Continuous wave Doppler-free
spectroscopy on the 𝐀 𝟐𝚺+ ← 𝐗 𝟐𝚷𝟑/𝟐
transition in thermal nitric oxide

Von der Fakultät Mathematik und Physik der Universität Stuttgart
zur Erlangung der Würde eines Doktors der Naturwissenschaften

(Dr. rer. nat.) genehmigte Abhandlung

vorgelegt von

Patrick Kaspar
aus Böblingen

Hauptberichter: Prof. Dr. Tilman Pfau
Mitberichter: Prof. Dr. Peter Michler
Prüfungsvorsitzender: Prof. Dr. Hans-Peter Büchler
Tag der mündlichen Prüfung: 6. Dezember 2022

5. Physikalisches Institut
Universität Stuttgart

2022



It was my choice or chance or curse
To adopt the cause for better or worse
And with my worldly goods & wit
And soul & and body worship it.

(Edgar Allan Poe, “To Isaac Lea”)



Ehrenwörtliche Erklärung

Schriftliche Bestätigung der eigenständig erbrachten Leistung gemäß §6 Ab-
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Die eingereichte Dissertation zum Thema

Continuous wave Doppler-free spectroscopy on the 𝐀𝟐𝚺+ ← 𝐗𝟐𝚷𝟑/𝟐 transition
in thermal nitric oxide
stellt meine eigenständig erbrachte Leistung dar.

Ich habe ausschließlich die angegebenen Quellen undHilfsmittel benutzt. Wörtlich
oder inhaltlich aus anderen Werken übernommene Angaben habe ich als solche
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Stuttgart den, . . . . . . . . . . . . . . . . . . . . . . . . . .

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geboren am . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,

dass das von mir eingereichte pdf-Dokument zur Dissertation

mit dem Thema

Continuous wave Doppler-free spectroscopy on the 𝐀𝟐𝚺+ ← 𝐗𝟐𝚷𝟑/𝟐 transition
in thermal nitric oxide
in Inhalt und Wortlaut der ebenfalls eingereichten Printversion meiner Dissertati-
onsschrift entspricht.

Stuttgart den, . . . . . . . . . . . . . . . . . . . . . . . . . .

(Datum)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(Unterschrift Doktorand)



Zusammenfassung

ImRahmen dieser Arbeit wurdeDopplerfreie Sättigungsspektroskopie in Stickstoff-
monoxid am A 2Σ+ ← X 2Π3/2 Übergang realisiert. Dieses hochauflösende Spektro-
skopieverfahren ermöglicht die direkte Auflösung von sogenannten “Lamb-dips”.
Dies ist gleichbedeutendmit der direkten Auflösung der Hyperfeinstruktur desMo-
leküls. Der Frequenzabstand zwischen einzelnen Hyperfeinübergängen wurde er-
mittelt und ermöglichte hierdurch die Bestimmung der Hyperfeinkonstanten des
A 2Σ+ Zustandes von Stickstoffmonoxid. Der Vergleich der neu bestimmten Kon-
stanten mit zuvor mit anderen spektroskopischen Techniken bestimmten Konstan-
ten zeigt hervorragende Übereinstimmung. Des Weiteren wird die Rydberganre-
gung von Stickstoffmonoxidmit einemdreiphotonigenAnregungsschema erläutert.
Diese ist Kernbestandteil der Entwicklung eines Laborprototyps zur Erprobung ei-
nes neuartigen Sensorprinzips zum Nachweis von Stickstoffmonoxid.

In dieser Arbeit werden zu Beginn die theoretischen Grundlagen, die für das Ver-
ständnis der weiteren Arbeit notwendig sind, eingeführt. Dies umfasst die Beschrei-
bung der Energiestruktur von zweiatomigen Molekülen. Diese kann in einen elek-
tronischen Anteil sowie die Vibrations- und Rotationsenergien aufgeteilt werden.
Im Anschluss daran werden die verschiedenen Kopplungsfälle für die Drehimpulse
innerhalb des Moleküls behandelt. Hier beschränkt sich die Arbeit auf die relevan-
ten drei Fälle, die als Hund’sche Fälle (a), (b) und (d) bezeichnet werden. Es folgt
darauf eine kurze Einführung zur Parität molekularer Zustände in zweiatomigen
Molekülen sowie ein Überblick über die verschiedenen Auswahlregeln für Dipol-
übergänge innerhalb des Moleküls. Die Beschreibung der eingangs erwähnten Hy-
perfeinstruktur erfolgt mittels eines effektiven Hamiltonoperators (effective Hamil-
tonian). Dieses Konzept wird zunächst kurz erläutert bevor dann die zugehörigen
Matrixelemente für Fein- und Hyperfeinstruktur aufgeführt werden. Die theoreti-
sche Einführung endet mit einem kurzen Abschnitt zu Rydbergzuständen.

I



Zusammenfassung

Als Nächstes wird das drei Photonen umfassende Anregungsschema für die Ryd-
berganregung von Stickstoffmonoxid vorgestellt. Es basiert auf drei Übergängen.
Zunächst werden die Atome vom X 2Π3/2 Zustand zum A 2Σ+ Zustand angeregt.
Hierzu wird Licht mit einer Wellenlänge von 226nm verwendet. Der nächste Über-
gang benötigt Licht bei 540nm und regt den H 2Σ+ Zustand des Moleküls an. Von
dort aus sind mit Wellenlängen zwischen 833nm und 835nm viele unterschiedli-
che Rydbergzustände erreichbar. Die detaillierte Struktur der Zustände X 2Π, A 2Σ+
undH 2Σ+ wird daraufhin diskutiert. Anschließend folgt die Einführung der beiden
verwendeten Spektroskopietechniken. Zunächst wird die optogalvanische Spektro-
skopie eingeführt. Diese Technik kann in drei Schritten erklärt werden. Als Ers-
tes wird mit schmalbandigen Lasern nach dem vorgestellten dreiphotonigen Anre-
gungsschema Stickstoffmonoxid in Rydbergzustände angeregt. Im zweiten Schritt
sorgen Stöße der angeregten Moleküle mit einem Hintergrundgas, wie zum Bei-
spiel Stickstoff, für die Ionisation der Rydbergmoleküle. Die dabei entstehenden
freien Ladungen werden im dritten Schritt mittels einer Spannung, die an die sich
in der Zelle befindlichen Elektroden angelegt wird, aus der Zelle abgezogen. Der
resultierende Strom kann dann mittels eines Tranzimpedanzverstärkers verstärkt
und in eine Spannung umgewandelt werden. Diese wird letztendlich gemessen.
Darauffolgend wird Dopplerfreie Sättigungsspektroskopie erläutert. Diese Spektro-
skopiemethode basiert auf einer gegenläufigen Strahlkonfiguration. Dabei wird ein
Laserstrahl mit Leistung nahe der Sättigungsintensität, der als “Pump” bezeichnet
wird, gleichzeitig mit einem gegenläufigen zweiten Laserstrahl mit deutlich gerin-
gerer Leistung, der “Probe” genanntwird, durch dasMediumgeschickt. Der stärkere
der beiden Laserstrahlen verringert dabei die Absorption des schwächeren Strahls,
wenn beide Laser mit derselben Geschwindigkeitsklasse interagieren. Dies ermög-
licht es, molekulare Übergänge aufzulösen, die sonst im Dopplerverbreiterten Li-
nienprofil verborgen blieben. Des Weiteren werden im Zuge dieser Einführung so-
wohl “Crossover-Resonanzen” als auch verschiedene Verbreiterungsmechanismen
kurz diskutiert.

Der Mittelteil der Arbeit ist technischer Natur. Hier werden zunächst kurz die ver-
wendeten Lasersysteme beschrieben. Außerdem wird schematisch erklärt, wie die-
se mittels “Transfer-Cavities” und dem Pound-Drever-Hall Verfahren frequenzsta-
bilisiert werden. Des Weiteren wird das im Rahmen dieser Arbeit entwickelte Gas-
mischsystem vorgestellt. Hierzu werden zunächst einige theoretische Konzepte in

II



Bezug auf die Berechnungen von Gasflüssen eingeführt. Diese waren für die Aus-
legung des Systems von Bedeutung. Im Anschluss werden Sicherheitsaspekte und
die einzelnen Komponenten des Setups diskutiert. Die Mischanlage verwendet vier
Massedurchflussregler, um Stickstoffmonoxid und Stickstoff miteinander zu mi-
schen.

Im letztenDrittel dieser Arbeit werden die spektroskopischen Ergebnisse diskutiert.
Zunächst wird der Messaufbau der optogalvanischen Spektroskopie erläutert. Es
werden einzelne Spektren qualitativ diskutiert, die die gelungene Rydberganregung
von Stickstoffmonoxid mittels des vorgestellten Anregungspfades belegen. Der Fo-
kus dieses Teils der Arbeit liegt jedoch auf spektroskopischen Resultaten, dieMittels
Dopplerfreier Sättigungsspektroskopie erzielt wurden.Hierzuwird zunächst der zu-
gehörige Spektroskopieaufbau eingeführt und erläutert. Daran schließt sich ein Ab-
schnitt an, der sich mit der Optimierung des entsprechenden Signals beschäftigt.
Nachfolgend erfolgt die Diskussion derHyperfeinspektren, die für verschiedeneGe-
samtdrehimpulsquantenzahlen 𝐽X desGrundzustands, für den P12ee Zweig des Spek-
trums zwischen den Zuständen X 2Π3/2 und A 2Σ+ gemessen wurden. Im Zuge der
Diskussion werden die Übergänge in den Spektren als Δ𝐹 = −1 Übergänge identi-
fiziert. Die Frequenzabstände zwischen den einzelnen Übergängen werden mittels
eines Fits bestimmt und mit theoretischen Berechnungen verglichen, die mit der
Software pgopher gemacht wurden. Mittels der Frequenzabstände werden dann
im Anschluss die Hyperfeinkonstanten des A 2Σ+ Zustandes neu bestimmt und mit
früheren Ergebnissen verglichen. Nachfolgend daran findet eine qualitative Diskus-
sion eines Spektrums statt, welches bei einem um zwei Größenordnungen niedrige-
ren Druck gemessen wurde. Die darin auftretenden zusätzlichen Linien werden als
“Crossover-Resonanzen” interpretiert. Jedoch stellt sich heraus, dass diese Interpre-
tation der Daten nicht vollständig durch theoretische Berechnungen gestützt wird.
Weitere Daten sind notwendig, um die korrekte Interpretation des Spektrums zu
bewerkstelligen. Im letzten Teil der Diskussion wird auf die Linienbreiten der ver-
schiedenen gemessenen Spektren eingegangen. Hier wird deutlich, dass eine weite-
re Verbesserung der Kontrolle über die Messparameter notwendig ist, um konkre-
tere Aussagen über das Verhalten der Spektren zu machen. Im Anschluss werden
mögliche weitereMessungen erläutert, die genaueren Einblick in die Eigenschaften
der untersuchten Zustände geben könnten.

III



Zusammenfassung

Die Arbeit endet mit einem Ausblick, der sowohl das Potential der Dopplerfreien
Sättigungsspektroskopie an Molekülen als auch die Möglichkeiten zur Verbesse-
rung des Spektroskopieaufbaus und des Gassensorprototyps beleuchtet.

IV



Abstract

Within the scope of this thesis the realisation of Doppler-free saturated absorption
spectroscopy for the A 2Σ+ ← X 2Π3/2 transition in a thermal gas of nitric oxide was
achieved. This high resolution spectroscopy technique enables the direct resolu-
tion of Lamb-dip spectra, i.e. the direct observation of the hyperfine structure of the
molecule. The frequency splitting between individual hyperfine spectra was mea-
sured and allowed the determination of hyperfine constants of the A 2Σ+ state of
nitric oxide [1]. The comparison of the newly determined constants to values previ-
ouslymeasuredwith different spectroscopic techniques, shows excellent agreement.
In addition, the Rydberg excitation of nitric oxide with a three photon excitation
scheme is presented. This is an essential part in the course of the development of a
laboratory prototype for the investigation of a new kind of gas sensing scheme for
nitric oxide.

In the beginning of this thesis a theoretical introduction to the energy structure of
diatomic molecules is given. This comprises the electronic, vibrational and rota-
tional energies. The different angular momentum coupling cases also denoted as
Hund’s cases are explained, followed by summaries concerning parity and dipole
transition rules. The description of the aforementioned hyperfine structure of the
molecules employs the effective Hamiltonian approach. The approach is briefly in-
troduced and the corresponding matrix elements for fine and hyperfine structure of
the molecule are given. It follows a very brief introduction of Rydberg states.

Next, the aforementioned excitation scheme is explained. It employs three transi-
tions starting from the X 2Π3/2 state the molecules are excited to the A 2Σ+ state with
light at a wavelength around 226nm. The second transition is at 540nm and ex-
cites the molecules to the H 2Σ+ state. From there several different Rydberg states
are addressable with wavelength between 833 - 835nm. The energy structure of the

V



Abstract

X 2Π, A 2Σ+ andH 2Σ+ state are discussed in detail. This is followed by the introduc-
tion of optogalvanic spectroscopy. This technique can be explained in three steps.
First, the aforementioned narrowband laser excitation to high Rydberg states takes
place. Second, collisions between the Rydberg molecules and the background gas
lead to ionisation of the Rydberg molecules. During the third and last step, the gen-
erated charges are detected by applying a small voltage to the onboard electrodes
of the cell. In addition, an introduction to Doppler-free absorption spectroscopy is
given. This technique is based on a counter-propagating beam configuration where
a strong laser beam denoted as pump beam reduces the absorption of a weaker laser
beam called probe beam, if both beams interact with the same velocity class. This
enables the resolution of the molecular energy structure that is otherwise hidden
within the Doppler-broadened line-profile.

