Cold polar molecules for novel collision experiments at low
energies
byBrian Sawyer
B.S., University of Arkansas, 2003

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulﬁllmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2010

iii
Sawyer, Brian (Ph.D., Physics)
Cold polar molecules for novel collision experiments at low energies
Thesis directed by Prof. Jun Ye

Research in the ﬁeld of cold polar molecules is progressing rapidly. An array of interesting
topics is being developed including precision measurement and fundamental tests, quantum phasetransitions, and ultracold chemistry. In particular, dipolar molecules in well-deﬁned quantum states
enable exquisite control of their interactions via applied electric ﬁelds. The long-range, anisotropicinteraction between dipolar molecules leads to new collision dynamics that could be used for novel
collective eﬀects, quantum state engineering, and information processing. The focus of this Thesisis the production of cold samples of neutral OH molecules via Stark deceleration and magnetic
trapping for novel collision experiments. This novelty results from our combination of cold externalbeams and trapped target molecules which facilitates observation of dipolar eﬀects as well as lowtemperature collision resonances. The large permanent electric dipole moment of OH allows us toprecisely tune the lab-frame velocity of the molecular packets from ∼ 500 m/s to rest. We havemagnetically trapped OH packets at the terminus of our Stark decelerator at a temperature of 70
mK and density of 106 cm−3 . With improved understanding of Stark deceleration, we optimize thedecelerator eﬃciency and its coupling to the magnetic trap. Our latest trap design uses permanentring magnets to create a three-dimensional magnetic quadrupole ﬁeld. Use of magnetically trapped
OH molecules for collision experiments with external beams allow us the unique opportunity ofobserving both elastic and inelastic collisions. In addition, the trap conﬁnement yields sensitivity
to diﬀerential elastic cross sections at low collision energies. This open magnetic trap has allowed
measurement of collision cross sections between trapped OH and external supersonic beams ofHe and D2 , the latter of which is of astrophysical interest due to the role of H2 -OH collisions inpumping interstellar OH megamasers. The combination of trapped OH molecules and temperaturetuned beams of He and D2 has facilitated measurement of the lowest-energy D2 -OH collision crossiv
sections yet reported. More recently, we report the ﬁrst observation of electric-ﬁeld dependent crosssections between two diﬀerent species of cold polar molecules - OH and ND3 - thereby demonstrating
control over molecular scattering in the cold regime. By combining for the ﬁrst time the production
techniques of Stark deceleration and buﬀer gas cooling, we increase the molecular interaction time by∼ 105 over traditional crossed-beam experiments to gain enhanced sensitivity at the characteristic
densities of cold molecule production.

Dedication

To Liz.

Acknowledgements

I have a number of people to thank for helping either directly or indirectly with the completionof my Ph.D. First oﬀ, I would like to thank Jun for giving me a spot in his cold molecule groupseven years ago. At the time, I was not completely sure which AMO group would be the best ﬁt,but I could not have picked a better experiment or group of people with whom to work. I wentfrom measuring OH spectra in ﬂames as an undergraduate to producing and trapping the coldest
OH samples in the world in the Ye group. I am also grateful to my undergraduate research advisor,
Dr. Raj Gupta at the University of Arkansas, for taking a chance on a freshman physics majorwith little to oﬀer besides enthusiasm. Going even farther back, my high school physics teacher
Jerrel Boast was instrumental in cultivating my interest in experimental physics. If the class had
been taught by someone else or if I had taken it before getting my act together, I can’t imagine
what I would be doing today.
I have worked with a number of great people during my time at JILA. I have to thank Eric
Hudson for welcoming me into the cold molecules group, showing me the ropes, and repeatedly
demonstrating that sometimes you just have to turn the knob and see what happens. I will neverforget the long days of bluegrass and jam music that I was forced to endure, and I’m sure I will
never be as good a ping-pong player as I was in 2005. Jason Bochinski ﬁrst introduced me tothe black art of supersonic beam production, and Heather Lewandowski still gives me advice on
ammonia detection and photolysis sources since becoming a faculty member. Ben Lev worked in
the group for two years as an NRC postdoc and was a major part of our ﬁrst magnetic trapping
experiment. Ben Stuhl joined the lab a few years after me, and we worked together on the trapping

vii
and collision experiments. He and I rarely run out of experimental issues to discuss, and I believe
our often diﬀerent points of view helped keep the experiments moving forward. He is also anamazing programmer who has improved a number of our data-taking and control codes. Mark Yeo
has done a great job bringing more advanced optics techniques into the cold molecule experiments,and he and I share a love of bacon and extreme metal. Matt Hummon recently joined our groupas an NRC postdoc, and he has brought with him a wealth of knowledge about cryogenics andbuﬀer gas sources. I know that I am leaving the OH experiment in good hands, and look forward
to reading some exciting papers in the near future.
Doing research at JILA is an amazing experience. I have to thank the staﬀ machinists Kim,
Hans, Tracy, Ariel, Blaine, Alan, Tom, Dave, and Todd for building pieces that, in many cases, areas artistically appealing as they are useful. The electronics shop staﬀ of Paul, Carl, James, Mike,and Terry also helped immensely with the construction and operation of many of our cold molecule
experiments. Our buﬀer gas source collaboration with the group of John Doyle at Harvard wasvery fruitful and culminated in our most recent OH-ND3 collision measurement. Dave Pattersonfrom John’s group is a very talented experimentalist and also a fun guy to hang with.
My parents always encouraged me to follow my own interests and never placed any speciﬁccareer expectations on me, and for that I am grateful. I can imagine that philosophy requires animmense amount of restraint, especially when your son says he wants to forego college or graduate
school for a career in the music industry. Nevertheless, my mom always expected the best out ofme and has remained supportive through everything. My dad has always been a free spirit, andthe combination of him and my mom helped to make me simultaneously cautious and rebellious. I
have enjoyed being the oldest of ﬁve kids. Layne, Max, Mike, and Will have always shown me anamazing amount of love and support, and I have missed them all since starting college and only
rarely ﬁnding time to get back to Arkansas.
When I ﬁrst moved to Colorado for graduate school in 2003 I only knew two or three people
in the Boulder/Denver area, but I was determined to start a band with talented, like-mindedmusicians. The only person who ever answered my online ad was Ryan Conner of Denver, andviii
we’ve been best friends ever since forming Cryogen in 2004. Since then, we’ve developed a solid
lineup with Aaron Lande, Mike Papajohn, and Chris Giggliotti. I want to thank all of theseguys for the countless hours of writing/rehearsing and the four albums we’ve released as a band.
As the only two guys who can stay awake in a moving vehicle, Mike and I had some interesting
political discussions at 4 am on tour between Salt Lake City, Milwaukee, Chicago, Indianapolis,
Des Moines, and Cincinnati. I want to thank Ryan, Mike, Aaron, and Chris for contributing tosome great extreme music and for allowing me to play shows with bands whose posters I used tohang on my wall in high school. They gave me something to do between the hours of 10 pm and 4
am that had nothing to do with cold molecule physics.
Finally, I would like to thank Liz for coming into my life as a rock star photographer slummingit shooting local bands on weeknights. I can’t imagine my life without her, as she has helped me
through some tough experiences. Her unwavering support and realism keep me grounded and help
me maintain a sense of perspective when I’m taking work or music too seriously. She has endured
my long hours away at the lab and on the road, but she understands and loves me in spite of myobsessive tendencies.

ix

Contents

Chapter
1 Introduction
1.1

1

Why Polar Molecules? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.1.1

Cold and Ultracold Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.1.2

Precision Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52 Molecular Spectroscopy
2.1

2.2

2.3

7

Hund’s Coupling Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2.1.1

Case (a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.1.2

Case (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.1.3

Case (c) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.1.4

Cases (d) and (e) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Spectra of Polar Symmetric Tops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.1

Hydroxyl Radical (OH) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.2

Ammonia (ND3 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

External Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3.1

Stark Eﬀect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.3.2

Zeeman Eﬀect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.3.3

Combined Electric, Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . 39

2.3.4

Diabatic vs. Adiabatic Level Crossings in Combined Fields . . . . . . . . . . 43

x
3 Production of Cold Molecular Samples
3.1

3.2

Supersonic Molecular Beams

Theory of Supersonic Expansions . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.1.2

Clustering of Carrier Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.1.3

Pulsed Supersonic Valves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.1.4

Production of OH Supersonic Beams . . . . . . . . . . . . . . . . . . . . . . . 55

Buﬀer Gas Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.2.1

The JILA Buﬀer Gas Beam Machine . . . . . . . . . . . . . . . . . . . . . . . 60

3.2.2

Translational and Rotational Cooling

3.2.3

The “Pulsing” Phenomenon and Beam Runtime . . . . . . . . . . . . . . . . 66

3.2.4

Electrostatic Velocity Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . 67

. . . . . . . . . . . . . . . . . . . . . . 63

73

Time of Flight Mass Spectrometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.1.1

4.2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.1.1

4 Molecular Detection Techniques
4.1

45

Calibration of Ampliﬁed Ion Signals . . . . . . . . . . . . . . . . . . . . . . . 75

Laser Induced Fluorescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.2.1

Calibration of LIF Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5 Stark Deceleration

83

5.1

Decelerator Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.2

Loss at Low Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.3

Distributed Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.4

Decelerator Overtones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.5

Quadrupole-Guiding Decelerator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6 Magnetic Trapping of OH
6.1

122

Magnetoelectrostatic Trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.1.1

Design and Operation of the Magnetic Quadrupole Trap . . . . . . . . . . . . 124

xi
6.1.2
6.2

Trap Dynamics in Combined Electric and Magnetic Fields . . . . . . . . . . . 126

Permanent Magnet Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.2.1

Loading the Permanent Magnet Trap . . . . . . . . . . . . . . . . . . . . . . 136

6.2.2

Current Conditioning of Magnetic Electrodes . . . . . . . . . . . . . . . . . . 138

7 Collision Studies with Trapped OH Molecules
7.1

7.2

7.3

7.4

139

He-OH, D2 -OH Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
7.1.1

Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

7.1.2

Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

7.1.3

Recent Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

Cold Dipolar Collisions between OH and ND3 . . . . . . . . . . . . . . . . . . . . . . 146
7.2.1

Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

7.2.2

Cold OH-ND3 Collision Apparatus . . . . . . . . . . . . . . . . . . . . . . . . 150

7.2.3

OH-ND3 Collision Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

7.2.4

Experiment/Theory Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

Semiclassical Estimate of Elastic Cross Sections . . . . . . . . . . . . . . . . . . . . . 160
7.3.1

Eikonal Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

7.3.2

Model OH-ND3 Scattering Potentials . . . . . . . . . . . . . . . . . . . . . . . 161

7.3.3

Elastic Cross Section Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 162

Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
7.4.1

Quantifying State-to-State Inelastic Cross Sections . . . . . . . . . . . . . . . 164

7.4.2

Orientation-Dependent OH-ND3 Collisions . . . . . . . . . . . . . . . . . . . 165

Bibliography

167

Appendix
A List of 250 ND3 REMPI Transitions

174

xii

TablesTable
2.1

OH Rotation and Fine-Structure Constants (obtained from Ref. [1]) . . . . . . . . . 15

2.2

OH Vibration Constants (obtained from Ref. [2]). The constant Te is deﬁned at theenergy minimum of the given electronic potential. . . . . . . . . . . . . . . . . . . . . 16

3.1

Values of the clustering constant k for a range of atoms and molecules (obtained
from Ref. [3]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.1

Typical and best detected OH (2 Π3/2 ) signals under various experimental conditions.
All numbers shown are for detection within the permanent magnet trap. Quoted
errors are standard deviations from a number of diﬀerent measurements. . . . . . . . 82

xiii

FiguresFigure1.1

Schematic view of dipolar scattering as a function of collision energy in units of thedipole energy, UD . At ultracold temperatures (U < UD ), the Born approximation isvalid and cross sections are independent of temperature. At energies above UD , thecross section decreases as the inverse of the relative collision velocity. This plot is
adapted from a more accurate numerical calculation presented in Ref. [4]. . . . . . .

2.1

4

Illustration of the ﬁve Hund’s Cases for angular momentum coupling in diatomic
molecules. (a) Both L and S are strongly coupled to the internuclear axis (L due
to electrostatic forces and S through the spin-orbit coupling), forming the quantum
number Ω = Λ + Σ along the axis. (b) Spin S is weakly coupled to the axis due
to small or absent (Λ = 0) spin-orbit mixing but instead couples to the resultant
K = Λ + N. (c) Neither L nor S are strongly coupled to the internuclear axis, butthese vectors add to form Ja whose projection on the axis is given by Ω. (d) Orbital
angular momentum L couples more strongly to nuclear rotation N than to the axis.
(e) The vector Ja =L+S results from strong spin-orbit mixing and couples to N. . . 10

2.2

Illustration of the rotational structure of the OH electronic and vibrational groundstate. The Λ-doublet splitting of each rotational level has been exaggerated forclarity. The (e,f) quantum numbers represent the symmetry of the given Λ-doublet
component, while the ± denotes the overall J-dependent parity of the state. . . . . . 14

xiv
2.3

Illustration of the dipole-allowed P1, Q21, and Q1 transitions for 2 Π3/2 OH moleculeswith corresponding measured LIF spectra of a supersonic beam. Generally, the∆v = 1 transition is used to allow ﬁltering of the 282 nm excitation light from the
313 nm ﬂuorescence photons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4

Ground state (X̃ 1 A′1 ) structure of deuterated ammonia showing the inversion doublet(expanded for clarity) of each rotational level. . . . . . . . . . . . . . . . . . . . . . . 20

2.5

Illustration of three intermediate two-photon transitions used in 2+1 REMPI spectroscopy of ND3 . Vibrational selection rules result from the diﬀerent (alternating)symmetries of the excited vibrational levels. The structure of ND3 transitions frompyramidal to planar after the two-photon excitation. The rotational structure is
excluded for clarity.

2.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Simulated 2+1 REMPI spectrum of 250 rotational transitions in ND3 . The simulated
rotational temperature is given for each plot. For consistency of presentation, aLorentzian width of 0.3 cm−1 has been assumed for all spectral lines. Note that at
high temperatures (& 100 K), Doppler broadening necessitates application of a Voigt
lineshape to correctly ﬁt experimental spectra. . . . . . . . . . . . . . . . . . . . . . 28

2.7

Stark shifts in both the OH and ammonia (NH3 /ND3 ) molecules. Coincidentally,
OH and ND3 have nearly identical parity doublets and permanent dipole moments
which gives them similarly large static polarizabilities. Partly for this reason, many
cold molecule experiments have made use of these two species. . . . . . . . . . . . . 30

2.8

Theoretical (black line) and measured (red point) doublet population ratios as afunction of external electric ﬁeld. For this measurement, we perform 2+1 REMPIof a state-selected beam of ND3 molecules while applying +950 V to one of the trap
magnets in the ionization region. Following Eq. 2.36, the ratio of lower- to upperinversion doublet populations allows accurate characterization of this applied electricﬁeld. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

xv
2.9

Stark shift of the 2 Π3/2 ground state of OH in the hyperﬁne basis. The two Λdoublet states repel as the ﬁeld is increased, creating strong- and weak-ﬁeld seeking
molecules. For electric ﬁelds above ∼1 kV/cm, the structure approaches the strongﬁeld limit of Fig. 2.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.10 Zeeman shift of the 2 Π3/2 ground state of OH. Since the Zeeman Hamiltonian only
couples states of like parity, the two Λ-doublet levels of OH show exactly the sameﬁeld dependence. Also, the lack of coupling between e and f states causes only
diabatic level crossings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.11 Zeeman shift of 2 Π3/2 OH molecules in the hyperﬁne basis. Both Λ-doublet statesshow the same eﬀect since the magnetic ﬁeld does not induce an interaction betweene and f. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.12 Level shifts for OH (2 Π3/2 ) in combined electric and magnetic ﬁelds as a function ofrelative ﬁeld angle β. The β = 0◦ , 180◦ shifts are identical and plotted together. To
keep track of the diabatic curves, it is useful to start from the far left of each graphwhere only the electric (upper plots) or magnetic (lower plots) ﬁeld dominates. Of
course, the adiabatic curves maintain energy ordering. . . . . . . . . . . . . . . . . . 42
2.13 (a) Diabatic levels for ground state OH molecules. Upper-doublet (black) and lowerdoublet (red) levels are completely uncoupled in this picture. (b) Color-coded adiabatic levels resulting from coupling between e and f parity states due to an external
electric ﬁeld. (c) Adiabatic transition probability (Pad ) for OH molecules within combined magnetic and electric ﬁelds. The probabilities shown are for coupling between
the | 23 , 32 , + 23 , e⟩ and | 32 , 32 , + 12 , f ⟩ states as a function of electric ﬁeld and ﬁeld angle
β. Note that extremely small electric ﬁeld magnitudes (< 10 V/cm) are suﬃcient to
induce adiabatic transitions in this system. . . . . . . . . . . . . . . . . . . . . . . . 44

xvi
3.1

(Top) Illustration of a typical supersonic beam assembly featuring a gas reservoir atpressure and temperature P0 and T0 , respectively and supersonic nozzle admitting
the gas into a vacuum chamber at background pressure Pb . The skimmer is shownat right. (Bottom) Plot of predicted beam velocities for diﬀerent noble carrier gases
initially at room temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2

Drawing of the PZT valve assembly showing the adjustable poppet, disk piezo translator, and valve housing. The Kr/H2 O bubbler system for production of OH via
pulsed electric discharge is also shown. (Inset) Measurement of the high voltage
pulse applied to the PZT. It is critical that the RC time constant of the voltage
pulse be & 40 µs to avoid cracking the piezoelectric crystal. . . . . . . . . . . . . . . 54

3.3

Comparison of a single-pulse electric discharge to a multi-pulse discharge of roughly
the same duration. The three separate OH pulses spatially combine in free-ﬂight tothe detection region and yield > 50% improvement in beam density. The above data
was taken 1 m downstream from the supersonic valve. . . . . . . . . . . . . . . . . . 57

3.4(Top) Illustrations of the pulsed electric discharge and photolysis sources for OH
supersonic beam production. Note the 40◦ exit nozzle cut into both sources forenhanced OH beam density. (Bottom) Plot of relative OH yield for the discharge andphotolysis sources measured within the source chamber. The photolysis precursor
for this data was 99% pure HNO3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.5

Illustration of our buﬀer gas beam assembly. The ND3 gas is fed down a heatedcopper tube at 285 K. This hot supply tube is thermally isolated from the cold
buﬀer gas cell via a polyetherimide (Ultem) insert and thin-walled epoxy-ﬁberglass
composite (G10) tube. The pre-cooled He gas (not shown) is fed into the side of
the cell. Charcoal sorb glued to copper plates (∼ 2000 cm2 total) acts as a vacuum
pump for He and ND3 gas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

xvii
3.6

Data comparing electrostatically guided ﬂux of a 15 K Ne/ND3 beam with that of
a 5 K He/ND3 under otherwise identical conditions. Flow rates for both conditions
were varied over the full range of the ﬂow controllers in the time shown, and thecell-to-guide distance was ﬁxed at 1 cm. . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.7

Measured spectra of a guided buﬀer gas source of ND3 under two diﬀerent buﬀer
gas ﬂow conditions. The red curve gives a para rotational temperature of 6.3 K for
buﬀer as ﬂows of 2.0 sccm He and 2.5 sccm ND3 . The black spectrum reﬂects thebetter rotational thermalization observed with ﬂows of 3.5 sccm He and 1.0 sccm ND3 . 66

3.8

Comparison of pulsing behavior with (a) < 5 mm spacing between the ammonia
inlet and cell wall and (b) the larger 3.8 cm x 3.8 cm x 4 cm cell with an inlet-to-wall
spacing of 1 cm. The greater distance between the warm ND3 inlet and the cold
cell wall in (b) leads to signiﬁcant improvement in beam stability over ∼ 1 hr. (c)Illustration of the buﬀer gas cell cross section showing regions of ND3 ice buildupthat eventually lead to pulsing as ice accumulates near the warm ND3 inlet tube. . . 68

3.9

Drawing of the buﬀer gas beam assembly and electrostatic velocity ﬁlter for cold ND3molecules. The ionization region and time-of-ﬂight mass spectrometer assembly forND+
3 detection are displayed at the terminus of the hexapole guide. Cold ND3 isionized 1 cm from the guide exit with the 317 nm pulsed REMPI laser.

. . . . . . . 69

3.10 Measured velocity distribution of electrostatically guided ND3 molecules. The process of obtaining g(v) from the measured f (vmin ) is described by Eq. 3.17. . . . . . . 72
4.1

Top-view illustration of our ToFMS for detection of guided ND3 molecules within
the OH-ND3 collision region. For initial ND+
3 extraction, a pulse of +950 V isapplied to the front magnet after the ionization laser. Ions are then accelerated tothe microchannel plate detector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

xviii
4.2(a) Measured (black points) and simulated (red line) ND3 spectra at 295 K. The
experimental 2+1 REMPI spectrum was taken within the OH-ND3 collision region.
The relative strengths and line positions for two of the |J ′′ , K ′′ ⟩ = |1, 1⟩ transitions
at the single-photon frequencies of 31505.6 cm−1 and 31510.9 cm−1 are shown inblue. (b) Measured (black points) and simulated (red line) spectrum of a guided
ND3 beam with a rotational temperature of 8 K. . . . . . . . . . . . . . . . . . . . . 78

4.3

(Left) Side-view illustration of the Stark decelerator, magnetic trap, and OH laserinduced ﬂuorescence detection assembly. Fluorescence photons are collected at 313
nm with a 2.5 cm-diameter lens placed 2.5 cm from the center of the magnetic trap.
(Right) Level structure showing the excitation and decay pathways used for LIF
detection of OH molecules.

5.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

Illustration of the Stark decelerator and pulsed supersonic molecular beam source.
A packet of OH is created at the valve using a pulsed electric discharge positioned
at the valve nozzle. After passing through the skimmer 10 cm downstream fromthe discharge plates, the packet is focused into the decelerator via a 5 cm hexapole
with an inner diameter of 6 mm. The decelerator rod pairs are spaced by 5.5 mm
center-to-center along the beam ﬂight (longitudinal) direction and are mounted 2
mm apart in the transverse dimension. . . . . . . . . . . . . . . . . . . . . . . . . . . 85

xix
5.2

(a) Simulation of the longitudinal Stark potentials for OH created by the two distinct
rod pairs within the decelerator. To slow a packet of molecules, we switch between
the red and black potentials to continually remove kinetic energy at a well-deﬁned
position relative to the rod pair (ϕ0 ). (b) The diﬀerence between the two Stark
potentials shown in panel (a), which represents the amount of kinetic energy removed
from OH molecules in one stage as a function of the ﬁxed slowing phase angle ϕ0 .
Stable acceleration occurs for −90◦ < ϕ0 < 0 while phase-stable deceleration isobserved for 0 < ϕ0 < 90◦ . We denote operation at ϕ0 = 0◦ as “bunching” sinceno net kinetic energy is removed from the molecular packet, but the molecules arenevertheless conﬁned within a moving longitudinal potential well. The form of the
kinetic energy diﬀerence closely follows that of a sine function as demonstrated by
the ﬁt curve (dashed line). (c) Longitudinal phase-space acceptance of a deceleration
sequence as a function of ϕ0 assuming a one-dimensional sine-function form for theslowing potential of panel (b). Note that the separatrices shown are incremented by10◦ starting from 0◦ at the outer edge. Molecules with longitudinal velocities andpositions within the separatrix for a given ϕ0 will be phase-stably decelerated. . . . . 87

5.3

Time of ﬂight data (black) and Monte Carlo simulation results (red) for Stark decelerated OH molecules showing the excellent agreement between ﬁnite-element calculations of slowing electric ﬁelds and real decelerating ﬁelds. Note that as the phaseangle is increased, the decelerated packet is continually removed from the central
background of unstable molecules in time of ﬂight. for clarity, the two bottom panels show only the decelerated packet. . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

xx
5.4

(a) Simulations of the phase-stable molecule number as a function of stage number
in our 142-stage decelerator. Note the dramatic decrease in number in the last
several stages for ϕo = 30.43◦ . This decrease is due to transverse overfocusing andlongitudinal reﬂection of these slow (14 m/s) molecules. (b) Simulated transverse(trace 1) and longitudinal (trace 2) fractional loss rates as a function of time within
the ﬁnal three stages at ϕo = 30.43◦ . The vertical dashed lines denote the times of
the given stage switches. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.5

Experimental (dots) and Monte Carlo simulation (open circles) results for total
molecule number as a function of ﬁnal velocity. The dashed curve is the expected
decelerator eﬃciency calculated from the one-dimensional (1D) theoretical model ofRef. [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.6

Monte Carlo simulation results for the longitudinal phase space of decelerated molecules.
The left column shows ϕo = 0◦ and 26.67◦ for S = 1 slowing, while the right column
shows ϕo = 0◦ and 80◦ for S = 3 deceleration. The factor of three between S = 1 andS = 3 phase angles ensures that molecules have roughly the same ﬁnal velocities.
The observed velocity diﬀerence at the higher phase angle occurs because the 142
stages of our slower is not a multiple of three. Note that, although the S = 3 phaseplot is more densely populated than that of S = 1 at ϕo = 0◦ , its phase-space acceptance decreases dramatically relative to S = 1 at the lowest velocities. All plots are
generated using an identical initial number of molecules, and therefore the densityof points is meaningful for comparison.

5.7. . . . . . . . . . . . . . . . . . . . . . . . . 98

Experimental results from changing the voltage on decelerator rods (squares) anddecreasing the eﬀective decelerator length (circles). Eﬀective slower length is modiﬁed by initially operating the decelerator at ϕ0 = 0◦ , S = 3, then slowing with S = 1
to vf inal = 50 m/s for the number of stages labeled. Both curves illustrate thattransverse/longitudinal couplings are strongly dependent on phase angle, and havea marked eﬀect on decelerator eﬃciency.

. . . . . . . . . . . . . . . . . . . . . . . . 100

xxi
5.8

Deceleration schemes. (a) Potential energy shift of polar molecules in the Stark
decelerator. The dotted (blue) curves show the potential energy shift when thehorizontal (circular cross section) electrodes are energized, while the dashed (red)curves show the potential energy shift when the vertical (elongated cross section)
electrodes are energized. Deceleration proceeds by switching between the two setsof energized electrodes. In panels (b)-(d), the thick black line indicates the potential
experienced by the molecules. The empty circles indicate a switching event. (b)Traditional (S = 1) operation at ϕo = 0◦ . For phase stability, the switching always
occurs when the molecules are on an upward slope, and as such the molecules arenever between a pair of energized electrodes. Thus, the maximum transverse guiding
is never realized. (c) First overtone operation (S = 3) at ϕo = 0◦ . By switching at
one-third of the S = 1 rate, the molecules are allowed to ﬂy directly between an
energized electrode pair, and thus, experience enhanced transverse guiding. (d)Optimized ﬁrst overtone operation (S = 3+) at ϕo = 0◦ : Initially, the packet rises
the Stark potential created by one set of electrodes. When the molecules reach the
apex of this potential, the alternate set of electrodes is energized in addition. In thisway, the molecules experience one more stage of maximum transverse guiding foreach slowing stage. Note that, to minimize the un-bunching eﬀect, the grounded-set
of electrodes is switched on when the molecules are directly between the energizedelectrodes.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

xxii
5.9

Comparison of deceleration using S = 3 versus S = 1. (a) Experimental ToF dataof decelerated OH packets with S = 3 (top) and S = 1 (bottom). Note the factor ofthree between S = 1 and S = 3 phase angles. (b) De-convolved, integrated molecule
number for S = 3 (trace 2) and S = 1 (trace 1) for the packets shown in panel (a).
(c) Simulated transverse loss rate per stage for S = 1 (trace 1) and S = 3 (trace 2)
deceleration. As expected, S = 1 results in larger transverse loss rates throughout.
(d) Calculated stable longitudinal phase-space area for S = 1 (trace 1) and S = 3(trace 2), with initial points scaled to the experimental ratio of 2.75. The above
panels highlight that the observed shortcoming of S = 3 deceleration is entirely due
to loss of longitudinal velocity acceptance at high phase angles. . . . . . . . . . . . . 105

5.10 Monte Carlo results for decelerated molecule number using S = 3 and ϕo = 20◦ versus
ﬁnal velocity. The number next to each data point is the number of stages used.
Because of transverse overfocusing and longitudinal velocity ﬁltering, essentially nomolecules survive below 100 m/s.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.11 (a) Experimental ToF data of decelerated OH packets produced using the S = 3+
modiﬁed overtone. Also shown for comparison is the experimental bunching packet
for operation at S = 1. (b) The de-convolved, integrated molecule number calculated
from S = 3+ (open squares), S = 3 (open circles), and S = 1 (dots) data. . . . . . . 107
5.12 Monte Carlo results of decelerated molecule number using S = 3+ and ϕo = 20◦
versus ﬁnal velocity. The number next to each data point is the number of stages
used. Because of transverse overfocusing, essentially no molecules survive below 100
m/s.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.13 Quadrupole-guiding decelerator. (a) Schematic of QGD. (b) Electric ﬁeld of quadrupole
guiding stage energized to ±12.5 kV. (c) Switching scheme for deceleration with the
QGD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

xxiii5.14 (a) The Stark shift of an OH molecule in the |2, ±2, +⟩ state inside the QGD. The
solid curve is the Stark shift due to the slowing electrodes, while the dashed curveis the Stark shift due to the electrodes that will be energized at the switching time.
(b) The change in the molecule’s kinetic energy as a function of position is shown(solid) as well as a ﬁt of Eq. 5.4, including up to n = 3 (dashed). The solid curve iscalculated from the subtraction of the two curves in panel (a).

. . . . . . . . . . . . 113

5.15 The left column is stable phase space of molecules decelerated inside the QGD.
The solid line is the separatrix predicted by the theory, while the points represent
positions of molecule in the 3D Monte Carlo simulations. The right column shows
the ToF spectra of OH molecules in the |2 ± 2, +⟩ state at the exit of this decelerator
which has 142 deceleration stages, along with 142 quadrupole stages.

. . . . . . . . 114

5.16 (a) The energy removed per stage as a function of phase angle for both traditional
deceleration and deceleration with a QGD. Both curves are calculated for OH in the|2, ±2, +⟩ state and scaled down by 1.76 cm−1 at ϕo = 90◦ . (b) The calculated longitudinal phase-stable area for deceleration versus energy loss per stage for traditional
deceleration and deceleration with a QGD is plotted on the left axis. The gain of
the QGD over traditional deceleration is plotted on the right axis. Note that for agiven energy loss the gain in phase stable area due to the larger volume of the QGD
is ≤1.5 for all 0◦ < ϕo < 80◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

xxiv
5.17 Simulations of the phase-stable molecule number as a function of stage number inthe QGD and S = 1 decelerator. The voltage on the quadrupole stages of the
QGD is held constant throughout the deceleration sequence. All simulation data
is for vf inal = 14 m/s. The traces shown are S = 1 deceleration at ϕo = 30.43◦ ,QGD operated with ±1 kV on the quadrupoles, and QGD operated with ±3 kV on
the quadrupoles. Note the decrease in stable molecule number in the last several
stages for the QGD results. This decrease is due to transverse overfocusing of theslow molecules through the ﬁnal few quadrupole stages, and suggests that scaling of
quadrupole voltage is necessary.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.18 Monte Carlo simulation results for the decelerated molecule number using traditional
S = 1 deceleration (ϕ0 = 30.43◦ ) and deceleration using a QGD (ϕ0 = 52.75◦ )with a dynamic voltage scaling of (v/vinitial )0.875 . For both curves, 142 stages ofdeceleration were used, and vf inal = 14 m/s. The diﬀerent phase angles chosen for
the two decelerators are a result of their diﬀering potential proﬁles for deceleration.
The vertical dashed lines represent the deceleration stage at the given velocity. Notethat when the quadrupole voltage within the QGD is scaled in this manner, weobserve a 40% gain in molecule number at 14 m/s, and a factor of 5 gain over S = 1
at higher velocities.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.19 S = 1 simulation results for vf inal = 80 m/s (ϕ0 = 5.22◦ ) plotted along with simulation results using the voltage-scaled QGD to the same vf inal (ϕ0 = 23.5◦ ). The
number labeling each vertical dashed line is the number of deceleration stages necessary to reach the given velocity. Note the large number of stages (803) used toreach 80 m/s, which suggests that a very long QGD may be employed for slowing
molecules with a poor Stark shift to mass ratio.

. . . . . . . . . . . . . . . . . . . . 120

xxv
6.1

(a) Illustration of the magneto-electrostatic trap (MET) assembly. The terminus of
the Stark decelerator (yellow, gray electrodes), shown to the left, couples stateselected cold OH molecules into the magnetic quadrupole (blue).

The double-

electrode structures within the electric quadrupole (red) allow for application of
uniform electric ﬁelds within the magnetic trap. The ring (green) supports a 1 inchlens for collection of laser-induced ﬂuorescence. (b) Plots of the magnetic ﬁeld distributions within the MET for both stopping and trapping of decelerated OH molecules.123
6.2

Schematic of the high-current switching circuit used to load Stark decelerated OH
into the MET. The actual switching electronics consisted of 24 of these driver circuits
operating 24 individual MOSFETs. Twelve MOSFETs in parallel carried the 2000
A current during the loading step while the other 12 where switched into the circuit
to carry the 1500 A trapping current.

6.3. . . . . . . . . . . . . . . . . . . . . . . . . . 127(a) Zeeman and (b) Stark eﬀects of the ground state structure of OH. (c) Magnetic
and (d) electric quadruple ﬁelds viewed from ẑ. (e) Side view of the MET conﬁguration with quadrupole E and B ﬁelds. (f) Adiabatic potential surfaces for various
components of OH ground state at the longitudinal MET center. The top surface
depicts the decelerated/trapped |J = 23 , mJ = 32 ⟩ state. . . . . . . . . . . . . . . . . . 129

6.4

Time-of-ﬂight data (points) and Monte Carlo simulation results (solid red line) for
two diﬀerent electric ﬁeld conﬁgurations. The magnetic ﬁeld switches from thestopping conﬁguration to quadrupole trapping at t = 2.65 ms. (a) Magnetic trap
only. (b) Stopping and trapping in the presence of combined electric and magnetic
quadrupole ﬁelds. Note the larger steady-state trap population when the electric
quadrupole is applied. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6.5

Measured (τ ) and de-convolved collisional (τc ) lifetimes of magnetically trapped OH
at a background pressure of 1 × 10−6 Torr (•, red line) and 4 × 10−8 Torr (◦, blueline). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

xxvi
6.6

(a) Illustration of the permanent magnet trap and Stark decelerator. (b) Results ofﬁnite-element calculations of the magnetic ﬁeld distribution within the permanent
magnet trap. The small “lobe” observed in the ρ-dimension appears at the 2 mm
inner radius of the magnet ring. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

6.7

Illustration of the trap loading sequence. (a) High voltage is applied to the surfacesof the two permanent ring magnets 1 µs after the ﬁnal deceleration stage (shownat left) is grounded. The OH packet is stopped directly between the two magnets
by the electric ﬁeld gradient in 400 µs. The stopping potential due to the applied
electric ﬁeld is depicted. (b) The magnet surfaces are grounded, leaving the packet
trapped within the displayed permanent magnetic quadrupole potential. . . . . . . . 137

6.8(a) Time-of-ﬂight data (circles with error bars) and three-dimensional Monte Carlosimulations (solid line) corresponding to OH trap loading. Stopping E-ﬁelds are
switched oﬀ at 400 µs, which leaves 50% of the stopped OH molecules trapped in thepermanent magnetic quadrupole. (b) Measurement of the lifetime of OH trapped
within the magnetic trap at a background pressure of 7.5 × 10−9 Torr. A singleexponential ﬁt (solid line) of the data yields 432 ± 47 ms. . . . . . . . . . . . . . . . 137

7.1

Illustration of the low-temperature He–OH, D2 –OH collision apparatus consisting ofa Stark decelerator and permanent magnetic trap for OH (background) as well as atemperature-controlled pulsed solenoid valve and skimmer assembly for He and D2
beams (foreground). Collisions between the external He or D2 beams and trapped
OH result in measurable OH trap loss. . . . . . . . . . . . . . . . . . . . . . . . . . . 141

xxvii
7.2(a) Measurement of He and D2 beam velocities as a function of valve nozzle temperature. The beam speeds were measured with a miniature microphone and fastionization gauge placed 13 cm apart within the magnetic trap chamber. The solidlines are ﬁts to the data indicating that, as expected, the measured beam velocity
is proportional to the square root of nozzle temperature. (b) Calibration of He andD2 beam density as a function of valve nozzle temperature. The relative density
between He and D2 is obtained by taking into account the diﬀerent sensitivity of thefast ionization gauge to these two species. . . . . . . . . . . . . . . . . . . . . . . . . 143

7.3(a) Time dependence of OH trap loss due to collisions with a supersonic He beam.
Trap density drops sharply over 1 ms upon He beam collisions, then remains constantover the time scale shown. The valve is triggered 20 ms after the magnetic trap
is loaded. (b) Total collision cross sections for He–OH (open circles) and D2 –OH(squares) as a function of Ecm . The decrease in the He–OH cross section at low
energy is attributed to reduced inelastic loss as Ecm drops below the 84 cm−1 splitting
between the J = 3/2 and J = 5/2 states of the OH molecule. . . . . . . . . . . . . . 145

xxviii
7.4(a) Rotational structure of OH and ND3 molecules. The parity-doublet splitting ofeach rotational level has been expanded for clarity, while the black and red parity
levels are the ones included in the OH-ND3 scattering calculation. The upper, red,parity states are those selected by the experimental apparatus for the cold collisions
described herein. The Stark and Zeeman shifts of ground state OH and ND3 arepresented in Chapter 2. (b) Illustration of the Jacobi coordinates used for OH-ND3
collision calculations. We deﬁne R as the distance between the molecular centers ofmass while the Euler angles (θA , ϕA ) and (θB , ϕB ) give the orientation of the OH
and ND3 axes, respectively, in the body-ﬁxed frame relative to R. The coordinate
ψB speciﬁes the orientation of the ND3 monomer within the OH-ND3 complex. The
geometry ϕA = ϕB = θA = θB = 0 corresponds to the OH axis lying in the σv
symmetry plane of ND3 with the O atom facing the plane of D atoms. (c) Contour′′

plot of the lowest adiabatic potential energy surface for the A state of the OH-ND3
collision complex at R = 3.1 Å and ϕA = ϕB = 0 as a function of θA and θB . The
color legend is scaled in units of cm−1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
7.5

Illustration of the combined Stark decelerator, magnetic trap, and buﬀer gas beam
assembly. The curved hexapole ﬁlters cold ND3 from the He buﬀer gas and guides
the continuous beam to the OH magnetic trap. (Inset) Closeup of the trap assembly showing the dual-species detection components. We detect OH and ND3 inthe collision region using laser-induced ﬂuorescence (LIF) and resonance-enhanced
multiphoton ionization (REMPI), respectively. Hydroxyl ﬂuorescence at 313 nm iscollected using a lens mounted 2.5 cm below the magnetic trap center. Ionized ND+
3molecules are accelerated to a microchannel plate (MCP) detector by placing 950
V on the front magnet, 0 V on the back magnet, and -1100 V on the acceleration
plates that make up a time-of-ﬂight mass spectrometer (ToFMS). . . . . . . . . . . . 149

xxix
7.6

(a) Translational energy spectrum of the guided continuous ND3 beam as measured
by 2+1 REMPI in the collision region. (b) Rotationally-resolved REMPI spectrum ofguided ND3 molecules showing diﬀerent J, K states. The upper red curve (oﬀset forclarity) was taken at buﬀer gas ﬂows of 2.0 and 2.5 sccm of He and ND3 , respectively.
The lower black curve displays the smaller rotational temperature observed at 3.5
and 1.0 sccm. Intermediate rotational levels in the excited B̃(v2 = 5) electronic state
are labeled in parentheses for each transition. . . . . . . . . . . . . . . . . . . . . . . 151

7.7(a) Semi-logarithmic plot of OH trap decay rates with (◦) and without (•) the colliding ND3 beam. The decay rate due solely to cold OH-ND3 collisions is γcoll =γon − γoﬀ . (b) Plot of all experimental runs measuring total cross sections with ()and without () a polarizing electric ﬁeld. Average cross sections are determinedfrom the weighted mean of all points and errors for the given E-ﬁeld condition. The
cross-hatched regions represent one statistical standard error. Individual error bars
for each experimental run are obtained by propagating the standard deviation fromexponential ﬁts of γon and γoﬀ to γcoll . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

xxx
7.8

(a) Plot of theoretical OH-ND3 elastic (σel ) and inelastic (σin ) cross sections overthe experimental collision energy range. (Inset) Theoretical and experimental trap
loss cross sections, with numbers in parentheses representing combined statisticalloss are shown for bothand systematic errors. Experimental loss cross sections σexp
loss
unpolarized and polarized colliding molecules. The σtheory
value includes the eﬀectof reduced elastic loss from glancing collisions due to trap conﬁnement. All crosssections are given in units of 10−12 cm2 . The quantities σ̄el and σ̄in are the theoreticalelastic and inelastic cross sections, respectively, in the absence of trapping potentials
and averaged over the experimental collision energy distribution of Fig. 7.6a. (b)Semi-logarithmic plot of the theoretical diﬀerential cross section for 5 K elastic OH2loss
ND3 collisions in units of Å /sr. This result is used to calculate σtheory
from the free-space elastic and inelastic cross sections of panel (a). The forward-peaked nature of
the elastic collision leads to a factor of 25 suppression of OH elastic loss at our ﬁnite
radial magnetic trap depth of 240 mK. . . . . . . . . . . . . . . . . . . . . . . . . . . 158
7.9

Plot of the Eikonal estimates for the elastic collision cross section between OH and
ND3 molecules. At the experimental relative velocity of 100 m/s, the electric ﬁeld is
expected to increase the elastic cross section by a factor of ∼ 6. . . . . . . . . . . . . 163

Chapter 1

Introduction

1.1

Why Polar Molecules?
Tremendous progress in the ﬁeld of laser cooling and trapping of neutral alkali atoms has al-

lowed observation of exotic ultracold phases of matter [6, 7, 8], realization of fundamental condensedmatter systems [9], and precision measurements [10]. Interestingly, these novel physical systems
rely on purely isotropic interatomic potentials. Nevertheless, the ability to exert control overatomic interactions via Feshbach resonances has proven invaluable [11]. Inspired by these successes, researchers began to imagine the possibilities for cold or ultracold systems in the presenceof anisotropic inter-particle interactions [12]. Such anisotropy has been realized in atomic species
with large magnetic dipoles [13], but polar molecules can also provide this anisotropy due to theirdipole-dipole interaction potential in the presence of an external electric ﬁeld that takes the form
U (r) =

1 µ2 [1 − 3(µ̂ · r̂)2 ]
4πϵ0
r3

(1.1)

where µ is the molecular permanent electric dipole moment, r is the intermolecular distance, and r̂ isa unit vector along this intermolecular spacing. Immediately, we see that an ensemble of molecules
mutually aligned in a homogeneous electric ﬁeld will experience attractive head-to-tail interactions
but repulsive side-to-side interactions. Furthermore, compared to the contact interactions withdelta-function potentials between colliding ultracold atoms, polar molecules display a long-range
r−3 potential that signiﬁcantly alters their collision dynamics. In an ultracold dipolar molecularsystem, this long-range interaction may even be exploited to perform quantum computation [14, 15]2or, in the not-too-distant future, direct simulation of quantum Hamiltonians [16]. In the following
sections, two applications of cold and ultracold neutral polar molecules are brieﬂy discussed –namely, dipolar collisions and precision measurements.

