Wesleyan University
Physics Department
Auto Ionization and Radiative Lifetime
Measurements on Some Na2 Excited States
by Laser Spectroscopy
by
Roy B. Anunciado
Faculty Advisor: Dr. Lutz Hüwel
A dissertation submitted to
the faculty of Wesleyan University
in partial fulfilment of the requirements
for the degree of Doctor of Philosophy
Middletown, Connecticut
February 2013
Acknowledgement
I express my utmost gratitude to my research advisor and mentor, Dr. Lutz Hüwel,
for his continuous guidance, support, motivation and tireless help he showed
during the course of my Ph.D. years. Thank you Lutz for your undying patience.
Lutz’s in-depth knowledge and skills on atomic and molecular physics as well as
in the lab has been very beneficial to me.
I would like to acknowledge my committee members, Dr. Stewart and Dr. Blümel
for their critical remarks and suggestions that helped strengthen my skills and this
manuscript. To Dr. Morgan, you inspired me in so many ways. Thank you to the
professors of the Physics Department for sharing your knowledge through courses
and academic discussions.
Special thanks to the machine shop people, David, Bruce and Tom for the
assistance whenever I need some parts for fabrication, to Merik for sharing his
expertise in electronics, and to Vacek for helping me finding ways and means in
those times when nothing seems to work. To Anna Milardo and Erinn Savage,
thank you for the assistance on all administrative work.
It is my pleasure to also acknowledge my current and old colleagues and friends in
Wesleyan University, especially to Shima, Saman, Susantha, Ramesh, Beatriz,
Fernando, Nam and Iulian for their support and warm friendship. My years of stay
at Wes won’t be that fun without you guys.
I am grateful to all my Filipino friends around the Middletown area, your
friendship means a lot to me. Thank you for treating me as a family, you provided
me a very comfortable environment in the area and in many situations; you
become the ways and means. I couldn’t ask for more with you guys. Many thanks
to my closest friends Ryan and Ben!
To my tatay and nanay, all these years you are my very source of inspiration. You
taught me how to dream, to hope and to believe. I thank my siblings ate, manong,
kokoy and bb, your endless support means a lot. I dedicate this thesis to you all,
the most important and special people in my life.
Above all, I bring back all the glory, honor, praises and adoration to God!
ii
Table of Contents
1
Introduction .......................................................................................................... 1
2
Experimental Set-up........................................................................................... 10
2.1
3
2.1.1
Vacuum system .............................................................................. 13
2.1.2
Molecular Beam Source ................................................................. 14
2.1.3
Characterization ............................................................................. 16
2.2
Laser System .......................................................................................... 20
2.3
Time-Of-Flight Spectrometer, Detection and Data acquisition ............. 23
2.3.1
Time-of-flight spectrometer ........................................................... 24
2.3.2
Detection ........................................................................................ 26
2.3.3
Data acquisition ............................................................................. 27
Non-Resonant Multi-Photon Ionization ............................................................. 32
3.1
Experimental Method............................................................................. 33
3.2
Summary of Findings ............................................................................. 36
3.3
Analysis of Data ..................................................................................... 41
3.3.1
Channel I: Direct Dissociative Ionization of Na2 .......................... 48
3.3.2
Channel II: 1-Photon Dissociation of Na2+ .................................... 50
3.3.3
Channel III: 2-Photon Dissociation of Na2 .................................... 53
3.3.4
Channel IV: Indirect Ionization via Repulsive Rydberg States ..... 58
3.4
4
Vacuum system and molecular beam source ......................................... 13
Summary ................................................................................................ 73
Radiative Lifetimes on Some Excited Na2 States .............................................. 75
4.1
Transition Probability ............................................................................ 77
iii
Table of Contents
4.2
The 21Σu+ and 41Σg+ of Na2 .................................................................... 81
4.3
Excitation Scheme ................................................................................. 84
4.4
Excited Atomic Na Lifetime .................................................................. 89
4.4.1
Lifetime Measurement of Na 3p2P ................................................ 90
4.4.2
Lifetime Measurement of Na 4p2P ................................................ 92
4.5
5
Excited Molecular Na2 Lifetimes........................................................... 98
4.5.1
Lifetime Calculations ..................................................................... 98
4.5.2
Lifetime Measurement of 21Σu+ ................................................... 108
4.5.3
Lifetime Measurement of 41Σg+ ................................................... 115
4.6
Discussion ............................................................................................ 124
4.7
Summary .............................................................................................. 128
Case Study: Shallow Well of the Na2+ 12Σu+ ................................................... 129
5.1
Introduction .......................................................................................... 131
5.2
ZEKE Background ............................................................................... 134
5.3
Experimental Method........................................................................... 143
5.4
Calculations.......................................................................................... 134
5.5
Experimental Data ............................................................................... 145
5.6
Discussion ............................................................................................ 151
5.7
Summary .............................................................................................. 152
6
Summary .......................................................................................................... 153
7
References ........................................................................................................ 157
iv
List of Figures
Figure 2.1 Schematic diagram of the general experimental set-up ........................ 12
Figure 2.2 Schematic diagram of the molecular beam source ............................... 14
Figure 2.3 REMPI: A1Σu+  X1Σg+ ....................................................................... 17
Figure 2.4 Schematic of the TOF spectrometer. .................................................... 23
Figure 2.5 Schematic of the MCP circuitry. .......................................................... 27
Figure 2.6 Laser Temporal Profile ......................................................................... 30
Figure 3.1 Simplified Na2 potential diagram. ........................................................ 34
Figure 3.2 Laser Power Dependence ..................................................................... 35
Figure 3.3 Typical TOF spectra. ............................................................................ 36
Figure 3.4 Comparison of normalized atomic peaks. ............................................ 39
Figure 3.5 Na2+ FWHM laser energy dependence. ................................................ 45
Figure 3.6 532 nm Na+ Simulation vs. Experiment ............................................... 46
Figure 3.7 355 nm Na+ Simulation vs. Experiment ............................................... 48
Figure 3.8 FCF distribution for the 11Ʃg+  X1Σg+. ............................................. 50
Figure 3.9 Channel II Experiment vs. Calculation. ............................................... 53
Figure 3.10 Simplified Na2 potential diagram ....................................................... 54
Figure 3.11 Recoil anisotropy for rapid dissociation ............................................. 57
Figure 3.12 Model .................................................................................................. 60
v
List of Figures
Figure 3.13 Rydberg Potentials.............................................................................. 62
Figure 3.14 FC density of Rydberg potentials ....................................................... 64
Figure 3.15 TOF: Constant and linear Γ(R) ........................................................... 67
Figure 3.16 TOF quadratic Γ(R) ............................................................................ 69
Figure 3.17 Potential width functions, Γ(R) .......................................................... 70
Figure 4.1 Two-level system.................................................................................. 77
Figure 4.2 Na2 21Σu+ and 41Σg+ States .................................................................... 83
Figure 4.3 Excitation Scheme 1 ............................................................................. 85
Figure 4.4 Excitation Scheme 2 ............................................................................. 87
Figure 4.5 Schematic diagram of the three-laser experiment. ............................... 88
Figure 4.6 REMPI: 3p2P State ............................................................................... 91
Figure 4.7 Population decay of 3p2P1/2 and 3p2P3/2 plotted in semi-log scale....... 92
Figure 4.8 REMPI: 4p2P ........................................................................................ 93
Figure 4.9 4p2P state radiation cascade.................................................................. 94
Figure 4.10 4p2P Population decay ........................................................................ 97
Figure 4.11 Simplified potential diagram of Na2 ................................................. 101
Figure 4.12 21Σu+ state calculation ....................................................................... 103
Figure 4.13 Calculated lifetime of 21Σg+ and 11Πg+ states ................................... 105
vi
List of Figures
Figure 4.14 41Σg+ state calculation ....................................................................... 107
Figure 4.15 Na2 1+1 REMPI................................................................................ 110
Figure 4.16 Na2 1+1 REMPI: 21Σu+ (v = 25) ← X1Σg+ (v=0) .............................. 110
Figure 4.17 Typical Na2 TOF spectra for three different probe laser delays. ...... 112
Figure 4.18 Population decay curves ................................................................... 114
Figure 4.19 Lifetime of 21Σu+ as a function of vibrational level, v. ..................... 115
Figure 4.20 Na2 1+1 REMPI: A1Σu+ (v = 20)  X1Σg+ (v = 0) ........................... 116
Figure 4.21 Double resonance Na+ yield spectrum ............................................. 118
Figure 4.22 Na+ TOF of 41Σg+ v = 52, J = 19 ...................................................... 120
Figure 4.23 Population decay curves of 41Σg+ (v = 43, 44, 45, 49). .................... 122
Figure 4.24 Experimental lifetime measurement of the 41Σg+ Na2 State ............. 123
Figure 5.1 Shallow well of the 12Σu+ state of Na2 ................................................ 133
Figure 5.4 Absolute energy level difference ........................................................ 135
Figure 5.5 A  X FCF ........................................................................................ 136
Figure 5.6 The shallow well of the 21Σu+ state with vibrational energy .............. 138
Figure 5.2 Schematic representation of the ZEKE process. ................................ 142
Figure 5.3 Excitation Scheme .............................................................................. 144
Figure 5.7 Yield Spectra ...................................................................................... 147
vii
List of Figures
Figure 5.8 Electron TOF comparison for two different laser excitation lines. .... 147
Figure 5.9 ZEKE signal ....................................................................................... 150
viii
List of Tables
Table 2-1 Typical heating up ................................................................................. 15
Table 3-1 Fragment kinetic energy estimates ........................................................ 41
Table 3-2 Two-photon excitation pathways. ......................................................... 56
Table 3-3 Three-photon excitation pathways. ....................................................... 56
Table 3-4 Angular distribution............................................................................... 71
Table 4-1 Summary of experimental 21Σu+ excitation lines ................................. 111
Table 5-1 Table of parameter values. .................................................................. 133
Table 5-2 Potential spectroscopic constants for X1Σg+ and (b) A1Σu+ states. ....... 136
Table 5-3 Spectroscopic constants of the Na2 21Σu potential. .............................. 137
Table 5-4 Calculated energy levels of the 12Σu+ state. ......................................... 138
Table 5-5 Line Assignments ................................................................................ 148
ix
ABSTRACT
Three different research projects were performed and documented in this
dissertation. With the help of a molecular beam, a linear time-of-flight (TOF)
spectrometer, and a pulsed laser system, experiments were performed that led to
(i) understanding of the different pathways of Na+ production in a non-resonant
multi-photon ionization using 355 nm and 532 nm photons, (ii) measurement of
vibrational lifetime dependence of the 21u+ and 41g+ states, and (iii) the study of
the shallow well of the Na2+ 12Σu+ state.
In the non-resonant multi-photon ionization project, flight time spectra reveal
several processes with Na+ photo-fragment energies ranging from 0 to 1.32 eV.
Emphasis lies on 2(or 3)-photon excitation followed by dissociative autoionization
along doubly excited states converging to the repulsive 12Σu+ state of Na2+. A
semi-classical model is employed to explain the observed fragment energy
distribution in this process which involves competition between electronic
(autoionizing) and nuclear (dissociative) degrees of freedom. A fit to experimental
polarization dependent TOF spectra was accomplished using model Rydberg
potential curves with position dependent autoionization widths Γ(R) and
appropriate fragment angular distributions.
The second project is about lifetime measurements of the 21u+ and 41g+ states.
With the aid of high resolution dye lasers used for excitation and using variable
delay pump-probe resonant ionization technique, the radiative lifetime of the
selective individual vibrational levels of the 21Σu+ state (v = 22 – 52) and of the
41Σg+ state (v = 43 – 65) of Na2 have been experimentally measured as a function
of vibrational quantum number. Calculation of lifetimes was also performed using
the Level 8.0 and BCONT programs. In general, it has been observed in both
experiment and calculation that a strong variation of lifetime exists especially as
the vibrational levels approach the potential barrier of the 21Σu+ state or the shelf
of the 41Σg+ state. Overall lifetime magnitude measured and calculated for the
21Σu+ state is in good agreement with the vibrationally averaged lifetime data
reported by Mehdizadeh. For the 41Σg+ state, the current work is the first account
of lifetime data. Again, the overall experimental lifetime trend agrees with the
calculation.
The third project relies on Zero Electron Kinetic Energy (ZEKE) technique.
Experiments using ZEKE was successfully demonstrated in the laboratory for the
12Σg+ state of the Na2+. Two rotational levels were successfully identified for v = 0
of the ground ion state. However, the attempt of this study to measure
experimentally the ro-vibrational levels and spacing of the 12Σu+ state of Na2+ was
unsuccessful. Calculations using LEVEL 8.0, the program used for calculating the
bound vibrational levels of a diatomic molecular potential was tested and the
shallow well of the 12Σu+ state with a depth of about 70 cm-1 was predicted to have
about 26 bound vibrational levels.
2
1 Introduction
An excited state of a molecular system is a quantum state that has a higher energy
than that of the ground state. When a molecule in a ground state absorbs energy,
an electron may be excited from the ground state to a higher energy and is
promoted to an electronically excited state. Any molecule in an excited state has a
finite lifetime. Its lifetime is usually short; even in the absence of de-exciting
collisions, spontaneous or induced emission of energy occurs shortly after the
molecule is promoted to the excited state, returning the molecule to a state of
lower energy, i.e. a less excited state or the ground state. If excited states cannot
decay by dipole-allowed one-photon emission, they are relatively long-lived; such
excited states are called metastable states.
A molecule in a very high excited state is called a Rydberg state. Rydberg states
are electronically excited states that have energies converging on the energy of an
ionic state. The ionization energy threshold is the energy required to remove an
electron from an ionic core of a molecule. When the energy absorbed by a
molecule is above the first ionization potential, an electron will be ejected and
could lead to ionization. In general, several other processes can occur:
AB  energy  AB   e 
(i ) Direct Ionization
 A  B  e  (ii ) Direct Dissociative Ionization
 AB **
Superexcitation
AB   e 
AB ** 
(iii ) Indirect Ionization
A  B  e  (iv) Indirect Dissociative Ionization
A B
(v) Neutral Dissociation
(vi ) other processes
(1.1)
Chapter 1: Introduction
The AB** is a transition state which is also called as doubly excited state or a
“superexcited” state. This is a neutral state which has an internal energy greater
than the energy of the first ionization potential. Other decay processes that can
occur include fluorescence and ion-pair formations. The current thesis work
explores decay mechanisms and processes associated with excited states of the
sodium dimer. In particular, the thesis documents the experimental determination
of lifetimes of singly excited states decaying by dipole allowed radiation and of
superexcited transition states decaying rapidly via autoionization. In addition,
calculation and modelling of these two processes are presented. Finally, an
experiment – though unsuccessful in the end – is described that attempts to
measure the vibrational levels of the shallow well of the first excited state of the
sodium dimer ion. This feature has been predicted to exist just below the threshold
for dissociative ionization of the sodium dimer molecule.
The alkali molecules are very good candidates for spectroscopic experiments.
They have relatively low ionization potentials, corresponding to low excitation
energies in the visible range. Since there is only one electron in the outermost
electron shell, alkali molecules can be treated - approximately – as “hydrogenlike” molecules, more specifically as heavy isotopes. The major difference is of
course the existence of extended cores formed by the inner electrons (as opposed
to the compact nucleus of hydrogen or its true isotopes. Because of the similarity,
the electronic structure of alkali molecules is relatively simple, and its theoretical
treatment can be based on pseudopotential methods1, 2 or ab-initio calculations3-5
allowing more refined comparisons between experimental data and results of
different calculated approximations.
2
Chapter 1: Introduction
The sodium dimer molecule Na2 in particular has been the subject of experimental
and theoretical studies. Early experiments were focused on the molecular
structure;6,
7
Verma et al.8 have provided an extensive and complete review of
work prior to 1983. Subsequently, experiments involving dynamical molecular
properties such as lifetimes9, transition dipole moment10, 11, and inter-nuclear and
ionization dynamics12-14 have been developed.
As mentioned above, this thesis contributes to three different aspects of sodium
dimer spectroscopy, namely lifetime measurements of single electron excited
states, autoionization lifetimes of doubly excited dissociated Rydberg states, and
an attempt to map out the first excited state of the dimer ion. The subsequent
sections, address these items in some more detail.
3
Chapter 1: Introduction
1.1 Lifetimes of Excited States
Time-resolved spectroscopy allows the determination of lifetimes of excited
states. Lifetime measurements are an important tool for the understanding of
molecular electronic structure as well as chemical reaction pathways. Lifetimes
are also directly related to the related quantities of transition probabilities, line
intensities, and oscillator strengths (f-values). In astrophysics, f-values are used to
determine relative abundance and concentration of the chemical elements in stars,
and accurate oscillator strengths of the transitions involved or the radiative
lifetimes of excited states are critical for stellar atmosphere modelling.15 Another
field of application of time-resolved spectroscopy is in the study of chemical
reactions. In collisions important information in the interpretation of collisional
investigations become available16. Zare et al.17 reported that transition moments
and radiative lifetimes can be used to determine orbital mixing coefficients in a
perturbed system.
Lifetimes of molecular excited are typically of the order of a few tens of
nanoseconds.17 One well establised technique used to study lifetimes is laser
induced fluorescence.17-20 In these types of experiments, the fluorescence, i.e.
emission, to a lower state from an upper state is measured as a function of time.
The excitation of molecules under investigation is carried out by using a spectrally
narrow, pulsed or temporally modulated laser.
In sodium dimer, lifetime measurements have been performed for only a few
electronic excited states. Lifetime data were reported for the first excited singlet
4
Chapter 1: Introduction
stare,21-23 the A1u+ state, with the most precise value of 12.45 ± 0.05 ns having
been reported only recently. 24 Lifetime data for the B1Πu+ state were published by
several groups. Lifetime measurements of sodium dimer using laser induced
fluorescence are also available for the 21u+, 25, 26 21Πg+ (v = 25, J = 20), 27 C1u+
(v = 9),25 51g+ (v = 10),28 and 61g+ (v = 77, J = 0 – 20) excited states.29 For the
lifetime measurement work reported here of the 21u+ and 41g+ states, a pumpprobe variable delayed ionization technique is used.
5
Chapter 1: Introduction
1.2 Non-Resonant Multi Photon Ionization
In the second experimental study reported here, photo-ionization and photodissociation of sodium dimers have been investigated using non-resonant multiphoton ionization (MPI). Photoionization can proceed in different ways. Direct
photoionization, the simplest ionization process, requires photo-absorption of one
or multiple photons such that the total absorbed energy is above the ionization
threshold and that the photoelectron is ejected promptly. Another ionization
process is so called auto-ionization. This happens when a neutral molecule is
excited to a quasi-bound eigenstate with higher energy than the first ionization
potential. In this case, the excited molecule can decay by photon emission (which
usually takes of the order of nanoseconds) or by interaction with the electronic
continuum and spontaneous ejection of an electron. In the pertinent project
presented here, it has been found that doubly excited Rydberg states can play an
important role in the auto-ionization process.
To every ro-vibrational state in molecular ions – there is a set of corresponding
neutral eigenstates converging to it.30 For high principal quantum numbers
(typically n = 10 and larger) these eigenstates are known as Rydberg states. Since
their radiative decay is much slower (see above), they decay via autoionization
and/or predissociation. These states are key players in processes such as
dissociative electron-ion recombination,31,
32
associative ionization,33 and
dissociative ionization.34-36
Dissociative electron-ion recombination, as the term suggests, is a process in
which a molecular ion captures an electron, resulting in a highly excited neutral
6
Chapter 1: Introduction
molecule. This highly excited molecule can dissociate into two neutral atoms.
Thus, dissociative electron-ion recombination plays an important role in any
plasma environment. For example, interstellar clouds whose composition can be
unravelled from the emission information as the excited atom radiatively decays
to its ground state.37
Associative ionization is the reverse process of dissociative electron-ion
recombination. In this process, two neutral atoms, at least one in an excited state,
are brought together and interact, producing a molecular ion and an electron. This
process, in particular for the sodium dimer, has been widely studied.38 Since the
process is exothermic for sodium, it is also active at the lowest temperatures
where it was found to have surprisingly large cross section.39 Therefore, in BoseEinstein Condensation (BEC) experiments, it can be an unwanted source trap loss.
Finally, in the dissociative ionization process, simultaneous ionization and
dissociation of the highly excited neutral molecule occur. The electron leaves the
molecule with an asymptotic energy such that the total energy is conserved.
Ionization in this process will result in molecular ions with certain energies. These
molecular ions are produced through indirect photo-ionization. If the kinetic
energy of the resulting ion is less than the local binding energy, then the molecular
ion is stable. However, when molecular ion energy is greater than the local
binding energy, the molecule will dissociate producing an atomic ion, a neutral
atom, and an electron. Electronic auto-ionization and ion-neutral fragmentation
compete during this process.
7
Chapter 1: Introduction
As mentioned above, radiative decay for Rydberg states is much slower than at
low values of n. At higher principal quantum number, n, electron-core interaction
is greatly reduced. The lifetime of high Rydberg states scales as n3. Using this
scaling, for n = 200, the Rydberg state has a lifetime of about 1 μs. However, in
Zero Electron Kinetic Energy (ZEKE) spectroscopy, it has been observed that
Rydberg lifetime obeys a scaling law of n4. It has been suggested that Stark
mixing of angular momentum states (l states) induced by electric fields increases
the lifetimes of very high Rydberg states. Chupka40 provided a very good
description in which the optical mixing of l states from high n, low l to high n,
high l drives the electron orbit to become more spherical, reducing further the
electron-core interaction, therefore reducing auto-ionization and predissociation
and leading to extended lifetimes. It has also been found that perturbation due to
presence of nearby atoms or molecules in the system leads to ml mixing and
increasing further the lifetime of high n Rydberg states.40 For the combined l, ml
mixing, the lifetime of high Rydberg states is estimated to scale as n5, an increase
by a factor of 100. For a Rydberg state with n = 200, the lifetime is expected to be
around 200 – 500 μs.
8
Chapter 1: Introduction
1.3 Thesis Structure
This thesis is divided into three distinct research projects. Following this general
introduction is a chapter dedicated to the general experimental apparatus and
methodology used for the projects discussed in this thesis. Specifics pertinent to
each experiment are detailed in the corresponding subsequent chapter. Chapter 3
discusses the study performed using non-resonant multiphoton ionization. Chapter
4 covers lifetime measurements of two excited states of Na2, while chapter 5
includes the investigation of the shallow well of the 12Σu+ state of Na2+. Chapter 6
concludes this document with a summary and closing comments.
9
2 Experimental Set-up
All experimental data presented in this dissertation were performed with a
supersonic molecular beam set-up already discussed in previous theses41, 42 which
incorporates nanosecond pulsed laser excitation and linear time-of-flight (TOF)
spectrometer operated in low field extraction mode which is based on the standard
Wiley-MacLaren TOF device.43 The axes of the three components form a
Cartesian coordinate system with the vertical z-axis along the TOF spectrometer,
the x-axis parallel to the sodium motion in the beam, laser in y direction and the
origin at the intersection of laser and molecular beam.
Sodium atoms and molecules in gas phase are produced in a supersonic expansion
by heating metallic sodium inside the source chamber up to 800 K. As the beam of
particles exits the nozzle (diameter of a tenth of a millimiter) of the source
chamber, it is further shaped by a set of apertures to a full angular divergence of
about 7 mrad and a cross section of about 2×2 mm at the site of the laser crossing.
One or more pulsed nanosecond laser crosses the molecular beam in the
interaction region perpendicularly from either a Nd:YAG or a dye laser depending
on the design of experiment. Photons absorbed from the laser by the sodium atom
or molecule leads to excitation and/or ionization resulting in Na+ or Na2+. Photoions are mass selected by the linear time-of-flight (TOF) spectrometer and are
detected by micro-channel plates (MCP) in Chevron configuration. The signal
from the MCP detector is fed into a 150 MHz digital storage oscilloscope (or
Chapter 2: Experimental Set-up
boxcar integrator) which is in turn GPIB interfaced to a lab computer using
LabVIEW code that both controls experimental parameters and collects the data.
The experimental setup can be divided into three major sections: (i) molecular
beam source and vacuum system, (ii) the laser system, and (iii) TOF mass
spectrometer, detection and data acquisition. Each section is explained in this
chapter. Besides the experimental hardware, characteristic properties of the
molecular beam are also discussed in 2.1.3.
11
Chapter 2: Experimental Set-up
Figure 2.1 Schematic diagram of the general experimental set-up
12
Chapter 2: Experimental Set-up
2.1 Vacuum system and molecular beam source
2.1.1 Vacuum system
It is crucial to maintain sufficiently low pressure of the vacuum system for the
experiment to work. High pressure will result in degradation of the sodium beam
density due to collisions with the background gas inside the chamber. For this
reason, a mechanical pump and two different diffusion pumps are necessary to
keep the pressure low throughout the experiment. The vacuum assembly consists
of the source chamber and the TOF chamber. The mechanical pump which is
connected to both chambers, pumps both chambers to pressure of about 103 torr.
At this pressure, the diffusion pumps, one on each chamber, are turned on to lower
the pressure further. The operating pressure of the source chamber is slightly
higher than that of the TOF chamber because of the relatively higher ambient
temperature and sodium contamination from the source. At normal operating
condition, a 10-7 torr is achieved for the TOF chamber while the source has about
an order magnitude higher pressure. Alkali resistant type diffusion pump oil (Kurt
J. Lesker Co, Diffoil 30) is used for the source chamber and silicone based
diffusion pump oil (Dow Corning 704) is used for the TOF chamber.
13
Chapter 2: Experimental Set-up
Figure 2.2 Schematic diagram of the molecular beam source
2.1.2 Molecular Beam Source
The molecular beam container (source) is made of stainless steel and is located
inside the source chamber. The source consists of two cylindrical containers,
referred to as reservoir and nozzle (see Figure 2.2). Solid metallic high purity
(99.99%) sodium is loaded quickly into the reservoir part of the source. The
source is sealed off with a knife-edge stainless steel metal cover pressed into
copper ring and tightened with six screws. Each part of the source is heated by
separate coaxial heaters (Ari # Bxd13b50-4T and Bxd13b62-4T for reservoir and
nozzle, respectively). With newly loaded sodium, reservoir and nozzle heating
currents are set to 0.5 A and 1.5 A, respectively. This setting is kept for about 12
14
Chapter 2: Experimental Set-up
hours to remove contaminants of the metallic sodium that entered during loading
of the source. The source is heated up at an average rate of about 4ºC/minute.
Table 2-1 summarizes the heating up procedure of the source to obtain a steady
molecular beam, cooling down procedure is also shown in the table. The nozzle is
heated at about 50 K higher than the reservoir to minimize the possibility of
clogging from condensing sodium. To achieve the required vapour pressure44 for
non-seeded hydrodynamic expansion, at least about 10 Torr, the source container
is heated to a temperature of about 800 K.
The beam is collimated further by a set of apertures upstream. The final beam
cross section is 2 x 2 mm2 at the interaction region where it is crossed by the laser
beam.
Heating Up
Time
(minute)
Nozzle
I (A)
Reservoir
I(A)
Nozzle
T (°C)
Reservoir
Front T (°C)
0
30
60
90
120
2.2
2.5
2.8
3.2
3.5
1.5
2
2.5
2.8
3.5
22
292
388
456
514
22
192
300
385
441
Cooling Down
Time
(minute)
Nozzle
I (A)
Reservoir
I(A)
Nozzle
T (°C)
Reservoir
Front T (°C)
0
30
60
90
2.5
2
1
0
0
0
0
0
568
417
345
260
530
311
256
190
Table 2-1 Typical heating up
and cooling down of sodium beam source.
15
Chapter 2: Experimental Set-up
2.1.3 Characterization
For the expansion conditions of the experiment, the molecular beam contains
mostly (ca. 90%) Na atoms while Na2 dimers provide the large majority of the
remainder44. With the help of several auxiliary experiments, the internal
temperature, speed distribution, and cluster composition of the molecular beam
properties have been established:
(i) Internal Temperature
Two color resonance enhanced multiphoton ionization (REMPI) via the A1u+
state reveals a rotational and vibrational temperature of 70  10 K and 120  20 K,
respectively, of the Na2 dimers. Figure 2.3 shows the experimental spectrum of a
single vibrational band (A1Σu+ (v = 13)  X1Σg+ (v = 0)) together with the
simulation using a Gaussian lineshape and the laser bandwidth used as the
linewidth according to Eq. (2.1.1).
I  ( Ascale gFFCF e

