Cold, Controlled, Ion-Molecule Reactions
byJames Greenberg
B.S., University of Arizona, 2014
M.S., University of Colorado Boulder, 2017

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
2020

This thesis entitled:
Cold, Controlled, Ion-Molecule Reactions
written by James Greenberg
has been approved for the Department of Physics

Prof. Heather Lewandowski

Prof. Eric Cornell

Date

The final copy of this thesis has been examined by the signatories, and we find that both thecontent and the form meet acceptable presentation standards of scholarly work in the above
mentioned discipline.

iii
Greenberg, James (Ph.D., Physics)
Cold, Controlled, Ion-Molecule Reactions
Thesis directed by Prof. Heather Lewandowski

Coulomb crystals in linear Paul ion traps (LIT) have proven to be a fruitful environment
to study gas-phase ion chemistry and spectroscopy of molecular ions. Recently, the capabilities ofsuch apparatuses have been extended by (radially) coupling them to a time-of-flight mass spectrometer (TOFMS). The TOFMS enables simultaneous detection of all constituent masses in theion trap. Thus, one can follow reactions both by the depletion of reactants, as well as growth of
product channels, allowing for a more complete description of reaction kinetics. We have designed,
built, and characterized a combined LIT-TOFMS apparatus for the purpose of studying cationmolecule reactions. This thesis provides an overview of ion trapping theory and basic reaction
kinetics/dynamics. A detailed description of the apparatus and four ion-molecule reactions are also
discussed. The first two ion-molecule reactions covered demonstrate quantum control over lasercooled Ca+ in the reactions Ca+ + NO and Ca+ + O2 . The third ion-molecule reaction elucidates
the isomer specific reaction mechanisms observed in the reaction C2 H2 + + C3 H4 . The last reaction
covered is CCl+ + C2 H2 , which reveals new pathways to ionic products important in chemistry of
the interstellar medium. The thesis concludes with future prospects to extend the capabilities of
the apparatus in pursuit of colder and more controlled ion-molecule reaction studies.

Dedication

To future graduate students who may read this. Don’t panic.

v

Acknowledgements

It may take a village to raise a child, but it takes an entire institute, and then some, to raise
an experiment. None of this work would not have been possible without the resources, expertise,dedication, and amazing people at JILA. First and foremost, thanks to my advisor, Heather, forbeing a great scientist, leader, and friend. Thanks to the Lewandowski group: Philipp, Katherine,
Ben, Olivia, Kyle, Jason, John, Yomay, Cam, and Maya. Thanks to the amazing JILA machine
shop who built the ion trap and many other parts: Hans, Kyle, Kim, Todd, and Calvin. Thanks to
the JILA electronics shop who helped make the LIT-TOFMS electronics a reality: Terry and Karl.
Thanks to the JILA administrative staff that keep the institute running, especially Agnieszka andKrista, for making my life so much easier. I also want to acknowledge our theory collaborators who
gave so much of their effort to the research as well: John, James, and Lam. Thanks to Mathias
and Barney for their advice and ideas. Thanks to MIT Tim for putting up with me in the lab foran entire week. Also thanks to the visiting professors who I got to know briefly: Bas, Tim, andGary. An enormous number of my friends and family also were critical to my success in grad school.
Thanks and much love to Greg, Bonnie, Mom, and Dad; my aunts, Mary, Kathy, and Susan alongwith my uncle George. I couldn’t have even made it to Boulder without my best friend, Christian.
As if the thesis wasn’t enough, Kevin and I conquered some mountains. Summer forever, Amy!
Thanks to my girlfriend Lotto for being so rad and supportive. So grateful for my fuzzy little
Mugen :3 And cheers to all my other friends I’ve made in Boulder: Doug, Nathan, Quynh, Chris,
Brendan, Kyle, Adam, Jeff, and all the members of STMP that I definitely don’t have room to
name. You all mean so much to me.

viContents

Chapter
1 Introduction1

1.1

Motivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Cold ion-molecule collision experiments . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.3

Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62 Ion trapping and ion-molecule reaction theory
2.1

7

Ion trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2.1.1

Pseudo-potential approximation. . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.2

Trapping stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.3

Ion kinetic energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2

Laser cooling of Ca+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3

Coulomb crystals and sympathetic cooling . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4

Time-of-flight mass spectrometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.5

Bimolecular reaction kinetics and dynamics . . . . . . . . . . . . . . . . . . . . . . . 21
2.5.1

Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.5.2

Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 Experimental Apparatus

26

3.1

Vacuum chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2

Linear ion trap – time-of-flight mass spectrometer (LIT-TOFMS) . . . . . . . . . . . 27

vii
3.3

Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.3.1

LC tank circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.3.2

Quench circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.3.3

HV switches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.4

TOFMS performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.5

Ion number calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.6

Calcium state population calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.7

Gas Handling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.7.1

Pulsed leak valve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.7.2

PZT valve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4 Quantum controlled reaction with Ca+
4.1

4.2

49

Ca+ + NO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.1.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.1.2

Ca+ + NO reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.1.3

Ion trap apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.1.4

Reaction Rate Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.1.5

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.1.6

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Ca+ + O2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.2.2

Ca+ + O2 Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.2.3

Ion Trap Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.2.4

Reaction Rate Constant Measurements

4.2.5

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.2.6

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.2.7

Ca+ + H2 O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

. . . . . . . . . . . . . . . . . . . . . 66

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4.2.8

Addendum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5 Isomer specific chemistry and isotopologue reactions of C2 H2 + + C3 H4

76

5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.2

Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.3

Propyne – HC3 H3

5.4

5.5

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.3.1

Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.3.2

Computational results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.3.3

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

Allene – H2 C3 H2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.4.1

Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.4.2

Computational Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.4.3

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6 Translationally cold trapped CCl+ reactions with acetylene (C2 H2 )

103

6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.2

Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.3

Computational Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6.4

Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.4.1

CCl+ + C2 H2 reaction measurements . . . . . . . . . . . . . . . . . . . . . . . 108

6.4.2

Modeling the CCl+ + C2 H2 reaction . . . . . . . . . . . . . . . . . . . . . . . 111

6.5

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.6

Supplementary information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

7 Conclusion

125

7.1

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

7.2

Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

ix
Bibliography

128

Appendix
A Electronics circuit diagrams

144

A.1 Ion trap driver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
A.2 HV switches and quench boards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
A.3 DC shim box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
A.4 RF secular excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
B Labview control and automation

151

B.1 Automation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
B.2 Labview control code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
B.2.1 Producer consumer architecture . . . . . . . . . . . . . . . . . . . . . . . . . . 153
B.2.2 Code hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
B.2.3 How to add a step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
B.2.4 How to add a module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
C MATLAB analysis code

159

C.1 OBE global fit and model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
C.2 Reaction model fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
C.3 TOFMS integrator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
C.4 Image fitting code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

xTablesTable
1.1

List of many cation-molecule reactions studied in linear Paul ion traps. 300K should
be understood as room temperature. Reactions studied in this thesis are the first
four entries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1

5

Typical trapping parameters and trap geometry specifics. Also included are some
calculated values (not measured) for 40 Ca+ using the typical trap parameters andthe equations found in Ch. 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2

Variables used to calibrate OBE fit parameters with typical trap parameters in Tab.
3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.1

Table of results for the two independent determinations of reaction rate constants
and branching ratios. The listed uncertainties represent the statistical uncertainty. . 59

4.2

Measured reaction rate constants for pairs of Ca+ excited state populations shownin Fig. 4.9. The fractional populations in the 2 P1/2 -state (fP ) and 2 D3/2 -state (fD )are shown with the corresponding measured effective rate constant (kef f ). The data
point numbers correspond to measurements shown in Fig. 4.9. . . . . . . . . . . . . 68

4.3

The resulting rate constants kP (kD ) for the three quantum states of Ca+ . Uncertainties are from the 90% confidence interval for the fit parameters, kP (kD ) . . . . . 68

4.4

The resulting reaction rates for the three quantum states of Ca+ . Uncertainties arefrom the 90% confidence interval for the fit parameters. . . . . . . . . . . . . . . . . 73

xi
5.1

Reaction rate constants for all measured isotopologue reactions. All rate constants
determined by fit of the loss of charged reactants. Uncertainties are derived fromthe 90% confidence interval of the fit and do not account for possible inaccuracies in
the neutral concentration measurement. . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.2

Branching ratios for the reaction C2 H2 + + HC3 H3 −−→ C5 H5 + + H and its isotopologues that led to C5 Hx Dy + (x + y = 5). Numbers are the branching ratios (in %)for each mass channel as determined by the fit of the model. Uncertainties are the
90% confidence interval of the branching ratio fit. Blank spaces indicate the mass
channel was either not populated or not statistically different from zero at the 10%
significance level. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.3

Branching ratios for the reaction C2 H2 + + HC3 H3 −−→ C3 H3 + + C2 H3 and its
isotopologues that led to C3 Hx Dy + (x + y = 3). Numbers are the branching ratios(in %) for each mass channel as determined by the fit of the model. Uncertainties
are the 90% confidence interval of the branching ratio fit. Blank spaces indicate themass channel was either not populated or not statistically different from zero at the
10% significance level. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.4

Estimated branching ratios for the reaction C3 H4 + + HC3 H3 −−→ products andits selected isotopologues. Numbers are the branching ratios (in %) for each mass
channel, as determined by the fit of the truncated model. Uncertainties are the 90%
confidence interval of the branching ratio fit. Blank entries indicate the mass channel
was not statistically different from zero at the 10% significance level. Detected massto-charge ratios that could not be assigned a unique chemical formula have been
omitted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.5Reaction rate constants for all measured isotopologue reactions. All rate constants
determined by fit of the loss of charged reactants. Uncertainties are derived fromthe 90% confidence interval of the fit and do not account for possible inaccuracies in
the neutral concentration measurement. . . . . . . . . . . . . . . . . . . . . . . . . . 96

xii
5.6

Branching ratios for the reaction C2 H2 + + H2 C3 H3 −−→ C3 H3 + + C2 H3 and its
isotopologues. Numbers are the branching ratios (in%) for each mass channel as
determined by the fit of the model. Uncertainties are the 90% confidence interval of
the branching ratio fit. Blank spaces indicate the mass channel was not statisticallydifferent from zero at the 10% significance level. . . . . . . . . . . . . . . . . . . . . . 96

5.7

Selected isotopologue data for C6 H5 + and C6 H7 + products from allene. Numbers
are the branching ratios (%) for each mass channel as determined by the fit of thetruncated model. Uncertainties are the 90% confidence interval of the branchingratio fit. Blank spaces indicate the mass channel was not statistically different fromzero at the 10% significance level. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.1

Results from the fits to the pseudo-first order reaction rate models for the formationof intermediate and secondary products from the reaction of CCl+ + C2 H2 . Included
are the rate constants for the single pressure measurement of C2 H2 ≈ 5(1) × 106 cm3 . 110

6.2

Results from the fits to the pseudo-first order reaction rate models for the formationof intermediate and secondary products from the reaction of CCl+ with C2 H2 (fromTable I in main text) and C2 D2 . Included are the rate constants for the single
pressure measurement of C2 H2 ≈ 5(1) × 106 cm3 and C2 D2 ≈ 7(1) × 106 cm3 . For
the C2 D2 case, rates extracted from the data are less certain and the majority of thereported error is due to the calcium ion number uncertainties, as the C3 D2 + productchannel overlaps with the Ca+ channel . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.3

Previously measured and calculated constants of CCl+ for comparison with current
computational results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

xiii
6.4

Electronic energies and vibrational zero-point corrections from calculations at the
CCSD/aug-cc-pVTZ level of theory with accompanying CCSD(T)/CBS single point
energies, all in Hartrees. Stationary point numbers correspond to those on the
CCl+ + C2 H2 potential energy surface in Figure 6.3. Structures 6 and 10-12 areenergetically similar isomers where the HCl moiety rearranges along the l -C3 H+
backbone. The other linear isomer of C3 H2 + is also included for reference. All
structures are illustrated in Figure 6.8. . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6.5

CCSD/aug-cc-pVTZ calculated geometries for stationary points on potential energy
surface represented in Figure 6.5. Structures for each point are depicted in Figure
6.8 and electronic and zero point energies are provided in Table 6.4 . . . . . . . . . . 122

xiv

FiguresFigure2.1

Schematic representation (not to scale) of the ion trap geometry. The coordinate
system is drawn for reference and orientation. To the left is an isometric view of thetrap and to the right is an end-on view. . . . . . . . . . . . . . . . . . . . . . . . . .

2.2

8Mathieu Stability diagram for experimentally accessible values of ax,y and qx,y . Stable trapping conditions are represented by the shaded region. The intercept on the
qx,y -axis is ∼ 0.908.

2.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Energy level diagram for 40Ca+ . Transition wavelengths are measured in air, thelifetimes are total excited state lifetimes, and gj are the Landé g-factors.

2.4. . . . . . 15

False-color fluorescence microscope images of two separate coulomb crystals withdifferent values of the plasma parameter. The top image shows the charged fluid
phase and the bottom image shows the crystalline phase.

