VRIJE UNIVERSITEIT
XUV laser studies of
Rydberg-valence states in N2
and H+H− heavy Rydberg states
ACADEMISCH PROEFSCHRIFT
ter verkrijging van de graad Doctor aan de Vrije
Universiteit Amsterdam, op gezag van de rector
magnificus
door
Maria Ofelia Vieitez Hornos
geboren te Buenos Aires, Argentina
promotor: prof.
promotor: prof.
promotor: prof.
copromotor: dr.
dr.
dr.
dr.
O.
W. M. G. Ubachs
C. A. de Lange
L. E. Berg
Launila
Reading committee:
Dr. ir. G.C. Groenenboom (Radboud University Nijmegen)
Prof. dr. R.A. Hoekstra (University of Groningen)
Prof. dr. Th. Lindblad (Royal Institute of Technology, Sweden)
Prof. dr. H.B. van Linden van den Heuvell (University of Amsterdam)
Vrije Universiteit
Amsterdam
The investigations described in this thesis were partly carried out in
the Laser Centre Vrije Universiteit (De Boelelaan 1083, 1081 HV Amsterdam, The Netherlands) and partly carried out at the Department
of Physics of the Royal Institute of Technology (AlbaNova University
Centrum, SE-10691 Stockholm, Sweden).
Contents
Contents
Introduction
Experimental: the XUV laser setup . .
Rydberg states . . . . . . . . . . . . .
Rydberg-valence state interactions . .
Laser induced breakdown spectroscopy
Outline of this thesis . . . . . . . . . .
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1 On the complexity of the absorption
molecular nitrogen
1.1 Introduction . . . . . . . . . . . . . . . .
1.2 The complexity of the spectrum . . . . .
1.3 Experimental . . . . . . . . . . . . . . .
1.4 Illustrative examples . . . . . . . . . . .
1.5 Conclusions and outlook . . . . . . . . .
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vii
vii
ix
xi
xii
xiv
spectrum of
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1
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2 Quantum-interference effects in the o 1 Πu (v = 1) ∼
b 1 Πu (v = 9) Rydberg-valence complex of molecular nitrogen
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Experiments . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Results and discussion . . . . . . . . . . . . . . . . . . .
2.4 Summary and conclusions . . . . . . . . . . . . . . . . .
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3 Interactions of the 3pπu c 1 Πu (v = 2) Rydberg-complex
in N2
51
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.2 Experimental Methods . . . . . . . . . . . . . . . . . . . . 55
Contents
3.3
3.4
3.5
CSE calculations . . . . . . . . . . . . . . . . . . . . . . . 58
Results and discussion . . . . . . . . . . . . . . . . . . . . 59
Summary and conclusions . . . . . . . . . . . . . . . . . . 80
4 Observation of a Rydberg series in a heavy Bohr atom 83
5 Spectroscopic observation and
H+ H− heavy Rydberg states
5.1 Introduction . . . . . . . . . . .
5.2 Experiment and observations .
5.3 Analysis . . . . . . . . . . . . .
5.4 Conclusion . . . . . . . . . . .
characterization of
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6 Elemental analysis of steel scrap metals and
by laser-induced breakdown spectroscopy
6.1 Introduction . . . . . . . . . . . . . . . . . . .
6.2 Experimental . . . . . . . . . . . . . . . . . .
6.3 Optimization of experimental parameters . .
6.4 Results and discussion . . . . . . . . . . . . .
6.5 Conclusions . . . . . . . . . . . . . . . . . . .
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93
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114
minerals
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117
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Samenvatting
131
Bibliography
137
List of publications
153
vi
Introduction
This thesis is based on a number of experimental investigations in the
field of laser spectroscopy that were carried out at two different institutes. The work began at the Royal Institute of Technology in
Sweden in the period 2003-2005, focusing on the technique of laserinduced breakdown spectroscopy (LIBS). Thereafter a number of studies
were performed with the Amsterdam extreme ultraviolet (XUV) laser
setup, starting with Rydberg-valence state interactions in the nitrogen
molecule. Afterwards it proceeded to the characterization of the socalled “heavy Rydberg states” in the H+ H− composite.
In the following sections of this chapter a few key concepts of the
experiments will be explained, as an introduction to the chapters contained in this thesis.
Experimental: the XUV laser setup
The majority of the experimental work presented in this thesis (Chapters 1 to 5) has been performed in the extreme ultraviolet (XUV) region
of the spectrum. A review on the nature of XUV light and a historical
perspective on its applications in spectroscopy can be found in [1]. The
presently used instrument, the Amsterdam narrowband and tunable extreme ultraviolet laser facility, and its applications to gas-phase atomic
and molecular spectroscopy has been described in a number of PhD thesis at the Vrije Universiteit [2]. Some of its main characteristics will be
detailed briefly below.
The tunable, narrowband XUV laser source is based on harmonic
upconversion. To achieve this, narrowband visible radiation is generated in a pulse-dye laser (PDL) pumped by the second harmonic of a
Nd:YAG laser. The visible radiation is frequency doubled using a non-
Introduction
linear crystal (BBO, KDP). The resulting ultraviolet (UV) output is
guided by dichroic mirrors (that filter the remnant of visible light) into
a vacuum chamber. The XUV radiation is generated by a third harmonic generation (THG) process; this is achieved by focusing the UV
light in an inert gas (Xe, Kr) [3, 4]. The need of a windowless vacuum
system arises as XUV radiation is absorbed in all materials below 110
nm, and at these wavelengths the use of cells sealed off by some optical material is prohibited. The solution is to use a pulsed gas jet as
non-linear medium [5]. The pulsed gas jet, in combination with differential pumping provides an solution because the density of inert gas in
the focus of the UV laser can be made large without having too much
absorption for the generated XUV radiation, as the molecular density
is restricted to the small path length across the molecular jet close to
the nozzle. The differential pumping provides the conditions of vacuum
outside the THG region, and the XUV radiation is not absorbed and can
propagate into another (differentially pumped) vacuum chamber to be
used for spectroscopy experiments. The efficiency of the tripling process
is rather low (10−6 -10−7 ) and therefore there is a strong remnant beam
of UV light that travels collinearly to the XUV beam generated.
The physics of the harmonic generation is well understood [3, 6,
7, 8, 9], and following the notation of [3], the equation that governs
the frequency generation process, in terms of the component i of the
polarization and the susceptibility tensor is:
(1)
Pi = ε0 χij Ej + PN L
(
)
(1)
(2)
(3)
= ε0 χij Ej + χijk Ej Ek + χijkl Ej Ek El
(1)
(1)
(2)
where χij is the linear susceptibility, χijk is responsible for the frequency
(3)
doubling generation and χijkl is responsible for the third harmonic generation. In isotropic homogeneous media (such as a gas jet) a reversal
in the sign of Ej and Ek must cause a reversal in the sign of Pi . This
(2)
condition results in χijk = 0. This means that in a gas jet it is not
possible to generate the second harmonic of the initial frequency. Note
that the above equation describes harmonic conversion in the perturbative regime (i.e. field densities of < 1012 W/cm2 ); these conditions
are typically met in a setup with nanosecond pulsed laser systems. The
amount of XUV light produced by these means is low because it relies
viii
Introduction
on a third-order perturbative process while the material density in the
focus (gas-phase) is low: hence the conversion efficiency is 10−6 or less.
The bandwidth achieved in the XUV range is determined by that
of the incident visible pulsed laser beam. For a Gaussian beam (the
√
ideal case) the XUV-bandwidth is ∆νXU V = 6∆νvis , but in practical
cases the bandwidth is slightly larger. In a scanning experiment the
wavelength can be calibrated in the visible domain by on-line monitoring
the Doppler-limited absorption spectrum of I2 in an absorption cell;
XUV frequencies relate to the visible by the exact relation νXU V = 6νvis .
The used excitation scheme is resonantly enhanced multiphoton
ionization, combined with a time-of-flight detection system (REMPITOF). The XUV+UV radiation is perpendicularly intersected by a
pulsed molecular beam. The XUV photon is resonantly absorbed by the
molecules in the beam (H2 , N2 ) and the UV photon excites the molecule
(non-resonantly) further above the ionization energy. The generated ion
is detected by a time-of-flight spectrometer that allows for mass identification. This scheme is particularly useful to study isotope effects.
In the experimental scheme described above, the spatial and temporal overlap (within 1 ns) of the XUV and UV laser pulses is assured by
the THG process. However, in Chapters 4 and 5 a slightly different experimental approach is used: only the XUV photons are selected and the
UV photon necessary for the non-resonant ionization is provided by the
frequency doubled output of a second PDL. In this case, the XUV frequency was kept constant, and the UV frequency was tuned. The XUV
radiation was kept fixed to an intermediate state of the H2 molecule
that has a short lifetime (> 0.5 ns), and therefore careful alignment and
precise triggering was necessary to achieve spatial and temporal overlap
of the two laser pulses.
Rydberg states
Most of this thesis work deals with Rydberg states, and in the following
paragraphs some basic ideas about them will be explained. Rydberg
states are excited states of the molecule (or atom) for which the energy
of the states can be expressed as:
En = IEion −
Rm
(n − µl )2
(2)
ix
Introduction
where n is called the principal quantum number having positive integer
values, µl is the quantum defect, which is a characteristic of a particular Rydberg series and it depends on the orbital angular momentum
quantum number l of the Rydberg electron. Rm is the mass-scaled
Rydberg constant
(
)taking into account the finite mass of the nucleus
M
Rm = R∞ me +M where R∞ = 109 737.318 cm−1 , M is the mass of
the molecule (or atom) and me is the mass of the electron. IEion is
the ionization energy of the neutral molecule (or atom) and is called the
Rydberg series limit.
In a pure Coulombic potential, classical mechanics gives the orbit
of the electron as ellipses, and the quantum defect is related on how
much the electron penetrates the core [10]. Due to the l(l+1)
centrifugal
r2
potential, in the higher l states the electron does not penetrate the core
as much, and therefore the quantum defect becomes smaller when the
l values increase. Also, the mean radius of the orbit is proportional to
the principal quantum number squared. In atomic units [11]:
{
[
]}
n∗2
1
l (l + 1)
hrin∗ ,l =
1+
1−
Z
2
n∗2
where Z is the charge of the ion core (Z = 1 for neutral molecules) and
n∗ = n − µl is the effective quantum number. This means that even for
rather “low” quantum number (n ∼ 10), the size of the radius of the
orbit is rather large. When the electron is at such a large distance, the
core electrons shield the excited electron of the charge of the nucleus,
so that the electric potential provided by the core is similar to that of
a hydrogen atomic ion. Therefore the energy dependence is hydrogenatom like (−1/n2 ), but the principal quantum number n∗ = n − µl
takes into account (via the quantum defect) the deviation from a purely
Coulombic potential.
Electronic Rydberg series can converge to excited rotational or vibrational states of the ion and even to electronically excited states of
the ion. In the high energy regions (close to the series limit), it becomes
crowded with Rydberg series converging to the various limits. In most
cases, Rydberg-Rydberg state interactions occur. Such high- n value
electronic Rydberg series are observed in Chapters 4 and 5 for the H2
molecule. In Chapters 1, 2 and 3, low- n Rydberg states of N2 were
studied. In the latter cases, the Rydberg states were heavily perturbed
x
Introduction
by neighboring valence states. In Chapter 4, a different Rydberg series
is studied: a Rydberg series of the H+ H− ionic pair. In this series, the
Rydberg electron is substituted by an H− composite particle, and in
spite of being a different type of state altogether, the energy spacing of
the levels and its quantum mechanical treatment still follows the simple
eq. (2), but with a new Rm value defined [12].
The description of Rydberg states, as well as the interactions between
them and with other bound states and the continua above them, are the
main subject of quantum defect theory (QDT) [10, 13].
Rydberg-valence state interactions
Rydberg-valence interactions and perturbations arise whenever the approximations (in most cases the Born-Oppenheimer (BO) approximation) used to derive wave functions associated with these states are not
sufficiently accurate. In the BO approximation an approximate Hamiltonian of the system (H BO ) is built, and its orthonormal solutions (φBO
j )
obey:
BO BO
hφBO
|φi i = EiBO
i |H
BO BO
hφBO
|φj i = 0
i |H
The total Hamiltonian of the system can be written as H T = H BO +
H negl [11] and the H negl is a neglected term in the BO approximation.
In most cases, the interactions are local and they involve only two states
(sometimes three). This means that the matrix element:
T BO
BO
negl BO
hφBO
|φj i 6= 0
i |H |φj i = hφi |H
(3)
where the term neglected (H negl ) in the H BO Hamiltonian couple the
different φBO
states to each other. In principle it is possible to express
j
any exact solution (Ψi ) of the total Hamiltonian (H T ) as an (infinite)
expansion of the approximated BO functions:
∑
Ψi =
cij φBO
j
j
If one term of the expansion is sufficient, this means that the BO approximation is reasonable. In the work presented in Chapters 1, 2 and
xi
Introduction
3, the states have to be expressed as expansions of two and sometimes
three terms of the BO expansion.
Usually these interactions (perturbations) are detected in terms of
irregularities in the quantum level structure, as the spacing between
recorded spectral lines becomes irregular. We also investigate the interactions in terms of constructive and destructive interference effects,
affecting the intensities of the lines in the spectrum. The quantum interference effect manifest itself by the intensity borrowing phenomenon
and unusual predissociation J- dependence of the linewidths.
The first three chapters deal with Rydberg-valence interaction between states, and specially homogeneous and heterogeneous interactions.
Homogeneous interactions are those where ∆Ω = 0 with Ω the value of
the projection of the total angular momentum (exclusive of nuclear spin)
Jz of the molecule onto the internuclear axis. It fulfills Ω = Λ+Σ, where
Λ is the value of the projection of the total electronic orbital angular
momentum Lz onto the internuclear axis, and Σ is the value of the projection of the total electron spin Sz onto the internuclear axis. Both Σ
and Λ values are (in most cases) labels of the eigenfunction. A heterogeneous perturbation is characterized by ∆Ω = ±1. The homogeneous
interactions dealt with here are of electrostatic nature, and therefore
negl |φBO i = H = constant [11]. The important feature in hethφBO
ij
i |H
j
erogeneous interactions is that the interaction matrix element depends
on the quantum number J. In all perturbations, the total angular momentum quantum number J remains defined (because J 2 commutes with
H), and the selection rule common to all perturbations is ∆J = 0.
Laser induced breakdown spectroscopy
In laser-induced breakdown spectroscopy (LIBS), the idea is to focus a
powerful pulsed laser onto a target, ablating the surface of this target
and creating a plume of plasma. This plasma emits light as the excited
atoms (and molecules) decay to their ground states. The main goal
is to record this emission spectrum and determine the composition of
the original target. Apart from being able to know which elements are
present in the target, the objective is to quantify the elemental abundance. Normally one uses samples of known composition to calibrate
the setup.
xii
Introduction
The calibration is done by determining the relationship between the
light intensity measured and the amount of various elements present in
the sample. In the calibration process, standard samples with known
chemical composition are measured. After selecting the standards, it
is necessary to select the spectral lines to use for the analysis of the
element of interest (the analyte). The desirable situation for spectral
lines used to detect the concentration of the analyte is that the total
(integrated) intensity of the line varies linearly with the concentration
of the element. In reality it is found that this intensity versus concentration curve is not linear for all concentrations. At high concentrations of
the analyte, re-absorption in the cooler, outer parts of the plasma takes
place, diminishing the intensity of some (more sensitive) lines. Therefore, a high sensitivity spectral line must be used for low concentrations,
while a low sensitivity line should be used for high concentrations. For
the purposes of detection, it means that different lines should be used for
calibration for different concentration ranges. Therefore a pre-existent
knowledge of the range of concentrations to be measured in the sample
must be established before LIBS studies take place.
Another major drawback of this detection system is what is called
the “matrix effect“. The matrix is the composition of the substance
in which the analyte is found. Usually, the matrix refers to the major
component or base element (for example, iron in steel). The introduction
of additional elements into the sample in large amounts (> 5%) can
affect the slope of the calibration curve. This is called interelement effect
or matrix effect. In many cases the matrix effect is independent of the
spectral line, i.e. the same matrix effect on many different spectral lines
of the same element in a given matrix can be found. The phenomenon is
not well understood from a physical point of view [14]. For the purposes
of detection, this means that for each specific study, the target’s main
composition should be known, in order to perform a calibration based
on reference samples of the same matrix composition.
What makes the LIBS detection system attractive is the fact that it
can be used in harsh environments, such as inside melting furnaces of
metals. Or in places where a fast selection of the target is important,
such as a scrap metal yard, where the value of the different pieces of
metal depends greatly on their specific composition. The paper presented here was part of a project together with Stena Metall AB, a
xiii
Introduction
metal recycling company present in Stockholm, in which the aim was
to assess the possibility to use LIBS to discriminate between metallic
pieces to be recycled according to their composition. This work was
used for the initiation of a study on a larger scale: Laser-Induced Breakdown Spectroscopy for Advanced Characterization and Sorting of Steel
Scrap (LCS) (EU Research Programme of the Research Fund for Coal
and Steel, Grant Agreement Number: RFSR-CT-2006-00035). The goal
of the latter project was to industrially evaluate the use of LIBS for
scrap analysis and sorting.
Outline of this thesis
Chapter 1 is an introduction to the problems and challenges that must
be dealt with when studying the spectrum of molecular nitrogen. From
a general perspective, it describes the contribution of the Amsterdam
XUV laser facility to the understanding of the complex N2 energy levels
and their interactions. This chapter also serves as an introduction to
the next two chapters.
Chapter 2 is about Rydberg-valence states interacting and perturbing each other. In this Chapter one exemplary case, prototypical for the entire N2 spectrum, is investigated in detail. The Rydberg state o 1 Πu (v = 1) is perturbed (homogeneous perturbation) at
low- J values by the valence state b 1 Πu (v = 9) and at high- J values
by the b0 1 Σ+
u (v = 6) valence state (heterogeneous perturbation). The
o 1 Πu (v = 1) ∼ b 1 Πu (v = 9) mixing is so pronounced that it has led
in the past to incorrect assignment of the spectral lines. The mixing
of states gives rise to effects of quantum interference on the oscillator
strength of the observed lines.
Chapter 3 deals with the perturbations of the Rydberg complex
3pπu c 1 Πu (v = 2) with other valence singlet states and with triplet
states. The perturbations with the singlet valence state b0 1 Σ+
u (v = 7)
results in a strong Λ-doubling between the e and f states of the c(v = 2)
and a P/R branch intensity anomaly for the b0 −X(7, 0) band. Strong local perturbations in energy and line width of the c(v = 2) are attributed
to a heterogeneous perturbation of the C 3 Πu (v = 17) state.
Chapter 4 is about the first spectral observation of a series that
was baptized “heavy Rydberg series”, a short for Rydberg series in a
xiv
Introduction
heavy Bohr atom. A heavy Bohr atom is a hydrogen atom where the
electron is replaced by a H− composite-like “particle”, forming the ionpair H+ +H− . The remarkable result is that the energy dependence of
the heavy Rydberg series is similar to that of a “regular” (electronic)
Rydberg series, but with rescaling of the Rydberg constant due to the
difference in mass between the electron and the H− .
Chapter 5 deals with the characterization of the heavy Rydberg series
excitation and observation mechanism, the “complex resonances”. The
linewidths and line positions of the heavy Rydberg series are analyzed.
The resulting quantum defects are related to the line positions using an
(over-simplified) quantum defect theory model.
Chapter 6 deals with using laser-induced breakdown spectroscopy
detection system to measure the trace amounts of nickel, copper and
other metals in steel targets. As an example of an industrial application, the concentration of copper in scrap metals is studied, which is
an important factor to determine the quality of the samples to recycle.
Another application of the LIBS method is the study of the nickel and
copper concentrations in a sample of iron-rich magma.
xv
Chapter 1
On the complexity of the
absorption spectrum of
molecular nitrogen
The spectral properties of molecular nitrogen are crucial to
a better understanding of radiative-transfer phenomena and activated N/N2 chemistry in the Earth’s upper atmosphere. Excited
states of N2 are difficult to access experimentally, and analysis of
its electric dipole-allowed spectrum is notoriously complex. In this
paper, we give an overview of these complexities and of the power
of extreme ultraviolet ionization spectroscopy in unraveling many
of the observed features. Some illustrative examples from our own
research will be discussed.
1. On the complexity of the absorption spectrum of N2
1.1
Introduction
The importance of molecular nitrogen as the most abundant species
in the Earth’s atmosphere is evident. The strong absorption bands in
the range 80 – 100 nm shield the Earth’s surface from the extreme
ultraviolet (XUV) part of the solar radiation [15]. In fact, even the entire
troposphere and stratosphere are free from this hazardous radiation that
penetrates only some ∼ 150 km above the Earth’s surface. Absorption
of the short-wavelength light leads to molecular dissociation, and for N2
this process is via predissociation, with ground- and excited-state atoms
as products. Clearly, an understanding of the spectroscopy of N2 in this
wavelength range is essential for a better understanding of radiativetransfer phenomena and activated N/N2 chemistry in the Earth’s upper
atmosphere. Similar processes are expected to take place in our solar
system in the upper atmospheres of Jupiter, Saturn and its moon Titan,
and Triton, the largest moon of Neptune [16].
Molecular nitrogen, N2 , together with the isoelectronic carbon
monoxide CO, is one of the most stable molecules in nature. The electronic configuration of homonuclear N2 in its X 1 Σ+
g ground state is:
(1sσg )2 (1sσu )2 (2sσg )2 (2sσu )2 (2pπu )4 (2pσg )2 ,
corresponding to a triple chemical bond. For 14 N15 N, the g (gerade)
and u (ungerade) symmetry assignments for the orbitals hold only in
approximation. The triple chemical bond explains the large dissociation
limit of N2 (78 714 cm−1 [17]). Removal of an electron from the highest
occupied molecular orbital leads to the lowest X 2 Σ+
g ionic state, with
configuration
(1sσg )2 (1sσu )2 (2sσg )2 (2sσu )2 (2pπu )4 (2pσg )1 ,
and an ionization energy of 125 666.959 cm−1 [18]. As a result, excited
electronic states of molecular nitrogen are high lying and not easily
accessible by normal experimental means.
Focusing on optical transitions involving the ground state, the
1 +
weak spin–forbidden A 3 Σ+
u −X Σg Vegard–Kaplan bands, the weak
1 +
symmetry-forbidden a0 1 Σ−
u −X Σg Ogawa-Tanaka-Wilkinson-Mulliken
and a 1 Πg − X 1 Σ+
g Lyman-Birge-Hopfield bands have been observed,
2
1. On the complexity of the absorption spectrum of N2
both in emission and absorption in the (far) ultraviolet (UV) [17]. The
weakness of these bands implies that N2 is optically transparent in the
visible and UV regions of the spectrum. The much stronger one-photon
electric-dipole-allowed absorption features in the N2 spectrum involve
1
transitions to valence and Rydberg states of 1 Σ+
u and Πu symmetry
from the ground state and are found in the extreme ultraviolet.
In this paper, we shall focus on the complexities of the electricdipole-allowed spectrum of molecular nitrogen and on the role that XUV
ionization spectroscopy can play in unraveling them. The N2 spectrum,
situated in the energy range just above 100 000 cm−1 , displays many
irregularities due to strong global vibronic Rydberg-valence interactions
between the singlet ungerade states. Other local and accidental perturbations in the rotational structure are also evident in many places,
generally arising from heterogeneous interactions that are usually significantly dependent on the isotopomer involved. All these interactions
strongly affect vibronic and rotational intensities, and can result in vibronic and rotational quantum interferences. Another key process is
predissociation, which is mediated through the spin-orbit interaction
with triplet states. This coupling between the singlet and triplet manifolds is another source of spectral complexity. The rate of predissociation in molecular nitrogen is often sufficiently slow not to wash out
the rotational structure in highly-excited states completely, but, at the
same time sufficiently fast to allow the observation of line broadening
of individual rotational transitions. Because of its excellent spectral
resolution, XUV laser spectroscopy is eminently suitable for resolving
this rotational structure and for determining the degree of line broadening and the corresponding rate of predissociation. Various illustrative
examples derived from our research in Amsterdam will be discussed.
1.2
The complexity of the spectrum
The dipole-allowed absorption spectrum of molecular nitrogen in the
XUV shows a very complex behavior. Initially, it was thought that the
many bands in the spectrum were due to transitions involving a large
number of excited electronic states [17], but later it was found that they
arose as a result of Rydberg-valence and Rydberg-Rydberg interactions
between a limited number of singlet ungerade states lying at excitation
3
1. On the complexity of the absorption spectrum of N2
c Πu
3
F Πu
11.0
4
−1
Potential Energy (10 cm )
1
o Πu
1
1
+
b’ Σu
1
10.0
1 +
c’ Σ
u
b Πu
3
G Πu
’3
C Πu
3
C Πu
9.0
1.0
1.5
2.0
2.5
Internuclear Distance (Å)
Figure 1.1: Potential-energy curves for the ungerade singlet states that govern
the dipole absorption spectrum, and the triplet (C, C 0 , F and G) electronic
states that govern the predissociation behavior of N2 . Full lines: 1 Πu states.
3
Dashed lines: 1 Σ+
u states. Dotted lines: Πu states.
energies just above 100 000 cm−1 [19, 20, 21]. There are two valence
1
states involved, one of 1 Σ+
u and one of Πu symmetry (designated as
1
b0 1 Σ +
u and b Πu , respectively). The relevant Rydberg states belong
either to the series converging on the lowest X 2 Σ+
g ionization limit (npσu
0
1
+
1
cn+1 Σu and npπu cn Πu , with principal quantum number n ≥ 3), or
the nsσg on 1 Πu series, converging on the A 2 Πu ionic limit of N+
2 . The
relevant electronic configurations of the lowest-lying singlet states are:
b0 1 Σ +
u
b 1 Πu
c04 or c0 1 Σ+
u
c3 or c 1 Πu
o3 or o 1 Πu
...(2sσu )2 (2pπu )3 (2pσg )2 (2pπg )1
...(2sσu )1 (2pπu )4 (2pσg )2 (2pπg )1
...(2sσu )2 (2pπu )4 (2pσg )1 (3pσu )1
...(2sσu )2 (2pπu )4 (2pσg )1 (3pπu )1
...(2sσu )2 (2pπu )3 (2pσg )2 (3sσg )1
valence
valence
Rydberg
Rydberg
Rydberg.
Potential-energy curves of these states, together with the C, C 0 , F and
G states of triplet character (to be discussed later), are presented in
Fig. 1.1. They have the following configurations:
4
1. On the complexity of the absorption spectrum of N2
C 3 Πu
C 0 3 Πu
F 3 Πu
G 3 Πu
...(2sσu )1 (2pπu )4 (2pσg )2 (2pπg )1
...(2sσu )2 (2pπu )3 (2pσg )1 (2pπg )2
...(2sσu )2 (2pπu )3 (2pσg )2 (3sσg )1
...(2sσu )2 (2pπu )4 (2pσg )1 (3pπu )1
valence
valence
Rydberg
Rydberg.
We note, in particular, that the configurations listed above for the b,
b0 , and C valence states are those predominating at smaller internuclear
distances R. As R increases, other configurations become important
[22], as evidenced by the unusual shapes of the potential-energy curves
for these states in Fig. 1.1.
A detailed understanding of the spectroscopy in this energy region
has long been hampered by the complex nature of the observed spectra. In a benchmark paper by Stahel et al. [23], a model of Rydbergvalence interactions was developed that provides a quantitative explanation for the energy-level perturbations, the seemingly erratic behavior
of the rotational constants, and the observed band intensities that deviate strongly from Franck-Condon predictions, due to vibronic quantuminterference effects. In particular, the homogeneous vibronic interac0
0
tions between states of 1 Σ+
u symmetry (the b valence and the c4 and
c05 Rydberg states) and between states of 1 Πu symmetry (the b valence
and the c3 and o3 Rydberg states) were treated [23]. These global perturbations have been crucial in understanding the key features of the
allowed optical absorption spectrum of N2 . Later, Spelsberg and Meyer
put forward a quantitatively improved model, based on ab initio calculations [24]. Edwards et al. [25] extended the model by incorporating
heterogeneous interactions to treat the mixing of states with different
symmetries. These rotationally-dependent perturbations are local in
character, and experimental methods to study such interactions usually
require rotational resolution. These perturbations may cause shifts in
rotational energy levels and can affect rotational transition intensities
and predissociation line widths. Similar to what has been observed in
the case of vibronic levels, such interactions may also give rise to rotational quantum interferences.
Photodissociation can occur directly, by photoexcitation from a
bound state to a repulsive state or to a bound state above its dissociation limit. Dissociation can also be indirect, when photoexcitation
takes place from a bound state to another bound state, which in turn
‘predissociates’ through a perturbative interaction with the continuum
5
1. On the complexity of the absorption spectrum of N2
of another electronic state. The importance of predissociation phenomena in the lowest-lying electric-dipole accessible states of the abundant
14 N and its rarer stable isotopomers, 14 N15 N and 15 N , has been ap2
2
parent for decades [26]. A diversity of experimental techniques were
exploited to chart the most prominent of the predissociation effects.
Studies of fluorescence excitation, induced either by electron collisions
[27] or by synchrotron-radiation absorption [28], revealed the subclass
of states that are subject to radiative decay, while the early XUV-laser
studies revealed the strongly varying predissociative behavior for vibrational levels within a single electronic manifold [29]. Complementary
techniques employing neutralization in fast ion beams allowed for monitoring of the photo-fragments from predissociating N2 [30, 31]. However,
a detailed quantitative understanding of the mechanisms underlying N2
predissociation was not achieved until the work of Lewis et al. [32, 33].
This work is based on a coupled-channel Schrödinger equation (CSE)
model, and it should be emphasized that it achieves spectroscopic accuracy, therefore allowing for a close comparison with experiment. In
essence, the homogeneous (∆Ω = 0) spin-orbit coupling provides an interaction between accessible singlet states and the triplet manifold. The
spin-orbit coupling takes place between 1 Πu and C 3 Πu , which in turn is
coupled to C 0 3 Πu above its dissociation limit. This is a form of accidental (or indirect) predissociation and can be interpreted as a perturbation
of a nominally bound rotational level by a predissociated level that lies
nearby in energy. The above pathway for predissociation in molecular nitrogen in the region below 105 000 cm−1 has been confirmed by a
large body of experimental spectroscopic evidence, based on an analysis
of global and local perturbations. We note that the present understanding of the predissociation mechanisms for the 1 Σ+
u states (with smaller
predissociation rates) is not as well developed.
Since predissociation of molecular nitrogen is such a key process,
experimental methods suitable for its detection are important. In this
paper, we shall show how ionization of N2 and its isotopomers in a
1 XUV + 10 UV ionization process is a very convenient way of monitoring predissociation with rotational resolution. The method allows
for the reliable detection of predissociation rates from experimentally
observed line broadening. The excited-state lifetimes in molecular nitrogen happen to be in the range where the rotational structure in the
6
1. On the complexity of the absorption spectrum of N2
spectra can still be discerned, while the associated line broadening often exceeds the instrumental line widths if narrow-band laser systems
are used. Moreover, the interplay between our XUV laser experiments
and the theory, as treated in the CSE model, forms a very powerful
and successful combination that has led to significant new insights. As
examples, the CSE model was able to explain the strongly varying lifetimes for the b 1 Πu (v = 1) state in the three N2 isotopomers [34], and
it predicted the location of the very weak spectral lines probing the
F 3 Πu (v = 0) state, that were indeed observed [35].
Since our 1 XUV + 10 UV laser experiments start with molecular nitrogen in its X 1 Σ+
g ground state, we are principally limited in what we
can learn to detailed studies of the singlet ungerade manifold. Our studies are concerned with rotationally-resolved interactions between states
that lead to perturbed term energies and transition intensities. These
perturbations can take the form of rotational quantum interferences,
similar to the vibronic quantum interferences discussed in [23]. However,
important information about the triplet manifold can also be collected,
both directly and indirectly, since spin-orbit coupling with triplets can
cause observable perturbations in the singlet manifold. When the focus
is on rotationally-resolved phenomena, these perturbations are usually
isotopomer-dependent. Hence, the experimental study of the different
stable isotopomers tends to be very informative.
As illustrative examples from our own research, we shall discuss the
analysis of homogeneous and heterogeneous perturbations in the singlet manifold, rotationally resolved quantum interferences in oscillator
strengths and predissociation line widths, and the direct and indirect
observation of triplet states through their interactions with the singlet
manifold.
1.3
Experimental
Details of the experimental method, including a description of the lasers,
vacuum setup, molecular-beam configuration, time-of-flight (TOF) detection scheme and calibration procedures, have been given previously
[36]. A skimmed and well defined pulsed molecular beam of nitrogen is
perpendicularly intersected by temporally and spatially overlapping the
XUV and UV laser beams. Nitrogen molecules are resonantly excited
7
1. On the complexity of the absorption spectrum of N2
Figure 1.2: Competing decay mechanisms in a 1 XUV + 10 UV ionization
experiment. The XUV photon excites the molecule with rate kabs . From the
excited state, the molecule can fluoresce to lower levels with rate kf luor , it can
undergo predissociation (with rate kpred ) or, via a UV photon, it can become
ionized. This ionization process is, in principle, non resonant and it occurs
with rate kion .
by the XUV photons and subsequently ionized by the intense UV light.
N+
2 ions are detected using a TOF mass selector.