The middle part of the thesis is very technical. It comprises a short overview over
the employed laser systems, their working principles and the frequency stabilisa-
tion setup that was employed as well as an overview about the gas mixing unit that
was developed within the scope of this work. To understand the gas mixing unit
the necessary theoretical concepts for gas flow calculations are given, before the re-
spective setup is explained in detail. The discussion includes safety aspects and a
description of the individual parts of the setup. The gas mixing unit employs four
mass flow controllers to generatemixtures of nitrogen and nitric oxidewith different
concentration of nitric oxide

In the last part of the thesis the spectroscopic results are discussed. First, the opto-
galvanic spectroscopy setup is explained and exemplary datasets are discussed prov-
ing the Rydberg excitation of nitric oxide with the described three photon excitation
scheme.

However, the main focus lies here on the spectroscopic results of the Doppler-free
saturated absorption spectroscopy. The corresponding setup is introduced, followed
by a section where the process of signal optimisation is stated. Then the Lamb-dip
data for different total angular momenta 𝐽X is discussed. Within the course of the
discussion the different transitions are assigned to be Δ𝐹 = −1 transitions and the
splittings between them are determined via a fit. The retrieved splittings are com-
pared to a theoretical calculation employing pgopher and subsequently used to de-
termine new values for the hyperfine constants of the A 2Σ+ state by fitting the data

VI



employing a wrapper program for pgopher. It follows a qualitative discussion of
spectra taken at two orders of magnitude lower pressure, showing additional spec-
troscopic lines. The additional lines might be attributed to crossover resonances.
However, the corresponding theory does not fully support this hypothesis. Thus,
further data would be necessary to clarify thematter. The final part of the discussion
deals with the linewidth of the hyperfine spectra and is followed by a short overview
on how the spectroscopic resolutionmay potentially be further improved andwhich
additional measurements on that transition could yield important information for
the general enhancement of the gas sensor prototype.

VII





Contents

Introduction 1

1 The energy structure of diatomic molecules 5
1.1 Electronic energy structure . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Vibrational energy structure . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Rotational energy structure . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Thermal distribution of vibrational and rotational energy levels . . . 12
1.5 Coupling of angular Momenta: Hund’s cases . . . . . . . . . . . . . 14

1.5.1 Hund’s case (a) . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.5.2 Hund’s case (b) . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.5.3 Hund’s case (d) . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.6 Parity of molecular states . . . . . . . . . . . . . . . . . . . . . . . . 19
1.7 Electronic transitions and transition rules . . . . . . . . . . . . . . . 21
1.8 The effective Hamiltonian approach . . . . . . . . . . . . . . . . . . 23

1.8.1 Fine structure effective Hamiltonian . . . . . . . . . . . . . 24
1.8.2 Λ-type doubling . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.8.3 Spin-rotation coupling . . . . . . . . . . . . . . . . . . . . . 25

1.9 Hyperfine structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.10 Rydberg states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2 Excitation scheme and spectroscopic techniques 33
2.1 Nitric oxide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2 Excitation of NO to a Rydberg state . . . . . . . . . . . . . . . . . . . 34

2.2.1 The A 2Σ ← X 2Π3/2 transition in nitric oxide . . . . . . . . . 35
2.2.2 The 3d-complex states H 2Σ+ and H′ 2Π . . . . . . . . . . . . 38
2.2.3 Excitation to high lying Rydberg states of nitric oxide . . . . 44

2.3 Optogalvanic spectroscopy . . . . . . . . . . . . . . . . . . . . . . . 44

IX



Contents

2.4 Doppler-free absorption spectroscopy . . . . . . . . . . . . . . . . . 46
2.4.1 Doppler Broadening . . . . . . . . . . . . . . . . . . . . . . . 47
2.4.2 Saturated absorption spectroscopy . . . . . . . . . . . . . . . 48
2.4.3 Additional broadening effects . . . . . . . . . . . . . . . . . 52

3 Laser setup and frequency stabilisation 55
3.1 Laser systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.1.1 Frequency-quadrupled Ti:sapphire laser . . . . . . . . . . . 55
3.1.2 Frequency-doubled fibre amplifier system . . . . . . . . . . 57
3.1.3 Tapered amplifier system . . . . . . . . . . . . . . . . . . . . 59

3.2 Frequency stabilisation of the individual laser systems . . . . . . . . 60
3.2.1 Pound-Drever-Hall technique . . . . . . . . . . . . . . . . . 61
3.2.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.3 Lock error estimation for the ground state transition . . . . . . . . . 65

4 A gas mixing unit for nitrogen and nitric oxide 69
4.1 Basic concepts for the description of gases . . . . . . . . . . . . . . . 70

4.1.1 The ideal gas law . . . . . . . . . . . . . . . . . . . . . . . . 70
4.1.2 The Van der Waals equation . . . . . . . . . . . . . . . . . . 71

4.2 Gas flow dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.2.1 Viscous or molecular flow - the Knudsen number . . . . . . 74
4.2.2 Laminar or turbulent flow - the Reynolds number . . . . . . 75
4.2.3 The Hagen-Poiseuille equation . . . . . . . . . . . . . . . . . 75

4.3 The setup and its components . . . . . . . . . . . . . . . . . . . . . 76
4.3.1 Safety aspects . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.3.2 Mass Flow Controllers . . . . . . . . . . . . . . . . . . . . . 78
4.3.3 Pipe and valve system . . . . . . . . . . . . . . . . . . . . . . 80
4.3.4 Vacuum system . . . . . . . . . . . . . . . . . . . . . . . . . 84

5 Rydberg excitation of nitric oxide 91
5.1 Optical setup and cell . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.2 Excitation of the H 2Σ+ state . . . . . . . . . . . . . . . . . . . . . . 94
5.3 Rydberg excitation of nitric oxide . . . . . . . . . . . . . . . . . . . . 96
5.4 Outlook: Optogalvanic spectroscopy . . . . . . . . . . . . . . . . . . 97

X



Contents

6 Doppler-free spectroscopy within the 𝜸00–band in nitric oxide 99
6.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.2 Optimisation of the spectroscopic parameters . . . . . . . . . . . . . 103
6.3 Determination of the hyperfine constants for the A 2Σ+ state . . . . 113
6.4 Discussion of crossover resonances . . . . . . . . . . . . . . . . . . . 122
6.5 Investigation of the linewidth of the spectra . . . . . . . . . . . . . . 127
6.6 Potential further investigations . . . . . . . . . . . . . . . . . . . . . 130

7 Conclusion and Outlook 131

Danksagung 133

Bibliography 137

Appendix 157

XI





Introduction

This theses was conducted within the scope of the development of a laboratory pro-
totype for the optogalvanic detection of nitric oxide. However, this work mainly
focuses on high resolution optical spectroscopy. To set this into perspective, first an
overview about the historic development behind the underlying sensor principle is
given.

The technique of optogalvanic spectroscopy is based on electrically contacted spec-
troscopy cells. They were originally developed for the work with alkali vapours,
opening up new possibilities to manipulate Rydberg atoms with electrical fields [2,
3]. In addition, the cell electrodes can not only be used to manipulate atoms inside
the cell but also to read-out small electrical currents [4, 5]. These currents originate
from the ionisation of the Rydberg atoms, i.e. they can be used to obtain information
on the population of the Rydberg state [4]. If it is known how efficient the Rydberg
state can be excited and how large the rate of ionisation is, it is possible to calcu-
late the number of atoms in the ground state. Henceforth, the technique can be
employed as a sensing scheme. This was further investigated in an idealised model
system [6]. The basic principle of the sensing scheme is based on three steps. First,
the atom or molecule in question is excited to a Rydberg state. Narrowband con-
tinuous wave lasers and a multi-photon excitation scheme ensure a high selectivity.
Second, the weakly bound Rydberg atoms or molecules are then ionised by colli-
sional ionisation with the background gas. Third, the generated charges are picked
up by the cell electrodes and a current is measured [7, 8].

The feasibility of this concept was studied in [9, 10], setting it into perspective to
other currently employed sensing techniques. Instead of alkali atoms nitric oxide
was detected. In contrast to them nitric oxide is an important molecule in the hu-
man body. The 1998 nobel price inmedicine was awarded to Furchgott, Ignarro and

1



Introduction

Murad for their “discoveries concerning nitric oxide as a signalling molecule in the
cardiovascular system” [11]. They were able to identify nitric oxide to be responsi-
ble for the process of vasodilation in the human body [12–14]. In addition, nitric
oxide regulates neurotransmission and the immune function of macrophages [15,
16]. Furthermore, it acts as a signalling molecule indicating inflammatory diseases
like cancer [17, 18] and asthma [19, 20]. Recently an extensive review on this field
was published by Lundberg and Weitzberg [21]. Consequently, the efficient detec-
tion of nitric oxide in breath gas samples is of high interest in medical research and
diagnostics.

Optogalvanic spectroscopy requires only small amounts of gas, since millilitre cell
volumes are easy to realise and can achieve a high bandwidth in the 100 kHz range.
Narrowband continuouswave lasers ensure high selectivity by exploiting theunique-
ness of the energy structure of the molecules.

The development of the respective laboratory prototype requires precise knowledge
on the involved states. The two involved states X 2Π and A 2Σ+ were investigated
quite thoroughly [22–28]. Also the hyperfine structure of the two states has been
investigated with different spectroscopic techniques [29–36]. However, the corre-
sponding Lamb-dips have never been resolved directly. The new state of the art
narrowband UV-laser system enables the direct resolution of the Lamb-dips. The
investigation of the A 2Σ+ ← X 2Π3/2 transition with Doppler-free saturated absorp-
tion spectroscopy, allows to determine if the transition can be saturated with the
corresponding laser system. The gained knowledge improves the quantification and
assessment of the aforementioned gas sensor prototype.

This thesis is split into six chapters. The first two chapters are mainly of theoretical
nature. In chapter 1 the theoretical concepts necessary to describe the energy struc-
ture of diatomic molecules are introduced. In chapter 2 the excitation path used
for the sensor prototype and the structure of the involved states are discussed. In
addition, optogalvanic spectroscopy and Doppler-free absorption spectroscopy are
introduced.

The middle part of the thesis is of technical character and starts with chapter 3.
Here, an overview on the employed laser systems and their frequency stabilisation
is given. It is followed by chapter 4, which deals with the required gas mixing unit

2



and vacuum system to safely mix and handle nitric oxide and nitrogen. This system
has been developedwithin the scope of this thesis. The chapter gives an overview on
the different setup components and the theoretical considerations that were made
before setting the system up.

In the final part of this thesis the focus lies on the spectroscopic results achieved
with the two introduced spectroscopic techniques. Chapter 5 presents first data on
the Rydberg excitation of nitric oxide with the new excitation scheme and narrow-
band laser setup as published in [8]. The data is discussed qualitatively. In chap-
ter 6 the main spectroscopic results are presented. It starts with an introduction to
the experimental setup. This is followed by a description on how to optimise the
experimental parameters to increase the signal quality. Subsequently the data on
the hyperfine structure is quantitatively evaluated as published in [1]. In the last
two sections crossover resonances and the width of the spectroscopic lines are dis-
cussed. The chapter ends with a short outlook, discussing the potential of further
investigations.

3





1 The energy structure of diatomic
molecules

Introduction

The energy structure of single atoms is described by quantummechanics and quan-
tum electrodynamics [37]. The electrons move around the nucleus in distinct vol-
umes, atomic orbitals, resulting from the absolute square of the electron wave func-
tions. Molecules are formed by twoormore atoms bound to each other by a chemical
bond. The simplest class of molecules are diatomic, i.e. consist of only two atoms
bound together. Since this thesis is focused on the spectroscopy of nitric oxide (NO),
this chapter will only deal with the energetic structure of diatomic molecules.

First the different contributions to the energy structure of diatomic molecules will
be discussed in section 1.1-1.3. To understand the finer details of the diatomic en-
ergy structure the different coupling schemes for the angular momenta within the
molecule, called Hund’s coupling cases, are discussed. In section 1.6 and section 1.7
parity and symmetry will be introduced and an overview about the types of elec-
tronic transitions and the dipole transition rules will be given. The chapter ends
with the introduction of the effective Hamiltonian in section 1.8 and the discus-
sion of the smallest energy terms corresponding to Λ-type doubling, spin-rotation
coupling and the terms describing the hyperfine structure, i.e. the coupling to the
nuclear spin. At the very end of the chapter a short introduction to Rydberg states
is given.

5



1 The energy structure of diatomic molecules

1.1 Electronic energy structure

This discussion relies closely on the explanations given by Herzberg [38]. The total
energy of a molecule depends on the potential and kinetic energy of the electrons
as well as on the corresponding energies of the nuclei [38]. The energy of a specific
electronic state is based on which of the molecular orbitals are occupied by elec-
trons. Transitions of electrons between different molecular orbitals can be induced
by dipole radiation. Compared to the vibrational and rotational levels, which will be
explained in the following sections, electronic energy levels have a larger distance
to each other. Electronic transitions are often observed at wavelengths of the visible
part of the electromagnetic spectrum or at the adjacent near-infrared or ultraviolet
part. Vibrational transitions are often observed in the infrared and rotational tran-
sitions are mostly in the microwave regime.