EVib
kTVib


)(
) S J "( P / R ) n0 (2 J ' 1)e kT
max
ERot
Rot
(2.1.1)
Here, Ascale is a scaling parameter, g is the nuclear spin degeneracy, FFCF is the
Franck-Condon factor, λ is the laser wavelength, SJ”(P/R) is the Hönl-London
factor, n0 is the Na2 density, J is the rotational quantum number, and the
exponential
terms
correspond
to
rotational
and vibrational
Boltzmann
distributions. The vibrational temperature is extracted by fitting multiple
vibrational bands not shown in Figure 2.3. At the temperatures mentioned above,
16
Chapter 2: Experimental Set-up
about 90% of Na2 molecules are in the vibrational ground state v = 0 and the most
probable rotational level is at J = 17.
180
Emission Intensity [arb. units]
Experiment
Simulation (T=70 K)
160
140
120
6204
6206
6208
6210
6212
6214
6216
6218
6220
6222
wavelength [Angstrom]
Figure 2.3 REMPI: A1Σu+  X1Σg+
Rotational population distribution of A1Σu+ (v = 13)  X1Σg+ (v = 0) characterized
with Trot = 70 K and Tvib = 120 K.
(ii) Speed Distribution
In the following chapter, the speed distribution plays an important role in the
simulation to model our experimental results. Different components of the sodium
beam have different speed distributions due to their mass difference45. However,
only the sodium dimer speed distribution is relevant since ions detected in the
non-resonant multi-photon ionization experiment, whether atomic or molecular,
are produced from Na2 molecules.
17
Chapter 2: Experimental Set-up
The ions TOF are influenced by the initial molecular speed distribution because:
(1) the molecular beam is not perfectly collimated therefore there is a vertical
velocity component of the ions due to the initial velocity of Na2. (2) The
horizontal electric field generated at the steering plate region helps to decide the
fate if the ions are detected or not, i.e. if the x-velocity component of an ion is
large, it takes large field in the steering plates to compensate to the negative xdirection. Therefore, the possibility of detecting the ions is determined by the ion
initial position and velocity, mass and kinetic energy, and the electric fields
applied.
Dimer ion yield as a function of the strength of the electric field in the steering
plate region have been measured, that is situated in the otherwise field free TOF
drift region and deflects ions in the x-direction. With the known geometry of the
set-up, the results can be used to find both the most probable speed and speed
ratio. A Maxwell-Boltzmann distribution is used in order to describe the velocity
distribution of the molecular beam. For the molecular beam with expansion
through the nozzle, the velocity distribution may be written in the form46
2
2
  v
 v 
 

f (v)  C   exp    S  
  vw
 vw 
 

(2.1.2)
where vw, C, and S stands for most probable velocity, normalization constant, and
the speed ratio respectively with:
vw 
18
2 k BT
m
(2.1.3)
Chapter 2: Experimental Set-up
where T is the translational temperature of the expanded beam. It is found out that
the most probable speed of 580 m/s and speed ratio of S = 4.1 ± 0.5 agrees
reasonably well with experimental data.47
(iii) Cluster Composition
One-photon ionization at photon energies just slightly above the adiabatic limit
can be expected to be both efficient and of low degree of fragmentation.
Experiments performed at 308 nm (equivalent to 4.02 eV) – too low for atomic
and dimer ionization, but sufficient to ionize trimers – yielded TOF mass spectra
showing only atomic and dimer ions through multi-photon ionization. Thus we
can neglect contributions by trimers and higher order clusters.
19
Chapter 2: Experimental Set-up
2.2 Laser System
The laser system consists of two Q-switched Nd:YAG lasers and two tunable dye
lasers. For the experiment described in chapter 3 (Non-Resonant Multi-Photon
Ionization), only one Nd:YAG laser is used while for the experiments described in
chapter 4 (Radiative Lifetimes of Excited Na2 States), all four lasers are used: the
first Nd:YAG laser pumps the two dye lasers, while the second Nd:YAG laser is
used as a delayed probe laser. For ZEKE experiments, only three lasers are used in
the experiment, one Nd:YAG laser pumping two dye lasers.
The Nd:YAG laser has a primary output of near IR (1064nm) light. Using nonlinear crystals (KD*P) for frequency doubling/tripling, 532 nm and 355 nm light
is produced. The first Nd:YAG laser (Spectra Physics, Quanta-Ray Pro Series
200) used for pumping the two dye lasers operates at 20 Hz repetition rate and has
an IR pulse width of about 10 ns (both 532 nm and 355 nm are slightly shorter).
Before each experiment, the Nd:YAG laser is kept running at operating conditions
for about 30 minutes in order to stabilize the laser output. At operating condition,
this laser can have a maximum power output of more than 30 W for the IR output
and about 12 and 7 W for 532 nm and 355 nm, respectively. Polarization of the
doubled and tripled output can be changed by turning the non-linear crystal 90º;
however the IR polarization is fixed at horizontal polarization. The other Nd:YAG
laser (Continuum YG-61) also operates at 20 Hz repetition rate. The output power
it can produce (normally using 532nm and 355 nm light) is less than the first
Nd:YAG laser and the polarization for the 532 nm and 355 nm are fixed to
vertical polarization. The line broadening of the YAG gain transitions defines the
20
Chapter 2: Experimental Set-up
upper limit of the bandwidth of a Nd:YAG laser, which is about 4 cm-1 at room
temperature48. However, the limit of the bandwidth for the Nd:YAG laser is set by
the laser resonator and typically found to be around 1 cm-1.49
The two dye lasers are commercial lasers (Lambda Physik, ScanMate Pro C-400)
in grazing incident configuration. They are pumped with 532 nm light from the
Nd:YAG (Quanta-Ray Pro Series) using a 50% beam splitter. The output of both
lasers has a bandwidth of 0.15 cm-1 but with an additional Improved Bandwidth
(IB) Module, a 0.08 cm-1 laser bandwidth can be achieved across the whole dye
tuning range. One of the two dye lasers is equipped with a Second Harmonic
Generation (SHG) Unit. The unit uses BBO type crystals and its tuning range is
205 nm to 420 nm. The tuning synchronization of the SHG unit is automatically
controlled by the electronics of the laser. Common dyes used in the experiments
are LDS 698 (645-745 nm), DCM (608-689 nm) and KR 620 (570-604 nm). All
dye concentrations are those specified in the ScanMate Pro dye chart.
During experiments, laser power is monitored regularly with a power meter
positioned near the entrance window of the TOF chamber. After passing through
an aperture near the entrance window, the laser beam has a diameter of about 0.3
cm. For 15 mJ laser energy, an estimate of the peak power density is therefore
obtained as:
 E 


0.015mJ
t 
P
 8
 2.1104W / cm2
A
10 s    0.152 cm2
21
(2.2.1)
Chapter 2: Experimental Set-up
where E is the energy per pulse, t the pulse duration, and A the beam cross
sectional area. Dye laser energy typically used in the experiment is in the range
0.05 – 0.3 mJ.
22
Chapter 2: Experimental Set-up
2.3 Time-Of-Flight Spectrometer, Detection and Data
acquisition
Figure 2.4 Schematic of the TOF spectrometer.
The TOF-Mass Spectrometer (TOF-MS) used in the experiments has a doublestage acceleration scheme as shown in Figure 2.4 with three regions, namely
extraction, acceleration, and field-free drift regions. Wiley and McLaren were the
ones who first introduced this type of TOF-MS.43 Linear TOF-MS provides
23
Chapter 2: Experimental Set-up
information on mass-resolved ion yields, ion kinetic energy, and angular
distributions.
2.3.1 Time-of-flight spectrometer
Ions are accelerated from their place of origin towards a detector in a TOF
spectrometer. Their flight time from the point of ionization to the detector is
measured and recorded. Flight times are affected by several factors: the ion mass,
initial kinetic energy, the distance travelled, and electric field strengths applied.
Figure 2.4 shows the schematic of the TOF spectrometer which can be divided
into three different regions. These three regions are the extraction, acceleration
and drift region. The electric fields in the extraction and acceleration region
provide acceleration in the positive (+) z direction in the TOF spectrometer. The
total flight time of the ion is the sum of the flight times in each of the three
regions.
Tflight  Textraction  Tacceleration  Tdrift
(2.3.1)
As the molecular beam enters the extraction region from the negative (-) x
direction and leaves in the positive x direction, the laser crossed the molecular
beam from the positive (+) y direction and leaves in the negative (-) y direction.
The ions are born in a small volume at the center of the extraction region.
Ionization occurs at the interaction region defined by the geometric overlap of
laser (3 mm diameter) and the molecular beam (2 x 2 mm). Ionization occurs
essentially instantly, electrons are removed towards the –z direction. Ions traverse
24
Chapter 2: Experimental Set-up
half the distance of the vertical length of the extraction region or about 0.9 cm.
The flight time of the ion in the extraction region is given by:
1
Textraction
(2mU 0 ) 2
qE d 1

[(1  e ) 2  1]
qEe
U0
(2.3.2)
where m, q and U 0 are mass, charge, and the initial kinetic energy, respectively of
the ion. Ee is the strength of the electric field in the extraction region and d is the
actual distance from the location of ionization to the end of the extraction region.
Ions arrive in the accelerating region and pick up more speed. They travel about
3.17 cm and experiences higher electric field in standard operation conditions, the
voltage across the acceleration region is kept four times the value of the extraction
region. The flight time of the ions in the acceleration region is given by:
1
Tacceleration
1
1
(2m) 2