For the detailed investigation of the N2 spectral features, a tunable
light source in the extreme ultraviolet region with sufficiently narrow
bandwidth is required. Tunability and narrow bandwidth are achieved
using two types of laser sources which deliver energetic pulses in the
visible wavelength domain. Harmonic generation in two steps, by frequency doubling in nonlinear crystals and subsequent frequency tripling
in gas jets, provides coverage of the XUV wavelength range, while the
bandwidth of the fundamental light sources is nearly retained in the
conversion process. When using a commercially available Pulsed Dye
Laser (PDL), the bandwidth in the XUV is ∼ 0.3 cm−1 full-width at
half-maximum (FWHM) and the absolute wavenumber uncertainty in
the XUV for this system is ± 0.1 cm−1 for fully-resolved lines. When
using a home-built Pulsed Dye Amplification system, a bandwidth of
∼ 0.01 cm−1 FWHM is attained and the absolute calibration uncertainty is ± 0.005 cm−1 . Wavelength calibration can be performed in
the visible range, since exact harmonics are produced in the nonlinear
optical conversion process.
The two-photon-ionization TOF experiment has some useful characteristics that are favorably employed. Mass separation can be combined
with laser excitation to separate the contributions to the spectrum of
8
1. On the complexity of the absorption spectrum of N2
the main 14 N2 isotopomer from those of the mixed 14 N15 N and 15 N2
species. Furthermore, by changing the relative delay between the N2
pulsed-valve trigger and the laser pulse, as well as varying the nozzleskimmer distance, the rotational temperature in the molecular beam
can be selected to measure independent spectra of cold (10 − 20 K) and
warm (up to 300 K) samples. This form of temperature tuning of the
gaseous sample aids in the assignment of the spectral lines.
In a 1 XUV + 10 UV two-photon-ionization experiment, the excited
state is populated by the XUV absorption process and depopulated by
decay mechanisms that all, in principle, lead to a shortened lifetime
(see Fig. 1.2). This occurs through (i) fluorescence, (ii) predissociation,
and (iii) through ionization by UV radiation. As we intend to measure
linewidths with our setup, excessive UV radiation that would lead to
depletion of the population of the intermediate state is avoided in our
experiment. Information on predissociation can be obtained using several complementary methods. The lifetime τ (s) of the excited level,
shortened due to predissociation, can be expressed as τ = (2πcΓ)−1 ,
with Γ the natural (Lorentzian) line width (in cm−1 FWHM). Hence, the
excited-state lifetime τ can be derived straightforwardly from line-width
measurements. The shortening of the lifetime due to predissociation will
not only cause line broadening, but also a decrease in signal intensity,
since we detect ions that result from ionization of a decaying excited
level. Using a rate equation model [37], it can be proved that the intensity of the signal is proportional to the lifetime of the excited state,
when the laser line width exceeds the natural width Γ.
Hence, predissociation can be monitored by (i) directly detecting the
broadening of the XUV transition [29]; (ii) carrying out a pump-probe
experiment on the excited state with a variable time delay between the
pulses [38]; (iii) measuring the decrease in the ionization signal which is
proportional to the decrease of the excited state lifetime; or (iv) measuring the decrease in the fluorescence signal. In our experiments, both the
decrease in the ionization signal and the broadening of the excited-state
line width are signatures of the occurrence of predissociation.
9
1. On the complexity of the absorption spectrum of N2
Figure 1.3: PDL-based XUV-source spectra of the b1 Πu − X 1 Σ+
g (9,0) and
14
(1,0)
bands
of
N
,
with
corresponding
line
assigments.
Two
o1 Πu − X 1 Σ+
2
g
separate scans are joined in the region marked with an asterisk (*), and their
relative intensities should not be compared. Note that several lines are blended
and many transitions are too weak to be observed.
1.4
Illustrative examples
In this section we shall treat a number of examples of (i) homogeneous interactions between states of the same symmetry and heterogeneous interactions between states of different symmetry; (ii) quantum-interference
effects occurring both in the oscillator strengths between electric-dipole
allowed transitions, and in the line widths between predissociating levels; and (iii) evidence for triplet states through their coupling with the
singlet manifold. Without trying to be exhaustive, we have selected
these examples from our research as representative of the current experimental and theoretical state-of-the-art in studying perturbation and
predissociation phenomena in molecular nitrogen.
1
01 +
The o 1 Σ+
u (v = 1) ∼ b Πu (v = 9) ∼ b Σu (v = 6) interaction complex in 14 N2
In the energy region between 107 000 and 108 000 cm−1 , the following
energy levels in 14 N2 are situated closely together, thus allowing for
10
1. On the complexity of the absorption spectrum of N2
Figure 1.4: Term values of the b1 Πu (v = 9) ∼ o1 Πu (v = 1) ∼ b01 Σ+
u (v = 6) eparity states, reduced such that the deperturbed b(v = 9) levels are on the zero
line. Clear anti-crossings for the b(v = 9) ∼ o(v = 1) and o(v = 1) ∼ b0 (v = 6)
levels are shown. Measured energy levels are displayed using black symbols,
predicted levels using grey symbols.
possible interactions: o 1 Σ+
u (v = 1) (Rydberg), and the valence states
1
0
1
+
b Πu (v = 9) and b Σu (v = 6). For 14 N2 at low J-values, the o(v = 1)
and b(v = 9) states cross, while at higher J-values this complex can
interact with the b0 (v = 6) state. In Fig. 1.3, the b 1 Πu −X 1 Σ+
g (9, 0) and
o 1 Πu − X 1 Σ+
(1,
0)
high-temperature
spectrum
is
presented,
showing
g
14
15
15
the P , Q, and R branches. For the isotopomers N N and N2 , the
relative positions of the o(v = 1) and b(v = 9) states are such that
significant interactions are not expected between these states.
Rotational levels associated with excited states can be of e or f parity
1
in the case of 1 Πu states, or only of e parity for 1 Σ+
u . Moreover, in Πu
states, Λ-doubling occurs. These issues are discussed in detail in many
places [11, 39]. In order to obtain the term energies and transition intensities, a detailed analysis must be carried out. First, the rotational transitions P , Q and R are assigned, guided by the nuclear spin statistics, the
combination differences (for the P and R branches) and the differences in
intensities from the cold and warm spectra. These experimental transition energies are then compared with theoretical values calculated using
ground- and upper-state term values which are parametrized employing
the usual spectroscopic parameters (rotational constant B, centrifugal
11
1. On the complexity of the absorption spectrum of N2
distortion parameters D and H). The rotational level structure of the
ground state is well understood and represented by the constants published by Trickl et al. [40] in the case of 14 N2 , and by the constants of
Bendtsen et al. [41] for the other isotopomers. By adjusting the upperstate rotational constants to minimize the differences with respect to the
experimental transitions, a least-squares fit of the rotational constants
and therefore of the term values is obtained.
The b 1 Πu (v = 9) and o 1 Πu (v = 1) levels in 14 N2 undergo an
avoided crossing in the rotational structure between J = 4 and J = 5.
Since both states have the same symmetry, this interaction is homogeneous (∆Ω = 0), electrostatic in character, J-independent and involves
both the e- and f -parity levels. Moreover, at high J an interaction that
1
only involves e-parity levels between b0 1 Σ+
u (v = 6) and o Πu (v = 1)
is apparent. This interaction is heterogeneous (∆Ω 6= 0), arises from
L-uncoupling and is therefore J-dependent. For the e-parity levels, a
complete three-state deperturbation was performed for each value of J
by diagonalizing the matrix


Tb9 (J)
Hb9o1
0
√


(1.1)
To1 (J)
Ho1b0 6 J(J + 1)  .
 Hb9o1
√
0
0
0
Ho1b 6 J(J + 1)
Tb 6 (J)
The diagonal elements are the term energies of the b (v = 9), o (v = 1)
and b0 (v = 6) states. The off-diagonal element Hb9o1 is the homogeneous
√
interaction between the 1 Πu states, and Ho1b0 6 J(J + 1) represents the
effective heterogeneous interaction matrix element between o (v = 1)
and b0 (v = 6). For f -parity levels, Eq. (1.1) reduces to a 2 × 2 matrix.
In Fig. 1.4, the e-parity term values, reduced such that the deperturbed b (v = 9) values lie on the zero line, are shown. The figure clearly
shows an avoided crossing between J = 4 and J = 5 for the b (v = 9)
and o (v = 1) states, with a maximum energy shift of 8.7 cm−1 at J = 4.
A second avoided crossing, this time between o (v = 1) and b0 (v = 6),
occurs between J = 24 and J = 25, with a maximum shift of 13.5 cm−1
at J = 25. A similar f -parity plot can be constructed for the coupling
between the Rydberg and valence 1 Πu states.
Moreover, around the rotational levels where the interaction takes
place, the wave functions are strongly mixed. For each state, the wave
12
1. On the complexity of the absorption spectrum of N2
function will be a linear combination of the unperturbed wave functions:
Ψ = c1 Φo(1) + c2 Φb(9) + c3 Φb0 (6) ,
(1.2)
where the values of the ci coefficients are the components of the eigenvectors of Eq. (1.1). Near J = 4, o (v = 1) and b (v = 9) exchange
electronic character, while the level of mixing of b0 (v = 6) is almost negligible. The wave functions of o (v = 1) and b (v = 9) are then expressed
as:
√
Ψo(1) = cΦo(1) + 1 − c2 Φb(9) ,
√
Ψb(9) = − 1 − c2 Φo(1) + cΦb(9) .
(1.3)
Also, near J = 25, the same happens for the e-levels of o (v = 1) and
b0 (v = 6), while the mixing of b (v = 9) is close to zero. For the intermediate J levels, a three-state problem can be solved.
Rotational quantum-interference effects
Oscillator strengths
Because both the b − X (9,0) and o − X (1,0) transitions carry oscillator
strength, and since both b (v = 9) and o (v = 1) levels of like symmetry interact, this leads to a classic situation where two-level quantuminterference effects are expected. As discussed by Lefebvre-Brion and
Field [11], the perturbed vibronic oscillator strengths for transitions
from a common level 0 (the X 1 Σ+
g ground state in our case) to upper (+) and lower (−) levels of the interacting pair are given by:
√
f+0 = c2 f10 + (1 − c2 )f20 ± 2c (1 − c2 )f10 f20 ,
√
f−0 = (1 − c2 )f10 + c2 f20 ∓ 2c (1 − c2 )f10 f20 ,
(1.4)
where f10 and f20 are the vibronic oscillator strengths for transitions to
the unperturbed levels 1 and 2, and c > 0 signifies the mixing coefficient
that corresponds to the amount of character of the unperturbed level
1 in the perturbed upper-level wave function, as explained previously.
Using the mixing coefficients that have been obtained from the deperturbation procedure in the previous section, we can now deperturb the
experimental oscillator strengths through the application of Eq. (1.4).
13
1. On the complexity of the absorption spectrum of N2
Oscillator Strength
0.020
0.015
0.010
0.005
0.000
0
5
10
15
20
J
Figure 1.5: Rotational dependence of band oscillator strengths (obtained
from synchrotron-based experiments [39]) in the mixed b − X(9, 0) and o −
X(1, 0) transitions of 14 N2 , demonstrating a strong quantum-interference effect
near J = 6, together with the results of a deperturbation analysis (see text).
Open squares: experimental (perturbed) oscillator strengths for transitions to
the higher-energy levels. Solid squares: experimental oscillator strengths for
transitions to the lower-energy levels. Dot-dashed line: deperturbed oscillator
strength for the o−X(1, 0) transition. Long-dashed line: deperturbed oscillator
strength for the b − X(9, 0) transition. Dashed curve: calculated perturbed
oscillator strength for the higher levels. Solid curve: calculated perturbed
oscillator strength for the lower levels.
Our 1 XUV + 10 UV experiments reveal perturbations in the intensity pattern of the observed transitions, but as the photon flux is not
measured, and the ionization cross sections are not known a priori, absolute oscillator strengths are not obtainable with our setup. Therefore,
recent synchrotron-based measurements are used [39] (see also [42]) and
shown in Fig. 1.5. As is apparent from the figure, transitions to the
higher-energy level, i.e., b (v = 9) for J ≤ 4 and o (v = 1) for J ≥ 5,
show a strong rotational dependence and peak around J = 6. At the
same time, oscillator strengths to the lower-energy level, i.e., o (v = 1)
for J ≤ 4 and b (v = 9) for J ≥ 5, show a minimum at J = 6, to the extent that this transition was too weak to be observed. Deperturbation of
14
1. On the complexity of the absorption spectrum of N2
Figure 1.6: 1 XUV + 10 UV ionization spectrum for the c3 1 Πu − X 1 Σ+
g (2,0)
band of 15 N2 , with corresponding line assigments. The asterisks (*) indicate
how large transition intensities would have been (on a relative scale) if not
affected by the singlet-triplet interaction.
the experimental oscillator strengths according to Eq. (1.4) leads to the
dot-dashed line in Fig. 1.5 for the o−X (1,0) transition, and to the longdashed line for the b − X (9,0) transition. In summary, the experimental
oscillator strengths of both bands show clear evidence of rotationallydependent constructive and destructive quantum-interference effects in
14 N . A full account of this work is presented in Ref. [39].
2
Predissociation line widths
If two energy levels predissociate via the same perturbative state, an
interaction between these levels may result in a quantum-interference
effect in the strength of the predissociation, in the same fashion as for
the oscillator strengths. The predissociation strength can be detected
via line broadening of the measured transitions. In this case, the width
interference is described by equations similar to Eq. (1.4), but with the
oscillator strength f replaced by the width Γ, and with 0 the perturbative state. The constructive and destructive interferences observed near
J = 6 for the oscillator strengths of the b − X (9,0) and o − X (1,0)
transitions support the notion of line-width interferences and associated predissociation rate modulations. This is actually observed for the
15
1. On the complexity of the absorption spectrum of N2
Figure 1.7: Energies of the c3 1 Πu (v = 2) f -parity state of 15 N2 , reduced to
be positioned on the zero-energy line. The straight lines indicate the crossings
with C 3 Πu (v = 17) as predicted by an extended model based on [32].
o (v = 1) and b (v = 9) levels. A complete description of this phenomenon is presented in Ref. [39].
Triplet-singlet interactions
As mentioned above, in principle our setup only allows the direct measurement of singlet ungerade states of N2 . Nevertheless, direct measurement of transitions of the triplet manifold are possible if these transitions become visible via intensity borrowing [11]. This is the case for
the F 3 Πu (v = 0) Rydberg state. Sprengers et al. presented the results
of an ultrahigh-resolution laser-spectroscopic study of the F − X(0,0)
transition in 14 N2 [35]. This dipole-forbidden transition became observable through the spin-orbit-induced intensity borrowing from the
dipole-allowed b − X(5,0) transition. This phenomenon was facilitated
by the near-degeneracy of the F (v = 0) and b(v = 5) levels in the 14 N2
isotopomer. Direct observation of the F state led to assignments of the
R, P and Q branches for the Ω = 0, 1, 2 triplet sublevels, together with
the predissociation widths of the transitions. The obtained rotational
parameters, together with the term values of the predissociating levels, suggest strong interactions among the F and G 3 Πu Rydberg and
16
1. On the complexity of the absorption spectrum of N2
Figure 1.8: 1 XUV + 10 UV ionization spectra of the Q(15) and Q(6) lines
15
of the c3 1 Πu − X 1 Σ+
N2 , recorded using the PDL-based XUV
g (2,0) band for
source and showing that Q(6) is a factor of two broader than Q(15), due to
the increase of predissociation. Note that the relative X-scales are the same in
each subfigure.
C 0 3 Πu valence states. Hence, the complexity of the triplet manifold is
somewhat similar to that of the singlet states.
Even when the fortuitous coincidence of energy levels does not allow
their direct observation through intensity-borrowing effects, the triplet
states may manifest themselves in an indirect way, via perturbations of
the singlet states. In the energy range 108 500 − 109 500 cm−1 , several
singlet states, i.e., the Rydberg states c3 1 Πu (v = 2) and c04 1 Σ+
u (v = 2)
0
1
+
1
and the valence states b Σu (v = 7) and b Πu (v = 11) are situated.
The predictions of the CSE model indicate a crossing between the singlet
c3 1 Πu (v = 2) state and the three components (Ω = 2, 1, 0) of the triplet
C 3 Πu (v = 17) valence state.
For 15 N2 , the crossing is predicted to occur at relatively low J. This
is favorable, because it means that, with the 1 XUV + 10 UV setup,
where the lower-J lines show the most intensity, this phenomenon can
be detected. For 14 N15 N and 14 N2 , the crossing takes place at higher
values of J, that are difficult to access with our XUV setup. In the
following, we shall focus on 15 N2 .
17
1. On the complexity of the absorption spectrum of N2
The c3 1 Πu −X 1 Σ+
g (2,0) band between J = 5 and J = 13 shows large
intensity deviations from those expected for a Boltzmann distribution.
These deviations are apparent in the P , Q, and R branches, shown
in Fig. 1.6. It was found that the c3 (v = 2) state interacts with the
b0 (v = 7) state via heterogeneous coupling and that the states exchange
electronic character around J = 20. At higher J values, this complex
possibly interacts with the b(v = 11) state, but the results at higher
J for all states did not allow for definitive conclusions. As mentioned
before, the heterogeneous interaction involves only e-parity levels, and
therefore does not show up in Q-branch transitions. Hence, this intensity
depletion is attributed to the crossing with the triplet state.
After the perturbations in the singlet manifold have been accounted
for, remaining spectral deviations can be ascribed to the interaction
with C 3 Πu (v = 17). In Fig. 1.7, the reduced f -parity levels of c3 (v = 2)
are shown. Similar results were obtained for the e-parity levels. Between J = 7 and J = 9, and also between J = 11 and J = 12, two
perturbations are clearly present. These J-value positions coincide approximately with the predicted crossings with C 3 Πu (v = 17). In principle, only CΩ=1 (v = 17) can interact via the homogeneous (∆Ω = 0)
spin-orbit coupling with c3 (v = 2). However, since the S-uncoupling
mechanism induces mixing between the three CΩ (v = 17) components,
all three states can interact to some extent with c3 (v = 2). The perturbation between J = 7 and J = 8 is assigned to CΩ=1 (v = 17), and the
one between J = 11 and J = 12 to CΩ=0 (v = 17). No clear indications
for a crossing with CΩ=2 (v = 17) are observed, but they are expected
to occur at lower J values.
In Fig. 1.8, the line widths for two transitions from the ground state
to levels with f -parity of the c3 1 Πu (v = 2) state in 15 N2 are shown.
Near J = 8 and J = 12, an increase in broadening, and hence in predissociation rates, is observed that is again ascribed to a coupling with the
crossing, strongly-predissociated CΩ (v = 17) states. As an example, the
line width of Q(6) is increased by predissociation and is approximately
double that of Q(15) which shows no broadening beyond the normal
Doppler width. Notably, the intensity depletions shown in Fig. 1.6 take
place at the same J-values at which the crossings with the triplet states
are predicted. Altogether, these results are taken as convincing cumulative evidence that the local interactions with the C 3 Πu (v = 17) triplet
18
1. On the complexity of the absorption spectrum of N2
state can be observed indirectly, through their perturbative effects on
the singlet manifold. Moreover, the predictions of the CSE model appear
to be very adequate in this respect.
1.5
Conclusions and outlook
In this paper, some of the complexities observed in the electric-dipoleallowed spectrum of molecular nitrogen are discussed. These complexities arise from global vibronic and local rotationally-dependent perturbations, causing energy level shifts, redistribution of intensities and intensity borrowing, and quantum-interference phenomena. Experimental
methods, such as XUV ionization spectroscopy, that can achieve rotational resolution are a suitable tool to study this important molecule.
The rate of predissociation in molecular nitrogen is often fast enough
to cause line broadening that exceeds the narrow-band XUV laser instrumental line width, but predissociation rates are often slow enough
not to remove the rotational structure completely. The triplet manifold
plays a key role in the predissociation processes in molecular nitrogen.
The continuous interplay with the theoretical CSE modeling that is essentially capable of spectroscopic accuracy has played a crucial role in
the detailed analysis and understanding of our experimental results.
The connection between high-resolution XUV spectroscopy of the
lowest ungerade states of molecular nitrogen and atmospheric chemistry is a strong one. For example, the effective emission from c04 1 Σ+
u −
X 1 Σ+
(0,0)
and
(0,1)
bands
in
the
Earth’s
airglow
are
unusually
weak.
g
The radiation of the (0,0) band, which is in fact the strongest emission
feature in the N2 spectrum [27], is radiatively trapped and undergoes
resonant fluorescent scattering under atmospheric conditions. In this
process the (0,1) band acquires some intensity also. This (0,1) band is
in accidental resonance with the transition b 1 Πu −X 1 Σ+
g (2,0). Since the
b(v = 2) level is strongly predissociated, the observed overall emission is
unexpectedly weak [43]. This remarkable coincidence in the complex N2
spectrum is strongly dependent on isotopomer, as are many other observed predissociation phenomena in molecular nitrogen [34]. So far, isotopic fractionation in nitrogen-containing atmospheres has been studied
principally from the perspective of gravitational phenomena. Dissociative recombination of N+
2 and electron-impact dissociation of neutral N2
19
1. On the complexity of the absorption spectrum of N2
produce kinetically-hot atomic nitrogen, that, depending on the planetary escape velocities, may lead to isotopic fractionation and strongly
varying 15 N/14 N ratios [16, 44]. The isotope-dependent predissociation
effects resulting from the complexities in the N2 spectrum will add to
this behavior, although the particularities of each case are to be explored
in more detail.
Although fundamental in character, studies of perturbation and predissociation processes in molecular nitrogen and their isotopomers are
crucial to a better understanding of radiative-transfer phenomena and
activated N/N2 chemistry in the Earth’s upper atmosphere. In a similar
spirit, this type of research has implications for other planetary systems
with nitrogen-containing atmospheres. For example, Liang et al. [45]
have already used the results of a CSE model of N2 photodissociation to
explain nitrogen isotope anomalies in HCN in the atmosphere of Titan.
Future work will attempt to expand both the experimental database
and the accurate theoretical modeling to higher energies. This requires
the incorporation into the model of a plethora of higher-lying Rydberg
and valence states of singlet as well as triplet character, and the interactions among them.
Acknowledgments
Part of this research was supported by Australian Research Council
Discovery Program Grant DP0558962.
20
Chapter 2
Quantum-interference effects in
the o 1Πu(v = 1) ∼ b 1Πu(v = 9)
Rydberg-valence complex of
molecular nitrogen
Two distinct high-resolution experimental techniques, 1 XUV
+ 10 UV laser-based ionization spectroscopy and synchrotronbased XUV photoabsorption spectroscopy, have been used to
study the o 1 Πu (v = 1) ∼ b 1 Πu (v = 9) Rydberg-valence complex of 14 N2 , providing new and detailed information on the perturbed rotational structures, oscillator strengths, and predissociation linewidths. Ionization spectra probing the b0 1 Σ+
u (v = 6)
state of 14 N2 , which crosses o 1 Πu (v = 1) between J = 24 and
J = 25, and the o 1 Πu (v = 1), b 1 Πu (v = 9), and b0 1 Σ+
u (v = 6)
14 15
states of N N, have also been recorded. In the case of 14 N2 , rotational and deperturbation analyses correct previous misassignments for the low-J levels of o(v = 1) and b(v = 9). In addition, a two-level quantum-mechanical interference effect has been
found between the o−X(1, 0) and b−X(9, 0) transition amplitudes
which is totally destructive for the lower-energy levels just above
the level crossing, making it impossible to observe transitions to
b(v = 9, J = 6). A similar interference effect is found to affect
the o(v = 1) and b(v = 9) predissociation linewidths, but, in this
case, a small non-interfering component of the b(v = 9) linewidth
is indicated, attributed to an additional spin-orbit predissociation
by the repulsive 3 3 Σ+
u state.
2. Quantum-interference effects on a Rydber-valence complex of N2
2.1
Introduction
The dipole-allowed absorption spectrum of molecular nitrogen in the
extreme ultraviolet (XUV) wavelength region was initially thought to
consist of a multitude of electronic band structures, until the true nature of the excited states was unravelled [19, 20, 21]. The apparent
complexity of the XUV spectrum is a result of Rydberg-valence mixing
between a limited number of singlet ungerade states lying at excitation energies just above 100 000 cm−1 . There are two valence states
1
01 +
involved, one of 1 Σ+
u and one of Πu symmetry (referred to as the b Σu
and b 1 Πu states), and there exist singlet Rydberg series converging
+
0
1 +
on the first X 2 Σ+
g ionization limit in the N2 ion (the npσu cn+1 Σu
and npπu cn 1 Πu series; principal quantum number n ≥ 3) and the
nsσg on 1 Πu series converging on the A2 Πu ionization limit. The vibrational numbering in the o3 1 Πu state, of relevance to the present study,
was determined by Ogawa et al. [46]. In the seminal paper by Stahel et al. [23], a model of Rydberg-valence interactions was presented
that provides a quantitative explanation for the energy-level perturbations, the seemingly erratic behaviour of the rotational constants, and
the observed pattern of band intensities which deviate strongly from
Franck-Condon-factor predictions. A comprehensive ab initio study by
Spelsberg and Meyer [24] later confirmed the main conclusions of the
Stahel et al. [23] model. In addition to these homogeneous perturbations
in which states of like symmetry are coupled, the effects of heterogeneous
perturbations, i.e., coupling between states of 1 Πu and 1 Σ+
u symmetry,
were also included in subsequent analyses [25], thereby improving the
quantitative agreement between theory and experiment. Recently, the
inclusion of spin-orbit interactions between the 1 Πu and 3 Πu states in a
coupled-channel model of the Rydberg-valence interactions has allowed
the complex isotopic pattern of predissociation in the lower vibrational
levels of the 1 Πu states to be explained [32], including rotational effects
[47].
In addition to the comprehensive theoretical studies describing
the overall excited-state structure for dipole-allowed transitions in N2
[23, 24], several semiempirical local-perturbation analyses have been
performed which focus on particular level crossings. Yoshino et al. [48]
examined a number of such local perturbations, most of which had one
22
2. Quantum-interference effects on a Rydber-valence complex of N2
of the o3 1 Πu (v) or o4 1 Πu (v) Rydberg states as a perturbation partner.
Yoshino and Freeman [49] treated a multi-level local perturbation involv1
ing the Rydberg states c05 1 Σ+
u (v = 0) and c5 Πu (v = 0) interacting with
1
1
+
a number of valence states of Πu and Σu symmetry. A well-known
perturbation, observed as a pronounced feature in N2 spectra involving
01 +
the c04 1 Σ+
u (v = 0) and b Σu (v = 1) levels, was analysed by Yoshino and
Tanaka, based on classical spectroscopic data [50], and later by Levelt
and Ubachs, based on XUV-laser data [51]. In the 15 N2 isotopomer,
several local perturbation studies have also been performed, e.g., for the
1
o1 Πu (v = 0) ∼ b01 Σ+
u (v = 3) crossing [36]. (In the on Πu (v) series, the
subscripts are commonly dropped from the state designation for n = 3.)
Due to differing isotopic shifts in the Rydberg and the valence states,
the accidental perturbations occur at different locations in the rovibronic
structure of the three natural isotopomers of N2 .
In this study, the o1 Πu (v = 1) ∼ b1 Πu (v = 9) Rydberg-valence complex of 14 N2 is examined using two different experimental techniques,
providing new and detailed information on the perturbed rotational
structures, oscillator strengths, and predissociation linewidths. Rotational and deperturbation analyses are performed which correct previous
misassignments [48] for transitions to the low-J levels of o(v = 1) and
b(v = 9), and elucidate the quantum-interference effects occurring in oscillator strength, between these two electric-dipole-allowed transitions,
and in linewidth, between these two predissociated levels.
In addition to recording the b − X(9, 0) and o − X(1, 0) bands for the
main 14 N2 isotopomer, these bands were also investigated for 14 N15 N
and a rotational analysis performed. In the case of the mixed 14 N15 N
isotopomer, the rotational structure of each transition is unperturbed
due to differing isotopic shifts. Since, for high J, the homogeneous
perturbation complex of the two 1 Πu states undergoes a heterogeneous
14 N by
interaction with the b01 Σ+
2
u (v = 6) state, as already noticed in
Yoshino et al. [48], the b0 (v = 6) level is also included in the present
study for both 14 N2 and 14 N15 N.
Christian Jungen has made significant contributions to the understanding of the excited states of N2 . In 1990, Huber and Jungen reported
a high-resolution jet absorption study of N2 in the region near 80 nm
[52], unravelling the Rydberg structure and following the vibrational
sequence of the b01 Σ+
u state even beyond the ionization potential. In
23
2. Quantum-interference effects on a Rydber-valence complex of N2
a subsequent study [53], he was part of a team investigating the nf Rydberg series in the last 6000 cm−1 below the ionization energy, based
on high-resolution spectra of 14 N2 and 15 N2 recorded with the 10.6 m
spectrograph at Ottawa and at the Photon Factory synchrotron facility
in Tsukuba. Finally, this led to the development of a comprehensive
multichannel quantum-defect analysis of the near-threshold spectrum of
N2 [54]. It is with great pleasure that we dedicate our present work to
Dr Jungen.
2.2
Experiments
Two distinct experimental techniques were employed in this work to
study the interaction between the b(v = 9) and o(v = 1) levels of
N2 . First, very-high-resolution laser-based ionization spectroscopy was
used to determine the energy perturbations. Second, high-resolution
synchrotron-based quantitative photoabsorption spectroscopy was used,
primarily to study quantum-interference effects in the corresponding oscillator strengths.
Laser-based 1 XUV + 10 UV two-photon ionization spectroscopy
was employed to study the excitation spectrum of N2 , initially in the
range λ = 92.9 − 93.5 nm. Details of the experimental method, including a description of the lasers, vacuum setup, molecular beam configuration, time-of-flight (TOF) detection scheme and calibration procedures, have been given previously [36]. Two different laser systems
were used, a pulsed dye laser (PDL)-based source, delivering an XUV
bandwidth of ∼ 0.3 cm−1 full-width at half-maximum (FWHM), and a
pulsed dye amplifier (PDA)-based source, delivering an XUV-bandwidth
of ∼ 0.01 cm−1 FWHM. The wavelength range was later extended to
λ = 92.59 − 92.87 nm, to also cover excitation of the b01 Σ+
u (v = 6) state,
which was investigated under similar molecular-beam conditions using
the PDL-based source. The PDL-based system and its application to
the spectroscopy of N2 has been described in Refs. [36, 55]. Briefly, the
sixth harmonic of a pulsed dye laser was employed, calibrated against
the reference standard provided by the Doppler-broadened linear absorption spectrum of molecular iodine [56]. The absolute wavenumber
uncertainty in the XUV for this system is ± 0.1 cm−1 for fully resolved
lines. The PDA-based system was used in a frequency-mixing scheme:
24
2. Quantum-interference effects on a Rydber-valence complex of N2
ωXU V = 3(ωPDA + ω532 ), where ω532 is the frequency-doubled output of
an injection-seeded Nd:YAG laser. It has been documented previously
how this frequency-mixing scheme produces a bandwidth of ∼ 0.01 cm−1
FWHM in the XUV [57, 58]. The efficiency for XUV production with
this mixing scheme is much lower than for the PDL-based system, and
signal levels decrease accordingly. Even though the molecular-beam densities were increased to compensate, this scheme could only be used for
calibration (and linewidth measurements) of a few low-J 00 lines. In view
of the lifetime broadening encountered in the b(v = 9) and o(v = 1) levels
of 14 N2 (see also [38]), the absolute accuracy of the corresponding energy
calibration was limited to ∼ 0.05 cm−1 . The two-photon-ionization TOF
experiment has some characteristics that were favourably employed in
the present study. Mass separation can be combined with laser excitation to separate the contributions to the spectrum of the main 14 N2
isotopomer from those of the mixed 14 N15 N species. Even in the case of
natural nitrogen, which contains only 0.74% 14 N15 N, resolved isotopic
lines at the bandheads of the o(v = 1), b(v = 9) − X(v 00 = 0) systems could be observed. However, in a later stage of the experiments,
isotopically-enriched 14 N15 N gas became available, yielding better quality spectra for the b − X(9, 0), o − X(1, 0), and b0 − X(6, 0) transitions
that were used in the final analysis. Furthermore, by changing the relative delay between the N2 pulsed-valve trigger and the laser pulse, as
well as varying the nozzle-skimmer distance, the rotational temperature
in the molecular beam could be selected to measure independent spectra
of cold (10 − 20 K) and warm (up to 180 K) samples. These options
helped greatly in the assignment of the spectral lines.