1.1.1

Cold and Ultracold Collisions

The expectation value of the molecular electric dipole moment is entirely dependent on themagnitude of an applied external ﬁeld (see Chapter 2). As such, polar molecules not only oﬀer
anisotropic interactions in an electric (E) ﬁeld; they present an opportunity for direct control of
the intermolecular potential through the application of E-ﬁelds as small as 1 kV/cm. However, therelative inﬂuence of the long-range portion of the intermolecular potential depends on the collisionenergy of the interacting molecules. In Ref. [4], Bohn et al deﬁne convenient energy (UD ) andlength (D) scales for two colliding polar molecules with dipole moments, µ1 /µ2 , and reduced mass,
M , as (in atomic units)
D = M µ1 µ2
UD =

1

.
M 3 µ21 µ22

(1.2)
(1.3)

Inserting appropriate values for the OH molecule (⟨µ⟩ = 1.0 D, M = 8.5 amu), one obtains
OH = 3.6 µK and D OH = 2400 a (a is the Bohr radius). At energies below U , the collision
UD
0
0
D

cross section is dominated by a single partial wave that is either an s (l = 0) or p (l = 1) wave
depending on the quantum statistics of the colliding molecules. In this ultracold regime, the elastic
cross section due to the dipolar interaction is energy-independent and scales as µ4 . As the collisionenergy is increased above UD , the molecular collision becomes semiclassical in nature as morepartial waves contribute to the cross section. We denote this as scattering in the cold regime. At
these higher temperatures, the elastic cross section decreases as U −1/2 and elastic collision rates
deﬁned as σv are energy-independent. Figure 1.1 provides a schematic depiction of the behavior ofcross sections in these two energy regimes.
The cold molecule labs at JILA are leading the ﬁeld with respect to cold and ultracold dipolar

3collisions. Recent experiments with ultracold samples of KRb molecules have demonstrated E-ﬁeld
dependent chemical reactions in the quantum collision regime [17], and future KRb experiments intraps of reduced dimensionality promise to directly probe the universal elastic scattering behavior
described in Ref. [4]. Also, as detailed in Chapter 7 of this Thesis and Ref. [18], we have demonstrated electric-ﬁeld enhancement of elastic collision cross sections between samples of polar OH
and ND3 molecules at a temperature of 5 K. The history of low-temperature dipolar molecularcollisions is just beginning to be written, and the future promises a host of new discoveries in theﬁeld of cold collisions and chemistry [12].

4

10

2

/ D ]

Ultracold

4

Cross Section [

Cold

1
2

/v

0.1
0.01

0.1

1

10

100

Collision Energy [U / U ]DFigure 1.1: Schematic view of dipolar scattering as a function of collision energy in units of thedipole energy, UD . At ultracold temperatures (U < UD ), the Born approximation is valid andcross sections are independent of temperature. At energies above UD , the cross section decreases
as the inverse of the relative collision velocity. This plot is adapted from a more accurate numericalcalculation presented in Ref. [4].

5
1.1.2

Precision Spectroscopy

Precision measurements represent another enticing application for cold polar molecules. With
the development of molecular Stark deceleration [19] and buﬀer gas cooling [20] techniques morethan a decade ago, it became possible to signiﬁcantly increase one’s interaction time with the species
of interest and, at the same time, reduce the sample temperature while simultaneously increasing
its rotational/vibrational state purity. Additionally, the rich internal structure of molecules provides a host of transitions from microwave to ultraviolet wavelengths with diﬀering sensitivity toa number of fundamental constants. This range of spectra within a given species can be used tocompare molecular transition frequencies over galactic timescales with the goal of constraining thetime evolution of fundamental constants [21]. By using diﬀerent molecular transitions as opposed
to diﬀerent atomic species, one can minimize systematic errors common to space-based measurements [22]. As described in the following sections, diﬀerent groups have made use of cold polar
species for precision molecular spectroscopy in both beams and trapped samples.

1.1.2.1

Cold Beams

Using Stark decelerated OH molecules coupled to a microwave cavity, our group has improved upon previous measurements of the main (∆F = 0)) and satellite (∆F = ±1) ground-stateΛ-doublet transitions by an order-of-magnitude [21, 23, 24]. By combining these measurementswith those of large-redshift OH megamasers of similar precision, we will be able to restrict the fractional variation of the ﬁne structure constant, α, to 10−6 over ∼ 1010 years. Furthermore, we havequantiﬁed the diﬀerential magnetic g-factor between the upper and lower Λ-doublet states [24]. The
100 mK temperature, low velocity, and high state purity of the Stark decelerated OH molecules
all contributed to these improved measurements of the Λ-doublet transitions. Through comparison
with our laboratory measurements, recent astrophysical OH spectra have hinted at possible temporal variation of the ﬁne structure constant over ∼ 2.9 × 109 yrs [25]. Additional astrophysicalatomic spectra have suggested possible spatial variation of α [26].

6With their internal electric ﬁelds as large as 100 GV/cm, heavy polar molecules can provide
enhanced sensitivity for fundamental symmetry tests like the search for an electron electric dipole
moment(eEDM) [27]. Two eEDM searches using beams of neutral polar molecules produced via
buﬀer gas cooling are currently underway. These experiments are separately probing metastablePbO (a(1)3 Σ+ ) [28] and ThO (H 3 ∆1 ) molecules [29]. Stark decelerators readily produce colder
molecular samples with larger state purity than buﬀer gas beams — generally at the expense ofmolecular density. However, given the large masses and small rotational constants of candidate
eEDM molecules, thus far only one such Stark deceleration experiment has been proposed [30].

1.1.2.2

Trapped Molecules

By conﬁning molecules in magnetic or electrostatic traps, one gains unprecedented interrogation times of ∼ 1 s with cold, state-selected molecular samples. The group of G. Meijer has directly
measured the 59 ms lifetime of vibrationally-excited (v = 1) OH molecules trapped within an electrostatic quadrupole, thereby reducing the uncertainty of the Einstein A-coeﬃcient to ∼ 3% [31].
In a similar experiment, the same group directly measured the decay ﬂuorescence from trapped
metastable CO molecules [32]. The group of J. Doyle has also performed direct vibrational lifetime
measurements on buﬀer gas cooled NH (3 Σ− , v = 1) molecules conﬁned within a magnetic trap [33].
These unique spectroscopy experiments with trapped cold molecules are likely just the beginningfor this relatively new research ﬁeld.

Chapter 2

Molecular Spectroscopy

Molecules possess rich internal structure that spans the electromagnetic spectrum from microwave to ultraviolet. In contrast to atoms, the ∼ 1Å separation of constituent nuclei allows
for rotations and vibrations at energies far below those of electronic excitations. While this complicated structure has precluded development of a general molecular laser cooling technique [34],it makes molecules ideal candidates for certain precision measurement [29, 35, 36, 37], quantum
information [38], and controlled collision experiments [17, 39, 40], as well as future experiments inquantum many-body physics [16]. The goal of this chapter is to acquaint the reader with the basic
molecular theory necessary to perform experiments with polar symmetric top molecules.

2.1

Hund’s Coupling Cases
Before delving into full molecular Hamiltonians, it is useful to review the Hund’s cases (a)-(e) that describe coupling of electronic and orbital angular momenta within diatomic molecules.
Although the ﬁve cases summarized here are idealizations, they provide a convenient starting point
for spectroscopy calculations. Generally, one will begin with a basis of angular momenta that isnearly diagonal considering appropriate coupling constants for the molecule of interest. Of course,
upon diagonalization of their Hamiltonian, many molecules will be best described by intermediate
coupling cases. For the following overview, I deﬁne angular momentum vectors as shown below:

8L,Orbital electronic angular momentumΛ,

Vector component of L along the internuclear axis

S,

Total spin electronic angular momentumΣ,

Vector component of S along the internuclear axis

N,

Nuclear rotational angular momentum

Ja , Vector sum of L, S in the absence of coupling to the internuclear axis

2.1.1

Ω,

Vector component of (L+S) along the internuclear axis

K,

Vector sum of Λ, NR,

Total angular momentum excluding nuclear, electron spin (J - S = L + N)

J,

Total angular momentum excluding nuclear spin (L + N + S).
Case (a)The ﬁrst Hund’s case deals with molecular states possessing a large spin-orbit coupling aswell as strong electrostatic coupling of L to the internuclear axis as illustrated in Fig. 2.1a. The
large spin-orbit coeﬃcient A ensures that S is also coupled to the internuclear axis. We may thendeﬁne the quantum numbers Λ and Σ as the projections of L and S along this axis, respectively.
These projections sum to form Ω, which assumes all values |Λ − Σ|, |Λ − Σ| + 1,...|Λ + Σ|. The
total angular momentum J is deﬁned as J = Ω + N and satisﬁes the condition J ≥ Ω, which isa direct result of Ω ⊥ N. The good quantum numbers are Λ, Σ, Ω, and J. As will be discussed
later, since Ω, Λ, and Σ may all take positive or negative values, these molecules possess so-calledΛ-doublets arising from a breaking of this projection degeneracy. Assuming a rigid rotor withrotational constant B = ~2 /2µr2 , where µ is reduced mass and r is internuclear separation, therotational energy of a given quantum state in this basis is
Erot = B⟨N2 ⟩= B[J(J + 1) − Ω2 ].
~2

(2.1)

Some good examples of Hund’s case (a) states are NO, ClO and BrO, all of which are 2 Πmolecules in their ground states satisfying A ≫ B. Our cold molecule of choice, OH, deviates from

9pure case (a) due to its small moment of inertia which yields A/B ≈ −7.5. In this intermediate
regime, Ω strictly ceases to be a good quantum number due to a non-negligible mixing of diﬀerentΩ states possessing the same J. Generally speaking, case (a) molecules move closer to case (b) as
J increases.

2.1.2

Case (b)When S is weakly coupled to the internuclear axis due to either weak or absent (Λ = 0)
spin-orbit mixing and Λ ̸= 0 is strongly coupled to the axis, a state is classiﬁed as Hund’s case(b). The simplest example of a case (b) molecule is one in a 2 Σ state. Molecular states with 2 Πsymmetry may also be included if J is suﬃciently large. Returning to hydroxyl, the electronically
excited A2 Σ+ state used for laser-induced ﬂuorescence detection is case (b). The good quantum
numbers for this case are Λ, N , K, and J, where K = Λ + N as shown in Fig. 2.1b. Case (b)molecules with Λ ̸= 0 exhibit what can be considered a ‘true’ Λ-doubling arising exclusively froma breaking of degeneracy between ±Λ projections.

2.1.3

Case (c)If the L · S term in the molecular Hamiltonian is suﬃciently strong so as to overcome any
electrostatic coupling of L to the internuclear axis, then a state exhibits case (c) coupling. Heavy
diatomics such as AsF and SO (both 3 Σ− ) are best described as case (c), where Λ and Σ are nolonger good quantum numbers but the projection Ω = Ja · R̂ = (L + S) · R̂ is still well-deﬁned.
As with cases (a) and (b), these molecules can possess parity-doublets known as Ω-doublets since
they originate from the diﬀerent Ω projections along R̂. The good quantum numbers are Ja , Ω,and J. In fact, one may consider case (c) a more general form of (a) and (b) where Λ and Σ arenot well-deﬁned.

10

(a)

(b)

J

J
K

N
L

S

Λ

S
N

L

Σ

Λ

Ω

(c)

(d)

J
Ja

N
S

S
L
R
J

N

L
Ω

Ja

(e)
J

N

Figure 2.1: Illustration of the ﬁve Hund’s Cases for angular momentum coupling in diatomic
molecules. (a) Both L and S are strongly coupled to the internuclear axis (L due to electrostaticforces and S through the spin-orbit coupling), forming the quantum number Ω = Λ + Σ along theaxis. (b) Spin S is weakly coupled to the axis due to small or absent (Λ = 0) spin-orbit mixing
but instead couples to the resultant K = Λ + N. (c) Neither L nor S are strongly coupled to theinternuclear axis, but these vectors add to form Ja whose projection on the axis is given by Ω. (d)Orbital angular momentum L couples more strongly to nuclear rotation N than to the axis. (e)The vector Ja =L+S results from strong spin-orbit mixing and couples to N.

11
2.1.4

Cases (d) and (e)In this section, I combine the discussion of Hund’s cases (d) and (e) molecules due to boththeir similarity and rarity. Both cases are characterized by a loss of Λ, Σ, and Ω as good quantum
numbers. Shown in Fig. 2.1d, case (d) molecules exhibit a strong L · N coupling term forming theresultant R. Total angular momentum is then deﬁned as J=R+S, and the good quantum numbers
are L, N , R, S, and J. Rydberg molecules, deﬁned by their large principal quantum number,possess very small valence-core interactions and may be considered case (d). If the spin-orbit termin the Hamiltonian is large enough to render L and S undeﬁned, we move to a ﬁnal case (e) couplingin which only Ja , N , and J are well-deﬁned. Molecules possessing case (e) states are rare, with the
only known example being the exotic vibrationally-excited HeKr+ .

2.2

Spectra of Polar Symmetric Tops
Molecular rotations may be described by three moments of inertia (Ix , Iy , Iz ) about threeorthogonal principal axes (x̂, ŷ, ẑ). If two or more of these moments are identical (e.g. all diatomics
and molecules of the form XY3 ), the molecule is a symmetric top. If all three moments are identicalas in C60 , a molecule is a spherical top. In this section, I will describe the structure of two symmetric
top molecules that we have used in cold collision experiments - namely OH and ND3 . These two
species provide a nice overview of molecular Hamiltonians since both require non-trivial calculations
to get an accurate view of their spectroscopy. For ND3 , this is especially true in the case of threephoton ionization which is commonly used for detection.
When a molecule’s electron distribution is asymmetric along a particular dimension, it possesses a permanent electric dipole moment and is dubbed ‘polar.’ The simplest examples of polar
molecules are the heteronuclear diatomics (e.g. OH, HCl, SrO, etc.), wherein electrons are more
likely to be found near the most electronegative atom. The most common unit for molecular dipolesis the Debye (D), where 1D = 0.39ea0 = 3.33 × 10−30 Cm (e is the electron charge and a0 = 0.5Å is
the Bohr radius). Perhaps a more familiar unit to physicists is the atomic unit ea0 , but the Debye

12
was originally deﬁned in cgs units as 10−18 statC cm, or the dipole moment of two opposite chargesof 10−10 statC = 3.3 × 10−20 C at a separation of 1Å. The actual expectation value of the dipole
moment is state-dependent and will be discussed later in the context of the molecular Stark eﬀect.

2.2.1

Hydroxyl Radical (OH)The hydroxyl radical is a unique molecule due to its combination of low mass and 2 Π groundelectronic structure. As described in the previous section, the molecule’s relatively large rotational
constant resulting from the small moment of inertia yields an intermediate coupling in the groundstate. While most closely resembling a Hund’s case (a) molecule, the 2 Π3/2 ground state displays
non-negligible mixing with the spin-orbit split 2 Π1/2 state which makes the description more complicated than for heavier 2 Π molecules like CH. Furthermore, OH possesses a small (∼1.7 GHz)Λ-doublet splitting resulting from coupling of the angular momentum projection Λ with nuclear
rotation of excited electronic states.
We begin with the following eﬀective Hamiltonian to describe the rotational, ﬁne structure,and Λ-doublet energies [41]:Ĥef f = Ĥrot + Ĥf s + ĤΛ(2.2)

B(J − S)2

(2.3)

where
Ĥrot ==BR2=

Ĥf s = AL · S + γR · S==

B(J 2 + S 2 − 2J · S)[]
AL · S + γ J · S − S 21122
ĤΛ = − q(J+
exp−2iϕ +J−
exp2iϕ ) + (p + 2q)(J+ S+ exp−2iϕ +J− S− exp2iϕ ).
2
2

(2.4)
(2.5)

In the above Hamiltonian, A is the spin-orbit coupling, B is the rotational constant, and γ isthe electron spin-rotation parameter. The values p and q are the Λ-doubling parameters resulting
from interactions between (S, R) and (R, R), respectively. As a result, for singlet states only
the q parameter is non-zero while, for triplet states, there is an additional o parameter describing
the (S, S) interaction. Note that the Λ-doubling parameters p, q, o are set by these interactions

13
across electronic states (e.g. Π ↔ Σ) and therefore depend on the shapes of the diﬀerent electronic
potentials.
We will now deﬁne a parity basis in order to obtain eigenvalues of Eq. 2.2. For a 2 Π molecule,
this basis consists of the usual case (a) quantum numbers (Λ, Σ, Ω̄ = |Ω|, J, MJ ) with the added
parameter ϵ = ±1 which will give the symmetry of each state. The overall parity of a state|ΛΣΩ̄JMJ ϵ⟩ is given by ϵ(−1)J−1/2 and we deﬁne(|JMJ Ω̄±⟩ =|JMJ Ω̄, ΛΣ⟩ + ϵ|JMJ Ω̄, −Λ − Σ⟩
√
2

)
.

(2.6)

Using the angular momentum operator identity A · B = Az Bz − 12 (A+ B− + A− B+ ), where A+ =(Ax + iAy ) and A− = (Ax − iAy ), we obtain the following 4 × 4 matrix using Eqs. 2.2 and 2.6:


a
a
0
0
 11 12





a
a
0
0
 21 22



(2.7)



 0
0
a
a
33
34 



0
0 a43 a44
where
a11 =
a12 =
a22 =
a33 =
a34 =
a44 =

[
]
1
7
A + B J(J + 1) −
2
4
√
[
)]
( )(
1 (
γ)
3
1
J−1/2 qa21 = − (J + )(J − ) B −− (−1)
J+
2
2
2
2
2
[
]
(
)
1
1
1
1
− A + B J(J + 1) +− γ − (−1)J−1/2 (p + 2q) J +
2
4
2
2
[
]
1
7
A + B J(J + 1) −
2
4
√
[
)]
( )(
3
1 (
γ)
1
J−1/2 qa43 = − (J + )(J − ) B −+ (−1)
J+
2
2
2
2
2
[
]
(
)
1
1
1
1
− A + B J(J + 1) +− γ + (−1)J−1/2 (p + 2q) J +
.
2
4
2
2

(2.8)

The above matrix represents a general result for 2 Π molecules. We can immediately see that theoﬀ-diagonal elements a12 = a21 and a34 = a43 couple the Ω̄ = 1/2 and Ω̄ = 3/2 states, andthat this coupling increases with J in a given molecule. This leads to a transition from case (a)to case (b) at large J. Also, the dependence of these terms on B tells us that light molecules

14

450

Energy [cm-1]

400
350

f e+

f+
e-

7/2

f e+

5/2

9/2

300
250
200

f+
e-

7/2

150
100

f e+

5/2

f+
e-

3/2

f+
ef e+

3/2
1/2

J

50
0

2

Π3/2

2

Π1/2

J

Figure 2.2: Illustration of the rotational structure of the OH electronic and vibrational groundstate. The Λ-doublet splitting of each rotational level has been exaggerated for clarity. The (e,f)
quantum numbers represent the symmetry of the given Λ-doublet component, while the ± denotes
the overall J-dependent parity of the state.

15
Table 2.1: OH Rotation and Fine-Structure Constants (obtained from Ref. [1])Frequency [cm−1 ]v=1
v=2Parameter

v=0

v=3

X 2Π

B
A
γ
p
q

18.5348734-139.050895-0.119190
0.235266041-0.03869315652

17.8239122-139.320532-0.113753
0.22467831-0.036939875

17.1224586-139.5881-0.10831
0.21392702-0.035175264

16.4278912-139.8431-0.10249
0.202883-0.03338660

A2 Σ+

B
D
γ

16.965060
0.002063218
0.22555

16.129332
0.002045047
0.216115.286447
0.00204786
0.206614.42222
0.00206501
0.1975will deviate from case (a) at lower J than their heavier counterparts. Specializing to OH, we
use the (v = 0) molecular constants of Table 2.1 to compute the ground rotational structure
shown in Fig. 2.2. All of the matrix elements and molecular constants of this section are basedon our initial choice of Ĥrot = BR2 = B(J − S)2 . Note that if a diﬀerent choice is made (i.e.
Ĥrot = BN2 = B(J − L − S)2 ), then the values of ﬁtted molecular constants will change if Λ ̸= 0.
Reference [42] gives the appropriate expressions for transforming spectroscopic constants between
these two regimes.

2.2.1.1

OH LIF Spectroscopy

Hydroxyl is readily detected using laser-induced ﬂuorescence (LIF) at ultraviolet wavelengths
2along the A2 Σ+
1/2 ← X Π3/2 electronic transition. To calculate the positions of the dipole-allowed∆R = {−2, −1, 0, 1, 2} = {O, P, Q, R, S} lines satisfying ∆J = 0, ±1, we must consider the molecular constants for the 2 Σ+ excited state, which is best described by Hund’s case (b). The vibration2changing A2 Σ+
1/2 (v = 1) ← X Π3/2 (v = 0) transitions are the most useful for ﬂuorescence detection

since the excitation and decay wavelengths are well-separated, allowing for eﬀective ﬁltering of thepump light. This separation arises from the large Franck-Condon overlap of ∆v = 0 transitions
2
relative to ∆v ̸= 0, which forces spontaneous decay primarily along A2 Σ+
1/2 (v = 1) → X Π(v = 1).

16Table 2.2: OH Vibration Constants (obtained from Ref. [2]). The constant Te is deﬁned at theenergy minimum of the given electronic potential.
Frequency [cm−1 ]
Te
ωe
ωe xe
X 2Π

0

3178.8

92.91

A2 Σ+

32684.1

3737.76

84.881

The relevant spectroscopic constants for the 2 Σ+ state of OH are given in Table 2.1, while the case(b) Hamiltonian for this excited state isĤef f= Ĥrot + Ĥf s= BN2 − D(N2 )2 + γN · S(2.9)

whose eigenvalues are simply[B − DN (N + 1)] N (N + 1) +γ
[J(J + 1) − N (N + 1) − S(S + 1)] .
2(2.10)The diagonal nature of Eq. 2.9 makes inclusion of the additional centrifugal stretching term (D)trivial. Since we are interested in transitions between diﬀerent vibrational levels in the 2 Π and2 Σ electronic states, we must also consider the vibrational structure of each electronic state whoseenergy is given by
1
1
Uvib = ωe (v + ) − ωe xe (v + )22
2

(2.11)

where v is the vibrational quantum number (v = 0, 1, 2, ...) and ωe , ωe xe are the vibrational spectroscopic constants given in Table 2.2.
To fully describe OH dipole transitions, we must also deﬁne the spectroscopic number F,
which is (confusingly) distinct from the hyperﬁne quantum number F . Levels are designated F = 1
by default, but the value F = 2 is used wherever there exists an angular momentum doublet, andin such cases the lower-energy partner is given the value F = 1 while the higher-energy component
is labelled F = 2. This label is aﬃxed to the spectroscopic terms {O, P, Q, R, S} as either F′ F′′17

A2Σ+1/2

1

1 3/2 -

1

1 1/2 -

F’=1
F’=2

0

0 1/2 +

F’=1

v’ = 1
R’ N’ J’ p’
P1 Q21 Q1

X2Π3/2 1 0 3/2 +v’’ = 0 R’’ N’’ J’’ p’’

F’’=1

Figure 2.3: Illustration of the dipole-allowed P1, Q21, and Q1 transitions for 2 Π3/2 OH moleculeswith corresponding measured LIF spectra of a supersonic beam. Generally, the ∆v = 1 transition
is used to allow ﬁltering of the 282 nm excitation light from the 313 nm ﬂuorescence photons.

18(F′ ̸= F′′ ) or just F′ (F′ = F′′ ), where prime and double-prime indicate upper and lower states,
respectively.
Combining Eqs. 2.10 and 2.11 with the constants of Tables 2.1 and 2.2, one can calculateline positions for all A2 Σ+ ← X 2 Π electric dipole transitions∗ . Figure 2.3 illustrates the P1, Q21,
and Q1 transitions along A2 Σ+ (v = 1) ← X 2 Π3/2 (v = 0) and shows corresponding experimental
spectra obtained from a supersonic beam of hydroxyl seeded in Kr. Even though the linewidth of
our pulsed LIF laser (∼ 2 − 3 GHz) is larger than the 1.7 GHz ground Λ-doublet splitting, thedipole parity selection rule causes the P and Q transitions to be split by 2B (∼32 cm−1 ), leaving
the Λ-doublet completely resolved. This is a nice feature of Σ ← Π transitions in molecules withlarge rotational constants. As we will see, ND3 also exhibits this feature of resolved parity-doublets
in its REMPI spectrum, albeit from a completely diﬀerent coupling mechanism. To calculate therelative strengths of these transitions, refer to Eq. 2.24 and replace the symmetric top quantum
number K with Ω.

2.2.2

Ammonia (ND3 )Ammonia is arguably the prototypical polyatomic molecule. A symmetric top in its normal(NH3 ) and fully-deuterated (ND3 ) forms, ammonia serves as a nice introduction to polyatomics andconcepts such as hindered motions, para/ortho symmetry, l-type doubling, and resonance-enhanced
multiphoton ionization (REMPI). Deuterated ammonia is ubiquitous in cold molecule experiments
due to its large static polarizability (see Section 2.3.1), low mass, and chemical stability in the gas
phase at standard temperature and pressure.
A full description of the group theory of polyatomic molecules is beyond the scope of thissection, but a detailed discussion can be found in Chapter 3 of Ref. [43]. The pyramidal structureof ground-state (X̃ 1 A′1 ) ammonia gives it a C3v point group symmetry, which means that it has
an axis with three-fold rotational symmetry. The v subscript indicates that it also possesses three∗

Note that, for large J, one must add the centrifugal distortion term DR4 = D(J − S)4 to the ground state
rotational Hamiltonian of Eq. 2.3 and recalculate the matrix elements.

19
vertical planes of symmetry passing through the vertical symmetry axis. The Mulliken symbol A
communicates that the ground state is singly degenerate and possesses C3 rotation symmetry, its
subscript tells us that the sign of the electronic wavefunction does not change upon reﬂection of
the molecule about its center, and the prime indicates an additional horizontal symmetry plane σh .
As with diatomics, the leading superscript gives the overall electron spin multiplicity 2S + 1.
For diatomic molecules, the moment of inertia about the symmetry (z) axis is vanishingly
small. As a result, rotational excitations in this dimension are not observed. This leads to only one
rotational constant, B, which corresponds to the total rotational energy about the two remaining,
identical x- and y-axes. However, polyatomic molecules can possess bonds in directions other thanthe z-axis, which necessitates consideration of the moment of inertia about this axis. Since we are
only considering symmetric tops here, the moments Ix and Iy are still identical and are denoted
IB . The third moment, Iz , is denoted IC in ND3 since it is larger than Ix = Iy (otherwise it would
be labeled IA ). This leads to the following rotational Hamiltonian:Ĥrot =
=

~2 ˆ2
~2 ˆ2
~2 ˆ2
Jx +
Jy +
J
2IB
2IB
2IC z
( 2
)
~2 2
~
~2
J +
−
Jˆz2
2IB
2IC
2IB

= BJ2 + (C − B)Jˆz2(2.12)

Urot = BJ(J + 1) + (C − B)K 2 .

(2.13)

whose eigenvalues are

The projection quantum number K can take the usual values K = −J, ..., J and, in the absence ofinteractions such as l-type doubling, ±K values are degenerate. Adding the requisite centrifugal
stretching terms with associated constants DJ , DJK , and DK to Eq. 2.12, we obtain the eigenvalues
Urot = BJ(J + 1) + (C − B)K 2 − DJ J 2 (J + 1)2 − DJK J(J + 1)K 2 − DK K 4 .

(2.14)The three identical deuterium nuclei in ND3 yield an additional symmetry which divides
rotational states into ortho and para categories according to their K projection. Since the D
nuclei are bosonic, the overall molecular wavefunction (including both spin and rotation) must be

20

60

3,1

3,0

3,2

50
3,3

Energy [cm-1]

40

30

2,1

2,0

2,2

20

10

1,1

0

1,0

0,0

para

J,K

ortho

ND3
Figure 2.4: Ground state (X̃ 1 A′1 ) structure of deuterated ammonia showing the inversion doublet(expanded for clarity) of each rotational level.

21
symmetric under exchange of deuterons. Therefore, we obtain two distinct “ladders” of rotational
states corresponding to ortho (K mod 3 = 0) and para (K mod 3 ̸= 0) symmetry as shown in
Fig. 2.4. Interconversion between ortho/para states is heavily suppressed due to the extremely
weak interaction between nuclear spin and molecular rotation. Experimentally, it can be useful tothink of p-ND3 and o-ND3 as distinct molecules each with their own rotational structure. This
viewpoint is especially useful when ﬁtting the rotational temperature of a supersonic beam ofammonia, which will often possess two distinct ortho and para rotational temperatures.
The pyramidal structure of the electronic ground state of ammonia allows for tunnelling of
the nitrogen atom through the plane of hydrogen/deuterium nuclei. This inversion occurs because
the Coulomb energy barrier corresponding to N lying in the H/D plane is ﬁnite. Despite thelarge vibrational constant of the NH3 (ND3 ) ground state measuring 950 cm−1 (748 cm−1 ) [2],the central potential barrier of 2072 cm−1 signiﬁcantly “hinders” the motion of the v2 = 0, 1
(v2 = 0, 1, 2) states and produces closely-spaced levels of opposite parity. The quantum nature ofthis tunnelling process is apparent in the isotope shift of the resulting inversion doublets, whichdisplays an exponential dependence of the inversion splitting on the molecular reduced mass. The
dependence of the inversion splitting on the rotational level is parametrized as [44][]∆Uinv = v0 − va J(J + 1) − K 2 − vb K 2(2.15)

where v0 = 0.7934 (0.05337) cm−1 , va = 5.05 × 10−3 (2.39 × 10−4 ) cm−1 , and vb = 1.998 ×
10−3 (9.61 × 10−5 ) cm−1 are the NH3 (ND3 ) inversion constants. Examining v0 , we see that merely
increasing the reduced mass by a factor of 1.7 reduces the tunnelling rate by a factor of ∼ 15.

2.2.2.1

ND3 2+1 REMPI Spectroscopy

We have thus far discussed one form of spectroscopic molecular detection in Section 2.2.1.1,
namely laser-induced ﬂuorescence. Fluorescence detection can be quite sensitive (limited to ≥100
molecules) with the correct choice of excitation and ﬂuorescence wavelengths in addition to goodstray light baﬄing. However, one can never detect 100% of the photons scattered into 4π sr. On

22
the other hand, since ions can be very eﬃciently guided and detected, state-sensitive molecular
ionization serves as a useful alternative to ﬂuorescence when the molecular structure allows foraccessible ionizing wavelengths, and the former is usually more sensitive.
Resonance-enhanced multiphoton ionization (REMPI) is commonly used for spectroscopy
and detection of neutral atomic and molecular species. By making use of a real (as opposed
to virtual) intermediate state, one can ionize the species of interest in multiple excitation steps.
This technique carries the advantage that by energetically separating ionization steps one can use
longer, more accessible wavelengths to ionize. Furthermore, the use of resonant excitation lightyields large enhancements in atomic/molecular absorption cross sections. When a multiphotontransition is used for the intermediate step, diﬀerent dipole selection rules apply and allow one toprobe transitions inaccessible by single photons. Generally, REMPI is characterized as p + 1, where
p is the number of photons used to reach the intermediate state and a single photon ionizes. In
the case of ammonia, 2+1 REMPI is most commonly used [45, 46], but 3+1 spectra have been
observed [47]. It is worth noting that OH can be ionized by REMPI [48, 49], but the wavelengths
for even the 2+1 step lay below 250 nm.
One electronic intermediate employed in 2+1 REMPI of ND3 is the B̃ 1 E ′′ Rydberg state.
In this case, the Mulliken symbol E denotes a doubly-degenerate symmetry class and the doubleprime indicates that the wavefunction is anti-symmetric with respect to a horizontal symmetry
plane. In short, the structure of B̃ is planar, which has the immediate consequence of removing
the inversion doublets common to the pyramidal X̃ state. Transitions between vibrational states′in the ground and intermediate levels are parametrized as 2vv′′ , where the number 2 indicates thatthe vibrations are of the v2 ’umbrella’ mode. Figure 2.5 illustrates three of the allowed ionization
pathways (240 , 250 , 260 ) from an inversion doublet in the ground electronic and vibrational state.
Vibrational selection rules require that the upper and lower states have the same symmetry, whichmeans that closely-spaced inversion doublets in the ground state connect to alternating vibrational
levels in the intermediate B̃ state. As with the aforementioned Σ ← Π transitions in hydroxyl, thisfeature allows one to easily distinguish between electrically high- and low-ﬁeld seeking states.

23

v’2 = 6~B 1E’’

v’2 = 5v’2 = 4

4

20

5

20

6

20

~

X 1A’1asv’’2 = 0

Figure 2.5: Illustration of three intermediate two-photon transitions used in 2+1 REMPI spectroscopy of ND3 . Vibrational selection rules result from the diﬀerent (alternating) symmetries ofthe excited vibrational levels. The structure of ND3 transitions from pyramidal to planar after thetwo-photon excitation. The rotational structure is excluded for clarity.

24
The Hamiltonian for the B̃ 1 E ′′ state of ND3 is more complicated than that of the groundstate due to l-type doubling. The complete Hamiltonian is [46]Ĥ = Tv + Ĥrot + ĤV R(2.16)

Ĥrot = BJ(J + 1) + (C − B)K 2 − DJ J 2 (J + 1)2 − DJK J(J + 1)K 2 − DK K 4(2.17)

1
2
2).
+ L̂2− Jˆ+ĤV R = −2CζlK + ηJ J(J + 1)lK + ηK K 3 l + q(L̂2+ Jˆ−2(2.18)

where

The spectroscopic term Tv gives the location of each v2 vibrational state, while the vibrationrotation term ĤV R breaks the degeneracy between ±K projections with the vibrational angularmomentum projection l. In the X̃ state, l = 0 since there is no vibrational angular momentum
along the molecular axis. However, the doubly-degenerate B̃ state possesses l = ±1. Note that theﬁrst three terms of Eq. 2.18 do not completely lift the degeneracy of K states — only the product±Kl are split. However, the last term of ĤV R mixes l = ±1 states and yields so-called “l-type′

doubling” [44]. In spectroscopic ﬁts of 2v0 transitions where v ′ > 4, the parameters ηJ and ηK of
Eq. 2.18 are constrained to zero due to their negligibly-small ﬁtted values [50].
To ﬁnd the eigenvalues of the above Hamiltonian, one can either build and numericallydiagonalize the full 2(2J + 1) × 2(2J + 1) matrix for each J ′ level in the excited state or only
2 . Thediagonalize the necessary 2 × 2 matrices that include the two-state coupling terms L̂2± Jˆ∓latter method allows for more simpliﬁed, scalable code and will be described here as in Ref. [46].
Keeping in mind that the quantum number l represents vibrational angular momentum about thesymmetry axis, we obtain|J, K⟩|l = −1⟩ |J, K + 2⟩|l = +1⟩⟨J, K|⟨l = −1|

U0

1
2 qf

⟨J, K + 2|⟨l = +1|

1
2 qf

U0′

(2.19)

25
where
U0 = Tv + BJ(J + 1) + (C − B)K 2 − DJ J 2 (J + 1)2 − DJK J(J + 1)K 2−DK K 4 + 2CζK

(2.20)

U0′ = Tv + BJ(J + 1) + (C − B)(K + 2)2 − DJ J 2 (J + 1)2

f

=

√

−DJK J(J + 1)(K + 2)2 − DK (K + 2)4 − 2Cζ(K + 2)

(2.21)[J(J + 1) − K(K + 1)][J(J + 1) − (K + 1)(K + 2)].