2.5. . . . . . . . . . . . . . . 17

False-color fluorescence microscope images of a bi-component coulomb crystal containing Ca+ and C2 H2 + ions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.1

Exploded schematic of the ion trap and detection optics. The ion trap has fourrods, which are broken into three segments each. RF potentials are applied to all 12
segments. In addition, small DC potentials are applied to form a trapping field along
the longitudinal axis. Directly above the ion trap sits a 10x microscope objective
to image the fluorescence from the Ca ions onto a EMCCD camera. The drift tube
(not shown) of the TOFMS is located directly below the ion trap.

. . . . . . . . . . 29

xv
3.2

Laser breadboard layouts for diode lasers used in the experiment. . . . . . . . . . . . 30

3.3

Optical beampaths of ionization lasers and cooling lasers used in this thesis. Two
flipper mirrors are used in conjunction with the shutters to send one of the two
ionization lasers into the chamber at a time or the radial cooling beam for the
calcium.

3.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Left: Schematic representation of the TOFMS. Shown in red and black are therepeller and extractor configuration of the trap rods for radial ejection of trappedions into the TOFMS. Right: The corresponding repeller and extractor voltages asa function of time, required to go from trapping to TOFMS. Vertical dotted lines
represent the timings of the quench and HV pulse control triggers. . . . . . . . . . . 33

3.5

A schematic diagram of the RF drive electronics. The diagram shows the threedifferent parts (marked by colored boxes) required for ion trapping and extraction
into the TOFMS (only single side of bifilar winding is shown). . . . . . . . . . . . . . 34

3.6

A single-shot TOFMS mass spectrum (right) of trapped, laser-cooled 40 Ca+ and cotrapped 39 K+ , 44 Ca+ , 85,87 Rb+ and 133 Cs+ . This Coulomb crystal (left) contained
986 ions with 594 of them being 40 Ca+ . Apart from these ions, no further mass
peaks can be identified. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.7

Dependence of mass resolution on trapped ion numbers. Pure 40 Ca+ Coulomb crystals of different sizes are trapped and detected in the TOFMS. We observe a decreaseof mass resolution with increasing ion number. . . . . . . . . . . . . . . . . . . . . . 39

3.8

Calibration of the MCP signal to ion number. To calibrate the MCP signal, we startby loading small crystals into our trap and counting the number of ions on the image.
Then, by extracting those ions into the TOFMS and integrating the resulting MCP
signal, we determine how the MCP signal relates to the number of ions in the trap.
Using this calibration, we can determine the absolute number of ions in the trap for
subsequent experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

xvi
3.9

Determination of the ion density in our ion trap. We correlate the measured ion
number from the TOFMS to the fitted Coulomb crystal volume taken from therecorded CCD images. From a fit to these data, we can determine the ion density in
our trap. The inset shows a Coulomb crystal with an ellipsoidal fit. . . . . . . . . . . 42

3.10 Five calibration data sets to constrain the parameters of the OBE. These five plots
are fit simultaneously (red lines) to determine the p-state population. The sixth plotis the global scaling of intensity to p-state population. NOTE: the detunings plotted
are defined opposite from the majority of the literature, i.e.: ∆model = −1 × ∆. Thisis a holdover from the base code taken from [1]. . . . . . . . . . . . . . . . . . . . . . 44
3.11 Schematic diagram of pulsed leak valve setup. The high precision all-metal leak valve
is backed by a solenoid actuated 3-way valve. This arrangement allows pulsed gas
admission into the UHV chamber. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.12 UHV chamber pressure reacting to opening and closing of the PLV. These tests were
performed with 10 PSIG of pure helium backing the valve. The plot on the right isa zoom in of the first pulse plotted to the left with markers to highlight the valve
opening time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.1

Comparison of reactant and product energies for the Ca+ + NO system. The exothermicity of the reaction depends on the quantum state of Ca+ . Excited states of Ca+
can be populated by excitation on the two cooling laser transitions at 397 nm and
866 nm. To achieve cooling, the 397 nm laser is red-detuned from resonance by anamount ∆. Only Ca+ in the P1/2 state is energetically allowed to react. The charge
exchange product channel barrier is overcome by the thermal energy of neutral NO
at 180 K.(Energies not to scale.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

xvii
4.2

NO+ and CaO+ measured ion numbers as a function of time. The ion numbers arenormalized by the initial number of Ca+ determined from the image captured at t =0. The error bars represent the standard error of the mean of measured ion numbers
from five experimental runs at each time point. The solid lines indicate fits to thedata using a pseudo first-order reaction rate model. Also shown are three false-color
fluorescence images of the Coulomb crystals taken immediately before ejecting theions into the TOFMS. The images were taken at t = 50 s, 200 s, and 320 s. The darkbands in the center and along the outer edges of the crystals indicate the presence ofions that are lighter and heavier than Ca+ respectively. These ions are the reaction
products. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.3

Measured reaction rates for the two product channels, NO+ and CaO+ , as a functionof Ca+ P1/2 state population. The NO concentration was 2.3(1)×108 cm−3 for all ofthese data. The initial number of Ca+ ions in the trap was around 600. The slopes ofthe fit lines are used to determine the 180 K reaction rate constants. The statisticaluncertainty in the P1/2 state population is smaller than the points. There is a anoffset for the reaction rate at zero excited state population. However, it corresponds
to only a 1 – 2% offset in the P1/2 state population, which is likely due to a small
systematic in determining the absolute state population from the florescence. . . . . 55

4.4

Measured reaction rates for product channels NO+ and CaO+ as a function of neutral
NO concentration in the chamber. The NO pressure in the chamber was varied
between 3.4×10−9 and 1×10−8 torr to change the NO concentration. For all of these
data, the P1/2 state population was 19.1(1.8)%. The slopes of the fit lines correspondto the 180 K reaction rate constants. . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.5

Generalized valence-bond (GVB) diagrams that illustrate the processes whereby 2 P
calcium atoms react with NO molecules to form CaO+ and nitrogen atom in their
ground states (top); and the charge exchange process that yields Ca atoms andground state NO+ (bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

xviii
4.6

Diagram of the Ca+ energy structure and product channel energy (energies are notto scale). The main laser cooling transition for Ca+ at 397 nm excites the ion to
the 2 P1/2 -state. The ion can decay to the ground 2 S1/2 - or the excited 2 D3/2 -state.
A second laser at 866 nm repumps population that accumulates in the dark 2 D3/2 state back into the cooling cycle. Here, ∆397 and ∆866 represent the detuning of thecooling lasers with respect to the two cooling transition frequencies. The reaction
with O2 is energetically possible in either of the two excited states, but not in the
ground state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.7

Predictions of the state distribution of the Ca+ ion from the 8-level OBE with afixed 866 nm detuning (+42 MHz). The state populations for the 2 S1/2 , 2 P1/2 , and2D

3/2 states are shown.

The vertical lines represent the parameters used for threemeasurements used to determine the rate constants presented in Section 4.2.4 (seeTable 4.2). To obtain the other state distributions shown in Table 4.2, we alsochanged the 886 nm laser detuning.
4.8. . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Example reaction measurement showing the growth of CaO+ product ions, normalized by the initial number of Ca+ ions (∼ 800). The fraction in the 2 P1/2 -state is 0.16
and 2 D3/2 -state is 0.22, with an O2 density of 8.9×107 cm−3 . The solid line is a fit
using Eqn. (4.6), where kef f is the only free parameter. The error bars correspondto the standard error of the mean of 7 measurements at each time point. . . . . . . . 67

4.9

Dependence of the effective rate constant, kef f , on the 2 P1/2 - and 2 D3/2 -state populations of Ca+ . The experimental measurements are shown as diamonds, where thecolor of the point represents the value of kef f . For all data shown, the O2 concentration was kept constant at 8.9×107 cm−3 . We perform a two-dimensional fit to
the data using Eqn. 4.7, which allows us to extract kP and kD . The contour lines
represent constant kef f values of the fit. . . . . . . . . . . . . . . . . . . . . . . . . . 69

xix
4.10 An energy-level diagram featuring electronic states of reactants, intermediates, andthe observed CaO+ + O(3 P) product channel. All energies calculated at the CCSD(T)level, apart from those levels involving excited states of molecular oxygen, that use
experimental excitation energies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.11 Dependence of the reaction rate on the 2 P1/2 - and 2 D3/2 -state populations of Ca+ forthe reaction Ca+ + H2 O. The experimental measurements are shown as diamonds,
where the color of the point represents the value of the rate. We perform a twodimensional fit to the data, which allows us to estimate the state dependent rates.
The contour lines represent constant reaction rate values of the fit. . . . . . . . . . . 74
4.12 Calibration data used to predict calcium p-state populations by coulomb crystal fluorescence. The red line is the output of the 3-level OBE model used in [2]. Retroactive
use of the 8-level OBE is not possible and thus the y-axis scaling of the p-state is
too large by as much as a factor of 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.1

Schematic render of the LIT-TOFMS apparatus. A cross-sectional view of the spherical octagon vacuum chamber reveals the ion trap at the center of our experiment.
The source of Ca+ is an effusive beam of neutral calcium, generated by an oven. The
source of molecular ions is a pulsed, supersonic gas jet of neutral acetylene seeded ina noble carrier gas. These neutral beams are intersected by pulsed ionization lasers
in the center of our trap to load atomic and molecular ions. Ca+ ions are addressedby the cooling lasers. To observe the laser induced fluorescence from Ca+ , a microscope objective (outside the UHV chamber) is implemented. A pulsed leak valve(also outside the UHV chamber) is used for the introduction of the neutral reactant.

5.2

79

Left: False-color fluorescence microscope image of a typical two-component Coulomb
crystal and its corresponding TOFMS trace (right). The TOFMS data were produced by a single trap loading, i.e., not averaged over multiple experimental cycles.
The mass spectrum shows ∼240 C2 H2 + ions were co-trapped with ∼600 Ca+ ions. . 80

xx
5.3

Primary products of the reaction C2 D2 + + DC3 D3 as a function of time. Each point
represents the mean and standard error of 12 measurements. Secondary reaction
products have been omitted for clarity. Ion numbers are normalized to the initial
number of acetylene ions in the trap (∼ 200 ions). . . . . . . . . . . . . . . . . . . . 82

5.4

Example detection of multiple orders of products from C2 H2 + + HC3 H3 . The productorder is apparent by the delay of onset in appearance and their mass is too large to
be a product of C2 H2 + + C3 H4 . C5 H5 + was first order, C6 H7 + second, and C7 H5 +was third. Ion numbers have been normalized to initial C2 H2 + number. Productchannels have been multiplied by 4 to enhance clarity of the differences in reaction
order. Note that not all measured product channels are shown in this plot. . . . . . 83

5.5Reaction model for propyne. Each arrow represents a bimolecular gas-phase reaction
between the ion and a neutral propyne. Vertically stacked ions are of the same reaction order. The exact relationship between higher order products was undetermined
and beyond scope of this work. This ambiguity is represented by the curly brackets.

5.6

84

Potential energy surface (PES) connecting the propyne and acetylene cation to experimentally observed primary products. The energy of each stationary point iscalculated using CCSD(T) in the scheme described in the main text. All energies
are relative to separated acetylene cation and propyne. Dashed lines between stationary points indicate reactions that occur via a barrierless process. Note that theinitial point of the PES is given for after the first charge exchange has taken place
already. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.7

Primary product of the reaction C2 D2 + + H2 C3 H2 as a function of time. Each point
represents the mean and standard error of 10 measurements. Secondary and higher
order products have been omitted for clarity. Ion numbers are normalized to theinitial acetylene in the trap 120 ions. . . . . . . . . . . . . . . . . . . . . . . . . . . 94

xxi
5.8

Example detection of multiple orders of products from C2 H2 + + H2 C3 H2 . Whichproduct order is apparent by delay of onset in appearance and their mass was toolarge to be a product of C2 H2 + + C3 H4 . C3 H3 + was first order, C6 H7 + second,and C7 H5 + was third. Ion numbers have been normalized to initial C2 H2 + number.
Second and third order product channels have been multiplied by 20 to enhance
the clarity of the differences in reaction order. Note that not all measured productchannels are shown in this plot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.9

Reaction model for allene. Each arrow represents a bimolecular gas-phase reaction
with the ion and a neutral allene. Vertically stacked ions are of the same reaction
order. The exact relationship between higher order products was undetermined andbeyond scope of this work. This ambiguity is represented by the curly brackets. . . . 95

5.10 Potential energy surface connecting the of allene and acetylene cation to experimentally observed primary products. The energy of each stationary point is calculatedusing the mHEAT+ method.[3] All energies are relative to separated acetylene cation
and allene.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.1

Ground-state structures of CCl+ and C2 H2 . . . . . . . . . . . . . . . . . . . . . . . . 104

6.2

Measured ion numbers of CCl+ (blue ×), l-C3 H+ (green ◦), c-C3 H2 + (black 2), andC5 Hx + (x = 2, 3, and higher; red ∗) as a function of time. Data are normalized by
the initial ion number of CCl+ (∼ 100). Each data point represents the mean andstandard error from four experimental runs per time point. The averaged data are
fit using a pseudo-first order reaction rate model (solid lines). . . . . . . . . . . . . . 109