[(U 0  qEe d ) 2  (U t ) 2 ]
qEe
(2.3.3)
U t  U 0  qEe d  qEa d a
where Ea is the strength of the electric field in the acceleration region and da is the
height of the acceleration region.
Ions travel further along the D = 34 cm long field free drift region where they
spend the majority of their flight time. The flight time of the ions in the drift
region is given by:
1
Tdrift 
(2m) 2 D
2(U 0  qEe d  qEa d ) 2
1
25
(2.3.4)
Chapter 2: Experimental Set-up
Two steering plates are situated in the drift region creating an electric field
oriented horizontally which pushes the ions to the negative (-) x direction to
counter the forward x-velocity of the ions due to the motion of their parent
molecules in the beam. Typically, steering plate voltage is set to 5 V with the
exception of steering plate voltage dependence experiment and some experiments
with higher extraction and acceleration fields that require no steering plate
voltage. The ions leave the drift region and are detected by the MCP.
2.3.2 Detection
At the end of the drift region, a dual micro-channel plate (MCP) detector converts
ions into the ultimately detected electric current. Figure 2.5 shows the schematic
diagram of the MCP. The MCP is a pair of resistance matched MCPs in high gain
Chevron configuration mounted between two ceramic rings. Each plate has an
active diameter of 18 mm, thickness of 0.43 mm and a nominal pore size of 12
μm. A voltage of 2000 V applied to the MCP results in a typical gain factor of 4 x
106. The copper anode collects the electrons and the current signal is fed to an
oscilloscope or boxcar using a 50  coaxial cable. The capacitor helps filter
electrical noise in the circuit.
26
Chapter 2: Experimental Set-up
to current measurement
(oscilloscope or boxcar)
1 M
50 M
6
5
1 M
1 M
4
copper anode
2 kV
3
copper ring
2
30 M
MCP
1
0
0.1 F
Figure 2.5 Schematic of the MCP circuitry.
2.3.3 Data acquisition
There are two different types of data acquisition used in the experiments presented
in this thesis. First is the TOF spectra data acquisition. This data is important to
study the details of the ion kinetic energy as well as the fragment angular
distribution due to different laser polarizations. This type of data collection uses
the TOF spectrometer in low field extraction mode. The output from the MCP is
fed to a digital oscilloscope (HP, LeCroy 9400A). The oscilloscope provides 125
MHz bandwidth, 100 megasamples/sec, 8-bit ADCs and is fully programmable
over RS-232-C or GPIB interface. The oscilloscope reads the analog signal and
converts to digitized data. The digitized data is read via GPIB by a LabVIEW
27
0.0
0.5
1.0
Chapter 2: Experimental Set-up
program running a PC with Windows OS. The timing resolution of the
oscilloscope is 10 ns, but finer resolution (< 10 ns) can be attained with
interleaved sampling mode. The LabVIEW program (data_collection.vi) reads
whatever data is displayed on the oscilloscope. The user can specify how many
samples will be read, can do averaging, yet keep individual raw data, and save
files individually or collectively. A second version of the program exists that
incorporates ion yield integration (Na+ and/or Na2+). A third version of the
program allows for automated laser polarization experiments using a polarization
rotator, controlled by LabVIEW. In this way, a TOF spectrum is recorded every
time the polarization has changed. An additional option in this version allows the
program to change the steering plate voltage systematically. This option is
incorporated to the TOF data acquisition and each TOF spectrum is again
recorded for each steering plate voltage applied.
The second type of data acquisition refers to the measurement of ion yield spectra
as a function of excitation energy, i.e. the measurement of the production of
atomic or molecular ions as the laser is tuned in wavelength. Here, the output from
the MCP is fed to a gated boxcar integrator (Stanford Research System 250). Ion
signal is integrated using proper delays and widths of the gates matching the
appropriate flight-time of the ion in question. Ion flight time is monitored using
the oscilloscope which is kept synchronized with the boxcar integrator by suitable
trigger. A fast pre-amplifier (Stanford Research System 255) is also used between
MCP and boxcar, nominally the signal from the MCP goes through a three-stage
amplification of 5 x 5 x 5 = 125. Analog output of the boxcar is then sent to a
National Instrument IO-board (NI PCI-1200) which is interfaced with the PC. A
28
Chapter 2: Experimental Set-up
LabVIEW program (dye_scan.vi) records the analog signal from the IO-board as
the laser is tuned to different wavelength. The same LabVIEW code also controls
continuous laser scanning over pre-determined intervals and/or adjusting the laser
to discrete, specific wavelength values.
For the one-laser experiment (chapter 3), a 20 Hz TTL pulse from the Nd:YAG
laser is used for triggering and timing of the experimental data acquisition. This
TTL pulse is sent to a digital delay generator (Stanford Research System DG 535)
to synchronize the actual laser pulse arrival at the interaction region with the start
of the data acquisition. A fast photo diode (Honeywell Co.) with rise/fall time < 1
ns is used to monitor the timing of the laser pulse. Sample temporal laser profiles
for a delay of 90 ns between laser pulses is shown in Figure 2.6 using 330 nm and
532 nm as pump and probe lasers, respectively.
29
Chapter 2: Experimental Set-up
Laser Profile
pump laser: 330 nm
FWHM: 9.2 ns
Intensity (arb. units)
probe laser: 532 nm
FWHM: 11.6 ns
90 ns peak-to-peak
0
20
40
60
80
100
120
140
160
time (ns)
Figure 2.6 Laser Temporal Profile
Temporal laser profile measured with a fast photo-diode using 330 nm and 532
nm as pump and probe lasers, respectively for a delay of 90 ns between the two
pulses.
For the three or four laser experiment described in chapter 4, it is critical to have
control of the timing of all the lasers, especially with the delayed probe laser
(Nd:YAG). The two dye lasers are pumped with the Quanta-Ray Nd:YAG laser,
therefore the timing of the dye laser is determined by this Nd:YAG laser. The
Continuum Nd:YAG laser on the other hand is a separate unit. To achieve
synchronization of the two Nd:YAG lasers, the lasers must be operated in external
control, remote trigger mode. This is achieved by providing 2 different TTL
pulses (4 total) to be used for triggering the flash lamps and the Q-switches of the
two Nd:YAG lasers. These 4 TTL pulses are taken from a pulse generator
(Stanford Research System DG645) whose output can be programmed in terms of
30
Chapter 2: Experimental Set-up
polarity, magnitude, pulse width, and relative delay. Actual timing of the laser
pulses is verified at the experimental chamber entrance by the same fast photo
diode mentioned above. The DG645 settings are then adjusted accordingly to meet
the desired delay of the probe laser pulse. In a normal data acquisition for lifetime
experiments, the DG645 is controlled by a LabVIEW code. Each TTL pulse
setting is already pre-set in the code and changed accordingly as the program
takes data. In addition, each laser can also be programmed to be on/off depending
on the type of data acquisition needed. The timing procedure is discussed further
in chapter 4.
31
3 Non-Resonant Multi-Photon
Ionization
This chapter is an experimental investigation of pathways for the production of
Na+ from ground-state Na2 in non-resonant multi-photon ionization with 355 nm
and 532 nm photons. Fragment energy and angular distributions were measured
using a molecular beam apparatus, nanosecond pulsed laser excitation, and a
linear time-of-flight (TOF) spectrometer. Comparison of experimental TOF
spectra with a semi-classical model, using a Monte-Carlo simulation, reveals that
dissociative auto-ionization along repulsive Rydberg states converging to the
12u+ excited state of the ion plays a major role. For excitation with 355 nm
photons, the entire Na+ flux can be accounted for by this channel. From the
comparison between experiment and simulation, autoionization lifetimes as a
function of internuclear distance are found to be of the order of a few tens of
femtoseconds. The lifetimes exhibit a local maximum of about 70 fs at a distance
of about 4.5 to 5 Å. In the 532 nm case additional features of the Na+ ion peak are
observed. They are characteristic of (i) multi-photon ionization into vibrationally
bound states of the dimer ion followed by one-photon excitation into the repulsive
first excited 12u+ state and (ii) photo-dissociation of Na2 molecules into ground
and excited state neutral Na atoms, followed by one photon ionization of the
excited fragment. Here again the Monte-Carlo simulation successfully reproduces
the measured spectra.
Chapter 3: Non-Resonant Multi-Photon Ionization
3.1 Experimental Method
All experiments were performed using a supersonic sodium molecular beam with
nanosecond pulsed laser excitation and a linear time-of-flight (TOF) spectrometer
as described in the previous chapter. Multi-photon ionization is achieved using
532 and 355 nm photons from a frequency doubled and tripled, Q-switched
Nd:YAG laser with a 10 ns pulse width. Leutwyler et al.50 measured the adiabatic
ionization energy of Na2 with a high degree of accuracy to be 4.88898 ± 0.00016
eV. Therefore using 355 nm (3.49 eV), two photons are required to ionize ground
state Na2. Ionization using 532 nm (2.33 eV) requires three photons. Figure 3.1 is
a simplified Na2 potential diagram showing the 355 nm and 532 nm photon
energy required for ionization.
The double Fresnel rhomb rotator is used to change laser power without affecting
laser beam pointing. In conjunction with the laser power adjustment, the direction
of the linearly polarized laser can be varied also using the Fresnel rhomb. Figure
3.2 panels (a) and (b) show the power dependence of the atomic and molecular ion
channels for 355 nm and 532 nm ionization, respectively, and using both
vertically and horizontally polarized light. Signal saturation for both polarizations
is observed at pulse energies higher than 60 mJ for 355 nm and about 100 mJ for
532 nm. All data presented in this dissertation were obtained at laser pulse
energies of about 2.0 and 5.0 mJ for the 355 and 532 nm wavelenghts,
respectively. For a 0.3 cm diameter laser beam, these correspond to average power
densities of 0.57 and 1.41 kW/cm2.
33
Chapter 3: Non-Resonant Multi-Photon Ionization
4
6x10
2
+
1 u
4
5x10
2
**
+
1
Na2 n u
1 g
4
4x10
Energy (cm-1)
1
5 g
1
2 g
1
3 g
4
3x10
1
4 g
1
B u
4
2x10
1
A u
4
1x10
1
X g
0
3
6
9
12
15
R (Angstrom)
Figure 3.1 Simplified Na2 potential diagram.
Simplified potential diagram showing only the relevant molecular states involved
in this study and non-resonant multi-photon ionization excitation scheme.
34
Chapter 3: Non-Resonant Multi-Photon Ionization
Laser Energy (mJ)
0
a)
30
60
90
+
Na
+
Na2
355 vertical polarization
6
Ion Yield (arb. units)
10
b)
5
10
355 horizontal polarization
+
Na
+
Na2
6
532 vertical polarization
10
5
10
532 horizontal polarization
4
10
0
70
140
Laser Energy (mJ)
Figure 3.2 Laser Power Dependence
Power dependence of Na+ in squares (black) and Na2+ in circles (red) ion yield
with (a) 355 nm and (b) 532 nm ionization, both with vertical polarization. Insert
graphs are for horizontal polarization case.
Photo-ions created in these experiments are mass selected by a Wiley-McLaren
type linear TOF spectrometer and are detected by a MCP detector in Chevron
configuration. The signal from the MCP detector is fed into a 150 MHz digital
storage oscilloscope which is interfaced via GPIB into a computer. Automation is
performed by a LabVIEW program which controls experimental parameters and
records the oscilloscope data.
35
Chapter 3: Non-Resonant Multi-Photon Ionization
3.2 Summary of Findings
Case: 355nm Na
+
Case: 532nm Na
532V
532H
355V
355H
4
Ion Signal (arb. units)
1x10
(a)
+
(b)
Case: 355nm Na2
+
+
4x10
5
3x10
5
2x10
5
1x10
5
355V
355H
~7X for 355nm Na
+
~1.5X for 532nm Na
(c)
3
5x10
0
0
15500
16000
16500
15500
16000
16500
22000
23000
24000
Time-of-Flight (ns)
Figure 3.3 Typical TOF spectra.
Typical TOF spectra summed over 5000 pulses. (a) 355 nm Na+ peaks (b) 532 nm
Na+ peaks (c) Na2+ peaks using 355 nm. Black line and red line + open circles
stand for vertical and horizontal polarizations, respectively.
Figure 3.3 displays typical time-of-flight spectra that were generated by summing
5000 linearly polarized laser pulses with 355 nm and 2.0 mJ energy in panel (a)
and (c), and with 532 nm and 5 mJ energy in panel (b). At these laser energies,
both atomic and molecular ion signals are in the unsaturated regime (see Figure
3.2). For ease of inspection, TOF spectra are rendered here and in other figures as
continuous lines connecting the actual data consisting of discrete points at 10 ns
intervals. In order to avoid clutter, the discrete data symbols are only shown for
36
Chapter 3: Non-Resonant Multi-Photon Ionization
the case of vertical laser polarization. Atomic ion peaks in panel (a) and (b) differ
significantly between the two wavelengths while molecular ion peaks are
essentially identical (panel c shows the 355 nm case). All three panels include
horizontal (black line) and vertical (red line + open circles) polarization. As can be
seen, the dimer peak does not change with polarization (also true for 532 nm
ionization), the 355 nm atomic peak changes very little, and the outer and mid
peaks of the 532 nm atomic peak change significantly. Atomic ion flight times are
smaller by a factor of
2 than the dimer ion flight times because of their 2:1
mass ratio. The overall dimer ion signal is larger than the integrated atomic ion
peak by a factor of about 7 for 355 nm and by about 1.5 for 532 nm. At the laser
power used in Figure 3.3, the observed full width at half-maximum (FWHM) of
the dimer ion peak is 30  5 ns for both wavelengths and clearly smaller than the
FWHM of the peaks located at the nominal atomic flight time. Directly ionized Na
atoms from the molecular beam should create mass peaks with about the same
FWHM as that of dimer ions. This has been verified by applying resonant
enhanced two-photon ionization via the Na(4p) states at 305 nm. Under these
conditions, atomic ion peaks with a width of about 25 ns are observed. The
absence of such a narrow feature and thus the lack of multiphoton ionization of Na
at either 355 or 532 nm can be attributed to cross sections that are significantly
smaller than those of the dimers. Upon focusing the YAG laser, an additional peak
emerges due to atomic ions at the mass m = 23 flight time. At the laser power
used in the current experiments, this contribution is entirely negligible and the ion
signal at the atomic flight time is due to molecular fragmentation. It has also been
verified in the simulation that a 10% signal coming from atomic Na ionization
37
Chapter 3: Non-Resonant Multi-Photon Ionization
will already produce a noticeable sharp peak in the spectra which does not have
counterpart in the experimental TOF data.
For 532 nm, but not for 355 nm, peaks exist on each side of the central peak of the
Na+ signal that are strongly power and polarization dependent (see Figure 3.4).
These features – labeled channel II and III – are indicative of atomic Na+ ions
generated by molecular fragmentation processes with narrow kinetic energy
distributions. Channel I is assigned as the direct dissociative ionization channel
via the Na2+ X2Σg+ continuum and is not present in the experimental data due to
very poor Franck Condon overlap from the ground state Na2. While the central
broad feature common to 355 and 532 nm ionization is the focus of this study,
these other peaks will be discussed in detail for completeness and to further
validate the modelling approach.
38
Chapter 3: Non-Resonant Multi-Photon Ionization
Case: 355 nm Na
+
Horizontal Polarization
105 mJ
10 mJ (x3.5)
Case: 532 nm Na
Vertical Polarization
150 mJ
40 mJ (x3.3)
Vertical Polarization
Horizontal Polarization
IV
+
IV
III
III
II
II
15500
16000
16500
15500
16000
16500
15500
16000
16500
Time-of-Flight (ns)
Figure 3.4 Comparison of normalized atomic peaks.
(a) 355 nm with horizontal polarization at two different laser energies. (b) 532 nm
ionization for two different polarizations at the same laser energies. (c) 532 nm
with vertical polarization at two different laser energies.
Specifically, the process generating the fast Na+ ions of channel II (see Figure
3.4b) is most prominent with vertically polarized light and saturates at relatively
low laser power. Channel III shows the same polarization dependence and is
easily observed already at low laser power.
Common to both 532 and 355 nm TOF spectra is a broad central peak (labeled
channel IV) appearing at the nominal flight time for Na+ ions and with a shape
that is similar for both wavelengths. Since the total energy of three 532 nm and of
39
Chapter 3: Non-Resonant Multi-Photon Ionization
two 355 nm photons is the same, the focus is to look for fragmentation channels
accessible in the Franck-Condon window corresponding to the absorption of these
photons. As will be shown later in detail, this central peak can be modelled by
excitation into doubly excited states converging to the repulsive 12Ʃu+ Na2+
potential leading to indirect dissociative ionization. Femtosecond-timescale
electron dynamics (leading to autoionization) along these curves competes with
repulsive nuclear motion (leading to vibrational excitation or dissociation),
creating the broad TOF peak in the process.
The laser power dependence of the 355 nm TOF spectrum shows a rather simple
behavior. As can be seen in Figure 3.4a, the shape of the normalized atomic TOF
peak is unchanged when the laser pulse energy is increased from 10 to 105 mJ for
horizontal polarization. For vertical polarization (not shown), a slight increase of
the wings is observed but no change of the center peak. In the 532 nm vertical
polarization case, generic shape of channels IV and III are unchanged when the
laser energy is increased by about a factor of 4 as seen in Figure 3.4c. The signal
due to channel II has a minor enhancement when the laser energy is increased.
This observation is also true for the horizontal polarization case of 532 nm (not
shown).
40
Chapter 3: Non-Resonant Multi-Photon Ionization
3.3 Analysis of Data
Apart from the information on ion mass distributions, TOF spectra contain in
principle also information about the underlying ion production processes. A wellknown example is the relation between the so called turn-around time t – the
flight time difference between the early and late peaks belonging to the same
fragmentation channel – and the vertical velocity component vz of the fragments
t 
2m
vz
qEex
(3.4.1)
where q/m is the charge-to-mass ratio of the fragment ion and Eex the electric field
in the TOF extraction region. Assuming a maximum initial kinetic energy Ekin
=½mvz2, the estimated energy release of the various features in our TOF spectra
(Figure 3.4) is summarized in Table 3-1, which lists results of such estimates
together with the ionization pathways having the closest energy match.
State
Total Fragment
Energy (eV)
Expected early-late peak
separation (in μs)
3s+3p
1.83
1.28
3s+4s
0.73
0.81
3s+3d
0.31
0.53
3s+4p
0.18
0.4
Table 3-1 Fragment kinetic energy estimates
and corresponding flight-time difference between the early and late fragments.
41
Chapter 3: Non-Resonant Multi-Photon Ionization
Equation (3.4.1) is not a sufficiently discriminating tool. Application of this
method is analogous to extracting information from broadened emission lines
from lower order moments like line position and linewidth only. However, the
detailed dynamics responsible for the broadening is only revealed by an analysis
of the entire lineshape. Equivalently, a simulation program is employed to
synthesize complete model TOF spectra that are then compared with the observed
data. The basic ideas and implementation principles of the simulation program
used here have been outlined elsewhere51-53. Briefly, the Monte Carlo program
calculates a large number of ion trajectories that are (1) constrained by the known
geometry and electric fields of the TOF spectrometer and (2) subject to initial
positions and velocities randomly chosen that are bound by distributions
determined by properties of the molecular beam and by the particular ionizationdissociation channel under consideration. For each simulated ion trajectory that
reaches the detector, a count is registered in the 10 ns wide time bin corresponding
to its flight time. With a sufficient number of trajectories a TOF spectrum is thus
synthesized. Typically, TOF simulated spectra are synthesized from over 106
individual trajectories.
Before applying the Monte-Carlo program to the interpretation of the atomic ion
peak, it is important to characterize the resolution achievable in the experiment.
Apart from fragmentation, the width of TOF ion peaks can be influenced by the
following sources and effects:
(i)
The digital oscilloscope recording the TOF spectra has a time
resolution of 10 ns.
42
Chapter 3: Non-Resonant Multi-Photon Ionization
(ii)
Ions starting at the top and bottom of the molecular beam, respectively,
are separated by 2 mm along the TOF axis. This results in a calculated
flight time difference of 24 ns for dimer ions and of 17 ns for atomic
ions obtained in resonant 305 nm ionization. Note that, here and below,
calculated TOF values are for the standard operating conditions which
are 20/80/5.25 V for the extraction, acceleration and steering plate
voltages, respectively.
(iii)
The residual 7 mrad divergence of the molecular beam leads to an
estimated most probable vertical velocity component of  3 m/s. This
in turn implies a flight time difference of about 2 ns and 1 ns for
molecular and atomic ions, respectively.
(iv)
Momentum transfer to the molecule or the atom by the ionizing
photons is negligible (and in any case perpendicular to the TOF axis).
However, upon multi-photon ionization of the sodium dimer in its
vibrational ground state, with either three 532 nm or two 355 nm
photons, the ejected electron has a kinetic energy of up to 2.1 eV. In
that case, the Na2+ ion acquires a recoil speed of about 10 m/s which
leads to an estimated flight time broadening of the Na2+ peak of about
9 ns. Atomic ions were created in an auxiliary experiment via 305 nm
resonant excitation into the 4p state and subsequent ionization by 355
nm photons. In that case, the photoelectron has a kinetic energy of
about 2.5 eV resulting in a recoil speed of about 22 m/s and an
estimated broadening of the Na+ peak of 10 ns.
43
Chapter 3: Non-Resonant Multi-Photon Ionization
(v)
Fringe electric fields in the TOF spectrometer can lead to vertical
accelerations. It is expected that the steering plates in the drift region
will have the largest contributions. However, because fringe fields
have both downward and upward components as the ion trajectory
cross this region, the effects should cancel to first order.
(vi)
Once the photoelectrons are swept to the bottom plate of the extraction
region, Coulomb repulsion between the remaining ions will also lead
to flight time broadening. This effect should be more pronounced with
larger ion yield, i.e. with higher laser power. Indeed, it has been
observed that a four- to five-fold increase of the Na2+ FWHM results
when the total ion yield is raised by a factor of 10 to 20 as shown in
Figure 3.5 for the case of 355 nm horizontal polarization.
The total expected RMS value for mechanisms (i) through (vi) is about 28 ns
for the Na2+ peak compared to an observed width of 30  5 ns at low laser
power. Corresponding numbers for the Na+ peak are 22 ns calculated versus
25 ns observed. It appears therefore that effects due to Coulomb repulsion and
fringe fields can be ignored for the experimental conditions used in this
current work. Hafferkamp54 calculated estimates of FWHM broadening due to
Coulomb repulsion and properly documented in his thesis.
44
Chapter 3: Non-Resonant Multi-Photon Ionization
120
100
FWHM (ns)
80
60
+
355 nm case horizontal polarization: Na2 FWHM
40
20
0
20
40
60
80
100
120
Laser Energy (mJ)
Figure 3.5 Na2+ FWHM laser energy dependence.
Na2 FWHM as a function of excitation laser energy for the case of 355 nm,
horizontal polarization.
+
Since the Monte-Carlo program reproduces the observed FWHM values for both
the resonant atomic and non-resonant dimer peak, the program is applied with the
same parameters to the interpretation of the ionization/fragmentation peaks in
panels (a) and (b) of Figure 3.3.
45
Chapter 3: Non-Resonant Multi-Photon Ionization
4
1x10
+
+
II: Na2 dissociation
IV
II: Na2 dissociation
IV
IIIa: Na (4p) dissociation
IIIb: Na (3d) dissociation
Na (4s) dissociation
IV: dissociative autoionization
IIIa: Na (4p) dissociation
IIIb: Na (3d) dissociation
Na (4s) dissociation
dissociative autoionization
III
3
9x10
II
III
III
a
Ion Signal (arb. units)
a
III
a
a
II
II
b
b
II
b
3
6x10
3
3x10
0
simulation
expt vertical polarization
15500
16000
simulation
expt horizontal polarization
16500
15500
16000
16500
Time-of-Flight (ns)
Figure 3.6 532 nm Na+ Simulation vs. Experiment
Simulation (red line) vs. experiment (open circles) of the atomic peaks of 532 nm
ionization for (a) vertical and (b) horizontal polarization. The inset shows
simulation of individual channels.
To this end, the initial molecular beam conditions need to be augmented with the
kinetic and angular distributions associated with the corresponding fragmentation
channels. The final outcome of this approach for the more complicated case of
532 nm ionization is presented in Figure 3.6 (left and right panel summarizes
vertical and horizontal polarization, respectively). In the insets, calculated TOF
spectra for each channel, indicated by their Roman numerals, are shown
separately. As can be seen from the main part of the figure, the sum of the
individual spectra (red line) agrees nicely with the experimental data (open
46
Chapter 3: Non-Resonant Multi-Photon Ionization
circles). It should be noted that the necessary scaling of the simulated spectra is
only done for vertical polarization; the other polarization case is then evaluated
without any further adjustment. Channel IV, the main topic of this study, will be
discussed in the context of the simulation for the 355 nm spectra where it is the
only active channel. The result for the 355 nm case for both vertical and
horizontal polarizations is shown in a semi-logarithmic plot in Figure 3.7. Note
that despite appearance to the contrary, the average background of the
experimental data in Figure 3.7 is the same as that of the simulation, since
negative values do not show up on a logarithmic scale. Clearly, the turn-around
time of about 1 μs of the fastest fragments is well reproduced by the model. This
quantity is not an adjustable parameter and instead follows directly from the TOF
operating parameters and the maximum kinetic fragment energy in this case.
47
Chapter 3: Non-Resonant Multi-Photon Ionization
355 nm Vertical Polarization
355 nm Horizontal Polarization
Experiment
Simulation
4
Ion Signal (arbitrary units)
10
3
10
2
10
15500
16000
16500
16000
Time-of-Flight (ns)
Figure 3.7 355 nm Na+ Simulation vs. Experiment
Simulation (red line) vs. experiment (open circles) of the atomic peaks of 355 nm
ionization for (a) vertical and (b) horizontal polarization.
In the following, details will be described of how each channel was incorporated
into the simulation program and what the specific findings for each channel are.
3.3.1 Channel I: Direct Dissociative Ionization of Na2
Absorption of three 532 nm (or two 355 nm) photons increases the total energy of
the molecule by 6.99 eV. The Na2 dimer has an adiabatic ionization potential of
4.89 eV and the dissociation energy of Na2+ X2Ʃg+ is 0.99 eV.55 Therefore, sodium
dimer ions can be produced in any vibrational level, and direct dissociative
ionization into Na (3s) and atomic Na+ ions with a maximum kinetic energy per
48
Chapter 3: Non-Resonant Multi-Photon Ionization
fragment of about 0.56 eV is possible. Because dipole moments for the transition
in question are not known, the Na2+/Na+ branching ratio is estimated by
calculating appropriate Franck-Condon factors between the Na2+ 12Ʃg+ and the
Na2+ 12Ʃg+ states (i.e. for photoelectron energies between 0 and 1.11 eV) and
Franck-Condon densities for excitation into the continuum states (photoelectron
energies between 1.11 and 2.11 eV). The calculation was performed on potential
curves obtained from the published data of Camacho et al. for the X1Σg+ state56
while Na2 12Ʃg+ and the Na2+ 12Ʃg+ potential curves from S. Magnier57
implemented with the help of the LEVEL Program of LeRoy58 taking the
measured state distribution of the neutral parent molecules (Tvib = 70 K) into
account. Figure 3.8 shows a calculation made using a vibrational distribution
which peaks at v = 6 and has more than 95% of its population below v = 15. The
results are similar in relative magnitude to cross sections for one-photon
ionization of Na2 into the lowest six vibrational levels obtained from a
calculation59 that incorporates appropriate electronic continuum wavefunctions.
Compared to the Franck-Condon factors found for the discrete Na2+ vibrational
levels, Franck-Condon densities for the Na+ continuum states up to the maximum
fragment energy are found to be several orders of magnitude smaller. Therefore,
channel I is not included in the Monte-Carlo simulation.
49
Chapter 3: Non-Resonant Multi-Photon Ionization
0.15
1
+
1
+
X g v = 0
X g v = 1
Franck-Condon Factor
weighted average
0.10
0.05
0.00
0
2
4
6
8
10
12
14
16
2
18
20
22
24
+
vibrational level of 1 g
Figure 3.8 FCF distribution for the 11Ʃg+  X1Σg+.
Red and purple curves are for X1Σg+ v = 0 and v = 1 contributions, respectively.
Black line is the weighted averaged FCF distribution taking into consideration the
relative population of v = 0 and v = 1 in the ground state of Na2.
3.3.2 Channel II: One-Photon Dissociation of Na2+
Channel II refers to the production of atomic Na+ ions by one-photon dissociation
of Na2+. The Na+ ions generated by this process have the highest kinetic energy
observed in the TOF spectra and are responsible for the prominent early and late
peaks (see Figure 3.4b and Table 3-1). This path requires four 532nm photons in
total: three to create Na2+ and one to dissociate it.
Na2  3hf  Na2  e
Na2  hf  Na   Na
50
(3.4.2)
Chapter 3: Non-Resonant Multi-Photon Ionization
Because of unfavourable Franck-Condon overlap in the second step, channel II is
not observed for 355 nm light. The Na+ fragment kinetic energy is uniquely
determined by the photon energy and the ro-vibrational level of the parent
molecular ion. Therefore, in order to incorporate an appropriate energy
distribution for this channel into the Monte-Carlo calculation, the Na2+ vibrational
distribution needs to be known. The distribution is expected to be the sum of a
near Franck-Condon part from channel II and a non-Franck-Condon fraction from
channel IV. The minimum fragment energy of 0.69 eV occurs for dissociation of
Na2+ molecules in the vibrational ground state. As mentioned above, parent Na2+
molecules are expected to have a broad vibrational distribution. However, a
theoretical investigation60 finds that the Na2+ photodissociation cross section for
532 nm photons is largest for v = 0 and becomes small for vibrational levels v > 2.
This has been further verified by calculating the Franck-Condon Density of the
Na2+ 12Ʃg+  Na2** n1Λu using BCONT program by LeRoy61. Incidentally, the
same study also indicates that photodissociation at 355 nm is negligible – at least
for lower vibrational levels – which explains the absence of the channel II feature
in the 355 nm TOF spectra. Figure 3.9a shows both experimental and simulation
results for the TOF spectrum region with the early channel II peak. The simulated
data only includes the contribution from direct production of Na2+ that can be
ionized by one 532 nm photon. It should be noted that the observed small peak
width is in good part due to the low extraction field used in the experiments. In
this mode, only fragments with small opening angle towards the vertical and
hence with a narrow range of vertical velocity components are detected.
51
Chapter 3: Non-Resonant Multi-Photon Ionization
The general angular distribution of dissociation fragments produced by up to three
photons is given by62
f ( ) 
1
[1   2  P2 (cos  )   4  P4 (cos  )   6  P6 (cos  )]
4
(3.4.3)
where θ is the angle between the laser electric field and the molecular axis while
Pn, (n = 2, 4, 6) is the nth order Legendre Polynomial, and n the associated
anisotropy parameter that depends on the number of photons and the symmetry of
the electronic states involved during the absorption process. For single photon
dissociation, 4 and 6 are 0. Since the transition described here is between two
states of -character, the theoretical value for 2 is 2.0 in the limit of zero rotation
of the parent molecule63. However, the best fit of the simulation (solid and dashed
lines) to the variation of integrated experimental early (red open circles) peaks
with polarization angle is obtained using β2 = 1.0 (see Figure 3.9b). Any initial
rotation of the parent Na2+ ions leads to a lowering of 2 compared to its nominal
value. In addition, the three-photon production of Na2+ may alter the isotropic
distribution of the neutral parent molecules, which also contributes to the lowering
of 2 value.
52
Chapter 3: Non-Resonant Multi-Photon Ionization
a)
b)
simulation
experiment
expt
simulation =1
8x10
3
6x10
3
4x10
3
2x10
3
simulation =1.5
Ion Signal (arbitrary units)
simulation =2
0
14800
15000
0
60
120
180
240
300
(degrees)
TOF (ns)
Figure 3.9 Channel II Experiment vs. Calculation.
(a) early part of the channel II TOF, (b) channel II Na+ yield as a function of laser
polarization angle, experiment (open circles) vs calculation (line).
3.3.3 Channel III: Two-Photon Dissociation of Na2
This channel is responsible for the shoulder peaks in the 532nm atomic spectrum
(see Figure 3.6) and is characterized by two-photon dissociation of the neutral
dimer into ground and excited state neutral atomic fragments, followed by onephoton ionization of the excited atom. The entire process thus requires three
photons:
Na2  2hf  Na*  Na
Na*  hf  Na   e
53
(3.4.4)
Chapter 3: Non-Resonant Multi-Photon Ionization
2 x 532 nm photon
3s + 4p
1
35000
5 g
3s + 3d
1
3s + 4s
-1
Energy (cm )
2 g
1
3 g
30000
1
4 g
25000
3
6
9
12
15
18
R (Angstrom)
Figure 3.10 Simplified Na2 potential diagram
for potentials involved in 2-photon dissociation process.
While absorption of two 532nm photons is not expected to lead to resonant
bound-bound Na2 excitation, bound-free transitions are possible. Based on the
observed turn-around times (see equation (3.4.1) and Table 3-1), it is anticipated
that dissociation into Na 3p, 4s, 3d, and 4p states plays a role. All these excited
atomic states can be ionized by one 532 nm photon, except the 3p state. Because
54
Chapter 3: Non-Resonant Multi-Photon Ionization
of 2-photon selection rules, only singlet states with g-symmetry need to be
considered.
For the angular distribution of ions produced in Channel III, equation (3.4.3) with
non-zero 4 has to be used. Corresponding -values for two-photon dissociation
have been calculated by Dixon64 and are tabulated in Table 3-2. A single 532 nm
photon will excite near resonant from the ground state of Na2 to the A1Ʃu+ state.
Therefore for two-photon dissociation, selection rules only permit final molecular
states of Ʃ or Π character. For a given excitation, the corresponding angular
distribution curves are shown in Figure 3.11 for two-photon linear polarization
process. For the current process, only ΣΣΣ and ΣΣΠ pathways are possible
corresponding to curves C and B in Figure 3.11. The 355nm photon energy of
28,271cm-1 lies below the 3s + 4s, 3s + 3d, and 3s + 4p limits. Therefore, channel
III is absent in the 355 nm spectra. It is energetically possible to dissociate into the
3s + 3p fragments via the A1Ʃu+ potential. However, the Frank-Condon overlap
for this channel can be expected to be very poor and indeed the 355 nm TOF
spectra show no trace of this pathway.
Therefore, ignoring triplet states, only the following potential curves will be
involved: 31Ʃg+ for the 3s + 4s limit; 41Ʃg+ and 21Пg for the 3s + 3d limit; and
51Ʃg+ for the 3s + 4p limit.
55
Chapter 3: Non-Resonant Multi-Photon Ionization
Two-photon dissociation
β2
β4
P(θ)
curve
ΣΣΣ
20/7
8/7
c4
C
ΣΠΣ
-10/7
3/7
s4
A
Σ  Σ  Π, Σ  Π  Π
5/7
-12/7
s2c2
B
ΣΠΔ
-10/7
3/7
s4
A
Table 3-2 Two-photon excitation pathways.
Excitation pathways and recoil anisotropy for rapid dissociation following twophoton excitation via two near-resonant intermediate states. (c = cos θ, s = sin θ)
P(θ)
curve
Σ-Σ-Σ-Σ
c6
D
Σ-Π-Σ-Σ, Σ-Σ-Π-Σ, Σ-Π-Π-Σ,
Σ-Σ-Π-Δ, Σ-Π-Π-Δ, Σ-Π-Δ-Δ
s4c2
B
Σ-Σ-Σ-Π, Σ-Σ-Π-Π, Σ-Π-Π-Π
s2c4
C
Σ-Π-Σ-Π, Σ-Π-Δ-Π, Σ-Π-Δ-Φ
s6
A
Three-photon Excitation
β2
β4
β6
Table 3-3 Three-photon excitation pathways.
Excitation pathways and recoil anisotropy for rapid dissociation following threephoton excitation via two near-resonant intermediate states. (c = cos θ, s = sin θ)
56
Chapter 3: Non-Resonant Multi-Photon Ionization
A
B
C
1.0
B
0.8
C
Two-photon linear polarization
0.6
P()
0.4
0.2
0.0
D
1.0
C
A
B
B
C
D
Three-photon linear polarization
0.8
0.6
P()
0.4
0.2
0.0
0
50
100
150
theta ()
Figure 3.11 Recoil anisotropy for rapid dissociation
following two-photon (top) and three-photon (bottom) excitation via a nearresonant intermediate states. (A) – (D) refer to the excitation
pathways detailed Table 3-2 & Table 3-3.
57
Chapter 3: Non-Resonant Multi-Photon Ionization
3.3.4 Channel IV: Indirect Ionization via Repulsive
Rydberg States
Channel IV corresponds to the broad central atomic ion peak shown in Figure 3.6,
present in the TOF spectra obtained with both 532 and 355 nm photons. This
process requires two 355nm or three 532nm photons and proceeds as follows:
Na2  n  hf  Na2**
Na2  e 