Photoabsorption spectroscopy on the o(v = 1), b(v = 9) − X(v 00 = 0)
systems of 14 N2 was performed at the 2.5 GeV storage ring of the Photon
Factory, a synchrotron radiation facility at the High Energy Accelerator
Research Organization in Tsukuba, Japan. Details of the experimental
procedure can be found in Stark et al. [42]. Briefly, a 6.65 m spectrometer with a 1200 grooves per mm grating (blazed at 550 nm and
used in the sixth order) provided an instrumental resolving power of
∼ 150, 000, equivalent to a resolution of ∼ 0.7 cm−1 FWHM. The spectrometer tank, at a temperature of 295 K, served as an absorption cell
with a path length of 12.51 m. A flowing gas configuration was used: N2
of normal isotopic composition entered the spectrometer tank through a
25
2. Quantum-interference effects on a Rydber-valence complex of N2
needle valve and was continuously pumped by a 1500 ls−1 turbomolecular pump. Absorption spectra were recorded at tank pressures ranging
from 1.8×10−5 torr to 7.5×10−5 torr, corresponding to N2 column densities ranging from 7.4×1014 cm−2 to 3.1×1015 cm−2 . The 92.88–93.14 nm
spectral region was scanned, in three overlapping portions, at a speed
of 0.005 nm min−1 . A signal integration time of about 1 s resulted in
one data point for each 8.7×10−5 nm interval of the spectrum. Signal
rates from the detector, a windowless solar-blind photomultiplier tube
with a CsI-coated photocathode, were about 50 000 s−1 for the background continuum; the detector dark count rate was less than 2 s−1 .
A signal-to-noise ratio of about 250 was typically achieved for the continuum level. The experimental absorption spectra were converted into
photoabsorption cross sections through application of the Beer-Lambert
law. Non-statistical uncertainties in the experimental cross sections are
estimated to be ∼ 10%, with contributions from uncertainties in the N2
column density, variations in the signal background, and scattered light.
2.3
Results and discussion
Energies
Using the PDL-based XUV source, rotationally-resolved spectra of
1
1 +
14 N were
the b1 Πu − X 1 Σ+
2
g (9,0) and o Πu − X Σg (1,0) bands in
recorded for mass 28 in the TOF spectrum at low and high rotational
temperatures, achieved by varying the timing of the N2 supersonic
beam. In Fig. 2.1, the two spectra are compared and line assignments
shown. High-rotational-temperature laser-excitation spectra were also
recorded for 14 N15 N using a molecular beam of isotopically enriched nitrogen. Figure 2.2 shows such a recording, displaying lines from the
14 N15 N.
1
1 +
b1 Πu − X 1 Σ+
g (9,0) and o Πu − X Σg (1,0) bands in
Using the narrower-bandwidth frequency-mixing PDA-based XUV
source, spectra of the b − X(9,0) and o − X(1,0) bands in 14 N2 were
recorded for some low-rotational lines. Recordings of the Q(2) line from
the b − X(9,0) band and the Q(1) line from the o − X(1,0) band in 14 N2 ,
obtained with this source, are shown in Fig. 2.3. Signal levels for the
transition to the broader b(v = 9, J = 2) level are low because of the
competition between predissociation and ionization in the 1 + 10 two-
26
2. Quantum-interference effects on a Rydber-valence complex of N2
R o(1) 24
0 2+3
20
4
2
5
6
10
12
1
8
R b (9) 0
4
4
5
21
Q o (1)
5
P o(1)
7
8
4
Q b(9)
1
2 P b(9)
P b’ (6)
(a)
*
(b)
14
N2
107560
107580
107600
107620
−1
107640
107660
Transition Energy (cm )
Figure 2.1: 1 XUV + 10 UV ionization spectra and line assignments for the
1
1 +
14
b1 Πu − X 1 Σ+
N2 in high [spectrum (a)]
g (9,0) and o Πu − X Σg (1,0) bands of
and in low [spectrum (b)] rotational-temperature molecular beams, recorded
using the PDL-based XUV source. Note that severe blending occurs in the
spectrum and several weak lines are not identified. In the region in spectrum
(a) marked with an asterisk, two separate scans are joined and intensities of
the two scans should not be compared.
photon ionization detection scheme and the rather low XUV and UV
intensity levels available with the frequency-mixing scheme employed.
Analysis of the linewidths and discussion of the predissociation phenomena in the excited states are deferred to Sec. 2.3. Transition energies
and line assignments for the o(v = 1) and b(v = 9) levels (see below) are
presented in Tables 2.1 and 2.2, for 14 N2 , and in Tables 2.3 and 2.4, for
14 N15 N.
1 +
14 N ,
An excitation spectrum for the b01 Σ+
2
u − X Σg (6,0) band of
which lies just above the overlapped b − X(9,0) and o − X(1,0) bands,
1 +
is shown in Fig. 2.4. Assignment of the lines in this 1 Σ+
u − Σg band,
containing only P and R branches, is straightforward, but the rotational
constants for the ground and excited states are such that, for J 00 ≤
9, the P (J 00 ) and R(J 00 + 3) lines overlap within the resolution of the
1 +
experiment. Observed line positions for the b01 Σ+
u − X Σg (6,0) band
14
14
15
are listed in Table 2.5 for N2 , and Table 2.6 for N N.
27
2. Quantum-interference effects on a Rydber-valence complex of N2
0
14
15
R 20
o −X (1,0)
N N
Q 17
1
P 16
2
b −X (9,0) 13
15
10
13
107365
107415
3
107465
107515
0
R
1Q
P
107565
−1
107615
Transition Energy (cm )
Figure 2.2: 1 XUV + 10 UV ionization spectrum and line assignments for the
14 15
1
1 +
N N, recorded using the
b1 Πu − X 1 Σ+
g (9,0) and o Πu − X Σg (1,0) bands of
PDL-based XUV source.
Q (1) o −X(1,0)
107628.4
14
N2
107628.8
Q (2) b −X(9,0)
107647.6
−1
14
N2
107648
Transition Energy (cm )
Figure 2.3: 1 XUV + 10 UV ionization spectra of the Q(2) b1 Πu − X 1 Σ+
g (9,0)
14
and Q(1) o1 Πu − X 1 Σ+
(1,0)
lines
of
N
,
recorded
using
the
PDA-based
2
g
XUV source and showing the different predissociation broadening observed
for b(v = 9) and o(v = 1).
28
2. Quantum-interference effects on a Rydber-valence complex of N2
0
21
18
14
1
R
P
N2 b’ − X (6,0)
107700
107800
107900
−1
108000
Transition Energy (cm )
Figure 2.4: 1 XUV + 10 UV ionization spectrum and line assignments for the
1 +
14
b01 Σ+
N2 , recorded using the PDL-based XUV source.
u − X Σg (6,0) band of
Term values and spectroscopic parameters were determined from the
experimental transition energies using least-squares fitting procedures.
The terms of the ground state X 1 Σ+
g were represented as
F (J 00 ) = B[J 00 (J 00 + 1)] − D[J 00 (J 00 + 1)]2 + H[J 00 (J 00 + 1)]3 , (2.1)
where B is the rotational constant, and D and H are the centrifugal
distortion parameters. The spectroscopic parameters of Trickl et al. [40]
14 N and 14 N15 N,
and Bendtsen [41] were used for the X 1 Σ+
2
g states of
1
respectively. The terms of the excited Πu states, where, for simplicity,
we denote the rotational quantum number by J, rather than J 0 (and the
same holds for v and v 0 ), were taken to have the form
Tf (J) = ν0 + B[J(J + 1) − 1] − D[J(J + 1) − 1]2 ,
(2.2)
for the f -parity [11] levels, where ν0 is the band origin, and
Te (J) = Tf (J) + ∆Tef (J),
(2.3)
for the e-parity levels, where the Λ-doubling is represented by
∆Tef (J) = q[J(J + 1) − 1],
(2.4)
29
2. Quantum-interference effects on a Rydber-valence complex of N2
Table 2.1: Observed transition energies (in cm−1 ) for the b1 Πu − X 1 Σ+
g (9,0)
band in 14 N2 . Deviations from transition energies calculated using the corresponding fitted term values are also shown (∆o−c = obs − calc). Wave numbers
given to three decimal places are from narrow-bandwidth PDA spectra, those
to two decimal places are from PDL spectra. Wave numbers derived from
blended lines are flagged with an asterisk (*) and those from shoulders in the
spectra by an s.
J 00
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
R(J 00 )
107 654.320
107 656.14*
107 656.01s
107 656.14*
107 635.07*
∆o−c
−0.075
0.26
−0.26
0.35
−0.48
107 611.14*
−0.23
107 588.42*
0.05
107 558.95
107 541.93
107 523.12
107 503.15
107 481.50
107 458.29
107 433.83
107 407.77
107 380.25
107 351.17
0.01
0.04
−0.19
−0.06
−0.10
−0.20
−0.05
−0.01
0.06
0.04
Q(J 00 )
∆o−c
107 650.44
107 647.840
107 644.33
107 640.01*
0.02
−0.082
−0.01
0.12
107 599.66
107 588.42*
107 575.52
107 560.96
107 544.59
107 526.84*
107 507.16
107 486.08
107 463.49
107 439.52
107 413.80
107 386.71
107 358.22
0.14
−0.03
−0.01
0.09
0.04
0.20
0.01
−0.04
−0.06
0.06
−0.07
−0.06
0.04
P (J 00 )
∆o−c
107 642.56
107 635.85*
107 628.63*
107 619.43*
0.10
−0.14
0.19
−0.56
107 567.76
107 552.85
107 535.81
107 517.26
107 497.27*
107 475.01
107 451.63
107 427.08*
107 400.09
107 372.12
107 342.58
0.04
0.17
0.01
0.07
0.35
−0.05
0.01
0.44
−0.04
0.01
0.00
and q is the Λ-doubling parameter. Rotational energy levels in the
b0 (v = 6) state, of 1 Σ+
u (e) symmetry, were represented by
T (J) = ν0 + B[J(J + 1)] − D[J(J + 1)]2 .
(2.5)
The b(v = 9) and o(v = 1) levels in 14 N2 give rise to an avoided
crossing in the rotational structure between J = 4 and J = 5, resulting from a J-independent homogeneous perturbation (∆Ω = 0, electro30
2. Quantum-interference effects on a Rydber-valence complex of N2
−1
Reduced term value (cm )
400
300
o (1) deperturbed
b (9) deperturbed
b’ (6) deperturbed
b (9)
o (1)
b’ (6)
e levels
200
100
40
20
0
14
N2
-20
0
20
40
60
80
100
0
0
100
200
300
400
500
600
700
800
J (J +1)
Figure 2.5: Reduced term values of the b1 Πu (v = 9) ∼ o1 Πu (v = 1) ∼
14
b01 Σ+
N2 (e-levels only), including the results of an
u (v = 6) crossings in
effective three-state deperturbation. The terms are reduced such that the deperturbed b(v = 9) levels lie on the zero line. The solid symbols represent
measured energy levels, the open symbols predicted levels.
1
o (1) mixing factors
14
N2
0.8
o (1)
0.6
0.4
b (9)
0.2
b’(6)
0
0
5
10
15
20
25
30
J
Figure 2.6: Calculated o(v = 1) mixing factors for the o1 Πu (v = 1), b1 Πu (v =
14
9), and b01 Σ+
N2 . Plotted is the absolute square |ci |2 of the
u (v = 6) levels of
coefficients ci representing the projection of the wave function Ψo(1) on a basis
of unperturbed wave functions (φo(1) , φb(9) , φb0 (6) ). The data pertain to the
e-parity component only.
31
2. Quantum-interference effects on a Rydber-valence complex of N2
static) between these valence and Rydberg states. Furthermore, at high
J, interaction between the b0 (v = 6) levels, of 1 Σ+
u symmetry, and near1
degenerate o(v = 1) levels, of Πu symmetry, becomes significant. This
interaction is heterogeneous (∆Ω 6= 0, L-uncoupling) and only involves
the e-parity levels of the o(v = 1) state. For the e-parity manifold, a
full three-state deperturbation analysis was performed for each J value
by diagonalizing the matrix


Tb9
Hb9o1
0
√


(2.6)
To1
Ho1b0 6 J(J + 1)  ,
 Hb9o1
√
Tb0 6
0
Ho1b0 6 J(J + 1)
where the diagonal elements are the term energies of b(v = 9), o(v =
1) and b0 (v = 6), given by Eqs. (2.2–2.5). The off-diagonal element
Hb9o1 is the two-level homogeneous interaction parameter between the
√
1 Π states, and H 0
u
o1b 6 J(J + 1) represents the effective heterogeneous
interaction matrix element between the o(v = 1) and b0 (v = 6) states.
Since the b0 (v = 6) state has 1 Σ+
u symmetry, and therefore only e-parity
levels, in the case of the f -parity manifold Eq. (2.6) reduces to a 2 × 2
matrix.
The resulting deperturbed spectroscopic parameters for o(v = 1),
b(v = 9), and b0 (v = 6), obtained from a comprehensive least-squares
fit for all available spectral lines pertaining to the three band systems,
are listed in Table 2.7. In the fitting procedure, the nominal uncertainty
in the absolute transition energy for fully-resolved lines of reasonable
strength was set at 0.1 cm−1 . In the case of weak or blended lines, the
uncertainty was set at an estimated value in the range 0.15 − 0.30 cm−1 .
The lines obtained with the PDA-based laser system have an uncertainty
of 0.05 cm−1 .
In Fig. 2.5, reduced experimental term values are plotted for the
b(v = 9), o(v = 1) and b0 (v = 6) levels in 14 N2 , based on the assignments
to be discussed below. An avoided crossing between b(v = 9) and o(v =
1) is clearly visible between J = 4 and 5, with a maximum energy shift
of 8.7 cm−1 at J = 4, as a result of the homogeneous interaction between
the two levels. A second avoided crossing occurs between J = 24 and
25, associated with the heterogeneous coupling between o(v = 1) and
b0 (v = 6), with a maximum shift of 13.5 cm−1 at J = 25. In Fig. 2.5,
only the e-parity levels of both 1 Πu states are displayed. For the f 32
2. Quantum-interference effects on a Rydber-valence complex of N2
parity components, a similar graph can be constructed, but in this case
there is no interaction with the b0 (v = 6) state. Note that the observed
levels in Fig. 2.5 are represented by solid symbols, the predicted levels
by open symbols. The mixing factors following from the diagonalization
procedure using Eq. (2.6) are presented in Fig. 2.6.
The present line assignments for the b(v = 9) ∼ o(v = 1) complex in
14 N differ from those in the previous study of Yoshino et al. [48]. (The
2
assignments for rotational lines accessing b(v = 9) given by Carroll and
Collins [21] are erroneous, as already discussed by Yoshino et al. [48].)
First, we have adopted a different convention for the labelling of the
levels before the crossing (J = 1−4), assigning the b(v = 9) and o(v = 1)
labels to the levels with the highest mixing factor for the corresponding
nominal level. Thus, as shown in Fig. 2.5, for J-levels before the crossing,
the higher term values belong to b(v = 9) and, after the crossing (J ≥ 5),
to o(v = 1), while the reverse is true for the lower terms. Yoshino et
al. [48], in contrast, assigned all the higher and lower term values as
o(v = 1) and b(v = 9), respectively.
Apart from this unimportant difference in nomenclature, we have
made some significant reassignments. For J ≥ 5 in the b(v = 9) ∼
o(v = 1) complex, the assignments of the current and previous (Yoshino
et al. [48]) studies agree, but, for lines accessing the J = 1 − 4 levels
of the excited states, the assignments differ. These reassignments were
facilitated in the present work by the recording of a number of spectra
under differing experimental conditions. Separate PDL-based spectra
for relatively low and high rotational temperatures aided considerably in
the assignment of lines, which may be blended in one or other spectrum,
to low or high J 00 . Furthermore, the PDA-based measurements were
performed with a rather low rotational temperature, so that only lines
originating from J 00 = 0 − 2 appeared strongly, with possible weak lines
from J 00 = 3 and 4. This conclusion was confirmed independently in
measurements of the b − X(10, 0) band in 14 N2 [58], recorded under the
same experimental conditions. Of course, the most convincing argument
for the present line assignments and analysis is the reproducibility of
all lines in the fitting procedure using Eqs. (2.1–2.6). With 163 lines
included in the fit, a χ2 of 93 is found.
The deperturbed rotational constant for 3sσg o 1 Πu (v = 1) in Table 2.7, B = 1.7325 cm−1 , is in agreement with expectation for a Ryd33
2. Quantum-interference effects on a Rydber-valence complex of N2
Table 2.2: Observed transition energies (in cm−1 ) for the o1 Πu − X 1 Σ+
g (1,0)
band in 14 N2 . Deviations from transition energies calculated using the corresponding fitted term values are also shown (∆o−c = obs − calc). Wave numbers
given to three decimal places are from narrow-bandwidth PDA spectra, those
to two decimal places are from PDL spectra. Wave numbers derived from
blended lines are flagged with an asterisk (*) and those from shoulders in the
spectra by an s.
J 00
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
R(J 00 )
107 632.73*
107 634.995
107 636.456
107 636.63*
107 654.66*
107 653.62
107 652.44
107 651.13
107 649.73
107 647.95*
107 645.87
107 643.34
107 640.52*
107 636.63*
107 632.73*
107 628.63*
107 623.31
107 617.84
107 611.71
107 605.09
107 597.73
∆o−c
0.10
−0.034
−0.029
−0.14
−0.13
0.03
0.05
0.00
0.03
−0.01
0.02
0.03
0.20
−0.22
−0.14
0.22
−0.08
−0.02
−0.06
−0.03
−0.11
107 579.56
0.10
Q(J 00 )
∆o−c
P (J 00 )
∆o−c
107 628.614
107 627.16
107 624.61*
107 620.81*
107 635.05*
107 629.81
107 624.61*
107 619.43
107 614.06s
107 608.51*
107 602.31
107 596.01
107 588.95
107 581.62
107 574.03*
107 565.44
107 556.54
107 547.25
107 537.36
107 526.84*
107 516.12
107 504.68
107 492.78
107 480.24
107 467.43
107 453.86
−0.039
0.08
0.05
−0.06
0.14
0.04
−0.01
0.01
0.02
0.13
−0.03
0.12
−0.03
0.01
0.28
0.03
−0.02
0.04
0.01
−0.14
0.02
−0.02
−0.01
−0.12
0.02
−0.08
107 620.81*
107 615.20
107 608.58
107 601.08
107 611.00
107 601.83
107 592.77
107 583.51
107 574.03*
107 564.49
107 554.45
107 543.94
107 533.03
107 521.59
107 509.70
107 497.24*
107 484.35
107 470.88
107 456.92
107 442.35
107 427.09*
107 411.26
107 394.50
107 375.87
0.12
0.07
−0.04
0.12
−0.06
−0.05
0.05
−0.01
−0.10
0.04
0.05
0.01
0.03
0.00
0.01
−0.05
−0.02
−0.04
−0.01
−0.02
−0.10
−0.04
0.08
0.33
berg state converging on the A2 Πu ionic state [B(A, v + = 1) = 1.716
cm−1 [59]], with the residual difference occurring because the effects of
34
2. Quantum-interference effects on a Rydber-valence complex of N2
Table 2.3: Observed transition energies (in cm−1 ) for the b1 Πu − X 1 Σ+
g (9,0)
band in 14 N15 N, from spectra obtained with the PDL-based XUV source. Deviations from transition energies calculated using the corresponding fitted term
values are also shown (∆o−c = obs−calc). Wave numbers derived from blended
lines are flagged with an asterisk (*) and those from shoulders in the spectra
by an s.
J 00
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
R(J 00 )
107 547.65s
107 548.47s
107 548.01s
107 546.27*
107 542.69
107 537.81
107 531.56
107 523.66*
107 514.70*
107 503.93
107 491.60
107 478.00
107 463.22
107 446.79
∆o−c
0.01
−0.12
−0.07
0.14
−0.03
−0.05
0.01
−0.13
0.13
0.03
−0.18
−0.20
0.06
0.12
Q(J 00 )
∆o−c
107 543.70
107 540.85
107 536.60*
107 530.62
107 523.66*
107 514.70*
107 504.60
107 493.09
107 480.18
107 465.46
107 449.48
107 432.00
107 413.18
107 392.91
107 371.77
−0.09
−0.04
0.06
−0.13
0.16
−0.10
−0.05
0.04
0.19
−0.03
−0.05
−0.12
−0.08
−0.03
0.60
P (J 00 )
∆o−c
107 529.29*
107 521.23
107 511.66
107 500.38
107 488.06
107 473.95
107 458.43
107 441.55
−0.06
0.08
0.15
−0.03
0.20
0.09
0.02
0.04
107 382.22
0.12
perturbations by more remote levels are still included in the two-leveldeperturbed value. The deperturbed rotational constant for b(v = 9)
is significantly lower, B = 1.2288 cm−1 , appropriate for a valence b1 Πu
state. The deperturbed D value for b(v = 9) in Table 2.7 is small
and negative, comparable to the value D = −5.5 × 10−6 cm−1 for 15 N2
reported in Ref. [36]. This negative D value is supported by the observations of Yoshino et al. [48], who find some even higher-J P - and
Q-branch lines accessing b(v = 9), up to J = 28, which exhibit a further
gradual shift upward in energy, by up to several cm−1 , for both e- and
f -parity components. It is likely that the negative D value for b(v = 9)
is a result simply of the multilevel perturbation by more distant levels.
The new assignments for the lowest-J levels of b(v = 9) and o(v = 1)
14
in N2 indicate a greater separation in energy than in Ref. [48] (espe35
2. Quantum-interference effects on a Rydber-valence complex of N2
Table 2.4: Observed transition energies (in cm−1 ) for the o1 Πu − X 1 Σ+
g (1,0)
band in 14 N15 N, from spectra obtained with the PDL-based XUV source. Deviations from transition energies calculated using the corresponding fitted term
values are also shown (∆o−c = obs−calc). Wave numbers derived from blended
lines are flagged with an asterisk (*) and those from shoulders in the spectra
by an s.
J 00
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
R(J 00 )
107 610.18*
107 613.09*
107 615.49*
107 617.07*
107 619.03*
107 619.03*
107 619.03*
107 619.03*
107 619.03*
107 617.07*
107 615.49*
107 613.09*
107 610.18*
107 607.57*
107 603.99*
107 599.36*
107 594.73
107 588.45*
107 582.49*
107 576.48s
107 569.11*
∆o−c
0.06
0.14
0.22
−0.02
0.63
−0.17
−0.46
−0.25
0.48
−0.25
−0.09
−0.24
−0.39
0.28
0.49
0.18
0.39
−0.50
−0.50
0.07
0.01
Q(J 00 )
∆o−c
P (J 00 )
∆o−c
107 606.26
107 605.29
107 603.99*
107 601.75
107 599.36*
107 596.19
107 592.64
107 588.45*
107 584.06s
107 578.97
107 573.51
107 567.60
107 561.40*
107 554.02*
107 545.27*
107 538.58s
107 529.29*
−0.01
0.03
0.24
0.02
0.15
0.01
−0.01
−0.17
−0.02
−0.08
0.00
0.13
0.46
0.11
−0.11
0.23
−0.55
107 599.36*
107 593.66
107 588.45*
107 582.49*
107 576.03*
107 569.11*
107 561.40*
107 554.02*
107 545.27s
107 536.60*
107 527.20
107 517.15
107 506.80
107 496.04
107 484.59
0.78
−0.05
0.11
0.02
−0.06
−0.09
−0.40
0.12
−0.22
0.02
0.04
−0.08
0.00
0.19
0.20
cially for J = 1f where the separation is larger by 3.4 cm−1 ) resulting
in significant differences between the two-level-deperturbed parameters
in Table 2.7 and previous values [48]. For example, the fitted homogeneous interaction matrix element, Hb9o1 = 9.47 cm−1 , should be compared with the previously accepted value of 8.10 cm−1 [48]. It has been
pointed out elsewhere [32] that it was the incorrect use of the deperturbed
experimental b(v = 9) and o(v = 1) spectroscopic parameters [21, 48] in
comparisons with the theoretical results of Stahel et al. [23] and Spels-
36
2. Quantum-interference effects on a Rydber-valence complex of N2
1 +
Table 2.5: Observed transition energies (in cm−1 ) for the b01 Σ+
u − X Σg (6,0)
14
band of N2 , from spectra obtained using the PDL-based XUV source. Deviations from transition energies calculated using the corresponding fitted term
values are also shown (∆o−c = obs−calc). Wave numbers derived from blended
lines are flagged with an asterisk (*).
J 00
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
a
R(J 00 )
108 000.93a
108 001.82
108 000.93a
107 998.65a
107 994.73a
107 989.23a
107 982.23*
107 973.35*
107 963.09*
107 951.11*
107 937.50*
107 922.44*
107 905.34*
107 886.63
107 866.31
107 844.29
107 820.67
107 795.15
107 768.04
107 739.27
107 708.79
107 676.76
∆o−c
0.013
0.08
−0.055
0.004
0.012
0.033
0.15
0.00
0.09
0.08
0.08
0.27
0.09
−0.04
−0.08
−0.13
−0.06
−0.17
−0.13
−0.03
0.08
0.27
P (J 00 )
∆o−c
107 994.49a
107 988.94a
107 982.23*
107 973.35*
107 963.09*
107 951.11*
107 937.50*
107 922.44*
107 905.84*
107 887.58
107 867.66
107 846.08
107 822.81
107 797.94
107 743.23
107 743.23
107 713.26
107 681.66
−0.045
−0.039
0.38
0.22
0.25
0.16
0.02
0.03
0.11
0.14
0.14
0.11
0.03
0.01
0.08
0.02
−0.05
−0.05
107 576.63*
107 538.23*
0.08
0.16
Data taken from PDA-based measurements [58].
berg and Meyer [24] which led to the large discrepancies, especially in
B, for these particular levels. Evidently, the present reassignments will
also have some effect in resolving those discrepancies.
37
2. Quantum-interference effects on a Rydber-valence complex of N2
1 +
Table 2.6: Observed transition energies (in cm−1 ) for the b01 Σ+
u − X Σg (6,0)
14 15
band of N N, from spectra obtained using the PDL-based XUV source.
Deviations from transition energies calculated using the corresponding fitted
term values are also shown (∆o−c = obs − calc). Wave numbers derived from
blended lines are flagged with an asterisk (*).
J 00
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
R(J 00 )
107 939.54*
107 940.32
107 939.54*
107 937.48
107 933.71*
107 928.46*
107 921.53*
107 913.15*
107 903.19*
107 891.68*
107 878.52*
107 863.96*
107 847.76*
107 830.03*
107 810.32
107 789.15
107 766.29
107 741.71
107 715.63
107 688.06
107 658.65
107 628.08
∆o−c
−0.03
−0.06
−0.14
0.03
0.01
0.05
−0.05
−0.05
−0.08
−0.08
−0.16
−0.05
0.03
0.20
0.02
0.02
−0.02
−0.13
−0.08
0.11
−0.01
0.00
P (J 00 )
∆o−c
107 933.71*
107 928.30*
107 921.53*
107 913.15*
107 903.19*
107 891.68*
107 878.52*
107 863.96*
107 847.76*
107 830.03*
107 811.15
107 790.35
107 767.99
107 744.19
107 718.35
107 691.17
107 662.39
107 631.92
0.32
0.27
0.38
0.40
0.36
0.29
0.11
0.07
−0.07
−0.18
0.13
0.09
0.08
0.23
−0.04
−0.03
0.02
0.03
107 565.96
−0.02
The major contribution to the splitting between the e- and f -parity
levels of o(v = 1) is due to the heterogeneous interaction with the
b0 1 Σ +
u (v = 6) state, as already discussed by Yoshino et al. [48]. Rotational levels in the higher-lying b0 (v = 6) valence state approach the
Rydberg o(v = 1) levels from above with increasing J, since b0 (v = 6)
has a smaller B value than o(v = 1). This is illustrated in Fig. 2.5.
For levels with J & 20, the e-parity components of o(v = 1) are pushed
down in energy, while the f -parity levels are unaffected. The R(24) line
from the o − X(1, 0) band, predicted from the perturbation analysis to
38
2. Quantum-interference effects on a Rydber-valence complex of N2
lie near 107 579.2 cm−1 , was found in the experimental spectrum as a
weak satellite, thereby pinpointing the culmination of the heterogeneous
interaction. The R(22) line of o − X(1, 0) is predicted to coincide with
the P (7) line in b − X(9, 0), while R(21) would coincide with Q(13)
of o − X(1, 0). R(23) is probably too weak to be observed under our
experimental conditions. Unfortunately, lines accessing the b0 (v = 6)
perturbation partner could only be followed up to R(21) and P (22),
below the crossing point. We note that the P (26) line in o − X(1, 0), reported by Yoshino et al. [48], does not match the present analysis and is
reassigned as the Q(19) line of b−X(9, 0). Despite the somewhat incomplete nature of the experimental data defining the b0 (v = 6) ∼ o(v = 1, e)
crossing, it has been possible to determine a heterogeneous interaction
matrix element Ho1b0 6 = 0.542 cm−1 , which is of a similar order to those
found by Yoshino et al. [48] for crossings involving other vibrational
levels of the same states: Ho0b0 3 = 0.29 cm−1 , Ho3b0 11 = 0.44 cm−1 , and
Ho4b0 14 = 0.60 cm−1 . The three-level-deperturbed Λ-doubling parameters q given in Table 2.7 for the o(v = 1) and b(v = 9) states, which
are negative and positive, respectively, agree in sign with those determined for 15 N2 by Sprengers et al. [36]. These residual Λ-doublings
are likely caused, directly or indirectly, by heterogeneous interactions
between o(v = 1) and other more remote levels of the b0 1 Σ+
u perturber.
14
15
In N N, the b(v = 9) level shifts further down in energy than the
o(v = 1) level following isotopic substitution. Therefore, the order of
the two states is reversed and the lower-lying b(v = 9) level, with the
smaller B value, does not cross the higher-lying o(v = 1). The same is
true for 15 N2 [36]. As a consequence, in the fitting procedure for 14 N15 N
it was not possible to determine a meaningful homogeneous interaction
parameter Hb9o1 , which was therefore set to zero. Furthermore, due to
the inferior signal-to-noise ratio for the 14 N15 N spectra, the highest-J
levels could not be observed, making it difficult to define the inherent
Λ-doubling parameters for both the b(v = 9) and o(v = 1) states. Accordingly, these parameters were also set to zero, as including them in
the fit of 132 lines did not improve the χ2 value of 93. Although it
was not possible experimentally to follow the b0 (v = 6) and o(v = 1)
levels of 14 N15 N to high enough J values to define the expected avoidedcrossing region, a significant reduction in χ2 was obtained by including the corresponding heterogeneous interaction parameter in the fitting
39
2. Quantum-interference effects on a Rydber-valence complex of N2
Table 2.7: Molecular parameters for the b1 Πu (v = 9), o1 Πu (v = 1) and the
14
b01 Σ+
N2 and 14 N15 N. All values are in cm−1 . 1σ statistical
u (v = 6) states of
uncertainties, resulting from the fit, are shown in parentheses, in units of the
last significant figure. Additional systematic uncertainties of order 0.05 cm−1
apply to the band origins ν0 .
b1 Πu (v = 9)a
Species:
B
D × 106
q × 103
ν0
Hb9o1
Ho1b0 6
14 N
2
Species:
B
D × 106
q × 103
ν0
Hb9o1
Ho1b0 6
14 N15 N
1.2288(4)
−3.7(8)
0.67(13)
107 647.64(4)
9.47(1)
–
1.199(1)
7(6)
0b
107 546.44(4)
0b
–
o1 Πu (v = 1)a
a
b01 Σ+
u (v = 6)
1.7325(3)
6.9(5)
−0.81(10)
107 636.43(3)
1.2005(1)
51.8(3)
–
107 998.51(1)
–
0.542(4)
1.6711(6)
2(2)
0b
107 608.45(4)
1.1647(4)
56(2)
–
107 937.24(3)
–
0.67(3)
a
Deperturbed parameters.
b
Fixed parameter. No statistically significant value found.
procedure, principally due to its ability to reproduce the o(v = 1)-state
Λ-doubling observed experimentally. The fitted spectroscopic parameters for 14 N15 N are given in Table 2.7. In the case of the heterogeneous
interaction between b0 (v = 6) and o(v = 1, e), the interaction parameter, Ho1b0 6 = 0.67 cm−1 , is of the same order as the value obtained for
14 N . A crossing of the b0 (v = 6) and o(v = 1, e) levels between J = 24
2
and 25 is predicted by the fit, with a maximum shift of ∼ 13 cm−1 at
J = 24. The B values given in Table 2.7 are slightly lower in the mixed
isotopomer due to the greater reduced mass.
Oscillator strengths
The Photon Factory experimental photoabsorption cross sections were
analyzed using a least-squares fitting procedure in which each line was
40
2. Quantum-interference effects on a Rydber-valence complex of N2
represented by a Voigt profile and account was taken of the effects
of the finite experimental resolution. The line oscillator strength and
the Lorentzian width component, corresponding to the predissociation
linewidth, for each Voigt line were parameters of the fit, while the Gaussian width component was fixed at the room-temperature Doppler width
of 0.24 cm−1 FWHM. The instrumental function was also defined by a
Voigt profile, with Gaussian and Lorentzian width components of 0.60
and 0.20 cm−1 FWHM, respectively, determined by analyzing scans over
1 +
the almost pure Doppler lines from the c04 1 Σ+
u − X Σg (0, 0) band [60].
The Voigt-model cross section was convolved, in the transmission domain, with the instrumental function, and compared iteratively with
the experimental cross section. In the case of weak lines, it was not possible to independently determine the predissociation linewidths, which
were fixed at realistic values interpolated from other known widths (see
Sec. 2.3), but, in any case, the fitted oscillator strengths were not very
sensitive to these adopted linewidths. In the case of overlapping lines,
generally the line-strength ratios were fixed at the values expected from
Hönl-London- and Boltzmann-factor considerations and an average oscillator strength determined. The fitting procedure is illustrated in
Fig. 2.7. In this example, it was possible to determine independent oscillator strengths for the stronger, partially-overlapped P (11) and Q(16)
lines from the o − X(1, 0) band, but only a common predissociation
linewidth. For the very weak P (8) line from the b − X(9, 0) band, however, only the oscillator strength could be determined.
