wave function)
levels. This behavior can be rationalized from the absence of any spectral features in
the photoionization spectra (not shown) in the corresponding spectral region. Channel
interactions thus do not significantly affect the intensity distribution of the PFI-ZEKE
photoelectron spectra in this spectral region.
The potential function of the A2 1/2 state was derived using the same model as for the
A1 3/2 state (see Eq. (7.2)). The extracted potential parameters are summarized in the
second column of Table 7.3. A comparison of the calculated Franck-Condon factors with
113
Chapter 7. The low-lying electronic states of ArXe + and KrXe+
Ion signal / arb. units
84
132
Kr
Xe
Electron signal / arb.units
0
+
A13/2
v =0
5
30
X1/2
10 15
*
*
*
*
*
*
*
0
+
v =0
5
95000
95500
10
15
20
96000
96500
-1
Wave number / cm
25
97000
97500
Figure 7.13: Comparison of the PFI-ZEKE photoelectron spectrum (lower panel) of the X 1/2 and
A1 3/2 vibrational progressions of KrXe+ with the photoionization spectrum (upper panel) of 84 Kr132 Xe.
Both spectra were recorded via the D 1 (v 0 = m + 1) intermediate level. The lines marked by an asterisk
correspond to impurity lines.
the intensity distribution of the PFI-ZEKE photoelectron spectra displayed in Fig. 7.15
suggests that the equilibrium internuclear distance of the A2 1/2 state is 3.45 ± 0.05 Å.
7.6.2
The A2 1/2 state of KrXe+
The PFI-ZEKE photoelectron spectra of the A2 1/2 state of KrXe+ measured via the
C 1 (v 0 = 19) and D 0+ (v 0 = m + 1) intermediate levels are presented in Fig. 7.16. The
assignment of the vibrational levels of the A2 1/2 state was derived from the isotopic
shifts following the procedure described in Section 2.7 from which the adiabatic ionization
energy corresponding to the A2 1/2 state is inferred to be 106545.9(12) cm−1 . Yoshii et
al. [206] reported the origin of this progression at 107389 cm−1 . They also observed
lines lying lower in energy in their TPEPICO spectra but assigned them to members
of the II(1/2u) ← X 0+ transition of Xe2 . We can exclude this interpretation because of
two reasons: First, excitation of the ionic states using a resonant two-photon sequence
prevented the spectra from having contributions originating from the ionization of the
corresponding homonuclear rare gas dimers. Second, none of these lines coincides with
positions of the vibrational levels of the II(1/2u) state of Xe+
2 presented in Chapter 6 (see
also Ref. [53]).
114
7.6. The A2 1/2 state of ArXe+ and KrXe+
(a)
*
Electron signal / arb. units
0
+
5
+
5
8
+
5
8
v =0
9
(b)
0
v =0
(c)
0
v =0
107600
107700
107800
107900
-1
Wave number / cm
108000
108100
Electron signal / arb. units Electron signal / arb. units
Figure 7.14: PFI-ZEKE photoelectron spectra of the A2 1/2 state of ArXe+ recorded via the C 1 (v 0 = 6)
(a), D 0+ (v 0 = 1) (b) and D 0+ (v 0 = 0) (c) intermediate levels. The line marked by an asterisk correspond
to an impurity line.
(a)
Re = 3.20 Å
Re = 3.30 Å
Re = 3.40 Å
0
+
v =0
5
8
(b)
Re = 3.45 Å
Re = 3.50 Å
Re = 3.60 Å
0
+
v =0
107700
107800
107900
-1
Wave number / cm
5
108000
8
108100
Figure 7.15: Intensity distributions of the PFI-ZEKE photoelectron spectra of ArXe recorded via the
v 0 = 1 (a) and v 0 = 0 (b) vibrational levels of the D 0+ state and comparison with Franck-Condon factors
for the A2 1/2 ← D 0+ (v 0 = 0, 1) transitions calculated for different equilibrium internuclear distances.
115
Electron signal / arb. units
Electron signal / arb.units
Chapter 7. The low-lying electronic states of ArXe + and KrXe+
*
(a)
0
+
v =3
5
15
11
*
(b)
*
0
19
+
v =0
106500
5
106750
11
107000
107250
107500
-1
Wave number / cm
*
15
18
107750
108000
Figure 7.16:
PFI ZEKE photoelectron spectra of the A2 1/2 state of KrXe+ recorded via the
C 0+ (v 0 = 19) (a) and the D 1 (v 0 = m + 1) (b) intermediate levels. The lines marked by an asterisk correspond to impurity lines.
7.7
The B 1/2 and C1 3/2 states of ArXe+ and KrXe+
Spectra of the B 1/2, C1 3/2 and C2 1/2 states of ArXe+ have not been recorded in
this work. The C 1 and D 0+ Rydberg states of ArXe located in the vicinity of the
Xe∗ ([5p]5 6s0 [1/2]1 ) + Ar(1 S0 ) dissociation limit are not suitable to access the three upper
states because the wave number of the third photon must be ≥ 48800 cm−1 which is close
to the limit reachable by frequency up-conversion with nonlinear crystals (see Section 3.2).
The study of the three higher lying electronic states of ArXe+ by resonance-enhanced twophoton excitation will first require an investigation of appropriate Rydberg states located
in the vicinity of e.g. the Ar∗ ([3p]5 4p) + Xe(1 S0 ) dissociation limits. These levels are likely
to be predissociative.
Transitions to the B 1/2 and C1 3/2 states of KrXe+ have only been observable via selected vibrational levels of the Rydberg state located in the vicinity of the
Kr∗ ([4p]5 5s[3/2]1 ) + Xe(1 S0 ) dissociation limit and are discussed below. The C2 1/2 state of
KrXe+ has not been investigated but information on its potential function could be derived
from the global treatment of all six low-lying electronic states of KrXe+ (see Section 7.9.3).
7.7.1
The B 1/2 and C1 3/2 states of KrXe+
The PFI-ZEKE photoelectron spectrum of the B 1/2 and C1 3/2 states of KrXe+ measured
via the v 0 = n + 6 vibrational level of the Rydberg state located below the
116
7.7. The B 1/2 and C1 3/2 states of ArXe+ and KrXe+
Kr∗ ([4p]5 5s[3/2]1 ) + Xe(1 S0 ) dissociation limit is presented in Fig. 7.17. The signal-tonoise ratio of this spectrum is not as good as for the other spectra, primarily because of
the large amount of ions produced by direct ionization into all the open ionization channels
at this total excitation energy. The absolute vibrational numbering of the C1 3/2 state was
derived from the partially resolved isotopic structure of the v + = 4 and v + = 5 levels using the procedure described in Section 2.7. Unfortunately, the isotopic substructure could
not be fully resolved, even in the highest observed vibrational bands. The analysis of the
isotopic shifts indicated that the lowest observed level is likely to be v + = 0. Although
Tsuji et al. [199] and Yoshii et al. [206] also proposed the same assignment, we cannot
exclude the possibility that the actual vibrational numbering differs from this assignment
by ±1.
B 1/2
+
Electron signal / arb.units
v =0
*
*
*
*
*
0
+
v =0
112000
5
9
112200
112400
-1
Wave number / cm
C13/2
112600
Figure 7.17: PFI ZEKE photoelectron spectrum of the B 1/2 and C1 3/2 states of KrXe+ recorded
via the (v 0 = n + 6) level of the Rydberg state located in the vicinity of the Kr∗ ([4p]5 5s[3/2]1) + Xe(1 S0 )
dissociation limit. The lines marked by an asterisk correspond to impurity lines.
The assignment of the strong line at ∼ 112425 cm−1 in Fig. 7.17 to the origin of the
B 1/2 state is supported by the fact that neither its width nor its position fits in the
vibrational progression of the C1 3/2 state. Yoshii et al. [206] also reported the v + = 0
level of the B 1/2 state to be at 112428 cm−1 . However, the assignment of an electronic
state on the basis of a single observed line must remain tentative.
The spectral features in the wave number region beyond the B 1/2 (v + = 0) state in
Fig. 7.17 could correspond to vibrational levels of the C1 3/2 and B 1/2 states. Unfortunately the low signal-to-noise ratio prevented an analysis of the isotopic shifts and an
117
Chapter 7. The low-lying electronic states of ArXe + and KrXe+
unambiguous assignment.
7.8
Spectroscopic constants for the ionic states
Analyzing the vibrational progressions associated with the three lowest electronic states
observed experimentally in terms of the standard expansion formula (Eq. (2.24)) led to
the set of adiabatic ionization energies Ei and harmonic (ωe+ ) and anharmonic (ωe x+
e ,
ωe ye+ ) vibrational constants summarized in Tables 7.4 and 7.5. In addition, values for
the dissociation energies D0+ were determined from Eq. (2.25) using the atomic ionization
energies of 97833.783 cm−1 and 108370.714 cm−1 corresponding to the formation of the
2
P3/2 and 2 P1/2 states of Xe+ [29, 30] and the dissociation energies of the X 0+ ground
states of ArXe (D0 = 117.3 cm−1 [40]) and KrXe (D0 = 151.3 cm−1 [40]).
Table 7.4: Adiabatic ionization eneries Ei , dissociation eneries D0+ and vibrational constants ωe+ , ωe x+
e
of the lowest three electronic states of ArXe+ . The values correspond to a mixture of all isotopomers
which is best represented by a fictive average isotopomer of ArXe with a reduced mass µ = 30.627 u.
State
Ei /(hc cm−1 )
X 1/2
96515.6(13)
96517 ± 2
96528 ± 97
96666 ± 137
D0+ /cm−1
ωe+ /cm−1
1435.5(14)
89.34(38)
1432 ± 8 88.14 ± 0.47
1445 ± 30
89.64
1420 ± 97
1129 ± 161
−1
ω e x+
e /cm
Reference
1.463(20)
1.46 ± 0.02
1.494
This work
[207]
[197] c
[202]
[201]
a,b
A1 3/2
97432.6(6)
97433 ± 3
518.5(7)
516 ± 9
548
48.79(76)
50.54 ± 1.60
48.49
1.30(13)
1.61 ± 0.26
1.267
This work
[207]
[197] c
a,b
A2 1/2
107638.3(16)
107636 ± 6
849.7(17)
850 ± 12
875
69.08(87)
68.39 ± 2.16
69.41
1.568(79)
1.49 ± 0.19
1.621
This work
[207]
[197] c
a,b
The uncertainties in Ei and D0+ include the full width at half maximum of the observed transitions
and potential errors in the wave number calibration and in the determination of the field-induced shifts of
the ionization thresholds.
b
The uncertainty in the vibrational constants represent one standard deviation in the fit.
c
An additional term describing the long-range behavior was also considered in Ref. [197].
a
The spectroscopic constants summarized in Tables 7.4 and 7.5 are compatible, with the
exception of the A2 1/2 state of KrXe+ , with the earlier results of Ng et al. [201], Dehmer
and Pratt [202], Huber and Lipson [197], Tsuji et al. [199] and Yoshii et al. [206, 207] but
are more precise.
The constants for the C1 3/2 state of KrXe+ were determined using the same formalism
118
7.9. The potential energy functions of the six low-lying electronic states of ArXe + and KrXe+
Table 7.5: Adiabatic ionization energies Ei , dissociation energies D0+ and vibrational constants ωe+ ,
+
+
ω e x+
e , ωe ye of the X 1/2, A1 3/2, A2 1/2, C1 3/2 and B 1/2 states of KrXe .
State
Ei /(hc cm−1 )
D0+ /cm−1
X 1/2
94821.6(10)
94811 ± 7
94875 ± 89
94827 ± 137
3163.5(10)
115.23(20)
3171 ± 7 121.39 ± 0.90
3105 ± 89
2984 ± 161
A1 3/2
97283.3(10)
97293 ± 2
701.8(10)
689 ± 2
41.00(30)
40.05 ± 0.86
40.72(29)
0.674(17)
1.03 ± 0.12
0.893(40)
A2 1/2
106545.9(12)
107387 ± 3
1976.1(12)
1132 ± 3
86.35(61)
64.41 ± 1.26
0.835(73)
1.03 ± 0.12
C1 3/2
112055.3(15)
112068 ± 1
1010.4(15)
995 ± 1
47.56(67)
44.20 ± 0.60
47.20(32)
0.691(62)
0.42 ± 0.06
0.800(33)
B 1/2
112426.6(19)
112428 ± 7
639.1(19)
635 ± 7
ωe+ /cm−1
30.50 ± 3.24
−1
ω e x+
e /cm
ωe ye+ /cm−1
Reference
1.182(14)
1.42 ± 0.03
0.00155(28)
This work a,b,c
[206] d
[202] d
[201] d
This work a,b,c
[206] d
[199] d
-0.0053(25)
This work a,b,c
[206] d
This work a,b,d
[206] d
[199] d
This work a,d,e
[206] d
The uncertainties in Ei and D0+ include the full width at half maximum of the observed transitions
and potential errors in the wave number calibration and in the determination of the field-induced shifts of
the ionization thresholds.
b
The uncertainty in the vibrational constants represent one standard deviation in the fit.
c
The values correspond to 84 Kr132 Xe.
d
The values correspond to a mixture of all isotopomers which is best represented by a fictive average
isotopomer of KrXe with reduced mass µ = 51.14 u.
e
Only the v + = 0 vibrational level of the B 1/2 state has been observed experimentally in this work.
a
with the atomic ionization energy of 112914.434 cm−1 corresponding to the formation of
the 2 P3/2 state of Kr+ [27] and the dissociation energy D0 = 151.3 cm−1 of the X 0+ ground
state of KrXe [40].