The Schrödinger equation for the energies 𝐸 of a diatomic molecule is given by
Herzberg [38]

1
𝑚𝑒

∑
𝑖
Δ𝑖Ψ +∑

𝑘

1
𝑀𝑘

Δ𝑘Ψ + 2
ℏ2 (𝐸 − 𝑉)Ψ = 0. (1.1)

Here Δ𝑖 is the Laplace operator acting on electron 𝑖 and Δ𝑘 the Laplace operator act-
ing on the nucleus 𝑘 with mass𝑀𝑘. All electrons have mass𝑚𝑒. 𝑉 is the respective
potential and ℏ the reduced Planck constant. The exact solution of equation 1.1 is
difficult. However, one can use a product wave function

Ψ = 𝜓𝑒(𝑥𝑖, 𝑦𝑖, 𝑧𝑖) ⋅ 𝜓𝑛(𝑥𝑘, 𝑦𝑘, 𝑧𝑘), (1.2)

to approximate the exact solution [38]. Here𝜓𝑒(𝑥𝑖, 𝑦𝑖, 𝑧𝑖) is the electronicwave func-
tion solving equation 1.3a which is the Schrödinger equation describing the elec-
tronsmoving in the field of the fixednuclei described by the potential𝑉𝑒. Thenuclear
wave function𝜓𝑛(𝑥𝑘, 𝑦𝑘, 𝑧𝑘) solves the Schrödinger equation 1.3b for themovement
of the nuclei under the influence of the potential 𝐸𝑒+𝑉𝑛. The potential energy of the
nuclearmotion𝐸𝑒+𝑉𝑛 depends on the solution of equation 1.3a and therefore on the
internuclear distance 𝑟. Thus a stable electronic state does only exist if 𝐸𝑒 + 𝑉𝑛 has
a minimum. The corresponding energy of the potential minimum is the electronic

6



1.2 Vibrational energy structure

energy 𝑇𝑒 of the state. The minimum of the lowest electronic state (i.e. groundstate)
is usually choosen as zero.

∑
𝑖
Δ𝑖𝜓𝑒 +

2𝑚𝑒
ℏ (𝐸𝑒 − 𝑉𝑒)𝜓𝑒 = 0, (1.3a)

∑
𝑘

1
𝑀𝑘

Δ𝑘𝜓𝑛 +
2
ℏ2 (𝐸 − (𝐸𝑒 + 𝑉𝑛))𝜓𝑛 = 0. (1.3b)

That the separation of the nuclear and electronic motion is legitimate was shown by
Born and Oppenheimer [39] and is often referred to as Born-Oppenheimer approx-
imation.

1.2 Vibrational energy structure

The minimum of the potential energy of the nuclear motion appears at a certain
equilibrium 𝑟𝑒 of the internuclear distance 𝑟. The nuclei vibrate around the equilib-
rium distance. The vibrational energy levels are quantised and for every electronic
state there exist usually several different vibrational states. The exact number of vi-
brational states depends on the width of the potential and on the respective energy
spacing between the vibrational levels. Radiative transitions between vibrational
states are of much lower energy than those between electronic states. They occur
at infrared wavelengths, so that early spectroscopic investigations relied on thermal
light sources like mercury-vapour lamps or Nernst-lamps [40]. Nowadays modern
infrared light sources like quantum-cascade lasers can be employed[41].

The exact energy levels can be obtained from the Schrödinger equation 1.3b. How-
ever, the solvability of equation 1.3b depends strongly on the form of 𝐸𝑒 + 𝑉𝑛 and is
in most cases not strictly possible. Therefore, it is feasible to approximate the term
𝐸𝑒 + 𝑉𝑛 with a potential 𝑉(𝑟) describing the oscillating movement of the nuclei. Of
course, the first idea that arises is the quadratic potential of an harmonic oscillator.
However, the harmonic oscillator potential is symmetric and yields equidistant en-
ergy levels. This is contradictory to experimental observations since a symmetric

7



1 The energy structure of diatomic molecules

potential will not allow dissociation and cannot explain the shrinking energetic dis-
tance between single vibrational energy levels [42, 43]. Therefore, an anharmonic
potential is necessary to appropriately describe the nuclear motion. A good empiri-
cal approximation for 𝑉(R) is the Morse-Potential [40] which allows to solve equa-
tion 1.3b exactly. It is given by

𝑉(𝐑) = De [1 − exp (−𝑎𝑀(𝑟 − 𝑟𝑒))]
2 . (1.4)

Here, De is the dissociation energy and 𝑎𝑀 a molecule specific constant. Figure 1.1
depicts a Morse Potential with its minimum at 𝑟𝑒. For 𝑟 < 𝑟𝑒 the potential is very
steep, taking into account the repulsive Coulomb potential between the two nuclei.
For 𝑟 > 𝑟𝑒 the potential asymptotically approaches the dissociation energy De. This
is the energy threshold above which the vibrational energy states form a continuum
(grey shading), so that themoleculewill separate into the individual atoms that once
formed it. The dissociation energy is sometimes also denoted by D0 and given with
respect to the lowest vibrational energy level v = 0. The yellow lines indicate a few
vibrational energy levels below the dissociation energy. For 𝑟 > 𝑟𝑒 the vibrational
energies can be calculated by solving equation 1.3b with 𝐸𝑒 + 𝑉𝑛 → 𝑉(𝑟). The vi-
brational energies can then be expressed in dependence of the vibrational quantum
number 𝑣 and several vibrational constants.

𝐺(𝑣) = 𝜔𝑒 (𝑣 +
1
2) − 𝜔𝑒𝑥𝑒 (𝑣 +

1
2)

2
+ 𝜔𝑒𝑦𝑒 (𝑣 +

1
2)

3
... (1.5)

The vibrational constants 𝑤𝑒,𝑤𝑒𝑥𝑒 and 𝑤𝑒𝑦𝑒 differ for each electronic state within
a molecule. The equation is dominated by the term linear in (𝑣 + 1/2). Terms of
higher order act usually only as small corrections, since the respective vibrational
constants are in most cases one or more orders of magnitude smaller than 𝑤𝑒.

1.3 Rotational energy structure

This section also follows the descriptions given in [38]. Molecules are not only able
to vibrate but also to rotate in space. The energy of the rotation is an additional con-

8



1.3 Rotational energy structure

re
Internuclear distance r

En
er
gy
E

De

v = 0

D0

Figure 1.1: Morse potential with its minimum at the equilibrium internuclear dis-
tance 𝑟𝑒. De denotes the dissociation energy and D0 the dissociation en-
ergy with respect to the lowest vibrational energy level, denoted as v = 0.
Higher vibrational energy levels are indicated as yellow lines. Above
the dissociation energy the grey shading represents the energetic con-
tinuum. After [40].

tribution to the total energy of the molecule. The spacing between single rotational
energy levels is considerably smaller than between vibrational or electronic energy
levels. Spectroscopic investigations of purely rotational transitions can therefore
rely on microwave spectroscopy [44]. For the theoretical description of the rotation
of diatomic molecules a number of different models exists. A detailed discussion of
the different models is given for example in [38, 45] and is summarised in the fol-
lowing paragraph. It will end with an expression for the rotational energy based on
the symmetric top model.

9



1 The energy structure of diatomic molecules

The simplest model is the rigid rotator where the two nuclei are considered to be
connected by a rigid, massless rod. The rotational energy is given by

𝐹(𝐽) = ℎ
8𝜋2𝑐0𝐼

𝐽(𝐽 + 1) = 𝐵𝐽(𝐽 + 1). (1.6)

Here, the rotational constant 𝐵 = ℎ/(8𝜋2𝑐0𝐼) given in cm−1 and the rotational quan-
tum number 𝐽 are introduced. The constant 𝐵 depends on Planck’s constant ℎ, the
speed of light 𝑐0 and the moment of inertia perpendicular to the internuclear axis 𝐼.
However, as already discussed in section 1.2 the nuclei vibrate periodically around
an equilibrium distance 𝑟𝑒. Consequently, the assumption of a fully rigid connection
between the nuclei is too simplified.

A better model is therefore the non-rigid rotator. Here, the massless rod is replaced
by amassless spring, introducing centrifugal forces resulting from the rotation. These
forces result in a correction term depending on the constant 𝐷𝑒 (not to be confused
with the dissociation energy De) often referred to as centrifugal distortion constant.
The rotational energy has then the form

𝐹(𝐽) = 𝐵𝑒𝐽(𝐽 + 1) − 𝐷𝑒𝐽2(𝐽 + 1)2. (1.7)

For the non-rigid rotator the vibrational and rotationalmovement are still treated in-
dependently of each other. The index 𝑒 refers to the equilibrium distance 𝑟𝑒 between
the two nuclei. The respective constants are often called equilibrium constants.

Further refinement of themodel leads to the vibrating rotator. Since the vibration of
the nuclei changes the internuclear distance, the moment of inertia of the molecule
changes alongside the vibrational motion. Henceforth the rotational energy and
thus also the constant𝐵 of themolecule depend on its vibrational energy state. Since
the vibrational motion is happening at a much higher frequency than the rotational
motion it is justified to use a mean value ̄𝑟 of the internuclear distance 𝑟 for a each
given vibrational energy level. As a consequence the rotational constant and also
its higher order centrifugal correction are given in dependence of the vibrational

10



1.3 Rotational energy structure

level 𝑣 as 𝐵𝑣, 𝐷𝑣. They are related to the equilibrium constants 𝐵𝑒 and 𝐷𝑒 by a series
expansion with the coefficients 𝛼𝑒, 𝛽𝑒 and 𝛾𝑒 [45]

𝐵𝑣 = 𝐵𝑒 − 𝛼𝑒 (𝑣 +
1
2) + 𝛾𝑒 (𝑣 +

1
2)

2
+ ... (1.8)

𝐷𝑣 = 𝐷𝑒 − 𝛽𝑒 (𝑣 +
1
2) + ... (1.9)

Here 𝑣 is the vibrational quantum number. The coefficients 𝛼𝑒, 𝛽𝑒 and 𝛾𝑒 are deter-
mined empirically.

So far the electrons were completely neglected in the treatment of the rotational en-
ergy. Even though themass of the electrons is very small compared to that of the nu-
clei it leads to a non-zero moment of inertia along the internuclear axis. The model
of the symmetric top includes this very small moment of inertia. It was treated first
by Reiche, Rademacher, Kronig and Rabi [46–48]. The electron cloud is considered
to be rigid, ignoring the motion of the individual electrons. The total angular mo-
mentum of the rotation of the symmetric top 𝐉 is no longer perpendicular to the in-
ternuclear axis. It’s perpendicular component is denoted𝐍 and resembles the pure
motion of the nuclei. The total angular momentum neglecting the electron spin, is
then given by the sum of 𝐍 and the angular momentum of the electrons along the
internuclear axis 𝚲 which is introduced within the scope of this model. Figure 1.2
illustrates the coupling of the vectors𝐍 and𝚲 to form 𝐉. The reversal of the rotation
of the electrons reverses also the direction of the vector 𝚲. Therefore for each value
of 𝐽 two different modes of motion exist [38]. For the less refined models: the rigid,
non-rigid and vibrating rotator 𝚲 does not exist, thus in their case 𝐉 = 𝐍.

The rotational energies for the symmetric top depend on the quantumnumber 𝐽 and
are given by

𝐹(𝐽) = 𝐵𝑣𝐽(𝐽 + 1) + (𝐴Λ − 𝐵𝑣)Λ2 − 𝐷𝑣𝐽2(𝐽 + 1)2 + ... (1.10)

The constant 𝐴Λ takes into account the moment of inertia parallel to the internu-
clear axis. It is defined in the same manor as the rotational constant to be 𝐴Λ =
ℎ/(8𝜋2𝑐0𝐼A). Since the moment of inertia perpendicular to the internuclear axis is

11



1 The energy structure of diatomic molecules

𝑁

Λ

𝐽

Figure 1.2: Schematic of the symmetric top. 𝚲 is the orbital electron angular mo-
mentum along the internuclear axis. 𝐍 resembles the motion of the
pure nuclei and couples with𝚲 to 𝐉, the total angular momentum of the
symmetric top. The whole diagram is rotating around 𝐉 indicated by the
curved arrow. After Herzberg [38].

much larger than the one along the axis, thus 𝐼 ≫ 𝐼A holds, the constant𝐴Λ is much
larger than 𝐵𝑣.

For the interpretation of the spectra of very light molecules, like for example H2, the
interaction between vibration and rotation requires a more detailed treatment [45].
The corresponding theory was developed by Dunham [49] and Pekeris [50].

1.4 Thermal distribution of vibrational and rotational
energy levels

In contrast to electronic states, where usually only the ground state is populated
the population of the vibrational and rotational energy levels is distributed. Con-
sequently, the spectroscopic strength of a transition is not only dependent on the
transition moment but also on the population of molecules in the initial state.

12



1.4 Thermal distribution of vibrational and rotational energy levels

The number of molecules 𝑁𝑣 in a particular vibrational state 𝑣 is given by [38]

𝑁𝑣 =
𝑁
𝑄𝑣

exp (−𝐺0(𝑣)ℎ𝑐0
𝑘𝐵𝑇

) . (1.11)

Here, 𝑁 is the total number of molecules, 𝑘𝐵 the Boltzmann constant and 𝑇 the
temperature. 𝑄𝑣 is the state sum of the vibrational levels which can be neglected
when the exponential terms decrease rapidly [38], which is often the case. It is given
by [38]

𝑄𝑣 = 1 + exp (−𝐺0(1)ℎ𝑐0
𝑘𝐵𝑇

) + exp (−𝐺0(2)ℎ𝑐0
𝑘𝐵𝑇

) + ... (1.12)

The energy term𝐺0(𝑣)ℎ𝑐0 is the vibrational energy referred to the zero-point energy
of the potential when 𝐺0(𝑣) is given in cm−1. Its relation to the equilibrium con-
stants can be found in [38]. Figure 1.3 shows the vibrational energy distribution for

0 1000 2000 3000 4000 5000 6000
Energy 𝐸/(ℎ𝑐0) (cm−1)

0

50

100

Po
pu
la
tio
n
(%
) v = 0

v = 1
v = 2

v = 3

293K
4400K

Figure 1.3: Population of the vibrational energy levels for 𝑇 = 293K and 𝑇 = 4400K
of the ground state X 2Π of nitric oxide. The blue and orange dots mark
the vibrational levels with quantum number 𝑣 = 0 − 3. The lines act
as a guides to the eye showing the population for continuous 𝑣 which of
course is not physically correct.

the ground state X 2Π (for the explanation of molecular term symbols see appendix
B) of nitric oxide at 293K and for comparison at 4400K. The populations were cal-
culated from the constants given in the appendix in table A.1. At room temperature

13



1 The energy structure of diatomic molecules

almost all of the molecules are in the lowest vibrational state, the contribution of
energy levels with 𝑣 > 0 is negligibly small. Only at 4400K the ratio between the
number of molecules in the vibrational ground and excited states is equal, i.e. 50%
of the molecules are still in the ground state.