Na  Na  Na  e
**
2
Na*  Na
(a )

(b)
(3.4.5)
(c )
where n = 2, 3 is the number of photons. Upon absorbing 2 or 3 photons, Na2
molecules are promoted into doubly excited states Na2** converging to the
repulsive 12Ʃu+ potential (see Figure 3.12 for a schematic representation). These
dissociative states are the analog to the well-known states in the hydrogen
molecule65. However, in contrast to the H2 case, no published information – based
on theory or experiment – is available. As long as the excited dimers are
energetically above the ion potential they can autoionize. This energy condition is
satisfied for atom-atom distances smaller than the critical value RC where the
dissociative Rydberg potential crosses the ion ground state potential X2Ʃg+. Once
the molecule is stretched beyond a distance RC, autoionization becomes
energetically impossible and the molecules dissociate into two neutral atoms
(channel 3.4.5c). The excited neutral atom can be ionized by a single photon. Ions
coming from this channel will have a unique and high fragment energy. However,
ions with the corresponding flight time are not observed in the TOF spectra.
Hence, either the majority of the Na2** molecules autoionizes before reaching
58
Chapter 3: Non-Resonant Multi-Photon Ionization
point RC, or the single-photon ionization cross section with 355 or 532 nm light is
very small for these excited neutral atoms.
In the following, a semi-classical point of view will be adopted. Immediately after
excitation onto the dissociative Rydberg state at distance RA (assumed to be the
equilibrium distance of the Na 2 molecule in the X1Σg ground state), the molecules
start to dissociate along the doubly excited state, converting potential energy into
kinetic energy. Depending on the distance R at which autoionization occurs, either
molecular (RA < R < RB) or atomic (RB < R < RC) ions will be produced (channels
3.4.5a and b, respectively). The distance RB dividing the two regimes is
determined by the condition that the kinetic energy acquired on the dissociative
Rydberg potential equals the binding energy of the Na2+ ion at RB. Between RB
and RC, the kinetic energy of the Na+ fragment increases from zero to the
maximum possible.
59
Chapter 3: Non-Resonant Multi-Photon Ionization
1x10
4
+
2
Na2 1 U
5x10
Na2  n  hf  Na2**
3
Na2  e 
Energy (cm )
i
(a )
Na2**  Na   Na  e  (b)
-1
Na*  Na
(c )
0
ii
+
2
Na2 X g
-5x10
3
iii
-1x10
4
RA
RB
5
RC
10
Internuclear Distance R (Angstrom)
Figure 3.12 Model
Critical points along the Rydberg potential.
The fragment energy distribution needed for the simulation is obtained from the
following semi-classical arguments35. If excitation occurs at time t = 0 and
distance RA, and if the characteristic autoionization lifetime  is independent of R,
then at times t > 0 the fraction f (t ) of dimers remaining in the Rydberg state is
given by:
f (t )  exp(t /  )
The time t molecules need to reach atom-atom distance R > RA is given by:
60
(3.4.6)
Chapter 3: Non-Resonant Multi-Photon Ionization
R
t  t ( R) 
dR
 u( R)
(3.4.7)
RA
u ( R) 
1 2 Etotal  VRyd ( R)
c

(3.4.8)
where u(R) is the relative speed of the two dissociating Na atoms at inter-nuclear
distance R, VRyd(R) is the Rydberg potential, Etotal = 8990 cm-1 (assuming initial v
= 0, J = 0 in the Na2) is the total excess energy of the molecule after absorbing two
355 nm (three 532 nm) photons, and μ is the fragment mass in atomic units.
If sodium dimers autoionize before reaching point RB, a molecular ion Na2+ is
created. Upon autoionization, any kinetic energy gained is converted into
vibrational energy of the molecular ion. The vibrational distribution associated
with these Na 2  ions is expected to be different from the Franck-Condon
distribution of channel II. Specifically, this process produces Na2+ with a broader
distribution of vibrational levels than those from the process described in channel
II. This vibrational distribution will be dependent on the autoionization lifetime as
well as the Franck-Condon overlap at a specific atom-atom distance the molecule
decides to autoionize. High vibrational levels of Na2+ from this channel are a
possible cause for the signal with even larger kinetic energy than the fast peak
from channel II shown in Figure 3.9. If the molecule auto-ionizes between point
RB and point RC (Figure 3.12), the kinetic energy of the molecular ion is larger
than the local binding energy and the excited molecule climbs out of the potential
well of the 12Σg+ curve and dissociates into Na+ and Na. This process of
simultaneous dissociation and ionization along the Rydberg Potential is called
61
Chapter 3: Non-Resonant Multi-Photon Ionization
dissociative autoionization. The energy left over after climbing up the potential
well is equal to the asymptotic fragment kinetic energy characteristic of this
process. Unlike channel II and channel III, which have discretely defined
fragment kinetic energy values, dissociative autoionization through Rydberg states
has a continuous energy distribution. If ionization happens at point RB, the
molecular ion will have just enough energy to climb the potential well, resulting in
Na+ with zero dissociation energy. If ionization occurs at point RC, the resulting
fragment will have the maximum fragmentation energy of 1.1 eV. This is the
reason why the dissociative autoionization peak is broader than that of the other
channels.
4
5.8x10
2
6p
5d
6s
5p
4d
5s
+
X u
4
6x10
2
+
X u
4
5.7x10
4
-1
V (cm )
5x10
v=1
v=0
4
5x10
4
5.6x10
4
4x10
2
+
X g
4
2
4
6
8
10
12
3.0
3.5
4.0
5.5x10
R (Angstrom)
Figure 3.13 Rydberg Potentials
a) Rydberg Potentials shifted from the 12Σu+ potential curve.
b) Rydberg states near the total photon energy with significant FC overlap
with the ground state Na2 v = 0 and v = 1.
62
Chapter 3: Non-Resonant Multi-Photon Ionization
The exact shape of the Rydberg potentials is not known and is therefore obtained
by shifting the 12Σu+ potential curve downward by the binding energy of the
appropriate asymptotic atomic Rydberg level. Six different Rydberg potentials
were considered that dissociate to atomic limits 5s, 6s, 5p, 6p, 4d and 5d as shown
in Figure 3.13a. Figure 3.13b zooms into the excitation region and shows the
Franck-Condon overlap region with the vibrational levels v = 0 and v = 1 of the
Na2 ground state. The appropriately weighted Franck-Condon Density from the
two contributions is shown in Figure 3.14. Based on this data, the Rydberg
potentials that dissociate to 4d and 5p states are clearly favoured. The 6s Rydberg
state, though shown in Figure 3.14 to have significant Frank-Condon overlap, will
not be considered because it produces only very poor agreement between the
experiment and simulated TOF. 5s, 6p and 5d will also not be considered due to
small Franck-Condon overlap with the ground state Na2.
63
0.20
0.15
-1
Franck-Condon Density (FCF/cm ) @ Total Photon Energy
Chapter 3: Non-Resonant Multi-Photon Ionization
0.10
0.05
asymptotic limit
5s
Rydberg state symmetry
6s

5p

6p
4d
5d

Figure 3.14 FC density of Rydberg potentials
Calculated FC density of Rydberg potentials shown in Figure 3.13 from the
ground state Na2 (weighted between v = 0 and v = 1) at total excess photon
energy using two 355 nm or three 532 nm photons.
Appropriate angular distributions according to Eq. (3.4.3) for a two-photon
dissociation process using 355 nm and a three-photon dissociation process using
532 nm are considered. Table 3-2 and Table 3-3 list the anisotropy parameters for
a given excitation process and the probability functions are shown in Figure 3.11.
The angular distribution of fragment photo-ions is observed experimentally by
changing the direction of the linearly polarized laser using a double Fresnel rhomb
rotator. Both vertical and horizontal polarization in the lab frame data will be used
to compare with the simulated TOF data.
64
Chapter 3: Non-Resonant Multi-Photon Ionization
The fragment energy distribution is calculated from a given potential energy
function and potential width function, Γ(R), which is proportional to the inverse of
the autoionization lifetime at atom-atom distance R. Given a finite number of
initial ions, the probability of autoionization, P(R), should also take into account
the decrease in Na2** population due to molecules that have already autoionized at
previous R values. The probability of autoionization can be written as66-68
( R )
( R ')
P( R) 
exp[  dR '
]
u ( R)
u ( R ')
RB
R
(3.4.9)
where Γ(R) is the potential width function or the autoionization width.
In the measured atomic TOF spectra, only molecules that ionize from RB to RC
contribute. That is why in the simulation, the exponentially decaying term is
integrated starting from RB and not from RA. In the simulated data, the probability
density function is divided into a discrete radial intervals between RB and RC and
the corresponding fragment energy with that particular internuclear distance is
recorded. The total fragment energy for an autoionization event at distance R (RB
≤ R ≤ RC) is given by the following equation
EexcessK .E. ( R)  Etotal  VRyd ( R)  Vg ( R)
(3.4.10)
where Vg(R) is the Na2 X2Σg+ potential.
Optimization of the simulated TOF spectra is achieved by the following
procedure:
65
Chapter 3: Non-Resonant Multi-Photon Ionization
i.
The energy distribution is calculated from a trial potential width
function, Γ(R) with adjustable parameters for each case of Rydberg
potential dissociating to 4d and 5p atomic limits as shown in Figure
3.13. Two different functional forms were tested, namely: linear and
quadratic functions.
ii.
Using the three possible angular distributions for the 355nm case
indicated in Table 3-2, the simulated TOF spectra are scaled by
different weight factors A, B, C such that A + B + C = 1 where A, B &
C correspond to the different excitation combinations as indicated in
the last column of Table 3-2.
iii.
For each linear combination of A, B and C, a χ2 calculation is
performed in steps of 0.01 from 0 to 1 according to
 2   ( I exp (i)  I sim (i  offset ))2
(3.4.11)
i
where C is not a free parameter (C = 1 – a – b) while the offset
corresponds to the number of the bin in the simulation matching the
first experimental point. The best χ2 value is recorded that corresponds
to the energy distribution used.
iv.
The step is repeated with a new energy distribution derived from the
functional form of the potential width function, Γ(R).
The optimization procedure is performed by Hafferkamp54 and documented in
details in his thesis. The linear function used for the Γ(R) has two adjustable
parameters in the form:
66
Chapter 3: Non-Resonant Multi-Photon Ionization
( R)  a  b( R  RB )
(3.4.12)
the latter being the simplest case of constant autoionization width. Positive,
negative, and zero slope values are considered. The other functional form for Γ(R)
considered is a quadratic function:
( R)  d  k ( R  R0 )2
9x10
3
(b)
(a)
Ion Signal (arb. units)
(3.4.13)
6x10
3
3x10
3
0
simulation (constant 
expt. horizontal polarization
simulation (constant )
expt. vertical polarization
15500
16000
16500
15500
16000
16500
simulation (linear )
expt. vertical polarization
15500
16000
16500
simulation (linear )
expt. horizontal polarization
15500
16000
16500
Time-of-Flight (x10 ns)
Figure 3.15 TOF: Constant and linear Γ(R)
Typical comparison of experimental TOF using 355 nm to simulation produced
with a constant (a) and (b) linear potential width functions.
with three adjustable parameters (R0, being the location of the extremum of the
parabola). Figure 3.15 shows the optimized comparison of experimental and
simulated 355 nm TOF spectra for constant and linear potential width functions.
67
Chapter 3: Non-Resonant Multi-Photon Ionization
For the constant Γ(R) case, the value was varied in steps of 0.005 from 0.001 to
0.5 in unit of 1/fs. This variation corresponds to an autoionization lifetime range
from 1000 fs to 2 fs. The minimum χ2 was found at Γ(R) = 0.101/fs, which
corresponds to an autoionization lifetime of 10 fs. As seen in Figure 3.15b for
both vertical and horizontal polarizations, using a constant potential width
function in the simulation, the generic shape of the TOF is reproduced. However,
the width is wider and the “wings” are underestimated. In the linear Γ(R) case, the
“a” parameter is varied in steps of 0.01 ranging from 0.01 to 0.06 while “b” ranges
from -0.015 to 0.25 in steps of 0.005. The resulting TOF in Figure 3.15b is not
that much of an improvement in comparison to the constant case. Adding a slope
to the constant function does not improve the quality of the fit between
experiment and simulated data.
68
Ion Signal (arbitrary units)
Chapter 3: Non-Resonant Multi-Photon Ionization
1x10
4
5x10
3
expt (horizontal polarization)
simulation
expt (vertical polarization)
simulation
0
15500
16000
15500
16500
16000
16500
Time-of-Flight (ns)
Figure 3.16 TOF quadratic Γ(R)
Comparison of experimental 355 nm TOF to simulation produced with quadratic
function potential width and weighted angular distributions that optimized the χ2
fit.
Using a quadratic function in Eq. (3.4.13), the best fit was found for parameters d
= 0.02, k = 0.06 and R0 = 5 Å. The comparison of the TOF spectra with
experiment is shown in Figure 3.16 for both vertical (a) and horizontal (b)
polarizations. The simulated TOF data for a quadratic potential width function
produces a much better fit to the experimental TOF data. It is evident that the
“wings” and the width of the peak are now reproduced reasonably well. For the
“wings” in the experimental TOF spectra to be reproduced, the potential width
function, Γ(R) needs to have large positive slope and hence smaller autoionization lifetime as it approaches RC in order to create fast ions observed in the
69
Chapter 3: Non-Resonant Multi-Photon Ionization
“wings”. The three different potential width functions that gave a χ2 minimized fit
to the experimental data are plotted in Figure 3.17 and the corresponding
optimizing weight factors for the fragment angular distributions are tabulated in
Table 3-4. For all three cases, the optimized angular distributions are very similar:
case A and B are weighted approximately evenly while case C contributes very
little. In addition, Hafferkamp also found that there is a distinct minimum in the χ2
contour plot and this has high sensitivity to angular dependence. This suggests
that all three angular distributions are contributing to the whole process and that
the dissociative Rydberg states in the experiment can have Σ, Π and Δ symmetry.
0.30
RB
RA
Constant
Quadratic
Linear
Potential Width Function, (R), (1/fs)
0.25
RC
0.20
0.15
0.10
0.05
0.00
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
Internuclear Axis, R(Angstrom)
Figure 3.17 Potential width functions, Γ(R)
Potential width functions, Γ(R), for three different cases (constant, linear and
quadratic) that produce the energy distribution necessary for best χ2 data between
experiment and simulated TOF spectra. At R < RB, Γ(R) is extrapolated from the
functional form established in the region between RB and Rc.
70
Chapter 3: Non-Resonant Multi-Photon Ionization
Γ(R) function
Case A
Case B
Case C
Constant
0.39
0.47
0.14
Linear
0.40
0.47
0.13
Quadratic
0.47
0.44
0.09
Table 3-4 Angular distribution
Fractional weights of three different angular distribution used in the simulation
that resulted in best fit for constant, linear and quadratic Γ(R) functions.
Between the 4d and 5p Rydberg potentials, simulations with fragments leading to
the 4d asymptote produce the best fit. In order to further improve the result, in the
future, a systematic modification of the potential curves converging to these
atomic limits can be incorporated in the simulation program to explore how the
shape of the Rydberg potential affects the shape of the simulated TOF data.
However, even in the absence of Rydberg potential shape dependence, a very
good agreement was achieved between experimental and simulated TOF.
The dissociative autoionization channel is open in both 532 nm and 355 nm
experiments. This is because the Franck-Condon window and the total photon
energy input are exactly the same for both three 532 nm and two 355 nm photons.
However, due to the selection rules, the final Rydberg states for the two different
excitations are different. This might account for the slight difference in the shape
of the TOF peak associated with this channel observed for 532nm and 355nm
photon experimental results. In the 532 nm case, a similar procedure was carried
out but using angular distributions appropriate for a three-photon excitation
71
Chapter 3: Non-Resonant Multi-Photon Ionization
process. The final result for 532 nm was shown in Figure 3.6. Here also, very
good agreement between simulation and the experimental TOF has been achieved.
72
Chapter 3: Non-Resonant Multi-Photon Ionization
3.4 Summary
The main goal of this work was an exploration of the different pathways of Na+
production in non-resonant multi-photon ionization with 355nm and 532nm
photons. The TOF mass spectrometer was used to obtain the flight times of Na+
ions experimentally, which carry information about the initial energy and angular
distribution of photo-dissociated ions.
The theoretical model was tested and validated through a simulation program
which produces TOF spectra starting with initial conditions compatible with the
experimental constraints and the various processes leading to the observed
products. Experimental TOF spectra of Na+ ions from 355nm and 532nm photoionization were compared with simulated TOF spectra. It was shown that
simulation and experimental results agree well. The angular dependence of the
detection efficiency was tested, and the dissociation angular distribution of the
Na+ fragments was studied. Theoretical models for three different pathways of
Na+ production, labeled channel I through IV are investigated in detail. Channel I
is the direct ionization of the ground state Na2 to the continuum of the 12Σg+ and
subsequent dissociation and is not observed in the experimental TOF spectra.
Channel II, one-photon dissociation of Na2+, produces the fastest observed Na+
fragments with the highest dissociation energy. Channel II is highly dependent on
the polarization angle of the laser and is absent from the 355nm photo-ionization.
Channel III, 2-photon dissociation and excited Na ionization, produces the
shoulder peaks observed in TOF spectra. Channel III is also highly dependent on
the polarization angle of the laser and absent from the 355nm photo-ionization.
73
Chapter 3: Non-Resonant Multi-Photon Ionization
Channel IV, dissociative autoionization, produces the broad center peak in the
TOF spectra. It was shown that Rydberg states of effective principal quantum
number n near 4, converging to the 12Σu+ potential of the sodium cation Na2+, play
a significant role in the production of atomic Na+ ions in the dissociative
autoionization channel. Channel IV is not sensitive to the polarization angle. Since
the exact shape of the Rydberg potential is not known, the simulated TOF data is
achieved by obtaining a shifted potential curve of the 12Σu+ of Na2+. The
asymptotic limit 4d + 3s produced the best fit simulated data to experimental
results. It has also been learned that the width function Γ(R) of the Rydberg
potential from critical potential points RB to RC follows a near quadratic function,
Γ(R) = 0.02 + 0.06 (R – 5). This corresponds to an auto-ionization lifetime range
of 4 to 50 fs along the RB to RC curve of the Rydberg potential. The parabolic
nature of the width function can be attributed to the interaction of neighboring
states, i.e. the ion-pair states as well as other double excited states which could
lead to complex resonances. The dynamics of this highly excited state which is in
femtosecond scale were described using semi-classical model and experimental
data taken using nanosecond lasers.
74
4 Radiative Lifetimes of Some Excited
Na2 States
In this chapter, details of experimental radiative lifetime measurements and
calculations of individual ro-vibrational levels in some excited states of Na2, the
21Σu+ (double well state) and the 41Σg+ (shelf state) are discussed. Calculations
were carried out using the LeRoy Level 8.0 program.58 Conventional lifetime
measurements use laser induced fluorescence spectroscopy. In this work, pumpprobe resonant ionization technique is employed. Ions generated are collected
with the aid of a linear time-of-flight (TOF) spectrometer. The population decay
of the excited state of interest is measured by delaying the fixed-frequency probe
laser relative to the pump laser.
Atomic excited states of Na, namely the 3p2P and 4p2P states, along with the first
excited state of Na2, A1Σu+ were used to test the method. This work measured
lifetimes for Na 3p2P and 4p2P averaged over spin-orbit splitting are 16.5 ± 0.2 ns
and 109.2 ± 4.1 ns respectively, both in good agreement with the known
values69-71 of 16.3 and 107.6 ns averaged over spin-orbit splitting. For the double
well 21Σu+ state, a two-photon scheme is used. Ground state Na2 produced in a
molecular beam is excited resonantly by the doubled output of a suitably tuned
dye laser and then ionized by a photon (532 nm) from a delayed Nd:YAG laser.
By adjusting the delay of the second laser, the population decay of the excited
state is observed and its lifetime extracted. Moreover, by tuning the pump laser to
different ro-vibrational levels, lifetime as a function of vibrational quantum
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
number was measured. Experimental data show a noticeable and systematic
variation, especially near the potential barrier. The overall magnitude of the
lifetime is consistent with the vibrationally averaged value of 52.5 ns reported by
Mehdizadeh25. For the 41Σg+ state, a double resonance technique via the A1Σu+ (v =
19, J = 20) state was employed followed by two-photon (1064 nm) delayed
ionization from a third laser. Here as well, the extracted lifetime shows vibrational
state dependence.
The following section briefly recounts the fundamental interactions of
electromagnetic fields with matter and the basic theory concerning spontaneous
lifetime (4.1). In the next section (4.2), the specific excited states of interest of Na2
are introduced. It will be followed by a discussion of the excitation scheme and
experimental set-up (4.3), a section covering tests of method by measuring the
known lifetimes of some excited Na atomic states (4.4), a brief description of how
the calculation was implemented, the results (4.5) and discussion (4.6) of the
experiment, and finally the summary of this project (4.7).
76
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
4.1 Transition Probability
When matter interacts with electromagnetic radiation such as in optical pumping
or laser excitation, one must be able to calculate the rate of radiative transitions
induced by such fields. This in turn requires knowledge of transition dipole
moment, absorption coefficient, Einstein A and B coefficients, etc. and the
relations between these quantities. A good discussion of these quantities can be
found in texts such as Steinfeld’s book on molecular spectroscopy.72
Einstein, in 1917, introduced the concept of stimulated emission and three
probability coefficients describing the rates of possible transitions between two
states |i> and |k>.73 Consider a two-level system with Ei > Ek such as the one
illustrated in Figure 4.1, populated by Ni and Nk particles respectively.
Ni
Aik Ni
Bik ρ(v) Ni
Bki ρ(v) Nk
Ei
Ek
Nk
Figure 4.1 Two-level system
Radiative processes connecting energy levels Ek and Ei.
Three radiative processes can occur between the two levels. In the figure, the rates
of these processes are expressed using the so-called Einstein coefficients. The
77
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
coefficient of spontaneous emission, Aik determines the probability to undergo
spontaneous transition from upper state |i> to a lower state |k> emitting radiation
with a frequency v in the process with an energy difference Ei - Ek = hv. When an
external electromagnetic radiation field of frequency v is present, particles initially
in lower state |k> can absorb a photon from the field and be excited to the upper
state |i>. The Einstein coefficient of induced absorption, Bki determines the rate of
this transition. It is also possible for the radiation field to stimulate particles in
state |i> to undergo a “forced” or so-called induced transition to lower state |k>
emitting a photon of energy hv. This process occurs with a probability
proportional to Bik, which is the Einstein coefficient of induced or stimulated
emission. These three processes are defined such that the rate of change in the
population of Nk and Ni is
dN k
dN
  i   Bki  ( ) N k  Bik  ( ) Ni  Aik Ni
dt
dt
(4.1.1)
where ρ(v) is the density of photons with a frequency v corresponding to the
energy difference hν = Ei – Ek.
From Eq. (4.1.1), using the blackbody radiation density  (v) 
8 hv3
1
,
3
hv / kT
c e
1
the following relations between the three Einstein coefficients can be derived74
Bki 
gi
Bik ,
gk
16 2  3
Aki 
Bki
c3
78
(4.1.2)
(4.1.3)
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
where gi and gk are statistical weights or degeneracy factors of the states |i> and
|k>, respectively.
It can be shown75 using quantum mechanical perturbation theory treatment of
spontaneous emission under certain assumptions, i.e. the electric dipole
approximation for weak fields, that the Einstein A-coefficient is
Aik 
32 3  3
| i | er | k  |2
3 4 0 c3
(4.1.4)
where Aik is the probability per unit time for spontaneous transition from |i> to a
state |k> of lower energy. In Eq. (4.1.4), ε0 is the permittivity of free space, ħ is the
reduced Planck constant, v the emission frequency, c is the speed of light, i and k
are the initial and final state radial wave functions and er is the dipole (or
transition dipole) moment function.
The lifetime τi of an excited state |i> is related to the Einstein coefficient of
spontaneous emission, Aik. In the case where there is no external field present,
ρ(v) = 0, the rate of change of the population in state |i> is given by:
dNi
  Ni  Aik
dt
k
(4.1.5)
The solution to equation (4.1.5) is:
Ni (t )  Ni (0) e