7.9
The potential energy functions of the six low-lying
electronic states of ArXe+ and KrXe+
7.9.1
A global potential model for the six low-lying electronic
states of the heteronuclear rare gas dimers
Whereas in the homonuclear rare gas dimer cations the two sets of three g and u states
can be treated separately (see Chapter 6 and Refs. [162, 53, 163]), a global treatment of
the low-lying electronic states of the heteronuclear rare gas dimer cations necessitates the
simultaneous treatment of all six states because of the absence of the g/u symmetry. Not
119
Chapter 7. The low-lying electronic states of ArXe + and KrXe+
only must the spin-orbit interaction of the two Σ and Π states be accounted for but the
charge-exchange interaction between states of the same Λ value must also be included.
Hausamann and Morgner [196] have proposed a model for a global treatment of all
six low-lying electronic states of the heteronuclear rare gas dimer ions which relies on a
description with a 4 × 4 coupling matrix of the potential energy functions of the two Σ and
the two Π states. The charge-exchange interactions are treated by the matrix


V1 − ∆V1 (R)
0
V12Σ (R)
0


Π


0
V
(R)
0
V
(R)
1
12
,

(7.3)
 V Σ (R)
0
V2 − ∆V2 (R)
0 


12
0
V12Π (R)
0
V2 (R)
where Vi (R), Vi − ∆Vi (R) (i = 1, 2) represent the potential energy functions of the two Π
and Σ states, respectively, and V12Σ (R) and V12Π (R) correspond to the Λ-conserving chargeexchange interactions between the two Σ and Π states, respectively. The spin-orbit interaction is treated separately for the Ω = 1/2 and Ω = 3/2 states using the matrix


√
0
2a1
0
0

√
2a1
a1
0
0 
1 

√ 
·
2 
2a2 
0
0
0


√
2a2
a2
0
0
(7.4)
for the Ω = 1/2 states where ai = 32 Ai (Ai is the atomic spin-orbit coupling constant of
atom i (i = 1: Xe; i = 2: Ar, Kr)) 1 . The ΠΩ=3/2 states only undergo an R-independent
shift. In Ref. [196] the potential functions of the Σ states were determined by diagonalizing,
at each value of R, the sum of matrices (7.3) and (7.4). This model offers the advantage of
simplicity but does not properly incorporate the effects of the charge-exchange interaction
on the spin-orbit coupling. Indeed, the spin-orbit interaction of the lower (upper) curves
are entirely determined by the Xe+ (Ar+ or Kr+ ) spin-orbit coupling constant. The model
therefore disregards, in the treatment of the spin-orbit interaction, the fact that the chargeexchange configuration interaction results in a significant mixing of the Xe+ + Rg and
Xe + Rg+ (Rg=Ar, Kr) configurations.
A more exact treatment of the spin-orbit and charge-exchange interactions necessitates
the inclusion of both interactions in a single 6 × 6 matrix set up in a Hund’s case (b)
basis set. The complete interaction matrix is presented in Table 7.6. The two 3 × 3
blocks along the diagonal describe the spin-orbit interaction within the Xe+ + Rg and the
Xe + Rg+ configurations and the two off diagonal 3×3 blocks describe the charge-exchange
interactions between states of the same Λ value.
The matrix elements V12Σ (R) and V12Π (R) describe the charge-exchange interaction be1
In Ref. [196] the multiplication factor 3/2 instead of 1/2 was reported which gives values of the spinorbit splittings of the atoms that are three times too large.
120
7.9. The potential energy functions of the six low-lying electronic states of ArXe + and KrXe+
Table 7.6: Complete interaction matrix for the six low-lying electronic states of the heteronuclear rare
gas dimers expressed in Hund’s angular momentum coupling case (b).
Σ1
Π−
1
Π+
1
Σ2
Σ1
VΣ1 (R)
a1
2
V12Σ (R)
Π−
1
a1
2
− a21
VΠ1 (R)
Π+
1
− a21
− a21
Σ2
Π−
2
Π+
2
V12Σ (R)
− a21
Π−
2
V12Π (R)
V12Π (R)
VΠ1 (R)
V12Π (R)
V12Π (R)
Π+
2
VΣ2 (R)
a2
2
a2
2
VΠ2 (R)
− a22
− a22
− a22
− a22
VΠ2 (R)
tween states of the same symmetry Λ correlating with the Xe+ (2 P) (subscript 1) and the
Rg+ (2 P) (subscript 2; Rg=Ar, Kr) dissociation limits, respectively. This interaction was
described semiempirically by Rapp and Francis [366] and later correlated with chargetransfer data in a large number of diatomic molecules by Olson et al. [367]. Their charge
transfer model postulates a two-state system with an off-diagonal coupling H12 , derived
from hydrogen-like orbitals, which, in the present case, can be written separately for the
Σ and Π states as
V12Λ (R) = DΛ · R · e−CΛ R ,
(7.5)
and describes the charge-exchange interaction with the two parameters CΛ and DΛ .
In our analysis, as in that of Hausamann and Morgner [196], the spin-orbit coupling
constants a1 and a2 are assumed to be independent of R and set to the value of the corresponding atomic dissociation limits: a1 = aXe = 32 AXe with AXe = [E(Xe+ , 2 P1/2 ) −
E(Xe+ , 2 P3/2 )]/hc = 10536.925 cm−1 [30]; a2 = aAr = 32 AAr with AAr = [E(Ar+ , 2 P1/2 ) −
E(Ar+ , 2 P3/2 )]/hc = 1431.5831 cm−1 [25] for ArXe+ and a2 = aKr = 23 AKr with AKr =
[E(Kr+ , 2 P1/2 ) − E(Kr+ , 2 P3/2 ]/hc = 5370.07 cm−1 [368, 27]2 for KrXe+ . This approximation is the generalization to the heteronuclear rare gas dimers of the approximation introduced for homonuclear dimers by Cohen and Schneider [18]. A recent ab initio calculation
for Xe+
2 revealed an R-dependence of the spin-orbit interaction constant (see Chapter 6),
and Chapter 6 describes in detail the consequences of the R-dependence and quantifies the
errors introduced by assuming the spin-orbit coupling constant to be R-independent. In
this first global treatment of the six low-lying electronic states of the heteronuclear dimer
cations the assumption of a to be independent of R represents a reasonable approximation.
The potential energy functions of the uncoupled Σi and Πi (i = 1, 2) states are expressed
2
When analyzing the KrXe+ data the new more precise value of AKr = 5370.294(44) cm−1 [28] was not
yet available.
121
Chapter 7. The low-lying electronic states of ArXe + and KrXe+
as
VΛi (R) = AΛi e
−bΛi R
− B Λi e
−bΛi R/βΛi
−
3
X
f2n (R, bΛi )
n=2
C2n,Λi
+ Vdissi
R2n
(7.6)
with Λi = Σi , Πi (i = 1, 2) (see Section 2.6 for a detailed discussion of this potential energy
function). Vdissi (i = 1, 2) is a constant used to relate the potential energies to the energy
of the X 0+ (v 00 =0) ground neutral state and is defined as
Vdiss1 = D0 (RgXe, X 0+ ) + Ei (Xe, 2 P3/2 ←1 S0 )/hc +
a1
2
(7.7)
a2
2
(7.8)
a2
2
(7.9)
with a1 = aXe for the Σ1 and Π1 states of ArXe+ and KrXe+ ,
Vdiss2 = D0 (ArXe, X 0+ ) + Ei (Ar, 2 P3/2 ←1 S0 )/hc +
with a2 = aAr for the Σ2 and Π2 states of ArXe+ and
Vdiss2 = D0 (KrXe, X 0+ ) + Ei (Kr, 2 P3/2 ←1 S0 )/hc +
with a2 = aKr for the Σ2 and Π2 states of KrXe+ .
Values of Vdiss1 = 101463.4 cm−1 and Vdiss2 = 127704.3 cm−1 for ArXe and
Vdiss1 = 101497.4 cm−1 and Vdiss2 = 114855.8 for KrXe are obtained from the literature values a1 = 32 AXe = 7024.617 cm−1 [30], Ei (Xe, 2 P3/2 )/hc = 97833.783 cm−1 [29],
a2(ArXe) = 32 AAr = 954.3887 cm−1 [25], a2(KrXe) = 32 AKr = 3580.05 cm−1 [368, 27]3 ,
Ei (Ar, 2 P3/2 )/hc = 127109.842 cm−1 [26], Ei (Kr, 2 P3/2 )/hc = 112914.434 cm−1 [27],
D0 (ArXe, X 0+ ) = 117.3 cm−1 [40] and D0 (KrXe, X 0+ ) = 151.3 cm−1 [40].
The four uncoupled Hund’s case (b) potential energy curves are correlated with the
Xe ( P) + Rg(1 S0 ) (Σ1 , Π1 ) and Xe(1 S0 ) + Rg+ (2 P) (Σ2 , Π2 ) dissociation limits (Rg=Ar,
Kr). Consequently, the long-range coefficients C4 and C6 are different for the two pairs
+ 2
of curves. The long-range behavior of the Σ1 and Π1 potentials is described in terms of
the interaction between a neutral Rg atom and a Xe+ ion, whereas for the Σ2 and Π2
potentials the interaction corresponds to that of an Rg+ ion and a neutral Xe atom. The
determination of the C4 and C6 coefficients is described in Section 2.6.1 and their values
are summarized in Table 7.7.
The potential energy functions Vj (R) of the X 1/2, A1 3/2, A2 1/2, B 1/2, C1 3/2 and
C2 1/2 states were then calculated from the VΣ1 (R), VΠ1 (R), VΣ2 (R) and VΠ2 (R) potentials
in a straightforward manner using the interaction matrix presented in Table 7.6. The
calculated transition wave numbers were determined by numerically solving the radial
Schrödinger equation
~2 d2
+ hc Vj (R) ψijv (R) = Eijv ψijv (R)
(7.10)
−
2µi dR2
3
When analyzing the KrXe+ data the new more precise value of A2(Kr) = 5370.294 cm−1 [28] was not
yet available.
122
7.9. The potential energy functions of the six low-lying electronic states of ArXe + and KrXe+
Table 7.7: All contributions to the C4 and C6 coefficients describing the long-range behavior of the
ArXe+ and KrXe+ ions in their low-lying electronic states. See Section 2.6.1 for details.
Xe+ + Ar Xe + Ar+ Xe+ + Kr Xe + Kr+
interaction
C4 /(Eh a0 4 )
5.537
13.6452
8.3685
13.6452
charge ↔ induced quadrupole
quadrupole ↔ induced dipole
dispersion (isotropic part)
dispersion (anisotropic part)
total isotropic part of C6
total anisotropic part of C6
C6,0 /(Eh a0 6 )
C6,2 /(Eh a0 6 )
C6,0 /(Eh a0 6 )
C6,2 /(Eh a0 6 )
C6,0 /(Eh a0 6 )
C6,2 /(Eh a0 6 )
25.105
166.11
102.7485
-3.2277
127.8535
162.8823
106.3
229.65
89.5993
-0.9156
195.8993
228.7323
47.775
251.055
146.2592
-4.5945
194.0342
246.4605
106.3
302.1036
135.2958
-2.5652
241.5958
299.5708
C6 for Σ states
C6 for Π states
C6,Σ /(Eh a0 6 )
C6,Π /(Eh a0 6 )
193.0065
95.2771
287.3922
150.1528
292.6184
144.7421
361.4241
181.6816
charge ↔ induced dipole
corresponding to the vibrational motion of the isotopomer with reduced mass µi in the
potential Vj (R). The indices i, j and v designate the isotopomer, the electronic state and
the vibrational quantum number of the ion, respectively.
7.9.2
Fitting procedure
The parameters of the model potentials were derived from the experimental data in a
nonlinear least-squares fitting procedure. The numerical procedure relies on a discrete
variable representation (based on the Gauss-Chebyshev quadrature of the 1st kind [350,
362, 363, 364]) on a grid with 701 equidistant grid points in the range of internuclear
distances R = 2.0 − 9.0 Å. In the fitting procedure all parameters describing the long-range
interaction were kept fixed at the values listed in Table 7.7. A total of 20 parameters were
refined, i.e. Re,Λ , De,Λ , βΛ , bΛ , for both pairs of Σ and Π states and CΛ and DΛ describing
the charge-exchange between states of the same Λ value.
In the first step of the fitting procedure the potential parameters of the two Π states were
derived directly from the measured positions of the A1 3/2 and C1 3/2 vibrational levels
because they have pure Π character. As initial values of the charge-exchange parameters
CΠ and DΠ the values estimated by Hausamann and Morgner [196] were used and held
fixed at the beginning. The other potential parameters were optimized. With only two
adjustable parameters for the weakly bound Π1 state (Re,Π1 , bΠ1 ; BΠ1 = 0 in Eq. (7.6)) and
four for the stronger bound Π2 state (Re,Π2 , De,Π2 , βΠ2 , bΠ2 ) it was not possible to reach a
good agreement between the calculated and measured vibrational levels of the A1 3/2 and
C1 3/2 states. Increasing the number of adjustable parameters of the Π1 state from two
to four (Re,Π1 , De,Π1 , βΠ1 , bΠ1 ) and adjusting the charge-exchange parameters CΠ and DΠ
yielded potential energy functions for the A1 3/2 and C1 3/2 states which provided excellent
123
Chapter 7. The low-lying electronic states of ArXe + and KrXe+
agreement between the calculated and measured vibrational levels and also satisfied the
boundary conditions related to the equilibrium internuclear distances of the A1 3/2 and
C1 3/2 states (see below).