6.3

The main features of the potential energy surface for CCl+ + C2 H2 calculated at the
CCSD(T)/CBS level of theory. The favored Cl loss product, dissociates without abarrier, forming c-C3 H2 + + Cl [8]. The HCl loss channel leading to l-C3 H+ + HCl[7] involves a few stationary points with lower energies than the dissociation limit of-0.1 eV. Here, the bare ‘+’ indicates infinite separation. . . . . . . . . . . . . . . . . . 112

xxii
6.4

Measured total ion numbers of the CCl+ + C2 H2 reaction as a function of time.
Each data point represents the mean and standard error from 4 experimental runs
per time point. The total ion data are fit using a 1st order polynomial giving a slopeof -0.04 with an uncertainty of 0.3 (the 95% confidence interval of the fit parameter)
indicating the number of ions are conserved over the reaction. . . . . . . . . . . . . . 115

6.5

Measured ion numbers of CCl+ (blue ×), C3 D+ (green ◦), C3 D2 + (black 2), andC5 Dx (x = 2, 3 and higher; red ∗) as a function of time for the reaction of CCL+ with
duetrated acetylene C2 H2 . The product masses shift as expected due to deuterium
substitution. Data are normalized by the initial ion numbers of CCl+ (∼ 100). Eachdata point represents the mean and standard error from 4 experimental runs pertime point. The average data are fit using a pseudo-first order reaction rate model(solid lines). C3 D2 + (m/z 40) overlaps with the Ca+ channel. To fit this channel,
the initial average Ca+ ion numbers are subtracted from the m/z 40 channel at eachtime step. There is some variability in the loaded numbers of Ca+ , making the ion
numbers and rates extracted from the 40 channel less certain. . . . . . . . . . . . . . 116

6.6

Measured total ion numbers of the CCl+ + C2 D2 reaction as a function of time.
Each data point represents the mean and standard error from 4 experimental runs
per time point. The raw total ion data are fit using a 1st order polynomial giving
a slope of -0.09 with an uncertainty of 0.2 (the 95% confidence interval of the fit
parameter) indicating the number of ions are conserved over the reaction. . . . . . . 117

6.7

Potential energy surface calculated at the CCSD/aug-cc-pVTZ level of theory for thereaction of CCl+ with C2 H2 . Energies (in eV) are from single point CCSD(T)/augcc-pVTZ calculations extrapolated to the complete basis set limit. Structures, energies, and geometries are given in Figure 6.8 and Tables 6.4 and 6.5, respectively. . . 121

xxiii
6.8

Structures of stationary points on the potential energy surface in Figure 6.7. Ineach case, energies are given relative to the CCl+ + C2 H2 limit [1]. Geometries werecomputed at the CCSD/aug-cc-pVTZ level. Structures 6 and 10-12 are the several
minima with different rearrangements of the HCl moiety along the C3 H+ backbone.
Also included is the l -C3 H2 + isomer energy. Geometries for each stationary point
are provided in Table 6.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

7.1

Schematic render of the partially constructed coupling between a TWRD and theion trap from this thesis. An in-vacuum molecular beam shutter separates the lineof site from the TWRD to the guiding hexapole. The guiding hexapole delivers themolecular beamline to the trap center. . . . . . . . . . . . . . . . . . . . . . . . . . . 127

A.1 Main RF driver for the ion trap.

Primary and secondary sides of the step-up

transformer are separated by the dashed line. The quench boards and high-voltage
switches are represented schematically as two and three-way switches, respectively.
Connections to the DC shim box are also labeled. . . . . . . . . . . . . . . . . . . . . 145
A.2 High voltage switches and quench circuit diagrams. The floating power supply is a
12VDC NIHM battery pack that rests on a glass plate to allow it to float up to 2kV. 147
A.3 Circuit diagra for DC shim box PCB. Buffer amps for potentiometer control voltages feed into the summing matrix via dividing resistors. The summing amps thengenerate the buffered DC voltages that are applied to the rods. . . . . . . . . . . . . 149
A.4 Secular excitation relay circuit schematic. . . . . . . . . . . . . . . . . . . . . . . . . 150
A.5 Time sequence of the secular excitation and control trigger. . . . . . . . . . . . . . . 150
B.1 Block diagram of all the control electronics in the experiment. . . . . . . . . . . . . . 152
B.2 Block diagram of all VI modules and what hardware they each control. . . . . . . . . 156

Chapter 1

Introduction

Collision studies are a common way to study physical interactions. The relationship between
experiment and theory in such studies often plays out as follows: experiments prepare precise initialconditions for a collision and then accurately measure the outcome. A model of the interaction
is built, and its predictions can be tested against the experimental data. This thesis describes
an apparatus that was designed, built, and used to measure the outcome of carefully controlledcollisions between ions and molecules, in the gas phase, in order to better understand the natureof their interactions. Atoms and molecules, whether neutral or charged, are complicated quantum
mechanical systems. Their interactions display a diverse range of behaviors stemming from intrinsically quantum phenomenon. For example, the rearrangement of electronic structure is often the
result of a collision. The ultimate goal of this research is to understand the mechanisms that drive
such reactions in pursuit of building/refining models of the underlying quantum interactions.

1.1

Motivation
With knowledge and mastery of small systems comes the push to understand larger andmore complicated ones. An enormous amount of work has been done studying the nature of
atomic collisions. The next logical step was to include molecules. Molecular collisions, while
not new, are enticing because they push the limits of known physics. Especially tantalizing are
the quantum control aspects of molecules with the realization that “more is different.” Froma chemistry perspective, the most fundamental process is the formation and breaking of bonds.

2
Studying isolated reactions between small molecules serves as an ideal laboratory to build upon
the foundational understanding of chemistry.
In a more applied direction, gas-phase reactions between ions and molecules play a role inthe chemistry of interstellar medium (ISM). The combination of ionizing radiation from stars andthe low molecular density found in the ISM is observed, through the use of spectroscopy, to produce many ionic species. Many-atom organic species ranging up to amino acids and the famous
buckminsterfullerene (C60 + ) have also been detected [4]. Because the long-range potential betweena charged and neutral molecule leads to a large reaction cross-section, ion-molecule reactions are
believed to contribute to the observed diversity. To this end, large databases of ion-molecule reactions have been built to help with modeling of the chemistry of the ISM [5, 6]. Other environments
where gas-phase reactions between ions and molecules play a role are planetary atmospheres [7, 8]
and combustion [9, 10].
Another striking attribute of the ISM is that it is cold. Interstellar clouds are in, or close
to, thermal equilibrium with the cosmic microwave background radiation (CMBR). Temperaturesfrom 3K to 10K are typical for interstellar chemistry. To probe this regime, we must produce cold
molecular samples in the lab.
In addition to understanding ISM chemistry, cold reactions dramatically reduce complexity.
Molecules are many atom quantum systems with many degrees of freedom to distribute thermalenergy. Less thermal energy available leads to fewer populated quantum states. As temperatures
reduce below 10K, one can begin to think about fully-quantum state resolved interactions.
Fully quantum-state resolved collisions would place the most stringent bounds on theoreticalinteraction models to date. So-called, quantum chemistry, demonstrates the ultimate level of control
on reactions. Quantum control has been achieved for many atomic systems and draws ever closer
with molecular collision experiments. A multitude of techniques that have been successful inachieving atomic control have been applied to molecular systems as well [11].

3

1.2

Cold ion-molecule collision experiments
Crossed-beam and merged-beam experiments have successfully demonstrated cold, ion-molecule

collisions spanning many energy regimes. These experiments typically produce a cold, neutral
molecular beam via supersonic expansion to collide with a guided ion beam in vacuum. The
collision energy can be adjusted by the relative angle of the the ion and neutral beams. The
collision energy of crossed beams, expressed as a temperature, is typically between 10–1000K [12].
Merged-beam experiments explore the collision energy regime below 10K [13, 14]. Product detection
typically employs a combination of spectroscopic, velocity-map imaging, and mass spectrometry
techniques. While these experiments have been very successful in studying both cold and controlled
ion-molecule reactions, they have very short reaction/interaction times. This limits detection sensitivity. Collisions studied inside an ion trap can dramatically extend the interaction time andthus increase sensitivity to measuring reactions. Ion traps also accumulate charged products whichallow for simultaneous tracking of reactants and products. The rest of this section will detail a few
experimental approaches that utilize ion traps for controlled collisions.
22-pole ion traps create cold ions via cryogenic buffer-gas cooling[15–17]. The large numberof poles reduces the radio frequency (RF) micromotion heating of the ion trap to allow for buffer
gas cooling of the trapped ions to low temperatures. Neutral species are added into the trap toreact with the ions that also are cooled by the buffer gas. Charged products of the reaction are accumulated in the 22-pole trap and can be identified via action spectroscopy [16, 18]. The relatively
high pressure of cryogenic buffer gas used in these systems is capable of stabilizing highly excitedreaction complexes. Chemistry involving cations typically produce such unstable complexes, which
are unlikely to stabilize in the ISM. Additionally, for thermodynamic reasons only recently elucidated [19], the ion temperatures cannot drop below 10K in these experiments. This fundamentally
limits the collision energy range available to 22-pole trap experiments. Replacing buffer gas coolingwith direct laser cooling creates the opportunity to study colder, cation-molecule reactions withouta large background gas pressure.

4Linear Paul ion traps (LIT) are particularly well-suited for direct laser cooling of ions because
the geometry allows for good optical access. While atomic cations are the only current candidates
for direct laser cooling, they sympathetically cool co-trapped molecular ions to translational temperatures <10 K [20, 21]. Sympathetic cooling does not address the internal degrees of freedom ofa co-trapped molecular ion, which means it will not stabilize highly energetic reaction complexes.
This makes the linear Paul ion trap an ideal workhorse for studying cation-molecule reactions relevant to the ISM [22, 23]. The translational energy of a neutral reactant typically sets the collisionenergy studied in these experiments.
One such realization of cold, controlled collisions involving a LIT are hybrid ion-atom traps.
These overlap a linear Paul ion trap with a neutral magneto-optical trap (MOT), sometimes called
MOTion traps [24–28]. Direct laser cooling of the MOT atoms achieves temperatures much lower
than 1K and afford some degree of quantum-state control. Collision energies can be carefully
controlled via ion or atom shuttling techniques up to about 1K [29–31]. Like in the 22-pole trap,
charged products of a reaction are accumulated in the trap and are investigated via a mixture ofspectroscopic and mass-spectrometry techniques. A drawback to this arrangement is the atomic
MOT species is quite specific. Additionally, a molecular MOT has not been considered, to myknowledge, which limits these studies to ion-atom collisions.
Another realization of LIT experiments introduce the neutral reactant as a room temperaturegas. This leads to average collision energies between the trapped ions and neutral gas on the order of
100–200K. As with all the ion trap studies mentioned so far, products of these reaction accumulate in
the trap. These studies can be separated into two different detection schemes for reaction products.
The first scheme utilizes optical detection with the help of secular mass excitation[32–34]. The
second scheme adds a time-of-flight mass spectrometer (TOFMS) to compliment optical detection
methods [35–38]. By using a room temperature gas as a neutral reactant, these experiments benefit
from being able to investigate a wide range of species. The disadvantage is that the chemistry does
not quite reach the temperatures of the ISM. Despite the higher collision energy, these experiments
have contributed a lot to the scientific community as evidenced by the number of ion-molecule5
Table 1.1: List of many cation-molecule reactions studied in linear Paul ion traps. 300K should
be understood as room temperature. Reactions studied in this thesis are the first four entries.
Reactants
Ca+ + NO
Ca+ + O2C2 H2 + + C3 H4
CCl+ + C2 H2
Mg+ + H2 /D2
Be+ /Mg+ + HD
Ba+ + O2 /N2 O /CO2
BaO+ /CaO+ + CO
H2 + /H3 + + H2 /HD /O2
Ca+ + H2 O
H3 O+ + NH3
Ca+ + N2 O
Xe+ + NH3 /ND3
Be+ + H2 O /D2 O /HOD
C+ + H 2 O
Ca+ + ND3 /H2 CO /CH3 CN
N2 H+ + CH3 CN
Ca+ + CH3 F
OCS+ + ND3
Ca+ + 3-Aminophenol
N2 + + N 2
N2 H+ + p/o-H2 O
Ca+ + CH3 F / CH2 F2 / CH3 Cl∗ Neutrals introduced via supersonic beam.