In case of the rotational levels the distribution is similar to that for the vibrational
levels but the degeneracy of rotational levels has to be taken into account. For a total
angular momentum 𝐽 there are 2𝐽 + 1 sublevels. These sublevels are energetically
degenerate in the absence of external fields. The degeneracy results in an additional
prefactor, so that the number of molecules 𝑁𝐽 with total angular momentum 𝐽 for
the rotational constant 𝐵 given in cm−1 is given by [38]

𝑁𝐽 =
𝑁
𝑄𝑟

(2𝐽 + 1) exp (−𝐵𝐽(𝐽 + 1)ℎ𝑐
𝑘𝐵𝑇

) . (1.13)

According to Herzberg [38] the state sum 𝑄𝑟 can be substituted by an integral, so
that equation 1.13 simplifies to

𝑁𝐽 = 𝑁 ℎ𝑐𝐵
𝑘𝐵𝑇

(2𝐽 + 1) exp (−𝐵𝐽(𝐽 + 1)ℎ𝑐
𝑘𝐵𝑇

) . (1.14)

The rotational constant for the ground state of nitric oxide is given by𝐵 = 1.696 cm−1

[35]. The resulting rotational energy distribution at 293K is depicted in figure 1.4.
The highest population is around 8% in the 𝐽 = 15/2 state. For comparison the
distribution is plotted for 50K and 1000K too.

1.5 Coupling of angular Momenta: Hund’s cases

The angular momenta in atoms and molecules couple to each other. This coupling
defines the energy structure of a molecule or atom and is therefore essential for the
understanding of the particular molecule or atom. For atoms there are two differ-
ent coupling schemes: LS-coupling sometimes also calledRussel-Saunders coupling
and jj-coupling [37].

14



1.5 Coupling of angular Momenta: Hund’s cases

1/2 7/2 13/2 19/2 25/2 31/2 37/2 43/2 49/2 55/2 61/2
Total angular momentum J

0

5

10

15

20

Po
pu
la
tio
n
(%
) 1000K

293K
50K

Figure 1.4: Population of the rotational energy levels in the ground state ofNOcalcu-
lated at different temperatures. At 293K (room temperature) the highest
population is in the 𝐽 = 15/2 level, which is marked by the orange verti-
cal line.

For molecules the situation is more complicated. In comparison to atoms, diatomic
molecules are not spherical symmetric and possess more degrees of freedom. There
are five distinct cases for the coupling of angular momenta in molecules. They are
named Hund’s cases after Friedrich Hund who also developed Hund’s rules to de-
termine the energetic ground state of atoms.

In this section Hund’s cases (a) and (b) will be discussed in detail. These two cases
are the most common in diatomic molecules. However, Hund’s cases are idealised
cases, thus they often do not fully fit the real physical situation. In fact for many
states of diatomic molecules the coupling scheme is not purely given by case (a)
and case (b) description, but by intermediate coupling between the two cases. A
transition from Hund’s case (a) to (b) occurs for example for the ground state of
nitric oxide [27, 51].

Since case (d) is appropriate to describe most Rydberg states it will be mentioned
briefly. Case (c) and (e) are not relevant for the description of nitric oxide and will
be omitted. The discussion follows [45].

15



1 The energy structure of diatomic molecules

1.5.1 Hund’s case (a)

The Hund’s case (a) description is usually appropriate when the spin-orbit energy
depending on the spin-orbit constant 𝐴 is significantly larger than the rotational
energy, i.e. 𝐴Λ ≫ 𝐵𝐽.

The coupling scheme for Hund’s case (a) is illustrated in figure 1.5. Electrostatic
forces couple the orbital angular momentum 𝐋 to the internuclear axis. The total
electron spin 𝐒 is coupled strongly to 𝐋, so that both vectors 𝐋 and 𝐒 precess fast
around the internuclear axis. Therefore, L and S are not good quantum numbers.
The projections on the internuclear axis 𝚲 and 𝚺 however, are well defined. To-
gether they form𝛀 = 𝚲+𝚺which subsequently couples to the angular momentum
of the rotation of the nuclei which is denoted𝐑. The resulting total angularmomen-
tum is denoted 𝐉. The orientation of 𝐋 and 𝐒 can be reversed, which also reverses
the projections on the internuclear axis. Henceforth, there are two-fold degenera-
cies in Λ and Ω, consequently named Λ- and Ω-type doubling. In many cases the
degeneracy is lifted resulting in an additional splitting of energy levels. This will be
discussed at a later point (see section 1.8.2).

Hund’s case (a) basis wavefunctions can, according to [45] be written in the form
|𝜂, 𝑣, Λ, 𝑠, Σ, 𝐽, Ω,𝑀𝐽⟩. Here 𝑀𝐽 is the component of 𝐽 along a space fixed axis [45],
analogously to atomic physics. The symbol 𝜂 denotes all quantum numbers which
are not explicitly given and 𝑣 is the vibrational quantum number.

1.5.2 Hund’s case (b)

ForHund’s case (b) the coupling between𝐋 and 𝐒 is onlyweak. The projection𝚲 of𝐋
on the internuclear axis is still a good quantum number but𝛀 is no longer defined.
𝚲 couples directly to 𝐑 forming the total angular momentum without spin 𝐍 (in
older works 𝐍 is sometimes denoted 𝐊, e.g. [38]). The total electron spin couples
then to𝐍 to form the total angular momentum 𝐉. The coupling scheme is depicted
in figure 1.6. The coupling between 𝐍 and 𝐒 results in a small splitting for each
𝑁 > 1 if 𝑆 = 1/2. This results in two series of 𝐽-levels: 𝐹1(𝐽) with 𝐽 = 𝑁 + 1/2 and
𝐹2(𝐽)with 𝐽 = 𝑁 −1/2. This effect is called spin-rotation splitting and will reappear

16



1.5 Coupling of angular Momenta: Hund’s cases

𝑅𝐽

𝐿

Λ

𝑆

Ω

Σ

Figure 1.5: Angular momentum coupling scheme for Hund’s case (a). After [45]

17



1 The energy structure of diatomic molecules

at the discussion of the effective Hamiltonian (see section 1.8.3). For larger values
of 𝑆 the splitting still exists but with a more complicated structure which is beyond
the scope of this work.

The correspondingwavefunctions forHund’s case (b) are |𝜂, 𝑣, Λ, 𝑁, 𝑆, 𝐽,𝑀𝐽⟩ accord-
ing to [45]. Hund’s case (b) is a good descriptionwhen the rotational energy is much
larger then the spin-orbit energy 𝐴Λ ≪ 𝐵𝐽. Σ-states (Λ = 0) correspond almost al-
ways to Hund’s case (b) coupling. In the case of Λ ≠ 0 Hund’s case (b) coupling is
only an appropriate decription for some light molecules.

𝐿

𝑅𝑁

𝑆𝐽

Λ

Figure 1.6: Angular momentum coupling scheme for Hund’s case (b). After [45]

1.5.3 Hund’s case (d)

In Hund’s case (d) neither Λ nor Σ is defined, since the coupling between 𝐋 and 𝐑
is stronger than the coupling of 𝐋 to the internuclear axis. Consequently 𝐋 and 𝐑

18



1.6 Parity of molecular states

couple to a resulting vector 𝐍, which then couples with 𝐒 to form the total angular
momentum 𝐉. This scheme results in 2𝐿 + 1 values of 𝑁 for each value of 𝑅 as long
as 𝑅 > 𝐿.

In Rydberg states (see section 1.10) the excited electron is only weakly interacting
with themolecular core due to its large distance to the core and the shielding effect of
the electron shell. Therefore, Hund’s case (d) is inmost cases appropriate to describe
Rydberg molecules.

𝑅

𝐿
𝑆

𝑁𝐽

Figure 1.7: Angular momentum coupling scheme for Hund’s case (d). After [45]

1.6 Parity of molecular states

Transition probabilities between individual molecular states are strongly linked to
parity. Therefore, the concept of parity shall be introduced following [45]. To de-
scribe parity inmolecules there are two operators that can be defined. Themolecule
fixed inversion operator 𝑖 is used to define the overall wave function symmetry with
respect to molecule fixed coordinates [45]. However, this operator does not give any
information about the total parity of a state. Amore general approach is an inversion
operator 𝐸∗ for arbitrary space-fixed coordinates (𝑥𝑖, 𝑦𝑖, 𝑧𝑖). A function 𝑓(𝑥𝑖, 𝑦𝑖, 𝑧𝑖)
of these coordinates is transformed as follows

𝐸∗𝑓(𝑥𝑖, 𝑦𝑖, 𝑧𝑖) = 𝑓′(𝑥𝑖, 𝑦𝑖, 𝑧𝑖) = 𝑓(−𝑥𝑖, −𝑦𝑖, −𝑧𝑖). (1.15)

19



1 The energy structure of diatomic molecules

As a consequence for a wave function Ψ it can be written

𝐸∗Ψ(𝑥𝑖, 𝑦𝑖, 𝑧𝑖) = ±Ψ(𝑥𝑖, 𝑦𝑖, 𝑧𝑖). (1.16)

Parity can then be defined to be positive if Ψ transforms to Ψ and to be negative if Ψ
transforms to −Ψ.

As discussed in the previous section for diatomic molecules Hund’s case (a) and
(b) are most common. Even if an intermediate case is present the description will
usually be done eitherwith case (a) or case (b) wavefunctions. Therefore, it is crucial
to know how case (a) and case (b) wave functions transform with respect to 𝐸∗. For
Hund’s case (a) basis functions the following holds true

𝐸∗ |𝜂, 𝑣, Λ𝑠, 𝑆, Σ, 𝐽, Ω,𝑀𝐽⟩ = (−1)𝐽−𝑆+𝑠 |𝜂, 𝑣, −Λ𝑠, 𝑆, −Σ, 𝐽, −Ω,𝑀𝐽⟩ . (1.17)

The number 𝑠 has to be introduced within the treatment of the electronic orbital
wave function. Because Σ± states have to be treated as special cases [45]. For Σ+
states as well as forΛ > 0, 𝑠 is an even number, for Σ− states it is odd. For wavefunc-
tions in Hund’s case (b) the relation is very similar

𝐸∗ |𝜂, 𝑣, Λ𝑠, 𝑁, Σ, 𝑆, 𝐽,𝑀⟩ = (−1)𝑁+𝑠 |𝜂, −Λ𝑠, 𝑁, −Σ, 𝑆, 𝐽,𝑀⟩ . (1.18)

From equation 1.17 and 1.18 one can see that Hund’s case (a) and (b) wave functions
are not eigenfunctions of 𝐸∗. To obtain wave functions with a well defined parity,
i.e. eigenfunctions of 𝐸∗ linear combinations of the respective wave functions have
to be formed. For wave functions in Hund’s case (a) basis they are given by [45]

|𝜂, Λ𝑠, 𝐽,𝑀;+⟩ = 1
√2

[|𝜂, Λ𝑠, 𝑆, Σ, 𝐽, Ω,𝑀⟩ + (−1)𝑝 |𝜂, −Λ𝑠, 𝑆, −Σ, 𝐽, −Ω,𝑀⟩] ,

(1.19)

|𝜂, Λ𝑠, 𝐽,𝑀;−⟩ = 1
√2

[|𝜂, Λ𝑠, 𝑆, Σ, 𝐽, Ω,𝑀⟩ − (−1)𝑝 |𝜂, −Λ𝑠, 𝑆, −Σ, 𝐽, −Ω,𝑀⟩] .

(1.20)

With exponent 𝑝 = 𝐽−𝑆+𝑠. The energy eigenvalues of twowave functions 1.19 and
1.20 are degenerate. However, Λ-type doubling lifts exactly this degeneracy causing

20



1.7 Electronic transitions and transition rules

Table 1.1: Overview of the parity labelling scheme introduced in [52]

𝐽 Parity Label 𝐽 Parity Label
Integer (−1)𝐽 𝑒 Half-Integer (−1)𝐽−1/2 𝑒
Integer (−1)𝐽+1 𝑓 Half-Integer (−1)𝐽+1/2 𝑓

a small energy splitting between the different parity levels. As long as this pertur-
bation stays small enough the two functions can still be considered to be eigenfunc-
tions of 𝐸∗. As a result of the parity conserving wavefunctions the parity alternates
between different J-levels. For example if for 𝐽 = 1/2 the parity |+⟩ is lower than
|−⟩ it will be vice versa for 𝐽 = 3/2. To avoid this, a different labelling scheme was
proposed [52]. Here the parity label is either 𝑒 or 𝑓. Following this notation the
lower and upper parity levels of any value of 𝐽 will always be denoted by either 𝑒 or
𝑓 and no longer alternate. Table 1.1 gives an overview about this labelling scheme
which is widely used nowadays. For Hund’s case (b) it follows that in Σ+ states, all
parity doublets have the same labels, i.e. 𝐹1 levels are all denoted 𝑒 and 𝐹2 levels are
all denoted 𝑓. In the case of Σ− states the labelling is reversed.

1.7 Electronic transitions and transition rules

Like for atoms the evaluation of the electric transition dipole moment yields a set
of rules determining if a certain transition is allowed or not. For molecules those
rules can be divided into general rules that will hold in any case and rules that will
only apply for a certain Hund’s case. The later only apply if both of the involved
electronic states belong to the same Hund’s case. Since some states may change
their Hund’s case depending on the rotational quantum number, the rules may even
change within a single electronic transition. Table 1.2 gives an overview on the dif-
ferent transition rules which hold either in general or for Hund’s case (a) and (b).
Transition rules for identical nuclei and nuclei with equal charge are omitted. They
can be found in [38]. The parity selection rules can also be rewritten in the 𝑒, 𝑓-

21



1 The energy structure of diatomic molecules

Table 1.2: Overview of selection rules and their validity for the different quantum
numbers and coupling cases in diatomic molecules as given by [38].