In a second step the potential parameters of the two Σ states were included in the
fitting procedure and optimized using the measured positions of the X 1/2, A2 1/2, B 1/2
and C2 1/2 states by keeping the parameters associated with the two Π states fixed. The
initial values of the charge-exchange parameters CΣ and DΣ were taken from Ref. [196].
In this way it was possible to describe any set of two of the four (X 1/2, A2 1/2, B 1/2,
C2 1/2) states satisfactorily, but never all four.
To derive a final set of potential parameters, all 20 potential parameters were adjusted
simultaneously and all vibrational levels of all six (five) states of ArXe+ (KrXe+ ) were
included in the fitting procedure. Because of the large number of parameters and the strong
correlation between several of them, in particular between De,Λ , βΛ , bΛ and DΛ , it was not
possible to determine a unique set of parameters corresponding to a global minimum in the
fit. Moreover, the initial phase of the fitting procedure had to rely on adjustments of several
parameters by hand. The parameter sets summarized in Tables 7.8 and 7.9 for ArXe+
and Table 7.10 for KrXe+ represent one possible solution which reproduces all known
spectroscopic data satisfactorily with physically plausible parameters. The correlations
mentioned above prevented the determination of meaningful statistical uncertainties for
several parameters [351]. In the tables these parameters are given with a sufficient number
of significant digits that the numerical results can be reproduced in a calculation.
ArXe+
Only the three lowest electronic states associated with the Ar + Xe+ dissociation limits of
ArXe+ were investigated experimentally in the present study. The measured positions of
the vibrational levels of the B 1/2, C1 3/2 and C2 1/2 states were taken from Huber and
Lipson [197]. For the B 1/2 state Yoshii et al. [207] observed two vibrational levels below
the lowest one detected by Huber and Lipson [197] and assigned them to v + = 0 and
v + = 1. Because neither of these two assignments can be considered unambiguous, two
data sets for the B 1/2 state were used in the fitting procedure, one containing only the
data of Huber and Lipson [197], the other this same data, but including the two vibrational
levels observed by Yoshii et al. [207]. These two data sets led to the potential parameters
listed in Tables 7.8 and 7.9, respectively, and to the dissociation energies and equilibrium
internuclear distances summarized in Table 7.11. The experimental data for all six lowlying electronic states of ArXe+ on which the fit procedure was based are summarized
in Tables C.13 - C.18 in Appendix C which also list the positions calculated from the
optimized potential functions.
The parameters describing the potential energy funcfions of all six low-lying electronic
124
7.9. The potential energy functions of the six low-lying electronic states of ArXe + and KrXe+
states of ArXe+ were optimized using the experimental value of the equilibrium internuclear
distance of the A1 3/2 state as a boundary condition. Because the strongest line in the
C (C1 3/2 → A1 3/2) emission band corresponds to the transition between the origins of
these two states [197], the equilibrium internuclear distance of the C1 3/2 state must be
similar to that of the A1 3/2 state. The optimal parameters are listed in Table 7.8 (using
the data set with the assignment for the B 1/2 state proposed by Huber and Lipson [197])
and Table 7.9 (using the data set with the assignment for the B 1/2 state proposed by
Yoshii et al. [207]).
Table 7.8: Parameters of the interaction potentials of the lowest electronic states of ArXe + determined
from the data set with the assignment for the B 1/2 state proposed by Huber and Lipson [197]. The
uncertainties represent 95 % confidence intervals. Parameters given without uncertainties were held fixed.
Σ1
Re,Λ /Å
De,Λ /cm−1
βΛ
bΛ /Å−1
C4,Λ /(cm−1 Å4 )
C6,Λ /(cm−1 Å6 )
AΛ /cm−1 c
BΛ /cm−1 c
a,b
a,b
CΛ /Å−1
DΛ /(cm−1 Å−1 )
Π1
Σ2
Π2
3.3690 ± 0.0013 3.7192 ± 0.0132 3.3099 ± 0.0353 3.7146 ± 0.0010
1229.5 ± 0.8
538.4 d
1351.5d
1203.4 d
1.4434 d
1.5318 d
1.5499 ± 0.0011
1.8999 d
3.2084 d
3.1974 d
3.2002d
3.1138 d
95294
95294
234837
234837
930172
459177
1385053
723644
7
7
7
8.056·10
6.865·10
2.317·10
8.770·107
2.770·106
821797
-869439
244289
Σ1 ↔ Σ 2
Π1 ↔ Π 2
1.9555 ± 0.0128
898392 d
2.3868 ± 0.3228
683662 d
a
This parameter was kept constant during the fit.
Determined from Table 7.7 using 1 Eh = 219474.6 cm−1 , 1 a0 = 0.529177211 Å.
Λ (R) c
= 0 and VΛ (Re,Λ ) = Vdiss − De,Λ . See Appendix A in
Determined from the conditions dVdR
b
R=Re,Λ
Ref. [162].
d
The uncertainty is not defined for this parameter.
The level positions calculated using the parameters derived from both data sets for
the B 1/2 state are compared with the experimental results in Tables C.13 - C.18 in Appendix C. The root-mean-square (rms) deviation of the fit (see Eq. (5.16)) based on the
assignment proposed by Huber and Lipson [197] amounted to 1.57 and the maximum absolute deviation to −9.5 cm−1 (for the C2 1/2 (v + = 0) level). The discrepancies between
observed and calculated positions are largest for the C2 1/2 state. For the assignment proposed by Yoshii et al. [207] the rms deviation amounted to 1.82, which is slightly higher
than with the other data set. The maximum absolute deviation between measured and
calculated positions amounted to −10.4 cm−1 (for the v + = 1 vibrational level of the B 1/2
125
Chapter 7. The low-lying electronic states of ArXe + and KrXe+
Table 7.9: Parameters of the interaction potentials of the lowest electronic states of ArXe + determined
from the data set with the assignment for the B 1/2 state proposed by Yoshii et al. [207]. The uncertainties
represent 95 % confidence intervals. Parameters given without uncertainties were held fixed.
Σ1
Re,Λ /Å
De,Λ /cm−1
βΛ
bΛ /Å−1
C4,Λ /(cm−1 Å4 )
C6,Λ /(cm−1 Å6 )
AΛ /cm−1 c
BΛ /cm−1 c
a,b
a,b
CΛ /Å−1
DΛ /(cm−1 Å−1 )
Π1
Σ2
Π2
3.3674 ± 0.0015 3.7280 ± 0.0010 3.5070 ± 0.0025 3.3260 ± 0.0056
1248.8 ± 152.5
538.9 d
1500.8 ± 1.1
1212.0 d
1.4962 d
1.5400 d
1.2003 d
2.0500 d
3.1597 ± 0.0022 3.1760 ± 0.0079 2.7310 ± 0.0052
3.4897 d
95294
95294
234837
234837
930172
459177
1385053
723644
7
7
7
6.569·10
6.596·10
3.528·10
6.523·107
1.859·106
791937
5.331·106
-172086
Σ1 ↔ Σ 2
Π1 ↔ Π 2
1.95730 ± 0.00031
902560 d
2.4186 ± 0.2625
712285 d
a
This parameter was kept constant during the fit.
Determined from Table 7.7 using 1 Eh = 219474.6 cm−1 , 1 a0 = 0.529177211 Å.
Λ (R) c
Determined from the conditions dVdR
= 0 and VΛ (Re,Λ ) = Vdiss − De,Λ . See Appendix A in
b
R=Re,Λ
Ref. [162].
d
The uncertainty is not defined for this parameter.
state). Whereas the equilibrium internuclear distances of the A1 3/2 and C1 3/2 states lie
close to the experimental values for the data set corresponding to the assignment proposed
by Huber and Lipson [197], this is not the case for the other data set, which leads to too
small values for the C1 3/2 state. A low rms deviation could only be reached with the data
set corresponding to the assignment proposed by Yoshii et al. [207] by significantly changing the equilibrium internuclear distances of the Σ2 and Π2 states (compare Tables 7.8 and
7.9).
KrXe+
Because no information on the internuclear separation of any of the electronic states of
KrXe+ could be extracted directly from the experiments, neither from a Franck-Condon
analysis nor from rotational constants, initial values for the equilibrium internuclear distances were chosen to be slightly larger than for ArXe+ . This choice was motivated by
the larger atom radii of Kr (Kr+ ) compared with Ar (Ar+ ). As in the case of ArXe+ the
strongest line in the C (C1 3/2 → A1 3/2) emission band corresponds to the transition
between the origins of these two states [199]. This experimental observation leads to the
conclusion that the equilibrium internuclear distance of the C1 3/2 state must be similar
126
7.9. The potential energy functions of the six low-lying electronic states of ArXe + and KrXe+
to that of the A1 3/2 state.
No data from the C2 1/2 state was included in the fit procedure because this state has
not been observed in the present work. The vibrational level positions of the C2 1/2 state
reported by Yoshii et al. [206] were not used because of the uncertainty in their assignment
(see Section 7.6.2). Because the Xe+ (2 P1/2 ) + Kr(1 S0 ) dissociation limit lies closer to the
Xe(1 S0 ) + Kr+ (2 P3/2 ) dissociation limit than to the Xe+ (2 P3/2 ) + Kr(1 S0 ) dissociation limit,
the effect of the charge-exchange interaction between states with the same Ω symmetry is
stronger than in the case of ArXe+ (see also discussion in Section 7.10). Slight changes
of the charge-exchange parameters CΛ and DΛ therefore greatly influence all six potential
energy functions. This behavior rendered the fitting procedure more difficult than in the
case of ArXe+ .
Table 7.10: Parameters of the interaction potentials of the lowest electronic states of KrXe + . The
uncertainties represent 95 % confidence intervals. Parameters given without uncertainties were held fixed.
Σ1
Re,Λ /Å
De,Λ /cm−1
βΛ
bΛ /Å−1
C4,Λ /(cm−1 Å4 )
C6,Λ /(cm−1 Å6 )
AΛ /cm−1 c
BΛ /cm−1 c
a,b
a,b
CΛ /Å−1
DΛ /(cm−1 Å−1 )
Π1
Σ2
Π2
3.70328 ± 0.00028 3.74500 ± 0.00042 3.8581 ± 0.0022 3.70968 ± 0.00089
1458.18 ± 0.43
729.8 d
1129.8 d
1037.0 d
1.73787 ± 0.00043
2.6615 d
2.4953 d
2.6987 d
3.0441 ± 0.0012 3.6946 ± 0.0078
2.8302 d
3.2588 ± 0.1636
144025
144025
234837
234837
1410240
697567
1741841
875593
8
8
7
1.156·10
3.628·10
4.194·10
1.022·108
1.091·106
19100
28737
5189
Σ1 ↔ Σ 2
Π1 ↔ Π 2
1.8221 ± 0.0056
1048192 ± 18595
2.3575 ± 0.5370
340778 d
a
This parameter was kept constant during the fit.
Determined from Table 7.7 using 1 Eh = 219474.6 cm−1 , 1 a0 = 0.529177211 Å.
Λ (R) c
Determined from the conditions dVdR
= 0 and VΛ (Re,Λ ) = Vdiss − De,Λ . See Appendix A in
b
R=Re,Λ
Ref. [162].
d
The uncertainty is not defined for this parameter.
The level positions calculated with the optimized potential parameters listed in Table 7.10 are compared with the experimental results in Tables C.21 - C.26 in Appendix C.
The rms deviation of the best fit amounted to 1.78 and the maximum absolute deviation
to 8.1 cm−1 (for the A1 3/2 (v + = 0) level). The discrepancies between observed and calculated positions are largest for the A1 3/2 state. The origin of the C2 1/2 state is calculated
at 117646.8 cm−1 which is ∼ 105 cm−1 above the position of the first vibrational level of
127
Chapter 7. The low-lying electronic states of ArXe + and KrXe+
the C2 1/2 state reported by Yoshii et al. [206]. However, the assignment in Ref. [206] is
not established with absolute certainty (see above).
7.9.3
Potential energy functions
The potential energy functions of the X 1/2, A1 3/2, A2 1/2, B 1/2, C1 3/2 and C2 1/2
states of ArXe+ and KrXe+ calculated from the Σ1 , Π1 , Σ2 and Π2 potential energy curves
derived above using the coupling matrix presented in Table 7.6 are displayed in Figs. 7.18,
7.19 and 7.20. The dissociation energies De and the equilibrium internuclear distances Re
of the six low-lying electronic states of ArXe+ and KrXe+ are summarized in Tables 7.11
and 7.12 where they are compared with other available data.
Potential energy functions of ArXe+
The dissociation energies De and the equilibrium internuclear distances Re of the X 1/2,
A1 3/2 and A2 1/2 states of ArXe+ derived using the two different data sets for the B 1/2
state are almost identical (see Table 7.11). The equilibrium internuclear distances of the
B 1/2, C1 3/2 and C2 1/2 states were shifted to lower values when using the data set of the
B 1/2 state with the assignment proposed by Yoshii et al. [207], whereas the dissociation
energies only slightly changed with the exception of that of the B 1/2 state. In the region of
energies and internuclear distances covered by the experimental data the potential energy
functions of the A1 3/2 and A2 1/2 states determined using the global potential model agree
with those determined directly (see Table 7.3) within the experimental uncertainties.
C2 1/2
C1 3/2
B 1/2
-1
Potential energy / (hc cm )
128000
120000
112000
A2 1/2
104000
A1 3/2
96000
3
4
X 1/2
5
6
Internuclear distance / Å
7
8
9
Figure 7.18: Potential energy curves of the X 1/2, A1 3/2, A2 1/2, B 1/2, C1 3/2 and C2 1/2 states of
ArXe+ using the data set with the assignment for the B 1/2 state proposed by Huber and Lipson [197].