The fitted line oscillator strengths were converted into band oscillator strengths by dividing by appropriately normalized 1 Π − 1 Σ HönlLondon factors and fractional initial-state populations, the latter determined from T = 295 K Boltzmann factors based on the N2 ground-state
term values, and taking into account the 2:1 rotational intensity alternation caused by nuclear-spin effects. No significant systematic differences
were found between the P -, Q-, or R-branch band oscillator strengths,
for either the b − X(9, 0) or o − X(1, 0) transitions, over the range of
rotation studied, J ≤ 20. The overall results, summarized in Table 2.8,
and illustrated in Fig. 2.8 (circles) represent weighted means over each
available branch.
Inspection of Fig. 2.8 reveals an interesting effect: transitions to the
higher-energy levels [b(v = 9) for J ≤ 4, o(v = 1) for J ≥ 5; solid
41
2. Quantum-interference effects on a Rydber-valence complex of N2
3.0
Cross Section (10
−16
2
cm )
o -X (1,0)
2.0
Q (16)
P (11)
1.0
b -X (9,0)
P (8)
0.0
107562
107564
107566
107568
−1
107570
Transition Energy (cm )
Figure 2.7: Experimental room-temperature photoabsorption cross section
for a region of the overlapping b − X(9, 0) and o − X(1, 0) bands of 14 N2 (open
circles), together with the fitted cross section (solid curve). The P (8) line from
the b − X(9, 0) band, probing J = 7, is anomalously weak (see text).
circles] peak in strength near J = 6, while those to the lower-energy
levels [o(v = 1) for J ≤ 4, b(v = 9) for J ≥ 5; open circles] decrease
rapidly in intensity in the same region, with transitions to J = 6 too
weak to be observed. This is a classic example of a two-level quantummechanical interference effect, in the case where transitions to both levels, of like symmetry, carry an oscillator strength, as discussed in detail
by Lefebvre-Brion and Field [11]. The perturbed vibronic oscillator
strengths for transitions from a common level 0 to the upper (+) and
lower (−) levels of the interacting pair, respectively, are given by [11]:
√
f+0 = c2 f10 + (1 − c2 )f20 ± 2c (1 − c2 )f10 f20 ,
√
f−0 = (1 − c2 )f10 + c2 f20 ∓ 2c (1 − c2 )f10 f20 ,
(2.7)
where f10 and f20 are the vibronic oscillator strengths for transitions to
the unperturbed levels, 1 and 2, and c > 0 is the mixing coefficient corresponding to the presence of the unperturbed level 1 in the perturbed
upper-level wave function. The sense of the interference effect, corresponding to the signs of the right-hand terms in Eq. (2.7), depends on
42
2. Quantum-interference effects on a Rydber-valence complex of N2
Table 2.8: Experimental (perturbed) band oscillator strengths f for the b −
X(9, 0) and o − X(1, 0) transitions of 14 N2 . Values flagged with an asterisk
(∗) are derived only from blended lines. 1σ statistical uncertainties are shown
in parentheses, in units of the last significant figure. Additional systematic
uncertainties of ≈ 10% are applicable.
J
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
fb−X(9,0)
0.0154(8)*
0.0147(3)*
0.0160(3)
0.0164(2)*
0.0005(2)*
0.0007(1)
0.0014(1)
0.0022(2)
0.0031(1)
0.0030(1)*
0.0036(2)
0.0037(1)
0.0037(1)
0.0039(2)
0.0043(1)
0.0039(3)
0.0042(2)
0.0041(3)
0.0042(5)
fo−X(1,0)
0.0029(6)*
0.0022(2)*
0.0017(3)*
0.0006(2)*
0.0181(3)
0.0170(2)
0.0178(3)
0.0172(2)
0.0161(3)
0.0149(2)
0.0145(2)
0.0139(3)
0.0138(2)
0.0143(2)
0.0132(2)
0.0135(2)*
0.0120(4)
0.0121(3)
0.0122(5)
0.0137(5)*
the sign of µ10 µ20 H12 , where the µi0 are vibronic transition moments
(fi0 ∝ µ2i0 ) and H12 is the coupling matrix element [11]. The present
case, where the higher-energy transitions are enhanced in strength by
the interference effect, corresponds to µbX(9,0) µoX(1,0) Hb9o1 > 0.
Using the f -level mixing coefficients determined as part of the
energy-level deperturbation procedure described in Sec. 2.3, together
with the experimental (perturbed) oscillator strengths from Table 2.8,
we have successfully deperturbed the oscillator strengths through the
application of Eq. (2.7). In contrast to the perturbed case, the deperturbed oscillator strengths display only slight J dependences, yielding
the following fits: fbX(9,0) = 0.0064(2) − 4.2(9) × 10−6 J(J + 1) (dashed
43
2. Quantum-interference effects on a Rydber-valence complex of N2
Oscillator Strength
0.020
0.015
0.010
0.005
0.000
0
100
200
J (J +1)
300
400
Figure 2.8: Rotational [J(= J 0 )] dependence of band oscillator strengths in
the mixed b − X(9, 0) and o − X(1, 0) transitions of 14 N2 , demonstrating a
strong quantum-interference effect near J = 6, together with the results of
a deperturbation analysis (see text). Solid circles: experimental (perturbed)
oscillator strengths for transitions to the higher-energy levels for a given J.
Open circles: experimental oscillator strengths for transitions to the lowerenergy levels. Solid line: deperturbed oscillator strength for the o − X(1, 0)
transition. Dashed line: deperturbed oscillator strength for the b − X(9, 0)
transition. Solid curve: calculated perturbed oscillator strength for the higher
levels. Dashed curve: calculated perturbed oscillator strength for the lower
levels. Open squares: sum of experimental oscillator strengths for a given
J. Long-dashed line: sum of calculated perturbed (or deperturbed) oscillator
strengths.
line in Fig. 2.8), foX(1,0) = 0.0116(3) + 1.1(11) × 10−6 J(J + 1) (solid line
in Fig. 2.8). Back-generating the perturbed oscillator strengths from
these fits yields the results shown in Fig. 2.8 for the upper and lower
levels (solid curve, dashed curve, respectively), which are seen to be in
excellent agreement with the measurements. Furthermore, the summed
deperturbed (or perturbed) fitted oscillator strengths (long-dashed line
in Fig. 2.8) are in good agreement with the summed experimental values
(open squares), neither exhibiting any strong J dependence, as would
be expected in the case of the simple two-level interaction described
by Eq. (2.7). While the b(v = 9) ∼ o(v = 1) crossing occurs between
J = 4 − 5, the intensity minimum for transitions to the lower of the
44
2. Quantum-interference effects on a Rydber-valence complex of N2
perturbed levels appears to occur just below J = 6. This difference is a
consequence of the deperturbed foX(1,0) significantly exceeding fbX(9,0) .
From Eq. (2.7), only in the specific case f10 = f20 will an intensity zero
√
occur at the level crossing (c = 1/ 2).
These results confirm that the b(v = 9) ∼ o(v = 1) level crossing in
14 N produces a simple two-level quantum-mechanical interference effect
2
in the corresponding perturbed oscillator strengths for transitions from
the ground state, yielding a virtually complete destructive interference
for transitions to the lower level with J = 6 [principally of b(v = 9)
character]. As will be seen in Sec. 2.3, the situation for the corresponding
predissociation linewidths is somewhat more complicated. Finally, we
note that an ostensibly similar deep minimum in oscillator strength has
been reported in 14 N2 for transitions to b(v = 8) near J = 12 [47]. This
latter case is distinguished, however, by no level crossing being involved,
the destructive interference effect being a multilevel phenomenon.
Predissociation linewidths
Lorentzian linewidth components for individual rotational levels of b(v =
9) and o(v = 1) in 14 N2 , dominated by the contribution of predissociation over radiation, were determined from both the PDA laser-based
ionization spectra and the Photon Factory photoabsorption spectra. Of
course, the PDA-based system was to be preferred for the measurement
of linewidths because of its superior resolution and Doppler-reduced
character, but, in practice, these advantages applied only at low J in
the cold spectra, since the room-temperature background gas contribution dominated the jet contribution for higher J. On the other hand,
the high signal-to-noise ratio of the photoabsorption spectra made them
suitable for the determination of Lorentzian linewidth components with
Γ & 0.15 cm−1 FWHM, albeit with significant uncertainty, even with
an experimental resolution of only ∼ 0.7 cm−1 FWHM. Predissociation
linewidths, determined from the PDA-based spectra by deconvolving the
instrument width from the observed widths as explained in Ref. [34],
and from the photoabsorption spectra, simultaneously with the oscillator strengths as described in Sec. 2.3, are summarised in Table 2.9
and Fig. 2.9. Since no significant systematic e/f parity dependence
was found for either the b(v = 9) or o(v = 1) level widths, the val-
45
2. Quantum-interference effects on a Rydber-valence complex of N2
ues presented represent weighted means over determinations from each
available branch.
The PDA-based system was used only to determine level widths for
b(v = 9, J = 1 − 2) and o(v = 1, J = 1 − 3), i.e., restricted to J-levels
below the crossing. The average predissociation linewidths in this region, Γb(v=9) ≈ 0.24 cm−1 FWHM and Γo(v=1) ≈ 0.05 cm−1 FWHM,
differ substantially (see also Fig. 2.3). They are equivalent to lifetimes
τ = 1/2πΓ of 23(5) ps and 105(30) ps, for b(v = 9) and o(v = 1),
respectively, which agree with previous time-domain pump-probe lifetime measurements [38]. In Ref. [38], the rotational structure of the
b(v = 9) ∼ o(v = 1) complex was not resolved and the lifetimes observed, determined mainly by predissociation, varied from < 50 ps to
110 ps. From the present study, it can be deduced unambiguously that
the < 50-ps component arose from b(v = 9) and the 110-ps component
from o(v = 1). The predissociation lifetimes of these two levels in 15 N2
have been determined with the same frequency-mixing PDA-based XUV
source in Ref. [58]. For b(v = 9) in 15 N2 , an f -parity lifetime of 46(7) ps
was found, a factor of two higher than in 14 N2 . The lifetime of o(v = 1)
is also isotope dependent: a lifetime of 27(6) ps was obtained for this
level in 15 N2 [58], a factor of four lower than in 14 N2 . All data pertaining
to 14 N15 N were obtained here with the PDL-based XUV source and do
not, therefore, yield reliable information on the excited-state lifetimes.
While it is not possible unambiguously to detect any systematic J
dependence in the PDA linewidths, when taken together with the Photon Factory results, it is evident from Fig. 2.9, despite considerable
uncertainty in the data, that the width of the lower-energy levels (open
circles) increases substantially as J increases, while the width of the
higher-energy levels (solid circles) decreases overall, exhibiting a maximum near J = 6. Because of line overlap and the weakness of the
corresponding transitions (see Fig. 2.8), we have been unable to determine widths for the lower-energy levels with J = 4 − 8.
In the case of a simple, two-level interaction where both (unperturbed) levels predissociate via the same route, i.e., coherently, a
quantum-interference effect in the widths would be expected, analogous
to that observed for the corresponding oscillator strengths. In this case,
the width interference is described by expressions similar to Eq. (2.7),
with the widths Γ replacing the oscillator strengths f [11]. The width
46
2. Quantum-interference effects on a Rydber-valence complex of N2
0.4
−1
Lorentz Linewidth (cm FWHM)
0.5
0.3
0.2
0.1
0.0
0
100
200
J (J +1)
300
400
Figure 2.9: Rotational dependence of predissociation linewidths for the mixed
b(v = 9) and o(v = 1) states of 14 N2 , together with the results of a deperturbation analysis (see text). Solid circles: experimental (perturbed) widths for the
higher-energy levels. Open circles: experimental widths for the lower-energy
levels. Solid line: deperturbed o(v = 1) widths. Dashed line: deperturbed
b(v = 9) widths. Solid curve: calculated perturbed widths for the higher levels. Dashed curve: calculated perturbed widths for the lower levels. Open
squares: sum of experimental widths for a given J. Long-dashed line: sum of
calculated perturbed (or deperturbed) widths.
maximum observed near J = 6 for the upper-energy levels in Fig. 2.9
supports the notion of a width quantum-interference effect associated
with the b(v = 9) ∼ o(v = 1) level crossing. However, despite an inability to fully monitor the lower-energy widths in this region, there is no
rapid decrease in the J = 1−3 level widths indicated by the PDA results,
contrary to the case of the oscillator strengths in Fig. 2.8. Thus, while
there may be a minimum in the lower-energy widths near the crossing
region, that minimum is unlikely to be as deep as the effectively zero
value exhibited by the corresponding oscillator strengths, and required
for a coherent two-level interaction.
An initial attempt to deperturb the level widths in Fig. 2.9, using the
width analogue of Eq. (2.7), together with the same mixing coefficients
used for the oscillator-strength deperturbation, failed because such a
47
2. Quantum-interference effects on a Rydber-valence complex of N2
Table 2.9: Experimental (perturbed) predissociation linewidths Γ (in cm−1
FWHM) for the b1 Πu (v = 9, J) and o1 Πu (v = 1, J) levels of 14 N2 . Values
flagged with an asterisk (∗) are derived only from blended lines. 1σ statistical
uncertainties are shown in parentheses, in units of the last significant figure.
Additional systematic uncertainties of ≈ 25% are applicable to the non-PDA
measurements.
J
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
a
Γb(v=9)
0.25(4)a
0.22(2)a
0.29(4)
0.23(2)*
0.13(5)
0.17(2)
0.15(4)*
0.17(3)
0.15(2)
0.19(3)
0.22(3)
0.26(2)
0.23(5)
0.27(4)
0.16(5)
Γo(v=1)
0.05(1)a
0.06(1)a
0.04(1)a
0.26(2)
0.33(2)
0.29(2)
0.30(2)
0.21(2)
0.21(2)
0.22(2)
0.15(2)
0.18(2)
0.18(2)
0.14(2)
0.22(2)*
0.14(3)
0.15(2)
0.12(4)
PDA measurements
totally destructive quantum interference for the lower-energy levels was
incompatible with the experimental widths. Thus, the implied nonzero
width minimum for the lower-energy levels suggests that an additional,
incoherent, and thus additive, width component is applicable to these
levels. If a J-independent value of ∼ 0.03 cm−1 FWHM is assumed
for this component, together with coherent two-level interference for the
remainder, then the realistic deperturbation shown in Fig. 2.9 results,
yielding the following fits: Γb(v=9) = 0.17(2)+2.3(8)×10−4 J(J +1) cm−1
FWHM (dashed line in Fig. 2.9), including the incoherent component,
and Γo(v=1) = 0.15(1) − 1.0(4) × 10−4 J(J + 1) cm−1 FWHM (solid
48
2. Quantum-interference effects on a Rydber-valence complex of N2
line in Fig. 2.9). Back-generating the perturbed widths from these fits
yields the results shown in Fig. 2.9 for the upper and lower levels (solid
curve, dashed curve, respectively), which, as for the oscillator strengths,
are in excellent agreement with the experimental measurements. The
summed deperturbed (or perturbed) fitted widths (long-dashed line in
Fig. 2.9) are in good agreement with the summed experimental values
(open squares), neither exhibiting any strong J dependence. It is of interest that the two-level-deperturbed low-J predissociation lifetimes implied by the above linewidth analysis, 31(4) ps and 35(3) ps, for b(v = 9)
and o(v = 1), respectively, are in much better agreement with the 15 N2
experimental lifetimes, 46(7) ps and 27(6) ps [58], than are the perturbed lifetimes. This indicates that the bulk of the large experimental
isotope effect is caused simply by the interference effect associated with
the two-level crossing in 14 N2 .
According to Lewis et al. [32], the lowest 1 Πu states of N2 are predissociated, ultimately, by the C 0 3 Πu state, which correlates with the
4 S + 2 D dissociation limit at ∼ 97 940 cm−1 . On this basis, the fact
that the higher-energy levels are enhanced in width by the interference
effect indicates that the matrix-element product Hb9C 0 Ho1C 0 Hb9o1 > 0,
where the bound-free vibronic elements relate to the unperturbed widths
2 ). The question remains as to the source of the incoherent
(Γi ∝ HiC
0
contribution to the b(v = 9) widths. We propose here that it is the
interaction with a repulsive state, namely the second valence state of
3 Σ+ symmetry, labelled 3 3 Σ+ by Minaev et al. [61], which provides the
u
u
additional incoherent predissociation of the b 1 Πu state. The potential4
2
energy curve for the 3 3 Σ+
u state, which also correlates with the S + D
limit, crosses that of the b 1 Πu state on its outer limb, near R = 1.75 Å,
at an energy of ∼ 107 300 cm−1 [62]. Thus, it is energetically possi1
ble for the 3 3 Σ+
u state to predissociate b Πu (v = 9), which lies near
−1
107 640 cm
(see Table 2.7). Furthermore, the spin-orbit coupling
(∆Ω = 0) between the b 1 Π1u and 3 3 Σ+
1u states has been estimated
−1
ab initio to be on the order of 7 cm [63], making this the likely incoherent predissociation mechanism supplying the additional ∼ 0.03 cm−1
FWHM width to b(v = 9).
49
2. Quantum-interference effects on a Rydber-valence complex of N2
2.4
Summary and conclusions
Two distinct high-resolution experimental techniques, 1 XUV + 10 UV
laser-based ionization spectroscopy and synchrotron-based XUV photoabsorption spectroscopy, have been used to study the o 1 Πu (v = 1) ∼
b 1 Πu (v = 9) Rydberg-valence complex of 14 N2 , providing new and
detailed information on the perturbed rotational structures, oscillator
strengths, and predissociation linewidths. Ionization spectra probing the
14 N , which crosses o 1 Π (v = 1) between J = 24
b0 1 Σ +
2
u
u (v = 6) state of
1
1
and J = 25, and the o Πu (v = 1), b Πu (v = 9), and b0 1 Σ+
u (v = 6) states
of 14 N15 N, have also been recorded. In the case of 14 N2 , rotational and
deperturbation analyses correct previous misassignments for the low-J
levels of o(v = 1) and b(v = 9). In addition, a two-level quantummechanical interference effect has been found between the o − X(1, 0)
and b − X(9, 0) transition amplitudes which is totally destructive for the
lower-energy levels just above the level crossing, making it impossible
to observe transitions to b(v = 9, J = 6). A similar interference effect
is found to affect the o(v = 1) and b(v = 9) predissociation linewidths,
but, in this case, a small non-interfering component of the b(v = 9)
linewidth is indicated, attributed to an additional spin-orbit predissociation by the repulsive 3 3 Σ+
u state. The experimental linewidths, together
with the corresponding interference effect, will provide a challenging test
for coupled-channel models of the predissociation dynamics for the 1 Πu
states of N2 .
Acknowledgment
The authors gratefully acknowledge the support provided for this research by the following bodies: the Molecular Atmospheric Physics
(MAP) Program of the Netherlands Foundation for Fundamental Research on Matter (FOM); the Australian Research Council’s Discovery Program (project number DP0558962); and NASA (grant
NNG05GA03G). The assistance of the staff of the Photon Factory is
also acknowledged.
50
Chapter 3
Interactions of the 3pπu c 1Πu(v = 2)
Rydberg-complex in N2
01 +
1 +
The 3pπu c1 Πu −X 1 Σ+
g (2, 0) Rydberg and b Σu −X Σg (7, 0)
14
14 15
15
valence transitions of N2 , N N, and N2 are studied using
laser-based 1 XUV + 10 UV two-photon-ionization spectroscopy,
supplemented by synchrotron-based photoabsorption measurements in the case of 14 N2 . For each isotopomer, effective rotational
interactions between the c(v = 2) and b0 (v = 7) levels are found
to cause strong Λ-doubling in c(v = 2), and dramatic P/R-branch
intensity anomalies in the b0 − X(7, 0) band due to the effects of
quantum interference. Local perturbations in energy and predissociation line width for the c(v = 2) Rydberg level are observed and
attributed to a spin-orbit interaction with the crossing, short-lived
C 3 Πu (v = 17) valence level.
3. Interactions of the 3pπu c 1 Πu (v = 2) Rydberg-complex in N2
3.1
Introduction
The level structure of the nitrogen molecule in the region of its dipoleallowed absorption spectrum (i.e., above 100 000 cm−1 ) is one of severe
complexity. First, the singlet states that can be accessed directly in
the extreme ultraviolet (XUV) region via allowed transitions behave
seemingly erratically. Numerous investigations have been performed to
unravel these singlet structures. Through the semiempirical work of Stahel et al.,[23] the lowest-energy dipole-allowed spectrum of N2 has been
explained by considering the homogeneous electrostatic interactions be1
tween the Rydberg and valence states of 1 Σ+
u and Πu symmetry. In particular, the corresponding calculated vibronic band strengths[23] showed
strong quantum-interference effects. Later, a quantitatively-improved
model of the N2 spectrum, based on ab initio calculations, was put forward by Spelsberg and Meyer.[24] Edwards et al.[25] extended the description by incorporating heterogeneous, rotationally-dependent interactions in the coupling model. Nevertheless, there remain many details
in the structure of the singlet states to be discovered and explained. The
theoretical and experimental study of rotationally-dependent perturbations in the Rydberg-valence singlet manifold of N2 and its isotopomers
(15 N2 and 14 N15 N) has been a topic of recent interest. Strong local
perturbations can give rise, not only to level shifts, but also changes in
intensities. By analogy with the vibronic case, such perturbations can
cause rotational quantum-interference effects, leading, in some cases, to
complete damping of intensity.[39]
Second, the singlet states show predissociation behavior that is also
erratic. Four decades ago, Carroll and Collins[21] noted that, of the
vibrational levels of the b 1 Πu valence state, only v = 1, 5 and 6 could
be observed in emission. The predissociation rates pertaining to the
b 1 Πu (v) levels have been determined experimentally from line-width
studies employing either synchrotron-radiation[42] or narrow-bandwidth
XUV laser-radiation[29, 64] sources. The XUV laser technique is especially suitable for the determination of excited-state lifetimes with rotational resolution. In addition, complementary pump-probe experiments
employing picosecond lasers have been carried out to determine lifetimes
experimentally.[65] Numerous additional investigations have been performed in order to produce a quantitative experimental database for the
52
3. Interactions of the 3pπu c 1 Πu (v = 2) Rydberg-complex in N2
predissociation rates of the singlet states of N2 , including translationalspectroscopic studies,[66, 31] and investigations focusing specifically on
the 15 N2 and 14 N15 N isotopomers.[36] Knowledge of the N2 predissociation behaviour and mechanism is a prerequisite for a detailed understanding of radiative-transfer processes, as well as stratospheric chemistry, in nitrogen-rich planetary atmospheres such as those of the Earth
and Titan.
Over the years, a large effort has been invested towards an understanding of the origin of the predissociation in the lowest-lying electricdipole-accessible singlet states of N2 .[26] Since the singlet states couple
via homogeneous spin-orbit interaction to the C 3 Πu state, which in turn
is coupled to the C 0 3 Πu state above its dissociation limit, a detailed understanding of the predissociation of the singlet states requires consideration of the interactions within the triplet manifold. A recent study
by Lewis et al.,[32] based on a coupled-channel Schrödinger equation
(CSE) model for the interacting 1 Πu (b, c, o) and 3 Πu (C, C 0 ) states, has
provided a quantitative explanation for the predissociation mechanism
of the 1 Πu states in the energy region below 105 000 cm−1 .
This model achieves spectroscopic accuracy and builds upon the information gathered over the years concerning low-lying vibrational levels of the C 3 Πu and C 0 3 Πu states, starting with the benchmark study
by Carroll and Mulliken.[67] This triplet data is reviewed in Ref. [32].
Additional information beyond C 3 Πu (v = 5) and C 0 3 Πu (v = 2) has
been obtained indirectly by examining the predissociative effect of these
states on the singlet manifold, guided by ab initio potential curves and
an extrapolation from the known lower levels. The CSE model is capable of explaining the predissociation rates for b 1 Πu (v ≤ 6), with rotational resolution, for the three isotopomers 15 N2 , 14 N15 N and 14 N2 .[47]
In particular, the b 1 Πu (v = 1) level of 14 N2 provides a good test for the
model. For low J, the predissociation rate is very small and decay is
mainly radiative, with an exceptionally long lifetime.[68] The predicted
predissociation rate increases steeply with J, in good agreement with
experiment.[33]
In order to extend our understanding of the predissociation phenomenon in N2 into the excitation region above 105 000 cm−1 , significant
CSE-model development is necessary, facilitated by new experimental
measurements. In particular, this requires knowledge of the locations of
53
3. Interactions of the 3pπu c 1 Πu (v = 2) Rydberg-complex in N2
higher vibrational levels of the C 3 Πu valence state, together with a consideration of the other triplet Rydberg states, F 3 Πu ,[35] and G 3 Πu ,[69]
that play the role of inducing further predissociation through their interactions with the C 3 Πu and C 0 3 Πu states and direct spin-orbit coupling
to the 1 Πu states.[70]
Electric-dipole transitions from the X 1 Σ+
g ground state of N2 to
the triplet manifold are spin-forbidden, thus limiting the information
that can be obtained directly on these states using normal spectroscopic
techniques. However, spin-orbit interactions cause coupling between the
singlet and triplet manifolds, leading to local perturbations in the singlet
states that can be observed using narrow-bandwidth spectroscopic techniques, providing useful information on the triplet states indirectly. For
example, the C 3 Πu (v) valence-state levels, which have low rotational
constants, are most likely to cross, and interact with, levels of the Rydberg states of N2 , which have significantly higher rotational constants.
Indeed, spectroscopic parameters for the C 3 Πu (v = 14) level of 15 N2
have been reported recently, determined by analyzing its perturbation
of c 1 Πu (v = 1) near 106 500 cm−1 , including the observation of a few
extra lines due to transitions to the perturbing level.[71, 72]
The 3pπu c 1 Πu state is a member of the 3p Rydberg complex of
3 +
N2 ,[11] which also includes the 3pπu G 3 Πu , 3pσu c0 1 Σ+
u , and 3pσu D Σu
states. The c and G states exhibit a (small) spin-orbit interaction,
while the G and D states, and the c and c0 states, exhibit strong rotational interactions which perturb the rotational constants and contribute to Λ-doubling in the Πu 3p-complex components, leading to a
good deal of spectral complexity. In the present study, we investigate
the spectroscopy and predissociation characteristics of the c 1 Πu (v = 2)
3p-complex member in the energetic region 108 500 − 109 500 cm−1 ,
for all three N2 isotopomers, using principally 1 XUV + 10 UV twophoton-ionization spectroscopy. This level is perturbed by singlet valence states occurring in this energy region, most notably b 1 Πu (v = 11)
and b0 1 Σu (v = 7), the latter of which contributes to observable Λdoubling effects and quantum-interference phenomena. Local perturbations in the c 1 Πu (v = 2) level, caused by these singlet levels, as well
as those due to C 3 Πu (v = 17), are studied, together with associated
predissociative effects.
54
3. Interactions of the 3pπu c 1 Πu (v = 2) Rydberg-complex in N2
3.2
Experimental Methods
XUV laser spectroscopy
Laser-based 1 XUV + 10 UV two-photon-ionization spectroscopy is the
primary technique employed here to study the excitation spectrum of
N2 , in the range λ = 91.3−92.3 nm. Details of the experimental method,
including a description of the lasers, vacuum setup, molecular-beam configuration, time-of-flight (TOF) detection scheme, and calibration procedures, have been presented previously.[55] Briefly, narrow bandwidth
XUV radiation is produced by frequency tripling the UV light from
a frequency-doubled pulsed dye laser (PDL) pumped by an injectionseeded Nd:YAG laser. Frequency tripling takes place in a free Xe-jet
expansion, where the UV power (25 mJ in 5 ns) is focused. The frequency in the XUV range is calibrated against a Doppler-broadened I2
linear-absorption spectrum,[56] recorded simultaneously using part of
the visible output of the dye laser. An absolute accuracy of ∼ 0.1 cm−1
for fully-resolved line positions is estimated.
A skimmed, pulsed N2 beam is perpendicularly intersected by
temporally- and spatially-overlapping XUV and UV laser beams. Nitrogen molecules are excited resonantly by the XUV photons and subsequently ionized by the intense UV light. N+
2 ions are detected using
a TOF mass selector. In addition to normal N2 , isotopically-enriched
samples of 14 N15 N and 15 N2 are also studied. The TOF technique employed allows for mass identification of the N+
2 ions, which can thus be
distinguished from ions arising from the background oil in the vacuum
system.
The experimental parameters were established to provide the highest
N2 rotational temperature in the interaction region. This was achieved
by increasing the N2 density (∼ 2×10−5 mbar) and therefore increasing
the population of the higher rotational levels through collisions between
the molecular beam and the higher-pressure background N2 . To achieve
the background pressure required, the nozzle-skimmer distance was kept
to a minimum. We thereby recorded spectra associated with two distinct
rotational temperatures, one belonging to the supersonic expansion of
the molecular beam (∼ 10 K) and one belonging to the background gas
at ambient temperature (∼ 300 K). Moreover, by changing the relative
delay between the N2 pulsed-valve trigger and the laser pulse, the rota55
3. Interactions of the 3pπu c 1 Πu (v = 2) Rydberg-complex in N2
tional temperature in the molecular beam could be selected to measure
independent spectra of “cold” (10 − 50 K) and “warm” (200 − 300 K)
samples. These options helped greatly in the assignment of the spectral
lines.
The instrumental contribution to the measured line widths depends
on the ambient gas pressure, the settings for measuring cold or warm
spectra, the wavelength range, the dye used, and the alignment of
the PDL.[55] In the case of the high-temperature spectra, an additional Doppler broadening amounting to ∼ 0.25 cm−1 full-width at
half-maximum (FWHM) must be accounted for in the line-width analyses. The instrumental width is determined from observed transitions
1 +
in the c0 1 Σ+
u − X Σg (2,0) band, because of its close proximity to the
c 1 Πu − X 1 Σ+
g (2,0) transitions that are measured under virtually the
same experimental conditions. The c0 (v = 2) level is known to be
slightly predissociated,[65] but not to the extent that it will induce
significant line broadening in the present experiments. Hence, the instrumental width can be taken to equal the observed width for these
transitions. The values obtained for the instrumental contribution to
the line width are 0.34±0.04 cm−1 FWHM for 15 N2 , 0.42±0.04 cm−1
FWHM for 14 N15 N and 0.40±0.02 cm−1 FWHM for 14 N2 . Note that the
instrumental width also includes the Doppler contribution. The differences between these values are not so much related to the isotopomers,
but rather to the specific experimental conditions employed.
Using this narrow-bandwidth XUV laser system, line positions, line
widths, and relative intensity variations can be measured with rotational
resolution. Information on the predissociation process can be obtained
through several complementary means. The lifetime τ (s) of the excited
level, shortened due to predissociation, can be expressed as:
τ=
1
,
2πcΓ
(3.1)
where Γ is the natural (Lorentzian) line width (in cm−1 FWHM). Under our experimental conditions, Doppler broadening is also present,
contributing to the effective instrumental profile which can be taken to
be Gaussian.[37] Thus, the measured line shape has a Voigt profile, with
a width of ∆νobs FWHM. The Lorentzian line-width component, principally due to predissociation, can be obtained by deconvolution of the
experimental Voigt profile using the appropriate Gaussian instrumen56
3. Interactions of the 3pπu c 1 Πu (v = 2) Rydberg-complex in N2
tal profile, of width ∆νinst FWHM. With some approximations,[73] the
following relation is obtained:
Γ = ∆νobs −
(∆νinst )2
.
∆νobs
(3.2)
Hence, using Eqs. (3.1) and (3.2), the excited-state lifetime τ can be derived straightforwardly from the line-width measurements. In addition,
the shortening of the lifetime due to predissociation will not only cause
line broadening, but also a decrease in signal intensity, since the detected
ions result from the ionization of a decaying excited level. The resulting
ionization signal is proportional to the lifetime of the excited intermediate state that is ionized.[37] Thus, the intensity depletion also provides
information on the predissociation behavior of the excited state, in fact
more sensitively, in many cases, than from direct line-width measurements.
XUV synchrotron spectroscopy
Relative ionization signals in the spectra obtained using the two-photonionization spectroscopic technique described in Sec. 3.2 are influenced,
not only by the aforementioned competition between predissociation and
ionization, but also by the inherent oscillator strengths of the transitions. Thus, successful interpretation of the ionization spectra will,
in some cases, require a detailed knowledge of the oscillator strengths,
the measurement of which requires a different experimental technique.