Tneutral (K)
300
300
300
300
300
≥300
300
300
300
300
300
300
300
300
10,100,300
<10
<10
2-4
300
2∗
2∗
2∗
5-243

Reference[2], Ch. 4[39], Ch. 4
Ch. 5[40], Ch. 6
[32]
[33]
[21]
[21]
[41]
[34]
[42]
[37]
[43]
[44, 45]
[46]
[47]
[47]
[48]
[49]
[50, 51]
[52]
[53]
[54]

reactions in Tab. 1.1 with room temperature reactant gasses. The level of control and specificity
afforded by ion trapping techniques allows for better understanding of a large library of ion-molecule
reactions studied via previous methods (e.g., ion cyclotron resonance or selected ion flow tubes).
The reaction studies presented in this thesis fill some of the holes in that library by creating purified
samples of highly reactive molecules.
LIT experiments have pursued a few additional techniques of achieving cold, controlled collisions. All these techniques come in the form of collisions between a neutral molecular beam andthe ions in the trap. Stark velocity-selected beams have been demonstrated [47, 48, 55] as well as
conformer selected beams [50, 51], and cold buffer gas beams [46]. Multiple groups are pursuingcollisions with a Stark [56] or Zeeman [57] decelerated beam, though at time of writing, this has

6
yet to be demonstrated.
The apparatus we have built and characterized is a new combined LIT-TOFMS, which adds
to a growing list of similar experiments. This list is growing because of the success the LIT-TOFMS
has in characterizing cation-molecule reactions relevant to the ISM. We intend to extend the limits of cold-controlled reactions by coupling the LIT-TOFMS to a traveling-wave ring decelerator(TWRD). This will deliver a beam of neutral radical molecules with a specific quantum state into
the ion trap with a tunable velocity, which will lead to collision energies from 1–300 K. With theultimate goal of quantum control in mind, we can also pursue stimulated rapid adiabadic passage(STIRAP) methods to produce trapped molecular ions in specific quantum states [58].

1.3

Outline of this thesis
This thesis covers the work done to produce cold, controlled, ion-molecule reactions. Somebasic theory is covered to understand the operation of a linear Paul ion trap and how to conceptualize the collisions/reactions that take place inside. Then, the physical apparatus is describedand characterized. Afterwards, there are investigations into four different ion-molecule reactions:
Ca+ + NO, Ca+ + O2 , C2 H2 + + C3 H4 , and CCl+ + C2 H2 . The reactions involving Ca+ display
quantum control of photochemistry. Reactions with C2 H2 + show isomer specific chemistry resulting from the different arrangements of the neutral reactant C3 H4 . The reaction of CCl+ uncovers
astrochemically relevant reaction mechanisms. The thesis concludes with a summary of the work
presented and a brief outlook on the future work: incorporation of the TWRD.
Since this experiment was built from the ground up over my time at JILA, there is a large
amount of additional technical information that is not covered in detail in the thesis. Schematics ofthe trap electronics, LabVIEW code for experiment control, and MATLAB code written for data
analysis have been relegated to the appendices.

Chapter 2

Ion trapping and ion-molecule reaction theory

This chapter summarizes the theory underlying our method of studying reactions in an ion
trap. The detailed theory of each section has been covered in the literature in far greater detail thancan be included here and will be referred to often. I will include a brief overview of the concepts
to contextualize the experimental implementation described in the next chapter. All equations arederived using a Gaussian CGS unit system.

2.1

Ion trapping
Earnshaw’s theorem guarantees that charged particles cannot be trapped in three dimensions

via static potentials [59]. Ion trapping is thus achieved through use of sinusoidal time-varying
quadrupole potentials. An intuitive understanding of this comes from the saddle point at thecenter of a static qudrupole potential. If the polarity is flipped, the saddle point rotates 90 degrees.
By sinusoidaly varying the voltage, the saddle point rotation produces a local minimum in two
dimensions. The third dimension is confined using a static potential.
There are many realizations of ion traps utilizing a variety of geometries to produce thedesired time-varying quadrupole field. Our ion trap is a linear Paul trap [60], which is particularly
well suited for laser cooling ions (covered in the next section). A schematic diagram of our trap
geometry is depicted in Fig. 2.1. While hyperbolic electrodes produce an ideal quadrupole potential,
cylindrical rods are easier to manufacture and produce an adequate approximation. This is true
provided the radius of rods, R, and inscribed radius, r0 , of the trap are related by R = 1.15 r0

8

2z0

y

r0
x

z

y

x

Figure 2.1: Schematic representation (not to scale) of the ion trap geometry. The coordinate systemis drawn for reference and orientation. To the left is an isometric view of the trap and to the rightis an end-on view.

[61, 62]. The four rods have three electrically isolated segments each, for a total of 12 rod segments.
Radio-frequency (RF) voltage is applied to all six segments in a diagonal pair of rods. The opposite
phase RF voltage (180°) is applied to the other diagonal pair of rods to produce the time-varying
quadrupole potential
φRF (x, y, t) = 2VRF sin(Ωt)

x2 − y 2
.
2r02

(2.1)

VRF is the amplitude (zero-to-peak) of the applied RF, Ω is the angular frequency of the RF,x, y, and t are two spatial coordinates and a temporal coordinate, respectively.
A DC voltage is also applied to the eight, longer electrodes on the ends of each rod (theso-called endcaps). This produces an approximately harmonic potential along the axial directions
with a small radial dependence described byκVEC
φEC (x, y, z) =z02


x2 + y 2
2
z −
.
2

(2.2)

κ (< 1) is a geometrical factor, VEC is the DC voltage applied to the endcaps, and z0 is half thelength of the center electrode. Thus the total electrical potential isΦ(x, y, z, t) = φRF + φEC .

(2.3)

We can now derive the equations of motion for an ion in the trap. Unfortunately, solutions
to the equations of motion cannot be obtained analytically for many ions in the trap. However,9insights from the single trapped ion picture that follow are a useful intuition to build on and provide
a correct description (to lowest order) of the center-of-mass motion of many ions. Computational
solutions for many ions are attainable where more accuracy is desired. The following derivation
borrows elements of similar derivations given in [1, 63].
We exclusively trap cations, as they are laser coolable. To date our ionization methods only
allow us to remove only a single electron, so we consider a particle of charge +e and mass m,~
F~ = m~a = e∇Φ.

(2.4)

Or, separated into three differential equations, (for the three spatial coordinates)∂2xe κVECe 2VRF+ −+sin(Ωt) x = 0
∂t2
m z02
m r02


e κVEC∂2ye 2VRF+ −−sin(Ωt) y = 0
∂t2
m z02
m r02


e 2κVEC
∂2z
+
z=0
∂t2
m z02

(2.5)

It is worth noting that the equation in the z direction is that of a harmonic oscillator. The equations
in every direction have been rearranged to match the form of the Mathieu equation, which has awell-known analytical solution. That form is∂2u+ (au − 2qu sin(2τ )) = 0,
∂τ 2

(2.6)

where u and τ are generalized coordinates; au and qu are dimensionless parameters. The variable
substitutions that cast Eq. (2.5) into the form of Eq. (2.6) areu = {x, y, z},
Ωt
,
2
az
e 4κVECax = ay = − = −,2m z02 Ω2τ=qx = −qy =e 4VRF; qz = 0.
m r02 Ω2

(2.7)
(2.8)
(2.9)
(2.10)The solution of Eq. (2.6) can be expressed as a Fourier series. As long as qu  1 and au  1,which is typical for our experiment, the first term in the series is an adequate approximation of the10
ion motion, given byquu(τ ) ' A(1)
cos(ω
τ
+
ϕ
)
1
+
cos(Ωτ
)uuu
2(2.11)

where
Ω
ωu =
4

r
au +

qu2
2

(2.12)(1)is the so-called secular frequency. Au is the amplitude of secular oscillation and ϕu is the corresponding phase. Both are parameterized by the initial conditions of the ion.
Eq. (2.11) depicts oscillations associated with two different timescales, aside from the casewhere qz = 0. The slower oscillation occurs with angular frequency ωu and is called the secularmotion. Secular motion can be intuitively understood as the harmonic motion caused by the timeaveraged rotating saddle point potential. The faster motion occurs with angular frequency Ω, thedriving RF frequency, is called micromotion. Micromotion is from the ponderomotive force that thesaddle exerts on the ion as it changes in time. There is no static potential analogy for micromotion;
it can only be caused by a dynamically changing potential. Additionally, if the amplitude of secularmotion of an ion is zero, so too is its micromotion. This occurs when ions are along the z-axis of
the trap, which is co-incident with the saddle point of the RF potential, also called the RF-node.

2.1.1

Pseudo-potential approximation

Since au  1 and qu  1 are typical trapping conditions, ωu  Ωu by Eq. (2.12). The timescales of the two oscillatory motions can thus be adiabatically separated. Doing so mathematicallyamounts to time averaging over the RF oscillation of the potential. This reduces the equations of
motion to purely harmonic potentials, which is known as the trap pseudo-potential. A detailed
derivation of the pseudo-potential can be found in [64]. The result isφ∗ (x, y, z) =2eVRF
(x2 + y 2 ) + φEC (x, y, z)mΩ2 r04(2.13)

This approximation is particularly useful for estimating the trap depth for our ions in the x andy-directions
P Ex,y ≤qx,y VRF,
4(2.14)

11
where the equality holds for an ion at the center of the radial potential. An ion with a large
amplitude of oscillation – away from the center of the trap – is less bounded. This is essentiallya statement of energy conservation E = P E + KE where E is a constant and KE is the kinetic
energy of the ion.

2.1.2

Trapping stability

One would assume that there are a specific range of frequencies that work best for trappingions. Intuitively, a saddle rotating too slowly will allow ions to escape. Likewise, rotating too
quickly will eject ions via the ponderomotive force. Between these limits is the stable trapping
regime. This can be quantitatively described by a stability analysis of the Mathieu equation (Eq.
(2.6)). The stability regime is parameterized by au and qu , which can be related back to real world
trapping parameters via Eq. (2.7). A detailed stability analysis can be found in [64]. The result of
which is often given in terms of another unitless parameter
r
β≈

au +

qu2
2

(2.15)

also called the mathieu characteristic parameter. Stable trapping in the region defined by 0 < βu <1. However, Eq. (2.15) only holds for au  1 and qu  1 and thus β  1. Fortunately, almost
every major programming package has the mathieu characteristic parameter built in, and a general
analytic formula for β is unnecessary. Using a computational resource for β (Mathematica), one
can produce the stability diagram relevant to our experiment seen in Fig. 2.2.
We do not need to consider stability in the z direction because the purely harmonic potential
has a permanent, stable minimum.

12

0.00
-0.05

ax,y

-0.10
-0.15
-0.20
-0.25
-0.30

0.0

0.2

0.4

0.6

0.8

1.0

|qx,y |

Figure 2.2: Mathieu Stability diagram for experimentally accessible values of ax,y and qx,y . Stabletrapping conditions are represented by the shaded region. The intercept on the qx,y -axis is ∼ 0.908.

2.1.3

Ion kinetic energy

From Eq. (2.11), we can calculate the kinetic energy averaged over one period of the secularmotion

1
1
1 2 2
2
2
KEu = mhu̇2 i = mA(1)u (ωu + qu Ωu ).
24
8(2.16)

1
qu2
2 2
KEu ' mA(1)).
u ωu (1 + 2
4
qu + 2au(2.17)

Applying Eq. (2.12) gives

In the x and y directions, we can further simplify with the condition ax,y  qx,y giving
1
2 2
KEx,y ' mA(1)x,y ωx,y ,2(2.18)

which shows that in typical trapping conditions, secular motion and micromotion contribute approximately equally to the kinetic energy of the ion. Furthermore, Eq. (2.17) shows that thecondition required for cold ions is a low amplitude of oscillation. This can be realized by directlaser cooling, which is discussed in the next section.

13

2.2

Laser cooling of Ca+
Doppler laser cooling is a technique utilized in virtually every cold atom experiment. In ourcase, we use it to cool calcium-40 cations, 40Ca+ , which are the workhorse ions present in the trap
for all of our ion-molecule reactions. A useful model for understanding Doppler laser cooling of
Ca+ is a two-level atom. I will provide some semi-classical results of the derivation found in [65]for the purpose of building an intuition and connecting with experiment.
A two level atom irradiated by a laser near its resonance frequency will scatter light fromthe beam. The spontaneously emitted light, having no preferential direction, means that there is a
momentum transfer from the laser beam to the atom. Given the proper conditions, the resulting
force felt by the atom can be guaranteed to be opposite the direction of the atoms velocity. The
net effect is a dissipative force that removes kinetic energy from the atom.
The quantum mechanical process behind the force is spontaneous emission. To build up to
the force, we must first define characteristics of the atom and of the laser irradiating it. To startis the spontaneous emission rate of the atom, Γ ≡ 1/τ , where τ is the excited state lifetime. The
excited state lifetime sets the saturation intensity of the process
Is =

πhc
,
3λ3 τ

(2.19)

where λ is the laser wavelength on resonance. The saturation parameter defines the relative interaction strength of the laser with the atoms=s0,
1 + (2∆/Γ)2(2.20)

where ∆ ≡ ν − ν0 is the laser frequency detuning from resonance, ν0 , and s0 is the on resonancesaturation parameter
s0 =

I
2|Ω|2
=
,
I0
Γ2

(2.21)

where Ω is the Rabi frequency, which is another way to parameterize the light-matter interaction
strength. Finally, the force an atom feels from a laser near resonance is given by
!
~k
sΓ/2
2∆~
v
·
1+
= F~0 − β~v ,F~ = ~~k1+s(1 + s)(∆2 + Γ2 /4)(2.22)

14
where ~k = 2πv is the velocity of the atom, and β is a damping coefficient to
λ k̂ is the laser’s k-vector, ~be defined next. The equation has been suggestively parameterized into two terms. The velocityindependent term, F0 , is the classical radiation pressure from the laser. The velocity dependent
term is the dissapative force responsible for laser cooling. It’s form is identical to that of a friction
term. Solving for β gives us the damping coefficient
β = −~k 2

4s0 (∆/Γ),(1 + s0 + (2∆/Γ)2 )2(2.23)

which depends only on the laser intensity (through s0 ), detuning, and excited state lifetime. We
can now see that for ∆ < 0 (red detuned), the force is guaranteed to be damping and lead tocooling.
In the experiment we use two, counter-propogating beams to cool. Assuming the k-vectors
are equal and opposite, the resulting force generated by the two beams is
F~tot = −βtot~v ,(2.24)

where βtot = 2β. Intuitively, the radiation pressure force from each side cancels and the damping
coefficient doubles.
In general, six laser beams are needed to produce the most efficient cooling in three dimensions. Because of the symmetry of linear Paul ion trap, only the radial and axial directions needcooling, which relaxes the number of beams needed to four. In practice, our lasers cool along theaxial direction of our trap. A slight misalignment in the overlap of the counter-propagating lasers
provides enough cooling in the radial dimension to further relax the condition to the two beams
used in the experiment. Radial cooling is dramatically enhanced for large ion clouds extending
off of the RF-node. The strongly interacting ions (through coulomb repulsion) effectively couple
the laser cooling to all transnational degrees of freedom. A notable exception to this scenario is
when few enough ions are trapped that they can all be arranged along the RF-node. In this case,an additional radial laser beam is added to help cool the resulting ion string. It is not strictly
necessary, due to imperfections in alignment of the axial beams, but it noticeably improves laser
cooling efficiency.