128
7.9. The potential energy functions of the six low-lying electronic states of ArXe + and KrXe+
129000
(e)
(a)
-1
Potential energy / (hc cm )
128000
127000
126000
C2 1/2
(b)
C2 1/2
(f)
B 1/2
C1 3/2
B 1/2
C1 3/2
(c)
(g)
108000
A2 1/2
A2 1/2
98000 (d)
A1 3/2
A1 3/2
(h)
X 1/2
97000
3
X 1/2
4
5
6
7
8
Internuclear distance / Å
9
3
4
5
6
7
8
Internuclear distance / Å
9
Figure 7.19: Potential energy curves of the X 1/2, A1 3/2, A2 1/2, B 1/2, C1 3/2 and C2 1/2 states of
ArXe+ using the data set with the assignment for the B 1/2 state proposed by Huber and Lipson [197]
((a)-(d)) and the assignment for the B 1/2 state proposed by Yoshii et al. [207] ((e)-(f)). The dashed
horizontal lines indicate for each state the positions of the highest observed vibrational level.
Table 7.11: Dissociation energies De and equilibrium internuclear distances Re of the lowest electronic
states of ArXe+ determined from the potential energy curves displayed in Fig. 7.19.
De /cm−1
Re /Å
a
b
X 1/2
A1 3/2
A2 1/2
B 1/2
C1 3/2
C2 1/2
Reference
1480.3
1481.4
1113
542.8
542.6
887
881.8
880.7
1032
1042.9
1155.0
1444
1199.0
1192.4
1581
1098.4
1119.1
1500
This work (H)
This work (Y)
[196]
3.154
3.164
3.81
3.711
3.721
3.97
3.451
3.462
3.86
3.714
3.660
3.84
3.719
3.346
3.76
3.724
3.558
3.84
This work (H)
This work (Y)
[196]
a
b
a
b
Using the data set with the assignment for the B 1/2 state proposed by Huber and Lipson [197].
Using the data set with the assignment for the B 1/2 state proposed by Yoshii et al. [207].
Potential energy functions of KrXe+
The values for the dissociation energies De and the equilibrium internuclear distances Re
of the X 1/2, A1 3/2, A2 1/2, B 1/2, C1 3/2 and C2 1/2 states of KrXe+ summarized in
Table 7.12 are reasonable when comparing them with the values for ArXe+ . The only
other reported values were obtained in a semiempirical calculation by Hausamann and
129
Chapter 7. The low-lying electronic states of ArXe + and KrXe+
Morgner [196] in 1985 using the then available ab initio data. More experimental and/or
ab initio data on the equilibrium internuclear distances of the six low-lying electronic states
of KrXe+ would be desired for comparison with the present results.
C2 1/2
-1
Potential energy / (hc cm )
120000
B 1/2
112000
C1 3/2
A2 1/2
104000
A1 3/2
96000
X 1/2
4
3
5
6
Internuclear distance / Å
7
8
9
Figure 7.20: Potential energy curves of the X 1/2, A1 3/2, A2 1/2, B 1/2, C1 3/2 and C2 1/2 states of
KrXe+ . The dashed horizontal lines indicate for each state the positions of the highest observed vibrational
level. For the C2 1/2 state the dashed-dotted line indicates the lowest observed vibrational level reported
in Ref. [206].
Table 7.12: Dissociation energies De and equilibrium internuclear distances Re of the lowest electronic
states of KrXe+ determined from the potential energy curves displayed in Fig. 7.20.
7.10
X 1/2
A1 3/2
A2 1/2
B 1/2
C1 3/2
C2 1/2
Reference
De /cm−1
3220.8
2912
732.2
1073
2018.4
1944
656.8
968
1034.3
1387
809.3
1178
This work
[196]
Re /Å
3.177
3.28
3.742
3.86
3.434
3.60
4.351
4.18
3.713
3.97
4.054
4.07
This work
[196]
Conclusion
The potential energy functions of the six low-lying electronic states of ArXe+ and KrXe+
derived in this chapter reproduce all known spectroscopic data satisfactorily, though not
perfectly, using a minimal number of adjustable parameters (only 20 parameters for six
130
7.10. Conclusion
potential functions). The potential model developed for this analysis describes the contributions of the spin-orbit, the long-range electrostatic and the charge-exchange interactions
in separate analytical contributions which are based on well-established physical properties
(spin-orbit coupling constants, ionization energies, polarizabilities, quadrupole moments)
of Ar, Kr, Xe, Ar+ , Kr+ and Xe+ . The values of the dissociation energies De derived
from these curves are in excellent agreement with those that can be estimated from the
spectroscopic results summarized in Tables 7.4 and 7.5, and for ArXe+ the equilibrium
internuclear separations are compatible with the available experimental data.
The rms deviations of 1.57 (1.78) for the best fits of the ArXe+ (KrXe+ ) data lies above
1 which is likely to be caused by the approximations made in the derivation of the potential
model and the interaction matrix, in particular by the assumption of R-independent spinorbit coupling constants (see also discussion in Chapter 6). Additional deficiencies of
the present potential functions are the facts that (1) for both investigated heteronuclear
dimers several potential parameters are strongly correlated, (2) two different experimental
assignments of the vibrational levels of the B 1/2 state of ArXe+ in Refs. [197] and [207] have
led to two different sets of potential parameters, (3) the assignment of the B 1/2 and C2 1/2
states of KrXe+ is still tentative and (4) no experimental information on the equilibrium
internuclear distance of any of the states of KrXe+ with which to compare the values derived
from the potential energy curves are available. These deficiencies could, in future, be
eliminated by additional spectroscopic measurements. Rotationally resolved measurements
would be particularly desirable because they would provide independent information on
the internuclear separations which are currently only poorly characterized experimentally.
High-level ab initio quantum chemical calculations should permit the distinction between
the two experimental assignments of the B 1/2 state in ArXe+ and provide new arguments
for an unambiguous vibrational assignment of the B 1/2 and C2 1/2 states in KrXe+ . The
potential energy functions derived in the present work offer a reference with which to
compare potential energy curves calculated ab initio.
The potential energy curves of the six low-lying electronic states of ArXe+ and KrXe+
are compared in Fig. 7.21 (see also Tables 7.8, 7.9 and 7.10). Although the charge-exchange
interaction parameters of ArXe+ and KrXe+ are of similar magnitude, the effects of the
charge-exchange interaction are much more pronounced in KrXe+ than in ArXe+ . Because
the ionization energy of Ar (∼ 127109.9 cm−1 ) lies almost 2 eV higher than that of Kr
(∼ 112914.4 cm−1 ), the upper three curves in KrXe+ lie closer to the lower three curves
than in ArXe+ which facilitates charge transfer. The larger spin-orbit coupling constant
of Kr+ (∼ 5370.3 cm−1 instead of ∼ 1431.6 cm−1 in Ar+ ) further increases the effects of
charge transfer. A comparison of the potential energy functions of ArXe+ and KrXe+
enables one to quantify the influence of charge exchange on the strength of the bonds. The
following conclusions can be drawn: (1) The charge-exchange interaction leads to much
larger binding energies for the X 1/2 and A2 1/2 states of KrXe+ compared with the same
131
Chapter 7. The low-lying electronic states of ArXe + and KrXe+
(a)
C2 1/2
128000
(b)
B 1/2
-1
Potential energy / (hc cm )
C1 3/2
120000
C2 1/2
B 1/2
112000
C1 3/2
A2 1/2
A2 1/2
A1 3/2
A1 3/2
X 1/2
X 1/2
104000
96000
3
4
5
6
7
8
Internuclear distance / Å
9
3
4
5
6
7
8
Internuclear distance / Å
9
Figure 7.21: Comparison of the potential energy curves of the X 1/2, A1 3/2, A2 1/2, B 1/2, C1 3/2 and
C2 1/2 states of ArXe+ (a) and KrXe+ (b).
states of ArXe+ . Indeed, the dissociation energies of the X 1/2 and A2 1/2 states are about
twice as large in KrXe+ than in ArXe+ . (2) The B 1/2 and C2 1/2 states, to which the
X 1/2 and A2 1/2 states are coupled, are shifted to higher energies by the charge-exchange
interaction, and their dissociation energies are reduced. The effects of the charge-exchange
interaction being stronger in KrXe+ , these two states are less strongly bound in KrXe+
than in ArXe+ . (3) The B 1/2 state is most strongly affected vy the charge-exchange
interaction because it lies closest to the A2 1/2 state. Whereas in ArXe+ the B 1/2 and
C1 3/2 states have similar equilibrium internuclear distances and dissociation energies, the
B 1/2 state of KrXe+ has a much weaker binding energy and a much larger equilibrium
internuclear distance than the C1 3/2 state. (4) The larger radii of Kr and Kr+ compared
to those of Ar and Ar+ have an influence on the equilibrium internuclear separation of the
electronic states of ArXe+ and KrXe+ . The effect of the smaller radius of Ar+ manifests
itself in smaller equilibrium internuclear distances of the B 1/2 and C2 1/2 states of ArXe+ .
(5) The impact of the larger atom radius of Kr is compensated by the larger polarizability
of Kr. Consequently, the X 1/2, A1 3/2 and A2 1/2 states have almost identical equilibrium
internuclear distances in ArXe+ and KrXe+ .
Comparing the dissociation energies of the ground electronic states of ArXe+ (De =
−1
1480.3 cm−1 ) and KrXe+ (De = 3220.8 cm−1 ) with Xe+
2 (De = 7937.4 cm ; see Chapter 6),
the correlation between the effect of charge transfer and bond strength becomes particularly
obvious. The large value of the dissociation energy of the I(1/2u) state of Xe +
2 (and
also of other homonuclear rare gas dimer cations (see Table 2.4)) can be regarded as a
132
7.10. Conclusion
manifestation of resonant charge transfer.
133
Chapter 7. The low-lying electronic states of ArXe + and KrXe+
134
Chapter 8
Outlook
The determination of the potential energy functions of the six low-lying electronic states
of Xe+
2 including, for the first time, the R-dependence of the spin-orbit coupling constant
allowed to identify and quantify the effects caused by this R-dependence. In future, it will
be necessary to also include the R-dependence of the spin-orbit coupling constant in the
+
treatment of the potential energy curves of Kr+
2 and Ar2 .
The potential energy functions of the six low-lying electronic states of the homonuclear
rare gas dimer cations are experimentally and theoretically well characterized with the
exception of Ne+
2 . The only experimental information available on the electronic states of
Ne+
2 have been obtained by microwave spectroscopy close to dissociation [168] and by lowresolution threshold photoelectron spectroscopy [174]. To complete our picture of the lowlying electronic states of the rare gas dimer cations new measurements of the energy level
structure of Ne+
2 would be highly desirable. Unfortunately, spectroscopic investigations of
Ne+
2 are very challenging. Attempts, in the context of this dissertation, at obtaining information on the low-lying electronic states of Ne+
2 by PFI-ZEKE photoelectron spectroscopy
following resonance-enhanced two-photon excitation via selected vibrational levels of the
+
C 0+
u intermediate state of Ne2 have remained unsuccessful. The B 0u Rydberg state of Ne2 ,
suggested as promising intermediate state in the two-photon excitation process because it
is predicted to be less predissociative and easily detectable by absorption spectroscopy [83],
could not be observed by photoionization spectroscopy. Other experimental approaches to
study the electronic states of Ne+
2 could include photoionization from metastable states of
Ne2 produced in a high-voltage discharge. The lifetime of the metastable a 3 Σ+
u state of
Ne2 is only 6.6 µs [369], too short for beam experiments in which the source of metastable
states must be separated from the photoexcitation chamber. Single-photon excitation from
the ground state would in principle be possible using synchrotron radiation. However, the
large bandwidth of synchrotron radiation compared to that of our VUV lasers will cause
problems, particularly in experiments aiming at observing the rotational structure of the
photoelectron spectra. Alternative strategies must be sought to solve this challenging
problem.
135
Chapter 8. Outlook
Compared to the homonuclear rare gas dimers, the ionic states of the heteronuclear
rare gas dimers containing Ne, Ar, Kr and Xe have been much less thoroughly investigated
experimentally and theoretically. The potential energy functions of the six low-lying electronic states of ArXe+ and KrXe+ presented in this work are a first step towards a global
description of the lowest electronic states of all heteronuclear rare gas dimer cations. In
order to improve our understanding of these systems high-resolution spectroscopic data is
needed (in particular for the three higher lying B 1/2, C1 3/2 and C2 1/2 states of ArXe+
and KrXe+ ). Rotationally resolved spectra provide information on the equilibrium internuclear distances of the different electronic states and would be particularly important for
comparison with theoretical predictions. Relativistic ab initio quantum chemical calculations could provide information on the R-dependence of the spin-orbit interaction and
facilitate the assignment of the B 1/2 and C2 1/2 states of ArXe+ and KrXe+ which are
difficult to identify in the spectra. Such calculations would also be of invaluable help in
future studies of other heteronuclear rare gas dimer cations.
The global model for the determination of the potential energy functions of the lowest
electronic states of ArXe+ and KrXe+ derived in this work can also be applied to NeAr+ ,
NeKr+ , NeXe+ and ArKr+ . Beside the first adiabatic ionization energies and the dissociation energies D0 of the X 1/2 ground electronic state [72] of the heteronuclear rare gas dimer
cations containing Ne no experimental information has been reported yet. Because of the
necessity to carry out species-selective experiments a resonant two-photon excitation sequence via intermediate Rydberg states of these heteronuclear rare gas dimers is mandatory.