Therefore, in the case of the b0 − X(7, 0) transition of 14 N2 , the present
ionization spectra are interpreted using oscillator strengths derived from
absolute XUV photoabsorption spectra obtained at the 2.5 GeV storage
ring of the Photon Factory, a synchrotron radiation facility at the High
Energy Accelerator Research Organization in Tsukuba, Japan. These
measurements form part of the extensive experimental campaign of
Stark et al.,[42, 74] who have described the apparatus in detail. Briefly,
a 6.65 m spectrometer with a 1200 grooves per mm grating (blazed
at 550 nm and used in the sixth order) provides an instrumental resolving power of ∼ 150, 000, equivalent to a resolution of ∼ 0.7 cm−1
FWHM. The spectrometer tank, filled with N2 of normal isotopic composition in a flowing configuration, serves as an absorption cell with a
57
3. Interactions of the 3pπu c 1 Πu (v = 2) Rydberg-complex in N2
path length of ∼ 12.5 m and a temperature of 295 K. Rotational-line oscillator strengths and Lorentzian predissociation line-width components
are obtained from the experimental photoabsorption cross sections using a nonlinear least-squares fitting procedure which takes account of
the finite instrumental resolving power. The line oscillator strengths are
converted into equivalent absolute band oscillator strengths using appropriate Hönl-London and Boltzmann factors. Predissociation linewidths
determined from these synchrotron-based spectra for the c(v = 2) level
of 14 N2 are also used in Sec. 3.4 to supplement the present laser-based
results which, for this isotopomer, do not extend to high-enough J values to access the principal region of interest, i.e., the crossing region
with C(v = 17).
3.3
CSE calculations
Due to the many interactions associated with levels of the 3p Rydberg
complex, interpretation of the corresponding spectra is difficult without
some recourse to the theoretical aspects. Therefore, although the present
work is, in principle, an experimental study, some CSE calculations have
been performed in order to supplement the analysis.
The CSE model employed in this work, an extension of the (1 Πu +
3 Π ) Rydberg-valence model of Lewis et al.[32] and Haverd et al.,[47]
u
which includes 1 Σ+
u Rydberg and valence states and their mutual elec1
trostatic interactions, together with 1 Σ+
u ∼ Πu rotational interactions,
is described and applied in Liu et al.[75] and Liang et al.[45] Briefly,
the coupled-channel Schrödinger equation is solved for the radial wavefunctions of a series of interacting diabatic electronic molecular states
defined by potential-energy curves and off-diagonal coupling parameters,
and the corresponding photodissociation cross section from the ground
state computed, yielding line positions and oscillator strengths. The formalism of the technique is described, e.g., by van Dishoeck et al.[76] and
Lewis et al.[77] A major advantage of this method is that, after optimization to the well-known singlet energy levels of the 14 N2 isotopomer,
reliable interpolations and extrapolations may be made to previously
unknown 15 N2 and 14 N15 N levels. Additionally, because unbound states
are included among the coupled channels, predissociation line widths can
be modeled, and so observed line-width perturbations of singlet states
58
3. Interactions of the 3pπu c 1 Πu (v = 2) Rydberg-complex in N2
Figure 3.1: 1 XUV + 10 UV ionization spectrum and line assignments for
15
the c 1 Πu − X 1 Σ+
N2 . The diamond-shaped symbols act as a
g (2,0) band of
guide to indicate the perturbed intensity pattern in the Q branch, where the
3:1 ratio between odd- and even-J 00 transition lines, arising from nuclear-spin
statistics, has been taken into account.
may enable the indirect determination of potential-energy curves for the
triplet manifold as well as singlet-triplet coupling parameters.
The role of the present CSE modeling is to act as an intelligent
extrapolator to support the positive identification of C(v = 17) as the
perturber of c(v = 2). A comparison of experimental versus CSE-model
band origins and rotational constants for the C 3 Πu (v ≤ 9) levels is given
in Lewis et al.,[32] showing excellent agreement. Further confidence in
the rather large extrapolation to v = 17 is found in the physically-based
ab initio C 3 Πu potential-energy curve used within the model, as well as
indirect verification of intermediate C(v) levels through the perturbative
effects they produce on allowed singlet transitions.[72]
3.4
Results and discussion
The c 1 Πu (v = 2) and b0 1 Σ+
u (v = 7) levels
01 +
Rotationally-resolved spectra of the c 1 Πu − X 1 Σ+
g (2,0) and b Σu −
15 N , 14 N15 N and 14 N were recorded, for a range
X 1 Σ+
2
2
g (7,0) bands in
59
3. Interactions of the 3pπu c 1 Πu (v = 2) Rydberg-complex in N2
Table 3.1: Observed transition energies (in cm−1 ) for the c 1 Πu − X 1 Σ+
g (2,0)
band in 15 N2 . Wave numbers derived from blended lines are flagged with an
asterisk (*). Transitions that have not been included in the singlet-singlet
deperturbation analysis are marked with a cross-hatch (#).
J 00
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
60
R(J 00 )
108 558.19
108 561.52∗
108 564.14
108 566.65
108 569.19∗#
108 570.61#
108 573.58∗#
108 574.19∗#
108 574.19∗#
108 573.58∗#
108 571.64#
108 569.18∗
108 565.94
108 561.78
108 556.52
108 549.93∗
108 541.75
108 531.61#
108 519.35∗#
108 503.29∗#
108 483.94#
Q(J 00 )
P (J 00 )
108 554.46
108 553.81
108 552.83
108 551.62
108 549.99∗
108 548.10#
108 545.88#
108 547.07
108 542.77
108 538.09
108 533.09∗
108 527.97#
108 522.37#
108 540.72#
108 537.13#
108 533.09∗#
108 530.30#
108 524.79#
108 519.35∗#
108 513.33
108 506.54
108 498.81
108 489.72#
108 478.67#
108 465.30#
108 448.68#
108 428.08#
108 403.45#
108 374.59#
108 336.58#
108 510.34#
108 503.29∗#
108 496.29#
108 488.13#
108 481.02#
108 471.40#
108 461.49
108 450.91
108 439.30
108 426.73
108 412.76
108 359.65#
3. Interactions of the 3pπu c 1 Πu (v = 2) Rydberg-complex in N2
1 +
Table 3.2: Observed transition energies (in cm−1 ) for the b0 1 Σ+
u −X Σg (7,0)
15
band in N2 . For explanation of the annotations, see caption to Table 3.1.
J 00
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
R(J 00 )
108 804.36∗
108 805.48∗
108 805.48∗
108 804.36∗
108 802.03
108 798.42
108 793.90
108 788.14
108 781.29
108 773.37
108 764.46
108 754.48
108 743.34
108 731.13
108 717.70
108 702.66
P (J 00 )
108 792.74
108 786.78
108 779.44
108 770.84
108 761.09
108 750.07
108 661.67
108 623.54
108 602.64
108 580.25
of rotational temperatures. The corresponding transition energies and
line assignments deduced from the spectra are presented in Tables 3.1
to 3.6.
15
A high-temperature spectrum of the c 1 Πu −X 1 Σ+
g (2,0) band in N2
is shown in Fig. 3.1, where, with the aid of the diamond-shaped symbols,
it can be seen that there is a spectacular deviation from a Boltzmann
distribution in the Q-branch rotational intensities. This effect, present
in all branches, made the line assignments somewhat difficult. Finally,
these were established by using combination differences for the P - and
R-branches, the 3:1 ratio between odd- and even-J 00 lines associated
with 15 N2 nuclear-spin statistics, and the “cold” and “warm” spectral
recordings. From Fig. 3.1, it can be seen that there is a significant
intensity decrease between Q(6) and Q(13), with the Q(8) line too weak
to be observed.
61
3. Interactions of the 3pπu c 1 Πu (v = 2) Rydberg-complex in N2
Table 3.3: Observed transition energies (in cm−1 ) for the c 1 Πu − X 1 Σ+
g (2,0)
band in 14 N15 N. For explanation of the annotations, see caption to Table 3.1.
J 00
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
62
R(J 00 )
108 628.39∗
108 631.61
108 634.53
108 637.04∗
108 639.39∗
108 641.37
108 642.83∗
108 643.76∗
108 644.64∗
108 644.64∗
108 644.64∗
108 643.76∗
108 642.83∗
108 639.39∗
108 637.04∗
108 632.75
108 628.39∗#
108 622.81∗#
108 615.57∗#
108 606.42∗#
108 595.48#
108 582.63∗#
108 566.99#
108 524.41#
108 504.47#
Q(J 00 )
108 624.43
108 623.72
108 622.81∗
108 621.52
108 619.89
108 617.94
108 615.57∗
108 612.77∗
108 609.76
108 606.42∗
108 602.54∗
108 598.23
108 593.62
108 588.44
108 582.63∗
108 576.32
108 569.16#
108 561.40#
108 551.90#
108 540.56#
108 526.83#
108 509.86#
108 489.18#
P (J 00 )
108 616.82
108 612.77∗
108 607.62
108 602.54∗
108 597.11
108 591.29
108 585.15
108 578.56
108 571.67
108 564.14
108 556.05
108 547.59
108 538.33
108 528.65
108 517.77
108 506.18
108 493.40#
108 480.93#
3. Interactions of the 3pπu c 1 Πu (v = 2) Rydberg-complex in N2
Transitions terminating on levels between J =5 and J =13 also
show a marked intensity decrease in the P and R branches. As will
be seen below, this intensity depletion is attributed to the effects of the
crossing of c(v = 2) by the predissociated v = 17 level of the triplet state
C 3 Πu . In the case of 14 N15 N, small intensity deviations are observed
for transitions to both e- and f -parity levels of c(v = 2) with J = 17
and 18. No unusual intensity behavior is observed in the case of 14 N2 ,
where J = 19 is the highest rotational level of the excited state probed
experimentally.
1 +
Table 3.4: Observed transition energies (in cm−1 ) for the b0 1 Σ+
u −X Σg (7,0)
14 15
band in N N. For explanation of the annotations, see caption to Table 3.1.
J 00
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
R(J 00 )
108 880.90∗
108 882.06∗
108 882.06∗
108 880.90∗
108 878.69
108 875.29∗
108 870.74
108 864.94
108 858.15
108 850.34
108 841.55
108 831.80
108 821.08
108 809.49
108 797.10
108 783.71
P (J 00 )
108 874.20
108 869.13
108 862.74
108 855.17
108 846.41
108 836.40
108 825.39
108 812.91
108 799.44
108 769.05
108 753.21
108 698.09
108 678.06
108 656.94
The overall singlet structure in the region of the v = 2 level of the
3p Rydberg complex of N2 is summarized in Fig. 3.2 for the e-parity
levels of each isotopomer, the plotted points representing a combination of measurements taken using the present apparatus and data from
the Harvard-Smithsonian molecular database.[78] This structure is com-
63
3. Interactions of the 3pπu c 1 Πu (v = 2) Rydberg-complex in N2
Figure 3.2: Rotational term values for b 1 Πu (v = 11), b0 1 Σ+
u (v = 7),
c 1 Πu (v = 2) and c0 1 Σ+
(v
=
2),
for
the
three
isotopomers
of
N
.
For clarity,
2
u
only the e-parity levels are displayed. The symbols in gray represent data from
the Harvard-Smithsonian molecular database.[78] Indicative C 3 Πu (v = 17) energy levels (lines) were obtained using the CSE model. The missing rotational
levels for C(v = 17) were difficult to determine due to excessive predissociation
and additional perturbation by the F 3 Πu (v = 2) level.
pletely analogous to that reported and discussed in detail by Sprengers
et al.[36] for the v = 1 level of the same Rydberg complex in 15 N2 , where
the interacting valence states are b(v = 8) and b0 (v = 4). In the present
case, the diabatic c 1 Πu (v = 2) and the lower-lying c0 1 Σ+
u (v = 2) levels
exhibit the first-order heterogeneous L-uncoupling interaction characteristic of p-complex members,[11] providing the root cause of the Jdependent Λ-doubling in c(v = 2). In addition, the higher-lying valence
levels, b 1 Πu (v = 11) and b0 1 Σ+
u (v = 7), with lower rotational constants,
interact homogeneously with the c 1 Πu (v = 2) and c0 1 Σ+
u (v = 2) Rydberg levels, respectively, more strongly as J increases. In the case of
the f -parity levels, c 1 Πu (v = 2) ∼ b 1 Πu (v = 11) is the only operative
interaction. In reality, in the case of the e levels, the four singlet states
64
3. Interactions of the 3pπu c 1 Πu (v = 2) Rydberg-complex in N2
form a Rydberg-valence complex, with mutual effective interactions arising from the basis couplings just detailed. For example, b0 (v = 7) and
c(v = 2), the principal levels studied here, exhibit an effective rotational interaction which culminates in an avoided crossing at intermediate J values, providing a further dimension to the J-dependence of
the c(v = 2) Λ-doubling. The observed Λ-doublings for the c(v = 2)
levels of the three isotopomers, shown in Fig. 3.3, are seen to exhibit
qualitatively similar behavior. In the case of 15 N2 , the Λ-doubling is
very similar to that reported for the c(v = 1) p-complex member of the
same isotopomer by Sprengers et al.[36] At low J, the effective c0 (v = 2)
contribution to the Λ-doubling is a little more important, with the e
levels higher in energy. As J increases, b0 (v = 7) approaches from above
more closely and its contribution dominates, with the e levels now much
lower in energy.
Also shown in Fig. 3.2 is the indicative rotational structure of the
3
C Πu (v = 17) valence level that is responsible for the local-intensity
perturbation of c(v = 2), computed using the CSE model. As can be
seen from the figure, C 3 Πu (v = 17) crosses c 1 Πu (v = 2) at significantly
different J values for each isotopomer. The model crossing points agree
with the observed positions of maximum intensity perturbation for both
15 N and 14 N15 N. However, for 14 N , the C 3 Π (v = 17) state is pre2
2
u
1
dicted to cross c Πu (v = 2) at J ≈ 23, too high to be accessible with
the current experimental setup.
One of the main aims of the present work is to examine the character
of the local perturbation of c(v = 2) by C(v = 17). In order to facilitate
this in the case of the e-parity levels, we attempt to first remove the
effects of the stronger perturbation by b0 (v = 7) by performing a twolevel deperturbation analysis. In the case of the f -parity levels, this
procedure is not necessary, the avoided crossing between b(v = 11) and
c(v = 2) occurring only at high J levels, beyond the range of interest
for the C(v = 17) crossing.
Effective two-level-deperturbed spectroscopic parameters have been
determined from the experimental transition energies for the c − X(2, 0)
and b0 − X(7, 0) bands, using a nonlinear least-squares fitting procedure,
and omitting the J = 5 − 13 and J = 17 − 18 c(v = 2)-state levels,
for 15 N2 and 14 N15 N, respectively, that correspond to the regions of
observed intensity perturbation. For 14 N2 , no unusual intensity behavior
65
3. Interactions of the 3pπu c 1 Πu (v = 2) Rydberg-complex in N2
Table 3.5: Observed transition energies (in cm−1 ) for the c 1 Πu − X 1 Σ+
g (2,0)
band in 14 N2 . For explanation of the annotations, see caption to Table 3.1.
J 00
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
R(J 00 )
108 698.17∗
108 701.57
108 704.51
108 707.06∗
108 709.54∗
108 711.55
108 713.09
108 714.16∗
108 714.69∗
108 714.69∗
108 714.69∗
108 714.16∗
108 712.03
108 709.54∗
108 707.06∗
108 702.84
108 698.17∗
Q(J 00 )
P (J 00 )
108 694.06
108 693.35
108 692.27
108 690.87
108 689.15
108 687.04
108 684.68
108 681.91∗
108 678.71
108 675.12
108 671.31∗
108 666.88
108 662.02
108 656.84
108 651.00
108 644.69
108 637.56
108 629.70#
108 620.80#
108 686.60
108 681.91∗
108 676.68
108 671.31∗
108 665.79
108 659.69
108 653.30
108 646.52
108 639.14
108 631.45
108 623.10
108 614.28
108 604.74
108 594.55
108 583.56
108 571.84
108 559.03
is observed, so all levels have been included in the analysis. The terms
of the ground state, X 1 Σ+
g , are represented by
F (J 00 ) = B[J 00 (J 00 + 1)] − D[J 00 (J 00 + 1)]2 + H[J 00 (J 00 + 1)]3 , (3.3)
where J 00 labels the rotational levels of the ground state, B is the rotational constant, and D and H are centrifugal distortion parameters.
The spectroscopic parameters of Bendtsen[41] and Trickl et al.[40] are
15 N , 14 N15 N and 14 N . The terms for
used for the X 1 Σ+
2
2
g states of
1
the excited c Πu state, where, for simplicity, we denote the rotational
quantum number by J, rather than J 0 (and the same holds for v and
v 0 ), are taken to have the form
Tf (J) = ν0 + B[J(J + 1) − 1] − D[J(J + 1) − 1]2 ,
66
(3.4)
3. Interactions of the 3pπu c 1 Πu (v = 2) Rydberg-complex in N2
1 +
Table 3.6: Observed transition energies (in cm−1 ) for the b0 1 Σ+
u −X Σg (7,0)
14
band in N2 . For explanation of the annotations, see caption to Table 3.1.
Information derived from shoulders in the spectra is marked with an s.
J 00
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
R(J 00 )
108 954.80
108 956.40∗
108 956.40∗
108 955.38
108 953.17
108 949.72
108 945.00
108 939.26
108 932.43
108 924.61
108 915.91
108 906.24
108 895.50
108 884.08
108 872.15s
108 859.21
108 845.84
P (J 00 )
108 948.19
108 942.84
108 936.35
108 928.52
108 919.67
108 909.34
108 897.80
108 885.20
108 871.63s
108 856.94
for the f -parity[11] levels, where ν0 is the band origin, and
Te (J) = Tf (J) + q[J(J + 1) − 1],
(3.5)
for the e-parity levels, where q is the Λ-doubling parameter.
The terms for b0 (v = 7), which has only e-parity levels, are represented by
T (J) = ν0 + B[J(J + 1)] − D[J(J + 1)]2 .
(3.6)
As already noted, the effective interaction between the b0 (v = 7) levels,
1
of 1 Σ+
u symmetry, and c(v = 2) levels, of Πu symmetry, is of a heterogeneous nature (∆Ω = 1, L-uncoupling) and only involves the e-parity
levels of the c(v = 2) state. For the e-parity manifold, a two-state deperturbation analysis was performed for each J value by diagonalizing
the matrix
(
)
√
Tb0 (7)
Hb0 (7)c(2) J(J + 1)
√
,
(3.7)
Hb0 (7)c(2) J(J + 1)
Tc(2)
67
3. Interactions of the 3pπu c 1 Πu (v = 2) Rydberg-complex in N2
Figure 3.3: Observed Λ-doubling [∆Te−f = Te (J) − Tf (J)] for the c 1 Πu (v =
2) Rydberg levels of isotopic N2 . The symbols in gray represent data from the
Harvard-Smithsonian molecular database.[78]
where the diagonal elements are the term energies of b0 (v = 7) and
c(v = 2), given by Eqs. (3.6) and (3.5), respectively. The off-diagonal
elements represent the effective heterogeneous interaction between the
c(v = 2) e-parity and b0 (v = 7) levels. In the weighted fitting procedure, the nominal uncertainty in the absolute transition energy for
fully-resolved lines of reasonable strength was taken to be 0.1 cm−1 .
In the case of weak or blended lines, the uncertainty was set to an estimated value in the range 0.15−0.30 cm−1 . For all three isotopomers, the
quality of the obtained fit was reasonable. The results of the two-level
deperturbation analysis for the three isotopomers are given in Table 3.7.
It should be noted that the two-level deperturbation performed here
is only an approximation, and that, consequently, the spectroscopic parameters in Table 3.7 have limited physical significance. For example,
from molecular-orbital configurational arguments, and experimentation
with the CSE model, the direct heterogeneous electronic coupling be1
tween the b0 1 Σ+
u and c Πu states is likely to be very small, so it is principally the indirect mechanism involving both the homogeneous b0 ∼ c0
and the heterogeneous c0 ∼ c electronic couplings that is responsible for
the complexity of the observed c(v = 2)-level Λ-doubling in Fig. 3.3 and
the anomalous D values for b0 (v = 7) in Table 3.7. Consequently, for
68
3. Interactions of the 3pπu c 1 Πu (v = 2) Rydberg-complex in N2
Table 3.7: Two-level-deperturbed molecular parameters for the c 1 Πu (v = 2)
14
and b0 1 Σ+
N2 , 14 N15 N and 15 N2 . All values are in cm−1 .
u (v = 7) levels of
Statistical uncertainties (1σ), resulting from the fit, are shown in parentheses,
in units of the last significant figure. Additional systematic uncertainties of
order 0.05 cm−1 apply to the band origins ν0 .
14 N15 N
15 N
2
c 1 Πu (v = 2)
B
1.821(1)
6
D × 10
60(4)
q × 103
38(2)
ν0
108 696.32(5)
1.764(1)
74(5)
38(2)
108 626.57(4)
1.720(3)
151(9)
17(3)
108 556.38(5)
b0 1 Σ +
u (v = 7)
B
1.327(2)
D × 106
−110(7)
ν0
108 952.02(5)
Hb0 (7)c(2)
2.84(4)
1.278(1)
−64(6)
108 877.89(5)
2.76(5)
1.243(2)
−11(9)
108 801.35(7)
2.07(8)
Species
14 N
2
large J, the effective deperturbation parameter Hb0 (7)c(2) will not be well
represented as J-independent if c0 (v = 2) is not also explicitly deperturbed concurrently. However, additional possible perturbation partners
could not be followed up to sufficiently high J, and an attempted three
state analysis [b(v = 11), b0 (v = 7) and c(v = 2)] gave inconclusive results, as did the further inclusion of c0 (v = 2). Thus, only the two-state
analysis is presented and the higher-J levels for which the model does
not apply are left out of the fit. This limitation does not affect our
main aim of studying the additional local perturbation of c(v = 2) by
C(v = 17).
P/R intensity anomalies in the b0 − X(7, 0) band
The effective heterogeneous interaction between c(v = 2) and b0 (v =
7) not only produces energy shifts, but also affects the intensities of
transitions accessing the corresponding e-parity levels. In particular,
the b0 −X(7, 0) ionization spectra for each isotopomer display significant
P/R-branch intensity anomalies, behavior characteristic of transitions
69
3. Interactions of the 3pπu c 1 Πu (v = 2) Rydberg-complex in N2
into states of mixed Σ and Π character, where each component of the
transition is dipole-allowed.
P/R anomalies and their causative quantum-interference mechanism
have been described in detail by Lefebvre-Brion and Field.[11] Briefly,
in the case of a transition from a 1 Σ initial state to a nominal 1 Σ level
which is perturbed by a lower-lying 1 Π level, thus corresponding to the
present situation, the J dependence of the perturbed R-branch intensity
is given by:
IR (J 00 = J − 1) ∝ Jc2J µ2k + (J + 1)(1 − c2J )µ2⊥
√
±2cJ J(J + 1)(1 − c2J )µk µ⊥ ,
(3.8)
while the P -branch dependence is given by:
IP (J 00 = J + 1) ∝ (J + 1)c2J µ2k + J(1 − c2J )µ2⊥
√
∓2cJ J(J + 1)(1 − c2J )µk µ⊥ ,
(3.9)
where µk and µ⊥ are the unperturbed 1 Σ − 1 Σ and 1 Π − 1 Σ vibronic
transition moments, respectively, and cJ ≥ 0 is the J-dependent mixing
coefficient corresponding to the amount of 1 Σ character in the perturbed
1 Σ wave function. In Eqs. (3.8 – 3.9), the upper signs apply when the
Σ ∼ Π vibronic interaction matrix element HΣΠ > 0, the lower when
HΣΠ < 0. The sense of the quantum-interference effect, corresponding
to the overall signs of the cross terms in Eqs. (3.8 – 3.9), depends on the
sign of the product µk µ⊥ HΣΠ . In the present case, where the P branch of
the b0 − X(7, 0) transition is observed to decrease in intensity as a result
of the interference effect, evidently, µb0 X(7,0) µcX(2,0) Hb0 (7)c(2) > 0. Note
that as this is an excited-state interaction, we have chosen to express the
intensity of the R and P branches in terms of the excited-state rotational
quantum number J.
To study this phenomenon experimentally, it is necessary to extract
information on the intensity of the transitions measured in the ionization spectra. Generally, it is risky to rely on intensities in a 1 XUV +
10 UV experiment. However, if one assumes that the ionization cross
section and XUV intensity are constant during the measurement, some
semiquantitative information can be extracted from the experimental
results.
70
3. Interactions of the 3pπu c 1 Πu (v = 2) Rydberg-complex in N2
1 +
Figure 3.4: P/R-branch intensity anomalies in the b0 1 Σ+
u −X Σg (7, 0) bands
15
14 15
of N2 and N N. Solid circles: Reduced R-branch intensities (from experimental ionization spectra, see text). Open circles: Reduced P -branch intensities. Solid curve: P -branch band oscillator strengths from CSE-model calculations. Dashed curve: R-branch band oscillator strengths. The experimental
reduced intensities have been normalized to the CSE results.
Relative rotational intensities in well-behaved, unperturbed vibronic
transitions of N2 are given by:
I ∝ gJi 00 SJJ 00 (2J 00 + 1) e−
B 00 [J 00 (J 00 +1)]
kT
,
(3.10)
where gJi 00 is the nuclear spin-statistics factor responsible for the wellknown rotational-intensity alternation in the homonuclear isotopomers,
SJJ 00 is the Hönl-London factor, B 00 is the rotational constant of the
ground state, k is the Boltzmann constant, and T is the absolute rotational temperature. In deriving Eq. (3.10), it is assumed that the ground
state has a Boltzmann population distribution. ¿From Eq. (3.10), by
plotting − ln[I/(gJi 00 SJJ 00 (2J 00 + 1)] against J 00 (J 00 + 1), the molecularbeam temperature may be determined from the slope of B 00 /(kT ).
In many cases, Eq. (3.10) will not apply and the measured ionization signal will be influenced additionally, not only by genuine intensity interference effects, e.g., the P/R anomaly discussed above, but
71
3. Interactions of the 3pπu c 1 Πu (v = 2) Rydberg-complex in N2
1 +
Figure 3.5: P/R-branch intensity anomalies in the b0 1 Σ+
u − X Σg (7, 0)
14
band of N2 . Solid circles: Reduced R-branch intensities (from experimental ionization spectra, see text). Open circles: Reduced P -branch intensities.
Solid curve: P -branch band oscillator strengths from CSE-model calculations.
Dashed curve: R-branch band oscillator strengths from CSE-model calculations. Solid squares: R-branch band oscillator strengths from synchrotronbased experiments. Open squares: P -branch band oscillator strengths from
synchrotron-based experiments. The experimental reduced intensities have
been normalized to the CSE results.
also by J-dependent competition between predissociation and ionization. In order to study these effects, it is instructive to consider the
“reduced intensity”, i.e., the actual intensity divided by that implied by
Eq. (3.10), where it is assumed that there is an independent estimate of
the molecular-beam temperature(s) available, e.g., from a separate scan
over a well-behaved band, taken under similar experimental conditions.
The reduced intensity thus serves to highlight any strongly J-dependent
interference or predissociation effects, with subsequent analysis required
to decide between these two possibilities.
In Fig. 3.4, the reduced intensities (circles) obtained with our 1 XUV
+ 10 UV setup for the P and R branches of the b0 − X(7, 0) bands
in 15 N2 and 14 N15 N are shown as a function of J, normalized against
the corresponding absorption oscillator strengths predicted by our CSE
model (curves). The molecular-beam temperature used to determine
the reduced intensities was obtained from c − X(2, 0) Q-branch lines,
72
3. Interactions of the 3pπu c 1 Πu (v = 2) Rydberg-complex in N2
measured under virtually the same conditions. For both isotopomers,
Fig. 3.4 shows a remarkable difference in the behavior of the P - and
R-branch reduced intensities, with the P -branch intensities exhibiting
deep minima at intermediate J values, in good agreement with the CSEmodel oscillator strengths. This is a classic example of a P/R intensity
anomaly due to the quantum-interference effect described by Eqs. (3.8)
and (3.9). However, we note that attempts to fit the experimental results in detail using these equations were of limited success, due to the
inherently multilevel nature of the N2 spectrum in this region, which is
embodied in the CSE results.
In Fig. 3.5, the reduced intensities (circles) obtained with our 1 XUV
+ 10 UV setup in the case of 14 N2 are shown, normalized against CSEmodel calculations (curves), and compared with absolute band oscillator strengths obtained from our synchrotron-based experiments.(These
synchrotron-based results have also been reported in Ref. [74]) As for the
other isotopomers, once again there is a strong P/R intensity anomaly.
It is also of note that the excellent relative agreement between the reduced ionization intensities and the absolute optical oscillator strengths
implies that J-dependent predissociation effects are not a factor for
b0 (v = 7), at least for J . 15.
The quantum-interference effect also influences the e-levels of c(v =
2). A pair of equations complementary to Eqs.(3.8–3.9) can be written
to describe the R- and P -branch intensities in the transition to the
lower-lying 1 Π level. In the present case, the R branch of the c − X(2, 0)
band will suffer the intensity decrease and the P -branch intensity will
be enhanced by the interference effect. However, this phenomenon is
barely visible in our reduced intensities, since it is both masked by the
strong intensity perturbations due to the crossing of c(v = 2) by C(v =
17), discussed in Sec. 3.4, and less prominent due to the much greater
oscillator strength of the c − X(2, 0) transition.
The c 1 Πu (v = 2) ∼ C 3 Πu (v = 17) interaction
3
Transitions from the X 1 Σ+
g ground state to the Πu manifold of N2
are forbidden by the ∆S = 0 selection rule. However, as mentioned
in Sec. 3.1, the 3 Πu states may spin-orbit couple to the 1 Πu states.
As a result, transitions to the 3 Πu states may become directly ob-
73
3. Interactions of the 3pπu c 1 Πu (v = 2) Rydberg-complex in N2
Figure 3.6: (a) Reduced terms ∆T for the c 1 Πu (v = 2) level of 15 N2 . Solid
circles: f -parity levels. Open circles: e-parity levels. Solid line: CSE prediction
for crossing by Ω = 1 sublevel of C 3 Πu (v = 17). Dashed line: CSE prediction
for crossing by Ω = 0 sublevel. (b) Predissociation widths for c 1 Πu (v = 2)
in 15 N2 . Solid circles: f -parity levels, from fitting Q-branch line profiles in
c − X(2, 0) ionization spectra. Open circles: e-parity levels, weighted average
of fits to P - and R-branch line profiles. Solid line: CSE-model widths for f
levels. Dashed line: CSE-model widths for e levels. (c) Reduced intensities (see
text), taking into account the high- (202 K) and low- (25 K) temperature components of the molecular beam. Solid circles: f -parity levels, from integrated
Q-branch intensities in c − X(2, 0) ionization spectra. Open circles: e-parity
levels, weighted average of P - and R-branch reduced intensities.
74
3. Interactions of the 3pπu c 1 Πu (v = 2) Rydberg-complex in N2
servable through their intensity borrowing from dipole-allowed transitions, and/or the 3 Πu states may make their presence felt through
energy-, intensity-, or line-width-perturbation of the 1 Πu states. In the
present case, based on CSE-model predictions, we propose that it is the
C 3 Πu (v = 17) level which is responsible for the energy perturbations
observed here in the c(v = 2) levels of 15 N2 and 14 N15 N. Furthermore, since the C 3 Πu state couples electrostatically to the dissociative
C 0 3 Πu state, leading to strong predissociation,[32] it is likely that the
local perturbation of c(v = 2) by C(v = 17) is also responsible for the
intensity anomalies observed in the corresponding ionization spectra,
through accidental predissociation effects. Due to the complexity of the
N2 structure, however, these conclusions are by no means obvious from
an examination of the experimental spectra alone. We attempt to justify
them below.
After the perturbations in the singlet manifold have been roughly
accounted for and removed, as in Sec. 3.4, the residual local energy
shifts can be examined. In Fig. 3.6(a), the reduced term values of the
c(v = 2) rotational levels are shown for 15 N2 . Both e- and f -parity
reduced terms exhibit a similar pattern, with local perturbations evident
between J = 7 and 9, and, more clearly, between J = 11 and 12. In
Fig. 3.6(b), the fitted Lorentzian line-width components (symbols) for
isolated rotational transitions in the c − X(2, 0) band of the ionization
spectrum of 15 N2 are presented, with only minor differences observed
between the line-width patterns for transitions into the e- and f -parity
levels of the c(v = 2) state. Considerable predissociation of the c(v = 2)state levels is observed in the region of the local perturbations, with
level-width maxima near J = 8 and 12.
In the case of blended lines, term values could be extracted from the
experimental spectra, but neither line-width nor intensity information
could be obtained reliably. Also shown in Fig. 3.6(b) are the results
of our preliminary CSE-model calculations (lines), which are in good
overall agreement with the observed predissociation pattern, but with
some discrepancy in the region of the higher-J crossing. Finally, in
Fig. 3.6(c), the reduced intensities for transitions to the e- and f -parity
levels of c(v = 2) in 15 N2 are shown. For J > 14, these reduced intensities are essentially J-independent. However, at lower J, significant
intensity depletion is observed, with deep intensity minima near J = 8
75
3. Interactions of the 3pπu c 1 Πu (v = 2) Rydberg-complex in N2
and 12. In the case of the missing J = 8f level, for example, the associated Q(8) line was too weak to be observed. As stated in Sec. 3.4, there
is no clear evidence of P/R intensity anomalies in the c − X(2, 0) spectrum, so the dramatic behavior of the reduced intensities in Fig. 3.6(c)
is likely another signature of predissociation. It is notable that the positions of the intensity minima occur for the same values of J where the
line widths and term values also show unusual behavior.
The patterns in Fig. 3.6, containing only two distinct regions of perturbation, are not directly suggestive of perturbation by a triplet level,
despite the CSE model indicating that C 3 Πu (v = 17) is primarily responsible. There are two reasons for this.