15

4 2P1/2

τ = 7.1ns
gj = 2/3

866.214nm
τ = 1.2s
396.847nm

4 2S1/2

gj = 4/5

3 2D3/2

gj = 2

Figure 2.3: Energy level diagram for 40Ca+ . Transition wavelengths are measured in air, thelifetimes are total excited state lifetimes, and gj are the Landé g-factors.

Eq. (2.24) would suggest Doppler laser cooling is able to bring the atoms to rest. Since thiswould correspond to a temperature of absolute zero, it would violate the second law of thermodynamics and surely cannot be the case. What is missing from the model is the fact that momentum/energy is removed in discrete quanta of photons, which results in a finite minimum velocity
attainable. The actual Doppler cooling limit associated with this minimum velocity, expressed asa temperature, reduces to the form
TD =

~Γ
.
2kB

(2.25)The best candidates for Doppler laser cooling will be closed cycle systems with very short excitedstate lifetime. The shortest excited state lifetimes occur in dipole allowed transitions, like that ofan electron going from the P to S orbitals in hydrogen. The alkaline earth column of the periodic
table is a sensible place to look for hydrogen-like cations, because the cations will only have one
valence electron.
Since Ca+ is not an ideal two-level system, we must consider its specific energy levels. We
choose the mass 40u isotope of calcium because it is naturally the most abundant (98%) and it has
no nuclear spin (no hyperfine structure). A graphic description of the lowest lying energy levels and16
all their properties tabulated in the NIST atomic spectra database is shown in Fig. 2.3. The main
cooling transition is the S1/2 to P1/2 . There is a 7.5% chance the P1/2 state will spontaneously
decay to the low-lying D3/2 state, which does not have a dipole-allowed transition back to the S1/2
state. To prevent shelving of ions in the D3/2 state, a single repump laser is necessary to close thecooling-cycle. In practice, only two lasers are needed (main cooling and repump) to cool Ca+ to
the Doppler limit.
Since Ca+ is constantly cycling between the S1/2 , P1/2 , and D3/2 states, a steady-state
population builds up in each of the three states when considering time-scales longer than the
excited state lifetime (ns). The steady-state population of the excited states can be predicted byusing the optical Bloch equations (OBE). The OBE are relevant to experiments where the quantum
state populations of Ca+ need to be known. The OBE are explicitly laid out for Ca+ in [54].

2.3

Coulomb crystals and sympathetic cooling
Coulomb crystals, also called Wigner crystals, are cold, single-component plasmas confinedby an external field. They are periodic (or quasi-periodic) structures that form when the kinetic
energy of the plasma becomes less than the potential energy of the ions in confinement. With muchless kinetic energy than the potential of the confining field, the ion locations become locked into
place. The overall structure of the ions is determined by the balance of the confining force on each
ion with that of the coulomb-repulsion between ions. This is precisely what occurs when Doppler
laser cooling trapped ions.
The transition from a cloud of trapped ions to a coulomb crystal is parameterized by theplasma parameter
Γpl =

PE
e2
=
KE
aws kb T(2.26)

where aws is the Wigner-Seitz radius, related to the zero-temperature plasma density n0 by
n0 =

2
VRF
1
= 4 3 .
424πmΩ r0
3 πaws(2.27)

It has been both experimentally and computationally verified that for Γpl & 2 the ion cloud will

17

Figure 2.4: False-color fluorescence microscope images of two separate coulomb crystals with different values of the plasma parameter. The top image shows the charged fluid phase and the bottom
image shows the crystalline phase.

undergo a phase-transition into a charged fluid phase. At this point, the boundary of the ion cloudis now clearly defined but individual ion sites are not resolvable and only short-range order exists.
This is because the ion cloud has too much kinetic energy to settle on a single structure. The
constant re-arrangement of ion sites is on a faster time-scale than that of the camera exposure.
For Γpl & 150 − 170 the ion sites become locked into place and the crystal structure is clearly seen
with long-range order. Solving Eq. (2.26) for T with an ion density of n0 = 1081estimates the
cm3

crystalline phase transition to occur at T ∼ 100mK. A visual example of the transition from ourapparatus is shown in Fig. 2.1.
When ions are added to a trap where a coulomb crystal is already present, the new ionsare cooled by elastic collisions with the crystal. Momentum transferred to the crystal is quickly
removed by the laser cooling in a process dubbed sympathetic cooling. The sympathetically cooled

18Cold Ca+ with C2H2 +Figure 2.5: False-color fluorescence microscope images of a bi-component coulomb crystal containing Ca+ and C2 H2 + ions.

ion will typically reach a kinetic energy on the same order of magnitude as the directly laser cooledions. Depending on how efficient the sympathetic cooling is, the sympathetically cooled ions will
also crystallize. This is one clever way of achieving cooling of molecular ions that cannot be directly
laser cooled. It should be stressed here that only the external degrees of freedom (translations) of
molecular ions are cooled. Internal degrees of freedom (vibrations and rotations) of molecular ionscannot be cooled by collisions between the ions as they do not ever get close enough to one anotherto do so.
Ions not addressed by the laser will not fluoresce and are often refered to as dark ions.
Their location can be inferred by the displacement of laser cooled Ca+ observed in fluorescencemicroscope images. When a coulomb crystal contains more than one species of ion it is said tobe multi-component, borrowing the term from multi-componenet plasmas. An example of a bicomponent crystal is depicted in Fig. 2.5. An immediately noticeable feature of the bi-component
crystal is that all the C2 H2 + (26u) are gathered at the core of this crystal. The structure is1
qualitatively predicted by the pesudo-potential approximation, Eq. (2.13), which shows φ∗ ∝ m.

In other words, the radial potential is stronger for lighter mass ions in our trap, and they will fallto the bottom of the well. Ions that are heavier than Ca+ (40u) will collect at larger radii. Because19
ions arrange radially by mass, energy transfer will be most efficient between ions of neighboring
masses.
The quantitative structure and energetics of multi-component crystal cannot be predicted
analytically. The system is inherently far from thermodynamic equilibrium. The RF trapping
field is driving energy into the system via micromotion, while laser cooling removes it. A steadystate structure and energy distribution are none-the-less eventually achieved and can be modeled
computationally. Quantitative insight can be gained from these models in the form of effective
heating and cooling rates as shown in [66]. These rates, however, are very dependent on thenumber and masses of constituent ions in a crystal. In order to fully describe a multi-component
crystal, one must be able to unambiguously identify the crystal’s constituent ions. The tool we use
for this task is a time-of-flight mass spectrometer, covered in the next section.

2.4

Time-of-flight mass spectrometry
Many methods have been developed for performing mass spectrometry on trapped ions. The

most applicable to the area of trapped ion-molecule reactions are secular mass excitation [32–34]
and time-of-flight mass spectrometry (TOFMS) [35–38]. The former relies on exciting the massdependent secular frequency of a trapped species; it is, in principle, non-destructive and can bepreformed in situ. The latter ejects ions from the trap with a large, static DC potential andmeasures the mass-dependent flight times required to reach an ion detector far away from the trap.
While the measurement is destructive (empties the trap), it accurately detects both the mass of
species in the trap and how many ions in each constituent mass were trapped.
The underlying principle of TOFMS comes from the motion of an ion in a constant electricfield. Consider and ion between two parallel plate-like mesh screens holding a potential difference
VT OF . An ion near the higher potential mesh will be pushed towards the mesh at lower potentialand ejected from the region of constant electric field with kinetic energy
1
KET OF = eVT OF = mv 2 .
2(2.28)

20
solving for the ion’s velocity v givesrv=

2eVT OF.
m(2.29)The time it would then take to reach a detector a distance x away would be
x
t= =x
v

r

m
.
2eVT OF(2.30)The important result is that the ion arrival time is proportional to the square root of its mass-tormcharge ratio: t ∝.
e
The resolution of a TOFMS is a measure of the relative separation of neighboring mass peaks.
It is defined as
R≡

m
t
=
∆m
2∆t

(2.31)and can be interpreted as the highest measurable mass m that can be distinguished from massdt∆t
m + 1. To estimate R, we take a derivative of Eq. (2.30) dv≈ ∆v
which leads to

1
R=
2∆v

r

2eVT OF.
m(2.32)In order to optimize the resolution of a TOFMS, one must primarily minimize ∆v and secondarily
maximize VT OF . ∆v can be interpreted as the initial spread in velocity (kinetic energy) of a cloudof ions. Laser cooling directly reduces this, leading to excellent TOFMS resolution. Also, the larger
VT OF , the better. This is typically constrained by the geometry of the meshes or the electronics
driving them.
Counter-intuitively, the free-flight distance of a TOFMS, x, does not affect its resolution. It
does however set the scale of t and ∆t. These values have an important practical concern basedon the response time of the ion detection system. By ensuring that these are long enough, one will
be able to measure both t and ∆t, and quantify the resolution. Additionally, the integrated signal
from the ion detector will be proportional to number of ions in each mass channel. The more data
points measured per mass peak, the more accurate that integral calculation will be.

21

2.5

Bimolecular reaction kinetics and dynamics
The extremely dilute environment of the ultra high vacuum (UHV) chamber ensures that the

only reactions we will study are bimolecular. The basics of bimolecular kinetics and dynamics arecovered very well in [67–69] and more specifically for ion molecule chemistry in [70].

2.5.1

Kinetics

Here is a quick overview forming the theory basis for our reaction data analysis. Consider
reaction of the formk

A+ + B −−→ C+ + ...

(2.33)

where A+ is an ionic reactant, B is a neutral reactant, C+ is the ionic product, and k is the reaction
rate constant. By charge conservation, one ionic product is guaranteed from this reaction and anyother neutrals produced will be lost from the trap. The probability that a neutral product will
participate further in the reaction is practically zero. The rate constant, k, is related to the rate ofreactiondA+= −k A+ B,
dt
dC+= k A+ B.
rC =
dt
rA =

(2.34)

It is always the case in our experiment that the neutral reactant is in excess, B(t) = const, andthus we can define an effective rate constantkef f ≡ kB.

(2.35)A neutral reactant in excess makes the kinetics identical to that of a unimolecular process and thus
the reaction is called pseudo first-order. Solving Eq. (2.34) gives−kef f t
A+ (t) = A+0e
−kef f t
C+ (t) = A+
1
−
e
0
+
+
where A+0 is the initial number of A ions and C0 = 0.

(2.36)

22
Parallel reactions are described by the modelk

A+ + B −−c→ C+ + ...

(2.37)k

A+ + B −−d→ D+ + ...
which give three first-order linear differential equationsdA+= −(kc + kd ) A+ B
dt
dC+
rC =
= kc A+ B
dt
dD+
rD =
= kd A+ B
dt
rA =(2.38)+++
along with the conservation of mass condition A+
0 = A +C +D . The pseudo first-order solutionsare−(kc +kd )Bt
A+ (t) = A+
0e

kc 
−(kc +kd )Bt
1−e
C+ (t) = A+0
kc + kd
kd −(kc +kd )Bt
D+ (t) = A+
.
1
−
e
0
kc + kd(2.39)The new scale factors are called branching ratios
BRc ≡

kc
kd
; BRd ≡.
kc + kd
kc + kd(2.40)

Because the neutral is always in excess, we often observe sequential reactions described by
the modelkA+ + B −−c→ C+ + ...