(2.22)For completeness, the eigenvalues and normalized eigenvectors of the above 2 × 2 Hamiltonian areU±=|ψ ± ⟩ =(U0 + U0′ ) ±√(U0′ − U0 )2 + q 2 f 2
2
1
qf
|J,
K⟩|l
=
−1⟩
+ (U ± − U0 )|J, K + 2⟩|l = +1⟩
2√.
(U ± − U0 )2 + 14 q 2 f 2(2.23)In order to properly simulate a REMPI spectrum, one also needs the relevant transition′

strengths for a given set of rotational 2vv′′ transitions. In the special case of single-photon transitions,these are known as the Hönl-London factors SJ [51]. However, Dixon et al derive a compact form for
all k-photon transition strengths based on generalized spherical tensor transition matrix elements⟨J ′ , K ′ (Ω′ ), M ′ |Tqk |J ′′ , K ′′ (Ω′′ ), M ′′ ⟩ that is better suited for numerical evaluation [52]. Quoting only
the result of their derivation, the transition strength (S) for a k-photon ro-vibrational transition is
given by
S(J ′ , J ′′ , K ′ , K ′′ , k, q) ∝∑q|⟨η2 |Tqk |η1 ⟩|2 (2J ′ + 1)(2J ′′ + 1) 

J′
K′

k

J ′′

−q

−K ′′

2

 .

(2.24)

The ﬁrst term involving η1 and η2 reﬂects the vibrational transition strength and includes the′necessary Franck-Condon factor. This term can be ignored for transitions in a given 2vv′′ manifold.
The J, K-dependent terms give the generalized Hönl-London factors for a k-photon transition withspherical tensor components q, where q = −k, .., k. The M ′ , M ′′ components of J ′ , J ′′ have beensummed in the above expression since we are assuming there are no external ﬁelds. The symmetryof the Wigner 3j-symbol of Eq. 2.24 immediately gives the k-photon selection rules. Specializing to26
2+1 REMPI of ND3 , previous experiments have found that the transition is completely carried by
2 tensor. This yields the following two-photon selection rules:
the T±1∆J = 0, ±1, ±2∆K = ∆l = q = ±1.

(2.25)

Equation 2.24 is suﬃcient to determine transition strengths at low J, but as l-uncoupling increases
it becomes necessary to transform to the basis given by Eq. 2.23. This new basis allows for“intensity mixing” between coupled |J ′ , K ′ ⟩|l = −1⟩ and |J ′ , K ′ + 2⟩|l = +1⟩ states. Using the
2 ||X̃⟩, wephase convention for reduced matrix elements of Ref. [52] wherein ⟨B̃||T12 ||X̃⟩ = −⟨B̃||T−1

obtain the transition matrix|X̃, l = 0⟩⟨B̃, l = −1|s(J ′ , J ′′ , K ′ , K ′′ , k, q)(2.26)⟨B̃, l = +1| −s(J ′ , J ′′ , K ′ , K ′′ , k, −q)wheres(J ′ , J ′′ , K ′ , K ′′ , k, q) ∝∑⟨η2 |Tqk |η1 ⟩√
(2J ′ + 1)(2J ′′ + 1) 

q

J′
K′

k

J ′′

−q

−K ′′



.

(2.27)

Combining the phase convention of Eq. 2.26 with the eigenstates and eigenvalues of Eq. 2.23, we′

can now accurately assign transition strengths to all 2vv′′ lines in NH3 and ND3 . Eﬀective rotationaltemperatures can be calculated by multiplying transition strengths by the Boltzmann factor
SIK (2J ′′ + 1) exp [−UJ ′′ K ′′ /kB Trot ](2.28)

where UJ ′′ K ′′ are the energies of the |J ′′ , K ′′ ⟩ ground states, (2J ′′ + 1) is the degeneracy weighting factor for each J ′′ level, kB is Boltzmann’s constant, Trot is the eﬀective temperature of therotational distribution, and SIK accounts for the degeneracy due to the three identical H (I = 21 )or D (I = 1) nuclei in ammonia. The factor SIK for three identical bosonic nuclei is deﬁned in27
Eq. 2.29 [44]. Note that, in the case of fermionic nuclei, one must ﬂip the two K = 0 cases.

Bose
SIK
=




4I2 + 4I + 3
K mod 3 = 0, K ̸= 0






 (2I + 3)(I + 1) K = 0, J even, lower inversion or J odd, upper inversion
2I2 − I
 4I2 + 4I

(2.29)
K = 0, J even, upper inversion or J odd, lower inversion
K mod 3 ̸= 0.

The eﬀect of diﬀerent rotational temperatures on the measured 250 ND3 spectrum is illustrated inFig. 2.6, where each rotational transition has been given a Lorentzian FWHM linewidth of 0.3
cm−1 . See Appendix A for a list of line positions and strengths of the 250 2+1 REMPI lines of ND3 .

28

Figure 2.6: Simulated 2+1 REMPI spectrum of 250 rotational transitions in ND3 . The simulated
rotational temperature is given for each plot. For consistency of presentation, a Lorentzian width of
0.3 cm−1 has been assumed for all spectral lines. Note that at high temperatures (& 100 K), Doppler
broadening necessitates application of a Voigt lineshape to correctly ﬁt experimental spectra.

29

2.3

External Fields

2.3.1

Stark Eﬀect

2.3.1.1

Strong Electric Fields - Fine Structure

When considering external electric ﬁelds of &1 kV/cm, one can ignore the hyperﬁne structureof polar molecules such as OH and ND3 since J is decoupled from the nuclear spin I at these ﬁeld
magnitudes. To calculate the molecular Stark eﬀect in this regime, we ﬁrst consider the Hamiltonian
ĤStark = −µ · E(2.30)

where µ is the permanent dipole moment vector and E is the external electric ﬁeld. Using theparity basis for a Hund’s case (a) or (c) molecule deﬁned in the previous section and assuming thatE lies along the z-axis, we compute
1⟨JMJ Ω̄ϵ| − µ · E|JMJ Ω̄ϵ′ ⟩ = − (⟨JMJ Ω| + ϵ⟨JMJ − Ω|)µE cos θ(|JMJ Ω⟩ + ϵ′ |JMJ − Ω⟩)2]1[= − ⟨JMJ Ω|µE cos θ|JMJ Ω⟩ + ϵϵ′ ⟨JMJ − Ω|µE cos θ|JMJ − Ω⟩2()µEMJ Ω 1 − ϵϵ′= −J(J + 1)2()
1 − ϵϵ′= −µef f E.
(2.31)
2
To arrive at the ﬁnal result of Eq. 2.31, one needs the so-called ‘direction-cosine matrix elements’
given in Ref. [44]. It is immediately clear that the external electric ﬁeld couples states of opposite
parity (ϵ ̸= ϵ′ ) to generate a non-zero Stark shift. Furthermore, in the absence of an electric ﬁeld
to couple these states, the expectation value of the dipole operator vanishes.
The above equation for a ﬁrst-order Stark eﬀect works well for symmetric top moleculeswith extremely small parity-doublet splittings (∆) such that ∆ ≪ µef f E. However, we can better
approximate the Stark shift at intermediate ﬁeld strengths if we fully diagonalize the above Hamiltonian in the parity basis. Including diagonal terms that represent the parity-doublet splitting, we30

Figure 2.7: Stark shifts in both the OH and ammonia (NH3 /ND3 ) molecules. Coincidentally, OH
and ND3 have nearly identical parity doublets and permanent dipole moments which gives themsimilarly large static polarizabilities. Partly for this reason, many cold molecule experiments havemade use of these two species.

31


have

−∆
2


ĤStark = whose eigenvalues are−µef f E√(±
UStark=±∆2−µef f E (2.32)+ (µef f E)2 .

(2.33)∆
2)2

The normalized eigenstates of the above Hamiltonian are therefore(∆|ψ ⟩ =+−+
2 + UStark√(∆|ψ ⟩ =)−
2 − UStark(2.34)++
UStark(2UStark
+ ∆)

)

√

|JMJ Ω̄+⟩ − µef f E|JMJ Ω̄−⟩|JMJ Ω̄−⟩ + µef f E|JMJ Ω̄+⟩−−
UStark(2UStark+ ∆).

(2.35)

Note that as the electric ﬁeld strength is increased, |ψ ± ⟩ transitions from a state of well-deﬁned
1parity ϵ(−1)J− 2 to a mixed parity state. When the Stark eﬀect overcomes the parity-doublet
splitting (E > ∆/2µef f ), the molecule enters the linear Stark shift regime and is often said to be‘polarized.’ When considering nonlinear symmetric top molecules such as ND3 , the above Stark
shift still applies, but you must replace the quantum number Ω with K, which represents the
projection of J along the molecular symmetry axis. The Stark shifts for ground state hydroxyl andammonia (NH3 , ND3 ) are shown in Fig. 2.7.
Equation 2.34 presents an opportunity to not only quantify state mixing due to an external
electric ﬁeld, but to measure the value of such a ﬁeld. In cases where the molecular sample resides ina single parity state (e.g. Stark decelerated beams), one can directly measure the state-mixing ratio
between the upper (|↑⟩) and lower (|↓⟩) doublets as a function of electric ﬁeld. Stark decelerated
and electrostatically-ﬁltered beams are selected for weak-ﬁeld seeking states. Therefore, we canmonitor the ratio of lower- to upper-doublet “population” as a measure of the local electric ﬁeld.
From Eq. 2.34, we predict|⟨↓ |ψ + ⟩|2 =|⟨↑ |ψ + ⟩|2∆

2µef f E
 .
√( )
∆ 2
2
+ (µef f E)
2 +2

(2.36)

32
The graph of Fig. 2.8 compares the prediction of Eq. 2.36 (black line) with a measured population
ratio of guided, state-selected ND3 molecules (red point). We ionize ND3 in the presence of acalculable electric ﬁeld and measure both lower- and upper-doublet populations of ground state
ND3 . The excellent agreement between measurement and theory suggests that this technique canbe employed to measure large electric ﬁelds (& 1 kV/cm) in the molecule detection region.

33

Lower:Upper Inversion Ratio

0.8

0.7

0.6

0.5

0.4

0.3

0.2

+950 V

Theory0.1

Measurement

0.0
0

2

4

6

8

10

12

14

Electric Field [kV/cm]

Figure 2.8: Theoretical (black line) and measured (red point) doublet population ratios as a functionof external electric ﬁeld. For this measurement, we perform 2+1 REMPI of a state-selected beam of
ND3 molecules while applying +950 V to one of the trap magnets in the ionization region. Following
Eq. 2.36, the ratio of lower- to upper-inversion doublet populations allows accurate characterizationof this applied electric ﬁeld.

34
2.3.1.2

Weak Electric Fields - Hyperﬁne Structure

For very weak electric ﬁelds (.0.5 kV/cm for OH) acting on molecules with a net non-zero
nuclear spin, we must include the hyperﬁne structure to accurately calculate Stark shifts. Whenforming the resultant F=J+I, we use Clebsch-Gordon coeﬃcients to transform from the basis of|JMJ Ω̄ϵ⟩ states to states with quantum numbers F and MF , where MF is the projection of F on
the electric ﬁeld axis. The transformation takes the form
|F MF ϵ⟩ =∑|JMJ Ω̄ϵ⟩|IMI ⟩⟨JMJ IMI |F MF ⟩(2.37)

MJ ,MI

and, for a 2 Π molecule such as OH, yields the oﬀ-diagonal matrix elements [53])(′1 + ϵϵ′ (−1)J+J +2Ω̄+1
′
′
′
′ ′
⟨F MF ϵ|ĤStark |F MF ϵ ⟩ = −µE(−1)J+J +F +F +I−MF −Ω̄+1
2√
× (2F + 1)(2F ′ + 1)(2J + 1)(2J ′ + 1)






′
′
′
1 F  F F 1  J 1 J  F.(2.38)× J′ J 1 −Ω̄ 0 Ω̄′−MF′ 0 MF
The expressions in parentheses and brackets are the Wigner 3j and 6j symbols, respectively. Given
the symmetry properties of the 3j symbols and the ﬁrst term in Eq. 2.44, we see that states musthave the same MF and Ω̄ projections and diﬀerent parity to be coupled by the electric ﬁeld. Wethen combine the above expression with diagonal matrix elements that include the OH Λ-doublet(∆Λ = 0.056 cm−1 ) and hyperﬁne (∆hf = 0.0026 cm−1 ) splittings. Numerically diagonalizing theresulting 16 × 16 matrix over a range of electric ﬁeld magnitudes, we obtain the Stark shift curves
for the X 2 Π3/2 ground state of OH shown in Fig. 2.9. As in the strong-ﬁeld limit, only the parityof the given state determines whether it is strong-ﬁeld seeking or weak-ﬁeld seeking, and ±MF
projections within a given F state are degenerate. For reference, the 8 × 8 upper-right and lowerleft oﬀ-diagonal blocks in the basis {|F MF ϵ⟩ = |1 − 1+⟩, |10+⟩, |11+⟩, |2 − 2+⟩, |2 − 1+⟩, |20+⟩,

35|21+⟩, |22+⟩, |1 − 1−⟩, |10−⟩, |11−⟩, |2 − 2−⟩, |2 − 1−⟩, |20−⟩, |21−⟩, |22−⟩} are both


√
3
1
0
0
0 − 10
0
0
0 
 −2




1
 0
0
0
0
0
−5
0
0 




√

 0
3
1
0
0
0
0
−
0


2
10




3

 0
0
0
−
0
0
0
0
5


 √
.


3
3
 − 1000
0 − 10
0
0
0 




 0
− 51
0
0
0
0
0
0 




√


3
3
 0

0
0
0
0
0
−
10
10




3
0
0
0
0
0
0
0
5

2.3.2
2.3.2.1(2.39)

Zeeman Eﬀect
Strong Magnetic Fields - Fine Structure

As with the discussion of external electric ﬁelds, we may treat the molecular Zeeman shift in
two distinct cases — strong and weak ﬁelds. The primary diﬀerence between electric and magnetic
ﬁeld eﬀects is that the Zeeman shift respects parity. Otherwise, both ﬁelds couple states with thesame MJ , MF projection quantum numbers. Deﬁning the magnetic ﬁeld (B) to be along the z-axis,
the molecular Zeeman Hamiltonian is
ĤZeeman = −µ · B= µB (L + ge S) · B()= µB B L̂z + ge Ŝz(2.40)

where µB is the Bohr magneton, ge is the electron gyromagnetic ratio (ge = 2 + α/π + ... ∼2.002),
and L̂z , Ŝz are the usual electronic angular momentum operators. Using the same parity basis as
before, we arrive at the following for the case Ω = |Λ| + |Σ|:⟨JMJ Ω̄ϵ|ĤZeeman |JMJ Ω̄ϵ′ ⟩ =µB B(Λ̄ + ge Σ̄)Ω̄MJ
δϵϵ′
J(J + 1)(2.41)

36

F=2

F=1

F=2

F=1

Figure 2.9: Stark shift of the 2 Π3/2 ground state of OH in the hyperﬁne basis. The two Λ-doublet
states repel as the ﬁeld is increased, creating strong- and weak-ﬁeld seeking molecules. For electricﬁelds above ∼1 kV/cm, the structure approaches the strong-ﬁeld limit of Fig. 2.7.

37
and for Ω = |Λ| − |Σ|, we get⟨JMJ Ω̄ϵ|ĤZeeman |JMJ Ω̄ϵ′ ⟩ =µB B(Λ̄ − ge Σ̄)Ω̄MJ
δϵϵ′ .
J(J + 1)(2.42)The Kronecker deltas of Eqs. 2.41 and 2.42 result from the fact that eigenstates of ĤZeeman musthave well-deﬁned parity. Excluding excited rotational levels, ĤZeeman is clearly diagonal in the|JMJ Ω̄ϵ⟩ basis. The strong-ﬁeld Zeeman shift for the ground (2 Π3/2 ) state of OH is shown in
Fig. 2.10. Note that for states where Λ = 2Σ and Ω = |Λ − Σ| (e.g. 2 Π1/2 , 3 ∆1 ), the magnetic
dipole does not completely cancel but reduces to ∼ αµB /π ∼ 0.002µB .
A special discussion is in order for the magnetic ﬁeld shifts of vibrational (v2 ) states within
the excited B̃ 1 E ′′ state of ND3 . Rotational levels within this manifold exhibit a linear Zeeman eﬀect
even in the absence of a net electron spin. This Zeeman shift results from the vibrational angularmomentum l = ±1 which eﬀectively takes the place of Λ in Eq. 2.42. The resulting expression forlarge magnetic-ﬁeld shifts in these levels is [54]⟨JMJ Kl|ĤZeeman |JMJ Kl⟩ = gL µB B

lKMJ
J(J + 1)(2.43)

where the g-factor for vibrational angular momentum, gL , is vibrational-state dependent and generally deviates from unity. For our intermediate REMPI state of interest (v2 = 5), gL = 0.738.

2.3.2.2

Weak Magnetic Fields - Hyperﬁne Structure

For magnetic ﬁelds inducing a Zeeman shift smaller than a molecule’s hyperﬁne splitting(B . ∆hf /µ ∼ 50 G for OH), it is important to include the hyperﬁne structure to get accurate
line shifts. Using the Clebsch-Gordon transformation of Eq. 2.37, we obtain the following matrix
elements including the coupling between diﬀerent F -sublevels:)(′ (−1)J+J ′ +2Ω̄
1
+
ϵϵ
′
′
(−1)J+J +F +F +I−MF −Ω̄+1⟨F MF ϵ|ĤZeeman |F ′ MF′ ϵ′ ⟩ = µB B(Λ̄ + ge Σ̄)2√× (2F + 1)(2F ′ + 1)(2J + 1)(2J ′ + 1)






′
′
′

1 F 
F F 1  J 1 J  F
×. (2.44)
 J′ J 1 −Ω̄ 0 Ω̄′−MF′ 0 MF

38= +3/2= +1/2= -1/2= -3/2

Figure 2.10: Zeeman shift of the 2 Π3/2 ground state of OH. Since the Zeeman Hamiltonian only
couples states of like parity, the two Λ-doublet levels of OH show exactly the same ﬁeld dependence.
Also, the lack of coupling between e and f states causes only diabatic level crossings.

39
Note that the Wigner 3j- and 6j-symbols show exactly the same couplings as in the Stark shiftexpression, with the only substantive diﬀerence between the two cases being the parity-coupling
term of the ﬁrst line. Specializing once again to the case of OH (2 Π3/2 ), we need only diagonalize
two distinct 8×8 Hamiltonians given by Eq. 2.44 multiplied by an extra prefactor of −µB B(Λ̄+ge Σ̄)and added to the diagonal zero-ﬁeld Λ-doublet and hyperﬁne energies given by:


∆Λ
0
0
0
0
0
0
0
± 2





∆Λ
 0

± 2
0
0
0
0
0
0




 0

∆Λ
0
± 2
0
0
0
0
0






∆Λ
 0

0
0
±
+
∆
0
0
0
0
hf
2



.


∆Λ
 0

0
0
0
± 2 + ∆hf
0
0
0




 0

∆Λ
0
0
0
0
± 2 + ∆hf
0
0






∆Λ
 0

0
0
0
0
0
±
+
∆
0
hf
2




∆Λ
0
0
0
0
0
0
0
± 2 + ∆hf
(2.45)The result of this calculation is shown in Fig. 2.11. In the absence of diﬀerential g-factors between
the two Λ-doublet levels, the e and f state Zeeman shifts are identical.

2.3.3

Combined Electric, Magnetic Fields

All of our magnetic trapping experiments have allowed for manipulation of OH within combined electric and magnetic ﬁelds. As such, we have observed interesting trap dynamics arisingfrom these crossed E- and B-ﬁelds [55]. Experimental trapping results will be discussed in a later
section, but we summarize here the mathematical formalism for treating a Hund’s case (a) moleculein combined ﬁelds [56]. This example is interesting because of the strong coupling of L and S tothe internuclear axis, which causes the axis to follow magnetic as well as electric ﬁeld lines. In thesame sense, the electron spin projection is inﬂuenced by external electric ﬁelds. It should be notedthat one still needs an electric ﬁeld to polarize a case (a) molecule since a lone magnetic ﬁeld will
not couple opposite-parity states, and will therefore still leave a vanishing expectation value for the

40

F=2

F=1

F=2

F=1

Figure 2.11: Zeeman shift of 2 Π3/2 OH molecules in the hyperﬁne basis. Both Λ-doublet statesshow the same eﬀect since the magnetic ﬁeld does not induce an interaction between e and f.

41
dipole moment.
Beginning with the combined-ﬁeld Hamiltonian, we get the expected
ĤE,B = −µe · E − µb · B(2.46)

where µe and µb are the electric and magnetic dipoles, respectively. Without loss of generality, we
let B lie along the z-axis and deﬁne β as the angle between E and B. Ignoring hyperﬁne structure
for simplicity, we obtain the following matrix elements for the 2 Π3/2 ground state of OH:− ∆ − 65 µB B
 2


0



0




0


 3
 5 µE cos β √
 − 3 µE sin β
 5


0


0



√

− 53 µE sin β003µE cos β5√− 53 µE sin β

1µE cos β5− 52 µE sin β−∆+ 25 µB B
200− 52 µE sin β− 15 µE cos β

0

0

−∆
+ 65 µB B
200− 53 µE sin β− 53 µE sin β00∆− 65 µB B
2

0

0

1
µE cos β5− 25 µE sin β00∆− 25 µB B
20− 25 µE sin β− 15 µE cos β− 53 µE sin β00∆+ 25 µB B
2− 35 µE cos β

0

0

0

0

0

0

−∆
− 25 µB B
2

0

0
√

0

√
− 53 µE sin β

√

√

0






√

3
− 5 µE sin β − 35 µE cos β 

.

0




0



0


∆
6
+ 5 µB B
20(2.47)

The above matrix elements were calculated using the general spherical tensor expressions for Stark
and Zeeman shifts in the ﬁne-structure parity basis of Eq. 2.6. The relevant expressions can befound in Appendix A of Ref. [56]. The spherical tensor components of a vector V = (Vx , Vy , Vz ) are1
V±1 = − √ (Vx ± iVy )2
V0 = V z==|V| cos θ.

1∓ √ |V| sin θe±iϕ2(2.48)(2.49)

Figure 2.12 displays the OH level shifts in the presence of both E- and B-ﬁelds. The “fullystretched” states corresponding to |JΩMJ ϵ⟩ = | 32 , 32 , + 32 , f ⟩, | 23 , 32 , − 23 , e⟩ remain the highest- andlowest-energy states, respectively, in combined ﬁelds. However, all other projections exhibit levelcrossings at intermediate ﬁeld values.

|E| = 3 kV/cm|B| = 1000 Gauss

β = 45οβ = 90ο

Figure 2.12: Level shifts for OH (2 Π3/2 ) in combined electric and magnetic ﬁelds as a function of relative ﬁeld angle β. The β = 0◦ , 180◦
shifts are identical and plotted together. To keep track of the diabatic curves, it is useful to start from the far left of each graph where
only the electric (upper plots) or magnetic (lower plots) ﬁeld dominates. Of course, the adiabatic curves maintain energy ordering.

β = 0ο,180ο

42

43
2.3.4

Diabatic vs. Adiabatic Level Crossings in Combined Fields

Interestingly, the curves of Fig. 2.12 show avoided crossings whose energy gaps depend on
relative ﬁeld angle, β, and electric ﬁeld magnitude. In principle, if OH molecules are trapped
within a magnetic quadrupole in the presence of an electric ﬁeld, MJ projections diﬀerent from thestretched | 23 , 32 , + 32 , f ⟩ and | 32 , 23 , − 32 , e⟩ states will traverse these avoided crossings many times overa typical trap lifetime (∼1 s). In the case of the experimentally-interesting | 32 , 23 , + 23 , e⟩ state, one
would prefer purely diabatic level crossings as shown in Fig. 2.13a to ensure an identical magnetic
trapping potential to the stretched | 23 , 32 , + 32 , f ⟩ state. If the crossings are traversed adiabatically(Fig. 2.13b), the overall trap depth is reduced and, at low magnetic ﬁelds, removed altogether.
To estimate the adiabatic (Pad ) and diabatic (Pdia ) transition probabilities for a given molecule
velocity (v) and magnetic ﬁeld gradient (dB/dx), we use the Landau-Zener probability [57]:Pad = 1 − Pdia
(
)
2
V12
= 1 − exp −2π~(dU/dt)√()(− 53 µE sin β)2= 1 − exp −2π~(dU/dB)(dB/dx)(dx/dt)√)((− 53 µE sin β)2= 1 − exp −2π 6~( 5 µB )(dB/dx)v)(
π µ2 E 2 sin2 β.
= 1 − exp −5 ~µB (dB/dx)v(2.50)For a magnetic trap gradient of ∼ 1 T/cm and OH velocity of 5 m/s, we obtain the adiabatic
transition probability curves of Fig. 2.13c. As either the angle between E and B or the electricﬁeld magnitude is increased, Pad increases dramatically. This result suggests that trap loss due toadiabatic level crossings will be a serious problem for OH in magnetic quadrupole traps within ahomogeneous E-ﬁeld. The rather stringent requirement that |E| < 10 V/cm places severe restrictions on tolerable stray E-ﬁelds in the trap region. Working in our group, Benjamin Stuhl andMark Yeo have recently observed preliminary evidence of this trap loss mechanism.

44

a

b

c

Figure 2.13: (a) Diabatic levels for ground state OH molecules. Upper-doublet (black) and lowerdoublet (red) levels are completely uncoupled in this picture. (b) Color-coded adiabatic levels
resulting from coupling between e and f parity states due to an external electric ﬁeld. (c) Adiabatic
transition probability (Pad ) for OH molecules within combined magnetic and electric ﬁelds. The
probabilities shown are for coupling between the | 32 , 32 , + 32 , e⟩ and | 23 , 32 , + 12 , f ⟩ states as a functionof electric ﬁeld and ﬁeld angle β. Note that extremely small electric ﬁeld magnitudes (< 10 V/cm)
are suﬃcient to induce adiabatic transitions in this system.

Chapter 3

Production of Cold Molecular Samples

Accurate assignment of quantum numbers and ﬁtting of spectroscopic constants to a roomtemperature molecular gas are very diﬃcult tasks. Most molecules possess rotational constants
that are below 100 K, which leads to a wide distribution of population across diﬀerent angularmomentum and vibrational states at room temperature. Furthermore, Doppler broadening at 300
K gives inhomogeneous linewidths of ∼ 10 GHz that can blend nearby rotational transitions inheavy molecules. In the context of molecular collision physics, cold samples of molecules promisea host of exciting novel collision dynamics including but not limited to external-ﬁeld control ofelastic/inelastic collisions and chemistry. Cooling molecular samples from 300 K to ∼10 K will
alleviate the above spectroscopy issues and allow for further cooling to these collisionally-interesting
temperature regimes, but begs the question “How can one produce gaseous molecular samples attemperatures far below their freezing point?”
Two primary molecular cooling methods – supersonic expansion and buﬀer gas cooling –have been developed to answer the fundamental question posed above. Both methods are rather
simple in principle compared with atomic cooling techniques such as Doppler laser cooling, but thepractical implementation of these cold molecule sources is anything but trivial. In this chapter, thetheory behind these two production techniques will be presented along with experimental results
and common pitfalls.

46

3.1

Supersonic Molecular Beams

3.1.1

Theory of Supersonic Expansions

A supersonic beam apparatus generally consists of a gas reservoir at temperature T0 andpressure P0 as well as an exit nozzle admitting this gas into a vacuum chamber at backgroundpressure Pb . For molecular beams, the reservoir contains some dilute mixture of the molecule ofinterest within a noble gas carrier (commonly He, Ne, Ar, Kr, Xe, or some mixture of these). The
use of a noble carrier gas insures minimum cluster formation and chemistry during the supersonic
expansion, although the increased polarizability of the heavier species can lead to clustering (seeSection 3.1.2). Pulsed beams include some valve actuation mechanism for both control of thetemporal beam extent and limitation of overall gas load in the vacuum chamber. Furthermore, amolecular skimmer placed downstream from the valve oriﬁce is commonly employed to separate thecold core of the molecular beam from the warmer gas envelope. The small diameter of the skimmer (usually ∼1 mm) also allows for large diﬀerential pressures between the supersonic expansion
chamber and subsequent vacuum sections, which is critical for crossed-beam collisions and trapping
experiments. A representative sketch of a supersonic beam assembly is shown in Fig. 3.1.
The thermodynamics of a supersonic expansion can be quite complicated. Especially in thecase of a pulsed molecular beam, large swings in vacuum pressure throughout the valve pulse andsubsequent gas expansion render simple analytical estimates of beam parameters such as terminal
velocity (v∞ ), ratio of beam speed to longitudinal velocity width (S∥ ), and rotational temperature(Trot ) imprecise. However, the expressions for an ideal, continuous isentropic expansion give a nicestarting point for predicting the behavior of pulsed beams and will be discussed here.
As a gas mixture expands into vacuum, the pressure diﬀerence P0 − Pb across the exit nozzle
accelerates the particles and converts the enthalpy of the reservoir gas into forward velocity. Enthalpy per unit mass (h) is the most valid quantity for an isentropic expansion since the pressure
diﬀerential does work on the molecules, thereby modifying the internal energy of the expanding
particles. If the ratio of stagnation pressure to background pressure (P0 /Pb ) exceeds the value47[(γ + 1)/2]γ/(γ−1) , the velocity of the exiting beam will exceed the speed of sound and undergo
further expansion within the vacuum region. The constant γ is the ratio of speciﬁc heats for thegas in question (γ = 5/3 for an ideal monatomic gas) [58]. Denoting the initial and ﬁnal enthalpyof the gas as h0 and hf , respectively, and deﬁning the ﬁnal beam velocity as v, we obtain1
h0 = hf + v2 .
2

(3.1)

Assuming that the speciﬁc heat at constant pressure, Cp , does not change throughout the expansion(this assumption is species-dependent), we may express the ﬁnal velocity as
(
)
∫ T0
2kB
γ
2
v = 2(hf − h0 ) = 2
Cp dT =
(T0 − T ).
m
γ−1
T
In most instances, we make the approximation that (T0 /T ) = ∞ and arrive at√()
2kB T0
γ
v∞ =
.
m
γ−1

(3.2)

(3.3)

One can see immediately from Eq. 3.3 that the experimentalist has two major adjustableparameters with which to tune the ﬁnal velocity of a supersonic molecular beam — the mass (m)and temperature (T0 ) of the noble carrier gas. The predicted velocities for diﬀerent carrier gas
species are given in the plot of Fig. 3.1. It is often desirable to produce beams at velocities between
those shown in Fig. 3.1. In these instances, one can either change the temperature of the gas∑
reservoir or mix diﬀerent carrier gas species to obtain an eﬀective mass (m̄) given by m̄ = i wi mi
where wi is the fraction of carrier-gas atoms of mass mi .
Another important parameter of the supersonic beam is the speed ratio S∥ . This gives thelongitudinal velocity width relative to the mean speed of the beam (S∥ = v∞ /∆v∥ ). When couplinga supersonic beam to a Stark decelerator with ﬁnite longitudinal velocity acceptance, one would
like S∥ to be as large as possible. Perhaps the most useful prediction for the parallel speed ratio isthe empirical expression [59]S∥ ∼ 5.4(P0 d[torr · cm])0.32 ,(3.4)

where d is the diameter of the valve exit nozzle. At typical values of P0 ∼ 1000 torr and d ∼ 0.1
cm, this expression predicts a 4% beam spread, which corresponds to a velocity width as small as

48

Pb

T0, P0Figure 3.1: (Top) Illustration of a typical supersonic beam assembly featuring a gas reservoir atpressure and temperature P0 and T0 , respectively and supersonic nozzle admitting the gas into a
vacuum chamber at background pressure Pb . The skimmer is shown at right. (Bottom) Plot of
predicted beam velocities for diﬀerent noble carrier gases initially at room temperature.

49
13 m/s for an ideal Xe beam at T0 = 300 K. Accurate prediction of the perpendicular temperatureof a supersonic beam is much more diﬃcult than for S∥ [58]. However, we have found that for abeam of OH molecules seeded in either Kr or Xe, T⊥ ∼ T∥ /2 where T∥ ∝ ∆v2∥ .
Proper placement of the skimmer is critical to maximizing beam ﬂux to subsequent experimental apparatus. Since a supersonic beam by deﬁnition possesses a mach number M larger than 1,
shock waves form in the envelope of the expansion and re-collide with the beam itself at a distance
L of

√
2
P0
L= d
.
3
Pb

(3.5)

The distance L deﬁnes the downstream location of the so-called “Mach disk,” and the ideal valveto-skimmer distance (VSD) is generally well below this upper limit. However, for pulsed sources,
the pressure ratio of Eq. 3.5 can easily exceed 107 , yielding a mach disk location > 4 m downstreamof a 1 mm nozzle! As one can imagine, other eﬀects can become important at distances far below
a meter. One such consideration is the transverse diameter of the expansion vacuum chamber. In
small chambers, the carrier gas can reﬂect from the outer wall and collide with the supersonic beam
before it has been extracted by the skimmer. In a small 11 cm-diameter source chamber, we haveobserved that skimmer throughput drops signiﬁcantly for VSD > 6 cm. By moving the source toa much larger chamber with a diameter of 61 cm, we ﬁnd optimal VSDs of 10 − 12 cm. In bothcases, skimmer throughput decreases monotonically as the VSD is decreased below 5 cm. This is
likely due to gas scattering from the skimmer assembly itself, as we have seen improvements of∼ 30% in throughput by cooling the skimmer below the freezing point of the carrier gas (Xe in thiscase). Nevertheless, the absolute beam density remains largest at the aforementioned higher valuesof VSD.
Finally, it is important to characterize the rotational temperature (Trot ) of a supersonicmolecular beam. It generally takes only 10-100 collisions to fully thermalize Trot to T∥ , but thedegree of rotational thermalization can vary depending on factors such as the carrier gas used(clustering), the rotational structure of the molecule, or whether additional heating mechanisms

50(e.g. OH discharge sources) are necessary for beam production. For molecules with small rotational
constants (hB/kB . T∥ ), the spectroscopically-detected rotational populations may be ﬁt to a
Boltzmann distribution given by(
Ai ∝ Si (2Ji + 1) exp−Ui
kB Trot)(3.6)

where Ai is the measured signal from state i, Si is the strength of the transition probed to detect
the given state, Ji is the total angular momentum of state i, kB is Boltzmann’s constant, and thelast term is the usual Boltzmann factor for level i of energy Ui . When the rotational constant islarge compared to T∥ , little or no population can be detected in higher-lying rotational states and a
two-state temperature considering just the ground and ﬁrst excited level is often quoted. This same
technique is generally used for vibrational temperatures, Tvib . Denoting the ground and excitedlevels as i = 0 and i = 1, respectively, we obtain∆Uln
Trot =kB(

A0 S1 (2J1 + 1)
A1 S0 (2J0 + 1)

)−1
(3.7)

where ∆U = (U1 − U0 ).
When molecules possess identical nuclei that divide rotational states into ortho and paralevels due to a restriction on the overall exchange symmetry, as is the case with homonuclear
diatomics and ammonia, it is often necessary to ﬁt diﬀerent temperatures to the distinct rotationalladders. This phenomenon results from the small coupling between nuclear spin and molecularrotation that impedes collisional transfer between ortho/para levels [60].

3.1.2

Clustering of Carrier Gases

In the quest for slow supersonic beams with ever-narrower velocity distributions, it is tempting
to take Eq. 3.4 at face value and use the highest stagnation pressure and largest nozzle diameter
possible with a high-mass (e.g. Kr or Xe) carrier. Unfortunately, the heavier noble gases also
possess larger polarizabilities. The result is a species-dependent rate of cluster formation thatscales roughly as (P0 d)/T02 . Gas clustering has two primary negative eﬀects: higher molecular

51
Table 3.1: Values of the clustering constant k for a range of atoms and molecules (obtained from
Ref. [3]).
Gas
He
Ne
Ar
Kr
Xe

k
3.85
185
1650
2890
5500

Gas
H2
D2
N2
O2
CO2
CH4

k
184
181
528
1400
3660
2360

rotational temperatures due to ineﬃcient cooling in cluster-molecule collisions and larger beam
velocities/widths from gas heating. Smith et al. ﬁt an empirical functional form to quantify theonset of clustering, which is determined by the Hagena parameter (Γ∗ ) given by [3]Γ∗ = k(d[µm]/ tan α)0.85
P0 [mbar]
T0 [K]2.29(3.8)

where k is a species-dependent constant and α is the opening half-angle of the valve nozzle. To
insure that clustering does not occur, it is best to operate the expansion with Γ∗ < 100. The
average cluster size for larger values of Γ∗ is then ∝ (Γ∗ )2.0−2.5 . Values of k for both atomic andmolecular species are given in Table 3.1. To give a sense of scale, our pulsed valve assembly for
OH molecules features a 1000 µm-diameter nozzle with a 20◦ half-angle and a carrier gas of Kr
at 295 K and 2100 mbar. Applying Eq. 3.8 gives Γ∗ ∼ 2000, which suggests that some amount ofclustering is unavoidable for noble gases heavier than Ne at standard stagnation temperatures andpressures.
Interestingly, Christen et al. have recently demonstrated that pure beams of CO and CO2
formed from stagnation temperatures and pressures above their respective critical points show lesscluster formation than seeded beams [61]. In fact, they ﬁnd that the pure supercritical molecularbeams have speed ratios exceeding 100. They attribute this aﬀect to an enhanced value of γ near
the critical point. Given the thermodynamic nature of clustering, it is not altogether surprisingthat moving beyond the critical point in the phase diagram (and hence avoiding phase transitions)would make cluster formation less energetically favorable. However, given the extremely large

52
critical points of many chemically-interesting molecules, this trick is not universally applicable.
Nonetheless, certain crossed beam experiments might beneﬁt greatly from the enhanced densitythat a pure supercritical molecular beam can provide.

3.1.3

Pulsed Supersonic Valves

Throughout the experimental history of pulsed supersonic beams, many diﬀerent valve actuators have been developed and tested [58]. Given the countless number of articles on new valves andmodiﬁcations to existing valves (see Review of Scientific Instruments), we restrict the discussion ofthis section to three speciﬁc valve designs that have been used in our OH supersonic beam experiments. Some pros and cons of each design are presented along with experimental data speciﬁc to
OH production.