First, only CΩ=1 (v = 17) can interact by spin-orbit coupling (∆Ω =
0) directly with cΩ=1 (v = 2). However, as the J-dependent S-uncoupling
mechanism (∆Ω = ±1) induces mixing between CΩ=1 (v = 17) and both
CΩ=0 (v = 17) and CΩ=2 (v = 17), the latter substates may also interact indirectly with c(v = 2). For the limiting case of a slowly rotating
molecule, the Ω = 1 component of the triplet perturber will strongly
spin-orbit couple to the c(v = 2) state, while the Ω = 0, 2 components
will couple only weakly, corresponding to the Hund’s case (a) limit. On
the other hand, at high J values, the Hund’s case (b) limit is approached,
where the Ω = 0, 2 components interact strongly with c(v = 2), but the
Ω = 1 component is only weakly coupled.[79] The local perturbation
culminating near J = 8 coincides with the CSE-predicted crossing by
CΩ=1 (v = 17), indicated by the solid line in Fig. 3.6(a), while that at
J = 11 − 12 is due to the Ω = 0 crossing,(The C 3 Πu state is normal for
the lower vibrational levels, but inverted in this region (Ref. [72])) indicated by the dashed line. The CΩ=2 (v = 17) sublevel would be expected
to cross c(v = 2) in the region of J = 5, but no evidence for a perturbation associated with this crossing is visible in Fig. 3.6(a). Part of the
reason for this relates to the low J value for the crossing, suggesting
that the Hund’s case (a) limit will be approached, therefore implying
only a weak indirect coupling between CΩ=2 (v = 17) and cΩ=1 (v = 2).
This phenomenon was also observed in the k 3 Π state of CO, which is
spin-orbit coupled to the E 1 Π state, causing predissociation in the same
manner as in the present study (see Fig. 12 of Ref. [80]).
Second, the characteristics of local perturbations in energy depend
also on the predissociation width of the perturbing level. For example,
76
3. Interactions of the 3pπu c 1 Πu (v = 2) Rydberg-complex in N2
Figure 3.7: (a) Reduced terms ∆T for the c 1 Πu (v = 2) level of 14 N15 N. Solid
circles: f -parity levels. Open circles: e-parity levels. Dashed line: CSE prediction for crossing by Ω = 0 sublevel. (b) Predissociation widths for c 1 Πu (v = 2)
in 14 N15 N. Solid circles: f -parity levels, from fitting Q-branch line profiles in
c − X(2, 0) ionization spectra. Open circles: e-parity levels, weighted average
of fits to P - and R-branch line profiles. Solid line: CSE-model widths for f
levels. Dashed line: CSE-model widths for e levels. (c) Reduced intensities (see
text), taking into account the high- (275 K) and low- (72 K) temperature components of the molecular beam. Solid circles: f -parity levels, from integrated
Q-branch intensities in c − X(2, 0) ionization spectra. Open circles: e-parity
levels, weighted average of P - and R-branch reduced intensities.
perturbation by a long-lived level results in a perturbation with a rapid
J-dependence and a maximum shift at the crossing, while perturbation
by a short-lived level may result in a barely-noticeable perturbation
with a slow J-dependence and zero shift at the crossing. In the case of
crossing by a triplet level, the visibility and character of the associated
perturbations will also depend on the relative predissociation widths of
the triplet sublevels. In the present case, the CSE model predicts that
the lowest C(v = 17) sublevel (Ω = 2) is very strongly predissociated,
with the level widths, also strongly J-dependent, decreasing significantly
for Ω = 1, and Ω = 0, respectively. Thus, primarily due to the large
width of CΩ=2 (v = 17) relative to the spin-splitting of the triplet state,
77
3. Interactions of the 3pπu c 1 Πu (v = 2) Rydberg-complex in N2
only the highest two of the expected three energy perturbations in the
c(v = 2) level are visible in Fig. 3.6(a).
For the same reasons, only two clear peaks are visible in the c(v = 2)
predissociation pattern in Fig. 3.6(b), and two minima in the associated intensity-depletion pattern in Fig. 3.6(c). In particular, the higherJ crossing, corresponding to the better defined local perturbation in
Fig. 3.6(a), is seen in Fig. 3.6(b) to be associated with the most rapid
variation in the computed predissociation line width which is borrowed
by c(v = 2) from the C(v = 17) perturber. It is notable that, while the
energy perturbations in the c(v = 2) level of 15 N2 caused by the crossing
with the short-lived C(v = 17) level are small, the corresponding indirect predissociations and intensity depletions are dramatic, despite the
weak spin-orbit coupling, because of the otherwise long inherent lifetime
of c(v = 2).
Finally, we emphasize that caution is necessary when trying to infer detailed information on the perturbing C(v = 17) level from the
experimental spectra, with simplistic level deperturbations of little use.
Detailed examination of the CSE model indicates that the 3 Πu manifold in this energy region is of great complexity, with the F 3 Πu and
G 3 Πu Rydberg states also having an active influence on the character
of the c(v = 2) ∼ C(v = 17) perturbation. In fact, the very strongly
predissociated F (v = 2) level, which also contains a significant G-state
admixture,[70] crosses C(v = 17) in the region of the latter level’s crossing with c(v = 2), for all isotopomers, leading, e.g., to the irregularities
in the computed nominal C(v = 17)-level energies for 14 N2 displayed in
Fig. 3.2. In addition, electrostatic interactions within the 3 Πu manifold
lead to strong J- and Ω-dependences in the C(v = 17) predissociation
line widths, as mentioned above, which further complicate interpretation
of the data. In particular, the determination of C(v = 17) sublevel rotational terms becomes impossible when the corresponding predissociation
line widths are comparable to, or larger than, the spin splitting.
The results for 14 N15 N, summarized in Fig. 3.7, are more equivocal.
For this isotopomer, the crossing of c(v = 2) by C(v = 17) is predicted
by the CSE model to occur in the region of J = 17. In Fig. 3.7(a),
no significant term-value shifts can be observed for the f -parity levels,
but there is evidence of a weak local perturbation between J = 17 and
18 in the e-parity levels. Overall, the fitted Lorentzian line widths in
78
3. Interactions of the 3pπu c 1 Πu (v = 2) Rydberg-complex in N2
Fig. 3.7(b) (symbols) are not conclusive, because of large error bars in
the crossing region, but are not inconsistent with the computed CSEmodel widths (lines), which predict a predissociation maximum in the
J = 17−18 region. The reduced intensities in Fig. 3.7(c), however, show
fairly clear depletion, displaying a single minimum near J = 18, which is
shaded towards lower J values, behavior consistent with that expected
from the computed line widths. Taken together, the experimental results
are consistent with the perturbation of c(v = 2) by C(v = 17) in the
J = 17−18 region, but imply such a strong predissociation of the C(v =
17) sublevels that (1) the associated energy shifts are only small, and
(2) the predissociation signatures collapse into a single broad feature.
This is supported by the CSE results, which suggest that it is the Ω = 0
sublevel of C(v = 17), indicated by the dashed line in Fig. 3.7(a), that
has the lowest predissociation width and is responsible for the only local
perturbation evident in the c(v = 2) terms, with the other sublevels so
broad that no corresponding shifts occur.
As stated in Sec. 3.4, for 14 N2 , the CSE model predicts the crossing
between c(v = 2) and C(v = 17) to take place in the region of J = 23
(see Fig. 3.2). Since, under our experimental conditions, such highrotational levels cannot be accessed with the laser-based system, the fact
that no energy-level shifts or intensity anomalies have been observed in
the present data for this isotopomer is hardly surprising. Nevertheless,
in Fig. 3.8, we are able to compare the synchrotron-based predissociation
widths for the c(v = 2) level of 14 N2 (circles) (These synchrotron-based
results have also been reported in Ref. [74]) with the CSE-model values
(lines). The CSE results indicate a marginally double-peaked structure,
with predissociation maxima near J = 20 − 21 and 23 − 24, in excellent agreement with the measurements. Once again, the short lifetime
ol the C(v = 17) perturber, together with the complex interactions at
play, combine to severely modify the expected triplet pattern of predissociation. For this isotopomer, it is the Ω = 2 sublevel of C(v = 17)
which has the lowest predissociation width and is responsible for the
first c(v = 2) predissociation maximum in Fig. 3.8, while the broader
Ω = 1, 0 sublevels contribute to the second maximum, with the Ω = 1
contribution likely to be smaller because of the case-(b) nature of this
high-J crossing.
79
3. Interactions of the 3pπu c 1 Πu (v = 2) Rydberg-complex in N2
1
Γ ( cm
−1
FWHM )
1.5
0.5
0
0
200
400
600
800
J (J +1)
Figure 3.8: Comparison between experimental predissociation line widths for
the c 1 Πu (v = 2) level of 14 N2 , derived from synchrotron-based photoabsorption
spectroscopy of the c − X(2, 0) band (Ref. [74]), and CSE-model calculations.
Solid circles: Experimental values from fits to Q-branch transitions. Open
circles: Weighted average of experimental values from P - and R-branch transitions. Solid line: CSE widths for f -parity levels. Dashed line: CSE widths
for e-parity levels.
3.5
Summary and conclusions
01 +
1 +
The 3pπu c 1 Πu − X 1 Σ+
g (2, 0) Rydberg and b Σu − X Σg (7, 0) valence transitions of 14 N2 , 14 N15 N, and 15 N2 have been studied using
laser-based 1 XUV + 10 UV two-photon-ionization spectroscopy, supplemented by synchrotron-based XUV spectroscopy in the case of 14 N2 .
Due to the inherent complexity and multilevel nature of the N2 structure
and spectrum in this energy region, few-level deperturbation techniques
were found to be of limited use and interpretation of the results by
means of coupled-channel Schrödinger-equation techniques was found
necessary.
For each isotopomer, effective rotational interactions between the
c(v = 2) and b0 (v = 7) levels were found to cause strong Λ-doubling in
c(v = 2), together with dramatic P/R-branch intensity anomalies in the
80
3. Interactions of the 3pπu c 1 Πu (v = 2) Rydberg-complex in N2
b0 − X(7, 0) band due to the effects of quantum interference. The P/R
intensity anomalies for all isotopomers, together with the synchrotronbased oscillator strengths for 14 N2 , were found to be well described by
the present CSE calculations.
Local perturbations in energy and predissociation line width for the
c(v = 2) Rydberg level were observed and attributed primarily to a spinorbit interaction with the crossing, short-lived C 3 Πu (v = 17) valence
level. However, the CSE calculations suggest that this is a complex
interaction, also involving the even shorter-lived F (v = 2) Rydberg level
in the same energy region. The characterization of these interactions
should further help to elucidate the predissociation mechanism for the
1 Π states of N ,[32] extending the applicability of the corresponding
u
2
CSE code towards higher energies.
Acknowledgment
This research was partially supported by Australian Research Council
Discovery Program Grants DP0558962 and DP0773050.
81
Chapter 4
Observation of a Rydberg series
in a heavy Bohr atom
We report on the realization of a heavy ”Bohr atom”, through
the spectroscopic observation of a Rydberg series of bound quantum states at principal quantum numbers n = 140 to 230. The
system is made heavy by replacing an electron inside a hydrogen
atom by a composite H− particle, thus forming a H+ H− Coulombically bound system obeying the physical laws of a generalized
atom with appropriate mass scaling.
4. Observation of a Rydberg series in a heavy Bohr atom
Bohr developed the atomic model by imposing an ad hoc quantization condition on the angular momentum of an electron orbiting in the
Coulombic potential of a positively charged nucleus [81]. The same solutions for the level energies are found by solving the Schrödinger equation
for a quantum system bound by a 1/r potential; Rydberg states are represented by [82]
En = Elim −
RA
(n − δ)2
(4.1)
where RA is the atomic Rydberg constant and n is the principal quantum
number. For atoms with an extended core, i.e all atoms other than
Hydrogen, a quantum defect δ should be included. The wide variety
of atomic and molecular realizations of Rydberg systems all have in
common that the fundamental scale factor, the Rydberg constant, is
essentially the same, because it is governed by the mass of the electron
bound to a much heavier core: RA = (µ/me )R∞ ; for all these systems
the reduced mass of the electron µ is close to me . Only few exotic
systems like muonium (µ+ e− ) [83] or positronium (e+ e− ) [84] provide a
different scale factor, the latter having exactly R∞ /2.
Ion-pair systems bound by their Coulombic 1/r potential provide a
realistic extension of the Rydberg concept to heavier systems. Initially
the process of ion-pair formation above threshold was investigated by
Chupka et al.., [85] measuring H2 + hν → H+ + H− . Hepburn and
coworkers used the existence of long-lived bound states immediately below the ion-pair limit in their threshold ion-pair production spectroscopy
(TIPPS) to determine accurate values of ion-pair dissociation thresholds, applied amongst others to the H+ H− system [86]. Later Reinhold
and Ubachs demonstrated the existence of bound heavy Rydberg states
of the H+ H− system through the probing of wave packet dynamics in
the densely structured quantum region several cm−1 below threshold
[87]. These time-domain observations of heavy Rydberg states, conceptually analogous to observations of electronic wave packets [88, 89],
reveal the characteristic signatures of such states without resolving individual quantum states.
Fig. 4.2 displays the fundamental discovery of the present study.
A long series of regularly spaced resonances is observed in an energy
84
4. Observation of a Rydberg series in a heavy Bohr atom
Figure 4.1: Experimental setup. An H2 molecular beam is perpendicularly
intersected with a beam of XUV-radiation, obtained via third harmonic generation of a pulsed UV-laser beam underneath the orifice of a pulsed valve in
krypton gas. The XUV-radiation is geometrically filtered from the incident
fundamental UV-beam by selecting the phase-matched kXU V -vector in a UVdark region imposed by a post placed in the incident laser beam (see inset).
H+ and H+
2 are extracted from the interaction region by a pulsed electric field
for signal detection.
window of some 2000 cm−1 . These resonances visibly follow the pattern
of a Rydberg-like series indicated and numbered in the Figure. However,
the Rydberg series is not as ’clean’ as usually observed in atoms; the
intensities, widths and line shapes vary across the series superimposed on
a structured continuum, while some unassigned resonances are present
as well. The series of quantum states with a Rydberg electron bound
+
+
by an H+
2 molecular core in quantum states v = 7 and N = 0, 2 or 4
(only even rotational quantum numbers for para-H2 ), expected in this
energy range, do not match the observed features. Alternatively, the
series can be fitted to an equation for heavy Rydberg states:
En = EIP −
Rh
(n − δ)2
(4.2)
85
4. Observation of a Rydberg series in a heavy Bohr atom
+
H2 (7,4)
+
H2 (7,0)
HR(n)
170
|
180
|
|
190
|
+
H2 signal (arb. units)
160
136400
|
+
H2 (7,2)
210
|
220
230
|
|
137600
136000
200
136800
Excitation Energy, cm
137700
137800
137200
-1
Figure 4.2: Characteristic spectrum after two-step laser excitation via C-X
(3,0) R(0) in the para-H2 molecule. The markers and dotted lines show the
predicted positions in the heavy Bohr atom between n = 161 and n = 230 in the
H+ H− system. In the range between n = 205−225 the series is interrupted and
overlaid by resonant structures, but appears again for n = 225. The Rydberg
+
series converging to the H+
= 7, N + = 0, 2 and 4) states of the ion are
2 (v
denoted with open triangles, full triangles and open circles, respectively.
where Rh denotes the Rydberg constant for the H+ H− heavy Bohr atom.
While in the experiment the X1 Σ+
g , v = 0, N = 0 ground level of the
neutral H2 molecule is taken as the zero energy level, the H+ H− (ionpair) dissociation limit EIP can be determined from a thermodynamic
cycle [12], thus including the values for the ionization energy of the
neutral H2 molecule, the dissociation energy of the H+
2 ion and the
electron affinity of the H-atom, yielding EIP = IE(H2 )+D0 (H+
2 )−EA(H)
and resulting in EIP = 139 713.83 cm−1 . The Rydberg constant for the
heavy Bohr atom H+ H− can be obtained by replacing the electron by
an H− particle, yielding the scaling factor (µ/me ) = 918.5761 [12]. Note
that the Bohr radius for the heavy system is ah0 = 5.7 × 10−14 m. Based
on the numbers for EIP and Rh the observed Rydberg series in parahydrogen can be assigned to principal quantum numbers n between 161
and 230 (with a gap between n = 205 − 225). A fit to Eq. (4.2) yields a
value for the quantum defect of δ = 0.05.
The observations displayed in Fig. 4.3 add to the findings. They show
that the heavy Rydberg series can be observed at even lower energies,
86
4. Observation of a Rydberg series in a heavy Bohr atom
where states of n = 140 − 150 are located, that the B1 Σ+
u intermediate
state of valence character can be used, as opposed to the C1 Πu state of
molecular Rydberg character, and that the series can also be observed
in ortho-H2 , as opposed to para-H2 in Fig. 4.2. A remarkable observation features at excitation energies above 135500 cm−1 , where the
assignment in terms of Rydberg states in the heavy Bohr atom ceases
to match. From the more detailed upper part of Fig. 4.3 it becomes
clear that in this energy range the ordinary or electronic Rydberg states
converging to the v + = 6, N + = 1, 3 limits in the ortho-H+
2 core are excited. Also, some effects of mutual interaction between these Rydberg
series, a well-known feature in molecular Rydberg series [90], seem to
be present. From the perspective of electronic Rydberg spectroscopy,
an intermediate state with a 2p-electron, 2pσ in case of the B1 Σ+
u state
1
and 2pπ in case of the C Πu state, should give rise to ns and nd-series,
as are indeed observed [91] in such excitation schemes in H2 . Considering the angular momenta in a Hund’s case (d) framework, mutually
interacting ns and nd Rydberg series converging to several allowed N +
series limits are expected. This is indeed observed in Fig. 4.3 with series converging to N + = 1 and 3, for v + = 6 in the ortho-ion core.
We observed the same phenomenon in para-hydrogen, when exciting
via the B1 Σ+
u , v = 12, J = 1 intermediate state in the energetic region
near 135 600 cm−1 (not shown here), where Rydberg series converging to
N + = 0, 2 and 4, for v + = 6 in the para-ion core are observed. However,
the observed spectral lines, indicated by HR(n) in Figs. 4.2 and 4.3, are
not associated with electronic states, but with excitations in the heavy
H+ H− system.
An elucidating connection can be made between the quantum numbers that characterize the heavy Bohr atom, principal quantum number n and angular momentum quantum numbers ` or J, and the usual
molecular quantum numbers v and J associated with an intra-molecular
potential-energy curve (in this case 1/R). This connection was established by Pan and Mies [92] to be n ↔ v + J + 1. It can be understood
by considering the number of radial nodes in the wave function, v in
the molecular system and n − ` − 1, or n − J − 1 in the Rydberg system. In the present experiment for ortho-hydrogen only a J = 1 state
is excited (J = 0 in the intermediate state), hence a single n series in
the heavy Bohr atom results through jumps in the v quantum number.
87
4. Observation of a Rydberg series in a heavy Bohr atom
135650
135700
135750
135800
135850
+
H2 (6,3)
+
+
H2 signal (arb. units)
H2 (6,1)
HR(n)
|
134500
160
150
140
|
135000
|
135500
-1
Excitation Energy (cm )
Figure 4.3: Lower panel: Characteristic spectrum after two-step laser excitation via B-X (12,0) P(1) in the ortho-H2 molecule, showing n = 140 − 162
Rydberg states in the heavy Bohr atom. In the upper panel the part of the
spectrum between 135 650 and 135 880 cm−1 is enlarged, showing the electronic
+
+
Rydberg series converging to the H+
= 1, 3) states of the ion.
2 (v = 6, N
In para-hydrogen (a J = 1 intermediate state) J = 0 and 2 states can
be excited, when applying the ∆` = ∆J = ±1 (Laporte) selection rule
in the heavy Bohr atom. Hence, two series might be expected, possibly
with different quantum defects δJ . Since this H− particle is in a unique
1 S state, there exists only a single series limit (neglecting hyperfine
0
structure).
In the potential-energy diagram (Fig. 4.4) a distinction is made between a short-range regime (R < 12a0 ), where the covalent states in the
H2 molecule and the electronic Rydberg states are located, and a long
range regime (R > 12a0 ). At ≈ 10a0 the ion-pair potential undergoes
a strong interaction (≈ 3000 cm−1 ) with a dissociative state, thus giving rise to the characteristic double-well states in H2 , the HH̄1 Σ+
g state
1
+
in the g-manifold [93] and the B”B̄ Σu state in the u-manifold [94].
Heavy Rydberg states with small principal quantum numbers, thus having bond lengths of the order of the scaled Bohr radius, are unphysical
because the H− -ion itself is larger. Therefore, the principal quantum
88
4. Observation of a Rydberg series in a heavy Bohr atom
number n of the lowest essentially pure ion-pair state in H+ H− is given
by R = n2 ah0 ≈ 12a0 ≈ 6 × 10−10 m, i.e., n ≈ 100.
All presently observed heavy Bohr atomic states fall in the energetic
range between the H(1s)+H(3`) and H(1s)+H(4`) dissociation energies
(see Fig. 4.4). Below n = 130 a perturbation in the heavy Rydberg series
is expected, by about 60 cm−1 at 36a0 , due to the avoided crossing with
H(3`) [95, 96, 97]. In this range and above n = 250, where another
crossing with H(4`) occurs (presumably with a strength of only a few
cm−1 ), the series in the heavy Bohr atom indeed dies out. Unresolved
heavy Rydberg states were shown to exist above the n = 4 threshold by
wave-packet experiments [12].
Photo-excitation of molecular states is governed by the Condon principle: Electronic orbitals are excited by the optical transition during
which the nuclei remain essentially fixed in space. As a result, the
strengths of transitions are governed by the square of the overlap integral of the initial and final ro-vibrational wave functions of nuclear
motion (the Franck-Condon factor):
∫
In = | Ψ∗n (R)ΨB,C (R) dR|2
(4.3)
By far the main part of the nuclear wave function of the heavy Rydberg
states is at large internuclear distance (e.g., the outer turning point for
n = 200 is R ≈ 90a0 ), while the lower state is confined to R < 6a0 . The
Franck-Condon factor of the transition should therefore be extremely
small; further, excitation from the C state (which is confined to R < 3a0 )
should be much weaker than from the B state, but this is not observed.
From this we conclude that the heavy Rydberg states are not directly
excited from the prepared intermediate B and C states via their FranckCondon overlap.
We invoke the mechanism of a complex resonance [98] to resolve this
issue. A superposition quantum state:
∑
∑
+
+ −
Ψc = ΨhR (n) +
Ψ(H+
Ψ(H, H∗ )
(4.4)
2 ,v ,J ;e ) +
v + ,J +
diss
with mixed character of electronic Rydberg channels (some 20 of
+
+
those) associated with a Ψ(H+
2 , v , J ) core representing the ionizationcontinua above each limit and a bound series below, dissociativecontinua (both n = 2 and n = 3 channels), and a series of Bohr atomic
89
4. Observation of a Rydberg series in a heavy Bohr atom
Figure 4.4: Potential-energy diagram of the H2 molecule with distinction between two separated regimes: for R < 12a0 the range of internuclear distances
where covalently bound states and the electronic Rydberg states converging to
+
+
H+
2 (v , J ) dominate (dashed lines representing the potential curves of gerade
symmetry in this region [93]), and the range R > 12a0 where H+ H− heavy
Rydberg states exist. The H+ H− Coulomb potential is shown in dark (red)
and extended to smaller R by dots. Intermediate energy levels in the B and
C states (at ∼ 105 000 cm−1 excitation) are indicated and their accessible
Franck-Condon ranges mapped (boxes).
90
4. Observation of a Rydberg series in a heavy Bohr atom
states ΨhR (n); in excitation the latter have a zero transition dipole moment. A crucial ingredient in the complex resonance model is a broad
interloper state for which the transition dipole moment is large; a low n
+
Rydberg state converging to a higher lying limit H+
2 (v ), hence a state
in which v + is significantly larger than 6, takes on this role. As shown
for another transition scheme in the H2 molecule [98], under such conditions quantum interferences via different but indistinguishable pathways
can lead to excitation of narrow series (in the example of [98] electronic
Rydberg states), while excitation of the associated bound states may
result in auto-ionizing decay. This closely matches the situation in the
present experiment, although the number of channels is much larger
than in an idealized example. The interpretation in terms of a complex
resonance explains the occurrence of underlying (possibly structured)
continua in the spectrum, the variation of widths over the bound series,
and the variation of asymmetry in the line-shapes exhibiting differing
Fano-parameters. At the same time it provides an explanation for the
fact that H+ H− heavy Bohr atomic states are excited, while signal is detected in another channel, such as H+
2 . A detailed quantitative analysis
is beyond the scope of the present Letter
It is interesting to note that the pp matter-antimatter protonium
system [99] exhibits nearly the same Rydberg constant as the H+ H−
system. Hence both systems have a similar energy level structure. In
H+ H− the electrons have the positive effect to mediate a complex resonance, thus providing optical access to the series of heavy quantum
states, a phenomenon which does not occur in pp.
In conclusion, we have observed a frequency-resolved series of quantum resonances in a heavy Rydberg system by two-step excitation.
In principle such series should be observable in any A+ B− diatomic
molecule, or in A+ (BC)− poly-atomic molecules [92], as long as the negative particle is bound (which would exclude e.g. N− ). In such systems a
heavy Rydberg series is associated with each quantum state (rotational,
vibrational, electronic and fine structure) in either of the two constituting particles. The presently studied H+ H− system is the simplest, not
only because H2 is the simplest neutral molecule, but in particular since
H− in the 1 S0 state has no internal structure and therefore only a single
Rydberg series limit exists, making the system truly a heavy Bohr atom.
91
Chapter 5
Spectroscopic observation and
characterization of H+H− heavy
Rydberg states
A series of discrete resonances was observed in the spectrum
of H2 , which can be unambiguously assigned to bound quantum
states in the 1/R Coulombic potential of the H+ H− ion-pair system. Two-step laser excitation was performed, using tunable extreme ultraviolet radiation at λ = 94 − 96 nm in the first step,
and tunable ultraviolet radiation in the range λ = 310 − 350 nm
in the second step. The resonances, detected via H+ and H+
2
ions produced in the decay process, follow a sequence of principal quantum numbers (n = 140 − 230) associated with a Rydberg
formula in which the Rydberg constant is mass scaled. The series converges upon the ionic H+ H− dissociation threshold. This
limit can be calculated without further assumptions from known
ionization and dissociation energies in the hydrogen system, and
the electronegativity of the hydrogen atom. A possible excitation
mechanism is discussed in terms of a complex resonance. Detailed
measurements are performed to unravel and quantify the decay of
the heavy Rydberg states into molecular H+
2 ions, as well as into
atomic fragments, both H(n = 2) and H(n = 3). Lifetimes are
found to scale as n3 .
5. Characterization of H+ H− heavy Rydberg states
5.1
Introduction
The presence of ion-pair states and ionic binding in molecules has long
been recognized. In diatomic molecules potential energy curves are
known whose outer limb of the potential curve closely overlaps with
the Coulombic 1/R attraction of the ionic constituents. Such electronic states were investigated in ICl [100] and in I2 [101]. Also, in
H2 molecular states were identified for which the outer limb follows the
H+ H− potential. In the low-energy region this holds for the well-known
1 +
1 +
EF 1 Σ+
g and the B Σu states. At higher excitation energies the HH̄ Σg
and the B”B̄1 Σ+
u state were identified with dominant ionic character
at large internuclear separation [102, 94]. At even higher energies in
the H2 molecule additional electronic states of ion-pair character were
predicted [95, 97], but experimental searches for bound levels are still
inconclusive [96].
The photo-excitation, and the photo-physics including ionization and
dissociation phenomena above the ion-pair threshold have been widely
investigated over the years. Suits and Hepburn published a review on
the spectroscopy and dynamics of ion-pair dissociation processes [103].
While many of the ion-pair phenomena have been observed in diatomic
molecules (see e.g. the work on Cl2 [104, 105, 106]), some of the studies
have focused on poly-atomics as well: the excitation mechanism of ionpair states in bromomethane was recently investigated [107]. Previously,
it had been demonstrated that ion-pair dissociation (in N2 O) can also
lead to an unstable fragment (N− ) which then autodetaches [108].
The H2 molecule is often chosen as the benchmark target for investigation of ion-pair phenomena in molecules. Already in the early
studies involving classical light sources by Chupka et al. [85] and by
McCulloh and Walker [109] a strong coupling was found in the H+ H−
continuum above 17.3 eV between the ion-pair channel and electronic
Rydberg series converging upon highly excited vibrational levels of H+
2.
−
In the ion-pair photoionization production curve (obtained via H detection) the structure of predissociating and autoionizing Rydberg series
was clearly observed. Pratt and coworkers applied a laser multi-step excitation scheme to investigate the same energy region in H2 [110]; they
found strong coupling of the H+ H− continuum to electronic Rydberg series converging upon v + = 9 in the H+
2 ion. In a subsequent study [111]
94
5. Characterization of H+ H− heavy Rydberg states
it was found that the observed resonances above the H+ H− threshold
decay via autoionization as well as dissociation involving H(n=3) and
H(n=4) dissociation products. Kung and coworkers [112] approached
the same energy region above the H+ H− limit via intermediate states
0
(C 1 Πu , v 0 = 2) and (B 1 Σ+
u , v = 12) of ungerade symmetry. They observed the same Rydberg series converging to v + = 9 in the H+
2 ion,
−
again employing H detection. This once more demonstrated the strong
coupling between the H+ H− ion-pair and the electronic Rydberg channels. This coupling is of importance for the effective oscillator strength
for quantum states in the H+ H− potential. The studies mentioned here
were all focused on the energetic region above the ion-pair threshold.
The energy region very near and just below the onset of the
+
H H− threshold was investigated by various laser spectroscopic methods. Shiell et al. applied the threshold ion-pair production spectroscopy
(TIPPS) technique to the H+ H− system [86], therewith demonstrating
the existence of long-lived high-n ion-pair, or heavy Rydberg states.
Reinhold and Ubachs provided further evidence for such high-n heavy
Rydberg states by producing wave packets of angular momentum states
in an electric field and following their time evolution before pulsed field
dissociation (i.e ion-pair formation) [87]. The temporal behavior in the
wave-packet dynamics can be quantitatively interpreted in terms of a
heavy Rydberg system, or a ”heavy Bohr atom”: a hydrogen atom,
in which the negatively charged electron is exchanged for a composite
H− particle, which is treated as a point-like entity [12]. It was postulated that heavy Rydberg systems follow the physical laws of electronic
Rydberg systems and that each system can be described by a single
mass-scaling parameter that defines all properties of heavy Rydberg
states. This was tested and quantitatively demonstrated in detecting
wave packet dynamics in the H+ F− system [113], which has a reduced
mass differing by a factor of two from the H+ H− system.
As a follow-up on a recent Letter on the observation of a spectrum
of energy-resolved heavy Rydberg states in the H+ H− system [114] we
here report on a fuller description of these remarkable features. Regularly structured resonances are detected that obey a generalized Rydberg
formula [12]:
En = EIP −
Rh
(n − δ)2
(5.1)
95
5. Characterization of H+ H− heavy Rydberg states
with n the principal quantum number observed over the range 140 −
230, EIP the ion-pair threshold, Rh the Rydberg constant of the heavy
system, and δ a quantum defect, which is usual for a non-point-like
system [92, 82]. Although the assignment of principal quantum numbers
to the observed heavy Rydberg states is unambiguous, there still exist
some open issues about the spectroscopic features. They concern the
excitation mechanism of heavy Rydberg states, their quantum defects,
and their interactions with electronic Rydberg states; these form the
subject of the present paper.
5.2
Experiment and observations
The range of excitation energies 134 000 − 138 000 cm−1 in the H2
molecule (above the v 00 = 0, J 00 = 0 ground state of para-H2 ) is investigated via two-step laser excitation in an experimental configuration as
schematically displayed in Fig. 5.1. The first step is induced by tunable extreme ultraviolet (XUV) radiation, which is produced by third
harmonic generation in a krypton gas jet. The XUV-laser is tuned and
fixed at some intermediate resonances in H2 . Here any strong line in
the spectrum of the Lyman and Werner absorption bands can be chosen; we performed the studies at XUV-wavelengths at which resonanceenhanced third harmonic (THG) in krypton can be utilized to deliver
abundant amounts of XUV radiation. Such resonances are found to exist at λ = 94.6, 95.1, 95.25, and 96.3 nm [9]. At the wavelength positions
1 +
of these THG resonances the low rotational states in the B1 Σ+
u - X Σg
(12,0) and C1 Πu - X1 Σ+
g (3,0) bands can be probed. A special experimental feature in the present study is the filtering of the powerful fundamental radiation (used in the third harmonic generation process) from
the XUV-harmonic by a method of spatially-selective phase-matching as
discussed in Ref. [115]. This is done to avoid non-resonant ionization of
the excited H2 molecule by the powerful UV pulses from the first laser.
In a second excitation step a tunable ultraviolet (UV) laser pulse in
the range 310 − 355 nm, obtained from a frequency-doubled pulsed dye
laser, is employed to transfer the population of the intermediate states
to the energetic region, where the heavy Rydberg states are expected.
Pulses from both lasers are spatially overlapped in a region where they
perpendicularly intersect a pulsed and skimmed molecular beam of pure
96
5. Characterization of H+ H− heavy Rydberg states
Figure 5.1: Experimental setup. An XUV laser beam, generated via third
harmonic generation in a jet of krypton, intersects a molecular beam of H2 .