Affected Parameter Selection Rule Application

Total angular momentum 𝐽 Δ𝐽 = ±1, 0 & 𝐽 = 0 ↛ 𝐽 = 0 General

Parity ± + ↔ − & + ↮ + & − ↮ − General
Total orbital angular

momentum projection Λ ΔΛ = ±1, 0 Case (a) & (b)

Σ±-states
Σ+ ↔ Σ+
Σ− ↔ Σ−
Σ+ ↮ Σ−

Case (a) & (b)

Total electron spin 𝑆 Δ𝑆 = 0 Case (a) & (b)

Total spin projection Σ ΔΣ = 0 Case (a)
Total electronic angular
momentum along the
internuclear axis Ω

ΔΩ = ±1, 0 &
if Ω = 0 → Ω = 0:
Δ𝐽 = 0 forbidden

Case (a)

Total angular momentum
without spin 𝑁

Δ𝑁 = ±1, 0 &
for Σ → Σ transitions
Δ𝑁 = 0 forbidden

Case (b)

Total orbital angular
momentum 𝐿 Δ𝐿 = ±1, 0 Case (d)

Rotation 𝑅 Δ𝑅 = 0 Case (d)

22



1.8 The effective Hamiltonian approach

labelling scheme. In this scheme Δ𝐽 = 0 transitions require 𝑒 ↔ 𝑓 and Δ𝐽 = ±1
require 𝑒 ↔ 𝑒 and 𝑓 ↔ 𝑓 [52].

1.8 The effective Hamiltonian approach

The interaction between the different angular momenta leads to a rich and complex
energy structure for diatomic molecules. One way to treat such a complex system
is perturbation theory. Here, the Hamiltonian 𝐻 is separated into a strictly solvable
part 𝐻0 and a perturbation 𝐻1 which is only a small correction to 𝐻0.[53].

The separation of the Hamiltonian may prove to be difficult or in some cases even
impossible. An alternative approach is the effective Hamiltonian. This is essentially
a large matrix including terms for all the different angular momentum couplings
and interactions within the molecule. The matrix can also include terms describing
the interactionwith external fields. It can be diagonalised to obtain the energy eigen-
values. This method requires enough computational power to diagonalise large and
complex matrices.

For the effective Hamiltonian it is crucial to choose an appropriate set of basis states
and to include only a single electronic state. Dealing with several electronic states
at the same time will overcomplicate the Hamiltonian. However, exceptions exist.
When different adjacent electronic states perturb each other, it is necessary to in-
clude all those states to describe the system correctly.

In general, the matrix elements of the effective Hamiltonian have the form

⟨𝜓𝑓 ||Hint || 𝜓𝑖⟩ . (1.21)

With the initial state𝜓𝑖, the final state𝜓𝑓 and the sub-HamiltonianHint describing a
certain type of interaction. The matrix elements of the effective Hamiltonian given
here are all given for Hund’s case (a) basis states.

23



1 The energy structure of diatomic molecules

For the Wigner-3j and Wigner-6j symbols the following notation is used

Wigner 3j-symbol: (𝑎 𝑏 𝑐
𝑑 𝑒 𝑓) ,

Wigner 6j-symbol: {𝑎 𝑏 𝑐
𝑑 𝑒 𝑓} .

1.8.1 Fine structure effective Hamiltonian

The fine structure of a diatomic molecule depends mainly on two terms. The spin-
orbit interaction and the rotational term. The spin-orbit matrix elements are fully
diagonal in case (a) basis description and according to [45, 54] given by

⟨𝜓 |HSO | 𝜓′⟩ = 𝛿𝜂𝜂′𝛿ΛΛ′𝛿ΣΣ′𝛿ΩΩ′ΛΣ
× [𝐴 + 𝐴𝐷(𝐽(𝐽 + 1) − Ω2 + 𝑆(𝑆 + 1) − Σ2)] .

(1.22)

The spin-orbit constant is denoted as 𝐴, while 𝐴𝐷 is the constant to correct for the
centrifugal distortion of 𝐴.

To include the rotational energies the corresponding matrix element is [45, 54]

⟨𝜓 |HROT | 𝜓′⟩ = 𝛿𝜂𝜂′𝛿ΛΛ′𝛿ΣΣ′𝛿ΩΩ′ [2𝐵ΛΣ + 𝐵(𝐽(𝐽 + 1) + 𝑆(𝑆 + 1) + Λ2 − 2Ω2]

− 2𝐵 ∑
𝑞=±1

(−1)𝐽−Ω+𝑆−Σ ( 𝐽 1 𝐽
−Ω 𝑞 Ω′) (

𝑆 1 𝑆
−Σ 𝑞 Σ′)

× √𝐽(𝐽 + 1)(2𝐽 + 1)𝑆(𝑆 + 1)(2𝑆 + 1).
(1.23)

Here, the first term is again strictly diagonal while the sum over 𝑞 may give some
off-diagonal contributions. To correct for the centrifugal distortion two additional
matrix elements can be included depending on the two correction constants 𝐷 and
𝐻[54]:

⟨𝜓 |HROTCD | 𝜓′⟩ = −𝐷 (𝐻ROT)
2 + 𝐻 (𝐻ROT)

4 . (1.24)

24



1.8 The effective Hamiltonian approach

Inmany cases it is justified to omit the (𝐻ROT)4 term or even both of the terms, since
their energy contribution is negligible compared to ⟨𝜓 |𝐻ROT | 𝜓′⟩.

1.8.2 Λ-type doubling

Λ-type doubling is an effect that does not occur in an ideal Hund’s case (a) or (b)
picture, where the coupling between the total electronic angular momentum and
the rotation of the nuclei is neglected. If the speed of rotation is large, which is the
case for many molecules, this effect has to be taken into account [38].

The additional interaction lifts the degeneracy between the two symmetry compo-
nents + and − for Λ ≠ 0 as already stated in section 1.6. The splitting between the
two parity components for each 𝐽 is small but increases with increasing 𝐽 [38]. The
effect is usually on the order of MHz to GHz. For its description the three parame-
ters 𝑜, 𝑝, 𝑞 are introduced. For Π-states Λ-type doubling has been treated by Brown
and Merer [55], who give the following relations to obtain the matrix elements for
Λ-type doubling in Π-states

⟨𝜓 |HΛ | 𝜓′⟩ = ⟨∓1, Σ = Σ′ ± 2, 𝐽 = 𝐽′, Ω = Ω′ |HΛ | ±1, Σ′, 𝐽′, Ω′⟩

= 1
2(𝑜𝑣 + 𝑝𝑣 + 𝑞𝑣)√[𝑆(𝑆 + 1) − Σ(Σ ± 1)][𝑆(𝑆 + 1) − (Σ ± 1)(Σ ± 2)],

⟨𝜓 |HΛ | 𝜓′⟩ = ⟨∓1, Σ = Σ′ ± 1, 𝐽 = 𝐽′, Ω = Ω′ ∓ 1 |HΛ | ±1, Σ′, 𝐽′, Ω′⟩

= − 1
2(𝑝𝑣 + 2𝑞𝑣)√[𝑆(𝑆 + 1) − Σ(Σ ± 1)][𝐽(𝐽 + 1) − Ω(Ω ∓ 1)],

⟨𝜓 |HΛ | 𝜓′⟩ = ⟨∓1, Σ = Σ′, 𝐽 = 𝐽′, Ω = Ω′ ∓ 2 |HΛ | ±1, Σ′, 𝐽′, Ω′⟩

= 1
2𝑞√[𝐽(𝐽 + 1) − Ω(Ω ∓ 1)][𝐽(𝐽 + 1) − (Ω ∓ 1)(Ω ∓ 2)].

(1.25)

1.8.3 Spin-rotation coupling

Spin-rotation coupling occurs for Hund’s case (b), when the total electronic spin
couples to the total angular momentum without spin. This has been discussed in

25



1 The energy structure of diatomic molecules

detail in section 1.5.2 and is decribed by the constant 𝛾. The corresponding matrix
elements in Hund’s case (a) basis are given by [54] to be

⟨𝜓 |HSROT | 𝜓′⟩ = 𝛾𝛿𝐽,𝐽′ [𝛿Σ,Σ′𝛿Ω,Ω′(Σ2 − 𝑆(𝑆 + 1)) + ∑
𝑞=±1

(−1)𝐽′−Ω+𝑆−Σ

×√𝐽(𝐽 + 1)(2𝐽 + 1)√𝑆(𝑆 + 1)(2𝑆 + 1)

× ( 𝐽 1 𝐽′
−Ω 𝑞 Ω′) (

𝑆 1 𝑆
−Σ 𝑞 −Σ′) ] .

(1.26)

Since spin-rotational coupling is especially important forHund’s case (b), thematrix
element shall also be given in Hund’s case (b) basis [45]. Note that in Hund’s case
(b) basis the matrix element is fully diagonal.

⟨𝜓 |HSROT | 𝜓⟩ = 𝛾(−1)𝑁+𝐽+𝑆 {𝑆 𝑁 𝐽
𝑁 𝑆 1}√𝑆(𝑆 + 1)(2𝑆 + 1)𝑁(𝑁 + 1)(2𝑁 + 1).

(1.27)

The spin-rotational splitting can also be corrected for the centrifugal distortion with
the corresponding correction parameter 𝛾𝐷. The correction term depends also on
the rotational Hamiltonian [54].

⟨𝜓 |HSROT | 𝜓′⟩ = HSROT +
𝛾𝐷
𝛾 HROT ⋅HSROT. (1.28)

1.9 Hyperfine structure

The smallest structural details of the energy landscape in diatomic molecules are
called hyperfine structure. Like in atoms the hyperfine structure results from the
coupling between the total angular momentum of the molecule 𝐽 and the nuclear
spin 𝐼. The two angular momenta form the total angular momentum 𝐹 = 𝐽 + 𝐼.
The situation gets more complicated if the nuclei forming a diatomic molecule both

26



1.9 Hyperfine structure

have a non-zero nuclear spin. Yet, this is not the case for the present treatment of
nitric oxide and as a result omitted.

To fully describe the hyperfine structure several different interactions have to be
taken into account. The matrix elements summarised in this section were taken
from [45], where also a more detailed derivation of the individual matrix elements
can be found. Typographical errors that came to the attention of the author have
been corrected. All matrix elements are given in Hund’s case (a) basis.

The first interaction taken into account is the spin orbit interaction for the nuclear
spin 𝐼 with the total angular orbital momentum of the electrons 𝐿. The correspond-
ing hyperfine constants is denoted 𝑎 and the matrix element is given by [45]

⟨𝜓 ||HOrbit
HFS || 𝜓′⟩ = 𝑎Λ(−1)𝐽′+𝐹+𝐼 {𝐼 𝐽′ 𝐹

𝐽 𝐼 1}√𝐼(𝐼 + 1)(2𝐼 + 1)

× (−1)𝐽−Ω ( 𝐽 1 𝐽′
−Ω 0 Ω′)√(2𝐽 + 1)(2𝐽′ + 1).

(1.29)

The magnetic hyperfine interaction results form the interaction of the nuclear spin
𝐼 and the total electron spin 𝑆. For its description either the constant 𝑏 or the Fermi
contact constant 𝑏𝐹 can be employed. The matrix element including the latter one
is given by [45]

⟨𝜓 ||HMagnetic
HFS

|| 𝜓′⟩ = 𝑏𝐹(−1)𝐽
′+𝐹+𝐼 {𝐼 𝐽′ 𝐹

𝐽 𝐼 1}√𝐼(𝐼 + 1)(2𝐼 + 1)

×
1
∑
𝑞=−1

(−1)𝐽−Ω ( 𝐽 1 𝐽′
−Ω 𝑞 Ω′)√(2𝐽 + 1)(2𝐽′ + 1)

× (−1)𝑆−Σ ( 𝑆 1 𝑆
−Σ 𝑞 Σ′)√𝑆(𝑆 + 1)(2𝑆 + 1).

(1.30)

27



1 The energy structure of diatomic molecules

In addition to the magnetic interaction, the dipole-dipole interaction between the
nuclear spin and electron spin has to be included. It is included via the following
matrix element [45]

⟨𝜓 ||HDipole
HFS

|| 𝜓′⟩ = √30𝑔𝑠𝜇𝐵𝑔𝑁𝜇𝑁
𝜇0
4𝜋(−1)

𝐽′+𝐹+𝐼 {𝐼 𝐽′ 𝐹
𝐽 𝐼 1}√𝐼(𝐼 + 1)(2𝐼 + 1)

×
1
∑
𝑞=−1

(−1)𝐽−Ω+𝑞 ( 𝐽 1 𝐽′
−Ω 𝑞 Ω′)√(2𝐽 + 1)(2𝐽′ + 1)

×
1
∑

𝑞1=−1

2
∑

𝑞2=−2
( 1 2 1
𝑞1 𝑞2 −𝑞) (−1)

𝑆−Σ ( 𝑆 1 𝑆
−Σ 𝑞1 Σ′)

× √𝑆(𝑆 + 1)(2𝑆 + 1) ⟨𝜂, Λ || 𝐶2
𝑞2(𝜃, 𝜙)(𝑟−3) || 𝜂′, Λ′⟩ .