3.1.3.1

Current Loop ValveThe current loop valve (CLV) described here is a commercial unit (Jordan TOF Products,
Inc. Model C-211) that uses very large currents (∼ 4000 A − 5000 A) passed between two parallel
conducting strips to open the valve nozzle. The CLV is capable of rather short gas pulses (<100µs) at repetition rates of up to 10 Hz. We have found the current loop valve to be extremely
stable over long durations, and this stability is reﬂected both in the shot-to-shot and week-to-weekmeasured molecular pulse densities. As such, we often use this system to debug new molecularbeam experiments (e.g. new Stark decelerators or magnetic traps). However, there are two major
drawbacks to this design as a pulsed beam source for cold, high-density molecular beam experiments:
low overall output ﬂux compared with other valve designs and time-dependent beam heating.
Despite the manufacturer’s claims that choke ﬂow is obtained within the 60 µs gas pulse, we see asmuch as a 50% reduction in OH production with this valve compared with the solenoid and piezo
valves. Furthermore, as described in Ref. [62], we see that the high currents within the actuatormechanism heat the leading edge of the gas pulse and produce faster/hotter molecular beams.
We employed this valve for both precision measurements of the OH Λ-doublet splitting [21, 24]

53
as well as the ﬁrst Stark deceleration of formaldehyde (CH2 O) [63]. However, the lower densities produced by the CLV were not suitable for subsequent magnetic trapping or collision experiments [64, 40, 18].

3.1.3.2

Solenoid Valve

The solenoid valve (General Valve Series 9 or 99) is arguably the workhorse of pulsed beam
experiments. Consisting of a stainless steel body surrounded by a solenoid wire wrap, this valve may
be used with a large variety of corrosive substances with little or no adverse aﬀects. The actuatormechanism consists of a PTFE-coated ferromagnetic armature within the stainless housing onto
which is attached a (usually plastic) poppet that seals against the valve oriﬁce. Since the drive
current is well-isolated from the inner valve chamber, we observe no time-dependent heating eﬀects
with the solenoid valve (SV). Another feature unique to the SV Series 99 is its ability to operate
at both cryogenic [65] and elevated [66] temperatures. In stock form, this valve produces gas
pulses no shorter than ∼ 200 µs due largely to its reliance on spring tension to close the armature.
Furthermore, we often observe an initial “bounce” in the gas pulse within 100 µs of the initial
voltage trigger wherein the gas density is momentarily reduced. At the end of the pulse, a closing“bounce” signiﬁed by a two-pulse structure can also be observed. Benjamin Stuhl, a graduate
student in our group, has found that this trailing “bounce” can be removed by applying a second
260 V (200 µs) pulse to the solenoid 120 µs after the initial 400 µs opening pulse.
One fundamental problem with the long-term stability of the general valve is the plastic
poppet tip that seals the nozzle. Since this tip seals against a stainless steel oriﬁce, it necessarily
deforms with each closing strike. Over time, the behavior of the valve (pulse width, output gasﬂux) will change unpredictably and necessitate removal and replacement of the poppet. For Stark
deceleration and trapping experiments, these vacuum breaks can cost the experiment days or weeks.
As a result of this long-term instability, we do not use the SV for our primary OH source. It is
worth noting that other Stark deceleration groups, speciﬁcally the group of G. Meijer at the Max
Planck Institute, use a modiﬁed general valve for many of their beam experiments [67].

54

Kr

PZT

PoppetTensionAdjustment

H2O

Figure 3.2: Drawing of the PZT valve assembly showing the adjustable poppet, disk piezo translator,and valve housing. The Kr/H2 O bubbler system for production of OH via pulsed electric discharge
is also shown. (Inset) Measurement of the high voltage pulse applied to the PZT. It is critical that
the RC time constant of the voltage pulse be & 40 µs to avoid cracking the piezoelectric crystal.

55
3.1.3.3

Piezoelectric Transducer (PZT) ValveAs a compromise between the long-term operational stability of the CLV and the high output
ﬂux of the SV, we make use of a modiﬁed version of a piezoelectric transducer (PZT) valve originallydeveloped by Proch and Trickl [68]. A drawing of our valve design is given in Fig. 3.2 showing
our piezo disk actuator (Polytec PI 286.23) and stainless steel poppet sealing against an o-ring at
the nozzle oriﬁce. Proch and Trickl’s primary improvement over the existing PZT valves was toimplement a disk piezo actuator with a tenfold increase in excursion of ∼ 100 µm. This allows
for choke ﬂow conditions and pulse widths as short as 150 µs. The use of a viton o-ring to seal
the exit nozzle removes the issue of poppet wear associated with the solenoid valves, while thecapacitive load of the PZT minimizes heating of the supersonic beam. To allow ﬁne-tuning of thevalve opening dynamics in situ, we have added a rod that inserts into the back of the steel poppet
and adjusts its tension against the exit oriﬁce (see Fig. 3.2).
When operating the PZT valve, it is critical that the piezo crystal is not exposed to corrosive
substances or temperatures outside of the range 230 K - 350 K, otherwise it will crack and fail toopen consistently. Furthermore, the output impedance of the voltage drive circuit must be large
enough that the 65 nF piezo does not crack due to stress. We ﬁnd that an RC time constant of∼ 40 µs and drive voltages below -600 V are suﬃcient to keep the valve running stably for months
at a time. The actual shape of the applied drive pulse is plotted in the inset of Fig. 3.2 along withthe corresponding 60 µs square-wave trigger.

3.1.4

Production of OH Supersonic Beams

It can be said that supersonic beam sources have a “personality” due to their sensitive dependence on many diﬀerent experimental conditions. Optimizing a pulsed valve assembly involves
tuning a large number of parameters that often interact with one another (e.g. valve pulse voltage/width and poppet backing tension). In the case of radical sources, the addition of a production
method just before the supersonic expansion further complicates beam optimization and can dras-

56
tically modify beam parameters from the predicted values of Section 3.1.1. In this section, two OH
beam production methods will be discussed and beam data from both will be compared.

3.1.4.1

Pulsed Electric Discharge of H2 O

The pulsed electric discharge of H2 O is currently our OH production method of choice due
to its stability, longevity, and ease of operation. An illustration of the discharge “stack” mounted
to our PZT valve nozzle is provided in the upper left of Fig. 3.4. The stack consists of alternatingplates of ceramic insulator (Macor) and highly-polished stainless steel. After the valve is opened,
we apply a voltage of ∼ −1 kV to the outer steel plate to strike a ∼ 10 µs discharge within theexiting mixture of Kr and H2 O vapor. Triggering the discharge voltage ensures that we create
OH at the densest point within the gas pulse and avoid unnecessary heating from continuous
operation. Furthermore, we are able to run at rather low discharge voltages by seeding the processwith positive ions from a nearby ionizing pressure gauge [62]. This results in a colder, slower OH
beam. We deduce that positive ions are responsible for this enhanced discharge eﬃciency sincechanging the polarity of the outer discharge plate and/or turning oﬀ the ionization gauge leave avery unstable discharge. Overall, we are able to reduce the necessary discharge voltage by a factorof 4-5 by seeding in this manner, thereby obtaining a colder, brighter OH pulse than is attainedwith unseeded continuous discharges. We measure beam densities of ∼ 109 − 1010 cm−3 in thesource chamber with this production technique. The 40◦ opening angle of the nozzle beyond thedischarge region has been shown to give an order-of-magnitude increase in beam ﬂux due to beamcollimation [69]. It is possible that our required VSD of 10 cm results from this enhanced densityof the carrier gas, which can lead to “skimmer clogging” and reduced throughput.
Since our OH beam must travel a total of 15 cm to the Stark decelerator, we are also able toimplement a multi-pulse discharge to increase beam ﬂux within the longitudinal velocity acceptanceof our deceleration sequence. To accomplish this, we break up the discharge over three short 6 µs
pulses each separated by 3 µs. The three distinct OH pulses spatially combine over the long ﬂight
path to the Stark decelerator and yield a > 50% enhancement in beam density with a mere 8%

57

18

16

Multi-Pulse (3 x 6 s)
13.5% FWHM

OH Signal [arb .u.]

14

12Single Pulse (15 s)

1012.5% FWHM
8

6

4

2

0

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

Time of Flight [ms]Figure 3.3: Comparison of a single-pulse electric discharge to a multi-pulse discharge of roughly
the same duration. The three separate OH pulses spatially combine in free-ﬂight to the detection
region and yield > 50% improvement in beam density. The above data was taken 1 m downstream
from the supersonic valve.

58
increase in T∥ . This result is displayed in the plot of Fig. 3.3.
3.1.4.2

Photolysis of HNO3

One alternative to the comparatively violent electric discharge is a photolysis source that uses
a pulse of UV photons to fragment a precursor molecule following the pathway X + hν → OH + Y.
For OH production, the most commonly used photolysis precursors are nitric acid (HNO3 ) andperoxide (H2 O2 ). Since photolysis produces the molecule of interest over the spatial and temporalextent of the excitation laser, the product molecular packet possesses a well-deﬁned origin. The
resulting beams are also translationally/rotationally colder than their counterparts produced fromdischarges. In fact, propensities for speciﬁc Λ-doublet and J states within the OH vibrational andelectronic ground state have been identiﬁed for both nitric acid [70] and peroxide [71] precursors.
We have produced OH beams from HNO3 photolysis at 193 nm using a pulsed ArF excimer
laser (Lambda Physik OptexPro). An illustration of the photolysis setup is given in Fig. 3.4. We
modiﬁed our existing PZT valve for use with the extremely corrosive nitric acid vapor by adding
an additional chemically-resistive polyimide chamber to isolate the gas from the piezo crystal. The
poppet was also machined from polyimide and passed through a double-sided o-ring through the
Kr/HNO3 chamber to seal against the valve oriﬁce. In place of the electric discharge stack, we
installed a UV-transmissive capillary with an inner diameter of 1 mm. The excimer laser was
focused onto the capillary with a combination of two orthogonal 50 cm cylindrical lenses to allow
for adjustment of the OH production volume as well as pulse intensity. As with the dischargesource, we cut a 40◦ full-angle cone into the end of the capillary cell for improved molecular beamcollimation [69]. The plot of Fig. 3.4 directly compares the density of this optimized OH sourcewith that of the optimized discharge source. The data for both curves were taken on the same daywith identical OH excitation laser beams and ﬂuorescence collection optics. The discharge source
produces a beam with a 6% larger velocity width than the photolysis source, but wins in density
by a factor of ∼ 2. Given the dangers of working with HNO3 and the smaller yield producedby photolysis from our PZT valve, we now use the pulsed electric discharge exclusively for OH

59

Seeded Electric
Discharge

Photolysis of
HNO3

193 nm ArF Pulse- HV

Figure 3.4: (Top) Illustrations of the pulsed electric discharge and photolysis sources for OH supersonic beam production. Note the 40◦ exit nozzle cut into both sources for enhanced OH beam
density. (Bottom) Plot of relative OH yield for the discharge and photolysis sources measured
within the source chamber. The photolysis precursor for this data was 99% pure HNO3 .

60production.

3.2

Buﬀer Gas Beams
Buﬀer gas cooling is based on the rather simple concept of thermalizing a hot atomic ormolecular sample with a cold inert gas (usually He). This thermalization occurs within a buﬀer
gas cell anchored to a low-temperature cryostat assembly and, when combined with magnetic traps
or coupled out as a beam, allows for a wide variety of atom-molecule [72, 73, 74] and moleculemolecule [18] cold collision experiments. In contrast with other cold molecule production techniquesthat rely on electric/magnetic dipole moments [19, 75, 76, 77] or ultracold alkali-atom precursors [78,79], buﬀer gas techniques are in principle applicable to any atomic or molecular species. However,as we will see in the case of cold ND3 beams, these sources have their own technical limitations.
Our speciﬁc goals in developing a continuous buﬀer gas source for molecular collision experiments
were (1) to obtain stable cold ND3 beam ﬂux for ∼ 1 hr, (2) to produce a rotationally-cold ND3
sample, and (3) to obtain guided, state-selected ND3 densities of > 108 cm−3 . As will be discussed,
the ﬁrst goal was the most diﬃcult to achieve, and the path to realizing goal 2 was not completely
clear.

3.2.1

The JILA Buﬀer Gas Beam Machine

There are many useful texts covering construction of cryogenic systems. Two of our mostcommonly-used references are works by Pobell [80] and Ekin [81]. Far from being exhaustive, thissection will only highlight certain techniques necessary for operation of our buﬀer gas beam. We
will discuss some useful materials (both thermally conductive and insulating) as well as basic buﬀer
gas cell design. We are grateful to D. Patterson, J. Rasmussen, and J. Doyle for their contribution
to our development of an ND3 buﬀer gas beam source at JILA.
Construction of a buﬀer gas system begins with a cryostat. The cryostat is an evacuatedtube into which (in our case) two cold stages are mounted. These stages may be cooled by either
cryogenic liquids (N2 ,He,Ne) or with a cryogen-free refrigerator (e.g. pulse tube). The ﬁrst stage

61

4 K Helium Bath or
Pulse Tube Refrigerator 2nd Stage

285 K
ND3

360 mW
HeaterEpoxy
Fiberglass
Composite

Radiation Shield (< 100 K)

Charcoal
Sorb

Figure 3.5: Illustration of our buﬀer gas beam assembly. The ND3 gas is fed down a heatedcopper tube at 285 K. This hot supply tube is thermally isolated from the cold buﬀer gas cell
via a polyetherimide (Ultem) insert and thin-walled epoxy-ﬁberglass composite (G10) tube. The
pre-cooled He gas (not shown) is fed into the side of the cell. Charcoal sorb glued to copper plates(∼ 2000 cm2 total) acts as a vacuum pump for He and ND3 gas.

62of the cryostat is used to cool a few-mm-thick copper or aluminum radiation shield whose purpose
is to absorb 300 K blackbody radiation from the outer vacuum can (OVC). In “wet” systems, thisﬁrst stage is usually cooled with liquid nitrogen at 77 K. In cryogen-free systems, the ﬁrst stage canbe as cold as 30-40 K. From the Stefann-Boltzmann law, the intensity, IB , radiated by a blackbody
at temperature T is
IB = σT 4(3.9)

where σ = 5.67−12 W/(cm2 K4 ). From this, we expect a heat load of ∼ 45 mW/cm2 from the300 K OVC∗ . For a cylindrical OVC with a radius of 10 cm and height of 50 cm, the radiation
shield must absorb ∼ 140 W of blackbody power. With its large heat of vaporization of 199 J/g,
liquid nitrogen under a 140 W heat load will only boil at a rate of ∼ 3 L/hr. Moving past theradiation shield to the second stage of the cryostat, the blackbody heat load drops by a factor of 230
due to the T 4 scaling of Eq 3.9. However, the second stage is generally cooled with liquid helium
whose heat of vaporization is a much smaller 20.7 J/g. This means that a 1 W heat load burns
1.2 L/hr of liquid helium and that the experimenter must be diligent in minimizing second-stage
heat sources. One relatively easy method for reducing blackbody eﬀects is to lower the emissivity(absorbtivity) of cold surfaces. This basically means make them “shinier” to the infrared radiation
at 300 K and below, and can be accomplished by either polishing cold metal surfaces or wrapping
the system in highly-reﬂective aluminized mylar sheeting (also known as “space-blanket”). The
emissivity of highly-polished stainless steel can reach as low as 0.1, which would correspond to anorder-of-magnitude reduction in blackbody heat load.
Our buﬀer gas source is illustrated in Fig. 3.5. Originally, we developed this source within
a liquid helium cryostat that holds 1.5 L of liquid helium and ∼ 3 L of liquid nitrogen for theradiation shield. More recently, for the OH-ND3 collision experiment [18], we have mounted our
buﬀer gas cell to the second stage of a cryogen-free pulse tube refrigerator (Cryomech PT-410) to
simplify daily operation. In both cryostats the second stage and buﬀer gas cell temperatures are∗

The cryostat is generally constructed with suﬃcient thermal breaks between the OVC and cooled stages suchthat heat transfer is radiation-dominated.

63
4.5 − 5 K under typical operating conditions.
To produce the buﬀer gas cooled beam of ND3 in He, we ﬂow warm ND3 down a heated 3/32”
copper tube into the back of the buﬀer gas cell. The warm ammonia tube must be well-isolated
from the cold cell to avoid unnecessary heating and, in the case of the “wet” cryostat, maximize
runtime before reﬁlling liquid helium. With a combination of polyetherimide (Ultem) and thinwalled (0.010”) epoxy-ﬁberglass-composite (G10) thermal breaks, we have reduced our cell heatload to < 1 W when we are not ﬂowing He. These thermal breaks are shown in Fig. 3.5. When
we ﬂow the pre-cooled He gas into the side of the cell, gas conduction from the warm ammonia
inlet to the inner cell walls raises the heat load to 1 W. We control both the He and ND3 gas ﬂow
rates with thermal ﬂow controllers (MKS M100B). The primary vacuum pump in the cryostat isa cryopump consisting of 4.5 K copper sheets onto which are glued pieces of activated charcoal.
When cooled to near the boiling point of He, the charcoal sorbs trap the gas and hold it until thetemperature is increased. We ﬁnd that ∼ 2000 cm2 of sorb is suﬃcient to keep the He pressure
stable in the cryostat over our beam runtime of ∼ 1 hr.
Our 4 cm-long buﬀer gas cell is machined from a solid 3.8 cm x 3.8 cm bar of oxygen-free highconductivity (OFHC) copper. A 2 cm-diameter bore is drilled down the center of the block fromback to front. Tapped, helicoiled screw holes are also added to these ends for mounting of the ND3
inlet and 6 mm exit aperture. A small hole is drilled into the side of the cell for mounting of the
pre-cooled He gas inlet. We see no clear diﬀerence between mounting the ND3 inlet from the sideor the back, but other groups have reported more eﬃcient molecular cooling with a side-mounted
molecule inlet [82]. Furthermore, we have measured ND3 beams produced in both 12-15 K Ne and
4.5 K He, with He being the clear winner by an order of magnitude with respect to ultimate guided
molecular density in the |J, K⟩ = |1, 1⟩ para ground state.

3.2.2

Translational and Rotational Cooling

Much of the language of buﬀer gas beam dynamics is similar to that of supersonic expansions
with the exception that, in the case of buﬀer gas beams, one generally prefers as few collisions as64possible to cool the molecule of interest. The last thing we want is to work hard to cool ND3 to 5 K
only to signiﬁcantly boost the beam to supersonic velocities as it exits the cell aperture. Patterson
and co-workers identify three ﬂow regimes each possessing diﬀerent characteristic beam velocities(characterized here with the molecular mean free path ℓ and cell aperture diameter d) [83]:Eﬀusive (ℓ > d) Relatively few cold molecules are extracted from the cell before they stick to thecold walls, and the mean speed of the molecular beam is just the most probable speed of√the molecule of interest given by 2kB T /mmolecule .
Hydrodynamic (ℓ ∼ d) As the mean free path approaches the aperture diameter, collisions within
the expanding beam can cool the rotational degree of freedom below the thermal temperature. Also, the forward beam velocity is “boosted” to the most probable speed of the buﬀer√
gas given by ∼ 2kB T /mbuf f er . Additionally, considerable increases in cell extraction arereported in this regime [84].
Supersonic (ℓ ≪ d) This is the usual case covered in previous sections where signiﬁcant rotational
and translational cooling occurs and the beam is considerably boosted to the supersonic√velocity of ∼ 5kB T /mbuf f er . This expression for the supersonic velocity assumes an ideal
monatomic buﬀer gas (γ = 5/3).
Also in Ref. [83], it is argued that a hydrodynamic Ne buﬀer gas source operating at 15 Kshould perform better than a 4.5 K He source with respect to both beam temperature and ﬂux at
low velocities. Interestingly, we have directly compared a 15 K Ne/ND3 beam with a 5 K He/ND3
in our system and determined that the guided ﬂux of the 5 K He/ND3 beam was larger by a factorof > 10. This signal corresponded to a guided density of ∼ 108 cm−3 , fulﬁlling the ﬁrst goal of theintroduction. The corresponding data is shown in Fig. 3.6 for a ﬁxed cell-to-guide distance of 1 cm.
One possible reason for the diﬀerent results could be scattering of the hydrodynamically-boosted Ne
beam from the nearby hexapole electrodes. This is plausible since our optimal He:ND3 ﬂow rates
were 1:6 sccm while no combination of Ne:ND3 ﬂows in the range (1-20):(1-15) sccm produced
comparable guided beam densities. If the particle density is such that hydrodynamic boosting

65
1.8
1.6
1.4
1.2
1.0

15 K
0.8

Ne/ND

3

0.6

5 K

Guided ND

3

REMPI Signal [arb. u.]

2.0

0.4

He/ND

3

0.2
0.0
-0.2
0

20

40

60

80

100

120

140

160

180

200

Time [min]Figure 3.6: Data comparing electrostatically guided ﬂux of a 15 K Ne/ND3 beam with that of a 5
K He/ND3 under otherwise identical conditions. Flow rates for both conditions were varied overthe full range of the ﬂow controllers in the time shown, and the cell-to-guide distance was ﬁxed at
1 cm.

occurs, one would expect an enhanced sensitivity of beam parameters to nearby boundaries. For
reference, 1 sccm = 4.4 × 1017 s−1 .
The data of Fig. 3.6 convinced us to focus primarily on He/ND3 beams for production of acold, dense beam suitable for molecular collision experiments. Moving the buﬀer gas cell closer toour hexapole guide (now 2.5 mm cell-to-guide), we gained an additional factor of 2 in guided beam
density.
The science and technology of buﬀer gas beams is suﬃciently new that only a handful ofexperimental groups are actively running such sources. As such, the subject of rotational coolingin buﬀer gas molecular beams has not been studied extensively. Prior to our work, no direct
spectroscopic determinations of rotational temperatures had been published, although Sommer
and co-workers had investigated ND3 rotational temperatures using a state-dependent electrostatic
guiding model [85]. In our system, we have demonstrated control over the rotational temperature ofa He/ND3 beam – varying the eﬀective para temperature from 5 K to > 6 K. This is accomplishedby adjusting the thermalization rate of ND3 within the buﬀer gas cell with the relative ﬂow rates
of He and ND3 . At He:ND3 ﬂows of 2.0:2.5 sccm, we measure a rotationally hotter beam that is

66





6.3 K





5.4 K











Figure 3.7: Measured spectra of a guided buﬀer gas source of ND3 under two diﬀerent buﬀer gas
ﬂow conditions. The red curve gives a para rotational temperature of 6.3 K for buﬀer as ﬂows of
2.0 sccm He and 2.5 sccm ND3 . The black spectrum reﬂects the better rotational thermalization
observed with ﬂows of 3.5 sccm He and 1.0 sccm ND3 .

consistent with Trot = 6.3 K. However, by increasing the ﬂow ratio to 3.5:1.0, we observe Trot = 5.4
K, which is only slightly above the temperature of the buﬀer gas cell itself. This supporting datais shown in Fig. 3.7. While the diﬀerence in Trot seems rather small, the similarly small rotational
constant of ND3 results in a relative factor of 1.7 in the |1, 1⟩ : |2, 2⟩ population ratio over thistemperature range.

3.2.3

The “Pulsing” Phenomenon and Beam Runtime

Ammonia freezes at a temperature of 195 K. Therefore, if you expect ice formation within
the 4.5 K buﬀer gas cell to play a major role in beam dynamics over long time periods, you would
be correct. In fact, our ﬁrst buﬀer gas cells were quite a bit smaller than our ﬁnal design of 3.8
cm x 3.8 cm x 4 cm and possessed exit apertures of ∼ 1 mm not unlike those used by the Rempegroup. However, with cells smaller than 2 cm we see frequent disruptions in the cold beam ﬂux
accompanied by large pressure bursts and cell temperature increases. These ﬂux dropouts can beseen in Fig. 3.8a. Having tried a host of diﬀerent ND3 tube mounting schemes, we always see this“pulsing” eﬀect with small cells. Furthermore, we observe that small apertures clog with ammonia
ice over time scales similar to those reported in Ref. [85]. While those authors use a heater to de-ice

67
their aperture, they still observe constant reduction in output ﬂux over the intervening 1-2 hrs. To
avoid this eﬀect, we use the aforementioned 6 mm-diameter exit aperture. While this likely leads
to more eﬀusive beam behavior due to an inability to build up He pressure in the cell, we see no
evidence of long-term icing with such a large exit nozzle.
We have found that the only solution to the “pulsing” eﬀect is to scale up the dimensions of
the cell so as to maximize the distance between the warm ND3 inlet and any cold surface. This
result suggests that ammonia ice accumulates on the cell wall over time and eventually bridges the
gap to the warm inlet (see Fig. 3.8c). The “pulse” most likely results from rapid vaporization of ice
at the warm ammonia tube and subsequent heating of the He gas within the cell. With a distanceof ∼ 1 cm between the ND3 tube and cell wall, we obtain 1 hr of continuous, stable runtime of thecold beam as shown in Fig. 3.8b.

3.2.4

Electrostatic Velocity Filtering

The beam emitted from a buﬀer gas cell consists mostly of coolant atoms. The cooled
molecular sample can also exhibit population in excited rotational levels. In order to gain control
over both the translational and rotational temperature of the cooled molecular beam and ﬁlter the
species of interest from the buﬀer gas, it is necessary to introduce a state- and velocity-selective
guide. Both magnetic octopole [86] and electrostatic quadrupole [87] guides have been employed inprevious buﬀer gas beam experiments. For our system, we chose to implement a curved electrostatic
hexapole to ensure maximum transverse guide depth and ﬁeld homogeneity. An illustration of ourbuﬀer gas cell and electrostatic guide are given in Fig. 3.9. Note that we use separate straight andcurved guide sections to allow isolation of the buﬀer gas source from the ultra-high vacuum of the
OH trap chamber via a thin (3 mm) gate valve. This feature is absolutely necessary since ND3 icemust be removed from the buﬀer gas cell after ∼ 1 hr of continuous beam operation. To accomplish
this, we warm the cryostat to room temperature over ∼ 4 − 6 hrs and pump out the He and ND3
released from the charcoal sorbs and buﬀer gas cell.
The transverse dynamics of conﬁned ND3 molecules is rather straightforward and, neglecting

68

a

b

c
280 K
ND3

ND3 Ice

5K
He

Figure 3.8: Comparison of pulsing behavior with (a) < 5 mm spacing between the ammonia inlet
and cell wall and (b) the larger 3.8 cm x 3.8 cm x 4 cm cell with an inlet-to-wall spacing of 1 cm.
The greater distance between the warm ND3 inlet and the cold cell wall in (b) leads to signiﬁcant
improvement in beam stability over ∼ 1 hr. (c) Illustration of the buﬀer gas cell cross section
showing regions of ND3 ice buildup that eventually lead to pulsing as ice accumulates near thewarm ND3 inlet tube.

69

Sta

rk

dec

ele

rat

317 nmREMPI laser

orHexapole guideTime-of-flightmass spectrometer(ND+3 detection)

Thin gatevalve

He/ND3 buffer
gas cellFigure 3.9: Drawing of the buﬀer gas beam assembly and electrostatic velocity ﬁlter for cold ND3molecules. The ionization region and time-of-ﬂight mass spectrometer assembly for ND+
3 detection
are displayed at the terminus of the hexapole guide. Cold ND3 is ionized 1 cm from the guide exit
with the 317 nm pulsed REMPI laser.

70
the ammonia inversion splitting, conforms to a harmonic oscillator model. Ignoring the ﬁnite extent
of the hexapole rods (generally a safe approximation), one can write the following expression for
the electric ﬁeld within a hexapole of inner radius r0 as a function of radial position r as⃗
|E(r)|
=

3V r2
.
r03

(3.10)

The absolute value of the two symmetric voltages applied to the hexapole rods is denoted by V
(V = 5 kV for applied rod voltages of ±5 kV). Of course, the potential energy of a molecule with
an eﬀective electric dipole moment µef f within the hexapole is (ignoring inversion splitting)2⃗ = 3µef f V r .
|U (r)| = |µ · E|
r03

(3.11)⃗ (r) which comes out toNow, we obtain the force on the molecule F⃗ (r) = −∇U
6µef f V r
F⃗ (r) = −r̂ = −krr̂
r03
6µef f V
for k =.
r03

(3.12)(3.13)

Using Hooke’s Law, we can now deﬁne a transverse oscillation frequency, ν, for the guided moleculesof mass m as
1
ν=
2π

√

k
1
=
m
2π

√(

6µef f V
mr03).

(3.14)

Specializing to ND3 , we set V = 5 kV and r0 = 3 mm to obtain ν = 1.47 kHz.
By bending the guide, we can ﬁlter fast species — both buﬀer gas atoms and molecules —from the cold ND3 beam. Using Newton’s Second Law, we can estimate the maximum guidable
velocity (vmax ) for a hexapole bend of radius R as√
6µef f V R
vmax =
mr02(3.15)

where, assuming the experimental condition of R = 13.5 cm, we get vmax = 186 m/s. More accurate
Monte Carlo simulations yield a maximum velocity of 150 m/s, but Eq. 5.2 gives the correct scaling
behavior in bend radius R. Given the following expression for the µef f of a symmetric top molecule:µef f = µKMJ
J(J + 1)

(3.16)

71
we see that all K = 0 (ortho) states of ND3 are unguided. Furthermore, the guide depth andtransverse oscillation frequencies increase as J = K increases, leading to more eﬃcient guiding
of excited, fully-aligned rotational states. These factors result in a non-thermal distribution ofrotational state populations at the exit of the bent guide.

3.2.4.1

Measurement of Beam Velocity

Measuring the velocity of a continuous beam is tricky. Since there is no well-deﬁned temporalpulse origin, one cannot simply take time-of-ﬂight measurements at diﬀerent distances from thesource, as is commonly done with supersonic beams. The ﬂow controllers used for introductionof He and ND3 into our buﬀer gas source cannot be gated on time scales fast enough for creating
pulses. However, our electrostatic velocity ﬁlter gives us a method for measuring the translationalenergy distribution. Not unlike recent work done in the group of G. Rempe [88], where pulses werecreated by switching an electrostatic guide, we switch our guide on for diﬀerent durations (∆t) andrecord the guided ﬂux of ND3 as a function of this switch time. Since our guide is 50 cm in path
length and includes a 90◦ bend, we can only detect molecules that have been guided throughout
the full ﬂight time. The guide length (L) and duration (∆t) allow us to set a minimum guidable
velocity deﬁned as vmin (∆t) = L/∆t. One can then deduce that the measured density as a functionof vmin , here deﬁned as f (vmin ), can be written as∫ ∞f (vmin ) =g(v′ )dv′(3.17)

vmin

where g(v) is the actual velocity distribution of the continuous guided beam. Since determination ofg(v) requires diﬀerentiation of f (vmin ), we ﬁrst ﬁt f with a high-order polynomial. This smoothes
and interpolates the curve so that small variations do not cause un-physical features in g. One canimagine ﬁtting f with an error function (Gaussian integral), but we ﬁnd that the guided velocity
distributions are not perfectly Gaussian.
The measured function f (vmin ) is displayed in Fig. 3.10a along with its associated velocity
distribution in Fig. 3.10b. The data is taken with hexapole voltages of ±5 kV. Combining this with

72
the guide bend radius of 13.5 cm, we see that most of the ND3 molecules lie below the predicted
150 m/s velocity cutoﬀ.

f (vmin)g (v)Figure 3.10: Measured velocity distribution of electrostatically guided ND3 molecules. The processof obtaining g(v) from the measured f (vmin ) is described by Eq. 3.17.

Chapter 4

Molecular Detection Techniques

4.1

Time of Flight Mass Spectrometry
Ionization with time-of-ﬂight mass spectrometry is a rather sensitive detection technique forneutral molecules or atoms. Using resonance-enhanced multiphoton ionization (REMPI) in concert
with a time of ﬂight mass spectrometer (ToFMS) allows for both internal state and charge-tomass ratio resolution. The most commonly used ToFMS design is based on that of Wiley and
McClarin [89] in which a sample is ionized and the product ions are subsequently accelerated
towards a sensitive ion detector (usually an electron multiplier device). Provided that stray electricﬁelds are well-shielded, ions may be guided to the detector with near-unit eﬃciency. This device
can therefore be used with even the smallest molecular densities when coupled with an eﬃcientionization technique. As described in Chapter 2, we use 2+1 REMPI to characterize our cold
guided beam of ND3 molecules. In order to detect ND3 within the experimental collision region(i.e. between the OH trap magnets), we use the magnets themselves to apply an electric ﬁeld forND+
3 extraction.
The design of our ToFMS is illustrated in Fig. 4.1. To ionize ND3 , we focus a 317 nm pulsed
laser between the two permanent ring magnets of our OH trap. The magnets are grounded forthe ionization step to avoid inducing Stark shifts on the REMPI transition and concomitant state
mixing of inversion-doublet levels. After the ND+
3 is produced, we pulse a voltage of +950 V onto
the front magnet to accelerate the ions out of the magnetic trap region. The large magnetic ﬁelds
within the trap do not have a signiﬁcant eﬀect on the ion ﬂight path since they are axial near the

74

REMPI
laser

+ 950 V0V

- 1.1 kV- 1.1 kV

Microchannel
PlatesFigure 4.1: Top-view illustration of our ToFMS for detection of guided ND3 molecules within the
OH-ND3 collision region. For initial ND+
3 extraction, a pulse of +950 V is applied to the front
magnet after the ionization laser. Ions are then accelerated to the microchannel plate detector.

75
magnet surface and because the period of cyclotron oscillations near the magnets is ∼ 1 µs, which
is 100x longer than the ion extraction time at the applied electric ﬁeld. After exiting the trap, ND+
3
is further accelerated by an additional charged plate 1 cm behind the back magnet. The region
between the two plates at -1.1 kV is ﬁeld-free and is generally installed to allow ions with diﬀerent
charge-to-mass ratios to spatially separate on their way to the microchannel plate detector. With asuﬃciently long drift region, one can achieve mass resolutions below one atomic mass unit, thereby
separating diﬀerent atomic or molecular species produced in the ionization region. Due to spaceconstraints, our ToFMS features a 2 cm drift region which is not suﬃciently long to reach this levelof mass discrimination (many drift regions are 50 cm to 1 m). Nevertheless, the purity of our ND3
source and ultra-high vacuum conditions give a negligible amount of background ions.

4.1.1

Calibration of Ampliﬁed Ion Signals

Determining absolute densities of neutral atomic or molecular samples from ampliﬁed ion
currents produced via REMPI is often diﬃcult. When multiple photons are used to excite an atom
or molecule to an intermediate electronic state, the saturation behavior of the given transition
makes ionization eﬃciency rather sensitive to excitation laser parameters such as waist size andtransverse beam proﬁle. The use of pulsed dye lasers for REMPI renders accurate determinationof these parameters unlikely. Microchannel plate (MCP) detectors, which boast current gains of∼ 106 −109 , are commonly used to amplify ion signals as part of a time-of-ﬂight mass spectrometer.
Unfortunately, the gain of a given MCP assembly can change over the lifetime of the unit and vacuum breaks or complete replacement of the MCP stack would all require careful re-characterizationof the system gain for reliable calibrations.
In order to accurately calibrate the density of our cold, guided ND3 buﬀer gas beam, wechoose to compare the ampliﬁed 2+1 REMPI ion yield from the cold beam to that of a calculable
thermal source of ND3 molecules introduced into the ionization region. This procedure makes thedensity measurement insensitive to long-term ﬂuctuations in MCP gain and laser size/proﬁle sinceboth the cold beam ion yield and the density scaling factor obtained from the thermal signal include

76
these eﬀects. Since REMPI laser parameters and buﬀer gas source ﬂux can change on a daily basis,
this density calibration is performed for each realization of the collision experiment to allow forconsistent averaging of many cross section measurements. We determine the density, n, of ND3
using
n=

Sb (ν)P×× fJK (T ) × Cpressure′′
Sth (ν )Coverlap (ν ) kB T

(4.1)

where Sb (ν) is the ampliﬁed ion signal for a given REMPI line of cold ND3 at a laser single-photon
energy of ν, Sth (ν ′ ) is the ampliﬁed ion signal from the thermal ND3 at pressure P and temperature
T , kB is Boltzmann’s constant, and fJK (T ) is the thermal fraction of ND3 molecules in the state
J, K at temperature T . The constants Coverlap (ν ′ ) and Cpressure allow one to correct for the requisite
overlap of transition lines in the hot spectrum and pressure diﬀerences at diﬀerent positions in thevacuum chamber, respectively. In principle, the frequencies ν and ν ′ need not be identical so longas the laser beam proﬁle and position are stable over the range (ν − ν ′ ).
It is necessary to simulate the 2+1 REMPI spectrum of ND3 in order to accurately assign avalue for Coverlap . Therefore, we calculate all positions and line strengths for B̃(v2 = 5) ← X̃(v2 = 0)
rotational transitions using J ′′ , J ′ < 20 and assuming that the two-photon transition strengths are
determined exclusively by the transition hyperpolarizability tensor T2±1 [52]. When describing
2+1 REMPI transitions, we use |J ′′ , K ′′ ⟩ and |J ′ , K ′ ⟩ to represent the ground X̃ 1 A′1 (v2 = 0) andintermediate B̃ 1 E ′′ (v2 = 5) molecular states, respectively. Molecular constants for the groundstate are obtained from Ref. [90] while constants for the intermediate level are taken from Ref. [50].
Accounting for mixing of transition strengths due to l-uncoupling (mixed K, K + 2 states) at highJ [46], we obtain the simulated spectrum of Fig. 4.2a shown in red. Accompanying the spectrum
simulation is the measured thermal spectrum (black points) taken at a pressure P = 2.5 × 10−7
Torr and laser pulse energy of 3 mJ. A 30 cm lens is placed one focal distance from the magnetic
trap center. When calibrating the cold ND3 beam, we restrict P to below 10−6 Torr to avoid
nonlinearities in the response of our calibrated quadrupole mass spectrometer (Stanford Research
Systems RGA200) and MCP detector.

77
The constant Coverlap (ν ′ ) ≤ 1 is the fraction of ions produced from the state J, K within the
background of other lines overlapping ν ′ in the thermal room-temperature spectrum. At T = 295
K, the most populous angular momentum states are J = 5 − 6. As such, ions resulting fromREMPI of the |1, 1⟩ state make up a fraction of the total ion yield at a given line position ν ′ . For
the OH-ND3 collision experiment described in Chapter 7, we used the B̃|2, 0⟩ ← X̃|1, 1⟩ line ata single-photon frequency of 31505.6 cm−1 for calibration of the cold ND3 density. The position
and relative ion yield of this transition is shown in blue in Fig. 4.2a. Integrating ion yield over therange of the REMPI laser FWHM of 0.3 cm−1 obtained from spectroscopy of the 5 K ND3 beam,
we obtain Coverlap (31505.6 cm−1 ) = 0.16. The discrepancy between the simulated and measured
295 K spectra at this line position is addressed in the following section concerning error estimationof Coverlap . For comparison, we include a measured (black) cold ND3 spectrum and corresponding
simulation (red) in Fig. 4.2b. The cold beam spectrum reﬂects a rotational temperature of 8 K.
To compute the Boltzmann fraction fJK (T ) in Eq. 4.1, we use the following expression forsymmetric top molecules [44]fJK (T ) =1 ∑S(I = 1, K)(2J + 1) exp (−UJK /kB T )
2Z

(4.2)

J,K

where Z is the partition function and S(I, K) is the degeneracy due to the three identical D
atoms each possessing nuclear spin I = 1. The remaining terms comprise the standard Boltzmann
fraction for a |J, K⟩ state of energy UJK . The extra factor of two multiplying the partition function
is necessary since the 250 transitions in ND3 connect only antisymmetric v2 = 0 levels with v2 = 5
intermediate rotational states, while our pressure measurement is sensitive to all states. Using thisformula, we obtain f11 (295 K) = 4.7 × 10−3 .
The constant Cpressure is the ratio of actual pressure in the magnetic trap to measured pressure
at the quadrupole mass spectrometer (QMS) when gas is leaked into the trap chamber. To quantify
this term, we physically move our QMS from its customary position between the OH magnetic
trap and turbomolecular pump to a port parallel to the trap itself. Using a calibrated mass
ﬂow controller, we inject Ne through the usual ND3 leak valve port and measure the pressure at

78

a

295 K

b

8K

Figure 4.2: (a) Measured (black points) and simulated (red line) ND3 spectra at 295 K. The experimental 2+1 REMPI spectrum was taken within the OH-ND3 collision region. The relative strengths
and line positions for two of the |J ′′ , K ′′ ⟩ = |1, 1⟩ transitions at the single-photon frequencies of31505.6 cm−1 and 31510.9 cm−1 are shown in blue. (b) Measured (black points) and simulated(red line) spectrum of a guided ND3 beam with a rotational temperature of 8 K.

each QMS position. Repeating this procedure multiple times, we obtain Cpressure = 1.4(1). Neongas is substituted for ND3 in this measurement due to their identical mass and neon’s chemicalcompatibility with the mass ﬂow controller. Under molecular ﬂow conditions, the identical mass of
Ne will yield the same pressure diﬀerentials as ND3 .