Counter-propagating is a second UV laser beam, further exciting the H2 to
+
heavy Rydberg resonances. Ions (H+
2 and H ) produced after decay are accelerated by extraction field plates and detected after time-of-flight selection.
H2 . Temporal overlap is also required in view of the short lifetimes of the
intermediate states (≈ 0.5 ns). Signal is detected by either monitoring
+ −
+
H+
2 ions generated from immediate decay of the H H resonances or H
ions, produced via dissociation involving H(n = 2) and H(n = 3) products; the latter are subsequently ionized by the UV laser. Time-of-flight
+
mass selection permits distinguishing signals from either H+
2 or H ions.
Pulse extraction takes place after both laser pulses so that excitation of
the heavy Rydberg states is warranted under field-free conditions.
In Fig. 5.2 a spectral recording of a heavy Rydberg series is shown
0
0
when using B1 Σ+
u , v = 12, J = 1 in para-hydrogen as an intermediate state. The spectrum is calibrated to a total excitation energy
00
00
(above X1 Σ+
g , v = 0, J = 0 in the neutral H2 molecule) by adding
the calibration of the second tunable UV-laser to the level energies
of the intermediate states. The R(0) line in the B-X(12,0) band is
at 103844.62 cm−1 [116], hence this is also the level energy of the
0
0
B 1 Σ+
u , v = 12, J = 1 intermediate state. This recording clearly shows
97
5. Characterization of H+ H− heavy Rydberg states
135500
135600
135700
135800
+
H2 (6,4)
10
+
+
H2 (6,2)
5
15
HR(n)
150
140
160
+
H2 signal (arb. units)
H2 (6,0)
10
5
0
134400
134800
135200
Excitation Energy, cm
135600
-1
Figure 5.2: Lower panel: Characteristic spectrum after two-step laser excitation via B-X (12,0) R(0) in the para-H2 molecule. The markers and dotted
lines show the predicted positions of the heavy Rydberg series between n = 134
and n = 161. In the range between n = 154 − 161 the series is interrupted. In
the upper panel, the part of the spectrum between 135 465 and 135 840 cm−1
is enlarged, showing in detail the corresponding resonances for the electronic
+
Rydberg series converging to the H+
= 6, N + = 0, 2 and 4) states; these
2 (v
states are denoted with open triangles, full triangles and open circles, respectively.
the series of resonances that can be assigned to heavy Rydberg states
of principal quantum numbers n = 135 − 154, indicated in Figure 5.2.
In the region above 135 500 cm−1 this series is interrupted and instead
electronic Rydberg series appear that converge to v + = 6, N + = 0, 2
and 4 in H+
2.
Fig. 5.3 shows a second spectral recording of the heavy Rydberg
series in para-H2 , now using the R(0) line in the C-X(3,0) band in the
first excitation step. Hence the C1 Πu , v 0 = 3, J 0 = 1 level of (-) or
(e) parity at 105660.80 cm−1 above the H2 ground state acts as the
intermediate state. This spectrum clearly shows resonances that can
be assigned as heavy Rydberg states with principal quantum numbers
n = 162 − 191. In this spectrum no signatures of electronic Rydberg
98
5. Characterization of H+ H− heavy Rydberg states
170
180
190
|
|
|
+
H2 signal (arb. units)
160
135800
136000
136200
136400
136600
Excitation Energy, cm
136800
137000
-1
Figure 5.3: Spectrum obtained in two-step laser excitation via C-X (3,0) R(0)
in the para-H2 molecule. The markers and connected vertical lines show the
predicted positions of the heavy Rydberg series between n = 162 and n = 192.
series are discernable, although electronic Rydberg series converging to
v + = 7 are expected in this energy range.
In addition to the excitation spectra displayed in Figs. 5.2 and 5.3,
both obtained in para-hydrogen, also spectra in ortho-hydrogen were
recorded. In such spectra, located in the energy range 135650 − 135880
cm−1 , using the P(1) line in the B-X (12,0) band of H2 , heavy Rydberg
series are observed as well; a recording is shown in Fig. 3 of Ref. [114]. In
those observations excitation of the heavy Rydberg series competes with
excitation of electronic Rydberg series converging to the v + = 6, N + = 1
and 3 limits in the H+
2 ion.
An important experimental result when studying Rydberg series is
a determination of quantum defects. Rearranging Eq. (5.1) yields an
expression for an effective quantum defect:
√
Rh
δn = n −
(5.2)
EIP − En
where En is an observed line position and n is the (integer) quantum
number assigned to that particular resonance. Determination of quantum defects can be accomplished by fitting of the spectral line shapes.
While in principle the quantum defect can be larger than 1 in a system
99
5. Characterization of H+ H− heavy Rydberg states
n
140
160
180
200
220
240
(a)
effective difference d(n)
0.4
0
-0.4
(b)
0.4
0
-0.4
via J=1, C(v=3)
via J=1, B(v=12)
140
160
180
n
200
220
240
Figure 5.4: (a) Calculated quantum defects δn , from the complex resonance
model, assuming the interlopers are the Rydberg states (n = 5, 6, 7) converging
+
+
to H+
= 0 (marked with full lines), N + = 2 (marked with dashed
2 (v = 9) N
+
lines) and N = 4 (marked with dotted lines). The electronic Rydberg states
are assumed to have zero quantum defect. (b) Observed quantum defects δn
obtained from spectra of para-hydrogen, using two different intermediate states:
0
0
1
0
0
B1 Σ+
u , v = 12, J = 1 and C Πu , v = 3, J = 1.
with an extended core (as for the s-series in alkali atoms), here only the
fractional part of the quantum defect δn is deduced. Problems in determining the values for δn lie in the treatment of the structured underlying
continuum that gives rise to background slopes, in the moderate signalto-noise ratio, and in the occurrence of asymmetric Fano-type line profiles. These phenomena lead to uncertainties in the determination of the
quantum defects. Analysis of the spectra obtained for para-hydrogen,
for which long series of heavy Rydberg states are observed, results in
values of quantum defects as plotted in the lower panel of Fig. 5.4.
Another feature associated with the heavy Rydberg resonances is
their width as observed in the spectral recordings. In many cases asymmetric line shapes were observed, and in other cases the underlying
100
5. Characterization of H+ H− heavy Rydberg states
via J’=1, B(v’=12)
via J’=1, C(v’=3)
30
-1
Line width (cm )
40
20
10
0
140
160
180
n
200
220
240
Figure 5.5: Linewidths Γ for some of the heavy Rydberg resonances observed
with good signal-to-noise ratio and symmetric line shapes as a function of principal quantum number n. The central (full) line represents a fit to a functional
form Γ ∝ n−3 with the outer (dashed) curves representing the 1σ uncertainty.
continuum was very structured, making it difficult to derive a reliable
value for the resonance width. For those transitions observed at reasonable signal-to-noise ratio and with a symmetric line profile the widths Γ
were determined and plotted in Fig. 5.5.
The spectra displayed in Figs. 5.2 and 5.3 are recorded by monitoring
+
H2 signal. Indeed, autoionization is the most significant decay channel,
forming H+
2 ions in all quantum states below the total excitation en+
ergy, i.e, v = 0 − 5. Detection of H+
2 is a means of recording the heavy
Rydberg states, although in the present study we cannot determine the
+
+
final ionic states H+
2 (v , N ). A competing channel (difficult to quantify but we estimate several 10 %) is photodissociation in which the
heavy Rydberg states decay into H(n=1) + H(n=2) and into H(n=1) +
H(n=3). The second UV-laser exciting the heavy Rydberg states further
ionizes both H(n=2) and H(n=3), producing H+ that can be detected
separately from H+
2 in the time-of-flight setup. To further distinguish
and characterize the dissociation channels the ion particle detector was
replaced by a velocity map ion-imaging setup [117] to investigate if both
dissociation pathways indeed occur. A result recorded upon excitation
101
5. Characterization of H+ H− heavy Rydberg states
of the n = 191 heavy Rydberg resonance at 136950 cm−1 excited via
the C1 Πu , v 0 = 3 intermediate state is shown in Fig. 5.6. This experiment indeed proves that both H(n=2) and H(n=3) are produced. The
ring structures can be unambiguously assigned to the kinetic energies
required for these channels. In the outer ring H(n = 2) atoms are detected that have higher kinetic energy from the dissociation process. In
the inner ring the H(n = 3) atoms, that have higher internal energy and
lower kinetic energy, are probed. No distinction is made between 2s and
2p atoms, or between 3s, 3p and 3d hydrogen atoms. The angular distributions can be interpreted to gain further understanding of the decay
mechanism; this is outside the scope of the present work, but will be
pursued in the future [118].
5.3
Analysis
Interpretation of observed features
In the spectra two distinct phenomena are apparent. First, series of
electronic Rydberg series are observed. From an intermediate state with
1
a 2p-electron, 2pσ in case of the B1 Σ+
u state and 2pπ in case of C Πu ,
ns and nd-series can be expected. Due to selection rules Rydberg series
in ortho-H2 can converge upon N + = 1 and 3, and series in para-H2
upon N + = 0, 2 and 4 for each v + vibrational quantum number [91].
The (v + , N + ) ionization limits can be calculated from the value of the
ionization energy of H2 (124417.48 cm−1 [119]) and the level energies in
the H+
2 ion [120].
0
0
In the spectrum of Fig. 5.2, excited via the B1 Σ+
u , v = 12, J = 1
intermediate state in para-H2 , three series are observed, converging to
v + = 6, N + = 0, 2 and 4 states in the H+
2 ion. These states show
the typical behavior of series interaction as often observed in electronic
Rydberg series in the H2 [91, 121] and HD [122] molecules. One of the
visible effects of interaction of a series with continua is that the Rydberg
states converging upon the v + = 6, N + = 4 limit clearly appear as
window resonances in Fig. 5.2. It is noted that it requires an nd Rydberg
series converging upon N + = 4 to be observed in excitation from an
intermediate state of total angular momentum J 0 = 1; selection rules
do not permit excitation of an ns series converging to N + = 4. Similar
102
5. Characterization of H+ H− heavy Rydberg states
Figure 5.6: Ion image representation of the kinetic energy distribution upon
excitation of a heavy Rydberg resonance. The XUV beam passes from left to
right, the second UV laser beam from right to left; polarizations are linear and
parallel and in the plane of the paper. The outer ring corresponds to H+ ions
of high kinetic energy in decay to a H(n = 2) fragment, whereas the inner ring
represents decay to H(n = 3). The off-center dot represents detection of H+
2
ions; the offset from the center is related to the fact that H+
2 are accelerated
to lower velocities in a typical velocity-map-imaging configuration than H+ .
Therefore they travel a longer time along the direction of the molecular beam,
which is from up to down in the figure. Note that we present raw data here,
which are not 2D → 3D inverted.
0
electronic Rydberg series were observed in excitation via the B1 Σ+
u,v =
12, J 0 = 0 state in ortho-H2 . Selection rules impose that only Rydberg
states converging to v + = 6, N + = 1 and 3 should be observable,
which is indeed the case. Again, qualitatively some features of series
interaction are visible in the spectrum of Fig. 3 in Ref. [114]. In the
Rydberg series converging to v + = 6, N + = 3 the Fano q-parameter is
found to vary as a function of the principal quantum number, indicative
of an interaction between multiple channels. However, the observations
on electronic Rydberg states are not the predominant issue in the present
study, and are left for future analysis.
103
5. Characterization of H+ H− heavy Rydberg states
The second and key feature in the spectra is the observed series of
broad resonances that follows Eq. (5.1), the heavy Rydberg series, as
indicated in the spectra of Fig. 5.2 and 5.3 for para-H2 , and in Fig. 3
of Ref. [114] for ortho-H2 . They represent the first spectroscopic observation (i.e. frequency-domain resolved resonances) of heavy Rydberg
states in a molecular system. The assignment in terms of heavy Rydberg states is unambiguous, since the spectral features follow the representation by Eq. (5.1) without invoking adjustable parameters. As
discussed by Reinhold and Ubachs [12] the crucial parameters can be
calculated in elementary fashion. The Rydberg constant Rh for the
H+ H− heavy Bohr atom can be obtained by replacing the electron by
an H− particle, yielding the scaling factor (µ/me ) = 918.5761. Hence,
via Rh = (µ/me )R∞ one obtains: Rh = 1.00802 × 108 cm−1 . The other
parameter in Eq. (5.1), the value for the ion-pair dissociation limit EIP ,
can be determined from EIP = IE(H2 ) + D0 (H+
2 ) − EA(H), yielding
−1
EIP = 139 713.83 cm . Based on these parameters, and choosing zero
quantum defects (δn = 0), the positions HR(n) of the heavy Rydberg
states in Figs. 5.2 and 5.3 can be calculated in a straightforward manner.
The positions of zero quantum defects are indicated in the figures with
dotted lines to guide the eye. The observed resonances clearly follow
the calculated regular structure of the heavy Rydberg states, making
the assignment unambiguous.
On quantum defects
There are some striking observations in the spectra, which require further discussion and explanation. Here we address three: the fact that
the heavy Rydberg states are only observed for principal quantum numbers n > 100, the issue of angular momentum quantum numbers and
selection rules, and the question as to where the oscillator strength for
exciting heavy Rydberg states originates.
Effective quantum defects for the heavy Rydberg series are derived
and plotted in Fig. 5.4. The Bohr radius in a two-particle quantum
system is defined by [12]:
a0 =
104
4π0 ~2
µe2
(5.3)
5. Characterization of H+ H− heavy Rydberg states
+
H2 (8,0)
140000
H2
+
200
130
+
H2 (5,0)
150
+
−
HH
H(n=4)
H(n=3)
120000
−1
Energy (cm )
H(n=2)
100000
C
~
~
~
~
B
20000
X
0
1
10
100
Internuclear distance (a0)
1000
Figure 5.7: Potential-energy diagram of the H2 molecule with distinction
between two separated regimes: for R < 12a0 the range of internuclear distances where covalently bound states and the electronic Rydberg states con+
+
verging upon H+
2 (v , N ) dominate (dashed lines representing the potential
curves of gerade symmetry in this region [102, 94, 95, 97, 96]), and the range
R > 12a0 where H+ H− heavy Rydberg states exist. The H+ H− Coulomb potential is extended to smaller R by dots. Intermediate energy levels in the B
and C states (at ∼ 105 000 cm−1 excitation) are indicated and their accessible
Franck-Condon ranges mapped (boxes).
with µ the reduced mass of the heavy Bohr system (918.5761 me ).
Hence, the Bohr radius ah0 of the heavy Bohr system (H+ H− ) becomes as
small as 5.76 × 10−14 m. This is much smaller than the extension of the
composite H− particle [123]; consequently, the model of heavy Rydberg
states breaks down at small internuclear separation, or at small principal quantum numbers n. The potential energy diagram (Fig. 5.7) shows
how the H+ H− Coulombic 1/R potential is affected by perturbations of
covalent states even at large internuclear separation. Near the H(n = 2)
dissociation threshold there is a large interaction, with an avoided crossing of some 3000 cm−1 at ≈ 10a0 . If the H+ H− potential can be considered pure for R > 12a0 , this means R = n2 ah0 ≈ 12a0 ≈ 6 × 10−10 m,
105
5. Characterization of H+ H− heavy Rydberg states
and the description in terms of heavy Rydberg states may be considered
valid only above n ≈ 100. It is noted that perturbations with covalent
states exist at internuclear separations of 36a0 and at some 250a0 , where
the H(n = 3) and (n = 4) dissociation curves cross [95, 97, 96]. These
perturbations are much weaker, but should be included in more refined
models of H+ H− heavy Rydberg states. The above considerations imply
that quantum defects in the heavy Bohr system are large, much larger
than in the s-series of alkali atomic systems. Pan and Mies [92] have
calculated such quantum defects for the Li+ I− ionically bound system
from first principles, and deduced that the lowest 177 angular momentum states are removed from the purely Coulombic spectrum. Previously Asaro and Dalgarno had found a number of 128 excluded levels in
the Li+ F− system [124]. In the experimental analysis we will deal, as
usual, with the fractional part of the quantum defects δn .
In the work of Pan and Mies [92] a relation was also established
between sets of quantum numbers as commonly used for Rydberg states
(n, `, m) and those usually invoked to describe molecular motion (v, J,
M ). The orbital angular momentum is either described as ` equivalent
to J, with projections of a magnetic quantum number m or M . A
connection for the principal quantum number can be established through
the relation n ↔ v + J + 1. Hence, the equation for the Rydberg energy
levels can be written as [92]:
Ev,J = EIP −
Rh
(v + J + 1 − δJ )2
(5.4)
where the J-dependence of the quantum defect is explicitly written. In
most of the spectra reported here, H2 is excited in its para-configuration
via a R(0) transition in the first step. Hence, from the intermediate
J 0 = 1 state, following the ∆` = ∆J = ±1 selection rule, in principle
two heavy Rydberg series can be excited, with J = 0 and J = 2 angular
momenta. These series may possess different quantum defects δ0 and
δ2 , since low angular momentum states are known to exhibit strongly
varying quantum defects; an example is the strong difference between
the s and p electronic Rydberg series in the Na atom. In our experiments
we only observe a single heavy Rydberg series for para-H2 . Clearly, this
requires an explanation.
Notably, Pan and Mies [92] have shown in a theoretical analysis of
the Li+ I− heavy Rydberg system, that for the low angular momentum
106
5. Characterization of H+ H− heavy Rydberg states
states the quantum defects δJ can vary as by much as 1 for each step in J.
If such a situation would also prevail in the para-H+ H− system, the two
heavy Rydberg series that are expected would approximately coincide.
The same would be true if both quantum defects δ0 and δ2 would differ
by zero or any integer number. Without any knowledge about quantum
defects, an explanation for the observation of only a single series must
remain tentative. In ortho-H2 , when using the P(1) transition in the
00
00
0
first step to excite the molecule from X1 Σ+
g , v = 0, J = 1 to a J = 0
intermediate state, only a single heavy Rydberg state with J = 1 is
expected and observed.
It is important to note that the heavy Rydberg series in both paraand ortho-H2 possess the same ion-pair dissociation limit. This is not
necessarily true for every heavy Rydberg series, but is a consequence
of the fact that both the H+ and H− fragments possess no internal
electronic structure. In addition, the distinction between para- and ortho-H2 remains unaffected even if the nuclear spins have an internuclear
separation of ≈ 12a0 or more (several hundred a0 in case of n = 200).
This once more underlines the fact that it is not the distance-dependent
interaction between the nuclear spins, but only the different symmetries
of the nuclear spin wave functions that cause the ortho-para distinction. Another point of interest concerns the fact that situations in which
both electrons are positioned on either nucleus A or B should be indistinguishable. If we define the corresponding ’localized’ wave functions
as A and B, the true degenerate solutions of the Schrödinger equation
√
are 1/ 2(ΨA ± ΨB ). However, even very small stray electric fields will
lift this degeneracy due to the large polarizability of heavy Rydberg
states [12].
Intensities via channel interaction: complex resonance
The mechanism for exciting molecular ion-pair states has been discussed
in the literature [101, 107] and Franck-Condon arguments play a role.
High-n states in H+ H− cannot be directly excited from the intermediate
0
1
0
states B1 Σ+
u , v = 12 and C Πu , v = 3, because their wave function density is confined to R < 6a0 and R < 3a0 , respectively. The internuclear
distances at which the intermediate states exhibit wave function density is mapped onto the energy region of the heavy Rydberg states with
107
5. Characterization of H+ H− heavy Rydberg states
color-(grey) shaded rectangles in Fig. 5.7. Intensities In for excitation of
heavy Rydberg states, based on Franck-Condon factors, should follow:
∫
In = | Ψ∗n (R)ΨB,C (R) dR|2
(5.5)
Similar to usual Rydberg wave functions of high principal quantum number, heavy Rydberg states exhibit a wave function with their density entirely located at large separation from the core, in this case the H+ H−
separation. In the language of molecular structure, the wave function is
concentrated at the outer turning point of the H+ H− potential. Inspection of Fig. 5.7 indicates that this is at R ≈ 50−100a0 . This implies that
direct excitation of the heavy Rydberg states should be Franck-Condon
forbidden. In order to provide an explanation for their excitation, the
model of a complex resonance is invoked.
The observation of Rydberg series to which a direct transition is
highly improbable has been reported for the atomic case [125]. For the
molecular case this phenomenon was observed and analyzed for the case
of H2 . Jungen and Raoult [98] developed a model of channel interaction
that explains how a Rydberg series with zero transition dipole moment
can borrow effective oscillator strength, which they baptized as a complex resonance. Later this phenomenon was modeled in basic form by
Giusti-Suzor and Lefebvre-Brion [126]. Before givivg details we consider
if simpler models might explain the observed features; interaction between a heavy Rydberg series (without oscillator strength) with a broad
continuum (which exhibits oscillator strength) would in a Fano-type
analysis lead to window resonances, contrary to what is observed.
The complex resonance model involves (at least) three channels: one
open channel represented by an ionization continuum, and two closed
channels, viz. an interloper (i.e. a strong resonance, excited with a large
transition dipole moment) and a (Rydberg) series of close-lying states.
Couplings are assumed between each of the closed channels with the
continuum, and between both closed channels. From this simple model
it is shown that an interference takes place between on the one hand the
direct interaction between both closed channels, and on the other hand
the indirect interaction between the closed channels via the continuum.
This interference effect can enhance or diminish the intensities of the
observed transitions to both interloper and members of the Rydberg
series, and also affects line positions, line shapes and widths. In general,
108
5. Characterization of H+ H− heavy Rydberg states
+
+
+
+
+
+
+
v =9, N =4
v =9, N =2
v =9, N =0
channel 5
channel 4
channel 3
H H
-
continuum
channel 1
channel 2
R25
R12
R24
R23
H2
+
R13
R15
R14
intermediate
state
Figure 5.8: Interaction diagram for the model of a complex resonance in the
case of the H+ H− series. The interloper states are assumed to be members of
+
+
the electronic Rydberg series converging upon the H+
= 0, 2 and
2 (v = 9, N
4) states of the ion (for the case of para-hydrogen; in case of ortho-hydrogen
N + = 1 and 3). The upward pointing arrows represent transition dipole moments giving rise to direct excitation, while the sideways pointing arrows refer
to channel interactions. Note that in this 5-channel model the electronic Rydberg series converging to v + = 6, as observed in the spectrum of Fig. 5.2 are
excluded; this is one of the many assumptions made.
an essential feature of the model is that any channel i, for which the
transition dipole moment Di is zero, can obtain an effective oscillator
strength σi due to interference, such that it can be excited and observed.
In our case, the simple model of three interacting channels as presented in Ref. [126] is not exactly valid, but can easily be extended to
a situation that involves more than one interloper. In a five-channel
description, graphically displayed in Fig. 5.8, we assume that the closed
bound H+ H− heavy Rydberg series (channel 2) possesses a zero transition dipole moment (D2 = 0). A further simplification of this model is
that the electronic Rydberg series as observed in Figs. 5.2 are left out.
As far as interlopers are concerned, they are assumed to be low-n
members of electronic Rydberg series that converge upon high v + levels
in the H+
2 ion. From the work of Chupka et al. [85], McCulloh and
Walker [109], Pratt and coworkers [110, 111], and Kung et al. [112] a
109
5. Characterization of H+ H− heavy Rydberg states
+ = 9)
strong coupling between the H+ H− continuum and the H+
2 (v
electronic Rydberg series is apparent. Therefore we hypothesize that
our interloper states are members of this particular v + = 9 Rydberg
series. Taking the oscillator strength to be continuous across the ionpair dissociation limit, we assume that this coupling persists below the
threshold.
Following angular momentum selection rules, from a p intermediate
state ns and nd electronic Rydberg states can be excited. In view of
the fact that p − nd transition dipole moments are larger than those for
p − ns, we limit ourselves to nd states. In a Hund’s case (d) coupling
scheme for para-hydrogen nd electronic Rydberg series can converge
upon three different limits (N + = 0, 2 and 4) of the ionic ground state.
Low-n members of these series will act as interloper states and will be
denoted by the closed channels 3, 4 and 5 in Fig. 5.8, respectively.
As for the open continuum channel, at an excitation energy of 137 000
−1
cm the system is above the v + = 5 level of the ion. Hence, all the
electronic Rydberg states converging upon the v + = 0 − 5 levels can
contribute via autoionization to a background continuum signal of H+
2.
We consider only a single continuum as channel 1 in our model. This is
another crude approximation of the true situation at hand.
Following a multi-channel quantum defect theory (MQDT) approach
similar to Ref. [126], we introduce a set of five adjusted channel wave
functions, including an open channel (the interacting continuum, ϕ1 )
and four closed channels: the H+ H− series (ϕ2 ) and three electronic
Rydberg (interloper) series converging to v + = 9, N + = 0, 2 and 4 (ϕ3 ,
ϕ4 and ϕ5 respectively). The total wave function of the system can be
written as:
ψ=
5
∑
Zi cos [π (νi + µi )] ϕi
(5.6)
i=1
with Zi the amplitudes, ν1 = −τ the open channel phase, and for i = 2
(
ν2 =
Rh
EIP − E
)1/2
(5.7)
the phase of the H+ H− series, where Rh is the Rydberg constant for the
heavy Bohr atom (100802075 cm−1 ) and EIP is the ion-pair dissociation
110
5. Characterization of H+ H− heavy Rydberg states
limit (139713.83 cm−1 ) [12]. For the three interloper channels i = 3, 4, 5:
(
νi =
Re
i
Elim − E
)1/2
(5.8)
and Re is the Rydberg constant for the electronic Rydberg system
i
Re = µ(H2+ + e− )/me (109704.3945 cm−1 ), and Elim
is the ionization
3 = 139831.09
limit of channel i, calculated using the values of [120] (Elim
4 = 139930.14 cm−1 , E 5 = 140157.49 cm−1 ). The effective
cm−1 , Elim
lim
quantum defects µi have been all set to zero. Note that in this 5-channel
model we assume that the three interloper states do not mutually interact, and the observed electronic Rydberg series converging to v + = 6
are not included.
Similar as in Ref. [126], where a calculation was performed for the
quantum defects of an electronic Rydberg series interacting with an
interloper, we derive for the quantum defects of the heavy Rydberg
series (channel 2):
2
2
2
tan(x2 ) = R23
cot(x3 ) + R24
cot(x4 ) + R25
cot(x5 )
(5.9)
where xi = π(νi + µi ) is the notation for the quantum defects and
Rij are the R-matrix elements representing the channel interactions.
This is an important result: the resonance peaks will be found at the
energies where this equation is fulfilled. The strength of the interaction
between the interlopers and the H+ H− series (R23 , R24 and R25 in our
notation), as well as the exact location of the interloper states greatly
affect the position of the observed H+ H− lines. This will be reflected in
the values for the quantum defects δn of the H+ H− series. These values
are calculated, assuming zero quantum defects (µi ) for the interloper
+
+
states converging to H+
2 (v = 9, N = 0, 2 and 4).
We have attempted to reconstruct the findings on the experimental quantum defects from this model; the small amount of experimental
information and the large error bars on the one hand, and the large
number of parameters on the other hand do not allow for a fit or a
one-to-one correspondence of the model. However, by scanning the pa2 = 0.01π, R2 = 0.01π and R2 = 0.03π
rameter space the solution of R23
24
25
somewhat mimics the observed trend in the observed quantum defects,
as is displayed in Fig. 5.4. This merely serves as an illustration on how
the complex resonance model might explain trends in observed quantum
111
5. Characterization of H+ H− heavy Rydberg states
defects of the heavy Rydberg series. By comparing Figs. 5.4 and 5.5 it
may be verified that at the locations of the interactions also resonances
in the observed lifetimes seem to occur.
In view of all the assumptions made and the limited amount of spectroscopic information, we cannot provide a conclusive model to quantitatively reproduce the entire spectrum of heavy Rydberg resonances,
and the electronic Rydberg series observed in the same energy region.
We have presented a general framework that gives some insight in the
variation of the quantum defects as in Fig. 5.4. Furthermore from the
generalized model of Ref. [126] several conclusions relevant for our purpose can be drawn. Whereas in the previous investigations of complex
resonances [98, 126] the model was applied to electronic Rydberg series, here it describes properties of heavy Rydberg series: (i) Even if
the members of a Rydberg series cannot be optically excited directly,
coupling to an interloper and a continuum can still lead to their observation; (ii) The spectral range covered by a complex resonance can
be much larger than the interloper width; (iii) The interaction strength
between the Rydberg series and the interloper is of great importance,
since it affects the widths and positions of the peak maxima of the series;
(iv) Decay of the Rydberg resonances can occur into various channels
including the continua. The latter is the subject of the next section.
Resonance widths and decay
The heavy Rydberg series are observed as short-lived resonances having typical widths of some 5 − 25 cm−1 , corresponding to a lifetime of
a picosecond or even a fraction thereof. The observed linewidths are
displayed in Fig. 5.5 as a function of principal quantum number. The
uncertainties in the data points are large, due to limited signal-to-noise
ratio and asymmetries in the line shapes, but nevertheless some interesting conclusions can be drawn from these observations.
Lifetimes of Rydberg states are known to scale as τ ∝ n3 [82]. This
is reflected in the observed scaling of the linewidths of Fig. 5.5 as n−3 .
Notwithstanding the large error margins the data in Fig. 5.5 somewhat
follow the expected scaling law. In addition there is at least indication
of an increased linewidth near n = 163. This resonance may hint at
the location of the effective interloper. If this is true the interferences
112
5. Characterization of H+ H− heavy Rydberg states
in the multichannel problem give rise to enhanced widths, where in
principle the complex resonance model allows for both the possibility of
enhancement or reduction of the linewidth.
At n ≈ 200 a linewidth of Γ(200) = 10 cm−1 is observed. Following the usual n3 dependence for Rydberg states, this would scale
to Γ(2000) = 0.01 cm−1 , which would correspond to a lifetime of 0.5
ns. However, in the studies on heavy Rydberg wave packets [12] for
n = 2000 a lifetime of 90 ns was actually observed, deviating by over
two orders of magnitude from the n−3 -scaling based on lower n-values.
In addition, for these higher-lying heavy Rydberg states decay into the
H(n = 4) dissociation channel was observed [12]; this was accomplished
by selectively ionizing the H(n = 4) fragments by laser pulses of 1064
nm. The presently observed heavy Rydberg states of n < 230 are below
the H(n = 4) dissociation threshold and therefore this decay channel is
closed. Nevertheless, the lifetimes in the range n = 130 − 230 are much
shorter than those in the higher energy region probed in the wave packet
experiments.
This discrepancy may be resolved by considering the fact that the
wave packet experiments were conducted in electric fields [87, 12]. As is
extensively discussed in the literature on zero-kinetic energy (ZEKE)
spectroscopy [127, 128] high-n Rydberg states are known to exhibit
longer lifetimes in the presence of electric fields. Heavy Rydberg states
have a large polarizability; even in small stray electric fields of < 0.1
V/cm at high n values full `-mixing (or rather J mixing) occurs [12]. In
the oscillatory motion of the angular wave packets population is transferred to high angular momentum states of the heavy Rydberg system.
This gives rise to longer lifetimes than expected from n3 scaling. Indeed,
as shown in Fig. 11 of Ref. [12], lifetimes were found to scale as n4 , a
clear indication of angular momentum mixing in the electric field.
Heavy Rydberg states in other systems
Heavy Rydberg states are expected to exist in other molecular systems
and a viable question is whether the presently observed rather short
lifetimes are typical for heavy Rydberg states, or whether they depend
on the specifics of the level structure and associated decay channels
of the H2 molecule. In H2 heavy Rydberg states can be observed in
113
5. Characterization of H+ H− heavy Rydberg states
the range where vibrational autoionization occurs (into v + = 0 − 5
vibrational levels) and dissociation into channels with H(n = 2) and
H(n = 3); this may seem an unfortunate situation giving rise to strong
decay and broad resonances. Experimentally H2 is the only system
where frequency-resolved heavy Rydberg series are observed. During the
course of the investigations it was attempted to register similar heavy
Rydberg series in D2 , however unsuccessfully: only electronic Rydberg
+
series were observed in the D+
2 and D signal channels. This absence
might be attributed to the lack of a suitably located interloper state in
D2 .
Stark wave packets were observed in both H+ H− and in the H+ F−
system. In the latter system lifetimes of typically an order of magnitude
larger were found [113] in the high n region where `-mixing is important.
Hence, the H+ F− system might be a candidate for observing frequency
resolved heavy Rydberg series. We note that the Rydberg constant for
heavy Rydberg series scales with µ/me , where µ is the reduced mass of
the molecule. Therefore, for all systems heavier than H2 the density of
states for heavy Rydberg states increases, and it will be more difficult
to observe them, in particular if the lifetimes are as short as in H+ H− .
Further it is noted that in poly-atomic systems there is a multitude
of series limits for heavy Rydberg states. Each bound quantum state
(rovibrational, spin-orbit) in the positively or negatively charged ionpair fragments provides such a limit. This will further enhance the
density of heavy Rydberg states. Finally we note that the pp̄ protonium
system, consisting of a proton electromagnetically bound to an antiproton should exhibit a series of quantum states of the same density
and spacing as in the H+ H− system, since the Rydberg constants of
those systems are similar. Experiments for detecting and investigating
protonium are under way [99, 129].