(2.41)k

C+ + B −−d→ D+ + ...
which give three first-order linear differential equationsdA+= −kc A+ B
dt
dC+
rC =
= kc A+ B − kd C+ B
dt
dD+
rD =
= kd C+ B
dt
rA =(2.42)

23+++
along with the same conservation of mass condition A+
0 = A + C + D . The pseudo first-ordersolutions are−(kc )Bt
A+ (t) = A+
0e(2.43)

kc  −kc Bte− e−kd Bt
kd − kckckd+−kc Bt−kd Bt+.
e+e
D (t) = A0 1 −kd − kc
kd − kc
C+ (t) = A+0

The kinetics of a sequential reaction falls into one of three different regimes: kc  kd , kc ∼ kd ,and kc  kd . When one of the rate constants is approximately 10 times less than the other, theassociated step in the reaction model is called the rate-limiting step.
In practice, a realistic reaction model may be a large combination of parallel and sequential
reactions. Since this will always result in a system of linear, first-order differential equations, ananalytic solution always exists. The solution in analytic form is far more cumbersome than thesystem of differential equations, however. When dealing with a complicated model, the data canbe directly fit by the system of differential equations. See App. C for a MATLAB script that does
just that.

2.5.2

Dynamics

The dynamics of gas-phase reactions are the short-range physics that describe the process ofreactants leading to products. Better understanding of reaction mechanisms and the better models
that result from them, are the overall goals of this research. That said, there are many usefulmodels that have already been developed. The textbooks mentioned at the start of this section are
the most useful references on the subject of dynamics. I will point out the most applicable modelto this work.
Most models aim to predict the temperature dependence of reaction rate constants k = k(T ).
One very famous and successful model is called the Langevin model. This models the long range
interaction between an ion and a molecule as a point charge and a polarizable dipole. The effective
interaction potential that results contains only a centrifugal barrier. Assuming that all interactions

24
that make it past that barrier will react delivers the striking result is that the rate constant isn’t
a function of temperature at allsk=

4π 2 e2 αµ(2.44)

1.
where α is the polarizability of the molecule and µ is the reduced mass µ1 ≡ m1ion + mneutral

Experimentally, however, it was observed that some ion-molecule reactions would accelerate at
low temperatures. The increase was attributed to a permanent dipole moment, µd , of the neutral
reactant increasing the interaction strength. This is due to the average orientation of the dipole
being biased towards the ion when fewer rotational states of the neutral are populated. It resultsin the aptly named average dipole orientation (ADO) model
2πek(T )ADO = √
µ


√

r
α + Cµd2πkB T(2.45)

where C is the dipole locking constant, usually C ∼ 0.1. A detailed derivation of this model canbe found in [46]. Comparing this model to our data requires us to define a temperature of ourreactions. For the purposes of the ADO model, one needs to define a rotational temperature. Thisis room temperature for our experiment because the neutral reactant is leaked into the vacuum
without any means to cool it.
Generally speaking there is much more to reaction dynamics than the rate constant predictedby the ADO model. For example, both branching ratios of products and isotopic substitution effects
are not predicted by ADO. These effects are details of what happens after reactants make it over thecentrifugal barrier of the long range potential and are also functions of temperature. Unfortunately,a temperature relevant to these dynamics is not well defined, because a multi-component coulomb
crystal system is decidedly far from thermodynamic equilibrium. Since it does reach a steady statewith constant energy E, however, we can consider a microscopic picture of the rate constantk = k(E) = σ(E)vrel(2.46)

where σ(E) is the cross-section for reaction and vrel is the relative velocity of reactants within
the center-of-mass frame. The microscopic picture separates the internal degrees of freedom of25
a molecule (electronic, vibrations, rotations, spin) from the external ones (translations). The
distinction is exactly what we need to connect our experiments to the temperature dependence ofvarious models. The internal degrees of freedom are in thermal equilibrium with the black-bodybackground of our room temperature UHV chamber, ∼ 300K. The external degrees of freedom canbe compared to a classical temperature by average kinetic energy of the system divided by the
Boltzmann constant
Treaction =2 iµhvrelhEcoll i
=
.
kB
2kB

(2.47)

Tneutral.
neutral
1 + mmion(2.48)

If vion ∼ 0 then vrel ∼ vneutral , andTreaction =For Tneutral ∼ 300K, Eq. 2.48 is an adequate approximation. If Tneutral ≤ 10K, then the ion kinetic
energy distribution must be modeled using molecular-dynamics simulations as demonstrated in [49].
The convolution of the neutral and ion velocity distributions is then used to tabulate the averagecollision energy.
To model the details of the reaction cross-section and its energy dependence requires quantum chemical calculations of reaction potential energy surfaces. Rice–Ramsperger–Kassel–Marcus(RRKM) theory can then be used to predict branching ratios and build up σ(E) from the microscopic picture [71]. Because ∼ 300K is already pushing the limits of theoretical accuracy forquantum chemical computations [3], we do not emphasize this analysis. As such, the details ofquantum chemical calculations required are beyond the scope of this thesis.

Chapter 3

Experimental Apparatus

A large part of my thesis work was designing, building, rebuilding, and improving the ion
trap apparatus. This chapter will describe the apparatus in detail, which has enabled the study ofthe molecule-ion reactions that follow in later chapters. What follows is adapted from [38]. I will
also include updates and improvements we have implemented since then.
Our experimental apparatus consists of a ultra-high vaucuum (UHV) chamber containing
a segmented linear quadrupole ion trap and a linear TOFMS, coupled (radially) to the ion trap.
We trap 40 Ca+ and perform laser cooling to form Coulomb crystals with temperatures < 1 K.
Molecular ions can be co-trapped and sympathetically cooled by the atomic ions.
Trapped ions are detected either by the fluorescence of the atomic ions or by ejecting all
trapped ions into the TOFMS. To detect mass spectra with high mass resolution over a wide mass
range, the trapping RF amplitude must be switched off within a couple RF periods before the HV
extraction pulses are applied to the trap electrodes. Here, dedicated RF drive electronics werebuilt and characterized. The next sections describe in detail these individual components of theapparatus.

3.1

Vacuum chamber
To facilitate high cooling rates and have Coulomb crystal lifetimes of several hours, thevacuum chamber is maintained at a base pressure of <5×10−10 torr using a 600 l/s turbo pump(Pfeiffer HiPace 700) backed by a dry roughing pump (Edwards XDS 35i). A titanium sublimation

27
pump is used in addition for enhanced pumping of certain residual reactive contaminants. Finally,to eliminate background water in the chamber, a cold finger filled with liquid nitrogen is used
occasionally.

3.2

Linear ion trap – time-of-flight mass spectrometer (LIT-TOFMS)
The central component of the experimental apparatus is a segmented, linear quadrupole ion

trap built from four rods (see Fig. 3.1). RF potentials of several hundred volts peak-to-peak
are applied to the rods to produce a time-averaged trapping potential in the radial dimension.
Additionally, each rod is divided into three electrically isolated segments, which allow for theapplication of individual DC voltages on each segment to confine the ions along the longitudinalaxis. The DC voltages can also be used to make small adjustments to the position of the ion cloudin three dimensions to center the ions on the node of the RF potential, thus minimizing RF heatingof the ions.
Our trap has an inscribed radius of r0 = 3.91 mm and a trap rod radius of R = 4.5 mm.
This ratio produces a nearly perfect radial quadrupole field in the center of the trap [61], reducing
the micromotion heating of the ions. The central segments of the ion trap have a length of 2×z0
= 7 mm, while the outer segments have a length of 6×z0 = 21 mm. These lengths give rise to an
approximately harmonic axial potential in the center of the trap. To trap the laser-cooled atomic
ions, the RF frequency applied to the electrodes is ΩRF ≈ 2π×3.5 MHz with an amplitude up to
V = 250 V. Axial confinement is achieved with DC potentials of VDC ≤ 10 V applied to the outer
segments of the rods (Fig. 3.1). A summary of typical trapping parameters is given in Tab. 3.1.

28
Table 3.1: Typical trapping parameters and trap geometry specifics. Also included are some
calculated values (not measured) for 40 Ca+ using the typical trap parameters and the equations
found in Ch. 2.
Trap parameters

Calculated trap values

RF frequencyΩRF

2π×3.552MHz

Mathieu q param|qx,y |

0.25

RF voltage

VRF

200V

Mathieu a param|ax,y |

0.0012endcap voltage

VEC

3.5V

radial frequency

ωx,y

2π×162kHz

endcap separation

2z0

7mm

radial trap depth

Dx,y12.6eV

inscribed raduis

r0

3.91mmlongitudinal frequency

ωz

2π×43.8kHz

geometric factor

κ

0.22

longitudinal trap depthDz

0.77eVrod radius

R

4.5mm

ion density

n0

4.58×107 1/cm3

To produce calcium atoms for loading into the trap, we use an effusive calcium beam generated
by a home-built, resistively heated Ca oven. The oven is made out of a thin-walled stainless steel
tube, crimped on both ends, with an 1 mm2 opening at one end. The oven is mounted 300 mm
from the trap center and uses a collimating aperture in the beam path to minimize Ca depositionon the trap rods. A beam flag is placed in the atomic beam path as a shutter to block the Ca
impinging on the trap after the loading process. This allows the Ca oven to be kept at a consistenttemperature at all times, while reducing the influence of the oven products on the trapped ions.
The atomic beam of 40 Ca is then ionized in the trap center using non-resonant photo-ionization at
355 nm (Continuum Minilite II, 10 Hz, 5 ns pulse length, up to 7 mJ/pulse, focused f = 20cm). In
addition to the Ca oven, we installed a set of three alkali atom sources dispensers (SAES) — K,
Rb and Cs — for calibration of the TOFMS spectra. A shield prevents direct line of sight between
the dispensers and the trap.
The trapped atomic ions are laser cooled on the 42 S 1 - 42 P 1 transition of 40 Ca+ using a fiber2

2

coupled 397 nm diode laser (NewFocus, 3.5 mW, 600 µm beam diameter). The ions are repumped
from a dark state on the 42 P 1 - 32 D 3 transition using a second fiber-coupled diode laser at 866
2

2

29

to camera

pulsed leak valve

microscope objective

shutter
Ca skimmer

Kimballchamber

Ca ovenion trap

PZT skimmer
coolingbeams

to TOFMS

ionization
beams

PZT valve

Figure 3.1: Exploded schematic of the ion trap and detection optics. The ion trap has fourrods, which are broken into three segments each. RF potentials are applied to all 12 segments.
In addition, small DC potentials are applied to form a trapping field along the longitudinal axis.
Directly above the ion trap sits a 10x microscope objective to image the fluorescence from the Caions onto a EMCCD camera. The drift tube (not shown) of the TOFMS is located directly below
the ion trap.

nm (NewFocus, 9 mW, 2 mm beam diameter). The two diode laser frequencies are measured bya wavemeter (High Finnesse/Ångstrom WSU-30), which is calibrated daily to an additional diodelaser at 780 nm locked via saturated absorption spectroscopy to a transition in 87 Rb (see Fig. 3.2).
The calcium cooling laser frequencies are locked via a slow LabVIEW controlled servo based on the
wavemeter readings.
For efficient cooling of ion crystals down to the single ion level, the 397 nm laser beam is
directed into the center of the trap from up to three directions (two counter-propagating beams

30

λ/2 wp→

New Focus Vortex
Laser TLB-6013

866nm

PBS

New Focus Vantage
Laser TLB-7100

780nm
Rb cell

397nm

→

→
λ/2 wplens

New Focus Vantage
Laser TLB-7100

4
Dual
Photo-diode96µscopeslide

PBS

Fibers to
Wavelength meter

Fibers to
Ion Trap

Legend→→λ/2 wpOptical isolator
Fiber in-coupler
Shutter
Beam Dump

Figure 3.2: Laser breadboard layouts for diode lasers used in the experiment.

along the axial dimension and sometimes one beam along ionization beam-path, acting to coolradial dimension). The axial beams are of equal intensity, while the radial beam has about 2% of
the total power. The radial beam is required to efficiently cool only small strings of ions, but isunnecessary for large crystals. The emitted fluorescence at 397 nm from 40 Ca+ is collected by a
10× microscope objective (Mitutoyo) and imaged onto an EMCCD camera (Andor iXon 897). This
allows real-time, non-destructive detection of ions stored in the trap. See Fig 3.3 for a schematiclayout of the laser and imaging system.
In addition to imaging the fluorescence to determine the number of Ca ions, we have incorporated a radially extracted TOFMS to the ion trap to greatly increase the accuracy of measurementsof non-fluorescing ions. The TOFMS consists of two stages of longitudinal electric fields that ex-

31
Prism-pairperiscope

216nm

355nm

Liop-Tec Dye Laser

355nm
LegendContinuum Nd:YAG
Powerlite 8010
Continuum Nd:YAG
Minilite

Shutter
Flipper-mirror

CameraThF

Fiber out-coupler

ThF

UV lens
F = 20cm

Beam Dump

UV lens
F = 20cm
90

90
Fibers fromBreadboard

BS

PMT

10

BS
10

Ion Trap

B-fieldcoil

ND filter
wheel

397nm

10x mitotoyu
objective

50/50 BSND filter
wheel

λ/2 wpDichroic

866nm
λ/2 wp

PBS

400nm
Band pass filter

Ion Fluorescence

ND filter
wheel

Figure 3.3: Optical beampaths of ionization lasers and cooling lasers used in this thesis. Two
flipper mirrors are used in conjunction with the shutters to send one of the two ionization lasers
into the chamber at a time or the radial cooling beam for the calcium.