4.1.1.1

Estimating the Error of Coverlap

To estimate the systematic error of our line overlap correction factor Coverlap (ν ′ ), we compare
the spectrum simulation results of Fig. 4.2a with an experimental measurement of the ratio ρexp =(Coverlap (ν1′ )/Coverlap (ν2′ )). To obtain ρexp , we calibrate the density of the cold ND3 at two diﬀerent
ν ′ corresponding to distinct REMPI transitions from the |1, 1⟩ antisymmetric ground state. The
positions and relative ion yields of these two transitions with ν1′ = 31510.9 cm−1 and ν2′ = 31505.6
cm−1 are plotted as the blue line in Fig. 4.2a. We compare this measured ratio with that predictedby the spectroscopy simulation (ρsim =0.73). Our measured ρexp of 1.38(7) yields a 47% fractional
uncertainty estimate in our determination of Coverlap . The imperfection in our spectrum simulation
at 31505.6 cm−1 is causing this large fractional uncertainty in the determination of absolute ND3
densities and, consequently, loss cross sections.

79

4.2

Laser Induced Fluorescence
The concept of laser induced ﬂuorescence (LIF) detection is much simpler than that of REMPI

and time of ﬂight mass spectrometry. In the case of OH, we excite the molecule along the 2 Σ+ (v =1) ← 2 Π3/2 (v = 0) electronic transition at 282 nm and detect the shifted ﬂuorescence photons
generated by decay along the 2 Σ+ (v = 1) → 2 Π3/2 (v = 1) transition at 313 nm. Making use of
diﬀerent vibrational pathways for excitation and decay allows for ﬁltering of stray 282 nm photons
from the pulsed UV laser and, due to the favorable Franck-Condon factor for (v = 1) → (v = 1)
electronic transitions, 70% of the excited OH molecules emit photons at 313 nm over the 750 ns
lifetime of the excited state.
In contrast with REMPI, we collimate the 282 nm LIF laser to a diameter of ∼ 3 mm tomaximize the interrogation volume within the OH magnetic trap. With this laser size, we observe
saturation of the ﬂuorescence signal over the full laser diameter at a pulse energy of ∼ 300 µJ. For
maximum signal-to-noise, it is critical to place light-collection optics within the vacuum chamberas shown in Fig. 4.3. We mount a short focal length lens (f = 2.5 cm) one focal distance from thetrap center. This light is then refocused onto a photomultiplier tube (PMT) outside of vacuum
using an identical lens placed 5 cm after the ﬁrst lens. To reduce the photon signal due to scatteredexcitation light, we employ a combination of color-glass and interference ﬁlters to block wavelengths
below 300 nm and heavily red-shifted ﬂuorescence above 400 nm. These red-shifted photons are
generated by UV surface scattering from diﬀerent materials near the magnetic trap. We obtainthe largest signal-to-noise ratios using a bialkali PMT mounted in a UV-transmissive glass housing(Hamamatsu R3788). While one can use a wavelength-sensitive PMT quantum eﬃciency to further
ﬁlter light below 300 nm (as with the Hamamatsu 1P21), we measure signal-to-noise gains of ∼ 2
using the former model.
Photomultiplier tubes exhibit rather large current gains of 106 − 107 , which can lead tononlinear eﬀects such as “after-pulsing” events due to large numbers of stray photons from theexcitation laser. As described in Eric Hudson’s Ph.D. thesis [23], we reverse-bias the individual

80
dynodes of the PMT to restrict electron multiplication to time periods after the laser ﬁre. Gating
the PMT in this manner drastically reduces the number of electrons produced during the 10 ns
laser pulse and thereby decreases the “after-pulse” counts during the OH ﬂuorescence lifetime.

81

OH
2 +
Σ1/2 (v = 1)

313 nm282 nm2

Π3/2 (v = 1)2

Π3/2 (v = 0)

Iris
Filters
Photomultiplier
Tube

Figure 4.3: (Left) Side-view illustration of the Stark decelerator, magnetic trap, and OH laserinduced ﬂuorescence detection assembly. Fluorescence photons are collected at 313 nm with a 2.5
cm-diameter lens placed 2.5 cm from the center of the magnetic trap. (Right) Level structure
showing the excitation and decay pathways used for LIF detection of OH molecules.

82
4.2.1

Calibration of LIF Signals

Since OH is detected by counting ﬂuorescence photons over the 750 ns lifetime of the excited
2 Σ+ (v = 1) state, underestimation of the OH density is all but guaranteed. The trick to accurately

calibrating LIF signals is to correctly account for all photon loss mechanisms between the excitation
laser and photomultiplier tube output — all of which increase the ﬁnal calibrated density of themolecular sample. The ﬁrst consideration is to conﬁrm that the photon counter is not saturating due
to overlapping photon pulses. With our multichannel scaler/averager (Stanford Research Systems
SR430), we can reduce the counting bin size to a minimum of 5 ns and check that every bin iscounting less than 1 photon per laser shot. Next, one must calculate the percentage of 4π steradians
subtended by the ﬂuorescence-collection lens (hereafter denoted as dΩ/4π). Finally, one needs thequantum eﬃciency (QE) of the PMT used for light collection. Assuming that all molecules withina laser volume, V , are excited, we may estimate the number density, n, of the molecular sample asn∼

S [photons / laser shot](V )(Losses)(QE)(FC)(dΩ/4π)(4.3)

where FC is the Franck-Condon factor for decay at the detected wavelength, S is the measured ﬂuorescence signal, and (Losses) represents combined losses due to imperfect UV transmission throughvarious optics and ﬁlters. Since the spatial proﬁle of pulsed UV lasers is generally not well-deﬁned,
estimating the exact laser excitation volume is fraught with error. Therefore, density estimates for
LIF detection are usually only trusted to within an order-of-magnitude. For reference, Table 4.1
gives our typical and best observed S values for OH under diﬀerent experimental conditions.
Table 4.1: Typical and best detected OH (2 Π3/2 ) signals under various experimental conditions. All
numbers shown are for detection within the permanent magnet trap. Quoted errors are standard
deviations from a number of diﬀerent measurements.
Condition
Background Scatter
Bunching (490 m/s)
Slowing (36 m/s)
TrappedTypical [photons/shot]
0.18 ± 0.07
7.6 ± 1.85
0.5 ± 0.1
0.23 ± 0.07

Best [photons/shot]
0.06
9.7
0.6
0.37

Chapter 5

Stark Deceleration

Recent development of cold polar-molecule sources promises to reveal many interesting, andhitherto unexplored, molecular interaction dynamics. The permanent electric dipole moment possessed by polar molecules provides a new type of interaction in the ultracold environment. This
electric dipole-dipole interaction (and control over it) should give rise to unique physics and chemistry including novel cold-collision dynamics [91, 92, 18] and quantum information processing [93].
To date, cold polar-molecule samples have been produced most successfully via three diﬀerent
mechanisms: buﬀer gas cooling [94, 73]; photo- and magneto-association [95, 96, 79]; and Stark
deceleration [97]. Buﬀer gas cooling achieves temperatures below 1 K through thermalization ofmolecules with a He buﬀer. This technique produces relatively large densities (108 cm−3 ) of polar
ground-state molecules. Ultracold atom association achieves the lowest molecular temperatures ofthese techniques (∼100 nK), but is limited to molecules whose atomic constituents are amenable
to laser-cooling. Stark deceleration exists as an alternative to these methods as the techniqueemploys well-characterized supersonic beam methods [58] to produce large densities of ground statepolar molecules (∼109 cm−3 ), albeit at high packet velocities. One limitation of this technique fortrapping of decelerated molecules is an observed drastic loss of slowed molecules at very low mean
velocities in both our own group’s work and the Berlin group of G. Meijer [67].
The ﬁrst successful implementation of Stark deceleration was reported in 1999 [19]. The
group of G. Meijer slowed a supersonic beam of metastable CO molecules from 225 m/s to 98 m/s.
In collaboration with their group, we subsequently demonstrated the ﬁrst Stark deceleration of a

84
free-radical species (OH) [75]. A number of chemically-relevant neutral polar molecules have been
Stark decelerated in the intervening years including: ND3 [98], H2 CO [63], NH [99], LiH [100], andSO2 [101].

5.1

Decelerator Operation
The key concepts of Stark deceleration are similar to those of charged-particle accelerators.

However, since we are interested in decelerating neutral polar molecules, our slowing forces must
⃗ rather than the electric ﬁelds necessary
be provided by electric ﬁeld gradients (F⃗ = ∇(⃗µ · E))⃗ The largest electric ﬁelds that can be applied toto apply forces to charged particles (F⃗ = q E).
polished stainless steel surfaces in vacuum are ∼ 100 kV/cm (∼ 10 MV/m) [102]. Assuming themolecule of interest has a permanent dipole moment of 1 D fully aligned with the local ﬁeld, one
can induce a linear Stark shift of ∼ 1 cm−1 at this electric ﬁeld magnitude. Supersonic molecularbeams are produced at mean velocities of ∼ 500 m/s (assuming a Kr carrier gas) which, for the
OH molecule, corresponds to an initial kinetic energy of 177 cm−1 . By combining an array of∼ 100 of these high-ﬁeld stages and successively removing 1 cm−1 of molecular kinetic energy per
stage, one can expect to be able to decelerate or even stop molecular pulses using the Stark eﬀect.
However, one must have the ability to switch the electric ﬁelds on time scales much shorter thana molecular transit time, otherwise the polar molecule will fully traverse the potential maximum
and exit with its initial kinetic energy unchanged. Assuming that each stage has a length of 1 cm,
this suggests that we must switch the electric ﬁelds much faster than 20 µs. While this conceptof Stark deceleration of molecular beams has existed for decades, the technology needed to switchhigh voltage pulses (±15 kV) on time scales of 1 µs did not exist until the 1990s.
Figure 5.1 illustrates the design of our Stark decelerator assembly. First, we create a supersonic beam of polar molecules at the pulsed valve by seeding the molecule of interest within a carriergas of either Kr or Xe. These heavier noble gases are employed since they produce slower beams
better matched to the slowing capabilities of our decelerator. After passing through a skimmer,
the molecular pulse enters an electrostatic hexapole with an inner diameter of 6 mm and length ofSkimmer

Stark Decelerator(142 Stages)

Figure 5.1: Illustration of the Stark decelerator and pulsed supersonic molecular beam source. A packet of OH is created at the valve
using a pulsed electric discharge positioned at the valve nozzle. After passing through the skimmer 10 cm downstream from the discharge
plates, the packet is focused into the decelerator via a 5 cm hexapole with an inner diameter of 6 mm. The decelerator rod pairs are spaced
by 5.5 mm center-to-center along the beam ﬂight (longitudinal) direction and are mounted 2 mm apart in the transverse dimension.

Pulsed Supersonic
Valve

Hexapole85

86
5 cm. To minimize discharge events between the hexapole and decelerator rods, we only charge thehexapole when the molecular packet is present. This hexapole is eﬀectively a mode-matching optic
that optimizes the transverse velocity and spatial widths of the packet to match the transverse
phase-space acceptance of the Stark decelerator. The Stark decelerator is mounted such that there
is a 3 mm gap between the hexapole and ﬁrst rod pair. The longitudinal center-to-center distance
between successive rod pairs is 5.5 mm, while the transverse distance between parallel rods comprising a single stage is 2 mm. These two parameters deﬁne the spatial acceptance of the decelerator
and, under identical electric ﬁeld distributions, the number of molecules within a decelerated packet
scales simply as the volume deﬁned by these two lengths. Although decelerators possessing a 4 mm
transverse width and 1 cm longitudinal stage spacing have been constructed [103], the two-foldincrease in required voltage (±20 kV) over the smaller design introduces many technical challenges.
The longitudinal velocity acceptance of a decelerator is determined by the electric ﬁeld gradients
therein. In our system, we apply ±12.5 kV to our rod pairs to obtain a maximum electric ﬁeld of
125 kV/cm between the rods.
To describe the deceleration sequence, it is necessary to ﬁrst deﬁne the phase angle ϕ =180◦ (z/L), where z is the longitudinal position and L = 5.5 mm is the stage periodicity of our slower.
The concept of a phase is natural for the deceleration process since the electric ﬁeld distribution
within our decelerator repeats over 142 stages. To remove kinetic energy from the molecular
packet, we switch between two distinct electric ﬁeld conﬁgurations provided by alternating rod
pairs in the decelerator. As shown in Fig. 5.2a, by allowing a molecule to traverse the left side of
the potential deﬁned by the central (red) curve, we convert some of its kinetic energy to potential
energy. By quickly grounding the rod pair that is providing this Stark potential and charging thetwo neighboring rod pairs, the molecule will have lost that kinetic energy and will again traverse
a potential hill given by the black curve. Repeating this process, we can sensitively control thelongitudinal speed of a molecular packet. We deﬁne the deceleration phase angle, ϕ0 , to be thepoint at which the electric ﬁelds are switched between these two conﬁgurations. The ϕ0 = 0◦ point
is deﬁned as the position where the two potentials of Fig. 5.2a cross. No net deceleration occurs

87

b

c

Longitudinal Velocity [m/s]aφ0 = 0o∆Figure 5.2: (a) Simulation of the longitudinal Stark potentials for OH created by the two distinct
rod pairs within the decelerator. To slow a packet of molecules, we switch between the red and black
potentials to continually remove kinetic energy at a well-deﬁned position relative to the rod pair
(ϕ0 ). (b) The diﬀerence between the two Stark potentials shown in panel (a), which represents theamount of kinetic energy removed from OH molecules in one stage as a function of the ﬁxed slowing
phase angle ϕ0 . Stable acceleration occurs for −90◦ < ϕ0 < 0 while phase-stable deceleration isobserved for 0 < ϕ0 < 90◦ . We denote operation at ϕ0 = 0◦ as “bunching” since no net kinetic
energy is removed from the molecular packet, but the molecules are nevertheless conﬁned within amoving longitudinal potential well. The form of the kinetic energy diﬀerence closely follows thatof a sine function as demonstrated by the ﬁt curve (dashed line). (c) Longitudinal phase-space
acceptance of a deceleration sequence as a function of ϕ0 assuming a one-dimensional sine-function
form for the slowing potential of panel (b). Note that the separatrices shown are incremented by10◦ starting from 0◦ at the outer edge. Molecules with longitudinal velocities and positions within
the separatrix for a given ϕ0 will be phase-stably decelerated.

88
for this “bunching” sequence. Furthermore, we set ϕ0 = 90◦ to be the point of maximum kinetic
energy loss. In a given deceleration sequence, ϕ0 is a constant, which means that we remove aﬁxed amount of energy per stage. Because of this, the frequency of high-voltage pulses to our
deceleration stages must be chirped as the packet velocity changes. For the speciﬁc case of OH,
the change in kinetic energy (∆KE) versus deceleration phase angle is plotted in Fig. 5.2b. This
energy removal per stage closely resembles a sine function as shown by the ﬁt (dashed curve).
The dynamics of Stark deceleration may be mathematically modelled using the following
second-order diﬀerential equation, which is that of a pendulum driven at constant torque [104]:d2 ∆ϕ Wmax π+[sin (∆ϕ + ϕ0 ) − sin ϕ0 ] = 0
dt2
mL2

(5.1)

where ∆ϕ = (ϕ − ϕ0 ) at time t, Wmax is the maximum amount of kinetic energy removed perstage, and m is the mass of the particle. This model represents an idealized one-dimensionalslowing potential and does not include transverse guiding eﬀects or the deviation of the longitudinalpotential from a sine wave. Nevertheless, we may solve Eq. 5.1 at ϕ0 for diﬀerent initial conditions to
obtain the phase space acceptance for diﬀerent deceleration sequences. The phase-space boundary
between stable and unstable molecule orbits is deﬁned by a separatrix. The separatrices for diﬀerent
ϕ0 are given in Fig. 5.2c in increments of 10◦ beginning from ϕ0 = 0◦ . The phase space acceptancedecreases monotonically as the molecular packet is slowed more aggressively, and the maximum
velocity acceptance, ∆vmax , may be computed as
√
[
(
)
]
2Wmax
π
ϕ0
cos ϕ0 −1 − ◦ sin ϕ0 .
∆vmax = 2
mπ
2
90

(5.2)

The above maximum velocity acceptance exists for ϕ = ϕ0 at the so-called “synchronous molecule”
position. Similarly, we may deﬁne the largest allowed forward excursion (∆ϕmax ) from the positionof the synchronous molecule at the synchronous molecule velocity (∆v = 0):∆ϕmax = 180◦ − 2ϕ0 .

(5.3)

To accurately model the dynamics of a deceleration sequence and simulate experimental timeof ﬂight proﬁles, we forego the approximations of Eq. 5.1 and use potential curves simulated from89
a three-dimensional ﬁnite element analysis of our electric ﬁeld distributions. There are a numberof commercial programs available for this application. We primarily use SIMION or COMSOL
and ﬁnd that a grid spacing of 0.1 mm along with linear interpolation between points is suﬃcientto accurately model the electric ﬁeld gradients within our slower. To generate a high-voltage
switching sequence for deceleration at ϕ0 , we simulate the longitudinal position versus time of asingle synchronous molecule ﬂying along the axis of the decelerator. We subsequently simulate
the ﬂight of a packet of ∼ 106 molecules through this slowing sequence and, given some detection
volume at the end of the decelerator, generate a time of ﬂight proﬁle. The results of these Monte
Carlo simulations are compared with actual deceleration data as shown in Fig. 5.3. Note that
we only scale the overall height of the curve to match the data for the ﬁrst bunching trace, andall subsequent simulations at diﬀerent ϕ0 have no free parameters. The experimental curves wereobtained from a decelerated OH beam initially seeded in a Xe expansion. The excellent agreement
between data and simulation ensures that we understand both the dimensions (and hence electricﬁelds) of our decelerator as well as the molecular dynamics of a slowing sequence.
The experimental signature of Stark deceleration is the removal of a phase-stable slice ofmolecules from the the broader molecular distribution. We can see from Fig. 5.3 that, as the phaseangle is increased, the stable packet lags further and further behind the unstable background of nondecelerated molecules. One can also see that multiple phase-stable packets are decelerated. Thisoccurs when the longitudinal spatial spread of the incoming molecular beam occupies more than one
stage length. Molecules one stage ahead of the synchronous molecule see one less deceleration stageand exit at a higher velocity while molecules loaded one stage behind the synchronous molecule see
one more stage and exit slower. This is most obvious for the slow peaks shown in the bottom twopanels of Fig. 5.3. In these instances, the molecules one stage behind the synchronous molecule are
stopped within the decelerator (not detected) and the packet one stage ahead is faster and denser
than the central packet of interest.
Since the magnitude and sign of a molecule’s Stark shift is state-dependent, a given deceleration sequence is unique to F, MF total angular momentum quantum numbers as well as the zero-ﬁeld

90Time of Flight [ms]Time of Flight [ms]Time of Flight [ms]Time of Flight [ms]Time of Flight [ms]Figure 5.3: Time of ﬂight data (black) and Monte Carlo simulation results (red) for Stark decelerated OH molecules showing the excellent agreement between ﬁnite-element calculations of slowing
electric ﬁelds and real decelerating ﬁelds. Note that as the phase angle is increased, the decelerated
packet is continually removed from the central background of unstable molecules in time of ﬂight.
for clarity, the two bottom panels show only the decelerated packet.

91parity of the state. In the case of OH, we decelerate only the ground |F, MF , p⟩ = |2, ±2, + state.
However, simply changing the timing of the deceleration sequence allows for selection of excitedrotational states as well as diﬀerent MF projections. Given that the Stark shift for a ﬁxed angular momentum state depends weakly on the vibrational quantum number for low-lying vibrational
levels, populations residing in these diﬀerent vibrational levels may be decelerated simultaneously
as demonstrated in Ref. [31].
Stark deceleration relies on conservative potentials and is therefore not technically a molecularcooling mechanism. The phase space density of the slowed packet does not increase as a result of
slowing — we only remove some fraction of the mean kinetic energy. In general, colder packets canonly be obtained at the expense of molecule number. We obtain OH densities of ∼ 107 cm−3 fromour decelerator at a velocity width corresponding to 130 mK in the moving frame of the slowed
packet. The detected molecule number at some distance from the decelerator exit is reduced as
ϕ0 increases due to both decreased stable phase-space area and spreading of the packet in freeﬂight to the detection region. Furthermore, we have identiﬁed additional loss mechanisms at thelowest velocities (< 50 m/s for OH) that reduce this detected molecule number [105]. These loss
mechanisms are discussed in the following sections.

5.2

Loss at Low Velocities
In Refs. [5, 106], the assumption is made that all motion parallel and transverse to thedecelerator axis is stable up to some maximum excursion velocity and position from the beam
center, enabling the derivation of an analytical solution to predict stable-molecule phase-spacearea. However, there are several important instances where the assumptions of this model become
invalid. In the case of very slow molecules (<50 m/s) ∗ , we identify two distinct phenomena
leading to reduced decelerator eﬃciency at the ﬁnal deceleration stages: transverse overfocusing
and longitudinal reﬂection. Transverse overfocusing occurs when the decelerated molecules’ speed∗

This lower velocity limit depends on√
the molecule of interest as well as decelerator electrode geometry. In general,
we expect this velocity limit to scale as µ/m, where µ is the eﬀective electric dipole moment and m is the mass ofthe given molecule.

10-1
φ0= 0o
10o
20o

10-2

30o

Fractional Loss Rate
[millisecond-1]

Phase Stable Population [a. u.]

92

(a)

0.025 (b)

140

141

142

0.020
0.015

(2)

0.010
0.005

(1)

30.43o0

Decelerator Stage NumberTime [ms]Figure 5.4: (a) Simulations of the phase-stable molecule number as a function of stage number
in our 142-stage decelerator. Note the dramatic decrease in number in the last several stages for
ϕo = 30.43◦ . This decrease is due to transverse overfocusing and longitudinal reﬂection of theseslow (14 m/s) molecules. (b) Simulated transverse (trace 1) and longitudinal (trace 2) fractionalloss rates as a function of time within the ﬁnal three stages at ϕo = 30.43◦ . The vertical dashedlines denote the times of the given stage switches.

becomes so low that the decelerating electrodes focus the molecules too tightly (transversely) andthey either make contact with the electrodes or are strongly dispersed upon exiting the decelerator.
Longer decelerators tend to exacerbate this eﬀect due to the fact that molecules can travel at
low speeds for many stages. Nonetheless, there are several motivating factors for constructing
a longer decelerator. First, longer decelerators allow less energy per stage to be removed andconsequently lead to larger longitudinal phase-space acceptance. Second, a longer decelerator may
allow deceleration of molecules possessing an unfavorable Stark shift to mass ratio. We will discuss
critical issues for use of such long decelerators for slow molecule production.
A second low-velocity eﬀect, which we have denoted “longitudinal reﬂection,” is a direct resultof the spatial inhomogeneity of the electric ﬁeld at the ﬁnal deceleration stage. As highlighted in thecontext of transverse guidance in Ref. [99], the longitudinal potential is largest for those molecules
passing–in the transverse dimension–nearest to the decelerator rods. However, the decelerator
switching sequence is generally only synchronous to a molecule on-axis traveling at the mean speedof the packet. As a result, when the mean longitudinal kinetic energy of the slowed packet becomescomparable to the Stark potential barrier at the last stage, molecules oﬀ-axis can be stopped or93
reﬂected, resulting in a spatial ﬁltering eﬀect. Furthermore, the longitudinal velocity spread of themolecular packet at the ﬁnal stage, if larger than the ﬁnal mean velocity, can lead to reﬂectionof the slowest portion of the packet. It is important that the phenomena of overfocusing andlongitudinal reﬂection be addressed since molecule traps fed by Stark decelerators require slow
packets for eﬃcient loading.
To illustrate these low-velocity eﬀects, the number of phase-stable molecules predicted bythree-dimensional (3D) Monte Carlo simulation is shown in Fig. 5.4(a) as a function of stage number
for increasing phase angle. All Monte Carlo simulation results presented in this Thesis are basedon three-dimensional models. The quoted phase-stable molecule number is determined at eachtime step by counting the number of molecules within the instantaneous three-dimensional spatial
acceptance deﬁned relative to the synchronous molecule. Quasi-stable molecules—those that do not
remain in the acceptance volume for the full slowing sequence—can initially lead to an over-estimateof the stable molecule number. However, we observe that such molecules exit the decelerator within
the ﬁrst 40 stages, therefore we reserve comparisons between deceleration schemes to the region
beyond stage 40. We deﬁne the deceleration phase angle, ϕ0 , exactly as in previous publications,where ϕ0 = (z/L)180◦ [5]. The length of one slowing stage is given by L (5.5 mm for our machine),
while the molecule position between successive stages is denoted as z. We deﬁne z = 0 to be exactly
between two slowing stages, therefore, ϕ0 = 0◦ (bunching) yields no net deceleration. Phase angles
satisfying 0◦ < ϕ0 < 90◦ lead to deceleration of the molecular packet, while the maximum energy isremoved for ϕ0 = 90◦ . The 3D simulation results displayed in Fig. 5.4(a) include both longitudinal
and transverse eﬀects. The molecules have an initial velocity (vinitial ) of 380 m/s, corresponding
to the mean velocity of a molecular pulse created via supersonic expansion in Xe. All simulations
and experimental data hereafter possess this vinitial unless otherwise noted.
As expected, a higher phase angle leads to a smaller number of decelerated molecules. However, there is a sharp loss of molecules in the last several deceleration stages for the highest phaseof ϕ0 = 30.43◦ . This value of ϕ0 produces a packet possessing a ﬁnal velocity (vf inal ) of 14 m/s.
This loss is attributed to transverse overfocusing and longitudinal reﬂection. These distinct eﬀects

94
are illustrated in Fig. 5.4(b), which displays the transverse (trace 1) and longitudinal (trace 2)
fractional loss rate of molecules traversing the ﬁnal three slowing stages at ϕ0 = 30.43◦ , vf inal = 14
m/s. The switching time for each stage is denoted by a vertical dashed line, which is labeledby the corresponding stage number. Longitudinal reﬂection of molecules is clearly the dominant
loss mechanism for the lowest ﬁnal velocity shown in Fig. 5.4(a). Nonetheless, there also exists a
non-negligible rise in transverse losses at the ﬁnal stage. That is, because the molecular beam ismoving very slowly in the last few deceleration stages, the transverse guiding ﬁelds of the decelerator electrodes have a greater focusing eﬀect on the molecules (see Eq. 3 of Ref. [75]) and focus
the molecules so tightly that they collide with a deceleration stage and are lost. In the case of ourdecelerator, this leads to loss of 20% of the decelerated molecule number between ϕo = 30◦ (50m/s) and ϕo = 30.43◦ (14 m/s). Such a dramatic loss is not predicted by analytical theory, as thestable phase-space area decreases by <1% over this range of ϕ0 . This number is calculated directly
after the decelerator is switched-oﬀ and is thus an upper bound, since experiments employing thesecold molecules require them to travel out of the decelerator where transverse spread can lead todramatic loss of molecule number.
Experimental evidence of this sudden decrease in molecule number at very low velocities isgiven in Fig. 5.5, which displays data from time-of-ﬂight (ToF) measurements along with corresponding Monte Carlo simulation results at various phase angles. The decelerated OH molecules
are in the weak-ﬁeld seeking |F = 2, mF = ±2, +⟩ state. The ﬁrst two quantum numbers of thestate denote its hyperﬁne components, while the third number indicates the parity of the state in
the absence of electric ﬁelds. The total detected and simulated molecule numbers are plotted inFig. 5.5 as a function of ﬁnal velocity, along with the theoretically expected decelerator eﬃciency(dashed line) [5]. Note that the sudden population decrease in both simulation and experimental
results is not reﬂected in the one-dimensional theoretical model, which does not account for thetransverse dynamics or ﬁeld inhomogeneities that cause such behavior. This eﬀect is detrimental
to the production of dense samples of cold molecules. The diﬀerence between experiment and3D simulation in Fig. 5.5 is indicative of an inability to account for imperfections in decelerator

95
construction. At low velocities, we observe smaller longitudinal velocity widths than predicted bysimulation. Small errors in simulated detection aperture also have larger eﬀects as ﬁnal speed isreduced. Nonetheless, the experimentally observed dramatic molecule loss is reproduced by 3D
simulation.
To remove the overfocusing eﬀect at low velocities, the transverse focusing of the last several
decelerator stages needs to be reduced. Diﬀerent types of transverse focusing elements may beinserted into the deceleration beam line to compensate for this phenomenon. This idea is discussed
in detail in Sections 4 and 5. We note that the proposed solutions, while successful in addressing the
detrimental longitudinal/transverse coupling eﬀects, do not mitigate the problem of longitudinal
reﬂection at low velocities.

5.3

Distributed Loss
As noted in previous work [99, 101, 55], coupling between transverse and longitudinal mo-tion throughout the deceleration sequence invalidates the assumptions made for 1D simulations in
Refs. [5, 106], thereby necessitating 3D Monte Carlo simulations. The fact that the transverse guidance of the molecular beam comes from the same electrodes that provide the deceleration meansthat the longitudinal and transverse motions are necessarily coupled. While this phenomenon iswell understood in the ﬁeld of accelerator physics [107], it was ﬁrst pointed out in the context ofStark deceleration in Ref. [99]. The result can be seen in the left column of Fig. 5.6, where thelongitudinal phase space of OH packets is shown versus increasing phase angle. In Fig. 5.6, the darklines represent the separatrix, partitioning stable deceleration phase space from that of unstablemotion as calculated from Eq. 2 in Ref. [5]. Each dot represents the position in phase space of asimulated molecule. In the absence of coupling between the longitudinal and transverse motions,
one would expect the entire area inside the separatrix to be occupied. Therefore, the structure inthese graphs is evidence of the importance of the transverse motion. The S parameter labeling eachcolumn of Fig. 5.6 was previously deﬁned in Ref. [99, 108]. S = 1 refers to a standard deceleration
scheme in which stages are switched sequentially, while S = 3, 5, 7... denote “overtone” sequences

96
in which the molecules traverse S − 2 charged stages before ﬁelds are switched.
In the left column of Fig. 5.6, the coupling of longitudinal and transverse motions is responsible for two eﬀects† : First, in the center of the stable area at ϕ0 = 0◦ –near the synchronous molecule–the density of stable molecules is less than in the surrounding area. This is because molecules that
oscillate very near the synchronous molecule experience little transverse guiding [99]. This eﬀect is
not dramatic and is only discernable for an exceedingly large number of stages. Furthermore, thiseﬀect is even less important for the increased phase angles typically used for deceleration, sincefor these switching sequences the synchronous molecule experiences more of the transverse guiding
forces than it does during bunching. The second eﬀect, which is much more evident, is the absenceof molecules at intermediate distances from the synchronous molecule as shown in the left columnof Fig 5.6. This so-called ‘halo’ is due to parametric ampliﬁcation of the transverse motion and issimilar to the eﬀects seen in cold molecule storage rings [106] as well as charged particle accelerators [109]. Essentially, the longitudinal oscillation frequency of a molecule in this region is matchedto the transverse oscillation frequency, leading to ampliﬁcation of the transverse and longitudinalmotion and consequent loss [110].
There is a compromise between decreasing longitudinal phase-space area and increasing transverse guidance for increasing ϕ0 . To demonstrate this eﬀect, we decelerate molecules to a ﬁxed vf inal
and vary the phase angle used to reach this velocity. This is done either by changing the voltage
applied to the decelerator rods or by modifying the eﬀective length of the decelerator itself for each
ϕ0 of interest. The experimental data shown in Fig. 5.7 is the result of varying the voltage applied
to our decelerator rods (squares) and the eﬀective decelerator length (circles). Both lowering thedecelerator voltage and using shorter lengths of the decelerator for slowing requires increasing ϕ0
to observe the same vf inal of 50 m/s. We are able to eﬀectively shorten the decelerator by initiallybunching the packet for a given number of stages before beginning a slowing sequence. Note that
we use S = 3 bunching to remove any transverse/longitudinal couplings during these ﬁrst stages,†

For these phase space simulations, the input molecular beam has longitudinal spatial and velocity distributions
that overﬁll the acceptance area.

Molecule Number [a. u.]

97

100

10-1
Experiment
3D Simulation
1D Model10-2

400

300 200 100
Final Velocity [m/s]0

Figure 5.5: Experimental (dots) and Monte Carlo simulation (open circles) results for total molecule
number as a function of ﬁnal velocity. The dashed curve is the expected decelerator eﬃciency
calculated from the one-dimensional (1D) theoretical model of Ref. [5].

98

Figure 5.6: Monte Carlo simulation results for the longitudinal phase space of decelerated molecules.
The left column shows ϕo = 0◦ and 26.67◦ for S = 1 slowing, while the right column shows ϕo =0◦ and 80◦ for S = 3 deceleration. The factor of three between S = 1 and S = 3 phase angles
ensures that molecules have roughly the same ﬁnal velocities. The observed velocity diﬀerence at
the higher phase angle occurs because the 142 stages of our slower is not a multiple of three. Note
that, although the S = 3 phase plot is more densely populated than that of S = 1 at ϕo = 0◦ , itsphase-space acceptance decreases dramatically relative to S = 1 at the lowest velocities. All plots
are generated using an identical initial number of molecules, and therefore the density of points ismeaningful for comparison.

99
then switch back to S = 1 slowing for the remainder of the decelerator. The phase stable region of
S = 1 for ϕ0 ≥ 40◦ is entirely contained within that of S = 3 at ϕ0 = 0◦ , therefore no artifacts frominitial velocity ﬁltering are present in this data. We observe that, contrary to the predictions of
the one-dimensional theory [5], a higher phase angle can lead to greater decelerator eﬃciency up tosome maximum ϕ0 . This is a direct consequence of distributed transverse/longitudinal couplings illustrated in Fig. 5.6. At even larger phase angles, the longitudinal phase-space acceptance becomesa limiting factor. The labels next to each data point correspond to either the voltage applied to thedecelerator rods (squares) or the number of utilized S = 1 slowing stages (circles). Figure 5.7 further illustrates that the transverse/longitudinal couplings outlined by Ref. [110] reduce decelerator
eﬃciency, and are highly dependent on phase angle.
The coupling between longitudinal and transverse motion is detrimental to eﬃcient operationof a Stark decelerator, reducing the total number of decelerated molecules. This eﬀect will be evenworse for decelerating molecules with an unfavorable Stark shift-to-mass ratio. Fortunately, thetransverse and longitudinal motions can be decoupled by introducing a transverse focusing element
to the deceleration beam line that overwhelms the transverse focusing provided by the decelerationelectrodes in analogy to the focusing magnets of charged-beam machines. This technique also
has the advantage of providing a larger stable region in the transverse phase space which further
enhances the decelerated molecule number. The remainder of this manuscript discusses methodsof implementing a transverse focusing element to decouple the longitudinal and transverse motion.
We note that while S = 3 bunching has been previously discussed [99, 108], the dynamics ofdecelerated packets under such operation have not been investigated. In Section 4 we discuss theeﬀectiveness of S = 3 deceleration as well as other modiﬁed “overtone” schemes. Section 5 presents
an improved design for a Stark decelerator that solves this problem of distributed longitudinal andtransverse loss and also reduces the previously described overfocusing losses at the ﬁnal stage.

Molecule Number [a. u.]100

vfinal = 50 m/s
φ0 [degrees]
Figure 5.7: Experimental results from changing the voltage on decelerator rods (squares) anddecreasing the eﬀective decelerator length (circles). Eﬀective slower length is modiﬁed by initiallyoperating the decelerator at ϕ0 = 0◦ , S = 3, then slowing with S = 1 to vf inal = 50 m/s for thenumber of stages labeled. Both curves illustrate that transverse/longitudinal couplings are strongly
dependent on phase angle, and have a marked eﬀect on decelerator eﬃciency.

101

5.4

Decelerator Overtones
The simplest method for introducing a transverse focusing element to the decelerator beamline is to let the molecules ﬂy through an energized deceleration stage without removing the ﬁeld.
In this manner, molecules experience the transverse focusing of the entire stage without theirlongitudinal motion aﬀected. Traditional longitudinal phase stability requires the switching of
the ﬁelds to occur on an upward slope of the molecular potential energy, i.e., faster molecules
are slowed more while slower molecules are slowed less than the synchronous molecule. Hence,
it is necessary to de-sample the bunching switching rate by an odd factor (3,5,7...): the so-called
decelerator overtones [108]. For convenience, we deﬁne the quantity S = vo /vSwitch , where vo isthe synchronous molecule velocity and the switching speed vSwitch is given as the stage spacing L
divided by the switching time-interval. Reference [108] considered only the bunching case. In thiswork, we generalize to the case of actual deceleration. However, the above deﬁnition of S is stillvalid. That is, S is constant despite the fact that both vo and vSwitch vary when ϕo > 0◦ . With
this deﬁnition we see that traditional deceleration can be described by S = 1, while the method ofde-sampling the switch rate by a factor 3 is described by S = 3. These two methods of deceleration
can be seen in Figs. 5.8 (b) and (c), where their respective switching schemes are shown for ϕo = 0◦ .
By switching at one-third the rate, the molecule packet ﬂies through a deceleration stage that isenergized and experiences enhanced transverse guiding.
Longitudinal phase space simulations of S = 3 slowing at various phase angles are shown in
the right column of Fig. 5.6. No structure is present in these plots. Also, the region of longitudinal
phase stability for S = 3–even at ϕ0 = 0◦ –is reduced compared to S = 1. This is because themaximum stable velocity, as calculated from Eqs. 2 and 6 of Ref. [5], depends on the spacing
between deceleration stages as L−1/2 , and thus, the separatrix velocity bound is reduced by a√factor of 3 ‡ . Nonetheless, the absence of coupling to the transverse motion leads to a larger
number of molecules for the ϕo = 0◦ case shown in the uppermost panel of Fig. 5.6. In a given‡
Physically, this is because the molecules ﬂy longer between deceleration stages, and thus, the velocity mismatch
can lead to a larger accumulation of spatial mismatch.