5.4
Conclusion
The first spectroscopic observation of frequency-resolved heavy Rydberg
states in a molecular system is reported. Bound quantum states in the
1/R potential in H+ H− are detected for principal quantum numbers in
the range 140 − 230, as predicted from a generalized Rydberg formula,
which is mass-scaled for the orbital motion of the H− particle replacing
114
5. Characterization of H+ H− heavy Rydberg states
the electron. The resonances decay via autoionization forming H+
2 ions
as well as via dissociation forming H(n = 3) and H(n = 2) products.
Lifetimes scale with n3 and are typically shorter than 1 ps for the quantum states investigated (n = 140 − 230). A complex resonance model
is invoked, reproducing the observed quantum defects, and explaining
the origin of the effective oscillator strengths of the heavy Rydberg series. In H+ H− the series could be observed in excitation from states
1
of 1 Σ+
u and of Πu symmetry, and both in para- and ortho-hydrogen.
Such heavy Rydberg resonances are hypothesized to exist in all A+ B−
diatomic molecules, as well as in polyatomic molecules.
115
Chapter 6
Elemental analysis of steel scrap
metals and minerals by
laser-induced breakdown
spectroscopy
The atomic emission of laser-induced plasma on steel samples
has been studied for quantitative elemental analysis. The plasma
has been created with 8 ns wide pulses using the second-harmonic
from a Q-switched Nd:YAG laser, in air at atmospheric pressure.
The plasma emission is detected with temporal resolution, using
an Echelle spectrometer of wide spectral range (300-900 nm) combined with an intensified charge coupled device camera. A plasma
temperature of 7800 ± 400 K is determined using the Boltzmann
plot method, from spectra obtained under optimized experimental
conditions.
As an example of an industrial application the concentration of
copper in scrap metals is studied, which is an important factor to
determine the quality of the samples to recycle. Cu concentrations
down to 200 ppm can be detected. Another application of the
laser-induced plasma spectroscopy method is the measurement of
the nickel and copper concentrations in an iron-containing sample
of reduced magma from the 1870s expedition to western Greenland
by Adolf Erik Nordenskiöld. Different spectral lines of nickel are
used for calibration, and their results are compared.
6. LIBS on scrap metals and minerals
6.1
Introduction
Laser-induced breakdown spectroscopy (LIBS) is a powerful analysis
method which has undergone a rapid development during the past years.
The power of LIBS lies in its non-intrusive character, permitting on-line
analyses to be made in harsh or otherwise inaccessible environments
[130],[131], [132], and [133].
The range of applications that utilizes LIBS is growing rapidly,
driven by the materials processing industries [134] and [135] but also by
the environmental monitoring fields [136], [137], and [138] The technique
can be used in a variety of more complex analyses such as determination of alloy composition, origin of manufacture (by monitoring trace
components), and molecular analysis (unknown identification). There
is also a wide range of applications for in vivo medical investigations
[139], analysis of biological and pharmaceutical samples [140], archaeological investigations [141], art restoration and paint pigment analysis
etc. There is also a possibility of using LIBS as an analytical technique
for corrosive or hazardous environments (such as space and nuclear reactors) preventing risk to the operator.
Although the sensitivity of the LIBS method cannot yet compete
with some other analysis methods, e.g. inductively coupled plasmamass spectrometry, the rapid development of dedicated LIBS systems is
constantly narrowing this gap, with sub-ppm levels as a clearly realistic
goal.
There are several kinds of LIBS-systems on the market. The ones
that can combine rapidity with high resolution make use of large spectrometers. These systems are already now capable of few-ppm sensitivity
for some elements [142], particularly in combination with double-pulse
excitation [143]. The portable ICCD-based systems, as the one used in
the present work [144], are in general somewhat less sensitive with lower
resolution. A portable system has been compared with a laboratory system by Wainner et al. [138]. They found that their laboratory system
was only about factor of 2 more sensitive than the portable one.
In the present work, we are focusing on the use of LIBS as an on-line
analysis tool in metal recycling industry, where there is an urgent need
to get rid of traces of certain elements in order to obtain better raw
materials for steel plants. As another application, we investigate the
118
6. LIBS on scrap metals and minerals
possibilities of LIBS in detailed spatial analysis of mineral surfaces, in
this case a lava sample found on Greenland for more than 100 years ago.
The potential of LIBS in the mining industry as a field tool for on-site
characterization of ore quality
constitutes yet another important use of LIBS in the future. We are
carrying out these studies as a preparation for the construction of a pilot
plant where the techniques can be applied in realistic environments, as
has been described by Stepputat et al.[145] . The first application is
being carried out in collaboration with Stena Metall AB, Sweden [146].
The aim is to identify scrap metal samples with small concentrations
of chromium and copper, for subsequent removal. The concentrations
studied in this work range from about 0.04% to 1.6% (Cr) and 0.09% to
about 0.7% (Cu).
The other application, which also shows the power of the LIBS
method, is the study of a lava sample. The lava sample originates
from the island of Disko in Greenland and is now in the possession
of the Swedish Museum of Natural History. During one of his expeditions to west Greenland in the 1870s, Adolf Erik Nordenskiöld collected
the frozen magma sample with the belief that the metallic piece was
a meteorite. Unfortunately for A.E. Nordenskiöld, the nickel contents
reveal the sample’s origin as terrestrial. Metallic iron on earth possesses
a nickel concentration in the range 1% to 4%, while meteorites contain a much higher nickel concentration (5-16%) [147]. Using the LIBS
method, the nickel concentration of the piece is studied using several
calibration spectral lines of nickel. The main focus of this study is to
observe if the different calibration lines will lead to similar results. For
this purpose, the nickel concentration as well as the copper content of
the lava sample is studied spatially.
6.2
Experimental
Fig. 6.1 shows the experimental setup schematically. A pulsed Nd:YAG
laser (Lumonics HY500) placed in air at atmospheric pressure creates
the laser-induced plasma. The laser produces 8 ns wide laser pulses
at its second harmonic wavelength (532 nm) of about 10 mJ, with an
adjustable repetition rate up to 10 Hz. The laser beam is focused perpendicular to the sample surface, producing plasma via a plano-convex
119
6. LIBS on scrap metals and minerals
Figure 6.1: Experimental setup used in LIBS experiment.
lens of 50 mm nominal focal length. Placing the focal plane 1 mm below
the sample surface stabilizes plasma intensity and avoids plasma formation in air as the crater deepens [148]. In order to maintain constant
conditions for the sample during different measurements, the sample is
displaced each time to guarantee a surface free of impurities coming
from sputtering of previous laser shots. The samples surfaces had been
etched using a mixture of nitric acid and alcohol at the Swedish Museum
of Natural History. A low laser repetition rate improves furthermore the
data reproducibility as it matches the spectrometer readout speed. A
fiber optic cable collects the emission from the formed plasma and transmits the collected light to the entrance slit of an echelle spectrometer
(Mechelle 7500, Multichannel instruments, Sweden [149]). Echelle grating spectrometers are designed to work at high orders and high angles
of incidence and diffraction. The Mechelle 7500 provides high constant
spectral resolution (7500) over a wide spectral interval (300-900 nm)
displayable in one single spectrum. This allows the acquisition of a
spectrum containing all used spectral lines generated in one single laser
shot. An intensified CCD camera (Andor) coupled to the spectrometer
finally detects the dispersed light.
For the construction of the calibration curves, reference steel samples
having representative contents of copper, nickel and chromium are used
and are summarized in Table 6.1.
120
6. LIBS on scrap metals and minerals
Table 6.1: Reference samples used to create the calibration curves.
Ref. samples
Cu, %
SS-CRM 408/2 0.694
SS-CRM 410/2 0.436
SS-CRM 402/2 0.302
SS-CRM 409/2 0.205
SS-CRM 401/2 0.101
SS-CRM 215/3 0.052
SS-CRM 416/1 0.0091
The mass contents of the
6.3
Ni, %
Cr, %
4.13
0.111
2.07
1.684
0.808 0.652
3.02
1.318
0.019 0.138
0.038 0.040
6.124 14.727
elements are given in wt.%
Optimization of experimental parameters
Experimental parameters such as delay time, gate time or number of
shots to average vary from experimental group or publication [150].
Therefore, special care is taken in this work to find the optimum experimental parameters. Internal normalization reduces the influence of
the sample properties and laser shot-to-shot variation [151]. Fig. 6.2
illustrates the improvement using internal normalization (Fe) on the
standard deviation of the integrated line intensity of the nickel line at
341.48 nm. The relative standard deviation (RSD) diminishes as much
as 50%. Moreover, special care is taken to determine the optimum number of single-shot spectra to be averaged. As a representative example,
Fig. 6.2 shows the behavior of the relative standard deviation of the
341.48 nm Ni integrated line intensity versus the number of averaged
laser shots. Here, the RSD value does not diminish significantly by increasing the number of averaged laser shots to more than 60. Therefore,
the reproducible average spectrum is acquired using the spectra generated in 50 or 60 single shot laser pulses. All spectra are improved by
subtraction of the dark current of the detector.
At early stages of the plasma evolution, the continuum emission
is overwhelming and covers many spectral lines. As the delay time increases, the continuum reduces strongly with a pronounced improvement
of the signal-to-noise ratio. However, the self-absorption (optically thick
conditions) of the lines emitted by the laser induced plasma increases
with the delay time with respect to the laser pulse [152]. The visual
121
6. LIBS on scrap metals and minerals
Figure 6.2: Relative standard deviation of integrated spectral line of nickel
at 341.48 nm. The value of the RSD reduces strongly using the normalization
procedure.
inspection of the spectra revealed that the self-absorption reversal of
the copper line became more pronounced after 3 µs. Thus, the time
interval chosen to perform the measurements is from 2 µs to 3 µs. This
gate time of 1 µs is chosen because the plasma temperature does not
decrease substantially within this time interval leading to changes in the
equilibrium conditions of the plasma.
6.4
Results and discussion
Calibration curves
The spectroscopic lines studied in this work are: in the case of copper
327.4 nm , chromium 427.5 nm (resonance lines intended to study low
concentrations) and 338.05 nm, 341.5 nm 346.16 nm, 351.03 nm, 351.5
nm in the case of nickel [153] and [154]. A plasma temperature of 7800
K is calculated using six low self-absorption iron peaks free of overlap in
the wavelength interval 379 to 443 nm by means of equation [155, 156]:
ln(
122
Ej
Iλ0
hcDN 10−2 x
)=−
+ ln[
]
qj Aij
kT
Z(T )
6. LIBS on scrap metals and minerals
The calculated temperatures in all measurements vary from 7400 to 8200
K, typical values for plasma generated in these experimental conditions
[151], [152] and [157].
As the expected copper and chromium concentrations are very low
(Cu 0.3%, Cr 0.1%), their resonance spectral lines are used to create
the calibration curves [132]. In the case of nickel, the expected concentrations are higher, about 1% to 5%. For these concentrations, the
resonance lines of nickel experience a high level of self-absorption and
they are therefore not the most suitable lines to use. Instead, spectral
lines free of overlap are used to create the calibration curves for nickel.
The experimental spectral lines shapes are mainly determined by the
instrumental line profile.
Calibration curves are the plots of the integrated line intensity versus
the (reference) sample elemental concentration. The integration of the
spectral line intensity is performed using the measured data points. It
results in a similar outcome compared to the integral using for instance
a Voigt function fit of this same spectral line profile. The difference between these two results is ten times smaller than the standard deviation
of the integrated line profile. Thus, the integral of the measured data
points is used for all the calibration curves. To calculate the standard
deviation of the integrated line intensities, 10 measurements are applied
to each reference sample.
Fig. 6.3 shows the spectra obtained for the copper line at 327.39 nm
for different elemental contents. In this case an interfering iron line at
327.44 nm makes the used integration method inaccurate. Under the
assumption that the line profile is symmetric, only the left half of the
measured intensity line is integrated. Fig. 6.3 also shows that the background (baseline) is the same for different concentrations. As a result,
the integration over the measured spectral line without correction for the
baseline value only leads to a constant value added to the integral. This
constant value is the same for all samples with different concentrations.
The variation of the integrated line intensity with the certified sample
elemental concentration is presented in Fig. 6.4, Fig. 6.5, Fig. 6.6 and
Fig. 6.7. In these calibration graphs, the element concentrations are
expressed as a percentage of the weight.
For most of the analyzed lines, the integrated line intensity grows
non-linearly with increasing certified concentration values. Fig. 6.4 and
123
6. LIBS on scrap metals and minerals
Figure 6.3: Spectra of the copper line at 327.39 nm, plotted for different
concentrations.
Figure 6.4: Calibration curve of copper spectral line at 327.39 nm. The
measurements are performed using 1 µs gate time and with 2 µs delay between
measurement and laser firing.
124
6. LIBS on scrap metals and minerals
Figure 6.5: Chromium calibration curve using the spectral line at 427.48 nm.
Figure 6.6: Calibration curves using nickel spectral lines at 338.05 nm and
346.16 nm. The dashed line shows the optically thin linear limit.
125
6. LIBS on scrap metals and minerals
Figure 6.7: Calibration curves for Ni spectral lines at 341.48 nm and 351.03
nm.
Fig. 6.5 show that the resonance lines of copper and chromium present
strong self-absorption, as the optically thin linear limit is only valid for
concentrations less than 0.101% in the case of copper, and 0.138% for
chromium, unfortunately leaving only three of our reference samples on
the linear range interval.
For most used nickel lines, the optically thin linear limit curve can be
used as calibration curve for concentrations up to 2% Ni. All calibration
curves are fitted using the relation y = a + bc(1 − exp(−x/c)) where y
is the integrated line intensity and x is the certified concentration value
(wt.%) [158]. The limit of detection (LOD) is calculated using the 3σ
criterium, using the standard deviation of the available samples with the
lowest concentrations (0.0091% Cu, 0.019% Ni, and 0.040%Cr). The
obtained values for the LOD of copper and chromium are 0.02% and
0.025% respectively. In the case of nickel, the different spectral lines
give the following limits of detection: 0.5% (341.5 nm), 1.3% (346.2 nm),
1.6% (351.5 and 338.1 nm) and 4% (351.0 nm). The LOD obtained are
not as good as for example in Ref. [143]. The cause for these poor values
lies probably in the unexpectedly large shot-to-shot variations.
126
6. LIBS on scrap metals and minerals
Scrap metal samples
The possibility to use LIBS for the on-line detection of low concentration of copper and chromium in scrap metal samples depends on the
minimum amount of laser shots. In the optimum case this would only
be one or two laser shots. Unfortunately, Fig. 6.2 shows that the relative standard deviation for an integrated line in one or two laser shots
is at least 35%. This fluctuation is unacceptable and the possibility of
measuring copper concentrations with only one laser shot is therefore
discarded. The fluctuation in the calculated concentration is greatly
reduced by averaging spectra. The results are presented for 10 and 50
single laser shots respectively. In the case of scrap metal #1, the copper
contents are found to be 0.12 ± 0.03% for 10 laser shots, and 0.10 ±
0.02% for 50 laser shots. The results for copper of scrap metal #2 show
a more clear discrepancy; as for 10 laser shots the copper concentration
is 0.13 ± 0.03 % while for 50 laser shots this value is 0.09 ± 0.02%. The
amount of chromium for both scrap metal pieces is not detected.
Disko-lava sample
A two dimensional study of the frozen magma piece is performed, and
the locations of the measurements are presented in Fig. 6.8. For each of
the measured points, the nickel and copper concentrations are obtained.
The matrix effect is not accounted for in these measurements and can
then affect their reliability. The composition of the lava sample obtained
by analysis methods other than LIBS is to our knowledge unavailable
and therefore the correctness of the LIBS technique cannot be tested.
Fig. 6.9 shows the results for nickel with their uncertainties using
different spectral lines for calibration for 2 of the measured points. The
uncertainties in the obtained concentrations are calculated using the
propagation of the uncertainties of the fitting parameters a, b, c of the
calibration curves.
All these calibration curves are checked by re-measuring the concentrations of the certified reference samples. The curves give correct
values within the calculated uncertainties. However, the accuracy of the
calculated nickel contents differs with the sample concentration. For example, using the reference sample with 3.02 % nickel leads to similar and
correct results (within the uncertainty interval) for all the chosen refer127
6. LIBS on scrap metals and minerals
Figure 6.8: Locations of the measurements performed on the lava sample.
Figure 6.9: Nickel concentrations obtained with measurements performed
onto the Disko sample.
128
6. LIBS on scrap metals and minerals
Table 6.2: Calculated concentrations (in wt.%) on the magma sample using
the copper spectral line at 327.39 nm and the average of the results of all nickel
calibration curves.
MeasurementNickel, using aver- Copper,
using
number
age of results with spectral line at
different peaks
327.39 nm
#1
2.88 ± 0.53
0.197 ± 0.56
#2
2.25 ± 0.44
0.239 ± 0.067
#3
0.86 ± 0.27
0.85 ± 0.35
#4
3.40 ± 0.60
0.245 ± 0.069
#5
1.33 ± 0.32
0.151 ± 0.046
#6
3.83 ± 0.71
0.168 ± 0.050
#7
3.30 ± 0.61
0.154 ± 0.047
#8
3.25 ± 0.60
0.129 ± 0.042
#9
3.20 ± 0.59
0.38 ± 0.11
#10
3.61 ± 0.68
0.173 ± 0.051
The mass contents of the elements are given in wt.%
ence lines. But when the nickel content of the certified sample is as low
as 0.808%, the results of the calibration lines differ more pronouncedly.
This behavior is also present in the nickel concentrations obtained for
the lava sample (see Fig. 6.9).
Nevertheless, the average value of all calculated concentrations is in
general closer to the certified value than using only one specific spectral
line. The first column in Table 6.2 shows the average of all the different
concentration values using the calibration lines present in Fig. 6.9. The
presented uncertainty in the average value comes from the propagation of
the uncertainties of each calculated concentration. The second column
of Table 6.2 is the calculated copper concentration for each sampled
point, using the calibration line at 327.39 nm.
6.5
Conclusions
After initial generation of calibration curves for nickel, copper and
chromium, their absolute concentration is obtained in steel samples for
two studies.
129
6. LIBS on scrap metals and minerals
The high RSD value of the integrated line intensity for 1 or 2 laser
shots prevents the use of LIBS for on-line monitoring of low concentrations of copper in scrap metal samples. A spectral average of 10 laser
shots leads to an acceptable uncertainty of the calculated concentration
of copper (or chromium), but the implementation for the use in an online system can be difficult. A higher spectrometer readout speed is
required as well as increasing the laser repetition rate.
In the case of the lava sample, the chosen nickel calibration curves behave differently with the measured Ni concentration. When the nickel
contents are approximately 3%, all the calibration lines show similar
results (within their uncertainty intervals). Consequently, any of the
calibration spectral lines would be a reliable indicator of the nickel concentration present. However, when the calculated concentration is low
(≈1%) each calibration curve gives a different result. At low contents
the average of the calculated results leads therefore to a more reliable
estimate of the sample concentration.
Acknowledgment
The authors would like to thank Dr. Dan Holtstam from the Department
of Mineralogy of the Museum of Natural History in Stockholm, Sweden
for having provided us with the lava sample studied in this work.
130
Samenvatting
Dit proefschrift is gebaseerd op een aantal experimentele onderzoeken
op het gebied van laserspectroscopie, die werden uitgevoerd bij twee verschillende instellingen. De werkzaamheden begonnen aan het Royal Institute of Technology of Sweden, waarbij de nadruk lag op onderzoek met
de techniek van Laser Induced Breakdown Spectroscopy (LIBS). Daarna
werden een aantal studies uitgevoerd met de extreem ultraviolet (XUV)
laser opstelling in Amsterdam. Allereerst werden de Rydberg-valentie
interacties in het stikstofmolecuul bestudeerd. Daarna werd overgegaan
tot de karakterisering van de zogenaamde ”zware Rydbergtoestandenı̈n
H+ H− .
De meerderheid van de experimentele werk gepresenteerd in dit
proefschrift (Hoofdstukken 1 tot 5) is uitgevoerd in het extreme ultraviolet (XUV) gebied van het spectrum. Het gebruikte instrument,
een narrow bandwidth en afstembare extreem ultraviolet laserbron, is
gebaseerd op harmonische generatie. Om dit te bereiken wordt zichtbare
narrowband straling gegenereerd in een pulse-dye laser, welke gepompt
wordt door de tweede harmonische van een Nd:YAG laser. De zichtbare straling wordt in frequentie verdubbeld met behulp van een nietlineair kristal. De resulterende ultraviolette (UV) output wordt vervolgens geleid door dichrode spiegels (dat het overblijfsel van zichtbaar
licht filtert) in een vacum kamer. De XUV straling wordt uitendelijk
gegenereerd in een vacuumkamer als derde harmonische van het UVlicht in een inert gas (Xe, Kr). De efficintie van de generatie van de
derde harmonische is vrij laag (10−6 -10−7 ) waardoor er een sterke hoeveelheid UV licht overblijft dat collinear reist aan de gegenereerde XUV
lichtbundel.
Het excitatie-schema dat hierbij wordt gebruikt heet resonant versterkt multifoton ionisatie, en wordt gecombineerd met een time-of-flight
Samenvatting
detectie systeem (REMPI - TOF). De XUV + UV-straling is loodrecht
doorsneden door een gepulseerde moleculaire bundel. De XUV resonant foton wordt geabsorbeerd door de moleculen in de bundel (H2 ,
N2 ) en de UV-foton wekt het molecuul (non-resonant) verder boven de
ionisatie-energie. De gegenereerde ionen worden gedetecteerd door een
time-of-flight spectrometer die massa identificatie mogelijk maakt. Dit
ontwerp is vooral handig om isotoopeffecten te bestuderen.
Hoofdstukken 1 tot 3 zijn gericht op spectroscopische studies van
moleculaire stikstof. De spectrale eigenschappen van moleculaire stikstof zijn cruciaal voor een beter begrip van stralingsoverdrachtverschijnselen en de chemie van aangeslagen N/N2 in de bovenste atmosfeer
van de aarde. De sterke absorptie banden in het gebied van 80 tot
100 nm vormen een schild voor het aardoppervlak voor het XUV deel
van de zonnestraling. Absorptie van licht bij de korte golflengten leidt
tot moleculaire dissociatie, en voor N2 verloopt dit proces via predissociatie, met atomen in de grondtoestand en aangeslagen atomen als
eindproducten.
Moleculaire stikstof is een van de meest stabiele moleculen in de
natuur, doordat de elektronische configuratie overeenkomt met een
driedubbele chemische binding. Deze drievoudige chemische binding
verklaart de hoge dissociatie en ionisatie energielimieten van het molecuul. Vandaar dat de elektronisch aangeslagen toestanden van moleculaire stikstof hoog liggen en te vinden zijn in het extreme ultraviolet
gedeelte van het spectrum. Daardoor zijn aangeslagen toestanden van
stikstof experimenteel moeilijk toegangelijk, en is de analyse van het
elektrische dipool-toegestane spectrum notoir complex.
Hoofdstuk 1 richt zich op de complexiteit van het elektrische-dipooltoegestane spectrum van moleculaire stikstof en over de rol die XUV
ionisatie spectroscopie kan spelen bij het ontrafelen van het spectrum.
Vanwege zijn uitstekende spectrale resolutie, is XUV laser spectroscopie
bij uitstek geschikt voor het waarnemen van deze rotatie-structuur en
voor het bepalen van de mate van lijnverbreding en de bijbehorende mate
van predissociatie. Diverse resultaten uit ons onderzoek in Amsterdam
worden besproken.
Hoofdstuk 2 gaat over het gebruik van twee verschillende hogeresolutie experimentele technieken, 1 XUV + 10 UV laser op basis van
ionisatie spectroscopie en synchrotron-gebaseerde XUV photoabsorp132
Samenvatting
tion spectroscopie. Beide technieken worden gebruikt bij het bestuderen van de o 1 Πu (v = 1) ∼ b 1 Πu (v = 9) Rydberg-valence complex van
14 N . De resultaten verstrekken nieuwe en gedetailleerde informatie over
2
de verstoorde roterende structuren, oscillatorkrachten, en lijnbreedtes
tussen deze twee genoemde predissociatie toestanden. Rotatie en deperturbatie analyses worden uitgevoerd die corrigeren voor onterechte
toewijzingen voor overgangen naar de lage niveaus van J o(v = 1) en
b(v = 9), en bieden opheldering voor het optreden van de kwantum
interferentie-effecten in oscillatorsterktes tussen deze twee elektrischedipool-toegestane overgangen, alsmede de lijnbreedte, tussen deze twee
niveaus.
Naast het bestuderen van de b − X(9, 0) en o − X(1, 0) banden voor
het belangrijkste 14 N2 isotopomer, werden deze banden ook onderzocht
voor 14 N15 N en werd een rotatie-analyse uitgevoerd. In het geval van
het gemengde 14 N15 N isotopomer, is de rotatiestructuur van elke onverstoorde overgang te wijten aan verschillende isotopische verschuivingen.
Aangezien voor hoge J, het homogene verstoringscomplex van de twee
1 Π staten een heterogene interactie ondergaat met de b01 Σ+ (v = 6)
u
u
staat (die o 1 Πu (v = 1) kruist tussen J = 24 en J = 25), is het b0 (v = 6)
niveau ook opgenomen in de huidige studie voor zowel 14 N2 en 14 N15 N.
In hoofdstuk 3 worden de 3pπu c1 Πu − X 1 Σ+
g (2, 0) Rydberg en
0
1
+
1
+
14
b Σu − X Σg (7, 0) valentie overgangen van N2 , 14 N15 N en 15 N2
bestudeerd met behulp van laser-gebaseerde 1 XUV + 10 UV twee-fotonionisatie spectroscopie, aangevuld met synchrotron-gebaseerde photoabsorption metingen in het geval van 14 N2 . Voor elke isotopomer, worden
effectieve rotational interacties tussen de c(v = 2) en b0 (v = 7) aangetroffen, welke sterke Λ-verdubbeling in c(v = 2) veroorzaken. Tevens worden dramatische P/R-branch intensiteitanomalien in de b0 − X(7, 0)
band waargenomen als gevolg van de effecten van kwantuminterferentie. Vanwege de inherente complexiteit en het multilevel karakter van
de N2 structuur en spectrum in dit energiegebied, blijken paar niveau
deperturbatietechnieken zeer beperkt bruikbaar te zijn en is de interpretatie van de resultaten door middel van coupled-channel Schrödingervergelijking (CSE) technieken noodzakelijk. Lokale verstoringen in energie en predissociatie lijnbreedte voor het c(v = 2) Rydbergniveau werden waargenomen en vooral toegeschreven aan een spin-baan wisselwerking met het kruisende, kortstondige C 3 Πu (v = 17) valentieniveau.
133
Samenvatting
Echter, de CSE berekeningen suggereren dat dit een complexe interactie
is, waarbij het F (v = 2) Rydbergniveau met dezelfde energie in de regio,
maar met een nog kortere levensduur, een rol speelt.
Hoofdstukken 4 en 5 gaan over de eerste spectrale waarneming en
spectroscopische karakterisering van een reeks gedoopt met de naam
”zware Rydbergtoestanden”. Een zware Rydbergtoestand is een toestand in een zwaar Bohr-atoom. Een zwaar Bohr-atoom is een waterstofatoom waar een elektron wordt vervangen door een zwaar H− deeltje,
die gezamenlijk het ionenpaar H+ +H− vormen.
De energie van Rydbergtoestanden worden beschreven door
En = Elim −
RA
(n − δ)2
waar RA de atomaire Rydberg constante is en n het belangrijkste quantum nummer. Voor atomen met een uitgebreide kern, dat wil zeggen
alle atomen behalve waterstof, moet een quantum gebrek δ worden
opgenomen. De grote verscheidenheid van atomaire en moleculaire realisaties van Rydberg-systemen hebben in gemeen dat de fundamentele
schaalfactor, de Rydberg constante, in wezen hetzelfde is. Het wordt
namelijk beheerst door de massa van het elektron dat gebonden is aan
een veel zwaardere kern: RA = (µ/me )R∞ . Voor al deze systemen licht
de gereduceerde massa van het elektron µ dicht bij me . Ionenpaarsystemen (zoals H+ +H− ) gebonden door de 1/r Coulombpotentiaal vormen
een realistisch uitbreiding van het concept zware Rydberg systemen.
In hoofdstuk 4 worden de eerste spectroscopische waarnemingen van
frequentie-opgeloste zware Rydbergtoestanden in een moleculair systeem gemeld. Gebonden quantumteostanden in de 1/r potentiaal in
H+ H− is waargenomen voor de voornaamste quantumgetallen in het
bereik 140 − 230, zoals voorspeld door de gegeneraliseerde Rydberg formule, welke schaalt met de massa voor de orbital motion (baanbeweging)
van het H− deeltje dat het electron vervangt.
Hoewel de toewijzing van de voornaamste quantumgetallen in de
waargenomen zware Rydbergtoestanden ondubbelzinnig is, zijn er nog
steeds een aantal open vragen over de spectroscopische eigenschappen. Zij betreffen het excitatiemechanisme van zware Rydbergtoestanden, hun quantumdefect, en hun interacties met elektronische Rydbergtoestanden; deze vormen het onderwerp van Hoofdstuk 5. Een
complex resonantiemodel wordt aangeroepen, dat is overgenomen van
134
Samenvatting
de waargenomen quantumdefecten, en een verklaring geeft voor de oorsprong van de effectieve oscillatorkrachten van de zware Rydberg-serie.
Gedetailleerde metingen worden uitgevoerd om het verval van de zware
Rydberg toestanden in moleculaire H+
2 ionen te ontrafelen en kwantificeren, alsmede in de atomaire fragmenten H(n = 2) en H(n = 3). Een
levensduur op schaal van n3 wordt gevonden.
Hoofdstuk 6 behandelt laser-geı̈nduceerde plasma spectroscopie als
detectiesysteem voor het meten van de sporen van nikkel, koper en
andere metalen in staalmonsters. Als voorbeeld van een industriële
toepassing wordt de concentratie van koper in schroot, een belangrijke factor bij het bepalen van de kwaliteit van te recyclen monsters,
bestudeerd. Koperconcentraties tot 200 ppm kunnen worden opgespoord.
Een andere toepassing van de laser-geı̈nduceerde plasmaspectroscopie methode is het meten van de concentraties van nikkel
en koper in een ijzerhoudende magmamonster. Verschillende spectraallijnen van nikkel worden gebruikt voor de kalibratie, en de bijbehorende
resultaten worden vergeleken.
135
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List of publications
Included in this thesis:
M.O. Vieitez, T.I. Ivanov, W. Ubachs, B.R. Lewis and C.A. de
Lange, On the complexity of the absorption spectrum of molecular nitrogen, J. Molecular Liquids 141, 110-117 (2008) (Chapter 1).
M.O. Vieitez, T.I. Ivanov, J.P. Sprengers, C.A. de Lange, W. Ubachs,
B.R. Lewis and G. Stark, Quantum-interference effects in the o 1 Πu (v =
1) − b 1 Πu (v = 9) Rydberg-valence complex of molecular nitrogen, Mol.
Phys. 105 (2007) 1543-1557 (Chapter 2).
M.O. Vieitez, T.I. Ivanov, C.A. de Lange, W. Ubachs, A.N. Heays,
B.R. Lewis and G. Stark, Interactions of the 3pπu c 1 Πu (v = 2) Rydbergcomplex member in isotopic N2 , Mol. Phys. 105 (2007) 1543-1557
(Chapter 3).
M. O. Vieitez, T. I. Ivanov, E. Reinhold, C. A. de Lange and W.
Ubachs, Observation of a Rydberg series in a heavy Bohr atom, Phys.
Rev. Lett. 101 (2008) 163001-(1-4) (Chapter 4).
M. O. Vieitez, T. I. Ivanov, E. Reinhold, C. A. de Lange and W.
Ubachs, Spectroscopic observation and characterization of H+ H− heavy
Rydberg states, J. Phys. Chem. A 113 (2009) 13237-13245(Chapter 5).
List of publications
M.O. Vieitez, J. Hedberg, O. Launila, and L. E. Berg, Elemental
analysis of steel scrap metals and minerals by laser-induced breakdown
spectroscopy, Spectrochim. Acta B. 60 (2005) 920-925 (Chapter 6).
Additional publications:
T.I. Ivanov, M.O. Vieitez, C.A. de Lange and W. Ubachs, Frequency
1 +
calibration of B 1 Σ+
u − X Σg (6,0) Lyman lines in H2 for comparison
with quasar data, J. Phys. B 41 (2007) 035702.
M. Roudjane, T.I. Ivanov, M.O. Vieitez, C.A. de Lange, W.U.L.
Tchang-Brillet and W. Ubachs, Extreme ultraviolet laser calibration of
D2 Lyman and Werner transitions Mol. Phys. 106, 1193-1197 (2008).
T.I. Ivanov, M. Roudjane, M.O. Vieitez, C.A. de Lange, W.U.L.
Tchang-Brillet and W. Ubachs, On a variation of the proton-electron
mass ratio HD as a probe for detecting mass variation on a cosmological
time scale Phys. Rev. Lett. 100, 093007 (2008).
T.I. Ivanov, E.J. Salumbides, M.O. Vieitez, P.C. Cacciani, C.A. de
Lange and W. Ubachs, Extreme ultraviolet laser metrology of O I transitions Monthly Notices Roy. Astron. Society Letters 389, L4-L7 (2008).
154