(1.31)

The remaining matrix element including 𝐶2
𝑞2 can be resolved with the following

definition which includes the spherical harmonic 𝑌 𝑘𝑞(𝜃, 𝜙) [45],

𝐶𝑘
𝑞 (𝜃, 𝜙) = √

4𝜋
2𝑘 + 1𝑌 𝑘𝑞(𝜃, 𝜙). (1.32)

Substituting thematrix element with the corresponding spherical harmonics allows
the identification of the dipolar hyperfine constants [45]

𝑡0 = 𝑔𝑆𝜇𝐵𝑔𝑁𝜇𝑁
𝜇0
8𝜋 ⟨

3 cos2(𝜃) − 1
𝑟3 ⟩

Π
, (1.33a)

𝑡±1 = 𝑔𝑆𝜇𝐵𝑔𝑁𝜇𝑁
𝜇0
4𝜋 ⟨

cos(𝜃) sin(𝜃)
𝑟3 ⟩

Π−Σ
, (1.33b)

𝑡±2 = 𝑔𝑆𝜇𝐵𝑔𝑁𝜇𝑁
𝜇0
4𝜋 ⟨

sin2(𝜃)
𝑟3 ⟩

Π
. (1.33c)

A different approach to treat the dipole-dipole interaction relies on the dipole-dipole
constants 𝑐 and 𝑑. The first one is diagonal in Λ while the second one in Λ/2. The

28



1.9 Hyperfine structure

constant 𝑐 also links 𝑏 and 𝑏𝐹 via the relation 𝑏 = 𝑏𝐹−(𝑐/3)[36]. The corresponding
matrix element is given in [54]

⟨𝜓 ||HAltDip
HFS

|| 𝜓′⟩ = 𝛿ΛΛ′(−1)𝐽′+𝐼+𝐹√(2𝐽 + 1)(2𝐽′ + 1)√𝐼(𝐼 + 1)(2𝐼 + 1)

× {𝐹 𝐽′ 𝐼
1 𝐼 𝐽} (

1
∑
𝑞=−1

(−1)𝐽−Ω ( 𝐽 1 𝐽′
−Ω 𝑞 Ω′) 𝑐

√30
3

× (−1)𝑞+𝑆−Σ√𝑆(𝑆 + 1)(2𝑆 + 1) ( 𝑆 1 𝑆
−Σ 𝑞 Σ′)

× ( 1 2 1
−𝑞 0 𝑞) ) − 𝑑(−1)𝐽′+𝐼+𝐹√(2𝐽 + 1)(2𝐽′ + 1)

× √𝐼(𝐼 + 1)(2𝐼 + 1) {𝐹 𝐽′ 𝐼
1 𝐼 𝐽} ( ∑

𝑞=±1
𝛿ΛΛ′−2𝑞(−1)𝐽−Ω

× (−1)𝑞+𝑆−Σ√𝑆(𝑆 + 1)(2𝑆 + 1) ( 𝐽 1 𝐽′
−Ω −𝑞 Ω′)

× ( 𝑆 1 𝑆
−Σ 𝑞 Σ′) )

+ 𝑐𝐼𝛿ΣΣ′𝛿ΛΛ′𝛿𝐽𝐽′
1
2[𝐹(𝐹 + 1) − 𝐼(𝐼 + 1) − 𝐽(𝐽 + 1)].

(1.34)

It also includes a term proportional to the constant 𝑐𝐼 which takes the nuclear spin-
rotation interaction into account.

29



1 The energy structure of diatomic molecules

The last interaction term that has to be discussed is the quadrupole term for the
nuclear spin. It is given by [45]

⟨𝜓 ||HQuadpole
HFS

|| 𝜓′⟩ = 𝛿ΣΣ′(−1)𝐽
′+𝐼+𝐹 { 𝐼 𝐽 𝐹

𝐽′ 𝐼 2} (
−𝑒𝑄
2 ) ( 𝐼 2 𝐼

−𝐼 0 𝐼)
−1

×
2
∑
𝑞=−2

(−1)𝐽−Ω ( 𝐽 2 𝐽′
−Ω 𝑞 Ω′)√(2𝐽 + 1)(2𝐽′ + 1)

⟨𝜂Λ || 𝑇2𝑞 (∇𝐄) || 𝜂′Λ′⟩ .

(1.35)

Here, the remainingmatrix element including the tensor of the electric field gradient
𝑇2𝑞 (∇𝐄) allows the defintion of its 𝑞 = 0,±2 components. These are given by [45]

⟨Λ || 𝑇20 (∇𝐄) || Λ′⟩ = −12 ⟨Λ
||||
∑
𝑖

𝑒𝑖
4𝜋𝜀0

3 cos2(𝜃𝑖)
𝑟3𝑖

||||
Λ′⟩ = −12𝑞0, (1.36a)

⟨Λ = ±1 || 𝑇2±2(∇𝐄) || Λ′ = ∓1⟩ = − 1
2√6

⟨Λ
||||
∑
𝑖

𝑒𝑖
4𝜋𝜀0

sin2(𝜃𝑖)
𝑟3𝑖

||||
Λ⟩

= − 1
2√6

𝑞2.
(1.36b)

The constants 𝑒𝑄 and 𝑞0, 𝑞2 can be cumulated to form the nuclear quadrupole con-
stants 𝑒𝑄𝑞0 and 𝑒𝑄𝑞2. Here, the first one is diagonal inΛ and the second one inΛ/2.
The constant 𝜀0 is the dielectric constant and 𝑒𝑖 the elementary charge of electron
𝑖.

1.10 Rydberg states

Rydberg states are already mentioned in the discussion of Hund’s case (d) in sec-
tion 1.5.3. They are atomic or molecular states where a single electron is excited,

30



1.10 Rydberg states

to a high principle quantum number [56]. Rydberg atoms or molecules can be de-
scribed as hydrogen like, since the inner electrons shield the excited electron from
the nucleus so that the other electrons are mainly influenced by the potential of a
single positive charge [56].

According to the Bohrmodel the energy of an electronwith principle quantumnum-
ber 𝑛 is given by [37]

𝐸𝑛 = − 𝑒2
4𝜋𝜀0

⋅ 1
2𝑎𝑜

⋅ 𝑍
2

𝑛2 = −𝑅𝑦𝑍
2

𝑛2 . (1.37)

With the elementary charges 𝑒, dielectric constant 𝜀0 and the Bohr radius 𝑎0 =
(4𝜋𝜀0ℏ2)/(𝑒2𝑚𝑒), including the electron mass 𝑚𝑒. The energy depends only on the
proton number 𝑍 and the principle quantum number. Here, the Rydberg constant
𝑅𝑦 = 13.6 eV is introduced. To describe Rydberg atoms or molecules equation 1.37
has to be altered. The principle quantum number is substituted by an effective prin-
ciple quantum number 𝑛′ = 𝑛 − 𝛿𝑛𝑙𝑗 . The quantum defect 𝛿𝑛𝑙𝑗 depends on the re-
spective Rydberg state. A thorough review on quantum defect theory was published
by Seaton [57].

Not every molecules seams to be suitable for the excitation to Rydberg states. And
according to Herzberg [58] along time the only known Rydberg molecules had been
He2. However, nowadays Rydberg states have been observed in many molecules,
like for example in H2S [59], H2O and D2O [60], ammonia [61] and even in large
hydrocarbons like tetracene (C18H12) [62]. A good overview, especially for the earlier
research on Rydberg molecules is provided by Herzberg [58].

31





2 Excitation scheme and
spectroscopic techniques

Introduction

In chapter 1 the general energy structure of diatomic molecules has been discussed.
Chapter 2 specifically focuses on the energy structure of nitric oxide and the two
spectroscopic techniques that were used within the scope of this thesis.

At first, nitric oxide is introduced and its general properties will be summarised in
section 2.1. Subsequently, the excitation scheme for the Rydberg excitation of nitric
oxide is presented and the structure of the involved electronic states is discussed in
section 2.2.1-2.2.3. The Rydberg excitation is crucial to employ optogalvanic spec-
troscoy which is described in section 2.3. The chapter ends with the description of
Doppler-free saturation spectroscopy and a brief overview on the respective broad-
ening mechanisms in section 2.4.

2.1 Nitric oxide

Nitric oxide is a colourless and odourless gas. It is toxic [63] and highly reactive
since it is a free radical. Its electron configuration in the ground state is

(1𝜎)2(2𝜎)2(3𝜎)2(4𝜎)2(5𝜎)2(1𝜋)4(2𝜋)1,

33



2 Excitation scheme and spectroscopic techniques

[45, 64] or rewritten in the hybridized picture

(1𝑠𝜎)2(1𝑠𝜎∗)2(2𝑠𝜎)2(2𝑠𝜎∗)2(2𝑝𝜎)2(2𝑝𝜋)4(2𝑝𝜋∗)1.

There are two stable isotopes of nitrogen 14N and 15N and three stable isotopes of
oxygen 16O, 17O and 18O. Thus, a total of six different isotopologues exist for nitric
oxide. 14N and 16O are by far the most abundant isotopes of nitrogen and oxygen
[65]. The most abundant isotopologue of nitric oxide is therefore 14N16O which is
subject to this study. It has a mass of 30u and a chemical bond length of 115 pm
[66]. Nitric oxide has a small permanent dipole moment, 𝐝NO ≈ 0.16D [66]. Its
total nuclear spin is 𝐼 = 1 resulting from the nuclear spin of 14N[67].

2.2 Excitation of NO to a Rydberg state

To perform optogalvanic spectroscopy in nitric oxide the molecules have to be ex-
cited to high lying Rydberg states. The addressed Rydberg states have to be so close
to the ionisation threshold, that the collisional energies are sufficient to ionise the
excited molecules. For the excitation to Rydberg states three transitions are driven.
The employed excitation scheme is similar to the scheme described in [68]. Doppler-
free spectroscopy was only performed on the lowest of the three transitions. An
overview about the excitation scheme is given in this section. The individual tran-
sitions are discussed in detail in the subsequent sections 2.2.1-2.2.3

Figure 2.1 shows the excitation scheme usedwithin the scope of this thesis. The first
transition from the X 2Π3/2 state to the A 2Σ+ state can not be avoided even though
thewavelength of 226nm is inconvenient to workwith. Since light in the deepUV is
difficult to generate and requires additional safety gear, to avoid skin exposure and
minimise the risk of skin cancer. It is not possible to avoid the A 2Σ+ state because
there is no electronic state in between the ground state X 2Π and A 2Σ+ which corre-
sponds to a single electron excitationwithin themolecule [69]. There are lower lying
electronic states where more than a single electron is excited within the molecule
[70]. The transition can be driven from both spin-orbitmanifolds of the ground state
since both of them are populated at room temperature [71].

34



2.2 Excitation of NO to a Rydberg state

The second transition in the excitation scheme is at a wavelength of around 540nm
between the A 2Σ+ state and the H 2Σ+ state. This transition requires a lot of power
due to the different l-character of the electronic wave function. Its decomposition
into partial waves for the corresponding states is given in [72]. The A 2Σ+ state has
94% s-character, 1% p-character and 4% d-character. The H 2Σ+ state has 38% s-
and 62% d-character. The H′ 2Π state could in principle be reached with the em-
ployed laser systems but is assumed to require even more power since it has 99% d-
and 1% p-character. A four-photon excitation scheme would be possible since there
are states in betweenA 2Σ+ andH 2Σ+ [73], that could be reachedwith infrared tran-
sitions. However, according to [74] these alternative excitation schemes may suffer
from predissociation and are therefore not suited for the use of optogalvanic spec-
troscopy. Especially, if the technique is used for sensing as it was explained in the
introduction. From the H 2Σ+ state a large number of Rydberg states are accessible.
Depending on which Rydberg state should be adressed the required wavelength is
between 833 and 835nm. This wavelength region is covered by several different
laser technologies. The choice of Rydberg state may offer a large optimisation po-
tential for the sensing application of optogalvanic spectroscopy.

2.2.1 The 𝐀 𝟐𝚺 ← 𝐗 𝟐𝚷𝟑/𝟐 transition in nitric oxide

Figure 2.3 shows a schematic of the energy level structure of the X 2Π ground state
of nitric oxide and the A 2Σ+ excited state.
The ground state is spin-orbit split by around 123 cm−1 [35]. It does neither belong
exactly to Hund’s case (a) nor to Hund’s case (b) [27]. It can be best described by
Hund’s case (a) as long as the rotational energy is small enough. For larger rotational
energies the Hund’s case (b) description is more appropriate. The respective transi-
tion from Hund’s case (a) to Hund’s case (b) can be estimated by 𝐽 (𝑎)→(𝑏) ≈ 𝐴/(2𝐵)
(for the respective constants see: table C.1.1) which yields that the transition occurs
at 𝐽 (𝑎)→(𝑏) ≈ 35.5 [51]. The transition between the two different coupling cases has
been studied in detail by Klisch et al. [75] using sub-millimetre wave spectroscopy.
The ground state shows Λ-type doubling for both spin-orbit manifolds, i.e. the de-
generacy of the two symmetry components ± is lifted. The splitting behaves differ-
ently for the two ground state manifolds. It can be calculated using the effective

35



2 Excitation scheme and spectroscopic techniques

E/(ℎ𝑐0)
in cm−1

0

123

44140

62705
62719

74721

X 2Π1/2

X 2Π3/2
X 2Π

A 2Σ+

H 2Σ+
H 2Π

𝑛𝑙 Σ

226nm

540nm

833-835nm

Figure 2.1: Excitation scheme for the Rydberg excitation of nitric oxide. For all elec-
tronic states only vibrational states with 𝑣 = 0 are addressed. The ro-
tational structure is omitted. The dashed line marks the first ionisation
limit.

36



2.2 Excitation of NO to a Rydberg state

Hamiltonian approach introduced in section 1.8 taking into account the matrix ele-
ments given in equation 1.22 and 1.23 for the spin-orbit and rotational fine structure
and in equation 1.26 and equation 1.25 for spin-rotation and Λ-type doubling. The
centrifugal distortion term in equation 1.22 can be neglected, thus 𝐴𝐷 is set to zero.
Note that in Hund’s case (a) the basis states are not parity conserving, i.e. do not
have a definite parity. Henceforth, the effective Hamiltonian has to be transformed
to the parity conserving base states which were introduced in section 1.6 (see. equa-
tion 1.19 and 1.20). Plugging in the respective constants from [35]which are given in
table C.1.1, yields the splittings that are plotted in figure 2.2. In theΠ1/2 component
the splitting is significantly larger then in the Π3/2 component and goes approxi-
mately linear with 𝐽X. For the Π3/2 manifold the splitting is proportional to 𝐽2X. The
hyperfine structure is not shown in figure 2.3 and will be discussed in more detail
in chapter 6.