The C and D Rydberg states of NeXe in the vicinity of the Xe∗ ([5p]5 6s0 [1/2]1 ) + Ne(1 S0 )
dissociation limit were reported to be repulsive [114] and are not suitable intermediate
states. The study of the six low-lying electronic states of NeAr+ , NeKr+ and NeXe+ by
resonance-enhanced two-photon excitation will first require the search for, and characterization, of appropriate Rydberg states.
The knowledge of the vibrational levels of the electronic states of the rare gas dimer
cations are a starting point for the spectroscopic study of rare gas trimers and higher
clusters. Rare gas trimers allow the investigation of the Renner-Teller effect, the JahnTeller effect and pseudorotation, which are not present in dimers. Recent calculations
[7, 370, 157] suggest that the low-lying electronic states of the trimers span a wide range of
structures: some are linear, others are bent or form equilateral or distorted triangles. So
far, no spectroscopic information is available on these cations with which these predictions
could be compared. Here also resonance-enhanced multiphoton excitation appears to be
the method of choice, but unfortunately, nothing is known on the excited electronic states
of the trimers that could be used as intermediate levels.
The potential curves of the rare gas dimer cations represent an important ingredient in
the theoretical treatment of the Rydberg states of the rare gas dimers. At present, the data
available on these Rydberg states are scarce. A global interpretation of these somewhat
136
fragmentary data remains a challenge. The MQDT treatment of the rare gas dimers will
necessitate the inclusion of channels associated with all six low-lying electronic states of
the cations and their interactions. To be really useful, initial calculations will have to be
accurate enough to make contact with the spectroscopic data. A treatment of the gerade
Rydberg states of Ne2 appears particularly attractive now that reliable potential energy
curves have been derived for several low-n states.
137
Chapter 8. Outlook
138
Appendix A
Physical constants
Table A.1: Physical constants from Ref. [371].
Constant
Symbol
Value
Speed of light in vacuu,
Electric field constant
Planck’s constant
Planck’s (~ = h/2π)
Elementary charg
Rydberg constant
Bohr radius
Hartree Energy
Electron mass
Avogadro constant
Atomic mass unit
Molar gas constant
Boltzmann constant
c
0
h
~
e
R∞
a0
Eh
me
NA
u
R
k
2.99792458 · 108 m s−1 (exact)
8.854187817 · 10−12 A s V−1 m−1 (exact)
6.62606896(33) · 10−34 J s
1.054571628(53) · 10−34 J s
1.602176487(40) · 10−19 C
10973731.568525(73) m−1
0.529177210859(36) · 10−10 m
4.35974394(22) · 10−18 J
9.10938215(45) · 10−31 kg
6.02214179(30) · 1023 mol−1
1.660538782(83) · 10−27 kg
8.314472(15) J mol−1 K−1
1.3806504(24) · 10−23 J K−1
Table A.2:
significant.
Energy conversion factors from Ref. [371] The last digit is given, but may be not be
1J
1K
1 eV
1 Eh
=
ˆ
=
ˆ
=
ˆ
=
ˆ
5.034117 · 1022 cm−1
0.6950356 cm−1
8065.545 cm−1
219474.6 cm−1
139
Appendix A. Physical constants
140
Appendix B
Fundamental data on rare gas atoms
and dimers
141
Appendix B. Fundamental data on rare gas atoms and dimers
Table B.1: Natural abundances p A Rg , masses m A Rg and nucelar spins I A Rg of the stable
isotopes A Rg with mass numbers A of He, Ne, Ar, Kr and Xe. All data from Ref. [371].
a
p
A
Rg /%
Rare gas
A
m
He
3
4
0.000134(3)
99.999866(3)
Ne
20
21
22
Ar
A
Rg /u
I
A
Rg /~
3.0160293191(26)
4.00260325415(6)
4.002602(2) a
1/2
0
90.48(3)
0.27(1)
9.25(3)
19.9924401754(19)
20.99384668(4)
21.991385114(19)
20.1797(6) a
0
3/2
0
36
38
40
0.3365(30)
0.0632(5)
99.6003(30)
35.967545106(29)
37.9627324(4)
39.9623831225(29)
39.948(1) a
0
0
0
Kr
78
80
82
83
84
86
0.355(3)
2.286(10)
11.593(31)
11.500(19)
56.987(15)
17.279(41)
77.9203648(12)
79.9163790(16)
81.9134836(19)
82.914136(3)
83.911507(3)
85.91061073(11)
83.798(2) a
Xe
124
126
128
129
130
131
132
134
136
0.0952(3)
0.0890(2)
1.9102(8)
26.4006(82)
4.0710(13)
21.2324(30)
26.9086(33)
10.4357(21)
8.8573(44)
123.9058930(20)
125.904274(7)
127.9035313(15)
128.9047794(8)
129.9035080(8)
130.9050824(10)
131.9041535(10)
133.9053945(9)
135.907219(8)
131.293(6) a
0
0
0
9/2
0
0
0
0
0
1/2
0
3/2
0
0
0
Relative atomic mass (generally called atomic weight) Ar = m(Rg)/u (see Ref. [371]).
142
Table B.2: Static polarizabilities αd , quadrupole polarizabilities αq , quadrupoles Θ, disperion coefficients
Rg−Rg
C6,0
, polarizability anisotropies ∆αd and mean polarizabilities ᾱd of Ne, Ne+ , Ar, Ar+ , Kr, Kr+ , Xe
+
and Xe .
Rg
αd /a0 3
Ne
Ne+
Ar
Ar+
Kr
Kr+
Xe
Xe+
2.6629 a
1.37 d
11.074 a
6.71 d
16.737 a
10.90 d
27.2903 a
19.1 e
αq /a0 5
6.422
50.21
Θ/ea0 2
∆αd /a0 3
ᾱd /a0 3
b
0.438
d
d
0.06
1.122
d
−0.21
1.476
d
−0.62
1.33
d
d
6.85
d
d
10.90
b
95.55
b
212.6
b
2.0
f
−1.8
g
19.1
d
h
Rg−Rg
C6,0
/Eh a0 6
6.383 c
2.152 d
64.30 c
30.37 d
129.6 c
66.29 d
285.9 c
165.08 i
a
From Teachout and Pack [21].
From Standard and Certain [215].
c
From Kumar and Meath [218].
d
From Medved et al. [216].
e
Approximated by 0.7 · αXe
d .
f
Derived by extrapolation using literature values for the quadrupoles of Ne + , Ar+ and Kr+ [216].
g
Deduced by comparing the literature values of these quantities for the Ne, Ar, and Kr neutral atoms
[21] with the values of their cations [216].
+
h Xe+
ᾱd ≈ αXe
.
d
Xe−Xe
i
Calculated using the Slater-Kirkwood formula [216, 217]: C6,0
= 34 α3/2 Neff with α = 19.1 and
Neff = 6.953.
b
Table B.3: Natural abundances p
A
0
NeA Ne of Ne2 .
p
A
µ
A
0
Ne-A Ne /%
0
Ne-A Ne /u
A
0
0
NeA Ne and reduced masses µ A NeA Ne of the stable isotopomers
A \ A0
20
21
20
21
22
20
21
22
81.87
0.49
< 0.01
22
16.74
0.05
0.86
9.9962201 10.2404550 10.4721628
10.4969234 10.7405207
10.9956928
143
Appendix B. Fundamental data on rare gas atoms and dimers
Table B.4: Natural abundances p
A
0
0
XeA Xe in percent of the stable isotopomers A XeA Xe of Xe2 .
A \ A0
124
126
128
124
126
128
129
130
131
132
134
136
< 0.01
< 0.01
< 0.01
< 0.01
< 0.01
0.04
Table B.5:
Xe2 .
Reduced masses µ
A
A \ A0
124
126
128
129
130
131
132
134
136
0.05
0.05
1.01
6.97
0.01
0.01
0.16
2.15
0.17
0.04
0.04
0.81
11.21
1.73
4.51
0.05
0.05
1.03
14.21
2.19
11.43
7.24
0.02
0.02
0.40
5.51
0.85
4.43
5.62
1.09
0.02
0.02
0.34
4.68
0.72
3.76
4.77
1.85
0.78
0
XeA Xe in atomic mass units of the stable isotopomers
124
126
128
61.952947 62.201540 62.936490
62.452137 63.193056
63.951766
A \ A0
129
130
131
124
126
128
129
130
131
63.177957
63.436499
64.201102
64.452390
63.416919
63.677425
64.447881
64.701108
64.951754
63.654680
63.917147
64.693451
64.948615
65.201186
65.452541
A \ A0
132
134
136
124
126
128
129
130
131
132
134
136
63.889992
64.154408
64.936520
65.193610
65.448093
65.701359
65.952077
64.355861
64.624155
65.417834
65.678756
65.937049
66.194121
66.448620
66.952697
64.814686
65.086827
65.891982
66.156708
66.418781
66.679631
66.937884
67.449440
67.953609
144
A
0
XeA Xe of
Table B.6: Natural abundances p
A
0
0
ArA Xe in percent of the stable isotopomers A ArA Xe of ArXe.
A \ A0
124
126
128
36
38
40
< 0.01
< 0.01
0.09
< 0.01
< 0.01
0.09
< 0.01
< 0.01
1.91
Table B.7:
ArXe.
Reduced masses µ
A
129
130
0.09
0.01
0.02 < 0.01
26.33
4.06
131
132
134
136
0.07
0.01
21.10
0.09
0.02
26.78
0.04
0.01
10.40
0.03
0.01
8.83
0
XeA Ar in atomic mass units of the stable isotopomers
A \ A0
36
38
40
124
126
128
129
130
131
132
134
136
27.875743
27.925961
28.073143
28.121085
28.168329
28.215140
28.261278
28.352077
28.440761
29.059407
29.113984
29.273991
29.326126
29.377510
29.428430
29.478624
29.577428
29.673956
30.216799
30.275815
30.448885
30.505292
30.560896
30.616004
30.670335
30.777304
30.881837
Table B.8: Natural abundances p
A \ A0
124
126
78
80
82
83
84
86
< 0.01
< 0.01
0.01
0.01
0.05
0.02
< 0.01
< 0.01
0.01
0.01
0.05
0.02
A
A
0
XeA Ar of
0
0
KrA Xe in percent of the stable isotopomers A KrA Xe of KrXe.
128
129
130
131
132
134
136
0.01
0.04
0.22
0.22
1.09
0.33
0.09
0.60
3.06
3.04
15.04
4.56
0.01
0.09
0.47
0.47
2.32
0.70
0.08
0.49
2.46
2.44
12.10
3.67
0.10
0.62
3.12
3.09
15.33
4.65
0.04
0.24
1.21
1.20
5.95
1.80
0.03
0.20
1.03
1.02
5.05
1.53
145
Appendix B. Fundamental data on rare gas atoms and dimers
Table B.9:
KrXe.
Reduced masses µ
A
0
XeA Kr in atomic mass units of the stable isotopomers
A
A \ A0
78
80
82
83
84
86
124
126
128
129
130
131
132
134
136
47.837147
47.985228
48.421442
48.564247
48.705322
48.845444
48.983883
49.257263
49.525603
48.582082
48.734818
49.184831
49.332181
49.477760
49.622368
49.765253
50.047449
50.324491
49.312963
49.470337
49.934100
50.085981
50.236049
50.385131
50.532448
50.823438
51.109163
49.673864
49.833554
50.304184
50.458328
50.610640
50.761956
50.911489
51.206873
51.496937
50.030124
50.192116
50.669575
50.825971
50.980513
51.134052
51.285788
51.585544
51.879927
50.734002
50.900592
51.391692
51.552583
51.711583
51.869564
52.025704
52.334197
52.637212
146
0
XeA Kr of
Appendix C
Experimental data
147
Appendix C. Experimental data
C.1
Ne2
Table C.1: Relative positions of rotational levels (in cm−1 ), assigned quantum numbers J and derived
1
0
rotational constants B for the 0+
g state associated with the Ne( S0 ) + Ne(4p [3/2]2 ) dissociation limit.
isotopomer
20
Ne2
v
J
Ecalc /(hc cm−1 )
(Ecalc − Eobs )/(hc cm−1 )
0
0.109(1)
0
2
4
6
0.000
0.656
2.187
4.593
0.092
−0.007
−0.006
0.039
1
0.113(1)
0
2
4
0.000
0.676
2.254
0.059
−0.002
−0.002
2
0
2
4
0.000
0.681
2.271
0.002
0.001
−0.006
0
2
4
0.000
0.655
2.182
0.009
−0.004
−0.030
3
4
0
2
4
20
Ne-22 Ne
B/cm−1
0.000
0.615
2.051
0.114(2)
0.109(1)
0.103(1)
−0.008
−0.007
−0.005
0
0.111(8)
0
1
2
0.000
0.222
0.665
0.070
−0.003
−0.005
0
1
2
0.000
0.246
0.739
−0.028
0.012
−0.045
1
148
0.123(20)
C.1. Ne2
Table C.2: Relative positions of rotational levels (in cm−1 ), assigned quantum numbers J and derived
rotational constants B for the 1g state associated with the Ne(1 S0 ) + Ne(4p0 [3/2]2 ) dissociation limit.