tract the ions from the trap and accelerate them into a longitudinal zero potential drift tube. The
entire length of the TOFMS is 540 mm. The trap electrodes themselves create the repeller andextractor fields for the TOFMS. This is accomplished by applying large DC potentials to the four
trap rods, where the two rods farthest away from the entrance of the drift tube (i.e., repeller rods)
are at one positive potential, and the two trap rods closest to the drift tube (i.e., extractor rods)
are at another, lower, positive potential (see Fig. 3.4, left, for more details). A drift tube, held at
ground potential, is placed between the ion trap and the ion detector to minimize the influence ofstray fields on the post-accelerated ions. The entrance into the drift tube, which is 22 mm fromthe center of the trap, is limited by a 2 × 5 mm elliptical hole located directly below the center32of the trap. Using a small opening reduces the fringe fields from the trap electrodes penetrating
into the drift tube. Directly after the entrance of the drift tube are a set of four deflection plates
for beam steering and a cylindrical Einzel lens for beam focusing. At the end of the drift tube,
a 40 mm diameter micro-channel plate (MCP) detector (double-stack in Chevron configuration,
Jordan-TOF) produces a TOF spectrum from the ions.
During an experimental run, the process to detect the ions is as follows. First, the RFtrapping field is rapidly quenched over ∼2 RF cycles. Second, after an optimized time delay, highvoltage is pulsed onto the four trap rods (see Fig. 3.4 for more details). This creates a twostage acceleration field in a Wiley-McLaren configuration [72], increasing the mass resolution of
the recorded spectra. Third, the ions pass through deflection plates and an Einzel lens, neitherof which are currently used. Finally, the ions hit the MCP detector creating a current that is
split by a 50/50 RF power splitter (MiniCircuits ZSC-2-1). The currents are converted to voltages
using two separate channels of a fast (1GHz) digital storage oscilloscope. The two channels of
the oscilloscope are set with different vertical scaling to allow for the dynamic range required tosimultaneously measure the large number of calcium ions (25 – 1500) on one channel, while still
having the sensitivity on the other channel to measure lower numbers of (molecular) product ions
(1 – 25).

3.3

Electronics
The electronics for operating a combined ion trap/TOFMS instrument need to perform three

distinct operations : (1) application of RF and small DC fields for ion trapping, (2) rapid quenchingof the RF voltage to create zero electric fields to prepare for ion extraction, and (3) rapid applicationof high DC voltages for ion extraction. First, during the normal trapping mode, the electronics
must produce two RF sine waves at the same frequency (a few MHz), but with a 180◦ phase shift
between the two signals. The amplitude of these signals must be variable up to 250 V. Second, for
optimal extraction of the ions into the TOFMS, the RF fields must be turned off within 1-2 RF
cycles. Otherwise, the RF fields will distort the extraction field, reducing the mass resolution of

33+2.00 KV
+1.84 KV

Quenchtrigger

MCP

HV pulse
trigger-3.5 KVTOFMS trace

Figure 3.4: Left: Schematic representation of the TOFMS. Shown in red and black are the repeller
and extractor configuration of the trap rods for radial ejection of trapped ions into the TOFMS.
Right: The corresponding repeller and extractor voltages as a function of time, required to go fromtrapping to TOFMS. Vertical dotted lines represent the timings of the quench and HV pulse control
triggers.

the TOFMS. Finally, to direct the ions into the drift tube with well-defined starting conditions,two HV pulses with fast rise times and different amplitudes must be applied to the four trap rods
to extract and accelerate the ions towards the MCP.
In the following, we present a simplified version of an electronic circuit to drive an ion trap
TOFMS with the schematic shown in Fig. 3.5. Complete circuit diagrams can be found in App. A.
The electronics are capable of generating two RF potentials 180◦ out of phase with an amplitudeof several hundred volts for a single resonant frequency in the 1-5 MHz range. The RF amplitude
can be quenched within 1-2 RF cycles and two fast HV pulses of different amplitudes up to 2.5 kV
can be produced. Unlike other published implementations of such a combined system, our design
can have an arbitrary time delay between the RF quench and the HV pulses. This preserves theWiley-McLaren condition [72] for the TOFMS.

34
Trap electrodes
DC in

Trap electrodes
DC in+ HV+12V

Trap electrodes

Transformer

RF Switch

RF pre-amp

TTL In+ HV+12V

Buffer capacitance

DC in

OI
OI

MIC

TTL InMIC+12V

RF generator

Quench circuit
MIC

Trap electrodes
DC in

LC tank circuit

Trap electrodes
DC in

HV switch

Trap electrodes
DC in

Figure 3.5: A schematic diagram of the RF drive electronics. The diagram shows the three different
parts (marked by colored boxes) required for ion trapping and extraction into the TOFMS (onlysingle side of bifilar winding is shown).

Our design can be divided roughly into three distinct sections: a LC tank circuit for generating
the RF trapping potentials, a quench circuit for rapidly switching off the RF, and an HV switch
for creating the extraction pulses for the TOFMS. In the following sections, the individual parts ofthe circuit are described in more detail.

3.3.1

LC tank circuit

For driving the ion trap with the required RF field, we use a LC tank circuit, as shown in
Fig. 3.5. The inductor, L, is a hand-wound transformer coil, while the ion trap electrodes and35
SHV cables form the capacitance of the circuit. Low voltage RF is generated by a signal generator(Stanford Research Systems, DS345) and amplified by a home-built RF pre-amplifier before driving
the primary side of a toroidal transformer core (Mircometal, T200-2). For the secondary side, we
use a center-tapped bifilar winding, which is required for the application of the two individual HV
pulses for ion extraction. The center-tap creates two RF signals with 180◦ phase shift. During
ion trapping operation, the center-tap is kept at ground potential and pulsed to HV for mass
analysis with the TOFMS. Each electrode is also AC coupled by individual blocking capacitors.
By choosing the appropriate values of L and C, the RF frequency is set to Ω ≈2π×3.552 MHz. To
apply individual DC voltages on each ion trap segment, a low-pass filter couples the DC to the RFon the trap side of the blocking capacitors.

3.3.2

Quench circuit

Fast quenching of the RF before ion extraction is required for high resolution TOFMS mass
spectra. If the RF is turned off by removing the input signal only, a long ring-down period of several
tens of microseconds will be observed due to the high quality factor of the LC tank circuit. To
reduce the turn-off time to within 1-2 RF cycles, we implement an active damping circuit (quench)
into our LC tank circuit. The high voltage sides of each secondary winding are connected to therespective center-tap by a diode-MOSFET chain. When the MOSFET closes it creates a lowresistance current path, effectively shorting the coil. This stops the RF output in minimal time.
Quenching is implemented on the secondary winding of each side of the center tap simultaneously
to reduce the quench time by nearly a factor of two. Additionally, the input RF is turned off via
an RF switch (Mini-Circuits ZYSW-2-50DR) on the primary side of the coil at the same time as
the quench. This further reduces the turn-off time of the RF potentials on the trapping electrodes.
The arrival time of the ions at the MCP depends sensitively on the phase of the RF cycle
when the quench is triggered, thus we trigger the quench circuit at a zero crossing of the RF signal.
This ensures that the ions always experience similar electric field distributions, reducing RF phasedependent jitter in the mass spectra. The timing of the trigger signal with respect to the zero

36
crossing introduces an overall 20-30 ns jitter in the arrival times of all ions within a single mass
spectrum.
Additionally, because the HV pulses are applied on the center-taps, the quench circuit must
be able to float up to 2 kV together with the coil during ion extraction. To accomplish this, the
MOSFETs (MTP2N50E) are controlled by a driver chip (MIC4429, MIC in Fig. 3.5) triggeredby an optocoupler (6N137, OI in Fig 3.5), where the entire circuit is powered by floating 12 V
rechargeable battery pack (Tenergy NiMH 12V 2000mAh). This configuration allows the circuit tofloat to the required HV during the extraction.
A quick addendum on floating power: Batteries are the ideal floating power supply for thiscircuit because it produces extremely low-noise DC power. DC-DC switch mode power supplies
were used previously but they introduce ∼ 100 kHz noise into the circuit. This noise would find its
way onto the trap rods and excite secular resonances of the trapped ions.

3.3.3

HV switches

After the RF signal is quenched, a fast rise time HV pulse is applied to extract the ions out of
the trap and accelerate them into the drift tube. To produce the HV pulse (2 kV), we use a homebuilt HV switch, shown in Fig. 3.5. The central elements are two MOSFETs (IYXS IXTL2N450)
in a push-pull configuration. The low-voltage side MOSFET is driven by a single MOSFET gate
driver (MIC4429, MIC in Fig. 3.5) and the gate of the high-voltage side MOFSET is driven by a
opto-coupled gate driver (MIC4420, MIC in Fig. 3.5), which is powered by the same battery packas the quench circuit. This floating design is similar to the quench circuit.
Two of these HV switches are used to apply different potentials to the repeller and extractor
rods. The two switches are connected directly to the center-tap of each of the bifilar windings.
During the RF trapping configuration, the high-voltage side MOSFET is open while the low-voltage
side MOSFET is closed, thus providing a ground connection for the center-tap of the secondary
windings.
The output of the combined quench and HV switch circuits is shown in the right half of

37
Fig. 3.4. The two phases of the RF are measured by attaching a pair of 100× oscilloscope probes
to two trap electrodes. Time = 0 indicates the MOSFET trigger to short the coil and switch off
the input signal. The RF amplitude then decays within two RF cycles. We measure a 100% to
10% decay time of 360 ns, which is nearly independent of the initial RF amplitude. We adjust
the time between the quench and HV pulse triggers to optimize the detection efficiency and mass
resolution of our TOFMS. The delay of the HV switch with respect to the quench trigger was
set to 600 ns. The HV pulses on repeller and extractor have identical turn-on characteristics andreach the design amplitude of 2 and 1.84 kV at the same time (the mismatch between objective
and measured amplitudes is due to the oscilloscope probe capacitance). The 10% to 90% rise timeof the HV pulses is measured to be 220 ns and displays a smooth increase in voltage. The rapid
quench and HV pulses coupled with a controllable delay between the the two operations leads to ahigh mass resolution in the TOFMS spectra.

3.4

TOFMS performance
In addition to loading Ca ions into the trap to be laser cooled, we can load in other atomic(or molecular) ions where they will be sympathetically cooled by interactions with the Ca ions.
Using SAES atom dispensers as the source of neutral atoms, we were able to ionize, co-trap, andcool various alkali atoms in our trap, including 39 K+ , 85,87 Rb+ , and 133 Cs+ . These ions serve asa convenient source of different mass ions to be able to calibrate our mass spectrometer. Fig. 3.6
shows a typical single shot (non-averaged) mass spectrum of a multi-component Coulomb crystal
and its corresponding fluorescence signal. Because the mass range of interest is centered around
40 u, we optimized the TOFMS voltages and timings to this region.
The mass spectrum in Fig. 3.6 shows several distinct ion peaks, which can be directly assigned
to the ionized atoms. The peak with the strongest intensity is created by 40 Ca+ . The smaller peakto the left is 39 K+ , while the very small peak on the right hand side of 40 Ca+ is another isotopeof calcium, 44 Ca+ . The relative peak heights of 40 Ca+ and 44 Ca+ represent the natural isotopic
abundance. At higher masses, 85 Rb+ , 87 Rb+ , and 133 Cs+ can be identified. Apart from these38

m/Dm ~1400

K+

85Rb

40Ca

87Rb

Rb+
39K

85Rb

Cs+

44Ca

87Rb

133Cs

Figure 3.6: A single-shot TOFMS mass spectrum (right) of trapped, laser-cooled 40 Ca+ and cotrapped 39 K+ , 44 Ca+ , 85,87 Rb+ and 133 Cs+ . This Coulomb crystal (left) contained 986 ions with
594 of them being 40 Ca+ . Apart from these ions, no further mass peaks can be identified.

masses, no further ions are present. The inset shows the resolving power of the TOFMS by theclear separation of the 85 Rb+ and 87 Rb+ peak.
We determine the mass resolution of our TOFMS and how it depends on the mass of the ion
being detected and on the number of ions in the trap. The mass resolution is defined as R = t/(2∆t)= m/∆m. In the following experiments, the mass resolutions from single, individual mass spectrum
are reported. This is motivated by the planned experiments on measuring chemical reactions,which require the recording of non-averaged mass spectra. For our current experimental triggeringsystem, we have a 20-30 ns jitter in the arrival time between two individual, subsequent mass
spectra. However, we can calibrate each mass spectrum to 40 Ca+ , thus reducing the uncertainty inthe accuracy of the mass identification. The residual uncertainty is 0.01 u for 85 Rb, as an example.
Using the definition of mass resolution from above, every peak in the mass spectrum in Fig. 3.6
has a mass resolution of R>1200, with the exception of 40 Ca+ . The ion number for 40 Ca+ is too