102

(a)

(b)

(c)

(d)

Figure 5.8: Deceleration schemes. (a) Potential energy shift of polar molecules in the Stark decelerator. The dotted (blue) curves show the potential energy shift when the horizontal (circularcross section) electrodes are energized, while the dashed (red) curves show the potential energy
shift when the vertical (elongated cross section) electrodes are energized. Deceleration proceeds byswitching between the two sets of energized electrodes. In panels (b)-(d), the thick black line indicates the potential experienced by the molecules. The empty circles indicate a switching event. (b)Traditional (S = 1) operation at ϕo = 0◦ . For phase stability, the switching always occurs when themolecules are on an upward slope, and as such the molecules are never between a pair of energizedelectrodes. Thus, the maximum transverse guiding is never realized. (c) First overtone operation(S = 3) at ϕo = 0◦ . By switching at one-third of the S = 1 rate, the molecules are allowed toﬂy directly between an energized electrode pair, and thus, experience enhanced transverse guiding.
(d) Optimized ﬁrst overtone operation (S = 3+) at ϕo = 0◦ : Initially, the packet rises the Stark
potential created by one set of electrodes. When the molecules reach the apex of this potential,
the alternate set of electrodes is energized in addition. In this way, the molecules experience one
more stage of maximum transverse guiding for each slowing stage. Note that, to minimize theun-bunching eﬀect, the grounded-set of electrodes is switched on when the molecules are directly
between the energized electrodes.

103
decelerator, S = 3 slowing requires a factor of three higher phase angle than S = 1 to reach thesame ﬁnal velocity. As illustrated in Fig. 5.6, this fact severely limits the practicality of S = 3 as adeceleration scheme, as it implies a dramatic reduction in velocity acceptance at the highest phaseangles.
Shown in Fig. 5.9(a) are experimental decelerated molecular packets for S = 3 (upper) andS = 1 (lower) versus increasing phase angle. For each successive packet, the S = 3 phase angle
increases by 10◦ in the range ϕo = 0◦ –60◦ , while the S = 1 phase angle increases by 10◦ /3. In
this manner, the packets are decelerated to roughly the same velocity. There is a slight diﬀerenceat the highest phase angles shown because the total number of stages in our decelerator (142) is
not an exact multiple of 3. In Fig. 5.9(b) the de-convolved total molecule number for each of thesepackets is plotted versus ﬁnal speed. While the S = 3 method dominates over S = 1 for small
phase angles, its eﬀectiveness decreases as the deceleration becomes more aggressive–by 224 m/s,
the S = 1 molecule number is already larger than that of S = 3. This behavior is expected since
the phase angle used to decelerate to 224 m/s is 60◦ , while the required S = 1 phase angle is only
20◦ . Although the S = 3 longitudinal phase bucket does not exhibit structure, it is so much smaller
in enclosed area than the S = 1 that its total molecule number is smaller. The simulation results ofFig. 5.9(c) and the theory results of Fig. 5.9(d) support this description. Figure 5.9(c) shows that,even at ϕo = 80◦ , the transverse loss rate per stage is at all times greater for S = 1 than S = 3.
However, the calculated longitudinal phase-space acceptance of Fig. 5.9(d) mirrors the behavior
observed experimentally in Fig. 5.9(b) when the initial points are scaled to the experimental ratioof 2.75. This scaling accounts for the aforementioned ‘halo’ in the slowed S = 1 packet, which
persists relatively unchanged over the range of S = 1 phase angles used (ϕo = 0◦ − 20◦ ). The factthat the theory curves of Fig. 5.9(d) cross at a higher velocity than the data of Fig. 5.9(b) suggests
there is increased transverse guiding of S = 3 slowing at high phase angles. Nevertheless, even with
142 stages of deceleration, S = 3 is unfavorable for velocities below 224 m/s due to reduced stablephase-space area. The peak in the loss rate observed in Fig. 5.9(c) at ∼20 stages is due to loss ofquasi-stable molecules near the beginning of the deceleration sequence and is not observed beyond

104
40 stages.
At this point, one would expect that operation at lower phase angles would permit realizationof the gain produced by the S = 3 method. This can be accomplished by naively using a longer
decelerator. A simulation of this kind is presented in Fig. 5.10, which plots the number of molecules
present after deceleration at ϕo = 20◦ versus ﬁnal velocity § . The number next to a data point
represents the number of deceleration stages used. Initially the decelerated molecule number isrelatively ﬂat versus ﬁnal velocity. However, after about 500 stages (180 m/s), the number ofdecelerated molecules begins to decrease and dramatically falls oﬀ after 550 stages (150 m/s). Veryfew molecules survive below 100 m/s. This is because for S = 3 the decelerated molecules must ﬂy
through an entire stage while experiencing a guiding force in only one dimension [see Fig. 5.8(c)].
Once the molecules are at slower speeds they can spread out in one transverse dimension or beover-focused in the other and collide with the rods. As the mean kinetic energy of the slowed packet
becomes comparable to the full potential height, the packet can be nearly stopped as it traverses
the intermediate charged stage. This has two consequences: (1) longer transit time leading to more
intense transverse focusing; and (2) velocity ﬁltering of the low-speed packet as slower molecules
are longitudinally reﬂected from this potential.
The transverse loss responsible for the extreme drop in molecule number for S = 3 deceleration also occurs in traditional deceleration, but to a lesser degree. Because of this decreasein molecule number at low speeds, the usefulness of slowing with S = 3 is generally limited toexperiments that do not require the lowest velocities, such as microwave spectroscopy and collisionexperiments [21, 24, 111]. We note also that our simulations predict no low-velocity gain whenusing slowing sequences containing combinations of deceleration at S = 1 and bunching at S = 3.
A natural extension of the above overtone deceleration is the use of what we have termed a“modiﬁed decelerator overtone,” denoted by an additional plus sign, i.e., S = 3+. Deceleration inthis manner is shown in Fig. 5.8(d) for ϕo = 0◦ . In this method, deceleration proceeds similarly to§

In this simulation, ϕo = 20◦ is chosen because it produces the most gain over S = 1 in our deceleration
experiments.

105

OH LIF Signal [a. u.]0o, (0o)
3.33o, (10o)

2.5

S=1
(S = 3)(a)6.67o, (20o)2.010o, (30o)1.5

13.33o, (40o)1.0

16.67o, (50o)0.520o, (60o)
0.0

2.5

2.6

2.7

2.8

2.9

3.0

3.1

3.2

Relative Molecule Number [a. u.]

Experiment
3.0

3.0
2.5

(2)

(b)
2.0
1.5
1.0

(1)

0.5
0.0

380 360 340 320 300 280 260 240 220

Final Velocity [m/s]Time [ms]Simulation(1) S = 1
(2) S = 3

(c)
(1)

(2)

Stage Number

Stable Phase Space Area [a. u.]

Transverse Loss Rate Per Stage [a. u.](1) S = 1
(2) S = 3

(1) S = 1
(2) S = 3

(d)

(2)

(1)

Final Velocity [m/s]

Figure 5.9: Comparison of deceleration using S = 3 versus S = 1. (a) Experimental ToF data ofdecelerated OH packets with S = 3 (top) and S = 1 (bottom). Note the factor of three between
S = 1 and S = 3 phase angles. (b) De-convolved, integrated molecule number for S = 3 (trace2) and S = 1 (trace 1) for the packets shown in panel (a). (c) Simulated transverse loss rate perstage for S = 1 (trace 1) and S = 3 (trace 2) deceleration. As expected, S = 1 results in larger
transverse loss rates throughout. (d) Calculated stable longitudinal phase-space area for S = 1(trace 1) and S = 3 (trace 2), with initial points scaled to the experimental ratio of 2.75. The
above panels highlight that the observed shortcoming of S = 3 deceleration is entirely due to lossof longitudinal velocity acceptance at high phase angles.

Phase Stable Population [a. u.]

106

100 202

301

400

S=3
φ0=20o

502

10-1
601

10-2
10-3
625

10-4
Final Velocity [m/s]

Figure 5.10: Monte Carlo results for decelerated molecule number using S = 3 and ϕo = 20◦ versus
ﬁnal velocity. The number next to each data point is the number of stages used. Because oftransverse overfocusing and longitudinal velocity ﬁltering, essentially no molecules survive below
100 m/s.

(a)

2.0

0o, 380 m/s, S = 1
S = 3+

0o, 325 m/s
10o, 315 m/s

1.5o

20 , 299 m/s
30o, 281 m/s

1.040o, 259 m/s

0.5
0.0

50o, 234 m/s
60o, 207 m/s

2.6

2.8

3.0

Time [ms]

3.2

3.4

Relative Molecule Number [arb.u.]

OH LIF Signal [arb. u.]

2.5

3.0

107

(b)

S=1
S=3
S = 3+

2.5
2.0
1.5
1.0
0.5
0.0

380 360 340 320 300 280 260 240 220

Final Velocity [m/s]

Figure 5.11: (a) Experimental ToF data of decelerated OH packets produced using the S = 3+
modiﬁed overtone. Also shown for comparison is the experimental bunching packet for operation at
S = 1. (b) The de-convolved, integrated molecule number calculated from S = 3+ (open squares),S = 3 (open circles), and S = 1 (dots) data.

conventional S = 3. However, S = 3+ sequences yield conﬁnement of the packet in both transverse
dimensions. This is achieved by charging all slower rods for the period in which the synchronous
molecule is between switching stages. In order to minimally disrupt the longitudinal dynamics ofthe synchronous molecule, the second set of slower rods is charged only when the molecule is at
the peak of the longitudinal potential from the ﬁrst rod set. The packet then traverses two charged
rod pairs before reaching the next switching stage, at which point the rod pair that was originallycharged is grounded. While this does create a slight anti-bunching eﬀect, i.e., molecules in front of
the synchronous molecule gain a small amount of energy, it provides an extra stage of transverse
guidance in comparison to S = 3. Experimental results of this method of slowing are shown in
Fig. 5.11 for comparison to deceleration using both S = 1 and S = 3. Fig. 5.11(a) displays a uniqueconsequence of the S = 3+ switching sequence, where operation at ϕo = 0◦ leads to deceleration.
Figure 5.11(b) shows that operation with S = 3+ provides slightly more molecules than S = 3, but
the loss of molecules due to decreased longitudinal velocity acceptance remains a problem. The
increase in molecule number for S = 3+ over S = 3 is due to the extra stage of transverse guidance,
which for the higher-velocity packets we measured leads to a larger transverse acceptance.
To determine whether the extra stage of transverse guidance would counter the overfocusing

108
eﬀects ¶ , we perform simulations of S = 3+ deceleration at ϕo = 20◦ for a varying number ofdeceleration stages. The results of these simulations, shown in Fig. 5.12, are similar to the resultsfor S = 3. Namely, as the decelerator length is increased and the molecules’ speed is reduced, there
is a marked molecule number loss for velocities below 200 m/s. In our simulations, we could notobserve any molecules below 100 m/s. Again, transverse overfocusing and longitudinal ﬁltering of
the molecular packet by the deceleration electrodes are responsible for large losses in the decelerator,
and this suggests that measures beyond modiﬁed switching schemes are required to overcome theseloss mechanisms.

5.5

Quadrupole-Guiding Decelerator
In addition to modifying the timing sequences of existing decelerators, it is possible, andperhaps preferable, to uncouple the longitudinal motion inside the Stark decelerator from thetransverse motion by redesigning the decelerator electrode geometry. One simple redesign, which
we call the quadrupole-guiding decelerator (QGD), is shown in Fig. 5.13(a). In this decelerator, aquadrupole-guiding stage (Fig. 5.13(b)) is interleaved between each deceleration stage. While it may
not be necessary to have a quadrupole-guiding stage between each deceleration stage (especially inthe beginning of the decelerator), it simpliﬁes the analysis and will be used here. The switchingof the electric ﬁelds inside a QGD is similar to a traditional decelerator operated with S = 1.
Figure 5.13(c) shows the potential energy experienced by a molecule decelerated at ϕo ≈ 45◦ and isrepresented by a thick black curve. In this panel, the quadrupole-guiding electrodes are omitted forclarity. Note that the quadrupoles are always energized and their center coincides with the ϕo = 0◦
position, while ϕo = 90◦ occurs between the deceleration electrodes.
Deceleration with a QGD enjoys the same longitudinal phase-stability as a traditional decelerator. In a QGD, the maximum stable excursion position ∆ϕmax and velocity ∆vmax will diﬀer
from that of the traditional decelerator because the decelerating electrodes are most likely further¶

While this may seem counterintuitive, in some cases when the transverse overfocusing is not too strong, theaddition of another focusing element can change the sign of the molecules transverse velocity, keeping the beam
conﬁned within the decelerator.

109

Molecule Number [a. u.]100

φ0=20o

142
217
284

10-1

10-2
331
300

250

200

150

Final Velocity [m/s]
Figure 5.12: Monte Carlo results of decelerated molecule number using S = 3+ and ϕo = 20◦
versus ﬁnal velocity. The number next to each data point is the number of stages used. Because oftransverse overfocusing, essentially no molecules survive below 100 m/s.

110

Decelerato r
Array

(a)

Skimme r

Hexapole

PulsedValve(b) Quadrupole Guide Stage-V

3(107) V/m
2(107) V/m
1(107) V/m

+V

0 V/m
+V

-V

(c)

QG stage

QG stage

Figure 5.13: Quadrupole-guiding decelerator. (a) Schematic of QGD. (b) Electric ﬁeld ofquadrupole guiding stage energized to ±12.5 kV. (c) Switching scheme for deceleration with the
QGD.

111
apart. In other words, to prevent high-voltage breakdown, the quadrupole stages require the sameinter-stage spacing as deceleration stages in a traditional decelerator. Thus, the decelerating electrodes for a QGD will be twice as far apart as in our traditional decelerator, which possesses aninter-stage spacing of 5.5 mm. Since the dependence of the decelerating force on ϕo is less steep,
the shape of the stable longitudinal phase space will change. Understanding the shape of the stable
longitudinal phase area is crucial for predicting the performance of the QGD, and can be derived
by examining the longitudinal forces inside the QGD. Shown in Fig. 5.14(a) is the on-axis Stark
shift of an OH molecule in the |2, ±2, +⟩ state inside the unit cell, deﬁned as 3 deceleration stagesof the QGD. The quadrupole rods are held at a constant ±12kV for this calculation, though theircontribution to the longitudinal potential is negligible. The solid line is the Stark shift due to theslowing stage centered at 11 mm, while the dashed line is the Stark shift of the stages which will beenergized when the ﬁelds are switched. The subtraction of these two curves, shown in Fig. 5.14(b)as a solid line, is the amount of energy removed each time the ﬁelds are switched, ∆KE. We
represent ∆KE as sum of sine-functions [108]∆KE(ϕ) =∑an sin(nϕ),

(5.4)

n=odd

where we have used the deﬁnition of the phase angle ϕ = (z/L)180◦ . A ﬁt of the ﬁrst three terms ofthis equation to the actual ∆KE for deceleration stages spaced by 11 mm is shown as a dashed line
in Fig. 5.14, resulting in the ﬁt values a1 = 1.221 cm−1 , a3 = 0.450 cm−1 , and a5 = 0.089 cm−1 .
Using this ﬁt, we derive the equation of motion of the molecules about the synchronous molecule
position as

d2 ∆ϕ
π
+
(∆KE(∆ϕ + ϕ0 ) − ∆KE(ϕ0 )) = 0,
dt2
mL2

(5.5)

where we have used the excursion of the molecule from the synchronous molecule ∆ϕ = ϕ − ϕo .
The maximum stable forward excursion of a non-synchronous molecule is exactly the same as a
traditional decelerator and is given as [5]◦∆ϕ+max (ϕo ) = 180 − 2ϕo .

(5.6)

112
We calculate the work done in bringing a molecule starting at this position with zero velocity to
the synchronous molecule position as∫ End

1
W (ϕo ) =F dx = −
π
Start

∫ 0

∑

(an [sin(n(∆ϕ + ϕo )) − sin(nϕo )])d(∆ϕ).

(5.7)

∆ϕ+
max (ϕo ) n=odds

Integrating this equation and setting it equal to the kinetic energy yields the maximum stable
excursion velocity:√∆vmax (ϕo ) = 2∑

n=oddsanmπ()
cos(nϕo )π− ( − ϕo ) sin(ϕo ) ,
n
2

(5.8)

where ϕo is now in radians.
Using Eqs. 5.5- 5.8 it is possible to solve for the longitudinal separatrix, which separates stabledeceleration from unstable motion inside the decelerator. These separatrices are shown (thick blacklines) along with the results of Monte Carlo simulations of a QGD in the left column of Fig. 5.15
for successive phase angles. The longitudinal phase space is shown with each dot representing theposition of a stable molecule. The lack of structure inside these separatrices is evidence of the lack of
coupling between the transverse and longitudinal modes. The right column, which shows simulated
ToF curves, reveals a single stable peak arriving at later times as ϕo is increased. These simulations
are for a single |2, ±2, +⟩ state of OH and do not exhibit the large background contribution of theother states of OH present in experimental ToF data.
From the comparison of the simulations with the analytical results represented by the separatrices, we see that the simple theory of Eqs. 5.5- 5.8 is quite accurate in describing the longitudinal
performance of the QGD. Thus, by numerically integrating the area inside these separatrices, we
can predict the longitudinal performance of the QGD relative to the traditional decelerator. As
seen in Fig. 5.16(a), the energy removed per stage of the QGD is smaller than that of a traditional
decelerator for all 0◦ < ϕo < 90◦ because the decelerator stage spacing (in this simulation) is twicethat of a traditional decelerator. For this reason, given the same number of deceleration stages,
QGD deceleration with the same ϕo as in a traditional decelerator leads to a faster beam. When
comparing the longitudinal acceptance of the two types of decelerators it is important to take the

113

2

(a)
U [cm-1]

1.5

1

0.5

0

0

10

20

30

40

z [mm]
2∆KE [cm-1]

(b)
1

0

0

5

10

15

20

25

z [mm]
Figure 5.14: (a) The Stark shift of an OH molecule in the |2, ±2, +⟩ state inside the QGD. The
solid curve is the Stark shift due to the slowing electrodes, while the dashed curve is the Stark shift
due to the electrodes that will be energized at the switching time. (b) The change in the molecule’s
kinetic energy as a function of position is shown (solid) as well as a ﬁt of Eq. 5.4, including up ton = 3 (dashed). The solid curve is calculated from the subtraction of the two curves in panel (a).

114
1.00

410

Vz [m/s]

400

0.75

390
0.50

380
370

0.25

360
350

0.00

-1

-0.5

0
z [cm]

0.5

1

4

5

6

7

5

6

7

5

6

7

5

6

7

Time [ms]

1.00

400

Vz [m/s]

390

0.75

380
0.50

370
360

0.25

350
340

0.00

-1

-0.5

0
z [cm]

0.5

1

4Time [ms]

1.00

350

Vz [m/s]

340

0.75

330
0.50

320
310

0.25

300
290

0.00

-1

-0.5

0
z [cm]

0.5

1

4Time [ms]

1.00

240
230

0.75

Vz [m/s]

220
210

0.50

200
190

0.25

180
170

0.00

-1

-0.5

0
z [cm]

0.5

1

4Time [ms]Figure 5.15: The left column is stable phase space of molecules decelerated inside the QGD. The
solid line is the separatrix predicted by the theory, while the points represent positions of moleculein the 3D Monte Carlo simulations. The right column shows the ToF spectra of OH molecules in
the |2 ± 2, +⟩ state at the exit of this decelerator which has 142 deceleration stages, along with 142
quadrupole stages.

115
limit of high phase angles where both values of ∆KE converge. Nonetheless, the longitudinal
phase-space acceptance of the QGD shows some gain over that of traditional deceleration as shownin Fig. 5.16(b). This gain at a ﬁxed energy loss per stage is primarily due to the increased physicalsize of the stable longitudinal phase space resulting from the larger deceleration stage spacing.
Since Fig. 5.16 compares only the total area inside the separatrix—and S = 1 deceleration
does not completely ﬁll this area due to the coupling eﬀects—this gain is actually an underestimate of the QGD longitudinal performance. Furthermore, these graphs do not include transverse
focusing eﬀects, which can only be properly included through detailed simulation. The results ofMonte Carlo simulations including these transverse eﬀects are shown in Fig. 5.17. The number ofdecelerated molecules versus ﬁnal speed is plotted for both the traditional decelerator operating
at S = 1 and the QGD operating at two diﬀerent quadrupole rod voltages, ±1 kV and ±3 kV.
While the QGD initially delivers more molecules, once the molecules are decelerated below 100
m/s, the decelerated molecule number falls oﬀ abruptly. This behavior is expected since, for thesesimulations, the voltage on the quadrupole-guiding stages is held constant throughout the slowing
sequence. As detailed in Eq. (3) of Ref. [112], the focal length of a transverse guiding element isdirectly proportional to the molecular kinetic energy. Therefore, as the mean speed of the packet
is decreased, the molecules are overfocused and collide with the decelerator electrodes. This can beprevented by lowering the voltage on the quadrupole-guiding stages during the deceleration process.
Figure 5.18 displays simulation results of deceleration with this dynamically scaled voltage compared to S = 1 slowing at ϕo = 30.43◦ . For this simulation, the quadrupole voltages are scaled by(v/vinitial )0.875 after each deceleration stage, where v is the instantaneous packet velocity directly
following each stage switch. The exponent of 0.875 is found empirically to produce the most gain
at vf inal = 14 m/s. For ease of simulation, the transverse forces are scaled by (v/vinitial )0.875 whenever the molecules are closer to a quadrupole-guiding stage than to a deceleration stage. While
this may be a poor approximation at the lowest speeds, it will likely lead to an underestimateof the decelerated molecule number since the transverse guidance of the quadrupole-guiding stage
extends beyond this regime. Even if it leads to an overestimate, proper control of the quadrupole

116

Figure 5.16: (a) The energy removed per stage as a function of phase angle for both traditional
deceleration and deceleration with a QGD. Both curves are calculated for OH in the |2, ±2, +⟩ stateand scaled down by 1.76 cm−1 at ϕo = 90◦ . (b) The calculated longitudinal phase-stable area fordeceleration versus energy loss per stage for traditional deceleration and deceleration with a QGD
is plotted on the left axis. The gain of the QGD over traditional deceleration is plotted on the rightaxis. Note that for a given energy loss the gain in phase stable area due to the larger volume of
the QGD is ≤1.5 for all 0◦ < ϕo < 80◦ .

Phase Stable Population [a. u.]

117

10-1

100

50

120

133

140

1 kV
S=1

10-3

3 kV

10-5

10-7
Velocity [m/s]
Figure 5.17: Simulations of the phase-stable molecule number as a function of stage number in the
QGD and S = 1 decelerator. The voltage on the quadrupole stages of the QGD is held constantthroughout the deceleration sequence. All simulation data is for vf inal = 14 m/s. The traces shown
are S = 1 deceleration at ϕo = 30.43◦ , QGD operated with ±1 kV on the quadrupoles, and QGD
operated with ±3 kV on the quadrupoles. Note the decrease in stable molecule number in thelast several stages for the QGD results. This decrease is due to transverse overfocusing of the slow
molecules through the ﬁnal few quadrupole stages, and suggests that scaling of quadrupole voltage
is necessary.

118
voltages may compensate any overfocusing introduced by the decelerating elements. As seen in
Fig. 5.18, dynamically controlling the voltage of the quadrupole-guiding stages leads to a factorof 5 increase in decelerated number for larger velocities (>80 m/s) and delivers about 40% more
decelerated molecules than traditional S = 1 deceleration provides at the lowest ﬁnal speeds (14m/s). Because the voltages applied to the quadrupole-guiding stages are relatively low, dynamic
control of them should be possible using an analog waveform generator and high-voltage ampliﬁer.
It should be noted that the optimal voltage scaling may vary among decelerators since the real focal
length depends sensitively on the electrode construction, and at low speeds the transverse focusingof the decelerator electrodes becomes signiﬁcant. This is, presumably, because the overfocusing of
the decelerator electrodes can be compensated to a certain degree by injecting molecules which are
already slightly overfocused. In other words, two focusing stages can overcome the overfocusingof a single stage. Thus, it may be possible to use adaptive algorithms to optimize the quadrupole
voltage or change the design of the decelerating electrodes so that they provide less transverse
focusing, and maximize the number of decelerated molecules beyond what is reported here [111].
One important advantage of the QGD over traditional decelerators is its inherent ability tosupport more deceleration stages. Because the molecules experience a tunable transverse focusing
element after each deceleration stage, there is little loss in eﬃciency by extending the number ofdeceleration stages. In fact, as seen in Fig. 5.19, there is essentially no loss until the molecules
are decelerated to the lowest speeds previously discussed. This low-velocity loss is due to theaforementioned transverse overfocusing and longitudinal reﬂection. The former loss mechanism
may be overcome, while the latter presents a fundamental limit. Even with this loss, the QGD
outperforms both S = 3 (Fig. 5.10) and S = 3+ (Fig. 5.12). Thus, the QGD is an ideal decelerator
for more eﬃciently producing cold molecules and, perhaps more importantly, the ideal decelerator
for slowing molecules with poor Stark shift to mass ratios, like H2 O and SO2 .
In summary, we identify two speciﬁc loss mechanisms—transverse over-focusing and longitudinal reﬂection—observed in the operation of Stark decelerators and perform initial experiments anddetailed Monte Carlo simulations to address them. While the use of decelerator overtones yields im-

Phase Stable Population [a. u.]

119

10-1

50

100

120

133

140

Voltage Scaling

-2

S=1

10

Velocity [m/s]
Figure 5.18: Monte Carlo simulation results for the decelerated molecule number using traditional
S = 1 deceleration (ϕ0 = 30.43◦ ) and deceleration using a QGD (ϕ0 = 52.75◦ ) with a dynamic
voltage scaling of (v/vinitial )0.875 . For both curves, 142 stages of deceleration were used, andvf inal = 14 m/s. The diﬀerent phase angles chosen for the two decelerators are a result of theirdiﬀering potential proﬁles for deceleration. The vertical dashed lines represent the deceleration
stage at the given velocity. Note that when the quadrupole voltage within the QGD is scaled inthis manner, we observe a 40% gain in molecule number at 14 m/s, and a factor of 5 gain overS = 1 at higher velocities.

120

Phase Stable Population [a. u.]

100

10-1

-2

10

200

400

600

700

800

Voltage Scaling

S=1

Velocity [m/s]
Figure 5.19: S = 1 simulation results for vf inal = 80 m/s (ϕ0 = 5.22◦ ) plotted along with simulation
results using the voltage-scaled QGD to the same vf inal (ϕ0 = 23.5◦ ). The number labeling eachvertical dashed line is the number of deceleration stages necessary to reach the given velocity. Notethe large number of stages (803) used to reach 80 m/s, which suggests that a very long QGD may
be employed for slowing molecules with a poor Stark shift to mass ratio.

121
provement over S = 1 deceleration at high to intermediate speeds (vf inal > 80 m/s), the loss at very
low velocities remains problematic. The QGD solves the problem of coupling between the transverse and longitudinal motions inside a Stark decelerator by introducing independent transverse
focusing elements. By dynamically controlling the focal length (voltage) of these guiding elements,
large improvements (factor of 5) in deceleration eﬃciency can be achieved for vf inal > 80 m/s,
which would prove useful for molecule collision studies [111]. Gains of 2 and 40% are also predicted
for vf inal = 50 m/s and 14 m/s, respectively. Electrostatic [103] and magneto-electrostatic [55]trapping experiments stand to beneﬁt from such gain for vf inal ≤ 50 m/s. Furthermore, it appears
that with dynamic control of the guiding stage focal length there should be no limit to the lengthof decelerator that can be built. This enables the deceleration of molecules with a poor Stark shift
to mass ratio. However, we note that none of the techniques described in this chapter mitigate
the longitudinal low-velocity loss due to reﬂection, which appears to be a fundamental componentof Stark deceleration. Building upon the strong correlation between simulation and experimental
results, we are conﬁdent that the simulations presented in this work provide a solid guideline for
future implementations of Stark deceleration.

Chapter 6

Magnetic Trapping of OH

6.1

Magnetoelectrostatic Trapping
Neutral polar molecules are readily trapped in inhomogeneous electric ﬁelds [103, 113], how-ever theory predicts that at high phase-space densities such traps will suﬀer from large inelastic
collision losses [114]. One possible solution is to trap strong-ﬁeld seeking states via time-dependent
electric traps [115]. In contrast, we are interested in magnetically trapping these neutral polar
molecules so as to have the freedom to apply external electric ﬁelds to control collision dynamics
inside the trap. Highly vibrationally excited, triplet KRb molecules produced via photoassociation
have been magnetically trapped at a temperature of ∼300 µK and density of 104 cm−3 [116]. Buﬀer
gas cooling has been used to load a magnetic trap with ground-state CaH and NH molecules attemperatures of hundreds of mK and densities of ∼108 cm−3 [117, 118]. In this section, we describe
our ﬁrst realization of a magnetic trap for hydroxyl radical (OH) molecules. This work represents
the ﬁrst observation of magnetically trapped OH and the lowest temperature (30 mK) yet achieved
for a magnetically-trapped polar molecule in its rovibronic ground state [64]. Moreover, we have
performed the ﬁrst study of trap dynamics based on a single molecule possessing large magnetic andelectric dipole moments in B- and E-ﬁelds that are inhomogeneous and anisotropic. Interesting polar molecule dynamics have been predicted at large ﬁeld strengths [119]. Accurate correspondence
between experimental data and simulations is achieved only by accounting for the molecular state
mixing induced by the crossed ﬁelds. Large magnetic (B) ﬁelds may serve to suppress inelastic collision losses [120] while at the same time sympathetic cooling of molecules via co-trapped ultracold

123

a

ρ

z

b
ρ=0

z=0

Figure 6.1: (a) Illustration of the magneto-electrostatic trap (MET) assembly. The terminus of
the Stark decelerator (yellow, gray electrodes), shown to the left, couples state-selected cold OH
molecules into the magnetic quadrupole (blue). The double-electrode structures within the electric
quadrupole (red) allow for application of uniform electric ﬁelds within the magnetic trap. The ring
(green) supports a 1 inch lens for collection of laser-induced ﬂuorescence. (b) Plots of the magnetic
ﬁeld distributions within the MET for both stopping and trapping of decelerated OH molecules.

124
atoms may become feasible [121, 122]. In addition, the magnetic quadrupole trap described here
permits the application of arbitrary external electric (E) ﬁelds to a large class of polar molecules—anecessary ﬁrst step towards our recent achievement of observing cold dipolar collisions between OH
and ND3 [18].
To produce an OH sample suitable for magnetic trapping, we begin with a cold beam ofground state (2 Π3/2 ) OH produced via a pulsed electric discharge through 2% H2 O vapor [62]seeded in 1.5 bar of Kr. A piezoelectric transducer valve [68] operating at 10 Hz provides asupersonic expansion of the OH/Kr mixture through a 1 mm nozzle. This results in a 490 m/s
molecular packet with a 15% longitudinal (ẑ) velocity spread. The transverse (ρ̂) velocity spreadof the beam is limited to 2% by the 3 mm diameter molecular skimmer. Compared with a Xe
expansion, the higher-velocity Kr expansion leads to less transverse beam loss within the current
decelerator design. Thus, despite the fact that the Kr beam requires a higher phase angle fordeceleration (smaller decelerator acceptance) it results in more decelerated molecules as describedin Chapter 5. After passing through the skimmer, the OH packet is spatially matched to thedecelerator entrance via an electrostatic hexapole. We utilize the 142 stages of our Stark decelerator
to slow a 130 mK packet (in the co-moving frame) of OH to 20 m/s at a phase angle ϕ0 = 47.45◦
for coupling into the magneto-electrostatic trap (MET). Decelerated OH packets are detected via
laser-induced ﬂuorescence (LIF) upon entering the MET, whose center lies 2.7 cm from the ﬁnal
pair of decelerator rods. The 282 nm pulsed LIF beam is oriented along the mutual longitudinal
axis (ẑ) of the decelerator and MET, and the resulting 313 nm ﬂuorescence is collected in a solid
angle of 0.016 sr.

6.1.1

Design and Operation of the Magnetic Quadrupole Trap

The MET is illustrated in Fig. 6.1a, and consists of two copper coils operated in an antiHelmholtz (ﬁeld-opposing) conﬁguration surrounded by an electric quadrupole. The center-tocenter magnetic coil spacing is 1.5 cm, providing a ﬁeld gradient of 6700 G/cm by passing 1500 A
through the 4 turns of each water-cooled copper coil. To calculate the magnetic ﬁelds within the125coils for both stopping and trapping of OH, we use the closed-form expression for the magnetic
ﬁeld due to a current loop of radius A:
Bρ =
Bz =

]
µ0 I z [ 2(A + ρ2 + z 2 )E(k 2 ) − α2 K(k 2 )22π α βρ]µ0 I 1 [ 2
2
2
2
2
2
(A
−
ρ
−
z
)E(k
)
+
α
K(k
)
2π α2 β

(6.1)
(6.2)

where we deﬁne
α2 = (A − ρ)2 + z 2(6.3)

β 2 = (A + ρ)2 + z 2(6.4)

k2 = 1 −

α2
β2

(6.5)

and the functions K and E are the complete elliptic integrals of the ﬁrst and second kind, respectively. This functional form for B makes numerical evaluation rather straightforward in anenvironment like MATLAB and is an exact expression. Placing four coils each at z = ±7.5 mm
with opposing currents of 1500 A, we produce the magnetic quadrupole ﬁelds (Bρ (z = 0), Bz (ρ = 0))of Fig. 6.1b (shown in black). The red curves of Fig. 6.1b are the same components of B but with
only the back coil at a current of 2000 A. This conﬁguration is used to initially stop the 20 m/s
OH packet entering from the Stark decelerator. Given that OH has a magnetic dipole of 1.2µB , thelongitudinal trap depth is ∼ 400 mK. As is always the case with magnetic quadrupoles, the radial
trap depth is one half of this value.
The electrodes surrounding the coils allow the application of an E-ﬁeld to the trapped sample.
The double- and single-rod electrodes that comprise the electric quadrupole allow the applicationof two distinct ﬁeld conﬁgurations. A uniform bias electric ﬁeld is created by grounding the singlerods while oppositely charging the opposing double-rods to ±3 kV. Alternatively, we generate aquadrupole E-ﬁeld by charging neighboring rods with opposite polarities of ±9 kV. Shielding by
the MET coils reduces the interior E-ﬁeld by a factor of ∼5. Figure 6.3 depicts the quadrupoleelectric ﬁeld superimposed on the magnetic quadrupole ﬁeld. Hydroxyl has both an appreciablemagnetic and electric dipole, and we can easily alter the geometry of the trapping potential by

126
applying electric ﬁelds of variable magnitude and direction to the magnetically trapped sample.
To trap the incoming OH packet, the coil closer to the decelerator (front coil) is grounded
while the coil further from the decelerator (back coil) operates at 2000 A for the duration of the
3.7 ms deceleration sequence. The eﬀective magnetic dipole of OH in its J = mJ = 32 state is 1.2µB ≃ 0.81 K/T, where µB is the Bohr magneton, allowing the back coil to stop molecules withvelocity ≤ 22 m/s. The front coil is turned on after the 20 m/s molecules are stopped by theback coil, which occurs 2.65 ms after exiting the ﬁnal deceleration stage. The trap switching is
synchronous to the 20 m/s molecules since OH possessing larger longitudinal velocities escape thequadrupole ﬁeld. For technical reasons, the two magnet coils are connected in series, causing thecurrent in both coils to decrease from 2000 A to 1500 A over ∼800 µs once the front coil is switched
into the circuit. To switch between stopping and trapping ﬁeld conﬁgurations, we use two banks oftwelve parallel MOSFETs each rated for 250 A (3000 A capacity per bank). The MOSFET banks
are mounted on separate water-cooled base plates, and they are all synchronously controlled by thedriver circuit depicted in Fig. 6.2.

6.1.2

Trap Dynamics in Combined Electric and Magnetic Fields

The OH ground state is best described by Hund’s case (a), in which the spin (|Σ| = 12 )and orbital (|Λ| = 1) angular momenta are strongly coupled to the molecular axis, yielding thetotal angular momentum projection |Ω| = 32 . The |Ω| = 12 state lies 126 cm−1 above this groundstate. The nuclear spin of the hydrogen atom (I = 12 ) leads to a hyperﬁne splitting of the groundstate into F = 2, 1 components. In the presence of large external ﬁelds (E>1.7 kV/cm or B>100G), the electron angular momentum and nuclear spin decouple [24], and one considers the total
angular momentum J, which includes nuclear and electronic orbital angular momentum as wellas electron spin. The projections mJ of J are deﬁned along the axis of the applied ﬁeld. The
ground state is further split into two opposite-parity states (f /e) by the coupling of electronic
orbital angular momentum to nuclear rotation. This small Λ-doublet splitting of 1.69 GHz, alongwith the 1.67 Debye electric dipole, is responsible for the rather large Stark shift experienced

INPUT

8

0VIN

R1
1K

5

R16
51

7

GATE CURRENT IS HIGH, INDICATINGSHORTED FET

+
10uF

3

NOTE: SYSTEM SHUTS DOWN WHEN

CONTROL BOARD

67
5 MIC4429C
4
U2

8

GND
2

OUT

C3
0.1uF

IN

U10
LM7812CT

1

1

6

3

7
2

I

3

C2
10uF

G
2

O

U8
HCPL-2400 8
2

+

1

U7
LM78L05ACZ

7

C12

6

x12

6

+
C8
10uF

12VDC

+

R6
100K

C4
0.1uF

R12
100K

C13
10uF

TR

2

TR OUTCV GND
5
1

2

5

C5
1uF

THR

C6
0.1uF

RS

DIS

V+

3

4

8

6

U4
555

C14
0.1uF

GND
1

OUT

RS

V+

R5
100K7

C15
100nF

CV
5

THR

DIS

U9
555

6

7

R10
100K

5

3

4

8

+

R4
1K

C10
10uF

4

C7
10uF

+

R14
1K

4

D1
RED

R7
1K

2

8
R15
120

3

G
S

D
Q4
MTH40N06

SHEET OF2

DATE:1/2/07

DN:YJ071

1

BUILT FOR:B.STUHLDRAWN BY:BUILT BY:PAUL B

OFF= GATE SHORT!

JILA ELECTRONICS LAB
C.M. TRAP COIL DRIVER

R851

SOURCE

GAIT

DRAIN

LEDs ARE NORMALLY ON

R2
120

R3
1K

1

DESIGN BY:PAUL B

C9
0.1uF

2

1

R13
51

R11
1K

2

12 CHANNELS ON MAIN BOARD
U1
PS2501-1

8

C11
0.1uF
6
7
5 MIC4452C
4
U6

1

6
7
5 MIC4452C
4
U3

1

3

4

R9
1K

2

2

3

D2
RED

1

U5
PS2501-1
4

3

A

B

C

D

Figure 6.2: Schematic of the high-current switching circuit used to load Stark decelerated OH into the MET. The actual switching
electronics consisted of 24 of these driver circuits operating 24 individual MOSFETs. Twelve MOSFETs in parallel carried the 2000 A
current during the loading step while the other 12 where switched into the circuit to carry the 1500 A trapping current.