isotopomer
20
Ne2
v
J
Ecalc /(hc cm−1 )
(Ecalc − Eobs )/(hc cm−1 )
4
0.148(1)
1
2
3
4
0.295
0.886
1.773
2.954
−0.032
−0.024
−0.015
0.017
5
0.122(1)
1
2
3
4
5
6
0.243
0.729
1.459
2.431
3.647
5.106
0.030
0.002
−0.019
0.031
0.014
0.058
2
3
4
0.615
1.231
2.052
−0.056
−0.008
−0.001
6
0.103(2)
7
1
2
3
20
Ne-22 Ne
B/cm−1
0.192
0.575
1.149
0.096(4)
0.002
0.003
−0.024
4
0.136(1)
1
2
3
0.272
0.815
1.630
0.004
0.000
−0.003
1
2
3
0.240
0.721
1.443
0.012
−0.016
−0.030
5
149
0.120(3)
Appendix C. Experimental data
Table C.3: Measured (Eobs ) and calculated (Ecalc ) term values (with respect to the dissociation limit
of the neutral ground state of Ne2 (see Eq. (5.5))) and rotational constants of the vibrational levels of the
1
0
0+
g state associated with the Ne( S0 ) + Ne(4p [3/2]2 ) dissociation limit. Values in parentheses represent
the statistical experimental uncertainty (one standard deviation).
v
Eobs /(hc cm−1 )
Ne2
0
1
2
3
4
5
6
163640.8(15)
163674.5(12)
163704.8(14)
163731.4(4)
163755.9(10)
163777.1(36)
163643.4
163674.3
163703.6
163731.2
163756.9
163780.3
163800.9
0.109(1)
0.113(1)
0.114(2)
0.109(1)
0.103(1)
0.123
0.118
0.114
0.109
0.103
0.096
0.088
Ne−22 Ne
0
1
2
3
4
5
6
163640.4(14)
163673.5(13)
163703.2(13)
163729.5(11)
163753.4(14)
163774.2(6)
163643.1
163673.3
163702.0
163729.1
163754.4
163777.5
163798.0
0.111(8)
0.123(20)
0.123
0.119
0.114
0.109
0.104
0.097
0.089
isotopomer
20
20
Ecalc /(hc cm−1 ) Bobs /cm−1
150
Bcalc /cm−1
C.1. Ne2
Table C.4: Measured (Eobs ) and calculated (Ecalc ) term values (with respect to the dissociation limit of
the neutral ground state of Ne2 (see Eq. (5.5))) and rotational constants of the vibrational levels of the 1 g
state associated with the Ne(1 S0 ) + Ne(4p0 [3/2]2 ) dissociation limit. Values in parentheses represent the
statistical experimental uncertainty (one standard deviation).
isotopomer
20
20
Ne2
Ne−22 Ne
v
Eobs /(hc cm−1 )
0
1
2
3
4
5
6
7
8
9
163377.5(14)
163448.5(14)
163508.6(10)
163559.9(13)
163603.2(11)
163638.8(11)
163665.1(16)
163684.0(9)
0
1
2
3
4
5
6
7
8
9
163505.0(11)
163555.9(11)
163599.2(11)
163634.8(10)
163661.3(18)
163680.7(15)
Ecalc /(hc cm−1 ) Bobs /cm−1
163373.6
163449.6
163511.3
163561.2
163601.6
163634.6
163661.9
163684.4
163702.5
163717.1
163372.6
163447.2
163508.1
163557.5
163597.8
163630.8
163658.2
163680.8
163699.3
163714.1
151
0.148(1)
0.122(1)
0.103(2)
0.096(4)
0.136(1)
0.120(3)
Bcalc /cm−1
0.207
0.190
0.173
0.156
0.140
0.125
0.112
0.100
0.087
0.078
0.208
0.191
0.174
0.157
0.141
0.127
0.114
0.102
0.090
0.079
Appendix C. Experimental data
C.2
Xe+
2
129
Table C.5: Measured positions ν̃obs of the vibrational levels of the C 0+
Xe132 Xe
u Rydberg state of
00
and 131 Xe132 Xe relative to the X 0+
g (v = 0) ground neutral state. The uncertainties include possible
errors in the calibration. DL indicates the position of the Xe∗ ([5p]5 6s0 [1/2]1 ) + Xe(1 S0 ) dissociation limit.
129
Xe132 Xe
131
Xe132 Xe
v0
ν̃obs /cm−1
ν̃obs /cm−1
14
15
16
17
18
19
20
21
22
23
24
25
26
76511.8(9)
76545.8(6)
76579.8(8)
76613.7(8)
76647.6(7)
76680.8(9)
76713.5(8)
76745.5(8)
76776.8(8)
76807.2(7)
76836.5(7)
76864.9(5)
76891.9(5)
76509.0(9)
76543.3(7)
76577.4(9)
76611.5(8)
76645.1(9)
76678.4(8)
76711.0(9)
76743.0(9)
76774.2(9)
76804.6(7)
76833.8(6)
76862.2(6)
76889.7(5)
DL
77371.04
77371.04
152
C.2. Xe+
2
Table C.6: Measured positions ν̃obs from Rupper et al. [53] and differences between measured and
calculated positions (∆ν̃ = ν̃obs − ν̃calc) of the vibrational levels of the I(1/2u) state of 131 Xe132 Xe+ relative
00
to the X 0+
g (v = 0) ground neutral state. The uncertainties include possible errors in the calibration and
in the determination of the field-induced shift of the ionization thresholds. DL indicates the position of
the Xe+ (2 P3/2 ) + Xe(1 S0 ) dissociation limit.
a
v+
ν̃obs /cm−1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
90147.5(17)
90269.8(20)
90390.6(35)
90510.5(17)
90629.9(30)
90748.1(14)
90866.0(13)
90982.5(11)
91097.4(19)
91325.2(30)
91438.1(24)
91550.2(19)
91660.6(28)
91771.2(28)
91879.5(22)
91987.5(28)
92094.1(28)
92305.7(30)
92409.8(29)
92512.3(30)
92614.6(25)
92715.8(30)
92816.3(21)
92915.4(14)
93013.6(23)
93110.7(22)
93207.7(28)
93302.8(30)
93396.9(13)
93490.5(23)
93583.3(17)
93674.9(20)
93855.1(14)
93944.0(14)
94031.8(33)
∆ν̃/cm−1
a
3.5
3.3
2.6
1.9
1.6
1.2
1.4
1.1
0.2
91212.0
−0.7
−0.7
−0.5
−1.1
−0.5
−1.2
−1.3
−1.8
92202.0
−1.5
−1.6
−2.3
−2.2
−2.3
−2.1
−2.3
−2.5
−2.7
−2.1
−2.4
−2.7
−2.6
−2.3
−2.2
93767.6
−2.0
−1.7
−1.5
v+
ν̃obs /cm−1
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
94118.2(26)
94204.3(14)
94288.7(14)
94372.5(17)
94455.2(12)
94537.5(23)
94618.4(15)
94697.9(13)
94777.2(23)
94855.5(17)
94932.2(18)
95008.4(15)
95083.7(20)
95157.2(19)
95230.6(15)
95303.3(16)
95374.2(18)
95443.9(18)
95513.9(31)
95581.7(20)
95649.6(23)
95715.1(17)
95781.0(14)
95845.5(37)
DL
∆ν̃/cm−1
−1.7
−1.2
−1.4
−1.3
−1.2
−0.6
−0.4
−0.6
−0.06
0.5
0.4
0.8
1.3
1.0
1.5
2.4
2.4
2.2
3.3
3.1
4.1
3.6
4.5
5.0
95903.5
95965.5
96026.6
96086.7
96145.8
96203.9
96261.1
96317.3
96372.4
96426.7
96479.9
96532.2
96583.5
96633.8
98019.78
When the transitions could not be observed, the calculated positions are listed.
153
a
Appendix C. Experimental data
Table C.7:
Measured positions ν̃obs and differences between measured and calculated positions
(∆ν̃ = ν̃obs − ν̃calc ) of the vibrational levels of the I(3/2g) state of 131 Xe132 Xe+ relative to the X 0+
g
(v 00 = 0) ground neutral state. The uncertainties include possible errors in the calibration and in the
determination of the field-induced shift of the ionization thresholds. DL indicates the position of the
Xe+ (2 P3/2 ) + Xe(1 S0 ) dissociation limit.
v+
ν̃obs /cm−1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
96219.9(20)
96278.0(31)
96334.7(36)
96389.9(27)
96445.0(22)
96497.7(39)
96550.4(47)
96602.2(26)
96652.4(38)
96701.7(11)
96750.1(18)
96797.0(17)
96844.2(25)
96889.4(11)
96935.0(10)
96978.9(30)
97021.2(28)
97062.3(10)
97179.3(11)
97217.3(38)
∆ν̃/cm−1
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
97357.2(19)
97389.3(19)
97420.9(15)
97450.9(17)
97480.3(14)
a
0.3
0.6
0.6
0.05
0.4
−0.5
−0.5
−0.3
−0.7
−0.9
−1.0
−1.6
−0.9
−1.2
0.01
0.5
0.4
0.1
97102.5
97141.9
−0.9
−0.2
97253.8
97289.1
97323.4
0.5
0.2
0.5
0.2
0.2
v+
ν̃obs /cm−1
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
97508.8(15)
97535.9(24)
97562.5(22)
97588.4(14)
97612.9(18)
97636.5(17)
97658.7(17)
97681.2(18)
97701.3(25)
97721.3(17)
97740.0(20)
97758.2(16)
97775.4(16)
97792.0(16)
97807.2(17)
97822.6(16)
97836.3(29)
97849.3(25)
97861.5(28)
97873.7(17)
97885.1(20)
97895.9(10)
97905.1(16)
DL
a
b
∆ν̃/cm−1
0.3
0.01
0.1
0.5
0.5
0.5
−0.02
0.7
−0.05
−0.01
−0.4
−0.4
−0.5
−0.4
−0.9
−0.4
−0.8
−1.1
−1.4
−1.1
−0.8
−0.5
−1.1
97915.4
97923.9
97932.0
97939.4
97946.4
97952.8
97958.8
98019.78
When the transitions could not be observed, the calculated positions are listed.
Data from Rupper et al. [53].
154
a
C.2. Xe+
2
Table C.8:
Measured positions ν̃obs and differences between measured and calculated positions
(∆ν̃ = ν̃obs − ν̃calc ) of the vibrational levels of the I(3/2u) state of 131 Xe132 Xe+ relative to the X 0+
g
(v 00 = 0) ground neutral state. The uncertainties include possible errors in the calibration and in the
determination of the field-induced shift of the ionization thresholds. DL indicates the position of the
Xe+ (2 P3/2 ) + Xe(1 S0 ) dissociation limit.
v+
ν̃obs /cm−1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
97577.6(15)
97600.0(15)
97621.4(16)
97642.2(12)
97662.2(16)
97681.2(19)
97699.8(23)
97717.6(15)
97766.5(19)
97781.6(17)
97795.7(15)
97809.5(28)
97822.6(16)
∆ν̃/cm−1
a
0.3
0.5
0.4
0.5
0.6
0.4
0.5
0.5
97734.1
97750.5
0.3
0.4
0.1
0.2
0.2
v+
ν̃obs /cm−1
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
97835.2(27)
97846.8(17)
97858.1(16)
97868.8(15)
97879.0(24)
97888.8(16)
97896.9(13)
97907.0(12)
DL
a
∆ν̃/cm−1
a
0.2
−0.1
−0.2
−0.3
−0.3
−0.3
−1.4
−0.09
97915.4
97923.2
97930.6
97937.5
97944.1
97950.2
97956.0
98019.78
When the transitions could not be observed, the calculated positions are listed.
Table C.9:
Calculated positions ν̃calc of the vibrational levels of the I(1/2g) state of 131 Xe132 Xe+
00
+ 2
1
relative to the X 0+
g (v = 0) ground neutral state. DL indicates the position of the Xe ( P3/2 ) + Xe( S0 )
dissociation limit.
v+
ν̃calc /cm−1
v+
ν̃calc /cm−1
0
1
2
3
4
97948.7
97952.4
97956.0
97959.5
97962.9
5
6
7
8
9
97966.1
97969.3
97972.4
97975.4
97978.2
DL
98019.78
155
Appendix C. Experimental data
Table C.10:
Measured positions ν̃obs and differences between measured and calculated positions
(∆ν̃ = ν̃obs − ν̃calc ) of the vibrational levels of the II(1/2u) state of 131 Xe132 Xe+ relative to the X 0+
g
(v 00 = 0) ground neutral state. The uncertainties include possible errors in the calibration and in the
determination of the field-induced shift of the ionization thresholds. DL indicates the position of the
Xe+ (2 P1/2 ) + Xe(1 S0 ) dissociation limit.
v+
ν̃obs /cm−1
0
1
2
3
4
5
6
7
8
9
107159.6(23)
107207.7(17)
107254.5(24)
107301.2(19)
107346.4(17)
107391.0(19)
107434.3(19)
107476.8(15)
107517.8(29)
107558.7(38)
∆ν̃/cm−1
a
−1.6
−1.4
−1.6
−1.0
−1.0
−0.7
−0.7
−0.7
−1.2
−1.0
v+
ν̃obs /cm−1
10
11
12
13
14
15
16
17
18
19
107603.2(29)
107643.9(16)
DL
∆ν̃/cm−1
b
b
a
3.8
5.7
107676.1
107713.1
107749.2
107784.3
107818.6
107852.0
107884.5
107916.1
108556.71
a
When the transition could not be observed, the calculated positions are listed.
Data from Rupper et al. [53], incorrectly assigned to the 131 Xe132 Xe+ isotopomer. The corrected
positions and deviations are 107598.5(10) cm−1 and −0.9 cm−1 for v + = 10 and 107637.0(5) cm−1 and
−1.2 cm−1 for v + = 11, respectively.
b
Table C.11: Measured position ν̃obs from Rupper et al.[53] and difference between measured and calculated position (∆ν̃ = ν̃obs − ν̃calc ) of the vibrational level of the II(1/2g) state of 131 Xe132 Xe+ relative to
00
the X 0+
g (v = 0) ground neutral state. The uncertainty includes possible errors in the calibration and in
the determination of the field-induced shift of the ionization thresholds. DL indicates the position of the
Xe+ (2 P1/2 ) + Xe(1 S0 ) dissociation limit.
a
v+
ν̃obs /cm−1
0
1
2
3
4
108132.3(13)
∆ν̃/cm−1
a
0.01
108154.9
108176.7
108197.7
108217.9
v+
ν̃obs /cm−1
∆ν̃/cm−1
5
6
7
8
9
108237.3
108255.9
108273.8
108290.8
108307.1
DL
108556.71
When the transitions could not be observed, the calculated positions are listed.