t
j
(4.1.6)
where Ni (0) is the number of atoms or molecules in levels |i> at time zero and
79
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
1
i
  Aik
(4.1.7)
k
The sum in equations (4.1.5) and (4.1.7) extends over all possible final states |k>
and therefore includes all accessible radiative decay channels. For a case where
radiative decay is allowed to the continuum states, Eq. (4.1.7) will include a term
integrated over all possible final states. The lifetime τi is the mean life, or the
lifetime of the state |i>.
There are many reasons for studying radiative properties:
(i)
the natural lifetime determines the fundamental limit of resolution
Δv = 1/2πτ in spectroscopic investigations;
(ii)
transition probabilities can be used for sensitive testing of atomic wave
functions since Aik is related to the matrix element of the electric dipole
operator er between the two wavefunctions;
(iii)
transition probabilities and the related oscillator strengths are of utmost
importance for astrophysics, e.g., for calculations of the relative
abundances of the elements in the sun and stars;
(iv)
the radiative properties of atoms and ions are also of great importance
in plasma physics, e.g., for temperature determination and for the
calculation of the concentrations of different constituents;
(v)
In laser physics, lifetimes and transition probabilities are decisive for
predictions of potential laser action in specific media.
80
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
4.2 The 21Σu+ and 41Σg+ States of Na2
Bound electronic states displaying two (or more) minima in their potential energy
curves provide some of the largest challenges for experimental and theoretical
investigations. It is a challenge for experimentalists dealing with systems
involving two minima because of the irregular spacing of vibrational bands,
making the analysis of spectroscopic data and constructing the potential curves
difficult. Because of poor Franck-Condon overlap, the outer potential well of the
state is also often difficult to access compared to “typical” molecular states with
single potential well. A double minimum potential indicates an abrupt change in
the electronic structure of the adiabatic state caused by a strong interaction with
some neighboring state, which must be properly taken into account in the
calculations, making the task more complicated for theoreticians.
In the sodium dimer molecule, the 21Σu+ state is one of the best characterized
examples of a double minimum state.76 It was first predicted by Valence and
Tuan77 and confirmed by Jeung.78 The state is formed by the avoided crossing of
two diabatic states. The first is a Rydberg state dissociating to Na (3s) + Na (4s),
which gives rise to the inner well. The second one, at large distance, has
considerable ionic character, resulting in the outer minimum. The strong mixing
of two electronic characters of this state is the reason why its radiative lifetime is
expected to vary with vibrational level, especially near the barrier between the two
wells. Lifetime measurements of this state have been published by Mehdizadeh25
and Radzewicz.26 The first group reported an average lifetime of 52.5 ns for a
non-specified vibrational distribution. The second group found a lifetime of 40 ns,
81
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
again for a vibrational distribution that was not further characterized. Both
experiments were conducted using photon counting fluorescence spectroscopy.
Because the 21Σu+ is populated by collision in both experiments, the reported
lifetime from the time-resolved measurements of the diffuse fluorescence band is
an average over a range of vibrational levels as mentioned above.
In the case of the 41Σg+ state, the “shelf” of the state is formed following the same
physical principles underlying the double well potential. The avoided crossing
with the ion pair potential curve gives rise to the unusual adiabatic potential curve
with a “shelf” rather than an outer well.79-82 The existence of the “shelf” in the
41Σg+ state gives rise to highly non-monotonic behaviour of the vibrational level
spacing as the levels increase83. The lifetime of this particular Na2 state has neither
been measured nor been predicted theoretically. Here as well, the radiative
lifetime is expected to change with vibrational quantum number especially near
the “shelf” of the state. The potential diagram for both states is shown in Figure
4.2. Included in this figure are also three negative ion pair Na+ + Na- potentials.
They were generated starting at their asymptotic limits provided by Buckman84
with a 1/R coulombic potential at small inter-nuclear distance, R, and an
additional term 1/R4 incorporated with the dispersion coefficient from Tsai.85 The
ion pair curve (b) is responsible for the ionic character of the outer well of the
21Σu+ state and the formation of the “shelf” in the 41Σg+ state.
82
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
4
3.8x10
(b)
+
Na + Na
1 0
2 1
(3p4s P + 3p D)
(c)
+
Na + Na
3 0
(3s3d F )
(a)
+
Na + Na
3 0
(3s3p P )
4
3.6x10
4
3.4x10
-1
V (cm )
1
+
4 g
4
3.2x10
1
+
2 u
4
3.0x10
4
2.8x10
4
2.6x10
2
4
6
8
R (A)
10
12
14
16
Figure 4.2 Na2 21Σu+ and 41Σg+ States
Potential Diagram of Na2 21Σu+ (double well) and 41Σg+ (shelf).
The red dashed curves are the ion pairs Na+ - Na-.
83
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
4.3 Excitation Scheme
The general experimental set-up was already discussed in Chapter 2. In brief, the
Na2 molecules were produced in a supersonic expansion by heating metallic
sodium to about 800 K. The pump and delayed probe lasers crossed the molecular
beam perpendicularly in the interaction region of the TOF chamber. Na2+ ions
produced in the interaction region are extracted by an electric field Ee = 50 V/cm
and further accelerated with an accelerating field Ea = 126 V/cm. Ions are
collected by a Chevron configuration microchannel plate (MCP) detector and read
into a 150 MHz digital oscilloscope (HP Lecroy 9400), recorded via GPIB
through a PC using LabVIEW software.
84
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
4
6x10
2
+
1 u
a)
Excitation Scheme
+
+
b)
-
Na + Na
1 0
2 1
(3p4s P + 3p D)
C u
3x10
4
3x10
4
+
1 g
4
5x10
+
-1
V (cm )
Na - 3s v=50
v=33
4
4x10
-1
V (cm )
delayed ionization
+
1
1
+
2 u
4
3x10
1
+
2 g
4
1
2 u
3s-4s
+
1 g
3s-3p
4
3x10
4
-
Na + Na
3 0
(3s3p P )
+
c)
1
+
1
+
15
X g
2 g
2x10
1
1
3x10
+
1 g
+
A u
4
1x10
10
5
3s-3s
1
+
X g
0
0
0
5
Dipole Moment (Debye)
2
1
10
R (Angstrom)
15
2
4
6
8
R(Angstrom)
10
12
Figure 4.3 Excitation Scheme 1
a) Na2 simplified potential diagram and schematic of the experimental excitation
scheme for the lifetime measurement of 21Σu+ state; b) the 21Σu+ potential with
some of the ion pair potentials; c) Transition dipole moment of 21Σu+ as a function
of inter-atomic distance R (Å) for the three lower states to which it can decay by
radiative transition.
A two-photon scheme is used to measure the lifetime of the 21Σu+ state as shown
schematically in Figure 4.3a. Included in the figure are also the three emission
channels of the 21Σu+ state. Since the excitation energy of the Na2 21Σu+ state is
around 29 500 cm-1, ground state Na2 is excited resonantly by a frequency doubled
output of a tunable dye laser that is operated with DCM dye with a fundamental
output in the range from 666 – 714 nm. The bandwidth of the doubled dye output
is about 0.2 cm-1 and its pulse duration is about 10 ns. Laser power is adjusted
85
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
such that two requirements are satisfied: (1) power must be sufficiently high to
populate the excited state by a single photon, but (2) not so high that two-photon
ionization (resonant or not) depletes the excited state population before the probe
laser arrives. Excited Na2 molecules are subsequently ionized by 532 nm photons
from a variably delayed Nd:YAG laser. Energy balance requires only one 532 nm
photon to ionize dimers in the 21Σu+ state. Therefore, the probe laser power can be
adjusted such that non-resonant multi-photon ionization of the ground state Na2 by
the probe laser alone is completely negligible. The timing of the excitation and
probe lasers is controlled by a pulse generator (SRS DG645), which also serves as
the master clock of the whole experiment. The pulse generator is programmed
(delay, amplitude and polarity) to output the various TTL pulses needed to
synchronize and operate the two separate lasers. The timing of the TTL pulses is
adjusted according to the actual arrival time of the laser pulses at the molecular
beam, which is verified using a fast photo-diode with rise and fall-time of less
than 1 ns situated at the entrance window of the experimental chamber. Data
presented in this work was obtained by summing typically 500 laser pulses and
using delay of excitation and probe lasers from 0 to 800 ns.
86
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
4
+
1 u
2
b)
+
+
1 g
4
1
+
V (cm )
1
-1
1
2x10
+
c)
1x10
1
+
1
+
A u
3s-3p
B u
4
1
4
3.0x10
4
2.8x10
4
3s-4s
+
9
+
A u
(v, J)
3.2x10
-
4
B u
4
Na + Na
3 0
(3s3p P )
4 g
3x10
3.4x10
+
2 x IR (delayed ionization)
4
4
4 g
+
Na - 3s
4x10
3.6x10
-
Na + Na
1 0
2 1
(3p4s P + 3p D)
-1
5x10
a)
6
4
1
+
X g
3
3s-3s
0
0
5
10
V (cm )
2
Excitation Scheme
4
15
R (Angstrom)
6
8
10
12
14
Dipole Moment (Debye)
6x10
0
16
R(Angstrom)
Figure 4.4 Excitation Scheme 2
Excitation Scheme: (a) Double resonance technique via the A1Σu+ state was used
in the case of 41Σg+ state lifetime measurement. The ionizing laser is delayed
relative to the second laser. b) The 41Σg+ potential with some of the ion pair
potentials; c) Transition dipole moment of 41Σg+ as a function of inter-atomic
distance R (Å) for two lower states it can decay by radiative transition.
For the 41Σg+ state, a double resonance multiphoton excitation technique via the
A1Σu+ state followed by delayed two-photon (1064 nm) dissociative ionization
from a third laser is employed. Figure 4.4a shows the potential curves involved in
the excitation scheme using 3-color, 3-laser and 4-photon excitation-ionization
and dissociation. The two pump lasers are tunable dye lasers operating in the
wavelength range of 620 nm and 615 nm for the first and second laser
respectively. The second pump laser is optically delayed relative to the first one
87
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
by about 10 ns. The reason for this is to avoid accidental resonant A1Σu+  X1Σg+
transitions by the second laser and the two pump lasers thereby switching roles.
The timing for the pump-probe delay laser experiment is controlled by a pulse
generator in a similar manner mentioned above for the 21Σu+ excitation scheme.
The schematic timing diagram of the three-laser experiment is shown in Figure
4.5. The second pump laser used for excitation on the 41Σg+  A1Σu+ transition
has fixed delay from the first pump laser so that the initial population of the 41Σg+
state is kept constant from run to run. The probe laser delay relative to the second
laser used in this data varies from 0 to 500 ns.
Figure 4.5 Schematic diagram of the three-laser experiment.
The second pump laser has fixed delay relative to the first pump laser of about 10
ns, while the probe laser is delayed systematically relative from the second pump
laser.
88
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
4.4 Excited Atomic Na Lifetimes
Excited states of atomic sodium have been studied extensively both in theory and
experiment. Atomic states are less complicated than molecules, since there are no
vibrational and rotational states involved. Also the cross section for excitation is
usually much larger and hence it is much easier to perform measurements.
To validate the experimental approach discussed in the previous section for
measuring lifetimes of excited states of Na2, the method was first tested on two
excited atomic states of sodium, namely the 3p2P and 4p2P states of Na. The first
excited state of Na is the 3p2P state. Since the 3p2P state is the first excited state,
this state can radiatively decay only to the ground state of Na. The state has a
corresponding binding energy84 of 41,449 cm-1 and the transition to the ground
state occurs at around 590 nm (16956.18 and 16973.37 cm-1 for the two spin-orbit
levels). A Rhodamine dye is used for the 3p  3s excitation, and frequencytripled 355 nm (28169 cm-1) photon from a delayed Nd:YAG laser (probe laser)
ionizes atoms in the 3p2P state. The 3p2P lifetime is known to be 16.299(21) ns for
the 3p2P1/2 state and 16.254(22) ns for the 3p2P3/2 state;69,
70
hence a direct
comparison can be done for benchmarking.
The 4p2P transition from the ground state of Na is around 330 nm (30,267.0 and
30,272.6 cm-1 for the two spin-orbit states, respectively). The frequency doubled
output of a suitably tuned dye laser using a DCM dye is used for the excitation to
the 4p2P state. A 532 nm (18,797 cm-1) photon from a delayed Nd:YAG laser is
enough to ionize the 4p2P population. The lifetime of the 4p2P state is 108.0 ns
and 107.1 ns for the 4p2P1/2 and 4p2P3/2 spin-orbit levels respectively.71 Dipole
89
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
allowed transitions from the 4p2P terminate in the 3s, 3d and 4s. The latter two
levels can cascade further downwards via the 3p level. A simple rate equation
model of this transition cascade will be discussed in section 4.5.2.
4.4.1 Lifetime Measurement of Na 3p2P
A sample yield spectrum of the 3p2P state is shown in Figure 4.6 where the two
peaks are due to spin-orbit splitting. By comparing the absolute line positions with
the literature values, absolute wavelength calibration of the laser can be achieved.
The FWHM of the measured lines is less than 0.2 cm-1, in good agreement to the
laser bandwidth, which is 0.15 cm-1. However, the line intensities do not follow
the expected 2:1 ratio for I3/2/I1/2 indicating that the transition is saturated. Power
broadening may also be responsible for the observed slight excess linewidth.
90
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
20
2
Na: 3s S1/2
3
3p P3/2
15
Intensity (arb. units)
2
Na: 3s S1/2
3
3p P1/2
10
5
-1
17.2 cm
0
16930
16940
16950
-1
laser wavenumber (cm )
Figure 4.6 REMPI: 3p2P State
Na 1+1 REMPI through the spin-orbit split 3p2P-line. This spectrum is also used
for absolute wavelength calibration of the pump laser.
Figure 4.7 shows the population decay of the two 3p2P spin-orbit states as a
function of the probe laser delay plotted in semi-log scale. Experimental data is
background subtracted. The lifetime was extracted by fitting the data to an
exponential function through the relation in Eq. (4.1.6) and found to be 16.5 ± 0.2
and 16.4 ± 0.2 ns for the 3p2P1/2 and 3p2P1/2 state respectively. Both are in very
good agreement with the literature values.
91
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
Ni(t) = Ni(0)e
10000
-t/
2
3 p1/2
2
3 p3/2
2
Ion Signal (arb. units)
exponential fit of 3 p1/2
2
exponential fit of 3 p3/2
2
Na 3 p1/2:
1000
Measured : 16.5 +/- 0.2ns
1
Literature :16.3ns
2
Na 3 p3/2:
Measured : 16.4 +/- 0.2ns
1
Literature : 16.3ns
30
40
50
60
70
80
probe laser delay (ns)
Figure 4.7 Population decay of 3p2P1/2 and 3p2P3/2 plotted in semi-log scale.
4.4.2 Lifetime Measurement of Na 4p2P
A 4p2P sample yield spectra is shown in Figure 4.6, again showing the well
resolved spin-orbit splitting. This time, the FWHM of the measured lines, about
0.45 cm-1 is significantly larger than the expected laser bandwidth. This is an
indication that the transition is again, saturated. It can also be seen that the line
intensities do not follow the expected 2:1 ratio for I3/2/I1/2. However, the line
positions can still be used to calibrate the laser to its absolute wavelength by direct
comparison to the literature values.
92
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
40
2
Na 4p P Lines
2
Intensity (arb. units)
30
Na: 3s S1/2
2
4p P1/2
2
Na: 3s S1/2
2
4p P3/2
20
10
5.6 cm
-1
0
30320
30325
30330
30335
-1
laser wavenumber (cm )
Figure 4.8 REMPI: 4p2P
Na 1+1 REMPI through the spin-orbit split 4p2P-line.
As mentioned earlier, a transition cascade takes place in the population decay of
the 4p2P to other atomic states of Na: 4s, 3d, 3p and 3s. To model the
experimental data, transition to these other atomic states should be properly
handled in the rate equation. Of course, only dipole allowed transitions need to be
considered. However, the simple model described below does not include the 3d
 4p decay. While shown in Figure 4.9, it is ignored in the rate equation model
because the branching ratio coefficient for the 3d  4p is negligible (0.02)
compared to the 3s  4p and 4s  4p transitions.71 Ignoring the 3d  4p
transition does not change the final result beyond the experimental uncertainties.
When the 4s state is populated via radiative decay from the 4p state, it can only
93
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
decay radiatively to the 3p state, which in turn can only decay to the 3s ground
state. Figure 4.9 shows the transition cascade described here. In the experiment,
Na+ ions are detected. Energy requirement for ionization using one 532 nm photon
will only ionize atoms in 4p and 4s states. The laser power of the ionizing laser is
adjusted such that two-photon ionization is negligible. This means the model
should only incorporate populations from 4p and 4s states even though 3p and 3s
states are also populated.
4
5x10
+
Na
4
4x10
4
4p1/2
k
-1
Energy (cm )
|3>
|2>
4s1/2
3x10
k
3p1/2
|1>
4
2x10
k
k
4
1x10
3s1/2
|0>
Figure 4.9 4p2P state radiation cascade.
Schematic diagram of the radiation cascade from the 4p2P state. Decay to 3d state
is not included due to small transition probability compared to 4s and 3s.
94
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
Let n3 be the population of the 4p state. Its rate of change can be written as
n3   A3n3  n3  n3 (0)e A3t
(4.4.1)
where n3(0) is the population at t = 0 and A3 = A32 + A30 is the total Einstein
coefficient of the 4p state. Approximately, time t = 0 corresponds to the time the
state 4p is initially populated. If n2(t) is the population of 4s at any given time t,
then the rate of change of n2 can be written as
n2   A32 n3  A21n2  A32 n3 (0)e A3t  A21n2
(4.4.2)
where A32 is the Einstein coefficient of the 4s  4p transition, and A21 is the
Einstein coefficient of the 3p  4s transition. Solving for n2(t) using equations
(4.4.1) and (4.4.2) will result in
n2 (t ) 
A32
n3 (0)(e A3t  e A21t )
A21  A3
(4.4.3)
It can be see that the equation in (4.4.3) is a function of three different rate
coefficients, A32, A21 and A3. The equation that will describe the Na+ population
should be
Na  (t )  c  n3 (t )  x.n2 (t ) 

x. A32
A t

Na  (t )  c n3 (0)[e  A3t 
(e 3  e  A21t )]
A21  A3


(4.4.4)
where x is a variable that accounts for the difference of the relative ionization
probability of the 4s state using 532 nm compared to the 4p state, and c is
normalization constant.
95
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
Figure 4.10 shows the experimental data (circles) with the simulation (red solid
line) according to the relation in equation(4.4.4). Here, n3(0) is set to 1 and
experimental data is normalized to 1 also. The corresponding Einstein coefficients
expressed in terms of lifetimes are the following:
A3  A4 p 
A32 
1
4p
1
 4 p 4 s
A2  A21  A4 s 
(4.4.5)
1
 4s
Initially, the fit is performed using χ2 minimization to the experimental data with
all four parameters free: x, A32, A21 and A3. The resulting fit resulted in unrealistic
values for the three parameters A32, A21 and A3. To get a reasonable agreement of
the parameters mentioned above, the fit in Figure 4.10 was carried out using χ2
minimization to the experimental data with free parameter τ4p and x while τ4s and
τ4p-4s are fixed to the known value 143 ns and 38 ns, respectively.71 As seen on the
quality of the fit in Figure 4.10, the model agrees reasonably well with the
experimental data. The lifetime extracted from the fit in Figure 4.10 is 111.52 ±
0.85 ns and 106.8 ± 4.02 ns for the 4p2P1/2 and 4p2P3/2 states, respectively, in very
good agreement with Lowe et al. The x coefficient is found out to be about 0.6.
This means that the 4p state is roughly twice as much likely to be ionized by 532
nm photon compared to the 4s state.
96
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
1.0
2
Na 4p P3/2
Literature: 107.1 ns
This work: 106.82 ± 4.02
Ion Signal (arb. units)
0.8
0.6
0.4
2
Na 4p P1/2
Literature: 108.0 ns
This work: 111.52 ± 0.845
0.2
100
200
300
400
500
probe laser delay (ns)
Figure 4.10 4p2P Population decay
Population decay of Na+ during the transition cascade of 4p2P1/2 to other Na
excited states fitted according to Eq. (4.4.4). Inset is for the 4p2P3/2 state.
The tests on atomic lifetime measurement show that the method (delayed pumpprobe ionization) presented here for lifetime measurement produce reliable result
in these cases. In particular, it was shown that the method can resolve both short
(~16 ns) and long (~110 ns) lifetimes.
97
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
4.5 Excited Molecular Na2 Lifetimes
This section covers the lifetime measurements and calculations of the 21Σu+ and
41Σg+ states of Na2. The discussion starts with the details on how the calculation is
implemented and the results of the calculations. Following are the details of the
experiments necessary to measure the lifetime of the molecular states of interest.
4.5.1 Lifetime Calculations
All the calculations presented here were done using the Level 8.0 and BCONT
programs written by Robert J. LeRoy.58 An extensive documentation of Level 8.0
can be found on the author’s website: http://leroy.uwaterloo.ca/programs/. It is
written in FORTRAN and runs in a UNIX or Linux operating system. The core of
the program calculates the solution to the one-dimensional Schrödinger equation:

2
2
d 2  v , J ( R)
dR 2
 VJ ( R) v , J ( R)  Ev , J  v , J ( R)
(4.5.1)
and finds the eigenvalues Ev , J and eigenfunctions  v,J(R) of a given potential
VJ ( R) in which μ is the effective or reduced mass of the system, J the rotational
quantum number. The effective one-dimensional potential VJ ( R) is a sum of the
rotationless (electronic) potential V (R) and the centrifugal term. The calculation is
performed through a scheme based on the Cooley-Cashion-Zare routines.86-88 The
procedure is an integration method for second-order differential equations.
Basically, the method consults the values of the two adjacent points of the wave
equation to generate a third, either to the right or to the left. Thus the integration
98
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
may be permitted to propagate from small to large values of R or vice versa. In the
Cooley procedure for finding the eigenvalues of Eq. (4.5.1), for any given trial
energy the numerical integration proceeds inward from RMAX and outward from
RMIN until the two solution segments meet at a chosen matching point Rx. The
discontinuity in their slopes at Rx is then used to estimate the energy correction
required to converge on the eigenvalue closest to the given trial energy. This
process is repeated until the energy improvement is smaller than the chosen
convergence criterion. This procedure usually converges very rapidly, and for a
single-minimum potential it is insensitive to the choice of the matching point rx, as
long as it lies in the classically-allowed region where the wavefunction amplitude
is relatively large.
The Einstein A coefficient given in Eq. (4.1.4) for the rate of spontaneous
emission from the initial-state ro-vibrational level |i> = (v’, J’) into final-state rovibrational level |f> = (v”, J”) can be written using the expression89, 90
Aif 
16 3  3 S ( J ', J ")
 v ', J ' M ( R)  v ", J "
3  0 hc3 2 J ' 1
Aif  3.1361891107
2
S ( J ', J ") 3
  v ', J ' M ( R)  v", J "
2 J ' 1
(4.5.2)
2
(4.5.3)
In equation (4.5.3), Aif has units s-1, M(r) is the dipole moment (or transition
dipole) function in units of Debye, v the emission frequency in cm-1. S(J’,J”) is
the the Hönl-London rotational intensity factor and Ψv’,J’ and Ψv”,J” are the unit
normalized initial and final state radial wave functions. The numerical value in
99
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
equation (4.5.3) is nothing but the fundamental constants in the preceding
equation.
The input Na2 potentials used in the program were taken from the published data
from different authors: X1Σg+ and B1Πu+ from Camacho et al.56; A1Σu+ from Qi et
al.91, 21Σu+ from Pashov et al.76, 11Πg+ and 21Σg+ from Barrow et al.92 and 41Σg+
from Tsai.85 This selection reflects the most recent and/or most reliable potential
data. The potentials are shown in Figure 4.11. All transition dipole moment
functions used in the present context are results of calculations performed by and
obtained from Sylvie Manier through private communication.93
100
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
60000
2
2
1 g
50000
+
1 u
+
+
Na - 3s
40000
1
1
2 u
30000
-1
V (cm )
4 g
1
+
3s-3d
3s-4s
+
+
1 g
1
20000
10000
3s-3p
+
2 g
1
A u
1
X g
+
+
3s-3s
0
0
10
20
30
R (A)
Figure 4.11 Simplified potential diagram of Na2
Simplified potential diagram of Na2 showing the potentials involved
in the calculation of Einstein coefficients, Aij.
For homonuclear molecules, the parity selection rule requires that even electronic
states combine only with odd, i.e. g
and g
g u
u. Thus in the absence of
collisions the 21Σu+state can decay to three different electronic states, namely
X1Σg+, 21Σg+ and 11Πg+ (see Figure 4.3a). In each of these electronic states,
population resulting from the decay of the parent state is distributed across
different ro-vibrational states as well as scattering states. The decay to the
rotational states the excited Na2 molecule obey the usual selection rules: (i) for Σ –
101
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
Σ transition, only ΔJ = ±1 are allowed and (ii) for Σ – Π transition, ΔJ = 0, ±1 are
allowed. For each specific initial ro-vibrational level |i> of the 21Σu+ state,
Einstein coefficients Aif to each ro-vibrational level in the final “g” symmetry state
are calculated. All these coefficients summed to obtain the total spontaneous
emission rate out of the particular initial ro-vibrational level of the 21Σu+ and
hence its lifetime i = Ai-1 =  Aif-1. What is currently missing in the calculation of
Einstein A-coefficients is the contribution from any bound-free transition. While
the BCONT58 program used for our calculations is designed to compute boundfree transitions, those originating from double well potentials such as the 21Σu+
state are beyond its capability. Note that because of the missing bound-free
transitions, the term “calculated lifetime” only refers to an upper bound. For
simplicity, it will be used in this sense for the rest of this document.
102
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
a)
7
-1
A, Eintein Coefficient (s )
1.5x10
7
1.2x10
6
9.0x10
1
+
1
+
1
+
1
+
1
+
1
+
X g <-- 2 u
2 g <-- 2 u
1 g <-- 2 u
6
6.0x10
6
3.0x10
0.0
90
b)
outer well and above the barrier vibrational levels
inner well vibrational levels
(v = 12, 16, 19, 22, 25, 28, 31, 33)
Lifetime, (ns)
80
70
60
50
40
barrier
30
0
5
10
15
20
25
30
35
40
45
50
55
vibrational quantum number, v
Figure 4.12 21Σu+ state calculation
a) Total Einstein Coefficient, Aij for three different transitions as a function of
vibrational quantum number,v of the 21Σu+; b) Calculated lifetime of 21Σu+ as a
function of vibrational quantum number v, J = 14.
From Figure 4.3c, it is clear that the transition dipole moment function favors the
21Σg+  21Σu+ transition. This is consistent with Figure 4.12a where the Einstein
coefficient Aij (summed over all ro-vibrational levels in each final electronic state)
is plotted as a function of the vibrational quantum number v of the 21Σu+ state for
the fixed rotational quantum number J = 14 which falls in the range of the
experimentally explored values. In the figure, open symbols correspond to the
103
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
inner well vibrational levels of the 21Σu+ state while the solid symbols refer to the
vibrational levels of the outer well and above the barrier. The barrier of the
potential is indicated as the dashed line located around v = 33. It has been verified
that the calculated lifetime for a given vibrational state has a weak rotational
dependence in the range J = 0 to 40, of the order of 10%. Furthermore, the overall
trend of lifetime as a function of vibrational quantum number is not affected by
this rotational dependence. Figure 4.12b summarizes the lifetime of the 21Σu+ state
as a function of vibrational quantum number. The vibrational level counting starts
from the true v = 0 which is the first vibrational level of the outer well of the 21Σu+
state. The calculation clearly shows that the lifetime of the double well exhibits
strong variation with vibrational quantum number. Starting from the first
vibrational level in the inner well, the lifetime increases slightly and reaches a
maximum halfway up the inner well, drops as it approaches the barrier and finds a
minimum near the potential barrier, at v = 33. The lifetime then increases
monotonically above the barrier. The calculated lifetime is not a smooth function
of vibrational quantum number due to the fact that there are three different
contributions coming from three transitions for which transition dipole moment
functions as well as the Franck-Condon Factors are significantly different.
Calculation for Einstein A coefficients was extended to the 21Σg+ and 11Πg+ states
using the same method outlined above. Due to selection rules, the 21Σg+ and 11Πg+
states can radiatively decay only to the A1Σu+ and B1Πu+ states. The lifetime of the
21Σg+ and 11Πg+ states are not known. Because the radiative transitions between
the states 21Σg+→ A1Σu+ have lower frequencies than transitions between the states
B1Πu+ → X1Σg+ (radiative lifetimes of the order of 7 ns) and A1Σu+ → X1Σg+
104
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
(radiative lifetimes of the order of 12.5 ns), Camacho et al. expected that the
lifetime of the 21Σg+ will be of the order of one-tenth of a microsecond. Following
the same argument, the lifetime of the 11Πg+ state is also expected to be in the
same order. Figure 4.13 shows the calculated vibrational quantum number lifetime
dependence of the 21Σg+ and 11Πg+ states. It can be seen that there is a strong
vibrational lifetime dependence of 21Σg+ for v < 10. The lifetime value decreases
by about a factor of 7 from v = 0 to v = 10. For the 11Πg+ state case, lifetime data
shows a weak dependence to vibrational quantum number, v. The mean lifetime
for 21Σg+ state for vibrational levels v = 17 – 44 is found to be 2.2 s and the mean
lifetime for 11Πg+ state for vibrational levels v = 0 – 24 is 4.2 μs. The solid blue
line in Figure 4.13 is an average experimental data for the slow decay channel.
35
1
+
2 g
1
1 g
Lifetime, (microseconds)
30
expt
25
20
15
10
5
0
0
10
20
30
40
vibrational quantum number, v
Figure 4.13 Calculated lifetime of 21Σg+ and 11Πg+ states
as a function of vibrational quantum number v, J = 15.
105
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
In the case of the 41Σg+, selection rules only allow radiative transition to the A1Σu+
and B1Πu+ states. For A1Σu+  41Σg+, ΔJ = ±1 are while for B1Πu+ 41Σg+, ΔJ = 0,
±1 are allowed. For a given electronic transition, e.g. A1Σu+ (v’, J’)  41Σg+ (v”,
J”), Aij is calculated for each allowed ro-vibrational transition and summed up to
obtain the spontaneous rate of emission from a specific ro-vibrational state of the
41Σg+ to all possible states it can decay to A1Σu+. The total rate of spontaneous
emission of a single ro-vibrational level (v”, J”) in the 41Σg+ state is the sum of the
rates of the two different electronic states that it decays. Again, similar to the
calculations for the 21Σu+, the Einstein A-coefficients contribution from any
bound-free transition is not included.
Figure 4.14a shows the Einstein A-coefficients for the two different electronic
states into which the shelf state can radiatively decay, plotted as a function of
vibrational quantum number, v. It is obvious that the contribution from the A1Σu+
 41Σg+ transition is the dominant one. In fact, the Einstein A-coefficient for the
B1Πu+ 41Σg+ contribution is about two orders of magnitude smaller. Figure 4.4c
demonstrates that for values 7Å < R < 13Å, the transition dipole moment function
favors the B1Πu+  41Σg+ transition. Outside this R range, A1Σu+  41Σg+
transition dominates. However, the Franck-Condon Factors for B1Πu+  41Σg+
transition is very poor therefore producing the very small Einstein A-coefficients.
106
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
7
6.0x10
a)
1 +
1 +
1
1 +
A u  4 g
+
-1
A, Einstein Coefficient (s )
B u 4 g
7
4.0x10
shelf (v=51)
7
2.0x10
0.0
Lifetime, (ns)
200
b)
160
120
80
40
0
10
shelf (v=51)
20
30
40
50
60
70
vibrational quantum number, 
Figure 4.14 41Σg+ state calculation
a) Total Einstein Coefficient, Aij for two different transitions as a function of
vibrational quantum number,v of the 41Σg+ state; b) Calculated lifetime of 41Σg+
state as a function of vibrational quantum number v, J = 19.
Shown in Figure 4.14b is the calculated lifetime of the 41Σg+ (v, J = 19) as a
function of vibrational quantum number, v = 10 - 70. The calculated data shows a
gradual increase of the lifetime from 17 ns at v = 10 to 39 ns at v = 45 followed by
a rapid rise and maximum at v = 51 just one level below the “shelf” located at v =
52. Beyond v = 52 level, the lifetime drops reaching a value near 100 ns for v =
107
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
70. The sharp lifetime peak at the shelf is also seen as a kink in the Einstein
coefficient Aij of the A1Σu+  41Σg+ transition shown in Figure 4.14a.
4.5.2 Lifetime Measurement of the 21Σu+ State
To measure ro-vibrational lifetimes of the double well potential, a 1+1 REMPI
scheme is necessary to identify the locations of each ro-vibrational level. Using
the frequency doubled output of the tunable dye laser in the range of 28,000 to
30,000 cm-1, it is possible to excite vibrational levels of the 21Σu+ state from v = 22
to v = 50. A UV photon (355 nm) from a Nd:YAG laser delayed by 10 ns relative
to the excitation laser is used as a probe to ionize the excited molecules. The
molecular ions are extracted and accelerated by applying electric fields in the
corresponding regions of the TOF spectrometer, with values Ee = 50 V/cm and Ea
= 126 V/cm respectively. These electric field settings are used to maximize the
collection efficiency of the ions. The steering plates this time are kept field-free.
A sample Na2+ ion yield spectrum is shown in Figure 4.15 showing three
vibrational bands (v = 25, 28 and 33) of the 21Σu+ state. The total yield spectrum
measurement extends from 28,470 to 29,640 cm-1. The band assignments are
carried out by comparing the vibrational band spacing of the experiment with the
calculated data using the Level 8.0 program. Almost all lines are rotationally
resolved, especially those with high J’s. Shown in Figure 4.16 is a comparison of
simulation and experiment for the assignment of absolute quantum numbers to the
observed ro-vibrational lines. The simulated data follows as outlined in Section
2.1.3 using Eq. (2.1.1). In Figure 4.16a, the chosen excitation line in the
108
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
experiment and the assigned ro-vibrational states from the calculation agree very
well. However, in this case, there are two ro-vibrational transitions that
correspond to this line, v = 25, J = 16 and 20. These two transitions come from
rotational P(J) and R(J) branching where the J refers to the upper level rotational
quantum number. Figure 4.16b is an example of a clean excitation line where the
observed transition corresponds to only one ro-vibrational level, v = 28, J = 25.
After the experiments had been performed, a careful analysis revealed that for a
few runs, excitation lines even a mixture of more than one vibrational level. The
assignment for the experimental ro-vibrational levels of the 21Σu+ state is
summarized in Table 4-1. For multiple assigned levels, column 5 gives the relative
contribution of each ro-vibrational level from the known values of Trot = 70 K.
109
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
v=25
v=33
Na2+ Signal (arb. units)
v=28
background signal
28742.0
28792.0
28842.0
28892.0
28942.0
28992.0
Laser Wavenumber (cm-1)
1
2
Na2+ Signal
(a)
Σu+
Figure 4.15 Na2 1+1 REMPI
(v = 25, 28 and 33)  X1Σg+ (v = 0) transition.
v = 25, J = 16 & 20
28791
28792
28793
28794
28795
28796
28797
Laser Wavelenght (cm-1)
(b) vv==28,
28,JJ==25
25
Experiment
Simulation
28869
28870
28871
28872
28873
Laser Wavenumber (cm-1)
Figure 4.16 Na2 1+1 REMPI: 21Σu+ (v = 25) ← X1Σg+ (v=0)
110
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
Run
Excitation
energy (cm-1)
v
J
relative contribution
Lifetime (ns)
Uncertainty
(ns)
1
28490.3
22
65
1
51.9
1.6
2
28630
22
40
1
52.9
3.6
28
42
0.25
20
0.39
51.9
2.9
16
0.37
25
1
49.7
2.5
24
0.4
45.9
2.2
27
0.6
39
35
0.36
37
27
0.52
41.5
1.9
42
47
0.12
38
8
9
0.17
0.28
40
25
0.34
42.4
3.5
42
36
0.09
43
39
0.11
43
23
0.69
43.1
2
44
30
0.31
43
8
0.18
44
19
0.48
40.2
2.3
47
36
0.18
49
45
0.15
9
0.27
48
3.3
10
0.73
5
0.25
25
0.48
42.1
3.6
26
0.26
11
0.62
45.3
4.5
10
0.38
8
7
9
8
0.37
0.63
0.62
0.38
41.3
1.5
40.5
4.2
12
0.38
13
11
21
22
0.62
0.35
0.42
0.24
45.8
4
44.4
3.5
51
14
0.34
0.26
0.4
54.2
5.5
52
22
23
3
28794
25
4
28870.6
28
5
28944
31
6
7
8
9
10
29071.5
29162.3
29264.9
29314
29344.2
44
45
11
29383.9
47
12
29406.8
46
13
29445.2
47
14
29476.6
48
15
29501.4
49
16
29537
17
29566.7
50
51
Table 4-1 Summary of experimental 21Σu+ excitation lines
with the assigned ro-vibrational states. Column 5 shows the relative contribution
for cases of multiple transitions.
111
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
A typical Na2+ TOF delayed ionization spectrum for the 21Σu+ state case is shown
in Figure 4.17 for three different probe laser delays. For very small delay, there is
an overlap of two Na2+ contributions: the first due to the pump laser alone, the
second coming from the presence of the probe laser (pump + probe). At longer
delays (>20 ns), the two contributions are clearly separated thereby giving a clean
signal of the latter contribution. Although the pump laser power is tuned low, twophoton ionization from the pump laser alone is unavoidable without sacrificing
signal-to-noise. However, the pump laser contribution to the signal is constant at
all delay times. Therefore, it can be subtracted from the overall integrated ion
yield and does not affect the lifetime data.
+
Na2 TOF for three different pump-probe delay
20 ns delay
150 ns delay
800 ns delay
+
Na2 Yield (arb. units)
Na2 due to pump laser only
+
+
Na2 due to pump + probe lasers
0.28
0.30
0.32
0.34
0.36
0.38
0.40
Time-of-Flight (s)
Figure 4.17 Typical Na2 TOF spectra for three different probe laser delays.
20 ns (red circle), 150 ns (blue triangle) and 800 ns (open square). Na2+ signal
comes from two contributions: (i) pump laser alone and (ii) pump + probe laser.
The former can subtracted for the total ion yield used in the analysis of the
population decay since it is a constant.
112
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
The population decay curves of some vibrational levels of the 21Σu+ state are
shown in Figure 4.18. The data shows both a fast and a slow decay channel. Two
different fitting procedures were carried out, using
(i)
single decay exponential with a constant background:

t
N (t )  N (0) e  C
(ii)