1/2 11/2 21/2 31/2 41/2
Total angular momentum JX

2000

4000

6000

Fr
eq
ue
nc
y
(M

H
z)

X 2Π1/2

3/2 11/2 21/2 31/2 41/2
Total angular momentum JX

0

500

1000

X 2Π3/2

Figure 2.2: Calculated Λ-type doubling for the two spin-orbit manifolds of the
ground state X 2Π of nitric oxide.

The A 2Σ+ state strictly belongs to Hund’s case (b). It shows the characteristic spin-
rotational splitting. Here states with parity +, e belong always to the F1 manifold,
i.e. 𝐽 = 𝑁 + 1/2 and states with parity −, f to the F2 manifold for which 𝐽 = 𝑁 − 1/2
holds.

37



2 Excitation scheme and spectroscopic techniques

There are twelve allowed branches for the A 2Σ+ ← X 2Π transition. Each branch
consists of transitions between different rotational levels in the ground and excited
state. Figure 2.3 shows only a single rotational transition for each of the branches.
The notation for the branches translates as follows: e.g. Q21fe is a branch with tran-
sitions of Δ𝐽 = 0 denoted by the letter Q (R → Δ𝐽 = 1, P → Δ𝐽 = −1). The letters
in the subscript denote that the transition is from an e-level in the lower state to
an f-level in the upper state. The numbers in the subscript give the respective F-
component, i.e. 21 denotes a transition from an F1 component in the lower to a F2
component in the upper state. Here, caution is necessary since the definition of the
F-components is different for Hund’s case (a) and (b).

The P12ee-branch is of particular interest to this thesis, since all measurements were
performed on different rotational transitions of this branch. The branch lies in the
outermost part of the total spectrum, and does only overlap with other branches for
𝐽 > 19.5, so that the line assignment is particularly easy. In addition, it has already
been experimentally shown that the excitation from the X 2Π3/2 via the A 2Σ+ to the
H 2Σ+ state is possible and able to generate enough population in the H 2Σ+ state for
quantitative measurements [76]. This is crucial for the application of optogalvanic
spectroscopy for sensing.

2.2.2 The 3d-complex states𝐇 𝟐𝚺+ and𝐇′ 𝟐𝚷

The second transition in the presented excitation scheme involves the H 2Σ+ state
which cannot be treated without considering the H′ 2Π state as well. The two states
lie only approximately 14 cm−1 away from each other [77]. A level scheme for the
transition from the A 2Σ+ to the H 2Σ+ and H′ 2Π state is shown in figure 2.4. The
branch in which the particular transition of the presented excitation scheme lies is
highlighted by a green arrow. Both the H 2Σ+ and H′ 2Π states are best described by
the Hund’s case (b) coupling scheme [68]. There are in total six branches from the
A 2Σ+ to the H 2Σ+ state and twelve for the transition to the H′ 2Π state. The small
letter in front of the capital letter defining the Δ𝐽-type of a transition is giving the
type of transition for the Δ𝑁, thus rR for example means Δ𝑁 = +1 and Δ𝐽 = +1.
The closeness of the H 2Σ+ state and H′ 2Π state hinders the straight forward cal-

38



2.2 Excitation of NO to a Rydberg state

X 2Π1/2(F1)

JX

0.5 + e
− f

1.5 − e
+ f

2.5 + e
− f

3.5 − e
+ f

4.5 + e
− f

X 2Π3/2(F2)

JX

1.5 − e
+ f

2.5 + e
− f

3.5 − e
+ f

4.5 + e
− f

5.5 − e
+ f

A 2Σ+NA JA

0 0.5 + e F1
1 1.5

0.5
− e F1
− f F2

2 2.5
1.5

+ e F1
+ f F2

3 3.5
2.5

− e F1
− f F2

4 4.5
3.5

+ e F1
+ f F2

R 2
1f
f

R 1
1e
e

Q
21
fe

Q
11
ef

P 2
1f
f

P 1
1e
e

R 2
2f
f

R 1
2e
e

Q
22
fe

Q
12
ef

P 2
2f
f

P 1
2e
e

Figure 2.3: Schematic of the A 2Σ+ ← X 2Π transition showing the fine structure of
both states and the transition with lowest JX as an example for every of
the twelve allowed branches in the corresponding spectrum of the tran-
sition. The P12ee branch investigated as part of this thesis is highlighted
in violet. In the ground state the splitting between the + and − symme-
try component of each 𝐽, is the aforementioned Λ-type doubling. After
[27].

39



2 Excitation scheme and spectroscopic techniques

A 2Σ+

NA JA

0 0.5 + e F1
1 1.5

0.5
− e F1
− f F2

2 2.5
1.5

+ e F1
+ f F2

3 3.5
2.5

− e F1
− f F2

4 4.5
3.5

+ e F1
+ f F2

H 2Σ+NH JH

0 0.5 + e F1
1 1.5

0.5
− e F1
− f F2

2 2.5
1.5

+ e F1
+ f F2

3 3.5
2.5

− e F1
− f F2

4 4.5
3.5

+ e F1
+ f F2

H′ 2ΠNH′ JH′

1 Π−
0.5 + e F2
1.5 + f F1

1 Π+
1.5 − e F1
0.5 − f F2

2 Π−
1.5 − e F2
2.5 − f F1

2 Π+
2.5 + e F1
1.5 + f F2

3 Π−
2.5 + e F2
3.5 + f F1

3 Π+
3.5 + e F2
2.5 + f F1

rR
11
ee

rR
22
ff

rQ
21
fe

pQ
12
ef

pP
11
ee

pP
22
ff

qR
12
ff

rR
22
ff

qQ
11
fe

qQ
22
ef

pP
11
ee

qP
21
ee

Figure 2.4: Level scheme for the transition between the A 2Σ+ and H 2Σ+, H′ 2Π
states, respectively. For the transition between A 2Σ+ and H 2Σ+ all six
branches are shown. From the twelve branches existing for the transi-
tions between A 2Σ+ andH′ 2Π six exemplary branches are shown. After
[68].

40



2.2 Excitation of NO to a Rydberg state

culation of the molecular energy levels by simple effective Hamiltonian matrix ele-
ments. In fact the two states are so close to each other that they perturb each others
energetic structure. Their treatment therefore requires a set of off-diagonal interac-
tion parameters that describe the perturbing interaction and shift the energy levels
of the individual states accordingly. The perturbation within the 𝐻 and 𝐻′ state is
heterogeneous and was already studied in the sixties by Huber, Miescher [78] and
Kovács [79]. The𝐻 and𝐻′ states have mostly d-character [75] and are also referred
to as 3𝑑𝜎 and 3𝑑𝜋 states, respectively. They belong to a set of states referred to as
3d-Rydberg complex which also includes the 3𝑑𝛿-state 𝐹 2Δ which is additionally
interacting with higher vibrational states of the 𝐵′ 2Δ state. In addition to the 3𝑑-
states also the close lying 4𝑠𝜎-state 𝐸 2Σ+ is involved forming an s-d supercomplex
[77]. The full system has been described by Bernard et. al. in [77].

If one wants to calculate the energy levels of the𝐻 and𝐻′ state it is sufficient to use
a 5 × 5-matrix description neglecting the interaction with the 𝐵′ 2Δ and 𝐸 2Σ+ [68].
In fact Bernard et al. were not able to accurately determine the interaction constant
linking the 4𝑠𝜎 and 3𝑑𝜎-state, since it seems to be too small to be determined from
their data [77]. The description is given in Hund’s case (a) basis in equation 2.1. The
off diagonal matrix elements including the interaction parameters 𝛼, 𝛽, 𝜂 and 𝜉were
retrieved from [77] while the diagonal elements and the matrix elements 𝑀12,𝑀34
can be found in [80].

⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝

𝐹 2Δ5/2 𝐹 2Δ3/2 𝐻′ 2Δ3/2 𝐻′ 2Δ1/2 𝐻 2Σ+

𝑀11 −𝐵F√𝑋 + 2𝐷F𝑋3/2 −𝛽√𝑌 0 0

... 𝑀22 𝛼 + 𝛽 −𝛽√𝑍 0

... ... 𝑀33 −𝐵H′√𝑋 + 2𝐷H′𝑋3/2 −𝜂√𝑍

... symmetric ... 𝑀44 𝜉 + 𝜂𝑀

... ... ... ... 𝑀55

⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠

(2.1)

41



2 Excitation scheme and spectroscopic techniques

The diagonal elements𝑀ii are given by the following equations

𝑀11 = 𝑇F + 𝐴F + 𝐵F(𝑋 − 2) − 𝐷F((𝑋 − 2)2 + 𝑋), (2.2)
𝑀22 = 𝑇F − 𝐴F + 𝐵F(𝑋 + 2) − 𝐷F((𝑋 + 2)2 + 𝑋), (2.3)

𝑀33 = 𝑇H′ + 1
2𝐴H′ + 𝐵H′(𝑋 − 1) − 𝐷H′((𝑋 − 1)2 + 𝑋), (2.4)

𝑀44 = 𝑇H′ − 1
2𝐴H′ + 𝐵H′(𝑋 + 1) − 𝐷H′((𝑋 + 1)2 + 𝑋), (2.5)

𝑀55 = 𝑇H + 𝐵H𝑥(𝑥 ∓ 1) − 𝐷H𝑥2(𝑥 ∓ 1)2. (2.6)

The upper sign in equation 2.6 refers to e-sublevels and the lower sign to f-sublevels.
The following abbreviations were used for clarity in equation 2.1 - 2.6

𝑋 = (𝐽 + 1/2)2 − Λ2, 𝑥 = 𝐽 + 1/2,
𝑌 = (𝐽 − 3/2)(𝐽 + 5/2), 𝑍 = (𝐽 − 1/2)(𝐽 + 3/2)
𝑀 = 𝐽 + 3/2, for f-sublevels, 𝑀 = − (𝐽 − 1/2), for e-sublevels.

The interaction parameters 𝛼, 𝛽, 𝜂 and 𝜉 are given in [77] and listed in table C.3.4.
Interactions between different vibrational levels of the states can be neglected due to
the large energetic distance amongst them [68]. Plugging in the respective constants
and interaction parameters, which are listed in table C.3.1, C.3.2 and C.3.3 in the ap-
pendix, yields twomatrices, one for e-parity sublevels and one for f-parity sublevels.
Diagonalisation of the two matrices gives then the corresponding eigenenergies for
the different states. In figure 2.5 the calculated values for a direct transition from
the X 2Π state to the H 2Σ+ state are compared to a large set of measured data given
in [78]. The energy values are given with respect to the X 2Π1/2 𝐽 = 1/2 level. The
ground state energies which are needed to calculate the different branches were cal-
culated like described in section 2.2.1 for the calculation of theΛ-type doubling. The
agreement between calculation and the plotted data is excellent.

42



2.2 Excitation of NO to a Rydberg state

62550 62600 62650 62700 62750 62800 62850
Energy 𝐸/(ℎ𝑐0) (cm−1)

0

5

10

15

20

To
ta
la
ng
ul
ar
m
om

en
tu
m
𝐽 𝑋

Data
Calculation

P11 P21P22 R11 R21R22

Figure 2.5: Comparison of calculated and measured data for the energy levels of the
H 2Σ+ state with respect to the lowest rotational state X 2Π1/2 𝐽𝑋 = 1/2.
The branches are those of the direct transition from the X 2Π1/2 to the
H 2Σ+. The measured data was retrieved from [78].

43



2 Excitation scheme and spectroscopic techniques

2.2.3 Excitation to high lying Rydberg states of nitric oxide

The excitation to high lying Rydberg states from the H 2Σ+ state should allow the
excitation of p,d and f states due to the d-character of the excited electron in the
H 2Σ+ state. However, since the H 2Σ+ state has also partial s-character transitions
into s-states may also be possible. The excitation targets Rydberg states of the lowest
Rydberg series converging to the ground state of the resulting cationNO+. However,
the exact assignment of principle quantumnumbers is not yet possible but subject of
ongoing research on the presented excitation scheme [81]. As this thesis focuses on
high resolution spectroscopy on the lowest transition in the excitation scheme and
on the general implementation of the Rydberg excitation, precise knowledge on the
addressed Rydberg states is at this point not necessary.

2.3 Optogalvanic spectroscopy

As described in the beginning, optogalvanic spectroscopy is the basic principle used
to realise a new type of gas sensor for nitric oxide as explained in the introduction.
The working principle of this technique is illustrated in figure 2.6.

Pure nitric oxide or a mixture of nitric oxide and one or several background gases
flow continuously through the spectroscopy cell. The cell is equipped with elec-
trodes which are later needed to retrieve the generated signal. The electrode con-
figuration inside the cell is depicted in a very simplified way in figure 2.6. Since the
electrode design may directly influence the measured signal it has been subject to
investigation [82].

The process of generating the signal can be explained in three steps. These steps
usually take place continuously and at the same time.

The first step is designated “Excitation” and shown on the top left of figure 2.6.
Gaseous nitric oxide shown in red and blue and a background gas, for example nitro-
gen, shown in grey flow through the cell. The nitric oxide in the mixture is excited
to a Rydberg state by continuous wave laser systems. The corresponding excita-
tion scheme has been explained in section 2.2 and is depicted in the top center of

44



2.3 Optogalvanic spectroscopy

External
amplifier

Uoff

In Out

1 Excitation

e−

External
amplifier

Uoff

In Out

2 Collision

e−

NO+

External
amplifier

U(t)

Uoff

In Out

3 Detection

226nm

540nm

833–835nm

X 2Π3/2

A 2Σ+

H 2Σ+

𝑛𝑙 Σ

Figure 2.6: Schematic of the working principle of the optogalvanic spectroscopy
which can be used for gas sensing as explained in the introduction. The
three steps “Excitation”, “Collision” and “Detection” happen simultane-
ous but are depicted separately for clarity.