156
a
C.3. ArXe
C.3
ArXe
Table C.12: Measured positions of the vibrational levels of the C 1 and D 0+ states of 40 Ar129 Xe relative
to the X 0+ (v 00 = 0) ground neutral state. The uncertainties include possible errors in the calibration.
DL indicates the position of the Xe∗ ([5p]5 6s0 [1/2]1 ) + Ar(1 S0 ) dissociation limit.
v0
0
1
2
3
4
5
6
7
DL
C1
D 0+
ν̃obs /cm−1
ν̃obs /cm−1
77201.1(12)
77090.4(9) 77222.9(13)
77117.9(12) 77239.0(12)
77137.2(11) 77250.0(11)
77159.9(11)
77184.1(7)
77204.4(10)
77228.6(10)
77302.34
157
77302.34
Appendix C. Experimental data
Table C.13:
Measured positions ν̃obs and differences between measured and calculated positions
(∆ν̃ = ν̃obs − ν̃calc ) of the vibrational levels of the X 1/2 state of ArXe+ relative to the X 0+ (v 00 = 0) ground
neutral state. The uncertainties include possible errors in the calibration and in the determination of the
field-induced shift of the ionization thresholds. DL indicates the position of the Xe+ (2 P3/2 ) + Ar(1 S0 )
dissociation limit.
v+
ν̃obs /cm−1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
96514.9(19)
96601.6(19)
96686.1(22)
96766.7(26)
96844.0(25)
96918.4(19)
96990.7(23)
97060.4(35)
97126.4(40)
97188.7(31)
97248.7(26)
97304.5(19)
97358.8(26)
97409.3(20)
97457.8(31)
DL
∆ν̃ H /cm−1
a,c
∆ν̃ Y /cm−1
−0.2
0.1
1.1
1.1
0.7
0.4
0.9
1.7
1.8
1.0
0.9
−0.6
−0.6
−1.7
−1.8
97505.5
97548.5
97588.8
97626.4
97661.3
97693.6
0.6
0.4
0.9
0.6
0.04
−0.4
−0.04
0.7
0.8
0.07
−0.01
−1.4
−1.4
−2.3
−2.3
97505.9
97548.9
97589.1
97626.7
97660.6
97693.8
97951.08
97951.08
a
b,c
Calculated positions using the data set with the assignment for the B 1/2 state proposed by Huber
and Lipson [197].
b
Calculated positions using the data set with the assignment for the B 1/2 state proposed by Yoshii et
al. [207].
c
When the transition could not be observed, the calculated positions are listed.
158
C.3. ArXe
Table C.14:
Measured positions ν̃obs and differences between measured and calculated positions
(∆ν̃ = ν̃obs − ν̃calc ) of the vibrational levels of the A1 3/2 state of ArXe+ relative to the X 0+ (v 00 = 0)
ground neutral state. The uncertainties include possible errors in the calibration and in the determination
of the field-induced shift of the ionization thresholds. DL indicates the position of the Xe+ (2 P3/2 ) + Ar(1 S0 )
dissociation limit.
v+
ν̃obs /cm−1
0
1
2
3
4
5
6
7
8
9
10
97432.6(10)
97478.9(11)
97522.4(13)
97563.4(11)
97601.7(10)
97637.6(14)
DL
∆ν̃ H /cm−1
a,c
∆ν̃ Y /cm−1
0.2
0.1
−0.09
−0.1
−0.3
−0.3
97671.2
97702.1
97730.6
97756.8
97780.7
-0.05
0.01
−0.1
−0.09
−0.2
−0.2
97671.1
97702.0
97730.5
97756.7
97780.6
97951.08
97951.08
a
b,c
Calculated positions using the data set with the assignment for the B 1/2 state proposed by Huber
and Lipson [197].
b
Calculated positions using the data set with the assignment for the B 1/2 state proposed by Yoshii et
al. [207].
c
When the transitions could not be observed, the calculated positions are listed.
159
Appendix C. Experimental data
Table C.15:
Measured positions ν̃obs and differences between measured and calculated positions
(∆ν̃ = ν̃obs − ν̃calc ) of the vibrational levels of the A2 1/2 state of ArXe+ relative to the X 0+ (v 00 = 0)
ground neutral state. The uncertainties include possible errors in the calibration and in the determination
of the field-induced shift of the ionization thresholds. DL indicates the position of the Xe+ (2 P1/2 ) + Ar(1 S0 )
dissociation limit.
v+
ν̃obs /cm−1
0
1
2
3
4
5
6
7
8
9
10
107637.9(22)
107704.2(19)
107767.2(19)
107827.1(20)
107883.3(17)
107936.5(22)
107986.6(20)
108033.9(19)
108077.6(13)
108119.1(13)
DL
∆ν̃ H /cm−1
a,c
∆ν̃ Y /cm−1
b,c
−2.3
−1.5
−0.9
−0.2
−0.1
0.06
0.1
0.4
0.01
0.3
108157.2
−3.2
−2.1
−1.1
−0.1
0.3
0.7
1.0
1.4
1.1
1.5
108155.9
108488.01
108488.01
a
Calculated positions using the data set with the assignment for the B 1/2 state proposed by Huber
and Lipson [197].
b
Calculated positions using the data set with the assignment for the B 1/2 state proposed by Yoshii et
al. [207].
c
When the transition could not be observed, the calculated positions are listed.
160
C.3. ArXe
Table C.16: Measured positions ν̃obs from Huber and Lipson [197] and differences between measured
and calculated positions (∆ν̃ = ν̃obs − ν̃calc) of the vibrational levels of the B 1/2 state of ArXe+ relative to
the X 0+ (v 00 = 0) ground neutral state. DL indicates the position of the Xe(1 S0 ) + Ar+ (2 P3/2 ) dissociation
limit.
Huber and Lipson
v+
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
DL
ν̃obs /cm−1
a
126217.6(20)
126274.6(20)
126328.9(20)
126381.8(20)
126433.3(20)
126482.8(20)
126351.0(20)
126575.6(20)
126619.7(20)
126660.4(20)
126701.6(20)
126738.5(20)
126775.3(20)
126808.7(20)
∆ν̃ H /cm−1
3.5
2.1
0.3
−0.7
−0.9
−0.9
0.1
−0.7
0.3
−0.4
1.5
1.0
2.3
2.1
126838.6
126868.8
126897.2
126923.9
126949.1
126972.7
126994.9
127227.14
Yoshii et al.
c
v+
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
DL
a
ν̃obs /cm−1
a
126095.1(50) b
126154.0(50) b
126217.6(20)
126274.6(20)
126328.9(20)
126381.8(20)
126433.3(20)
126482.8(20)
126351.0(20)
126575.6(20)
126619.7(20)
126660.4(20)
126701.6(20)
126738.5(20)
126775.3(20)
126808.7(20)
∆ν̃ Y /cm−1
c
−8.3
−10.4
−5.4
−4.9
−4.9
−4.1
−2.7
−1.2
1.0
1.6
3.8
4.1
6.8
7.1
9.0
9.3
126831.1
126861.0
126889.2
126915.9
126941.1
127227.14
Data from Huber and Lipson [197] using the value of 107704.2 cm−1 for the A2 1/2 (v + = 1) level
measured in this work.
b
Vibrational positions observed by Yoshii et al. [207].
c
When the transitions could not be observed, the calculated positions are listed.
161
Appendix C. Experimental data
Table C.17: Measured positions ν̃obs from Huber and Lipson [197] and differences between measured and
calculated positions (∆ν̃ = ν̃obs − ν̃calc) of the vibrational levels of the C1 3/2 state of ArXe+ relative to the
X 0+ (v 00 = 0) ground neutral state. DL indicates the position of the Xe(1 S0 ) + Ar+ (2 P3/2 ) dissociation
limit.
v+
ν̃obs /cm−1
0
1
2
3
4
5
6
7
8
9
10
126065.7(20)
126130.2(20)
126192.5(20)
126252.3(20)
126310.0(20)
DL
a
∆ν̃ H /cm−1
b,d
∆ν̃ Y /cm−1
c,d
3.7
2.2
0.9
−0.5
−1.7
126368.3
126422.7
126474.8
126524.6
126572.3
126617.7
−2.3
−2.3
−2.0
−1.7
−1.0
126365.8
126417.9
126467.8
126515.5
126560.9
126604.3
127227.14
127227.14
a
Data from Huber and Lipson [197] using the value of 97432.6 cm−1 for the A1 3/2 (v + = 0) level
measured in this work.
b
Calculated positions using the data set with the assignment for the B 1/2 state proposed by Huber
and Lipson [197].
c
Calculated positions using the data set with the assignment for the B 1/2 state proposed by Yoshii et
al. [207].
d
When the transitions could not be observed, the calculated positions are listed.
162
C.3. ArXe
Table C.18: Measured positions ν̃obs from Huber and Lipson [197] and differences between measured and
calculated positions (∆ν̃ = ν̃obs − ν̃calc) of the vibrational levels of the C2 1/2 state of ArXe+ relative to the
X 0+ (v 00 = 0) ground neutral state. DL indicates the position of the Xe(1 S0 ) + Ar+ (2 P1/2 ) dissociation
limit.
v+
ν̃obs /cm−1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
127581.9(20)
127644.8(20)
127705.5(20)
127763.8(20)
127819.9(20)
127873.5(20)
127924.5(20)
127974.3(20)
128021.2(20)
128066.2(20)
128108.6(20)
128147.0(20)
DL
a
∆ν̃ H /cm−1
b,d
∆ν̃ Y /cm−1
c,d
−9.5
−7.3
−5.0
−2.9
−0.9
0.6
1.7
3.8
5.0
6.3
7.0
5.7
128179.3
128215.1
128249.0
128281.1
128311.4
128339.9
128366.8
128391.9
128415.5
7.8
4.6
2.4
0.8
−0.3
−1.1
−1.8
−1.0
−0.8
−0.1
0.3
−1.1
128185.9
128221.5
128255.0
128286.5
128316.3
128344.2
128370.3
128394.8
128417.8
128658.73
128658.73
a
Data from Huber and Lipson [197] using the value of 107637.9 cm−1 for the A2 1/2 (v + = 0) level
measured in this work.
b
Calculated positions using the data set with the assignment for the B 1/2 state proposed by Huber
and Lipson [197].
c
Calculated positions using the data set with the assignment for the B 1/2 state proposed by Yoshii et
al. [207].
d
When the transitions could not be observed, the calculated positions are listed.
163
Appendix C. Experimental data
C.4
KrXe
Table C.19: Measured positions of the vibrational levels of the C 0+ and D 1 states of 84 Kr132 Xe relative
to the X 0+ (v 00 = 0) ground neutral state. The uncertainties include possible errors in the calibration.
DL indicates the position of the Xe∗ ([5p]5 6s0 [1/2]1 ) + Kr(1 S0 ) dissociation limit.
C 0+
D1
v0
ν̃obs /cm−1
v0
ν̃obs /cm−1
16
17
18
19
20
21
77018.7(10)
77055.9(9)
77089.4(11)
77119.4(10)
77146.2(7)
77169.8(5)
m
m+1
m+2
m+3
77289.6(8)
77307.3(7)
77318.1(7)
77330.9(8)
DL
77336.34
DL
77336.34
Table C.20: Measured positions of the vibrational levels of the Rydberg state of 84 Kr131 Xe correlated
with the Xe(1 S0 ) + Kr∗ ([4p]5 5s[3/2]1) dissociation limit relative to the X 0+ (v 00 = 0) ground neutral
state. The uncertainties include possible errors in the calibration. DL indicates the position of the
Xe(1 S0 ) + Kr∗ ([4p]5 5s[3/2]1) dissociation limit.
v0
ν̃obs /cm−1
v0
ν̃obs /cm−1
n
n+1
n+2
n+3
n+4
n+5
n+6
n+7
n+8
81005.4(7)
81011.9(6)
81017.8(7)
81023.2(7)
81028.3(5)
81033.0(5)
81037.4(7)
81041.6(6)
81045.6(5)
n+9
n + 10
n + 11
n + 12
n + 13
n + 14
n + 15
n + 16
n + 17
81049.2(8)
81052.5(7)
81055.5(6)
81058.2(6)
81060.7(7)
81062.6(4)
81064.2(4)
81065.3(3)
81066.1(2)
DL
81068.07
164
C.4. KrXe
Table C.21:
Measured positions ν̃obs and differences between measured and calculated positions
(∆ν̃ = ν̃obs − ν̃calc ) of the vibrational levels of the X 1/2 state of 84 Kr132 Xe+ relative to the X 0+ (v 00 = 0)
ground neutral state. The uncertainties include possible errors in the calibration and in the determination
of the field-induced shift of the ionization thresholds. DL indicates the position of the Xe+ (2 P3/2 ) + Kr(1 S0 )
dissociation limit.
v+
ν̃obs /cm−1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
94822.0(8)
94934.8(18)
95044.9(22)
95152.5(15)
95258.9(18)
95362.0(11)
95463.3(15)
95562.4(15)
95659.2(11)
95753.3(14)
95845.5(12)
95935.4(14)
96023.2(14)
96107.5(15)
96192.0(14)
96272.6(14)
96351.9(10)
96428.5(13)
96501.9(18)
96573.7(15)
∆ν̃/cm−1
a
0.5
0.5
0.1
−0.5
0.01
−0.5
−0.5
−0.4
−0.3
−0.7
−0.6
−0.5
−0.3
−1.3
0.2
0.0
0.8
1.1
0.4
0.4
v+
ν̃obs /cm−1
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
96643.6(12)
96711.0(12)
96776.3(16)
96839.5(13)
96900.3(15)
96959.3(11)
97016.5(15)
97071.3(16)
97124.0(14)
97174.6(17)
97223.2(14)
97270.0(16)
97314.6(7)
97357.7(21)
DL
a
∆ν̃/cm−1
0.6
0.5
0.5
0.5
0.2
0.2
0.5
0.5
0.4
0.2
−0.01
−0.1
−0.4
−0.4
97399.2
97438.6
97476.1
97511.9
97545.9
97578.3
97985.08
When the transitions could not be observed, the calculated positions are listed.