A

B

C

D

+15VIN

8

127

128
by OH molecules [21]. Fig. 6.3a and b show the Zeeman and Stark eﬀects of the OH groundstate respectively. External electric ﬁelds can thus dramatically alter the potential conﬁning themagnetically trapped molecules. Fig. 6.3c-e illustrate the ﬁeld geometries within the MET. In theplane perpendicular to ẑ, Fig. 6.3c shows the radial B-ﬁeld characteristic of a magnetic quadrupole,
while Fig. 6.3d depicts the electric quadrupole ﬁeld introduced by the electrode geometry. Fig. 6.3e
illustrates both B (blue) and E (red) ﬁelds along ẑ.
The top potential surface of Fig. 6.3f represents the energy shift of the decelerated |J =3
3
2 , mJ = 2 ⟩ state (E-ﬁelds couple e and f , therefore they are no longer good quantum numbers)

within the combined quadrupole E- and B-ﬁelds present in the MET. The transverse proﬁles of−→the four deepest adiabatic potentials are depicted at the longitudinal trap center, where B · ẑ = 0.
The remaining four surfaces (not shown) are simply inverted relative to those displayed, and are
therefore anti-trapping. The three-dimensional MET potentials are calculated by diagonalizing aneﬀective Hamiltonian that includes both Stark and Zeeman terms. Coupling between the ground|J = 23 , Ω = 23 ⟩, and excited | 52 , 32 ⟩, | 12 , 21 ⟩, | 32 , 12 ⟩ states is included, for a total of 64 hyperﬁnecomponents. The |J = 32 , Ω = 32 ⟩, | 23 , 12 ⟩ coupling is the most critical, increasing the trap depthof the top surface of Fig. 6.3f by 11%. Addition of the remaining states modiﬁes the potential by<0.1%. The square shape of the top surface is a result of the varying angle between E and B in
the MET. The potentials from each ﬁeld directly sum where the ﬁelds are collinear, but the trap
shallows from this maximum value as the angle between E and B increases.
The Stark decelerator slows only those OH molecules in the electrically weak-ﬁeld seeking(EWFS) component of the ground state, i.e., the upper Λ-doublet. Of these EWFS molecules, only
those with components J = 32 , mJ = ± 32 are synchronously slowed to 20 m/s. Because the positive
mJ state is magnetically weak-ﬁeld seeking (MWFS), only 50% of the decelerated molecular packet
is trappable by the MET. The molecules experience a >100 G magnetic ﬁeld at the last deceleration
stage emanating from the back coil, which preserves the quantization of the decelerated state.
Figure 6.8 displays typical time-of-ﬂight data resulting from the trapping sequence along withresults from Monte Carlo simulation of the trap dynamics. The large initial peak is the decelerated

129

ρ

ρ

300
250
E [mK]

200
150
100
50

ρ

0
−5

z

0
X [mm]

5−50
Y [mm]

5

Figure 6.3: (a) Zeeman and (b) Stark eﬀects of the ground state structure of OH. (c) Magnetic and(d) electric quadruple ﬁelds viewed from ẑ. (e) Side view of the MET conﬁguration with quadrupole
E and B ﬁelds. (f) Adiabatic potential surfaces for various components of OH ground state at thelongitudinal MET center. The top surface depicts the decelerated/trapped |J = 32 , mJ = 32 ⟩ state.

130

a)
LIF signal [a.u.]

200
150
100
50
0
0

2

4

6

8

10

12

Time from decelerator exit [ms]b)
LIF signal [a.u.]

200
150
100
50
0
0

2

4

6

8

10

12

Time from decelerator exit [ms]Figure 6.4: Time-of-ﬂight data (points) and Monte Carlo simulation results (solid red line) for two
diﬀerent electric ﬁeld conﬁgurations. The magnetic ﬁeld switches from the stopping conﬁguration
to quadrupole trapping at t = 2.65 ms. (a) Magnetic trap only. (b) Stopping and trapping in thepresence of combined electric and magnetic quadrupole ﬁelds. Note the larger steady-state trap
population when the electric quadrupole is applied.

131
OH packet traversing the detection region at the center of the trap. All subsequent peaks are theresult of oscillation in the ρ̂ dimension, with a frequency of ∼650 Hz. Fluorescence is gathered inρ̂, and oscillations in ẑ are not detectable since our detection region spans the entire visible trap
length. The magnetic quadrupole trap is unable to conﬁne all the molecules stopped by the backcoil alone due to the 25% current drop, as is clear from the diﬀerence in signal height at ∼6 ms
versus the steady-state level at, e.g., 14 ms. Note the larger steady-state population of Fig. 6.8(b)from the addition of a conﬁning transverse electric quadrupole. Also visible is the shorter coherencetime compared to Fig. 6.8(a), which results from trap distortion visible in Fig. 6.3f. An estimate ofthe density of trapped OH, as measured by LIF, gives ∼ 3×103 cm−3 . By comparison, the densityof a packet “bunched” at a phase angle of ϕ0 = 0◦ is 107 cm−3 , while the 20 m/s packet is 104 cm−3 .
Monte Carlo analysis of the velocity distribution of the trapped OH yields temperatures of ∼30
mK. With our improved knowledge of the trap dynamics due to accurate Monte Carlo simulations,
we expect to be able to trap 2-3 times more OH through modiﬁcation of deceleration phase angle
and trap switching times for optimal coupling of the slowed beam into the MET.
Higher decelerator phase angles should, in principle, produce a slower OH packet. However,
below 20 m/s the OH population drops sharply and no further deceleration is observed. We attribute
this eﬀect to a reduction in the eﬀective aperture seen by slow molecules as they traverse the ﬁnal
deceleration stage. The OH closest to the ﬁnal decelerator rods see a larger longitudinal potential
than those on-axis, and at suﬃcient phase angles, are stopped or reﬂected and hence cannot be
observed in the MET region. Numerical models elucidate this eﬀect, which may be mitigated by
new decelerator designs [55].
The 10 Hz repetition rate of the Stark decelerator currently limits our trap interrogation
time to 100 ms. Maintaining continuous currents of 1500 A presents technical challenges due toheating in both the switching transistors and cabling. To mitigate this, we operate the switching
MOSFETs on a water-cooled copper plate and also run chilled water through the MET coils.
Water-cooled cables are also necessary for such high-power operation. Not surprisingly, the largest
ohmic heating in the system occurs at copper interfaces. Figure 6.5 shows electric ﬁeld-free OH

132

1.2

(

LIF Signal [a.u.]

1.0

c

= 98

5 ms

= 500

130 ms)

0.8

0.6

0.4

0.2

(
0.0

0

c

5

= 16

3 ms

= 20

5 ms)

10

15

20

25

30

35

40

45

50

55

60

65

70

75

80

Trap Time [ms]Figure 6.5: Measured (τ ) and de-convolved collisional (τc ) lifetimes of magnetically trapped OH ata background pressure of 1 × 10−6 Torr (•, red line) and 4 × 10−8 Torr (◦, blue line).

133
trap lifetimes measured in two diﬀerent background pressure regimes: 1 × 10−6 and 4 × 10−8 Torr,
both dominated by N2 at 295 K. Non-ideal power supply control causes the trap current to decrease
from 1500 A to 1160 A at a rate of 2000 A/s. This lowers the trap depth, resulting in a loss of
12 ± 2 s−1 . After accounting for this loss mechanism, we ﬁnd background collision-limited lifetimesof 20 ± 5 ms and 500 ± 130 ms for the two pressure regimes, respectively. These lifetimes allow an
estimate of the OH-N2 cross section of 500 ± 100 Å2 .
We have therefore magnetically conﬁned ground state hydroxyl radicals in a MET that allows for the application of a variable E-ﬁeld to the trapped polar molecules. Diagonalization of aneﬀective Hamiltonian modelling the energy shift of OH in crossed E- and B-ﬁelds was necessary
for accurate Monte Carlo simulation of the trap dynamics. Future experiments could quantify
temperature dependent non-adiabatic transition rates between the diﬀerent surfaces depicted inFig. 6.3f akin to Majorana transitions observed in atomic magnetic quadrupole traps [56]. Furthermore, higher densities within the MET will facilitate the study of cold OH-OH collisions in variable
electric ﬁelds. As rather modest electric ﬁelds are expected to modify collision cross sections between polar molecules by as much as 103 [114], crossed-beam experiments exploiting the MET
described here will beneﬁt from both low center-of-mass collision energies and tunable E-ﬁelds inthe interaction region.

6.2

Permanent Magnet Trap
The MET described in the previous section is insuﬃcient for molecular collision experiments

due to its low density of trapped OH molecules (∼ 103 cm−3 ) and the technical diﬃculties associatedwith operating a 2000 A switching system. Magnetic traps are still attractive for dipolar collisionexperiments since the external electric ﬁeld may be varied, but it turns out that trapping withpermanent magnets represents a much better route to conﬁning a Stark decelerated OH beam.
Given the large magnetic ﬁelds produced by rare-earth magnets of ∼ 1 T, permanent magnets area natural choice to replace the bulky water-cooled coils and switching circuitry of electromagnets.
The diﬃculty with permanent magnets is precisely the inability to turn them oﬀ – speciﬁcally134
when it comes to loading a slow packet of molecules. Nevertheless, we solve this problem by using
the magnets themselves as “stopping” electrodes by charging them to ±12 kV. This allows us toalmost double the incoming OH velocity from the previous trap and stop molecules from a velocityof 36 m/s. Furthermore, the reduced size of the magnets allows us to match the volume of thetrap exactly to the volume of the stopped packet, thereby minimizing any density reduction due tospatial mismatch.
A pulsed supersonic beam of OH radicals is produced as described in the previous section
by striking an electric discharge through a mixture of 27 mbar H2 O and 1.5 bar Kr. The resulting
OH beam consists of rotationally cold, 2 Π3/2 molecules whose center longitudinal velocity is 490
m/s. The packet passes through a 3 mm diameter skimmer and is then coupled via an electrostatic
hexapole into the 142-stage Stark decelerator. The Stark decelerator slows weak-ﬁeld seeking
OH molecules residing in the |J = 3/2, mJ = ±3/2, f ⟩ state, where J represents total angularmomentum and mJ is the projection of J along the electric (E) ﬁeld axis. One diﬀerence from
our MET loading procedure is that the Stark decelerator is now operated at a phase angle ϕ0 of
50.352◦ in order to slow a 120 mK portion of the OH packet to 36 m/s. When slowing to velocities
below 50 m/s, we observe maximum decelerator eﬃciency for 45◦ ≤ ϕ0 ≤ 55◦ . Operation atthese intermediate phase angles increases transverse packet conﬁnement and reduces the coupling
between transverse and longitudinal motion [99, 105].
We stop and conﬁne the decelerated 36 m/s OH packet within a permanent magnetic trap
whose center lies 1 cm from the ﬁnal decelerator rod pair. This trap, depicted in Fig. 6.6a, represents
a marked improvement over our previous MET in both design simplicity and ultimate trapping
eﬃciency [64]. The magnetic trap is constructed by mounting two Ni-coated NdFeB permanentring magnets in a ﬁeld-opposing orientation such that a magnetic quadrupole ﬁeld is produced
between them as shown in Fig. 6.6b. The N42SH rating of these magnets ensures an operating
temperature of up to 150◦ C and a residual magnetization of 1.24 T. However, we do measure
reduced magnetization (∼ 20%) when baking above 100◦ . The inner and outer radii of the magnets
measure 2 mm and 6 mm, respectively, while their thickness is 4 mm. The center-to-center magnet

135a282 nm
LIF laser

Permanentring magnets
OH fluorescencecollection lens

Stark deceleratorb(ρ = 0)(z = 0)Figure 6.6: (a) Illustration of the permanent magnet trap and Stark decelerator. (b) Results ofﬁnite-element calculations of the magnetic ﬁeld distribution within the permanent magnet trap.
The small “lobe” observed in the ρ-dimension appears at the 2 mm inner radius of the magnet
ring.

136
spacing of 7 mm in this magnetic trap yields a longitudinal magnetic (B) ﬁeld gradient of 2 T/cm.
This longitudinal separation matches the extent of the molecular packet entering the trap region
from the Stark decelerator and therefore maximizes the trap density.

6.2.1

Loading the Permanent Magnet Trap

Figure 6.7 illustrates the trap loading sequence used with the magnetic trap of Fig. 6.6a.
In Fig. 6.7a, the Ni coatings of the ring magnets are charged to ±12 kV precisely 1 µs after theﬁnal deceleration stage is grounded. At this point, the magnets become high-voltage electrodes
and serve as a ﬁnal Stark-slowing stage for the 36 m/s molecules. In addition to the large stopping
potential between the magnets, there exists a smaller potential between the ﬁnal decelerator rod
pair and the ﬁrst trap magnet. This barrier reﬂects the small number of molecules with longitudinalvelocity less than 25 m/s. However, the barrier’s transverse E-ﬁeld gradient serves to conﬁne theslow molecules as they enter the trap region. The OH packet is brought to rest directly between
the magnets in 400 µs, at which point the high voltage is switched oﬀ. Those hydroxyl radicals
occupying the weak-magnetic-ﬁeld-seeking |3/2, 3/2, f ⟩ state (50% of the stopped molecules) arethen conﬁned within a magnetic quadrupole trap measuring kB × 480 mK deep in the longitudinal
dimension, where kB is Boltzmann’s constant. The trap potential is shown in Fig. 6.7b. We notethat a permanent magnet was used to reﬂect a molecular beam [123], but here we present the ﬁrst
realization of a permanent magnetic trap for neutral molecules.
Typical time-of-ﬂight data and corresponding three-dimensional Monte Carlo simulation results are displayed in Fig. 6.8a. Decelerated and trapped molecules are detected via laser-induced
ﬂuorescence (LIF). Lenses mounted in-vacuum allow for a ﬂuorescence collection solid angle of∼0.1 sr. In Fig. 6.8(a), the large peak at 400 µs is the stopped OH packet imaged at trap center.
Transverse oscillation of the trapped packet is observed in both data and simulation over 2 ms.
The number and density of trapped OH are measured to be >103 and ∼ 106 cm−3 , respectively. A
temperature of 70 mK is estimated from Monte Carlo simulation, also consistent with the molecular packet delivered by the Stark decelerator. Due to the large quadrupole B-ﬁeld present in the

137

a

b
0V

12 kV

-12 kV

0V

0V

0V

Figure 6.7: Illustration of the trap loading sequence. (a) High voltage is applied to the surfaces of
the two permanent ring magnets 1 µs after the ﬁnal deceleration stage (shown at left) is grounded.
The OH packet is stopped directly between the two magnets by the electric ﬁeld gradient in 400µs. The stopping potential due to the applied electric ﬁeld is depicted. (b) The magnet surfaces
are grounded, leaving the packet trapped within the displayed permanent magnetic quadrupole
potential.

a

b

+-

Figure 6.8: (a) Time-of-ﬂight data (circles with error bars) and three-dimensional Monte Carlosimulations (solid line) corresponding to OH trap loading. Stopping E-ﬁelds are switched oﬀ at 400µs, which leaves 50% of the stopped OH molecules trapped in the permanent magnetic quadrupole.
(b) Measurement of the lifetime of OH trapped within the magnetic trap at a background pressureof 7.5 × 10−9 Torr. A single-exponential ﬁt (solid line) of the data yields 432 ± 47 ms.

138
trap, only a fraction of the OH sample is detected as the longitudinal 5 GHz Zeeman shift neareach magnet is larger than our LIF laser linewidth. This eﬀect is included in the trap density
estimates. Figure 6.8b displays the observed trap lifetime of 432 ± 47 ms, limited by collisions withbackground gas. The trap chamber pressure of 7.5 × 10−9 Torr consists of equal parts H2 O and Kr.
Having improved on our original MET trap densities by three orders of magnitude, we can now
use this trap for low-temperature [40] and cold [18] collision studies between external atomic andmolecular beams. Also, we note that this trapping technique was recently adopted for dissipative
multiple-loading of Stark decelerated NH [124, 125].

6.2.2

Current Conditioning of Magnetic Electrodes

As with the Stark decelerator electrodes, we must prepare the permanent ring magnets forpulsed application of large electric ﬁelds by current conditioning. We ﬁnd that no additional
material coatings beyond the outer Ni layer are necessary or desirable. In fact, even electropolishing the magnets results in larger discharge currents than are obtained by simply picking thesmoothest samples from a large batch. Here, the most likely culprit is unintended surface scratches
throughout the electro-polishing process and subsequent cleaning. We have attempted coating themagnets with additional layers of Cu with the hope of obtaining a smoother surface, but it is likelythat the lower work function of Cu led to enhanced discharging. The conditioning process consistsof slowly raising the voltage on the magnets to ±15 kV while monitoring the current draw through100 MΩ limiting resistors. After this initial current conditioning, we remove the limiting resistors
and slowing raise the voltage while switching the magnets with a ∼ 500 µs pulse. We ﬁnd that
if the magnets are left for days without being switched, they must be reconditioned for reliable
operation∗ . Therefore, we often reduce the magnet voltages to ∼ ±9 kV for switching overnight.

∗

This is likely due to a long-term accumulation of paramagnetic contaminants onto the magnet surfaces fromthe vacuum chamber. Pulsing high voltage over time may serve to keep the surfaces clean through micro-discharge
events.

Chapter 7

Collision Studies with Trapped OH Molecules

Research in the ﬁeld of cold polar molecules is progressing rapidly. Magneto-association work
utilizing laser-cooled alkali atoms has yielded ∼100 nK samples of KRb molecules in their vibrational ground state [79]. A number of other groups have produced vibrationally- or electronicallyexcited molecules via Feschbach and photoassociation techniques [95]. These systems are promising for the development of polar-molecule quantum computers [126] and for investigation of theanisotropic dipole-dipole interaction at ultracold temperatures [39, 127, 128]. In addition, cryogenic
buﬀer gas cooling methods relying on thermalization of polar molecules with cold He have allowed
for magnetic trapping of NH [118] and CaH [117] at temperatures of ∼ 1 K. Stark decelerationof supersonic molecular beams readily produces state-selected, ∼ 100 mK samples at densities of
106 –108 cm−3 . As this technique allows complete control over a beam’s center-of-mass velocity,
Stark decelerated molecules have been used in crossed-beam collisions with atomic species [111]and precision spectroscopy experiments [129, 21, 24]. Spontaneous decay of long-lived vibrational
excited states has also been directly observed in trapped OH [31] and NH [130] molecules. In thischapter, we describe our recent work extending experiments with trapped, cold molecules to include
molecule-molecule collisions with magnetically-conﬁned OH. As will be discussed, collisions between
external beams and trapped molecules display unique dynamics that complement free-space inelastic collision experiments. Such beam-trap collisions display sensitivity to integral elastic, integral
inelastic, and even diﬀerential elastic cross sections.

140

7.1

He-OH, D2 -OH Collisions

7.1.1

Motivation

Collisions between hydrogen (H, H2 ) and larger polar molecules (OH, H2 O, CO, H2 CO, SiO)
are of astrophysical interest due to their possible role as pumps for interstellar masers [131]. In
particular, speciﬁc emission lines of interstellar OH (1720 MHz) and H2 CO (4.8 GHz) masers havebeen attributed to collision-induced inversion by H2 [132, 133]. In the following discussion, we
present collision cross sections for D2 –OH and He–OH collisions within a permanent magnet trap
conﬁning Stark-decelerated OH molecules. By directly comparing D2 –OH cross sections with thoseof the more theoretically tractable He–OH, we will aid molecular collision theory at energy scales
below 100 cm−1 .

7.1.2

Experiment

The open structure of our permanent magnet trap allows for low center-of-mass energy (Ecm )collision studies between the trapped OH and external molecular or atomic beams. As large electricﬁelds are used only for initial trap loading, there is no risk of voltage breakdown when pulsing such
beams through the trap. Furthermore, though all He–OH and D2 –OH collisions reported herein
were observed with the magnets grounded, a relatively small polarizing electric (E) ﬁeld (fewkV/cm) can be applied to the magnets after trap loading without loss of conﬁned molecules—
thereby enabling our more recent investigation of dipolar collisions between OH and ND3 [18].
Operation of our Stark decelerator and permanent magnet trap are described in Chapters 5
and 6, respectively. For this external beam collision experiment, we decelerate and magneticallytrap a 70 mK portion of an OH supersonic beam initially seeded in a Kr carrier gas. After baking
the trap chamber to remove H2 O vapor, the OH trap lifetime of ∼ 500 ms is limited primarily byresidual Kr gas from the supersonic expansion. However, when running the external He or D2 beam,
this lifetime drops by an order of magnitude due to the low pumping speed of the turbomolecularpump with regard to light species.

141

Permanent
Magnets
Stark Decelerator

SkimmerTemperature-Controlled
Solenoid Valve
Figure 7.1: Illustration of the low-temperature He–OH, D2 –OH collision apparatus consisting ofa Stark decelerator and permanent magnetic trap for OH (background) as well as a temperaturecontrolled pulsed solenoid valve and skimmer assembly for He and D2 beams (foreground). Collisions between the external He or D2 beams and trapped OH result in measurable OH trap loss.

142
To study collisions between trapped OH and He/D2 beams, we place a pulsed solenoid valve(General Valve Series 99) and 1 mm diameter skimmer assembly, as shown in Fig. 7.1, such thatthe skimmed atomic or molecular beam passes directly between the trap magnets. The solenoid
valve rests in a bath of liquid nitrogen while its 1 mm output nozzle is heated via 25 turns ofmanganin wire to allow for tuning of the beam velocity. The measured velocities of the He andD2 beams as a function of output nozzle temperature are displayed in Fig. 7.2a. The velocities
are determined using a combination of a miniature microphone-based pressure sensor and fastionization gauge (FIG) placed 13 cm apart in the trap chamber. As expected, we observe thatthe beam velocities scale as the square-root of the nozzle temperature. This is veriﬁed by ﬁtting
the velocity distributions of Fig. 7.2a. The Ecm of the He–OH and D2 –OH systems can therefore
be tuned to minima of ∼60 cm−1 and ∼145 cm−1 , respectively. The relatively low velocity of
the He beam is most likely a result of the low stagnation pressure used (2.0 bar) along with theextended exit oriﬁce of > 1 cm that could lead to a more isothermal as opposed to isentropic beam
expansion. Our observation of enhanced He beam velocities at higher stagnation pressures supports
this assertion. The stagnation pressure for the D2 beam is 2.7 bar. To calibrate the relative densitiesof the colliding beams, we again use the FIG and obtain the temperature-dependent density curvesof Fig. 7.2b. The uncertainty in the inter-species calibration of the ionization gauge is ∼10%. The
measured 8% velocity spread of the supersonic He beam gives a collision energy resolution of 9
cm−1 at the lowest nozzle temperature.
Figure 7.3a displays the time dependence of OH trap loss as a supersonic beam of He traverses
the magnetic trap. Trap density drops sharply over the ﬁrst ∼1 ms after the solenoid valve is ﬁred,then remains constant over the time scale shown. Although the partial pressure of the scattering
gas in the trap chamber rises as the supersonic beam scatters from the chamber walls, we measure
an OH trap lifetime of >50 ms following the initial collision. Such vastly diﬀerent time scales allow
us to diﬀerentiate between trap loss due to the supersonic beam and that resulting from backgroundgas collisions. Trap loss at a given nozzle temperature is measured by repeatedly comparing OH
population 1 ms before and 2 ms after the solenoid valve is triggered. The total cross sections

143

a

b

Figure 7.2: (a) Measurement of He and D2 beam velocities as a function of valve nozzle temperature.
The beam speeds were measured with a miniature microphone and fast ionization gauge placed 13
cm apart within the magnetic trap chamber. The solid lines are ﬁts to the data indicating that,as expected, the measured beam velocity is proportional to the square root of nozzle temperature.
(b) Calibration of He and D2 beam density as a function of valve nozzle temperature. The relative
density between He and D2 is obtained by taking into account the diﬀerent sensitivity of the fastionization gauge to these two species.

of Fig. 7.3b are determined from the trap loss data by normalization to the corresponding beam
ﬂux of Fig. 7.2b measured with the fast ionization gauge at a given nozzle temperature. We also
ﬁnd that the trap loss scales linearly with the beam ﬂux, conﬁrming that we are operating in the
single-collision regime.
An important advantage inherent to using trapped molecules as the scattering target is theability to directly measure the absolute collision cross section. To do this, we use a leak valve to
ﬁll the trap chamber with a known pressure of He gas, then directly measure the OH trap lifetimeat that pressure. The temperature of the chamber walls is 298 K, placing the Ecm of the thermal
He–OH collisions at ∼250 cm−1 . The data point for He–OH collisions at 230 cm−1 from Fig. 7.3b
is then scaled to this absolute cross section measuring 127 ± 18 Å2 .
For OH molecules in their ground electronic and vibrational state, the energy splitting of thetwo lowest lying rotational levels, J = 3/2 and J = 5/2, is 84 cm−1 . In a crossed-beam experiment,
an abrupt decrease in the Xe–OH inelastic cross section was observed as Ecm was tuned below
this value [111]. The He data of Fig. 7.3b possesses this feature. Because the magnetic trapping
potential is sensitive to the internal state of the OH molecule, we cannot diﬀerentiate between trap

144
loss due to elastic or inelastic collisions. Nevertheless, such a sudden decrease of the total crosssection below 84 cm−1 is indicative of threshold behavior. The collision cross section of D2 –OH islarger than that of He–OH. This is understandable since the D2 beam is an ortho/para mixture
and contains a large fraction of frozen-in J = 1 population. The ratio of the two cross sections at
230 cm−1 is consistent with previous H2 –OH and He–OH pressure-broadening measurements madeat a temperature of 298 K [134]. The enhanced collision cross sections for D2 –OH result from thequadrupole moment of the D2 J = 1 state that can interact strongly at long range with the OH
dipole. Yet another striking feature of the collision data is the pronounced peak in the D2 –OH
cross section for Ecm ∼ 305 cm−1 . Although more theoretical consideration is warranted, the 300
cm−1 J = 3 ← J = 1 transition of the 1 Σ+g D2 molecule may be contributing to the inelastic crosssection at this energy. Another possible explanation is collision-induced decay of J = 2 molecules
present in an imperfect D2 supersonic expansion.
To summarize, we demonstrate a new permanent magnetic trap design that conﬁnes a dense
sample of cold OH molecules as a cold collision target. Collision experiments consisting of velocitytunable supersonic beams of He and D2 intersecting the magnetically-trapped cold OH molecules
yield absolute collision cross sections over an energy range of 60 cm−1 to 230 cm−1 and 145 cm−1 to510 cm−1 , respectively. Threshold behavior is observed in He–OH collisions, and an enhancementof inelastic cross sections is seen in the D2 –OH system near 305 cm−1 . Having demonstrated theusefulness of this permanent magnetic trap for collision studies, future goals include using colder
continuous beams of polar molecules as the colliding partner for OH. With the ability to apply asuﬃciently strong polarizing ﬁeld within the magnetic trap [135], we aim to reach suﬃciently low
Ecm to observe dipole-dipole interactions.

7.1.3

Recent Developments

Kirste and co-workers have recently reported relative state-to-state inelastic cross sections
between Stark decelerated OH and supersonic beams of He and D2 [136]. With respect to D2 –OH
collisions, they do not observe enhancement of the OH state-to-state inelastic cross section near

145

a

b

Figure 7.3: (a) Time dependence of OH trap loss due to collisions with a supersonic He beam. Trap
density drops sharply over 1 ms upon He beam collisions, then remains constant over the time scale
shown. The valve is triggered 20 ms after the magnetic trap is loaded. (b) Total collision crosssections for He–OH (open circles) and D2 –OH (squares) as a function of Ecm . The decrease in the
He–OH cross section at low energy is attributed to reduced inelastic loss as Ecm drops below the
84 cm−1 splitting between the J = 3/2 and J = 5/2 states of the OH molecule.

146
the collision energy of 305 cm−1 . The primary diﬀerence between our experiment and that of
Kirste et al is that our collision experiment involves trapped OH molecules, while their collisions
take place in free space. In this way, we are sensitive to elastic as well as inelastic loss fromthe trap, while the results of Ref. [136] are not sensitive to elastic collisions at all. One possible
explanation for both experimental results is that the inelastic collision near 300 cm−1 preferentially
excites the D2 molecule along the J = 3 ← J = 1 rotational transition, leaving the OH state
unchanged. Such a collision will certainly cause OH trap loss in our system due to momentum
exchange from the D2 molecule. Since no rigorous quantum scattering calculations for D2 –OH
have been performed to date, we cannot make concrete predictions as to the relative propensities
for rotational transitions. However, future experiments sensitive to the internal state of the colliding
D2 molecule will invariably answer these intriguing questions.

7.2

Cold Dipolar Collisions between OH and ND3

7.2.1

Motivation

For half a century, scattering of crossed atomic or molecular beams under single-collisionconditions has remained the primary technique for investigation of inelastic and reactive dynamics
at collision energies above ∼1 kcal/mol (∼500 K) [137]. Experiments using electric ﬁeld-aligned
neutral polar molecules can probe the steric asymmetry of atom-molecule [138, 139, 140] andbi-molecular [141, 142] potential energy surfaces (PESs), but their large center-of-mass collisionenergies have precluded the observation of electric ﬁeld modiﬁcation of these potentials. However,in the cold collision regime where only tens of scattering partial waves contribute, one can move beyond molecular orientation and directly modify intermolecular dynamics via the long-range dipole
interaction. The relatively low number densities (106 -108 cm−3 ) of cold molecule production techniques have thus far limited gas-phase collision experiments between state-selected distinct polar
species to >300 K. We combine for the ﬁrst time the methods of Stark deceleration, magnetic
trapping, and buﬀer gas cooling to enhance molecular interaction time by ∼105 and overcome this

147

b

a

~~

θ, φ







ψ





c









 θ, φ Diatomic
axis


























Ω






ND3











OH







θ

Energy [cm-1]

C3 axis











    θ

Figure 7.4: (a) Rotational structure of OH and ND3 molecules. The parity-doublet splitting of eachrotational level has been expanded for clarity, while the black and red parity levels are the ones
included in the OH-ND3 scattering calculation. The upper, red, parity states are those selected by
the experimental apparatus for the cold collisions described herein. The Stark and Zeeman shiftsof ground state OH and ND3 are presented in Chapter 2. (b) Illustration of the Jacobi coordinates
used for OH-ND3 collision calculations. We deﬁne R as the distance between the molecular centersof mass while the Euler angles (θA , ϕA ) and (θB , ϕB ) give the orientation of the OH and ND3 axes,
respectively, in the body-ﬁxed frame relative to R. The coordinate ψB speciﬁes the orientationof the ND3 monomer within the OH-ND3 complex. The geometry ϕA = ϕB = θA = θB = 0
corresponds to the OH axis lying in the σv symmetry plane of ND3 with the O atom facing the′′plane of D atoms. (c) Contour plot of the lowest adiabatic potential energy surface for the A stateof the OH-ND3 collision complex at R = 3.1 Å and ϕA = ϕB = 0 as a function of θA and θB . The
color legend is scaled in units of cm−1 .

148density limitation. Here we present the ﬁrst observation of cold (5 K) heteromolecular dipolar collisions and describe the ﬁrst theoretical study of scattering in the many partial-wave regime withchemically-relevant OH and ND3 molecules.
The last decade has seen tremendous progress in the production of cold and ultracold
polar molecules [12]. Application of these production techniques to the study of novel atommolecule and molecule-molecule scattering has led gas-phase collision physics to new low temperature regimes. Speciﬁcally, crossed beam experiments using rare gas atoms and Stark decelerated OH(2 Π) molecules have allowed for the study of inelastic scattering at translational energies comparable to that of OH rotation (∼100 K) [111, 143, 136]. By measuring collisional loss of a magneticallytrapped OH target with incident beams of He or D2 , we have determined total trap loss cross sections at similar energies [40, 144]. Buﬀer gas cooling has been employed in the study of elastic andinelastic collisions of He-NH (3 Σ) [73, 72] and He-TiO [145]. Cold reactive collisions have been observed using a velocity-ﬁltered room-temperature beam of CH3 F colliding with trapped Ca+ atomic
ions [146]. In the ultracold regime [79], bimolecular chemical reactions between ground-state KRb
molecules at temperatures of a few hundred nanoKelvin have been directly controlled via external
electric ﬁelds [39, 17], opening the door for experimental probes of quantum many-body eﬀects ina dipolar molecular gas [16].
To observe dipolar collisions between state-selected OH and ND3 , we constructed a novel cold
collision apparatus combining Stark decelerated and magnetically trapped OH with a continuous
buﬀer gas cooled beam of ND3 , allowing interaction times of ∼1 s. In contrast to traditional scattering experiments, our measurement is sensitive to both inelastic and elastic collisions. Furthermore,
fewer than 100 partial waves contribute to the 5 K collision — placing the OH-ND3 interaction in
an intermediate regime between gas-kinetic and quantum scattering. Theoretical OH-ND3 crosssections computed from ab initio potentials are in good agreement with the measured zero-ﬁeld
trap loss cross section. Given the generality of buﬀer gas cooling and electro/magnetostatic velocity
ﬁltering, our apparatus allows for a large class of electric ﬁeld dependent cold molecular collision
studies.

149

Hexapole guide

Sta
r

kd

ece

ler
ato
r

+

ND3 ToFMS
assembly
Magnetic
trap

He, ND3 buffer
gas source

MCP

OH fluorescence,
ND3 ionization lasers

OH fluorescencecollection lensFigure 7.5: Illustration of the combined Stark decelerator, magnetic trap, and buﬀer gas beam
assembly. The curved hexapole ﬁlters cold ND3 from the He buﬀer gas and guides the continuous
beam to the OH magnetic trap. (Inset) Closeup of the trap assembly showing the dual-species
detection components. We detect OH and ND3 in the collision region using laser-induced ﬂuorescence (LIF) and resonance-enhanced multiphoton ionization (REMPI), respectively. Hydroxyl
ﬂuorescence at 313 nm is collected using a lens mounted 2.5 cm below the magnetic trap center.
Ionized ND+
3 molecules are accelerated to a microchannel plate (MCP) detector by placing 950 V
on the front magnet, 0 V on the back magnet, and -1100 V on the acceleration plates that makeup a time-of-ﬂight mass spectrometer (ToFMS).

150
Two readily coolable species, hydroxyl and ammonia, are ubiquitous in atmospheric andastrophysical spectroscopy. Large interstellar clouds of OH and NH3 (some co-located) have been
detected in both absorption and maser emission along microwave parity-doublets [147]. In the
troposphere, hydroxyl-ammonia reactions are the dominant mechanism for removal of NH3 [148].
We choose to investigate electric ﬁeld-dependent cold OH-ND3 collisions due to the large static
polarizability of both species and the near-degeneracy of the energy splittings of the OH Λ-doublet(0.056 cm−1 ) and ND3 inversion-doublet (0.053 cm−1 ). The former allows for ﬁeld orientationand precise control over the internal and external degrees of freedom of both molecules, while thelatter has been shown to cause large enhancements in inelastic rates at room temperature [149].
Experimentally-relevant rotational and parity-doublet states of OH and ND3 are illustrated inFig. 7.4a.

7.2.2

Cold OH-ND3 Collision Apparatus

The experimental setup consists of two distinct cold molecule apparatus as shown in Fig. 7.5:
a Stark decelerator/magnetic trap assembly [64, 40] and a buﬀer-gas cooled continuous molecularbeam source. A supersonic beam of OH molecules seeded in Kr is decelerated and trapped at atemperature of 70 mK and density of ∼106 cm−3 as described in Ref. [40]. Only those OH molecules
residing in the 2 Π3/2 |F, MF , p⟩ = |2, +2, f ⟩ ground state are phase-stably Stark decelerated andmagnetically trapped. Here F , MF , and p denote the hyperﬁne quantum number, its projection on
the external electric ﬁeld axis, and the parity of the state, respectively. To detect trapped OH, themolecules are excited at the 282 nm A2 Σ+ (v = 1)←X 2 Π3/2 (v = 0) transition. Fluorescence fromthe 313 nm A2 Σ+ (v = 1)→X 2 Π3/2 (v = 1) decay is collected by a lens mounted to the magnetic
trap assembly.
We employ buﬀer gas cooling and Stark velocity ﬁltering to generate a continuous eﬀusive
beam of cold ND3 molecules at a density of 108 cm−3 and mean velocity of 100 m/s. Our measured
beam ﬂux of 1011 s−1 is comparable to that of previous experiments [150, 83]. Cold ND3 is generated
in a copper cell ﬁlled with He buﬀer gas at 4.5 K. A 6 mm-diameter aperture is cut into the front

151

b



a





















Figure 7.6: (a) Translational energy spectrum of the guided continuous ND3 beam as measured
by 2+1 REMPI in the collision region. (b) Rotationally-resolved REMPI spectrum of guided ND3molecules showing diﬀerent J, K states. The upper red curve (oﬀset for clarity) was taken at buﬀer
gas ﬂows of 2.0 and 2.5 sccm of He and ND3 , respectively. The lower black curve displays thesmaller rotational temperature observed at 3.5 and 1.0 sccm. Intermediate rotational levels in the
excited B̃(v2 = 5) electronic state are labeled in parentheses for each transition.

152of the cell directly opposite the ND3 inlet to allow the cold He/ND3 mixture to escape. An
electrostatic hexapole consisting of six 3 mm-diameter steel rods and possessing an inner diameterof 6 mm is mounted 2 mm from the cell aperture. This straight hexapole guide is 20 cm longand mounts to a 3 mm-thick gate valve which isolates the OH trap vacuum from the cryogenic
vacuum. A curved hexapole is mounted just beyond the gate valve and follows a 90◦ bend at aradius of 13.5 cm. The hexapole guide terminates 1 cm from the permanent magnetic trap center.
The curved hexapole ﬁlters cold ND3 from the He buﬀer gas and sets an upper bound on theaccepted forward velocity for weak-ﬁeld seeking states (∼150 m/s at ±5 kV for |J, K⟩ = |1, 1⟩).
The quantum numbers J and K denote the total molecular angular momentum and its projection
on the ND3 symmetry axis, respectively. Figure 7.6a displays the measured energy distribution ofguided |1, 1⟩ ND3 molecules in the OH-ND3 center-of-mass frame. To quantify the continuous beam
velocity, the hexapole is switched so that guided ND3 ﬂux can be measured over diﬀerent guiding
durations. After including the guide path length, we diﬀerentiate a ﬁt of the resulting curve toobtain the ND3 velocity distribution of Fig. 7.6a. Guided ND3 is detected in the collision region
using 2+1 resonance-enhanced multiphoton ionization (REMPI) [50] and subsequent ion detection.
To accomplish this, ND3 in the |1, 1⟩ state is resonantly ionized between the trap magnets using
a focused 317.4 nm, 10 ns laser pulse. To accelerate ions to a microchannel plate (MCP) detector(see Fig. 7.5), the trap magnets are charged to a potential diﬀerence of 950 V. After extraction
from the trap, ions enter a 2 cm long ﬁeld-free region and are subsequently detected by the MCP.
Our dual-detection scheme permits characterization of the state purity of both OH and ND3 .
Due to the state selectivity of Stark deceleration and magnetic trapping, perfect OH rotational,
Λ-doublet, and hyperﬁne state purity is achieved. Vibrationally excited (v = 1) OH are phasestably decelerated with v = 0 molecules, but <5% of OH are initially produced in the v = 1
state [75]. We also characterize the ND3 rotational distribution within the collision region. The
rotational structure of ND3 is shown in Fig. 7.4a while Fig. 7.6b displays the measured rotational
spectra of guided ND3 molecules for two diﬀerent buﬀer gas ﬂows – 2.0 sccm He and 2.5 sccm ND3(upper curve) and 3.5 sccm He and 1.0 sccm ND3 (lower curve). A standard cubic centimeter per

153
minute (sccm) is equivalent to 4.4 × 1017 s−1 . As the ratio of He to ND3 ﬂow is increased, thetwo-state (|1, 1⟩, |2, 2⟩) rotational temperature drops from 6.3 K to 5.4 K. With the larger He:ND3
ﬂow ratio, the relative guided populations of anti-symmetric (weak-ﬁeld seeking) |1, 1⟩, |2, 2⟩, |3, 3⟩,
and |4, 4⟩ states are 87%, 9%, 4%, and <1%, respectively. This ground state fraction is similar
to that measured by depletion spectroscopy of a buﬀer gas cooled H2 CO beam [150]. States with
K = 0 do not exhibit a ﬁrst-order Stark shift and are not guided. Also, at this ﬂow and guidevoltage (±5 kV), no population is observed in those states where J̸=K. Our choice of 3.5/1.0
sccm for the collision experiment is a compromise between maximizing |1, 1⟩ ﬂux while minimizing
rotational temperature.
To quantify total trap loss cross sections between OH and ND3 , it is necessary to measure bothcollision-induced OH trap loss and absolute colliding ND3 density. Loss of trapped OH is measured
as shown in Fig. 7.7a over a period of 160 ms. First, the background-gas limited trap decay rate
(γoﬀ ) is measured with the ND3 ﬂow oﬀ. This decay rate is determined by a single-exponential ﬁt to
the data. The value of γoﬀ is set by the presence of residual Kr in the trap chamber and is typically
2-3 s−1 at a pulsed-valve repetition rate of 5 Hz. The cold ND3 beam is then turned on and the loss
rate due to background gas plus cold ND3 (γon ) is measured. The collision-induced loss rate (γcoll )is the diﬀerence of the two rates. This diﬀerential measurement makes γcoll insensitive to day-to-day
variations in background pressure, and yields a single-measurement fractional error of ∼30-40% in
the available runtime of 1 hr. To investigate the eﬀect of an external electric ﬁeld on the crosssection, we apply two distinct electric ﬁeld conﬁgurations to the colliding molecules. We deﬁne thepolarizing electric ﬁeld for a given molecule as Epol = ∆/2µ where µ is the molecular permanent
dipole moment and ∆ is the parity-doublet splitting (Epol =1.7 kV/cm for OH and 2.0 kV/cm for
ND3 ). For both ﬁeld distributions, E > 2 kV/cm throughout the collision region, polarizing bothcolliding species. One E-ﬁeld distribution is produced by charging the trap magnets themselves andlowers the OH radial trap depth by ∼20% due to the relative orientations of E- and B-ﬁelds within
the collision region. The alternate ﬁeld distribution is created by charging a stainless steel cup
placed between the rear trap magnet and time-of-ﬂight mass spectrometer. The resulting distinct

154

a

b


Figure 7.7: (a) Semi-logarithmic plot of OH trap decay rates with (◦) and without (•) the colliding
ND3 beam. The decay rate due solely to cold OH-ND3 collisions is γcoll = γon − γoﬀ . (b) Plotof all experimental runs measuring total cross sections with () and without () a polarizing
electric ﬁeld. Average cross sections are determined from the weighted mean of all points anderrors for the given E-ﬁeld condition. The cross-hatched regions represent one statistical standard
error. Individual error bars for each experimental run are obtained by propagating the standard
deviation from exponential ﬁts of γon and γoﬀ to γcoll .