(4.5.4)
and double exponential decay with zero background:
N (t )  N1 (0) e

t
1
 N 2 (0) e

t
2
(4.5.5)
The two fitting procedures lead to similar fast decay channel time constants which
correspond to the 21Σu+ state lifetimes. The slow decay is believed to come from
the 21Σg+ and 11Πg+ states which are populated by radiative decay of the 21Σu+ state
(see Figure 4.3) and are subsequently ionized by a single 532 nm photon which is
energetically possible for vibrational levels with v > 20. The lifetime data
presented here is taken from procedure (4.5.5) to get an order of magnitude
estimate of the average lifetimes of the 21Σg+ and 11Πg+ states. From the double
exponential decay fit, the vibrationally averaged lifetime of the slow decay
channel was found out to be about 3.8 μs. This is shown in Figure 4.13 as a solid
line. Note that contrary to the calculated lifetimes of 21Σg+ and 11Πg+ states in the
figure, this long decay lifetime is not vibrationally resolved and is only shown for
order magnitude comparison. This averaged long decay lifetime extracted from
the fit of the experimental data is in good agreement with the calculated lifetimes
of the 21Σg+ and 11Πg+ (particularly for higher v) states.
113
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
decay curve of v = 22
decay curve of v = 25
1 = 51.9 ± 2.9 ns
2 = 3.8 ± 4.7 s
2 = 7.3 ± 1.5 s
decay curve of v = 28
decay curve of v = 31
+
Na2 Yield (arb. units)
1 = 52.9 ± 3.6 ns
0
200
1 = 49.7 ± 2.5 ns
1 = 45.9 ± 2.2 ns
2 = 5.9 ± 7.1 s
2 = 3.5 ± 2.2 s
400
600
800 0
200
400
600
800
probe laser delay (10 ns)
Figure 4.18 Population decay curves
for some vibrational levels of 21Σu+ fitted with double exponential. The fast decay
is the lifetime of the 21Σu+ while the long decay is believed coming from the 21Σg+
and 11Σg+ populated by the radiative decay of the 21Σu+.
Both experimental (squares) and calculated (diamonds) lifetime data of the 21Σu+
as a function of vibrational quantum number, v, is shown in Figure 4.19. The
vertical error bars in the experimental data come entirely from the exponential
decay fitting while the horizontal bars indicate the fact that some measured
lifetimes originate from a mixture of different vibrational levels. Data rendered in
open and solid symbols correspond to vibrational levels of the inner well below
the potential barrier and/or vibrational levels of the outer well below the barrier
and above the barrier, respectively. The dashed line indicates the location of the
potential barrier around v = 33. As seen in Figure 4.19, lifetime dependence with
114
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
vibrational level is evident especially as the levels approach the potential barrier.
Above the barrier, calculated lifetimes are relatively higher than measured. Both
experiment and calculated data follow the same trend below and above the barrier.
80
Experiment
Experiment (inner well vibrational levels)
Calculation
Calculation (inner well vibrational levels)
Lifetime, t (ns)
70
60
50
40
0
5
10
15
20
25
30
35
40
45
50
55
vibrational quantum number, v
Figure 4.19 Lifetime of 21Σu+ as a function of vibrational level, v.
4.5.3 Lifetime Measurement of the 41Σg+ State
Excitation into the 41Σg+ state requires a resonant two-photon scheme via the
A1Σu+ state. As mentioned in section 4.3, an auxiliary 1+1 REMPI experiment is
performed on the A  X system to identify ro-vibrational levels in the A state.
This is necessary to determine which ro-vibrational level is to be used as an
intermediate state for the excitation unto the 41Σg+ shelf state. While scanning the
115
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
“pump” dye laser through the A  X transitions, a 355 nm photon is used to
ionize the molecule when the pump laser hits resonance in the A state. An
extraction and acceleration electric fields of 50 V/cm and 400 V/cm were applied
to accelerate the ions into the MCP detector. Yield spectra were taken using a
Boxcar integrator (SRS 250) gated to the Na2+ peak of the TOF mass spectrum. A
typical yield spectrum is shown in Figure 4.20 showing only part of the v’ = 20 –
v” = 0 band. All the lines are identified by direct comparison with the calculated
values using the Level 8.0 program. The line v = 20, J = 19 is chosen as the
intermediate state to meet criteria of clean single ro-vibrational level and
maintaining high transition probabilities for A1Σu+  X and 41Σg+  A1Σu+.
Ion Signal (arb. units)
A (v = 20, J = 19) - X (v = 0, J = 18)
16820
16825
16830
16835
Figure 4.20 Na2 1+1 REMPI: A1Σu+ (v = 20)  X1Σg+ (v = 0)
With the first dye laser tuned to the v = 20, J = 19 level of the A1Σu+ state, the
laser power is lowered to reduce multi-photon absorption which could lead to
ionization. The next step is to map out the vibrational levels of the shelf state. The
116
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
UV laser is blocked and a second dye laser is scanned through the region for
excitation from the excited A state level to the 41Σg+ state. In order to avoid
inadvertent role switching, the second dye laser is delayed by about 10 ns relative
to the first one by an optical delay set-up. Whenever the second laser energy
matches the resonance condition, another photon coming from the same dye laser
is enough to ionize the populated excited state. The second laser power is adjusted
such that the two requirements are satisfied: (i) it must be sufficiently high to
allow two-photon absorption from a single laser pulse but (ii) not so high that
three photon contribution (resonant or not) overpower the double resonance
signal. Yield spectra were taken in similar fashion as the 1+1 REMPI except that
the Boxcar is gated to the Na+ peak of the TOF mass spectrum. In principle, Na+
and Na2+ ion yield spectra produce the same progression, however in the actual
experimental data, the Na2+ yield spectra is more crowded. As the probe laser is
scanned, even in the absence of the pump laser, A  X transitions occur because
the probe laser wavelength lies within the A  X transition range. When A  X
resonance condition is met, absorption of another photon from the same laser is
enough to ionize the excited molecule in the A state therefore competing with the
Na2+ yield, making the molecular ion yield spectra congested. The atomic ion
yield spectrum is simpler, since Na+ is produced only through dissociative
ionization of molecules excited to the 41Σg+ state.
A sample double resonance yield spectra is shown in Figure 4.21. The Na+
progression shows v = 40 through 69 of the 41Σg+ state. Due to selection rules,
only J = 19 and 21 are allowed, evident in the figure as pairs of peaks for each
117
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
vibrational level. The shelf of the state is located between v = 51 and 52 which
also gives the least spacing between vibrational levels, consistent with the
calculation. In addition, 21Πg+  A transitions have been observed. They can be
seen in Figure 4.21 as line triplets which correspond to P, Q, and R branching.
Although the 11Πg+  A transition is present, the line intensities corresponding to
this transition are much less and the vibrational spacing is different to that of the
41Σg+ transitions. The majority of the 41Σg+  A transitions are clean and isolated
excitation lines and were used for lifetime measurement.
Figure 4.21 Double resonance Na+ yield spectrum
of 41Σg+ via the A1Σu+ (v=19/20, J=20) state followed by
delayed dissociative ionization.
After identifying the ro-vibrational levels of the shelf state, a pump-delayed probe
experiment is performed. At this time, the second dye laser is tuned to a specific
ro-vibrational level of the 41Σg+ state. Therefore the two dye lasers are now
operating at a fixed wavelength, with the second one still optically delayed by
about 10 ns relative to the first. The second dye laser energy (typically 80 μJ) is
118
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
adjusted to meet two criteria: (i) low enough to minimize absorption of two
photons resulting in dissociative ionization and (ii) high enough to yield
significant resonant A1Σu+  41Σg+ excitation. Even after laser power fine tuning,
some background Na+ is still present. However, this ion signal is time independent
and constant and thus can be subtracted from the overall integrated ion yield. A
Nd:YAG laser with output of 1064 nm wavelength photon is used to carry out the
task of ionization and dissociation of the molecules in the 41Σg+ excited state.
Energy balance requires two IR photons for dissociation. Using two IR photons,
the possibility of contamination in the atomic ion yield from lower energy states
A1Σu+ and B1Πu+, which are populated by radiative decay of the 41Σg+ is greatly
reduced. With 532 nm or 355 nm, two-photon absorption from these two lower
states can lead to dissociative ionization. TOF ion yield spectra summed over 500
pulses are recorded from the oscilloscope using GPIB connection to the computer
via the LabVIEW software.
Atomic and not molecular ions were chosen for the lifetime measurement. This is
done to avoid other channels that produce molecular ions therefore contaminating
the lifetime data. In Figure 4.22, Na+ TOF spectra are shown for three different
probe laser delays (50, 200, and 500 ns). It is clear that there is an overlap of the
two Na+ contributions (the first due to the two excitation dye lasers alone, the
second when both excitation and probe lasers are present). Both contributions
have clear separation only at very long delay, i.e. t ≥ 500 ns. The contribution
when excitation and probe lasers are present has an early and late component. This
is so because when the excited molecule in the 41Σg+ state absorbs two IR photons,
119
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
there is an extra fragment kinetic energy release during the dissociation process.
This extra kinetic energy will manifest itself as early and late fragment arrival
peaks in the TOF Na+ spectra. The peak-to-peak separation of early and late Na+
in the TOF spectrum is about 120 ns. This time difference in the TOF spectra
corresponds to an excess energy of about 0.2 eV. When the excited molecule
absorbs two IR photons (EIR = 1.17 eV), which corresponds to 2.33 eV, it ionizes
and dissociate. The produced Na2+ requires 2.1 eV to dissociate with zero extra
kinetic energy. In here, Na+ contribution from pump laser alone is constant and
can be subtracted.
3
6.0x10
+
Na TOF for three different pump=probe laser delay
500 ns delay
200 ns delay
50 ns delay
3
4.0x10
+
Na due to probe (2 x IR)
peak-peak ~ 120ns
0.2eV excess energy
3
2.0x10
+
Na Yield (arb. units)
+
Na due to
pump-pump only
0.0
0.5
0.6
0.7
0.8
0.9
1.0
Time-of-flight (s)
Figure 4.22 Na+ TOF of 41Σg+ v = 52, J = 19
for three different probe laser delays.
120
1.1
1.2
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
The lifetime of each ro-vibrational state is extracted from the corresponding
population decay curves. Figure 4.23 shows such curves for vibrational levels v =
43, 44, 45 and 49 of the 41Σg+ state. Lifetime is extracted using the expression
given in Eq. (4.1.6) with an additional constant term for the background ion
signal. Here, a single exponential decay function is sufficient to extract lifetime
data plus a constant for the background Na+. Although the 41Σg+ state radiatively
decays to lower electronic states A1Σu+ and B1Πu+, the IR probe laser is not
enough to ionize these states and they do not contribute to the Na+ ion yield. At
least for case of radiative decay to A1Σu+ state, an auxiliary experiment is
performed as a quick check using a tunable dye laser for A1Σu+  X excitation.
When the IR laser is used as probe to ionize the excited molecules in the A1Σu+
state, TOF spectra do not yield any ions, whether Na+ or Na2+.
121
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
decay curve of v = 43
decay curve of v = 44
= 22.5 ± 1.4ns
Na Yield (arb. units)
= 23.9 ± 1.1ns
decay curve of v = 49
decay curve of v = 45
= 28.3 ± 1.8ns
+
= 20.9 ± 1.4ns
0
30
60
90
120
150
180
0
30
60
90
120
150
180
probe laser delay (ns)
Figure 4.23 Population decay curves of 41Σg+ (v = 43, 44, 45, 49).
Vibrationally resolved experimentally measured lifetimes of the 41Σg+ state are
shown in Figure 4.24 as solid squares of the 41Σg+ as a function of vibrational
quantum number, v. The error bars originate from the exponential least square
fitting. For the lowest value of v investigated, (v = 43), the measured lifetime is
about 21 ns. Up to the shelf of the 41Σg+ state the lifetime data follows a nearly
linearly increasing trend by a factor of almost 2. Above the shelf (v ≥ 52) up to the
highest vibrational level measured, v = 64 the lifetime data is about constant
within the error bars. Also shown in Figure 4.24 are the calculated lifetimes of the
41Σg+ state shown in solid red diamond. The insert graph shows the calculated
lifetime including those above the shelf.
122
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
60
experiment
calculation
Lifetime,  (ns)
50
40
30
20
10
shelf
D
0
25
30
35
40
45
50
55
60
65
vibrational quantum number, 
Figure 4.24 Experimental lifetime measurement of the 41Σg+ Na2 State
as a function of vibrational level, v = 43 – 64. Red diamond and inset graph are
calculated lifetime data.
123
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
4.6 Discussion
The order of magnitude of the experimental lifetime data for the 21Σu+ state is
consistent with the vibrationally averaged value of 52.1 ns reported by
Mehdizadeh25 for an unspecified range of vibrational levels above the potential
barrier. In this work, vibrationally resolved lifetime values for the 21Σu+ state have
been measured and calculated. Clear variation with vibrational quantum number
for inner well levels up to the barrier at v = 33 as well as for levels above up to v =
52 is observed (see Figure 4.19). Because they are inaccessible from the Na2
ground state due to poor Franck-Condon overlap, levels in the outer well below
the barrier are not part of the experimental data. Calculated lifetimes in the outer
well increase monotonically with vibrational quantum number v, become
approximately constant as levels (v > 20) approach the barrier at v = 33, and
increase again above the barrier, in nearly linear fashion, up to the highest level
calculated of v = 55. For these above-the-barrier levels, calculated lifetimes are
systematically and increasingly larger than corresponding measured values. One
possible source for this discrepancy is the lack of bound-free transitions of 21Σg+
 21Σu+, 11Πg+  21Σu+ or X1Σg+  21Σu+ in the calculation. Because of this
deficiency, the present calculation is only a lower limit of the overall transition
probabilities. In other words, including these transitions should reduce the
calculated lifetimes. Until such calculations have been performed, it remains
unclear whether this is the only cause for the observed mismatch between
experiment and theory.
124
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
The main reason for the variation of lifetime of the 21Σg+ is the fact that the state is
formed by more than one diabatic states. As mentioned in section 4.3, these states
have different electronic characters. The inner well has an electronic character of a
Rydberg state with an asymptotic limit to Na (4s) + Na (3s). The outer well of the
potential has substantial ionic character. As seen in Figure 4.19, both calculated
and measured (v > 20) lifetimes for the inner well vibrational levels below the
barrier follow a decreasing trend, quite different to the calculated data of the
equivalent outer well vibrational levels below the barrier. This observation of two
different lifetime trends further supports the claim that the two potential wells,
inner and outer, although associated with the same electronic state, have different
electronic character. Above the barrier of the potential, the lifetime data
monotonically increase in roughly linear fashion. This suggests that the different
electronic characters of the state above the barrier are not anymore apparent or at
least do not dominate. The variation in lifetime trend below the barrier can be also
attributed to the fact that the 21Σg+ can radiatively decay to three different
electronic states with lower energy. The transition probability for these three
different electronic transitions depend on the three different transition dipole
moment functions as seen in Figure 4.3c. Apart from the dipole moment function,
Franck-Condon factors for each individual rotational transition to a range of
vibrational distribution within a specific electronic transition should also be
accounted. The intrinsic variation of the dipole moment functions and FranckCondon Factor distribution of the three different electronic transitions result in
both a non-monotonic, variation of lifetime on a larger scale as well as a noiselike changeability from one vibrational level to the next.
125
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
The long, second exponential decay measured in the experimental data is
consistent with what is likely the contribution from both the 21Σg+ and 11Πg+
states. The mean lifetime of these two states extracted from the fit of the
experimental data is 3.8 μs. This is in qualitative agreement with the calculated
combined average lifetime of the two states (v = 20 – 40) which is 3.4 μs. This
long lifetime comes from the ro-vibrational distribution of the two states
mentioned above. It is not resolved vibrationally nor is it clear what the relative
contributions of the two electronic states are.
For the case of the 41Σg+ state, an overall increasing trend of the lifetime has been
observed both in experiment and calculation (see Figure 4.24) as the vibrational
level increases up to, v = 51, one level below the shelf of the state. In the
calculation, a rapid increase was observed that sharply peaks at v = 51. The
experiment does not show this behavior. Another discrepancy observed is the
lifetime trend above the shelf. The calculation shows a decreasing trend that
approaches a value of around 100 ns at the highest calculated level, v = 70. Again,
the experiment does not have this behavior. Instead, above the shelf, the measured
lifetime stays roughly constant.
The averaged calculated lifetime below the shelf is 40 ns, larger by about 10 ns
compared to experiment. Above the shelf, calculation settles down to a lifetime
value which is about twice that of the experimental measurement. Following the
same argument as in the case of the 21Σu+ state, the absence of bound-free
contribution in the calculation of the radiative decay process is likely one of the
reasons for these discrepancies. Here as well, the lifetime magnitude agreement is
126
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
expected to improve between experiment and calculation once the bound-free
contribution is incorporated.
The sharp peaking in the calculated lifetime at v = 51 may be partly due to the
same cause mentioned above. As outlined in section 4.5.1, the decay of the 41Σg+
state is dominated by the A1Σu+  41Σg+ transition. Close inspection of the
transition dipole moment function of the A1Σu+  41Σg+ transition shows that the
abrupt decrease of dipole moment function with very small minimum value to R
about 8Å can cause the calculated lifetime to rapidly increase. The Einstein A
coefficient is proportional to the matrix elements of the transition dipole moment.
Thus, if the dipole moment is small, the Einstein A coefficient is small. And since
lifetime is inversely proportional to the Einstein A coefficient, for a small value of
Einstein A coefficient, lifetime is large.
Regardless of the discrepancies seen between experiment and calculated lifetime,
the lifetime variation for vibrational levels below and above the shelf is different.
Yet again, this is an indication that two electronic characters of the 41Σg+ state are
present. For vibrational levels below the shelf, its electronic character is close to
that of the Rydberg state dissociating to the asymptotic Na(4s) + Na(3s) limit. In
contrary to the 21Σu+ state case, the 41Σg+ state does not have an outer well. The
lifetime trend above the shelf suggests that the effect of each character of the two
electronic states involved in forming the state, add in accord to the lifetime data
that cannot be distinguished both in experiment and calculated data.
127
Chapter 4: Radiative Lifetimes on Some Excited Na2 States
4.7 Summary
The main goal of this study is to experimentally measure the dependence of the
Na2 21Σu+ and 41Σg+ lifetimes with vibrational level number, v. With the help of
high resolution dye lasers used for excitation and using variable delay pump-probe
resonant ionization technique, the radiative lifetime of the selective individual
vibrational levels of the 21Σu+ state (v = 13 – 52) and of the 41Σg+ state (v = 43 –
62) of Na2 have been experimentally measured as a function of vibrational
quantum number. Calculation of lifetimes was also performed using the Level 8.0
program. In general, it has been observed in both experiment and calculation that
there is a strong variation of lifetime exists especially as the vibrational levels
approach the potential barrier of the 21Σu+ state or the shelf of the 41Σg+ state.
Overall lifetime magnitude measured and calculated for the 21Σu+ state is in good
agreement with the vibrationally averaged lifetime data reported by Mehdizadeh.
For the 41Σg+ state, the current work is the first account of lifetime data. Although
the lifetime calculation does not include the bound-free contribution in the
radiative decay process, the overall experimental lifetime trend agrees with the
calculation. It is expected that the agreement between experiment and calculation
will improve when the bound-free contribution is incorporated in the calculation.
Vibrational lifetime dependence of the 21Σg+ and 11Πg+ states have been
calculated. The slow decay channel measured in the 21Σu+ lifetime measurement is
believed to come from these states. Indeed, the averaged measured lifetime agrees
well with the calculated data.
128
5 Case Study: Shallow Well of the Na2+
12Σu+ State
The goal of this work is to calculate and experimentally measure ro-vibrational
energy levels and spacing of the Van der Waals (shallow well) of the 12Σu+ state
of Na2+. Calculations are realized using the Level 8.0 program. From these
calculations, the shallow well of the 12Σu+ state with its potential depth of 70 cm-1
is expected to have about 26 vibrational levels.
The experimental part is performed using Zero Electron Kinetic Energy (ZEKE)
spectroscopy. Ground state Na2 from a molecular beam is resonantly excited using
a frequency doubled output of a tunable dye laser in the vibrational levels 70 – 80
of the 21Σu+ state of Na2. The outer turning points of these high vibrational levels
of the double well state were expected to have good Franck-Condon overlap with
the 12Σu+ well. Using a second tunable dye laser operating near 625 nm excited
Na2 molecules were promoted to high lying Rydberg states converging to the
12Σu+ state. Ions are generated in the last step by applying an appropriate delayed
pulsed field. By scanning the second laser, ro-vibrational levels of the 12Σu+ state
can be mapped out in principle. Prompt ions and electrons are swept out during
the delay time by a small constant DC field.
ZEKE methodology was tested first and successfully implemented on the ground
state 12Σg+ of Na2+. For the 12Σu+ state case, experiments were performed
subsequently but yielded no results. The details of both experiment and
Chapter 5: Case Study: Shallow Well of Na2+ 12Σu+
calculation will be discussed here as well as plausible explanations why the
experiment was unsuccessful.
130
Chapter 5: Case Study: Shallow Well of Na2+ 12Σu+
5.1 Introduction
Excited states of neutral molecular sodium have been extensively studied both in
theory and experiment for the past few decades. Precise molecular potentials and
spectroscopic constants are readily available in the literature. The measurement of
Kusch et al.94 for the X1Σg+ state resulted in vibrational levels up to v = 45 while
simultaneously measuring the B1Πu+ state up to v = 28. The work of Jones et al.95
measured the dissociation energy of the X1Σg+ state to be 5,942.6880(49) cm-1.
The A1Σu+ state was measured by Kaminsky96, 97 to up to v = 44 and have been
expended to v = 70 by Gerber et al.98 A more recent work by Qi et al.91 on the
A1Σu+ state led to a more accurate potential spectroscopic constants. Tsai et al.99
measured 31Σg+, 41Σg+, 51Σg+, 61Σg+, 21Πg+, and 21Πg+ using OODR technique.
Stwalley also contributed to the continuing precise and complete spectroscopic
measurements especially for long-range Na2 molecules.100
In contrast, spectroscopy of the Na2+ ionic molecule in general, the low-lying Na2+
potentials, 12Σg+ and 12Σu+, in particular are not known to the same degree of
completeness, accuracy and precision. The ground state ion 12Σg+ has been
measured experimentally only up to 8 vibrational levels via autoionization
resonances55. No experimental measurement exists for the 12Σu+. The potential
curve is only based on theoretical calculations of several groups.14, 101-106 Thus the
main motivation of this work was to measure experimentally the energy levels and
energy level spacing of the shallow well of the 12Σu+.
According to the calculations, the 12Σu+ state of Na2 is repulsive for R less than
about 9.2 Å. At larger R, it becomes attractive due to the long-range charge131
Chapter 5: Case Study: Shallow Well of Na2+ 12Σu+
induced dipole interaction. This long-range interaction produces a shallow well
with a minimum at around 11 Å and depth of about 70 cm-1. The potential Vu(R)
can be approximated by an analytical expression representing the sum of
polarization and exchange interaction. The specific form used here is adopted
from Johann et al.:106
2 1   R  1
1