45



2 Excitation scheme and spectroscopic techniques

figure 2.6. The laser beams are set up in such a fashion that the two longer wave-
lengths are counter-propagating with respect to the UV-beam. In the second step,
“Collision”, the excited Rydberg molecules collide with the background gas or other
nitric oxide molecules. The Rydberg electron is so weakly bound that the collisional
energy is high enough to ionise the molecule. This process generates charges inside
the cell. The generated charges have to be retrieved to generate a signal. This is the
last step: “Detection”. To retrieve the charges from the cell a small potential has to
be applied to the electrodes, so that the charges are separated and collected by the
respective electrode. One electrode is connected to a transimpedance amplifier, so
that the current generated by the retrieved charges can be amplified and converted
to a voltage signal U(t).

In fact, this method is not limited to Rydberg states. Charges are already generated
if only the first transition of the excitation scheme is driven [71]. However, as ex-
plained in [71] the ionisation mechanism is different. For the employment of this
spectroscopy technique for sensing applications the Rydberg excitation is crucial,
since it generates larger currents and increases the selectivity of the proposed sens-
ing scheme [10]. The application of this technique as a sensing scheme has been
investigated in a proof of concept study [9] and demonstrated in a model system [6].
The three photon excitation has been demonstrated and analysed in [8, 71]. The us-
age of an external amplifier has the disadvantage that itmay pick up additional noise
by the cables connected to the readout electrodes. It is in principle possible to use
on-board amplifiers which are directly put onto the spectroscopy cell [83]. However,
this requires a more elaborate electronic design and special fabrication techniques
for the corresponding cell. This is subject of ongoing research (F. Munkes, PhD
Thesis, in preparation) and will not be further discussed here.

2.4 Doppler-free absorption spectroscopy

Optogalvanic spectroscopy can be applied for gas sensing and is very useful for the
spectroscopy of higher lying excited states particularly Rydberg states were the gen-
erated current gives a large signal. However, to resolve the hyperfine structure of
a given transition or state, optogalvanic spectroscopy provides not the necessary

46



2.4 Doppler-free absorption spectroscopy

resolution. To resolve the hyperfine structure Doppler-free saturated absorption
spectroscopy can be employed. This technique allows a resolution well below the
Doppler width by using two counter propagating laser beams. In this section first
Doppler broadening will be briefly discussed before Doppler-free saturation spec-
troscopy is explained. The explanations follow [37].

2.4.1 Doppler Broadening

Doppler broadening is an inhomogeneous broadeningmechanism that results from
the thermal movement of atoms or molecules interacting with the laser. A more
detailed treatment can be found in [37]. The effect is exactly the same for atoms
andmolecules. Even though in the following explanation the termmolecule will be
used exclusively.

The thermal movement of molecules leads to a frequency shift of the absorbed laser
lightwhich depends on the velocity 𝑣 of themoleculeswith respect to the laser beam.
Movement of amolecule towards the laser beamwill result in a blue shift of the laser
light in the rest frame of the molecule. For movement away from the laser beam the
laser light is shifted to the red. Qualitatively, the effect is analogous to the Doppler
effect experienced for sound waves. For a laser beam of frequency 𝜔L the frequency
shift in the reference frame of a moving molecule is given by

𝜔Red = 𝜔L − 𝑘𝑣, 𝜔Blue = 𝜔L + 𝑘𝑣. (2.7)

Only the velocity component along the wave-vector 𝐤 is contributing to the Doppler
shift. For simplicity the problem is treated in one dimension, so that the scalars
𝑘 and 𝑣 can be used. Consequently (𝜔 − 𝜔0)/𝜔0 = 𝑣/𝑐 holds with the resonance
frequency 𝜔0 in the rest frame of the molecules.

The velocities in an ensemble of thermal molecules are Maxwell-Boltzmann dis-
tributed, thus not all the molecules have the same velocity. The corresponding ve-
locity distribution is given by [37]

𝑓(𝑣)d𝑣 = √
𝑚

2𝜋𝑘𝐵𝑇
exp (− 𝑚𝑣2

2𝑘𝐵𝑇
) d𝑣. (2.8)

47



2 Excitation scheme and spectroscopic techniques

Here,𝑚 the mass of the molecules, 𝑘𝐵 is the Boltzmann constant and 𝑇 the temper-
ature.

From the thermal distribution and the relation between frequency and velocity the
line-shape function can be derived. It is Gaussian and given by [37]

𝑔D =
𝑐
𝜔0

⋅ √
𝑚

2𝜋𝑘𝐵𝑇
exp (− 𝑚𝑐2

2𝑘𝐵𝑇
(𝜔 − 𝜔0

𝜔0
)
2
) . (2.9)

The corresponding frequency shift of the laser in the reference frameof themolecule,
can then be calculated from the full width at half maximum of equation 2.9 and is

Δ𝜔𝐷 = 𝜔0
𝑐 √

8 log(2)𝑘𝐵𝑇
𝑚 . (2.10)

From equation 2.10 one can deduce that heavier atoms or molecules will allow nar-
rower linewidth than lighter atoms or molecules. Nitric oxide has a mass of only
30u [66]. Plugging in the corresponding numbers for the ground state transition
at roughly 226nm and a temperature of 293K results in a Doppler broadening of
𝜔226 ≈ 2𝜋 × 3GHz. This has been experimentally confirmed during investigations
on different transitions between the X 2Π1/2 and A 2Σ+ state with our narrow band
continuous wave laser system [7].

2.4.2 Saturated absorption spectroscopy

To understand the working principle of saturated absorption spectroscopy the ab-
sorption cross section for Doppler broadening is required. The following derivation
summarises the work of [37]. First, it is necessary to take into account that the ab-
sorption of a molecule at rest is not arbitrarily sharp but has a certain minimum
linewidth Γ also denoted as natural linewidth or decay rate. Γ broadens the absorp-
tion line homogeneously, thus affects every molecule in the same way. The corre-
sponding line shape function is Lorentzian. The absorption coefficient 𝜅(𝜔) is given
by [37]

𝜅(𝜔) = ∫𝑁𝜎(𝜔 − 𝑘𝑣)𝑑𝑣. (2.11)

48



2.4 Doppler-free absorption spectroscopy

It depends on the number density of the molecules which is denoted as 𝑁(𝑣) and
the absorption cross section 𝜎(𝜔 − 𝑘𝑣). Plugging in the Lorentzian absorption line
shape and the number density yields [37]

𝜅(𝜔) = 𝑔2
𝑔1
𝜋2𝑐2
𝜔20

𝐴21𝑁∫𝑓(𝑣) Γ/(2𝜋)
(𝜔 − 𝜔0 − 𝑘𝑣)2 + (Γ2/4) . (2.12)

The Gaussian function 𝑓(𝑣) here is convoluted with the Lorentzian line shape func-
tion. The convolution of a Gaussian and Lorentzian function results in a line shape
profile denoted as Voigt profile. However, the Doppler width is in almost any case
way larger then the Lorentzian linewidth Γ. Only for low temperatures this does not
hold true. Therefore, it is possible to treat the Lorentzian absorption line shape in
the integral as a Dirac delta function with argument 𝜔 − 𝜔0 − 𝑘𝑣. The result of the
integration is, that the integral turns into 𝑔𝐷. The Doppler broadened absorption
cross section is then identified as [37]

𝜎(𝜔) = 𝑔2
𝑔1
𝜋2𝑐2
𝜔2 𝐴21𝑔𝐷(𝜔) (2.13)

The two factors 𝑔2 and 𝑔1 take into account the degeneracy of the two involved states.
Thus if a state is non-degenerate 𝑔𝑖 = 1 holds. 𝐴21 is the Einstein-A coefficient for
spontaneous emission.

Now, that the absorption cross section for a Doppler broadened transition has been
introduced, the technique of saturated absorption spectroscopy can be explained.
This exploits the physical principle that absorption cannot increase limitless but will
saturate for a certain laser intensity [37]. Assume again two states with number
densities𝑁1(𝑣) for the lower and𝑁2(𝑣) for the higher energy level. The difference in
population between the two levelswill not changemuch as long as the laser intensity
stays low enough. Themajority of themolecules will be in the lower state. However,
if the laser intensity equals the saturation intensity or is at least close to saturation,
the difference in the number density is high enough that 𝑁1(𝑣) ≈ 𝑁(𝑣) is not valid
any more. Henceforth, equation 2.11 changes to

𝜅(𝜔) = ∫(𝑁1(𝑣) − 𝑁2(𝑣))𝜎(𝜔 − 𝑘𝑣)𝑑𝑣. (2.14)

49



2 Excitation scheme and spectroscopic techniques

Figure 2.7: Schematic of the working principle of Doppler-free saturated absorption
spectroscopy. The laser beam is divided into two beams which are sent
through the sample counter propagating. The probe beam is depicted
with half the linewidth of the pump beam. As a detector a photodiode is
usually sufficient.

The change in the absorption coefficient can be probed by a laser beam with con-
siderably less intensity than the saturation intensity. Saturated absorption spec-
troscopy therefore uses two laser beamsof the same laser source. A simplified scheme
of the experimental setup is depicted in figure 2.7. The laser beam is split into two
beams which need to be overlapped in the cell containing a gas of molecules. The
best results are achieved when both beams overlap perfectly. One of the two beams
has to be set to an intensity close to the saturation intensity, it is denoted as pump
beam and will change 𝜅(𝜔) as discussed previously. The second beam has to be con-
siderably weaker in intensity than the pump beam and is denoted as probe beam.

The pump beam basically burns a hole in the population density 𝑁1(𝑣) of the lower
states. The probe beam does also excite some of the particles in the cell to the ex-
cited state but only few compared to the pump beam. The situation is illustrated
in figure 2.8. Since both laser beams have the same frequency molecules with a ve-
locity component along the optical axis will experience different Doppler shifts for
the two beams, i.e. absorb different frequencies. This is depicted in column (a) and
(c) of figure 2.8. However, for molecules that do not move or that have only velocity
components that are perpendicular to the optical axis both laser beams are absorbed
at the same frequency. Here the probe laser will sense the change in the optical ab-
sorption coefficient. The absorption of the probe laser is reduced due to the large
amount of molecules that are excited by the pump laser. The reduced absorption of
the probe laser leads to a dip with Lorentzian line shape within the Doppler broad-

50



2.4 Doppler-free absorption spectroscopy
𝑁 2
(𝑣
)

𝑁 1
(𝑣
)

Velocity 𝑣

(a) (b) (c)

Po
pu
la
tio
n
D
en
sit
y

Figure 2.8: Schematic of the effect of the pump beam drawn in red and the probe
beam drawn in yellow on the population densities𝑁1(𝑣) and𝑁2(𝑣) of the
ground and excited state in a two level system. After [37].

ened line. These dips are named Lamb-dips. The width of the Lamb-dips depends
on the homogeneous broadening mechanisms such as the natural linewidth, power
and pressure broadening. Saturated absorption spectroscopy is an ideal tool to di-
rectly resolve hyperfine transitions that would otherwise be hidden in the Doppler
broadened linewidth. Doppler free saturated absorption spectroscopy can be ap-
plied to a multitude of different atoms and molecules as long as narrow band lasers
are available for the respective transition [84–87].

If there are two adjacent transitions with a common ground state but different ex-
cited states and the separation between the transitions is less than theDopplerwidth,
so called crossover resonances occur. Assume the population density of the second
excited state to be𝑁3(𝑣). If the pump beam is on resonance with the transition from
the ground state to the second excited state it will burn a hole in the ground state
population 𝑁1(𝑣) by increasing the population density 𝑁3(𝑣). If at the same time
the probe laser is on resonance with the other transition i.e. increasing𝑁2(𝑣), its ab-
sorption will decrease because it is only sensitive to changes in the population of the
ground state. The situation were the pump and probe laser are on resonance with
two different transitions corresponds to the two lasers being absorbed by different

51



2 Excitation scheme and spectroscopic techniques

velocity classes. The problem is strictly symmetric, thus the crossover resonancewill
appear exactly in the middle between the two actual transitions [37]. This usually
allows to easily identify crossover resonances in a spectrum.

2.4.3 Additional broadening effects

In addition to the homogeneous broadening Γ denoted as the natural line width and
the Doppler broadening further broadening effects occur. Namely these are transit
time broadening, pressure broadening and power broadening. A short summary of
these effect is given here. More detailed information is given in textbooks like [37,
88, 89].

Transit time broadening is sometimes also referred to as time-of-flight broadening.
It takes into account that the interaction time of a molecule with the laser is finite.
Due to their thermal movement molecules or atoms typically will not stay in the
volume illuminated by the laser beam. The resulting broadening is given in [89]
and is

Γ𝑇 =
4 ̄𝑣(𝑇)
𝑑 √2 log(2). (2.15)

Here, 𝑑 is the diameter of the laser beam and ̄𝑣(𝑇) is the mean thermal velocity
which can be derived from the Maxwell-Boltzmann distribution, given in equa-
tion 2.8, by calculating its expectation value. It is given by

̄𝑣(𝑇) = √
8𝑘𝐵𝑇
𝜋𝑚 . (2.16)

Here,𝑚 is the mass of the molecule, 𝑇 the temperature and 𝑘𝐵 the Boltzmann con-
stant.

Pressure broadening is often also referred to as collisional broadening. Collisions
between particles lead to a decrease in the lifetime of the excited state, since they
may induce transitions to the ground or other excited states. The corresponding
decay rate can be estimated from the mean free velocity of the particles which is
given in equation 2.16 [89]

ΓPress = 𝑁𝜎 ̄𝑣 (2.17)

52



2.4 Doppler-free absorption spectro