165
a
Appendix C. Experimental data
Table C.22:
Measured positions ν̃obs and differences between measured and calculated positions
(∆ν̃ = ν̃obs − ν̃calc ) of the vibrational levels of the A1 3/2 state of 84 Kr132 Xe+ relative to the X 0+
(v 00 = 0) ground neutral state. The uncertainties include possible errors in the calibration and in the
determination of the field-induced shift of the ionization thresholds. DL indicates the position of the
Xe+ (2 P3/2 ) + Kr(1 S0 ) dissociation limit.
a
v+
ν̃obs /cm−1
0
1
2
3
4
5
6
7
8
9
97282.8(15)
97323.0(15)
97361.4(13)
97398.6(16)
97434.0(14)
97468.3(14)
97501.2(11)
97532.0(12)
97563.0(10)
97590.4(16)
∆ν̃/cm−1
a
8.1
6.2
4.1
2.5
0.8
−0.4
−1.4
−2.9
−2.6
−4.5
v+
ν̃obs /cm−1
10
11
12
13
14
15
16
17
18
19
97619.4(16)
97645.4(13)
97670.3(16)
97693.5(12)
97715.8(7)
97736.5(15)
DL
∆ν̃/cm−1
−3.2
−3.6
−3.6
−4.0
−3.9
−4.2
97760.4
97778.9
97796.2
97812.5
97985.08
When the transitions could not be observed, the calculated positions are listed.
166
a
C.4. KrXe
Table C.23:
Measured positions ν̃obs and differences between measured and calculated positions
(∆ν̃ = ν̃obs − ν̃calc ) of the vibrational levels of the A2 1/2 state of 84 Kr132 Xe+ relative to the X 0+
(v 00 = 0) ground neutral state. The uncertainties include possible errors in the calibration and in the
determination of the field-induced shift of the ionization thresholds. DL indicates the position of the
Xe+ (2 P1/2 ) + Kr(1 S0 ) dissociation limit.
v+
ν̃obs /cm−1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
106546.4(18)
106630.4(12)
106713.4(20)
106794.0(29)
106873.9(14)
106952.1(12)
107026.8(16)
107101.4(12)
107173.8(20)
107243.5(12)
107312.0(30)
107377.5(10)
107441.6(18)
107503.5(14)
107563.3(18)
∆ν̃/cm−1
a
−0.4
−1.4
−1.7
−2.3
−1.8
−1.1
−2.0
−1.0
−0.2
−0.1
0.8
0.8
1.4
2.0
2.5
v+
ν̃obs /cm−1
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
107620.9(16)
107676.5(16)
107730.0(15)
107781.4(23)
107830.4(20)
DL
a
∆ν̃/cm−1
a
3.0
3.5
4.2
4.8
5.2
107871.7
107916.1
107958.4
107998.7
108036.9
108073.1
108107.3
108139.6
108170.0
108198.6
108522.01
When the transitions could not be observed, the calculated positions are listed.
Table C.24:
Measured positions ν̃obs and differences between measured and calculated positions
(∆ν̃ = ν̃obs − ν̃calc ) of the vibrational levels of the B 1/2 state of KrXe+ relative to the X 0+ (v 00 = 0) ground
neutral state. The uncertainties include possible errors in the calibration and in the determination of the
field-induced shift of the ionization thresholds. DL indicates the position of the Xe(1 S0 ) + Kr+ (2 P3/2 )
dissociation limit.
a
v+
ν̃obs /cm−1
0
1
2
3
4
112426.6(19)
∆ν̃/cm−1
a
1.3
112457.4
112488.4
112518.5
112547.5
v+
ν̃obs /cm−1
∆ν̃/cm−1
5
6
7
8
9
112575.5
112602.6
112628.7
112653.8
112677.9
DL
113065.73
When the transitions could not be observed, the calculated positions are listed.
167
a
Appendix C. Experimental data
Table C.25:
Measured positions ν̃obs and differences between measured and calculated positions
(∆ν̃ = ν̃obs − ν̃calc ) of the vibrational levels of the C1 3/2 state of KrXe+ relative to the X 0+ (v 00 = 0)
ground neutral state. The uncertainties include possible errors in the calibration and in the determination
of the field-induced shift of the ionization thresholds. DL indicates the position of the Xe(1 S0 ) + Kr+ (2 P3/2 )
dissociation limit.
a
v+
ν̃obs /cm−1
0
1
2
3
4
5
6
7
8
9
112055.5(14)
112101.5(14)
112145.8(14)
112189.5(14)
112232.1(21)
112272.5(20)
112312.0(15)
112350.3(30)
112385.1(19)
112421.2(10)
∆ν̃/cm−1
a
0.4
0.2
−0.3
−0.08
0.5
0.2
0.3
0.6
−1.2
−0.5
v+
ν̃obs /cm−1
∆ν̃/cm−1
a
10
11
12
13
14
15
16
17
18
19
112455.8
112488.7
112520.2
112550.6
112579.8
112607.7
112634.6
112660.3
112684.8
112708.3
DL
113065.73
When the transitions could not be observed, the calculated positions are listed.
Table C.26:
Calculated positions ν̃calc and observed positions ν̃obs from Yoshii et al. [206] of the
vibrational levels of the C2 1/2 state of KrXe+ relative to the X 0+ (v 00 = 0) ground neutral state. DL
indicates the position of the Xe(1 S0 ) + Kr+ (2 P1/2 ) dissociation limit.
v+
ν̃calc /cm−1
v+
ν̃obs /cm−1
0
1
2
3
4
5
6
7
8
9
117646.8
117686.3
117724.4
117761.1
117796.6
117830.8
117863.7
117895.3
117925.7
117954.9
0
1
2
3
117542(25)
117586(25)
117620(25)
117662(25)
DL
118435.80
118435.80
168
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Curriculum Vitae
Curriculum Vitae
Name
Date of birth
Place of origin
Nationality
Oliver Zehnder
21 May 1974
Birmenstorf
Swiss
Education
09/2003 - 03/2008 PhD thesis in the group of Prof. F. Merkt
at the Laboratory of Physical Chemistry, ETH Zurich
10/1994 - 03/1999 Studies of physics at ETH Zurich
08/1990 - 01/1994 Matura at Kantonsschule Baden
Work experience
09/2003 - 12/2007 Teaching assistant in several lectures and laboratory courses
at the Laboratory of Physical Chemistry, ETH Zurich
11/2002 - 08/2003 Researcher at IT’IS (Foundation for Research on Information
Technologies in Society), Zurich
04/2001 - 10/2002 Optical engineer at GigaTera Inc, Dietikon
09/1999 - 03/2001 Research and teaching assistant at the Institute of
Quantum Electronics, ETH Zurich
Languages
German, English, French, Italian
199
Curriculum Vitae
Publications
Publications
1. “Potential energy curves of diatomic molecular ions from high-resolution photoelectron
spectra. II. The first six electronic states of Xe+
2”
P. Rupper, O. Zehnder and F. Merkt
J. Chem. Phys. 121(17), 8279 − 8290 (2004)
2. “Autoionizing even 2p51/2 nl0 [K 0 ]0,1,2 (l0 = 1, 3) Rydberg series of Ne: a comparison of
many-electron theory and experiment”
I.D. Petrov, V.L. Sukhorukov, T. Peters, O. Zehnder, H.J. Wörner, F. Merkt and H.
Hotop
J. Phys. B: At. Mol. Opt. Phys. 39(16), 3159 − 3176 (2006)
3. “Spectroscopic characterization of the potential energy functions of Ne2 Rydberg states
in the vicinity of the Ne(1 S0 ) + Ne(4p0 ) dissociation limits”
E. Kleimenov, O. Zehnder and F. Merkt
J. Mol. Spectrosc. 247(1), 85 − 99 (2008)
4. “The low-lying electronic states of ArXe+ and their potential energy functions”
O. Zehnder and F. Merkt
J. Chem. Phys. 128(1), 014306 (2008)
5. “Spectroscopic study and multichannel quantum defect theory analysis of the Stark
effect in Rydberg states of neon”
M. Grütter, O. Zehnder, T.P. Softley and F. Merkt
J. Phys. B: At. Mol. Opt. Phys. 44(11), 115001 (2008)
6. “On the R-dependence of the spin-orbit coupling constant: Potential energy functions
of Xe+
2 by high-resolution spectroscopy and ab initio quantum chemistry”
O. Zehnder, R. Mastalerz, M. Reiher, F. Merkt and R.A. Dressler
J. Chem. Phys. 128(23), 234306 (2008)
7. “The low-lying electronic states of KrXe+ and their potential energy functions”
O. Zehnder and F. Merkt
Mol. Phys. 106(9-10), 1215 (2008)
200
Curriculum Vitae
Poster presentations
1. “High-resolution photoelectron spectroscopy study of the energy level structure and
potential curves of Xe+
2”
8th EPS Conference on Atomic and Molecular Physics
Rennes, France, July 6-10, 2004
2. “High-resolution photoelectron spectroscopy study of the first six electronic states of
Xe+
2”
28th International Symposium on Free Radicals
Leysin, Switzerland, September 4-9, 2005
3. “High-resolution photoelectron spectroscopy study of the energy level structure and
potential curves of Xe+
2”
SCS Fall Meeting
EPF Lausanne, Switzerland, October 13, 2005
4. “Stark effect in Rydberg states of neon”
GRC on Electronic Spectroscopy and Dynamics
Les Diablerets, Switzerland, September 10-15, 2006
5. “Stark effect in Rydberg states of neon”
SCS Fall Meeting
University of Zurich, Switzerland, October 13, 2006
6. “High-resolution photoelectron spectroscopy study of the energy level structure of
ArXe+ and KrXe+ ”
SCS Fall Meeting
EPF Lausanne, Switzerland, September 12, 2007
201
Danksagung
Danksagung
Ich möchte mich ganz herzlich bei allen bedanken, die in irgendeiner Form zu dieser Arbeit
beigetragen haben:
Prof. Frédéric Merkt für die Möglichkeit, in seiner Gruppe eine Dissertation durchzuführen.
Insbesondere dein Enthusiasmus für die Wissenschaft, deine Freude über gelungene Spektren und deine nimmermüde Unterstützung habe ich sehr geschätzt. Auch die Ausflüge
zum Hotel Weisshorn, die Weihnachtsfeiern und die Post-SCS-Wanderungen werden mir
immer in bester Erinnerung bleiben.
Prof. Markus Reiher für die Uebernahme des Koreferats und die gute Zusammenarbeit
beim Xe+
2 -Projekt.
Hansjürg Schmutz für die immer fachkundige und prompte Hilfe bei Problemen mit der
Elektronik.
René Gunzinger für die perfekte Anfertigung jeglicher Art von Bauteilen für den experimentellen Aufbau.
Iréne Müller für die Abnahme administrativer Arbeit.
Remigius Mastalerz für die ab initio Rechnungen, welche mir zur Bestimmung der Potentialkurven von Xe+
2 eine grosse Hilfe waren.
Prof. Rainer Dressler für die Unterstützung bei der Zuordnung des Linienwaldes in den
Xe+
2 -Spektren.
Der ganzen Gruppe Merkt (Alex, Anna, Christian, Edward, Evgueni, Hans-Jakob, Jinjun,
Konstantina, Lorena, Martin, Matthias, Monika, Sandro, Stephen, Tom, Urs) für die gute
Stimmung und die Unterstützung beim Lösen von Problemen jeglicher Herkunft.
Anna und Edward für die gemeinsame Zeit im E 203. Ihr ward die besten Bürofreunde,
die ich mir vorstellen kann. Ich habe es sehr genossen mit euch. Die Gespräche über die
wirklich wichtigen Themen werden mir sehr fehlen.
Monika für die MQDT-Rechnungen der Stark-Spektren in Neon. Das wöchentliche Rudern
bei Martin, der nur lockere Trainings macht und die Diskussionen über Fussball und die
Welt werde ich sehr vermissen.
Evgueni für die Mithilfe beim Ne2 -Projekt. Obwohl wir nicht gefunden haben, wonach wir
eigentlich suchten, konnten wir dank deinem Modell Potentialkurven beschreiben.
Tom und Stephen für die Weihnachtsbeleuchtung.
Alex für das Mitrudern.
202
Danksagung
Konstantina für die griechischen Spezialitäten, besonders die Delikatess-Pistazien.
Urs für den IT- und Latex-Support.
Patrick Rupper für die Einführung in die Geheimnisse der Scanmates und Excimerlaser.
David Cisana für die gelungene Gestaltung des Umschlags.
Meinen Eltern für die unendlich grosse Unterstützung. Ohne euch wäre diese Ausbildung
nicht möglich gewesen.
203