Vu ( R)  Vx ( R )  V p ( R )  A2 R  e
2
  1
1 1
1   3  2  
  R
 
  3

1  1 1 
B 
 1   3  2     14  26  38  7 
  R 2 R R R R 
  2
2
(5.1.1)
where the parameter values that produces best fit to the theory points of Magnier57
that are tabulated in Table 5-1. The potential expression (5.1.1) is shown in Figure
5.1 together with the discrete data points from 5 different calculations.
132
Chapter 5: Case Study: Shallow Well of Na2+ 12Σu+
1
1
Figure 5.1 Shallow well of the 12Σu+ state of Na2
FIG. 1 taken from Delahuty et al.36 of the potential-energy curve of the shallow
well of the 12Σu+ state of Na2 according to six different calculations. The solid line
is the analytical expression in Eq. (5.1.1) fitted to the theory points.
Table 5-1 Table of parameter values.
Table I taken from Delahuty et al.36 for the table of parameter values for the
analytical expression given in Eq.(5.1.1).
133
Chapter 5: Case Study: Shallow Well of Na2+ 12Σu+
5.2 Calculations
To guide the experiment, calculations were performed of the shallow well of the
12Σu+ state to predict vibrational levels and spacing. First, test calculations were
executed on the ground and first excited state of Na2 since highly precise values of
molecular potentials and spectroscopic constants are available. The objective of
these test runs is to examine what to expect in terms of level of precision and
agreement to the known data because the program will be implemented for a
potential where spectroscopic data is not available.
The calculations are accomplished using Level 8.0 program written by Robert J.
LeRoy.58 The core of the program calculates the solution to the one-dimensional
Schrödinger equation and finds the eigenvalues and eigenfunctions of a given
input potential. The input potential can be in a functional form or a list of discrete
points. RKR potential for the Na2 ground state X1Σg+ is taken from Camacho et
al.56 while the A1Σu+ potential is from Qi et al.91 Franck-Condon factors for the
A1Σu+  X1Σg+ were obtained from Stwalley through private communication.
Figure 5.2 is a plot of the absolute difference between the calculated energy levels
using Level 8.0 and the published energy levels from the reference potential up to
v = 65 and 50 for the (a) X1Σg+ and (b) A1Σu+ state, respectively. A very good
agreement has been established and the absolute maximum discrepancy observed
is only a fraction of a wavenumber (0.2 cm-1) for both the X1Σg+ and A1Σu+ states.
An additional check was performed by comparing the published spectroscopic
molecular potential constants ωe and ωexe with the calculated energy levels. The
134
Chapter 5: Case Study: Shallow Well of Na2+ 12Σu+
potential of a molecule can be characterized as an anharmonic oscillator with
energy values given by90
1
1
1
Evib  G (v)  e (v  )  e xe (v  ) 2  e ye (v  )3  ...
2
2
2
22
26
G (v  1)  G (v)  3e ye v 2  ( e ye  2e xe )v  (e  2e xe  e ye )
4
8
(5.4.1)
where ωe is the harmonic vibrational level spacing and the constant ωexe << ωe and
ωeye << ωexe are coefficients of anharmonic higher order terms (ωeye is sometimes
negligibly small). The comparison of spectroscopic constants is tabulated inTable
5-2. Here again, the agreement for both ωe and ωexe, for both X1Σg+ and A1Σu+
states is excellent.
A absolute Energy level difference (ref - this work)
0.05000
0.2000
0.00000
delta E (ref - this work)
delta E (ref - this work)
X absolute Energy level difference (ref - this work)
0.2500
0.1500
0.1000
0.0500
0.0000
0
10
20
30
40
50
-0.05000
-0.10000
-0.15000
-0.20000
0
10
20
30
40
50
60
70
-0.0500
-0.25000
v
v
Figure 5.2 Absolute energy level difference
(in cm ) between reference potential and this work. (a) for X1Σg+ and (b) for
A1Σu+
-1
135
60
Chapter 5: Case Study: Shallow Well of Na2+ 12Σu+
X1Σg+
A1Σu+
Camacho et al this work
Kaminsky et al
this work
ωe [cm-1]
155.969025
155.968925
116.6058
116.5961
ωexe [cm-1]
0.36398
0.36393
-0.3521
-0.35195
Table 5-2 Potential spectroscopic constants for X1Σg+ and (b) A1Σu+ states.
Axis Title
A surface plot is also shown on Figure 5.3 for the absolute difference of FranckCondon Factors from Stwalley’s group for the A1Σu+  X1Σg+ transition. Once
more, a very good agreement is achieved except for some higher v” v’
transitions. These differences correspond only to few percent.
delta Franck-Condon Factor
-70
-70--60
-60
delta FCF (x10-4)
-60--50
-50--40
-50
-40--30
-40
-30--20
-20--10
-30
-10-0
0-10
-20
-10
41
35
29
0
10
23
A 1 3
5 7 9
17
11 13 15
17 19 21
23 25 27
11
29 31 33
35 37 39
5
41 43 45
47 49 51
53 55 57
X
59 61 63
65 67 69
Figure 5.3 A  X FCF
Absolute difference of Franck-Condon Factors of AX from
Stwalley’s group and this work.
136
Chapter 5: Case Study: Shallow Well of Na2+ 12Σu+
Similar calculations were done on the Na2 21Σu+ potential. Since there is limited
data available of the 21Σu+ molecular potential, comparison can only be made on
the spectroscopic constants of the inner and outer well of the 21Σu+ potential to the
published data of Pashov et al.76 and calculated data from Magnier.57 Table 5-3
summarizes the resulting data. Here again, for ωe a very good agreement is
achieved. There is no available data for ωexe and hence no comparison can be
made.
Spectroscopic constants for 21Σg+
inner well
ωe [cm-1]
ωexe [cm-1]
outer well
Magnier
Pashov
Magnier
Pashov
et al
this
work
et al
et al
this
work
et al
105.5
106.26
105.3
52.3
52.5
52.75
0.52
-0.152
Table 5-3 Spectroscopic constants of the Na2 21Σu potential.
From Eq. (5.1.1), fitted points from Magnier57 for the 12Σu+ ion potential are used
as an input to calculate vibrational levels and level spacing of the shallow well.
The calculation shows that the shallow well has about 26 vibrational levels with
ωe = 8.1 cm-1 and ωexe = 0.3 cm-1. Table 5-4 lists the corresponding energy levels
for each vibrational quantum number, v. The shallow well of the potential is
shown in Figure 5.4 with the corresponding vibrational levels.
137
Chapter 5: Case Study: Shallow Well of Na2+ 12Σu+
v
Energy (cm-1)
v
Energy (cm-1)
0
47405.1
13
47461.2
1
47412.7
14
47462.7
2
47418.9
15
47464.0
3
47425.1
16
47465.1
4
47430.9
17
47466.0
5
47436.4
18
47466.9
6
47441.2
19
47467.6
7
47445.6
20
47468.2
8
47449.6
21
47468.7
9
47452.7
22
47469.2
10
47455.3
23
47469.6
11
47457.6
24
47469.9
12
47459.5
25
47470.2
13
47461.2
26
47470.4
Table 5-4 Calculated energy levels of the 12Σu+ state.
47,510
47,490
Energy (cm-1)
47,470
47,450
47,430
47,410
47,390
8
9
10
11
12
13
14
15
16
R (Angstrom)
Figure 5.4 The shallow well of the 21Σu+ state with vibrational energy
138
Chapter 5: Case Study: Shallow Well of Na2+ 12Σu+
5.3 ZEKE Background
Over the past 25 years, since its invention, Zero Electron Kinetic Energy (ZEKE)
photoelectron spectroscopy has become a widely used threshold photoelectron
method and has turned into a common tool for high-resolution studies of
molecular ions and clusters. The technique was pioneered in 1984 by MüllerDethlefs, Sanders, and Schlag.107 Since its advent, several review articles have
been written on the topic.108-110 The method relies on detecting the zero kinetic
energy electrons produced by field ionization of high n Rydberg states. Using this
technique, two to three orders of magnitude in spectral resolution of the ion can be
gained compared to the conventional photoelectron spectroscopy (PES).
ZEKE spectroscopy, a modified version of photoelectron spectroscopy and
originally termed as threshold photoelectron spectroscopy, was first used to
resolve rotational states at the photoionization threshold of NO+.107 Among the
first cationic systems studied using ZEKE are H2S
111
, H2O
112
, NH3 113 and I2 114
while for anionic systems, Sin (n=2-4) iron oxide115, silver and gold clusters116, 117,
as well as argon iodine clusters118 were investigated. ZEKE has proven to be also
effective on study of radicals and transition state structures.119, 120 But most of the
systems ZEKE technique was applied to are molecular clusters. Benzene-Ar was
the first cluster being studied with this technique.121 Since its advent, results from
ZEKE spectroscopy have yielded many of the electronic properties such as high
accuracy ionization potentials, ro-vibrational ion structure, molecular and cluster
geometries, and binding energies.
139
Chapter 5: Case Study: Shallow Well of Na2+ 12Σu+
The three features of ZEKE spectroscopy that make it attractive compared to
conventional PES are the following. (i) First is the improved resolution. In a
typical PES experiment, resolution is of the order of 20 cm-1 (~ 2.5 meV) while in
ZEKE experiments, resolution is often a fraction of a wavenumber. (ii) Secondly,
though not exclusive to ZEKE spectroscopy, another advantage of this technique
is the ability to selectively prepare excited state from which ionization is to occur.
This is done by preparing intermediate states using resonant excitation. Finally,
(iii) ZEKE spectroscopy has very high sensitivity. This might not be a necessary
improvement from a conventional PES, nevertheless one does not need to
sacrifice sensitivity over resolution.
In theory, ZEKE spectroscopy employs a simple mechanism. Figure 5.5 shows a
schematic representation of the ZEKE process. The first laser (pump laser) is
tuned to an excited ro-vibrational molecular state M* of interest while the second
laser (probe laser) is tuned and scanned through the ionization potential (IP). As
the probe laser is tuned near the threshold ionization, high n Rydberg states are
populated. This mechanism is the principal signal in ZEKE experiments.112, 122-124
These Rydberg states are known to have long lifetimes scaling as n3. By applying
a delayed pulsed field, atoms or molecules in these states can be ionized and
simultaneously extracted towards the detector. A small sweeping field is applied
first to separate prompt ions and electrons produced by direct ionization or
autoionization from the neutrals. This sweeping field will also ionize Rydberg
states by ΔE below ionization threshold depending on the magnitude of the
electric field given by:30
140
Chapter 5: Case Study: Shallow Well of Na2+ 12Σu+
E  4F 1/2
(5.2.1)
where F is the magnitude of electric field in V/cm. For a 1V/cm field, this
amounts to 4 cm-1 corresponding to n values of about 160 – 170. A delayed pulsed
field of much larger amplitude is then applied to ionize all the remaining longlived Rydberg molecules. This delayed pulsed ionization scheme differentiates
direct ions and electrons from the ZEKE signal.
Both ions and electrons can be detected using separate detectors. This scheme is
called ZEKE/MATI where MATI stands for Mass Analyzed Threshold Ionization
spectroscopy. It relies on detection of ions rather than electrons. Experimentally,
MATI will yield mass resolved information and is significant for studies of large
cluster molecules.
As seen in Figure 5.5, ZEKE spectroscopy will produce peaks for bound levels of
the ion contrary to conventional photoelectron spectroscopy which only shows
steps for each ion-channel that opens up.
141
Chapter 5: Case Study: Shallow Well of Na2+ 12Σu+
Figure 5.5 Schematic representation of the ZEKE process.
The pump laser is tuned to a M* resonance, and the probe laser is scanned through
the IP. As the probe is tuned through the cation state, the ionic ro-vibrational
states will be mapped out.
142
Chapter 5: Case Study: Shallow Well of Na2+ 12Σu+
5.4 Experimental Method
The details of the general experimental set-up were already described in chapter 2.
Ground state Na2 produced in a supersonic beam expansion are excited using
pump and probe tunable dye lasers that perpendicularly crossed the molecular
beam in the interaction region of the TOF chamber. Unlike the experiments
described in previous chapters, electrons are detected in this study. For this
purpose an additional MCP detector was installed. In practice, both ions and
electrons can be collected during the experiment. Electron/ion signals are fed
simultaneously into a 150 MHz digital oscilloscope (HP Lecroy 9400) and a gated
boxcar integrator (SRS 250). Both data are sent via GPIB interface to a computer
and recorded using LabVIEW software.
The schematic of the laser excitation scheme is shown in Figure 5.6. The pump
laser is tuned to the intermediate 21Σu+ state. This intermediate state is selected
because of its large outer turning point for vibrational levels v > 70.
Wavefunctions at these vibrational levels provide a good Franck-Condon overlap
to the shallow well potential of the 12Σu+ ion state. The probe laser is delayed by
about 10 ns to avoid accidental resonances with the A1Σu+ as it is tuned near but
below the ionization threshold of 12Σu+ ion state. In doing this, high n Rydberg
levels of the 12Σu+ ion become populated. As discussed in the ZEKE background,
these Rydberg states have long lifetimes. These Rydberg states are pulsed field
ionized with an appropriate PFI separation delay (nominally 3.5 μs) and the
electrons are extracted to the detector. By scanning the probe laser from below the
adiabatic ionization potential over a series of ionization limits in the 12Σu+ ion
143
Chapter 5: Case Study: Shallow Well of Na2+ 12Σu+
state and applying delayed PFI scheme, ro-vibrational levels of the 12Σu+ ion state
can be mapped out.
Figure 5.6 Excitation Scheme
ZEKE experimental excitation scheme using 21Σu+ state as the intermediate
excited state.
144
Chapter 5: Case Study: Shallow Well of Na2+ 12Σu+
5.5 Experimental Data
As a first test of our ability to obtain reliable ZEKE data, I performed an
experiment to measure the vibrational levels of the Na2 12Σg+ ground state ion.
The first step is performed through resonant excitation in the A1Σu+  X1Σg+
transition using a tunable dye laser and Rhodamine 101 dye with operating
wavelength of 611 – 662 nm. The pump laser is kept to the excitation of the rovibrational levels v = 13, J = 19 and 23 of the A1Σu+ state. For the same laser
frequency within the laser bandwidth, two rotational levels 19 and 23 that come
from the P(J) and R(J) branching during the A1Σu+ ← X1Σg+ excitation can occur.
The probe laser is scanned through the v = 0 of the 12Σg+ state using a second
tunable dye laser in the range 23,280 – 23,520 cm-1 with a Stilbene 420 dye with
lasing wavelength of 404 – 460 nm. Before ZEKE was performed, Na2+ yield
spectra were measured. This was realized by applying a constant DC field of 25
V/cm in the extraction region of the TOF spectrometer. Ions are further
accelerated to the MCP detector with an acceleration field of 200 V/cm. A sample
spectrum is shown in Figure 5.7a as a function of the laser wavenumber. The
sharp peaks seen in the figure are believed to come from high lying Na2 molecular
states that are below but close to the Na2+ ground state. Potential candidates for
these high lying Na2 molecular states are the 61Δg+, 71Δg+ and 81Δg+ states. High
ro-vibrational levels of these molecular states are above the Na2 ionization
threshold and autoionization can occur. When the laser energy meets resonance
condition as the laser is scanned above the Na2 ionization threshold, these states
autoionize to the Na2 12Σg+. Na2+ signal will then appear as sharp peaks in the
yield spectrum. From the linewidth information, it is possible to extract the upper
145
Chapter 5: Case Study: Shallow Well of Na2+ 12Σu+
limit of the autoionization lifetime of these states. Using a FWHM of 0.3 cm -1, it
was found out that the lower limit of the autoionization lifetime of these Na2 states
that produces the sharp peaks in the experimental data is in the order of 10 ps. For
this particular study, the transitions at which these peaks come from were not
identified for the reason that the main task of this preliminary experiment is only
to demonstrate ZEKE experiment in our laboratory.
When ZEKE was performed, low voltage sweeper field was applied and pulsed
field ionization was implemented by applying a 50 V pulse of 1 μs-width and
delayed 3.5 μs from the probe laser. The majority of the peaks in Figure 5.7a are
no longer observed proving that they are due to prompt ions. Figure 5.7b shows
the electron signal as a function of the laser wavelength. Although not shown in
the figure, apart from the four peaks shown in the figure, no other peaks were
observed in the probe laser scan range of 23280 – 23520 cm-1.
146
Chapter 5: Case Study: Shallow Well of Na2+ 12Σu+
70
+
Na2 Signal (arb. units)
60
a)
50
40
ionization threshold
30
20
10
0
-10
70
-
e Signal (arb. units)
REMPI
23300
23350
23400
23450
23500
ZEKE
b)
60
50
40
30
no observed peaks in this region
20
10
23300
23350
23400
23450
23500
-1
probe laser (cm )
Figure 5.7 Yield Spectra
(a) Na2+ ion yield and (b) threshold electron spectrum vs. laser wavelength
-1
2.0x10
4
1.5x10
4
1.0x10
4
-
e Signal (arb. units)
23402.1 cm excitation
-1
-1
23402.1 cm excitation (1 cm detuned)
probe laser arrives here
5.0x10
3
0
100
200
300
400
500
TOF (x10 ns)
Figure 5.8 Electron TOF comparison for two different laser excitation lines.
147
Chapter 5: Case Study: Shallow Well of Na2+ 12Σu+
Figure 5.8 shows the TOF spectra of the photoelectron signal for the probe laser
excitation at 23401.2 cm-1 (top trace) and 23402.2 cm-1 (bottom trace). Prompt
electrons arrive about 50 ns after the laser excitation. The background change at
around 3.2 μs in the TOF data is due to the application of the pulsed field. In the
23401.2 cm-1 excitation which is detuned 1 cm-1 from observed ZEKE signal, it
can be seen that only prompt electrons are available. It is also noticeable that the
prompt electron signal intensity is smaller compared to the 23402.2 cm-1 case. In
the 23402.2 cm-1 case, the ZEKE signal arrives at about 3.5 μs after the prompt
electrons which corresponds to the delay of the PFI which is the hallmark of a
ZEKE signal.
Assignment of the ZEKE signals is accomplished by comparing calculated energy
levels of the Na2 12Σg+ with the total photon energy used for excitation.
Calculation was performed using the spectroscopic numbers published by Bordas
et al.55 It was found out that the first and second peaks correspond to v = 0, J =20
and J = 24 of the Na2 12Σg+, respectively. The other two peaks were not identified.
The line assignment is summarized in Table 5-5 below.
Experiment
-1
-1
assignment
calculation
Δ (Calc – Expt)
(cm-1)
Probe Laser (cm )
Total energy(cm )
23402.2
39585.8
v = 0, J = 20
39586.7
0.9
23424.3
39607.9
v = 0, J = 24
39607.1
-0.8
23484.0
39667.6
?
23513.6
39697.2
?
Table 5-5 Line Assignments
148
Chapter 5: Case Study: Shallow Well of Na2+ 12Σu+
The corresponding ZEKE signal was measured as a function of PFI amplitude and
delay. The integrated ZEKE signal is plotted in Figure 5.9. From the figure can be
seen that the intensity of the ZEKE signal decreases as the PFI amplitude
increases. This behaviour is a contradiction to the expectation that the higher the
PFI amplitude, the signal should increase. There is no clear explanation yet why
this happens. It is suspected that as the PFI amplitude is increased, the amplitude
ramp somehow affects the Rydberg levels in a strange way which is not
understood here in details. Another observation in the data is when the PFI delay
is increased, the ZEKE signal decreases. This is qualitatively inconsistent with the
established long lifetime of high n Rydberg levels. Possible reason for this is due
to collision. Although there have been no indications from previous experiments,
it is possible that collisions do occur in the molecular beam, in particular with
molecules excited to high n Rydberg levels with their significantly enlarged
electron orbit size.
149
Chapter 5: Case Study: Shallow Well of Na2+ 12Σu+
160
1 s delay
2 s delay
4 s delay
-
e Signal (arb. units)
140
120
100
80
60
40
20
0
10
20
30
40
50
PFI Amplitude (Volts)
Figure 5.9 ZEKE signal
of the 23402.2 cm-1 excitation as function of PFI amplitude for three different PFI
delays.
ZEKE experiment was performed on the 12Σu+ state using the 21Σu+ as an
intermediate state. Several ro-vibrational levels of the 21Σu+ were tested but
generated no ZEKE signal. More than a few settings of the PFI delays and
amplitudes were also explored but again, no meaningful data would be extracted
and the author opted not to show any data for obvious reasons.
150
Chapter 5: Case Study: Shallow Well of Na2+ 12Σu+
5.6 Discussion
One possible reason why no ZEKE signal from the 12Σu+ state is observed may be
due to direct ionization of the excited intermediate Na2 molecules to the 12Σg+ ion
state. If the probe laser used in the experiment ionizes the majority, if not all, of
the excited molecules, no opportunity is left for excitation to high n Rydberg
states approaching the shallow well of the 12Σu+ state. Another possible origin why
ZEKE is unsuccessful in the 12Σu+ state can be attributed to a very small transition
probability. The transition dipole moment function for the 21Σu+  12Σu+
transition is not known. Although calculations show non-zero Franck-Condon
overlap for this transition, it might not be sufficient for the transition to be
observed within the sensitivity of the current experimental apparatus.
For future ZEKE experiments on the 12Σu+ state, the author proposes the 41Σg+
(shelf) state to be used as intermediate state. Some vibrational levels just above
the shelf of the state provide a good Frank-Condon overlap to the shallow well of
the 12Σu+ state. In addition, previous experiments from our group36 have shown
that the 12Σu+ state can be directly accessed from the 41Σg+ state with good
efficiency. Therefore, access to the corresponding Rydberg states should also be
strong. The only drawback for using the 41Σg+ state as an intermediate state is that
the whole experiment requires three dye lasers. Since the shelf state is of “g”
symmetry, selection rules do not permit one-photon excitation for the X1Σg+ 
41Σg+ transition. Hence, an intermediate state is necessary that is of “u” symmetry
which requires an additional tunable dye laser for resonant excitation. For this
process, the A1Σu+ or the B1Πu+ states are good candidates.
151
Chapter 5: Case Study: Shallow Well of Na2+ 12Σu+
5.7 Summary
LEVEL 8.0, the program used for calculating the bound vibrational levels of a
diatomic molecular potential was tested on known potentials of Na2 which
returned very good agreement. The shallow well of the 12Σu+ state with a depth of
about 70 cm-1 was predicted to have about 26 bound vibrational levels.
ZEKE experiment was successfully demonstrated in the laboratory for the 12Σg+
state of the Na2+. Two rotational levels were successfully identified for v = 0 of
the ground ion state. However, the attempt of this study to measure experimentally
the ro-vibrational levels and spacing of the 12Σu+ state of Na2+ was unsuccessful.
152
6 Summary
I have conducted three distinctly different experiments using sodium molecular
beam, linear time-of-flight mass spectrometer, and pulsed nano-second lasers.
These projects were (i) Non-Resonant Multi-Photon Ionization, (ii) Radiative
Lifetimes of Excited Na2 States, and (iii) Case Study: Shallow Well of the Na2+
12Σu+ State. Several auxiliary experiments were accomplished for the
characterization of the sodium molecular beam. It was established through
resonant enhanced multi-photon ionization on the first excited state of Na2, A1Σu+
state, that the molecular beam has an internal rotational and vibrational
temperature of about 70 and 120 K, respectively. It was also inferred from
steering plate voltage dependence experiments that the sodium dimer has a most
probable speed of 580 m/s and a speed ratio S = 4.1. For the molecular beam
expansion conditions, about 90% of the molecular beam consists of Na atoms
while the majority of the rest, is made of Na2 dimers. With the temperatures
mentioned, 90% of the Na2 population is in the vibrational ground state v = 0 with
the most probable rotational level at J = 17.
The first project, which is discussed in chapter 3, was implemented using a nonresonant multi-photon ionization experimental scheme. Atomic and molecular
sodium ions were produced using 532 and 355 nm light from a frequency doubled
and tripled Nd:YAG laser. One of the objectives of this study is to identify the
excitation pathways for the production of Na+. From the experimental TOF data,
kinetic energies and angular distributions of atomic ions were identified. The 355
Chapter 6: Summary
nm Na+ TOF data is the simpler case while a more complicated structure arose in
the 532 nm case. Only one pathway of Na+ production is identified in the 355 nm
case while four different channels (I – IV) were assigned in the 532 nm case.
Channel I is the direct ionization of ground state Na2 to the 12Σg+ continuum and
subsequent dissociation, however this was not observed in the experiment.
Channel II is assigned to the one-photon dissociation of Na2+ which is produced
by three-photon ionization of Na2 using 532 nm light. Channel III corresponds to
two-photon dissociation of Na2 followed by one-photon ionization of excited Na
atom. Channel IV is the dissociative autoionization process. This channel leads to
the production of Na+ that is responsible to the broad center peak of the Na+ TOF
spectra. This is the same channel responsible for the Na+ production of 355 nm
excitation. Na+ production via dissociative autoionization is the main focus of the
first project. It was shown that Rydberg states approaching the12Σu+ of Na2+ ion
state play an important role in the production of Na+ in this channel. A semiclassical model was implemented to understand the details of dissociative
autoionization process and reproduce experimental data. Calculated TOF spectra
were generated using a Monte Carlo simulation in conjunction with appropriate
energy and angular distributions. Very good agreement was established between
experimental and simulated TOF data for both 355 and 532 nm experiments.
Potential width of the dissociative Rydberg states as a function of atom-atom
distance was generated that best fit the experimental TOF data. It was found that
corresponding autoionization lifetimes of these Rydberg states are in the order of
tens of femtoseconds. The potential width function that best reproduced the
experiment follows a quadratic function with minimum around 5 Ǻ.
154
Chapter 6: Summary
The second project is concerned with radiative lifetime measurements of two
excited states of Na2, namely the 21Σu+ and 41Σg+ states. Experiments were carried
out with the help of tunable dye lasers. REMPI technique was used to map out rovibraitonal levels of 21Σu+ state. Population decay curves of the ro-vibratonal
levels of the 21Σu+state were measured by delayed ionization with 532 nm light
from an externally controlled Nd:YAG laser. Lifetime as a function of vibrational
quantum number v was measured. To within order of magnitude, the average
measured lifetime of the 21Σu+state was consistent with values reported elsewhere.
Lifetime calculations were also executed with the help of Level 8.0 program.
Systematic variation of lifetime with vibrational level was observed in both
experimental and calculated data especially near the potential barrier of the 21Σu+
state. Calculated lifetime values are larger in comparison to the experiment. This
discrepancy can be accounted for –at least in part- to the absence of the boundfree transition contribution in the calculation. But even in the absence of this
contribution in the calculation, the generic trend of the variation of lifetime data in
the experiment was reproduced.
The lifetime measurement of the 41Σg+ state was conducted using double
resonance spectroscopy technique via the A1Σu+ state. Individual ro-vibrational
levels of the 41Σg+ state were mapped out while the tunable probe dye laser was
scanned through the levels below and above the shelf of the 41Σg+ state. Population
decay of the shelf state was undertaken with variably delayed IR photons from a
separate Nd:YAG laser. Dependence of lifetime data with vibrational quantum
number v was observed especially near the shelf of the state. Lifetime calculations
were also completed using the same program used in the 21Σu+ state calculations.
155
Chapter 6: Summary
Again, contributions from bound-free transitions are absent which could account
for the magnitude discrepancy between experiment and calculation. The
agreement between the two data is expected to improve if the bound-free
contributions will be incorporated in the calculation. The lifetime values below the
shelf of the state for both experiment and calculation follow the same trend.
However, the calculated lifetimes above the shelf of the state are at least about
twice as large as the experimental values. In the calculation, there is sharp peaking
of lifetime near the shelf of the state which is not observed in the experiment. This
characteristic in the calculated lifetime data can be attributed to the transition
dipole moment function which rapidly decreases at distances larger than about 8Å.
ZEKE experiment was successfully demonstrated in the lab for the 12Σg+ state of
the ground state Na2+ ion. The main aim of the experiment was to do ZEKE
spectroscopy of the shallow well of the 12Σu+ state of the Na2+ ion. The 12Σu+ state
is repulsive in nature, but due to due to the long-range charge-induced dipole
interaction, it becomes attractive at large internuclear distance and exhibits a
minimum around 11Å. This shallow well was predicted to have a potential depth
of about 70 cm-1. Calculations implemented here predict that this shallow well has
26 bound vibrational levels with harmonic level spacing of about 8 cm-1 and an
anharmonicity constant of 0.3 cm-1. ZEKE was performed on this shallow well of
the 12Σu+ state but the experimental attempt led to a null result. The reasons for
unsuccessful experiments were discussed in the previous chapter.
156
7
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