155
distribution between the grounded magnets increases the radial trap depth by ∼40%. Polarizationof OH and ND3 under both ﬁeld conﬁgurations is experimentally veriﬁed in the collision region
through spectroscopic measurement of the population of cold ND3 molecules in the strong-ﬁeld
seeking (unguided) |1, 1⟩ state. This population is measured as a direct result of inversion-state
mixing under the polarizing electric ﬁelds and is not detected in the absence of either electric ﬁeld
conﬁguration. We observe no diﬀerence in collisional loss between these two conﬁgurations, andthus the changes in trap loss arise directly from the enhanced dipolar interactions between OH-ND3
under the applied E-ﬁeld.
loss , we use REMPI to calibrate the absolute
To determine the value of the loss cross section, σexp

density of the continuous ND3 beam within the collision region for each measurement of γcoll .
Conversion of MCP output current to absolute molecule density is notoriously diﬃcult due tounknown ionization volume/eﬃciency and uncertain MCP gain. To sidestep these issues, a leakvalve and calibrated quadrupole mass spectrometer are used to admit a known pressure of 295 K
ND3 gas into the trap chamber. The measured ND3 pressure is then scaled down by the Boltzmann
fraction of molecules in the |1, 1⟩ anti-symmetric state at thermal equilibrium (4.7x10−3 ), and an
absolute ND3 density is assigned to the observed MCP output current. Provided the cold beam ismuch larger than the ionization volume (true in our system), this calibration procedure is insensitive
to the above problems. We also correct for overlap of REMPI transitions in the 295 K ND3 spectrum
as described in Chapter 4. The relatively small rotational energy splittings of ND3 displayed inFig. 7.4a lead to a congested spectrum at room temperature. Failure to account for overlap oftransitions through the intermediate B̃(v2 = 5) state would lead to an underestimate of the cold
beam density and concomitant overestimate of cross sections. Finally, a total trap loss cross section−1
loss = γ
is determined for each experimental run as σexp
coll (n0 vrel ) , where n0 is the measured density

of ND3 and vrel is the mean relative molecule velocity.

156
7.2.3

OH-ND3 Collision Theory

A complete theoretical description of the OH-ND3 collision at 5 K is complicated by the 2 Πsymmetry of ground-state hydroxyl. Interactions including Π molecules are characterized by multiple electronic potentials, avoided crossings, and conical intersections. As such, scattering calculations based on ab initio potential energy surfaces have thus far been limited to Σ molecules [151, 152]or collisions between Π molecules and rare gas atoms or closed-shell diatomics with simple rotational structure [153, 154]. A full description of the theory results summarized here will be publishedelsewhere [155].
To describe the OH-ND3 collision complex, we calculate the two lowest adiabatic PESs of
A′ and A′′ symmetries using a spin-Restricted Coupled Cluster method with Single, Double, andnon-iterative Triple excitations (RCCSD(T)) as implemented in MOLPRO [156]. An augmented
correlation-consistent triple zeta basis (aug-cc-pVTZ) is used for both N and O atoms [157]. As
illustrated in Fig. 7.4b, we deﬁne R as the distance between the molecular centers of mass while
the Euler angles (θA , ϕA ) and (θB , ϕB ) give the orientation of the OH and ND3 axes, respectively,in the body-ﬁxed frame relative to R. A third angle, ψB , gives the orientation of the ND3 monomer
about its C3 symmetry axis. Figure 7.4c shows the lowest adiabatic PES of A′′ symmetry as afunction of the orientation angles θA and θB for R = 3.1 Å and ϕ = 0. Following the diagonalline deﬁned by θA = θB , a barrier of ∼1000 cm−1 is observed between head-to-tail ND3 · · ·OH and
D3 N· · ·HO conﬁgurations, demonstrating the large anisotropy of the OH-ND3 interaction.
To elucidate the dynamics of cold OH-ND3 collisions, we developed and implemented arigorous quantum scattering approach based on the close-coupling (CC) expansion of the wave
function of the collision complex using the total angular momentum representation in the body-ﬁxed
coordinate frame. The matrix elements of the Hamiltonian are evaluated analytically by expanding
the angular dependence of the ab initio PES in spherical harmonics and retaining only the lowestorder terms, including the isotropic term V000 (R) and the dipolar term V112 (R). The model PES
produced in this way correctly reproduces both the isotropic and long-range anisotropic parts of the

157
OH-ND3 interaction, which are of crucial importance for cold collisions. Test calculations show thatshort-range anisotropic terms V110 (R) and V101 (R) have a minor inﬂuence on collision dynamics
above 1 K.
The CC equations are solved numerically to produce converged S-matrix elements and scattering cross sections for collision energies between 0.5 and 8 cm−1 , which are convoluted with thebeam distribution function shown in Fig. 7.6a to enable direct comparison with experimental data.
The cross sections for trap loss are evaluated by integrating the diﬀerential cross section (DCS)
over a restricted angular range [θmin , π]. The cutoﬀ angle θmin serves to subtract contributions
from forward-scattered elastic collision products that do not have enough kinetic energy to leave
the trap.

7.2.4

Experiment/Theory Results

The measured total trap loss cross sections are displayed in Fig. 7.7b. Each point represents
a distinct cross section measurement and the average results for zero and non-zero electric ﬁeld
are determined as the weighted mean of each data set. The cross-hatched areas illustrate one
statistical standard error centered at the weighted mean of each set. In addition to the statisticalerror, we estimate that the ND3 leak pressure calibration, 295 K REMPI line-overlap correction,
loss add a common 50% systematic uncertainty toand our use of vrel = 100 m/s in obtaining σexp

both absolute measurements. We attribute the largest contribution to systematic error to theline-overlap correction factor described in Chapter 4.
Figure 7.8a shows the calculated cross sections for elastic scattering and inelastic relaxation
in OH-ND3 collisions as functions of collision energy (EC ). The cross sections are extremely large,
exceeding typical gas-kinetic values by orders of magnitude. This dramatic enhancement is a direct
manifestation of the dipolar interaction, which induces direct couplings between the opposite parity
levels. Because of the long-range couplings, both elastic and inelastic cross sections are dominatedby collisions occurring at large impact parameters. A partial-wave analysis of the cross sections at
EC = 5 K reveals contributions from as many as 90 partial waves, and the dependences σel (Jtot ) and158a

σ








σ


σ

 

-

σ

σ

 
-

-

σ

b

Figure 7.8: (a) Plot of theoretical OH-ND3 elastic (σel ) and inelastic (σin ) cross sections over theexperimental collision energy range. (Inset) Theoretical and experimental trap loss cross sections,with numbers in parentheses representing combined statistical and systematic errors. Experimental
loss are shown for both unpolarized and polarized colliding molecules. The σ loss
loss cross sections σexp
theoryvalue includes the eﬀect of reduced elastic loss from glancing collisions due to trap conﬁnement.
All cross sections are given in units of 10−12 cm2 . The quantities σ̄el and σ̄in are the theoreticalelastic and inelastic cross sections, respectively, in the absence of trapping potentials and averaged
over the experimental collision energy distribution of Fig. 7.6a. (b) Semi-logarithmic plot of the2theoretical diﬀerential cross section for 5 K elastic OH-ND3 collisions in units of Å /sr. This result
loss
is used to calculate σtheory
from the free-space elastic and inelastic cross sections of panel (a). The
forward-peaked nature of the elastic collision leads to a factor of 25 suppression of OH elastic loss
at our ﬁnite radial magnetic trap depth of 240 mK.

159
σin (Jtot ) peak at Jtot = 30.5 and Jtot = 50.5, respectively, where Jtot is the total angular momentumof the collision complex. At EC > 1 K, the cross sections decrease monotonically with increasing−2/3

EC , following the dependence σ(EC ) ∝ EC

. This result can be obtained analytically using theLangevin capture model [158] for a purely dipolar potential, demonstrating that the variation of
the cross sections with collision energy is determined by the dipolar interaction.
A more detailed theoretical analysis in the absence of external ﬁelds shows that the inelastic
cross section associated with both OH and ND3 changing their parity-doublet state is 0.34 × 10−12
cm2 , more than 6 times larger than that for only one partner changing state. This propensity
rule follows from the symmetry properties of the matrix elements of the dipolar interaction in the
scattering basis. The inset table of Fig. 7.8a compares the measured and theoretical trap loss
loss
cross sections. The σtheory
number is calculated using the OH-ND3 elastic diﬀerential cross section(DCS) of Fig. 7.8b and includes the recently-elucidated eﬀect of trap conﬁnement on measured
collisional loss at low momentum transfer [40, 159, 144]. Since glancing elastic collisions can leave
loss must be compared with σ loss′
scattered OH trapped, the measured σexp
theory = σ̄el + σ̄in , where σ̄in is′ is the velocity-averaged elastic loss cross section
the velocity-averaged inelastic cross section and σ̄el
′ /σ̄ = 0.04). The forward-peaked structure of thethat includes the trap loss suppression eﬀect (σ̄elelOH-ND3 DCS is responsible for this large reduction in elastic loss. Our measurement validates
the large theoretical prediction of σ̄el + σ̄in = 1.0 × 10−12 cm2 attributed to the OH-ND3 dipolarinteraction.
loss by a factor of 1.4(3) when both molecules are polarized
We observe enhancement of σexp

by an external E-ﬁeld. This signals an increase of the total cross section (σ̄el + σ̄in ) since experimentally we observe no dependence of loss on trap depth. More work is required to compute a
loss
from the ab initio PESs. However, we can estimate ﬁeld-dependent elasticﬁeld-dependent σtheory

cross sections using the semiclassical Eikonal approximation as described in Section 7.3. The re′ /σ̄ predicts a factor of 1.2
sult of this calculation combined with the above values of σ̄in and σ̄elelenhancement of the measured loss cross section when both molecules are polarized. This suggeststhat the elastic cross section plays a dominant role in the measured increase in trap loss.

160

7.3

Semiclassical Estimate of Elastic Cross Sections

7.3.1

Eikonal Approximation

When treating scattering problems in the limit where the collision energy is larger than the
scattering potential, the semiclassical Eikonal approximation is valid [160]. This treatment assumes
that the center-of-mass follows a straight-line path over the interaction length of the scattering
potential, V (R), where R is the distance between colliding partners. Within this framework, we
may write the scattering amplitude f (θ) for scattering angle θ asf (θ) = −ik

∫ ∞

[
]
dbbJ0 (kbθ) e2i∆(b) − 1

(7.1)

0

where b is the impact parameter, k is the collision wavevector, and the phase term ∆(b) is deﬁnedasM∆(b) ≡ −
2k~2

∫ +∞

√
V ( b2 + z 2 )dz.

(7.2)

−∞

In the above deﬁnition, M = M1 M2 /(M1 + M2 ) is the reduced mass of the scattering partners
with masses M1 , M2 . Knowledge of the scattering amplitude allows computation of the diﬀerential(dσ/dΩ) and integral (σ) elastic cross sections:dσdΩ= |f (θ)|2
∫
dσ
σ =
dΩ
dΩ

(7.3)

∫ π
=

2π

dσ
sin θdθ.
θmin dΩ(7.4)

Note that we have included the term θmin in the integration limits of Eq. 7.4. For collisions infree space where detection is sensitive to scattering over all θ, the minimum scattering angle iszero. However, if the detection of scattering products has a limited angular resolution or one issensitive to collisional trap loss due to some minimum amount of momentum transfer [159, 144],
then 0 < θmin < π. For our collisional trap loss meaurements, we can precisely deﬁne the minimum
scattering angle as [159]

(

MT UT
θmin = arccos 1 −M 2 v2

)
(7.5)

161
where MT is the mass of the trapped species, UT is the trap depth, and v is the relative velocityof the scatterers. Specializing to the parameters of our OH-ND3 collision experiment, we obtainθmin = 13◦ .

7.3.2

Model OH-ND3 Scattering Potentials

The intermolecular potential of the OH-ND3 complex is rather anisotropic as evidenced by
Fig. 7.4. However, to estimate the semiclassical scattering cross section for unpolarized molecules,
it is useful to ascribe an isotropic and attractive Van der Waals potential of the form
V (R) = −

C6
R6

(7.6)

where the magnitude of the C6 coeﬃcient can be estimated from the dynamic polarizability according to [161]
3
C6 =
π

∫ ∞
αOH (iω)αN D3 (iω)dω.

(7.7)

0

The individual polarizabilities α(iω) can be computed from the energy level structure (Ej ) andtransition dipole moments of the given atom or molecule asα(iω) =2 ∑ (Ek − Ej )|⟨j|µ|k⟩|2.
3(Ek − Ej )2 + ω 2

(7.8)

j

Performing the sum over rotational and electronic states of OH and ND3 , we ﬁnd that thedominant contribution to the C6 coeﬃcient originates from their near-degenerate parity-doublet
splittings. Working in atomic units, we obtain C6 = 21, 380 a.u. with only a 0.01% contribution
from states other than the ground Λ and inversion doublets.
When an external electric ﬁeld is applied within the collision region to polarize both OH and
ND3 , we assume the interaction potential takes an anisotropic dipolar form given by [4]V (R) = ⟨µOH ⟩⟨µN D3 ⟩[1 − 3 cos2 Θ]
R3

(7.9)

where Θ is the angle between the molecules’ symmetry axes and ⟨µOH ⟩ = 1.0 D, ⟨µN D3 ⟩ = 0.77 D
are the expectation values of the OH and ND3 permanent electric dipoles, respectively.

162
7.3.3

Elastic Cross Section Estimates

The quickest way to obtain elastic collision cross sections for θmin = 0 is to apply the opticaltheorem to the scattering cross section f (θ). This gives the rather simple result [160]σ=4πImf (θ = 0) = 4πk

∫ ∞dbb[1 − cos 2∆(b)].

(7.10)

0Inserting the Van der Waals potential yields the cross section (in atomic units)(

σC6)2 ∫ ∞
3πM C6 51= 4π×dxx[1 − cos 5 ]
8k
x
0
(
)2
3πC6 5≃ 4π(0.602).
8v(7.11)(7.12)

Bohn and co-workers use the Eikonal approximation with the dipolar potential of Eq. 7.9 to arriveat the following compact form for the dipolar elastic cross section [4]:σdip =

8π ⟨µOH ⟩⟨µN D3 ⟩.
3v

(7.13)The relative velocity dependence of both the C6 and dipolar cross sections are shown in Fig. 7.9. At
the relative velocity of 100 m/s, the elastic cross section enhancement due to the dipolar interactionis ∼ 6.
Integrating the unpolarized cross section over the experimental collision energy range of
Fig. 7.6a, we obtain σ̄Eik = 0.50 × 10−12 cm2 . The velocity-averaged dipolar cross section isdip′ /σ̄ ) = 0.04
σ̄Eik
= 3.0 × 10−12 cm2 . Including the values of σ̄in = 0.57 × 10−12 cm2 and (σ̄elelobtained from the ab initio PESs, semiclassical theory predicts a factor of 1.2 enhancement in themeasured loss cross section when both species are polarized — in agreement with the measured
value of 1.4(3). It is possible that σ̄in will not change in the presence of the external E-ﬁeld since
the dipole potential contributes weakly at short-range.

163

10

1E-11

8
)

6

4

C

6

(

v

-2/5

)

50

2

0

1E-13

0

dipoleelastic

/

1E-12

C6

v

2

[cm ]

Dipolar (-1

100 150 200 250 300 350 400 450 500
Relative Velocity [m/s]

Figure 7.9: Plot of the Eikonal estimates for the elastic collision cross section between OH and
ND3 molecules. At the experimental relative velocity of 100 m/s, the electric ﬁeld is expected toincrease the elastic cross section by a factor of ∼ 6.

164

7.4

Future Directions

7.4.1

Quantifying State-to-State Inelastic Cross Sections

The collision measurement described in Section 7.2 is a ﬁrst step towards full experimental
characterization of the OH-ND3 elastic and inelastic cross sections within a variable electric ﬁeld.
We are most sensitive to the integrated total OH trap loss due to collisions with the external
ND3 beam. However, it may also be possible to directly detect product states from the inelastic
collisions. It will be diﬃcult to measure the inelastic collision products of ND3 since they will not
be trapped and, hence, will not accumulate over the > 100 ms measurement duration. Without theability to accumulate these scattered molecules, this state-to-state measurement will be eﬀectively
identical to a traditional crossed-beam scattering experiment. In this case, we may estimate thenumber of detected ions from the collision products, N↓ , residing in the lower-inversion state for asingle laser shot asN↓ = nT nB σγ→γ ′ Vdet vrel τdwell Def f S↓

(7.14)

where Vdet is the laser detection volume, Def f is the combined ionization/detection eﬃciency, S↓
is the transition strength of the relevant REMPI line, σγ→γ ′ is the inelastic state-to-state crosssection, τdwell is the dwell time of the scattered ND3 molecule within the detection volume, and nT(nB ) is the target (beam) molecular number density. Comparing with the measured signal fromthe guided upper-inversion state, N↑ = nB Def f Vdet S↑ , we haveN↓ /N↑ = nT vrel σγ→γ ′ τdwell

S↓
.
S↑

(7.15)

Inserting realistic parameters into Eq. 7.15, we expect N↓ /N↑ ∼ 10−8 for a state-to-state crosssection of ∼ 10−13 cm2 . This would constitute a rather heroic measurement at our current OH and
ND3 densities.
One possible alternative to an impulsive measurement of ND3 products is to look for accumulated inelastic products of OH. Certain angular momentum projections within the lower Λ-doublet
of OH (e.g. |J, Ω, MJ , p⟩ = |3/2, 3/2, +3/2, e⟩) will remain within the magnetic trap if not given

165
suﬃcient kinetic energy to overcome the conﬁning potential. Because of the symmetry of the excited 2 Σ+ state of OH used for LIF detection, these lower-Λ-doublet molecules are registered asloss when detecting along the Q-branch transitions. Nevertheless, ﬂuorescence detection along theP1 branch will permit direct measurement of these accumulated products. An experiment of thiskind would be the ﬁrst state-to-state inelastic scattering measurement using accumulated, trapped
cold molecules.

7.4.2

Orientation-Dependent OH-ND3 Collisions

As described in Section 7.2.4, zero-ﬁeld ab initio theory predicts a strong propensity for bothOH and ND3 to ﬂip their parity-doublet state in an inelastic collision. A state-to-state detection
experiment like the one described in the previous section would help quantify this inelastic crosssection. However, one can also envision changing the parity of the trapped OH sample (f → e)
before colliding with the guided ND3 molecules. This would change the energetics and overall
symmetry of the collision complex. In addition, under an applied electric ﬁeld the OH and ND3dipoles would be anti-aligned with respect to the local electric ﬁeld vector. To the degree that
we are sensitive to collisions at diﬀerent relative OH-ND3 angles, this may yield diﬀerent dipolar
collision behavior from what is observed under the current conditions [18].
To accomplish the parity-state transfer of trapped OH molecules (|3/2, 3/2, +3/2, f ⟩ →|3/2, 3/2, +3/2, e⟩), we will implement adiabatic rapid passage (ARP) using microwave ﬁelds coupled directly to the trap magnets. The ARP technique of sweeping the microwave frequency acrossthe e → f transition will allow > 90% transfer eﬃciency of the trapped molecules despite thediﬀerent Zeeman shifts at diﬀerent positions within the permanent magnet trap [162]. Having
previously performed direct microwave spectroscopy to determine the diﬀerential Landé g-factor of
the OH Λ-doublet transition (∆g = 1.265(7) × 10−3 ) [24], we must sweep the microwave frequency∼ ±20 MHz relative to the Zeeman-shifted transition center frequency to achieve eﬃcient population transfer∗ . Production of these trapped |3/2, 3/2, +3/2, e⟩ molecules will also permit study of∗

The Monte Carlo simulations for the in-trap OH microwave transfer were performed by Mark Yeo.

166
non-adiabatic transitions in combined electric and magnetic ﬁelds as described in Chapter 2.

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Appendix A
List of 250 ND3 REMPI Transitions

For reference, a list of 250 REMPI transition frequencies and relative line strengths for ND3
molecules is provided in the table below. The line strengths include only the angular momenta of
the upper and lower states and do not account for nuclear or J degeneracy factors which ultimately
must be included to simulate a thermal spectrum. To facilitate direct comparison with experimental
spectra, the transition frequencies are given as single-photon energies in units of cm−1 . Only theground state (X̃ 1 A′1 (v2 = 0)) angular momentum quantum numbers J ′′ and K ′′ are given since
K ′ levels in the intermediate B̃ 1 E ′′ (v2 = 5) state are mixed due to l-type doubling. Calculation ofthese transition frequencies and line strengths for ND3 is detailed in Chapter 2.

J ′′

K ′′

Transition Frequency [cm−1 ]Strength

State Energy [cm−1 ]1

131496.612

0.9

8.32021

131505.576

1.44058.32021

131497.52

1.05958.32021

131519.023

0.486078.32021

131510.944

2.11398.32021031492.55

1.8

10.33731031501.194

1

10.33731031515.341

3.2

10.33732231490.7

1

22.8376

175
2231499.662

2.142922.83762231513.1

1.426322.83762231497.003

2.573722.83762231531.006

0.2863922.83762231514.911

2.570722.83762

131481.848

1

28.88462

131486.33

0.5

28.88462

131495.294

0.40045

28.88462

131487.238

2.099628.88462

131508.741

1.994228.88462

131500.661

0.0058184

28.88462

131526.671

0.82356

28.88462

131518.517

3.176428.88462

031482.113

30.90032031491.39

0.7142930.90032031504.106230.90032031523.265

4.285730.90033

331488.838

2.5

43.61083

331502.281

2.916743.61083331520.19

1.267943.61083

331496.06

4.282143.61083

331542.555

0.1809743.61083

331518.455

2.852443.61083231475.279

2

53.67873231497.679

1.836853.67873231481.583

2.829953.67873231515.586

2.374253.6787

176
3231499.49

0.42579

53.67873231537.952

0.59133

53.67873231521.823

3.942

53.67873131470.912

1.6

59.71963

131479.877

1.059559.71963

131471.82

1.440559.71963

131493.324

0.30658

59.71963

131485.244

1.676859.71963

131511.253

2.626859.71963

131503.099

0.1232459.71963131533.665

1.033659.71963

131525.35

4.133

59.71963031466.852

1.2

61.73323031475.496

4

61.73323031489.643

0.46667

61.73323031506.26

3

61.73323031530.571

5.333361.73324431486.542

4.2

70.64874431504.468

3.436470.64874431526.851

1.112470.64874431494.687

6.087670.64874431553.673

0.12195

70.64874431521.578

3.041770.64874

331468.281

2.5

84.72424

331481.724

0.35

84.72424331499.634

3.181484.72424331475.503

3.323184.7242

177
4

331521.998

2.379884.72424331497.898

1.320284.72424331548.802

0.4203184.72424331524.704

4.525184.72424231463.692

2.857194.77834231477.129

0.27261

94.77834231461.033

1.327494.77834231495.036

1.5862

94.77834231478.941

2.902194.77834231517.402

3.186994.77834231501.273

0.013098

94.77834231544.218

0.85275

94.77834231527.996

5.001894.77834

131459.331

2.0996100.81084

131451.274

0.40045

100.81084

131472.778

1.7093100.81084

131464.698

2.2407100.81084

131490.708

0.2783100.81084

131482.554

1.3172100.81084

131513.119

3.3564100.81084

131504.805

0.34365

100.81084

131540.01

1.1235100.81084

131531.409

5.131

100.81084031455.429

2.2857102.82174031468.146

5

102.82174031487.304

0.35065102.82174031507.643

4

102.8217

178
4

0

31537.244

6.3636102.82175531483.808

6

103.9645531506.224

3.8077103.9645531533.083

0.98001

103.9645531492.883

7.9486103.9645531564.364

0.086284

103.9645531524.28

3.1775103.9645431460.852

2.8

122.0295

5

431478.778

1.2

122.02955431501.161

4.2777122.0295

5

431468.997

3.6761122.02955

431527.982

2.2625122.0295

5

431495.887

2.5089122.02955431559.223

0.30645

122.02955

431527.172

4.9683122.0295

5

331456.046

3.7333136.0808

5331473.955

0.00036435

136.0808

5331449.825

1.1996136.0808

5

331496.32

3.0564136.0808

5331472.22

3.9103136.0808

5331523.124

3.3543136.0808

5331499.026

0.31709

136.0808

5331554.351

0.65768

136.08085331530.206

5.7709136.08086631480.631

7.8571143.5732

6631507.545

4.0857143.5732

6631538.887

0.87012143.5732

179
6

6

31490.643

9.8442143.5732

6631574.632

0.063348

143.5732

6631526.56

3.2795143.5732

5231451.46

3.4643146.1176

5231435.363

0.26908

146.1176

5231469.366

0.80643

146.1176

5231453.271

2.3936146.1176

5231491.733

1.4247146.1176

5231475.603

2.5804146.1176

5231518.548

3.9762

146.1176

5231502.326

0.052408

146.1176

5231549.798

1.0482

146.1176

5231533.398

5.9847146.1176

5

131447.114

2.5039152.1398

5

131439.034

0.96276

152.1398

5

131465.043

2.478

152.1398

5

131456.889

2.722

152.1398

5

131487.455

0.27463

152.1398

5

131479.14

1.051

152.1398

5131514.345

4.1903152.1398

5

131505.745

0.50965

152.1398

5131545.706

1.1101152.1398

5131536.655

6.1976152.1398

5031443.436

3.3333154.1472

5031460.053

6

154.1472

5031484.364

0.28205

154.1472

5031508.244

5

154.1472

180
5

0

31543.265

7.3846154.1472

6531452.987

3

165.6068

6531475.403

2.3571165.6068

6531502.262

5.1594165.6068

6

531462.061

3.9406

165.60686531533.543

2.1103165.6068

6531493.459

3.8897165.6068

6531569.22

0.22937

165.6068

6531529.23

5.3135

165.6068

6431447.975

4.3636

183.6349

6431470.358

0.19864183.6349

6

431438.194

1.0871

183.6349

6

431497.18

4.4269183.6349

6

431465.084

4.7238183.6349

6

431528.42

3.3298183.6349

6

431496.369

0.95594183.6349

6431564.057

0.50759183.6349

6431532.008

6.4067183.6349

7731477.004

9.75

189.4968

7731508.427

4.3015189.49687731544.262

0.7788189.4968

7731487.961

11.7629

189.49687731584.477

0.047859

189.4968

7731528.415

3.359

189.4968

6331443.167

4.5503

197.6572

6

331419.036

0.19515

197.6572

6

331465.531

0.21815

197.6572

181
6

3

31441.431

2.3676

197.6572

6

331492.336

2.8879197.6572

6

331468.238

3.781

197.6572

6

331523.562

4.26

197.6572

6331499.418

0.025678

197.6572

6

331559.19

0.86313

197.6572

6331534.93

6.8512

197.6572

6231438.588

3.9467207.67346231422.493

0.70781207.6734

6231460.955

1.4693207.6734

6231444.825

3.1878207.67346231487.77

1.3239207.67346231471.548

2.2216207.6734

6231519.02

4.7903207.6734

6231502.621

0.20967207.6734

6231554.688

1.1699207.6734

6231537.996

6.9729207.6734

6131434.272

2.7933213.6831

6131426.118

1.6612213.6831

6131456.683

3.3866

213.6831

6131448.369

2.9706

213.6831

6131483.573

0.2825

213.6831

6131474.973

0.84867

213.6831

6131514.934

5.1221

213.6831

6131505.884

0.59219213.6831

6131550.75

1.0216

213.6831

6131541.053

7.3213213.6831

182
7

6

31444.682

3.1429215.4719

7

631471.596

3.7143215.4719

7

631502.938

5.875

215.4719

7

631454.693

4.146

215.4719

7631538.683

1.9552215.4719

7631490.611

5.402

215.4719

7631578.799

0.17562215.47197631530.881

5.5891215.4719

6031430.872

4.3636215.6864

6031451.2117

215.68646031480.811

0.23636215.6864

6031508.0466

215.68646031548.613

8.4

215.68647

531439.48

4.8352237.4521

7

531466.339

0.75781237.4521

7531426.139

0.99219237.4521

7531497.62

5.6357237.4521

7531457.536

5.3852237.4521

7531533.297

3.217

237.4521

7531493.308

1.8485237.4521

7531573.342

0.39627

237.4521

7531533.41

6.9322237.4521

8831472.918

11.6667241.7592

8831508.865

4.4737241.7592

8831549.204

0.70227241.7592

8831484.83

13.6977

241.7592

8831593.899

0.036986

241.7592

183
8

8

31529.84

3.4227241.7592

7

431434.457

5.4112

255.4368

7

431402.293

0.14921255.4368

7

431461.279

0.0023134

255.4368

7

431429.183

2.2834255.4368

7

431492.519

4.3728255.4368

7431460.468

4.8483255.4368

7431528.156

4.3284255.4368

7

431496.107

0.29062255.4368

7431568.161

0.69688

255.4368

7431536.053

7.6168255.4368

7331429.647

5.1647269.4255

7

331405.547

0.54958

269.42557

331456.452

0.6928269.4255

7

331432.353

3.3429269.42557331487.678

2.7431

269.42557

331463.534

3.4492269.42557

331523.306

5.1444269.42557331499.046

0.016275

269.42557

331563.31

1.0213269.42557

331538.841

7.8757269.4255

8731435.93

3.25

271.6448

8731467.353

5.2083271.64488731503.188

6.4641271.6448

8731446.887

4.31

271.64488731543.403

1.8086271.6448

8731487.341

7.0081271.6448

184
8

7

31587.963

0.13703271.64488731532.124

5.8138271.6448

7231425.083

4.3277279.41777231408.953

1.2767279.4177

7231451.898

2.2404279.4177

7231435.676

3.7596279.4177

7231483.148

1.2626279.4177

7

2

31466.748

1.8935279.4177

7231518.816

5.6524279.4177

7231502.124

0.37139279.4177

7231558.879

1.2189279.4177

7231541.751

7.9968279.41777131420.818

2.9488285.41317131412.504

2.5017285.41317

131447.708

4.4402285.41317

131439.108

3.0241285.41317131479.069

0.29515285.41317

131470.019

0.69022285.41317

131514.885

6.1319285.41317

131505.188

0.6002285.41317

131555.129

0.89057

285.41317

131544.569

8.4771285.41317031417.732

5.3846287.41167031441.612

8

287.41167031476.632

0.20362

287.41167031507.03

7

287.41167031553.267

9.4118287.4116

185
8

6

31430.558

5.2

297.5479

8631461.9

1.5873297.5479

8631413.655

0.91265297.5479

8631497.644

6.6854297.5479

8631449.573

5.9304297.54798631537.761

3.0667297.5479

8631489.843

2.9333297.54798631582.215

0.31337297.54798631534.417

7.3708297.5479

9931468.364

13.6

300.3891

9931508.851

4.6143300.3891

9931553.708

0.63749300.3891

9931481.241

15.6443300.3891

9931602.896

0.029109

300.3891

9931530.829

3.4748300.3891

8531425.332

6.1028319.4676

8531385.131

0.11864319.4676

8531456.612

0.12163319.4676

8531416.528

2.1819319.4676

8531492.29

5.7704319.4676

8531452.3

5.7792319.4676

8531532.335

4.2747319.4676

8531492.402

0.80859319.4676

8531576.715

0.56267

319.4676

8531536.784

8.2794319.4676

9831426.723

3.3333334.1494

9831462.67

6.8

334.1494

186
9

8

31503.009

6.9561

334.14949831438.635

4.4439334.1494

9831547.704

1.6742334.1494

9831483.645

8.6834334.1494

9831596.715

0.10863334.1494

9831532.958

6.0005334.14948431420.296

6.1863337.4038

431388.2

0.44223337.4038

431451.536

0.20525

337.4038431419.485

3.3662337.4038431487.173

4.2667337.4038431455.124

4.6211

337.4038

431527.177

5.2884337.4038431495.07

0.044956

337.4038431571.52

0.85714337.4038431539.273

8.7218337.4038331415.488

5.6447351.3534

8331391.39

1.0339351.35348331446.714

1.3262351.35348331422.57

4.1202351.35348331482.342

2.6317351.35348331458.082

3.0722351.3534

8331522.346

6.0375351.35348331497.877

0.11254351.3534

8331566.697

1.1255351.3534

8331541.901

8.8955351.35348231410.948

4.6067361.3184

187
8

2

31394.726

1.9647361.31848231442.198

3.1207361.31848231425.798

4.1412361.31848231477.865

1.2269361.31848231461.173

1.6065361.31848231517.929

6.5715361.31848231500.8

0.49519361.31848231562.36

1.2033361.31848231544.624

9.0634361.31849731421.205

5.4902363.9417

9731457.039

2.6211363.94179731400.738

0.8456363.9417

9731497.254

7.5952363.94179731441.193

6.386

363.94179731541.815

2.9038363.9417

9731485.976

4.1659363.94179731590.682

0.25082363.94179731535.034

7.7414363.9417101031463.33

15.5455

365.4195101031508.376

4.7312365.4195101031557.769

0.58208

365.4195101031477.182

17.5997

365.4195101031611.464

0.023255

365.4195101031531.374

3.5182365.41958

131406.766

2.9634367.2975

8

131398.166

3.4866367.2975

8

131438.127

5.6235367.2975

188
8

1

31429.076

2.9181367.2975

8131473.943

0.30811367.2975

8131464.246

0.56426367.2975

8131514.187

7.1933367.2975

8131503.626

0.55668

367.29758

131558.824

0.7459367.29758131547.172

9.6401367.2975

8031404.009

6.4

369.2906

8031431.244

9369.29068031471.811

0.17895

369.2906

8031505.177

8

369.2906

8031557.201

10.4211369.2906

9631415.792

6.6675389.7644

9631367.547

0.097201389.76449631451.536

0.52094

389.7644

9631403.465

2.0791389.76449631491.652

7.047

389.76449631443.735

6.5883389.7644

9631536.107

4.1547389.76449631488.309

1.5362389.7644

9631584.862

0.45674

389.76449631537.133

8.8524389.764410931417.051

3.4

403.014110931457.538

8.4636403.014110931502.396

7.3722403.014110931429.928

4.5554403.014110931551.583

1.5525403.0141

189
10

9

31479.517

10.4111403.014110931605.057

0.087252

403.014110931533.379

6.1578403.01419531410.538

7.0461411.61649531370.454

0.3657411.61649531446.215

0.0071994

411.61649531406.225

3.3261411.61649531486.26

5.7691411.61649531446.327

5.7001411.61649531530.641

5.2938411.61649531490.71

0.29707411.61649531579.32

0.71208

411.61649531539.317

9.4827411.61649431405.49

6.7857429.49679431373.438

0.86131429.49679431441.126

0.64629429.49679431409.077

4.287

429.49679431481.131

4.159

429.4967

9431449.023

4.2606429.4967

9431525.473

6.2351429.49679431493.226

0.0012281

429.49679431574.12

0.97857

429.49679431541.627

9.7851429.496710831411.416

5.7263436.6569

10831451.755

3.8114436.6569

10831387.381

0.78859436.6569

10831496.45

8.3865436.6569

190
10

8

31432.392

6.7718436.6569

10831545.462

2.7407436.6569

10831481.704

5.5139436.6569

10831598.747

0.20291436.6569

10831535.261

8.058

436.6569

111131457.802

17.5

436.8882111131507.43

4.83

436.8882111131561.378

0.53423436.8882

111131472.643

19.5619436.8882

111131619.598

0.01881436.8882

111131531.465

3.555

436.88829331400.689

6.013

443.4044

9331376.545

1.634

443.40449331436.316

2.0846443.40449331412.057

4.7154443.40449331476.32

2.5501443.40449331451.852

2.7029443.40449331520.672

6.957

443.40449331495.876

0.23391443.40449331569.339

1.1749443.40449331544.07

9.9342443.4044

9231396.188

4.7781453.3399231379.788

2.7709453.3399231431.855

4.1125453.3399231415.163

4.3541453.3399231471.918

1.2069453.3399231454.79

1.3584453.339

191
9231516.35

7.5469453.3399231498.613

0.56829453.3399231565.116

1.1364453.3399231546.575

10.1675

453.3399131392.126

2.8491459.2999

9131383.075

4.6019459.2999

9131427.942

6.9022459.2999

9131418.245

2.6978459.2999

9131468.186

0.31847459.2999

9131457.625

0.46388459.2999

9131512.823

8.2807459.2999

9131501.171

0.48598459.2999

9131561.807

0.60727459.2999

9131548.835

10.7927459.2999

9031389.695

7.4118461.28699031420.093

10461.2869

9031466.33

0.15966

461.2869

9031502.464

9461.28699031560.388

11.4286

461.286910

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