39

1200

Mass resolution

1000
800
600
400
200
0
0

200

400
600
Ion number

800

1000

Figure 3.7: Dependence of mass resolution on trapped ion numbers. Pure 40 Ca+ Coulomb crystalsof different sizes are trapped and detected in the TOFMS. We observe a decrease of mass resolution
with increasing ion number.

great to observe equally high mass resolutions. Coulomb explosion in the drift tube means themass resolution is expected to decrease with increasing ion mass and ion numbers. To investigate
this relationship in our system, we plot the measured mass resolution as a function of ion number
for 40 Ca+ . The results of this are plotted in Fig. 3.7

3.5

Ion number calibration
Mass spectra must be calibrated to allow extraction of absolute numbers of ions present at

each mass for a single shot. We use information from the fluorescence images of the crystals tocalibrate our mass spectrometer. The trapped calcium ions represent a deterministic ion source,
as the number of ions can be directly counted for small crystals using the fluorescence images. To
determine the transfer function of our TOFMS, we start by loading a small crystal (1 –40 ions)
into the trap and counting the number of ions in the resulting fluorescence image via a MATLAB

40
script. Then, we eject the ions into the TOFMS and record a spectrum. We integrate the peakin the spectrum corresponding to 40 Ca+ , which produces a number with units of nVs. The resultsof this procedure are shown in Fig. 3.8. The plot shows a linear relationship between the numberof ions in the trap and the integrated signal from the MCP. The fitted slope of 1.73±0.07×10−3
nVs/ion is then used to determine the number of ions during subsequent reaction experiments.
This calibrated ion number is also valid for larger Coulomb crystals, as long as the MCP showsa linear response with the increasing ion numbers. To test this linearity, we loaded pure 40 Ca+
crystals with different sizes into the trap. As Coulomb crystals have nearly a constant ion density,
we expect a linear relationship between the size of the Coulomb crystal (as determined by the CCD
image) and the integrated MCP signal. From the measured data, we have seen little deviation fromlinear behavior up to 1500 ions (See Fig. 3.9).
Using a single trapped ion, we can also measure the probability of detecting a ion. We extract
a single ion into the TOFMS and observe a signal from the MCP about 38±3% of the time. Thisis the transfer efficiency of the entire system. The largest contribution to the decrease in detection
efficiency is the quantum efficiency of the MCP detector, which is 50–55% [73].
To use the LIT-TOFMS as a platform for studying chemical reactions, one must be able todetermine the initial number of trapped ions before the reactions. Although the number of ionsloaded into the trap is relatively reproducible, there is some variation and a direct measurementof the initial trap number will help to reduce the uncertainty in the reaction measurements. Since
the TOFMS measurements are destructive, we need to use data from the fluorescence images todetermine the initial number of ions. We calculate the number of ions by multiplying the density inthe trap by the volume of the Coulomb crystal. The Coulomb crystal volume is determined by anellipsoidal fit of the CCD image. The density is then found by fitting a line to the measurements of
the integrated MCP signal (calibrated ion number) vs. the volume of the crystal (Fig. 3.9). Fromthese data, we calculate the ion density for our ion trap to be 7.8±0.2×107 ions cm−3 . This density
depends on only the trapping potential and is fixed for any one reaction experiment.

41

Slope of fit:

0.00173

𝑛𝑉𝑠
ion

Figure 3.8: Calibration of the MCP signal to ion number. To calibrate the MCP signal, we startby loading small crystals into our trap and counting the number of ions on the image. Then, byextracting those ions into the TOFMS and integrating the resulting MCP signal, we determine how
the MCP signal relates to the number of ions in the trap. Using this calibration, we can determine
the absolute number of ions in the trap for subsequent experiments.

3.6

Calcium state population calibration
Doppler laser cooling enables quantum control of the electronic degree of freedom in Ca+ . We

can directly manipulate the steady-state populations of the three states in Ca+ accessible by our
cooling lasers (see Fig. 2.3). We utilize this control in the experiment to reveal any quantum-state
dependence in reactions involving Ca+ . In practice, quantum control is accomplished by changing
the cooling laser detunings from resonance. To be confident in the state populations generated bya given pair of detunings, the state populations must be calibrated.
Our calibration procedure is closely based off of the procedure in [1]. We even use their Matlab
code in our analysis (see App. C). Our extension to their procedure begins by identifying the fiveexperimentally controlled variables that affect the state populations and the five free parametersin OBE used to predict the state populations. These variables and parameters, along with their42𝑛 = 8 × 107

ions
cm3

Figure 3.9: Determination of the ion density in our ion trap. We correlate the measured ion number
from the TOFMS to the fitted Coulomb crystal volume taken from the recorded CCD images. Froma fit to these data, we can determine the ion density in our trap. The inset shows a Coulomb crystal
with an ellipsoidal fit.

typical values, are summarized in Tab. 3.2. Since the effects of each of the variables on thestate populations are highly coupled, our setup cannot calibrate each of the five fit parameters
individually. Instead, we collect a lot of data and globally fit all five parameters. The data that arecollected attempts to explore as large of the variable space as possible by only varying one variable
at a time, while holding the values of all the other variables constant. The procedure is essentiallya principle component style analysis.
The dependent variable in the analysis is the coulomb crystal fluorescence intensity as measured by the camera. This is measured by taking a background subtracted image of the coulomb
crystal and summing up the value of every pixel. Since the camera counts are recorded on an
absolute scale, the counts will be proportional to the fluorescence intensity, which in-turn, is proportional to the P1/2 -state population. As long as all the calibration data are taken with the same
Coulomb crystal, a global scale factor between the OBE P1/2 -state and the measured fluorescence43
can be defined. The fit procedure does this automatically, but it will not be the same from crystal
to crystal if the ion numbers are different. This is only a practical concern for calibration. Oncecalibrated, the OBE model calculates the state-populations that apply to any number of ions. It
cannot predict, however, the corresponding fluorescence. An example dataset with global fits generated by the OBE are shown in Fig. 3.10. This fit corresponds to the parameters given in Tab.
3.2.
Table 3.2: Variables used to calibrate OBE fit parameters with typical trap parameters in Tab. 3.1
Experimental variables

OBE fit parameters

397nm laser power

P397

1.7mW

397nm power correction factor

c397

0.6

866nm laser power

P866

13.2mW

866nm effective intensity

c866

1.0

397nm detuning∆397-30MHz397nm effective linewidth

Γ397

10 MHz866nm detuning

∆866

20MHz

866nm effective linewidth

Γ866

8 MHz

866nm polarizationαπ
4B-field magnitude

Bmag

1.9 Gauss

3.7

Gas Handling
The two other major labeled components in Fig. 3.1, not discussed thus far, concern intro-

duction of gases into the UHV chamber. They are the pulsed leak valve and a piezo-transducer
driven (PZT) valve.

3.7.1

Pulsed leak valve

The pulsed leak valve is for delivering a neutral reactant to the trap center. Since the valve
opening does not have a direct line of sight to the ion trap center, gas is delivered by flooding thechamber. The neutral reactant gas is nominally in thermal equilibrium with the UHV chamber(∼ 300K). The pulsed leak valve is the combination of two valves: a high-precision, all-metal leakvalve (Duniway VLVE-2000) backed by a solenoid actuated, 3-way valve (Parker XM30VBG45A).

44397 detuning scan

10

5

0
0

50

866 detuning scan

12

P-State Population (%)

P-State Population (%)

15

100

10

8

6

4
-100

150

-50

0

(MHz)
397
866 power scan

10
8
6
4
2
0
0

5

10

1.6

1.8

8
6
4
2

0.8

1

1.2

1.4

397 nm power in front beam before chamber (mW)

scan

15

P-state population (%)

P-State Population (%)

150

10

0
0.6

15

866 nm power before chamber (mW)
12100397 power scan

12

P-State Population (%)

P-State Population (%)

12

50

(MHz)
866

10
8
6
4
2
0

10

5

0
0

0.5

1

1.5

(rad)

2

2.5

3

0

2

2.5

BG subtracted fluorescence

0.5

1

1.5

108

Figure 3.10: Five calibration data sets to constrain the parameters of the OBE. These five plots
are fit simultaneously (red lines) to determine the p-state population. The sixth plot is the global
scaling of intensity to p-state population. NOTE: the detunings plotted are defined opposite fromthe majority of the literature, i.e.: ∆model = −1 × ∆. This is a holdover from the base code takenfrom [1].

45

Figure 3.11: Schematic diagram of pulsed leak valve setup. The high precision all-metal leak valve
is backed by a solenoid actuated 3-way valve. This arrangement allows pulsed gas admission into
the UHV chamber.

The 3-way valve connects the gas-inlet of the leak valve to either a gas reservoir, or a rough vacuum
pump. The arrangement allows the leak valve to be left at a constant opening. When the 3-way
valve is opened, the inlet of the leak valve becomes pressurized by the reservoir gas, and the UHVchamber is flooded with a small, constant gas pressure. When the 3-way valve is closed, the gas
is pumped away from the inlet of the leak-valve allowing the UHV chamber to return to its basepressure. Functionally, the opening and closing of the 3-way turns on or off the presence of a neutral
reactant in the chamber, thereby pulsing the leak valve on and off, hence the name: pulsed leakvalve.
The chamber reaction time to switching the on/off state of the valve is quite rapid. A steadystate pressure is typically reached in < 2 s. The UHV chamber pressure is consistent enough that
we consider our reactions to take place at a constant density that remains in excess. This is thebasis of the pseudo first-order approximation (see 2.5.1). Fig. 3.12 shows typical valve behavior
with helium gas. The pulsed leak valve behavior deteriorates when gases not chemically compatible
with the 3-way valve are used. In these cases, diluting the gases in a noble carrier gas can return46

PLV with 10 PSIG He

10-9

4.5

4.5

4

UHV chamber pressure (torr)

UHV chamber pressure (torr)

4

3.5

3

2.5

2

1.5

X 48.01
Y 4.337e-09

X 12.55
Y 4.116e-09

3.5

3

2.5

2

X 11.26
Y 1.267e-09

1.5

1Zoom on first peak

10-9

1
0

100

200

300

400

500

600

10

time (s)

20

30

40

50

time (s)Figure 3.12: UHV chamber pressure reacting to opening and closing of the PLV. These tests were
performed with 10 PSIG of pure helium backing the valve. The plot on the right is a zoom in of
the first pulse plotted to the left with markers to highlight the valve opening time.

normal operation of the valve. At time of writing, we are upgrading the 3-way valve to an all-teflon
3-way valve (Valcor SV75P48T-1) to mitigate future chemical compatibility issues.
Liquids with large vapor pressures at room temperature can also be used as neutral reactantsin our experiments. A liquid sample is delivered into the UHV chamber via mixture of its vapor
pressure with a noble gas in the line pressurizing the 3-way valve. Liquids can contain significant
quantities of impurities (typically water) that are removed via freeze-pump-thaw cycles. The liquidsample holder is frozen via submersion in liguid nitrogen, pumped on via a rough vacuum pumpwhile frozen, and then allowed to thaw after closing the connection to the pump. Repeating thiscycle 3-5 times noticeably improves sample purity in our experiments. When gas impurities arepresent, a coil filled with molecular sieve (Sigma-Aldrich 3Å) is added to the line between gas
reservoir and 3-way valve. The coil is externally cooled by a cold bath to freeze out gaseous
impurities. All of our gas lines are stainless-steel or teflon.

47
The constant density of gas is inferred from changes in the UHV chamber pressure as measured by a hot-cathode ion gauge. These gauges are not calibrated in the 10−10 to 10−8 torr regime
we typically operate in. The accuracy of the readings can be off by upwards of a factor of two
or more, depending on the gauge, controller, and gas being measured. This seriously impacts theaccuracy of reaction rate constants measured in the apparatus. However, relative uncertainties
in rate-constants across experiments with the same gas are unaffected. Reaction branching ratios
are also insensitive to inaccurate density measurements of neutral gases. In order to improve theaccuracy of absolute measurements of rate-constants, one will need to overcome the challenge ofmeasuring density at an extremely low absolute pressure.

3.7.2

PZT valve

The method we utilize to introduce many different species of molecular ions into the trap
is via a supersonic beam. A dilute gas mixture is prepared of our molecule of interest in a noble,
atomic gas (typically 2-6% in He or Ar). The gas mixture pressurizes a PZT valve that actuatesa Kalrez O-ring to form a seal with the UHV chamber. 150µs pulses of -200V to -400V are sent
at 10Hz to the PZT valve to open it. The supersonic expansion into vacuum cools the moleculeof interest to ∼ 4K, while giving it a large longitudinal velocity. A molecular beam is formedby skimming the supersonic expansion. The beam exits the skimmer and has direct line of sightwith the ion trap center. At the trap center the gas pulse is intersected by a focused laser pulse
to produce molecular ions that are immediately trapped. Between the ionization laser and themolecular beam, there is a large parameter space that can be explored for loading optimization. A
detailed description of PZT valve behavior and the resulting molecular beam can be found in [74].
In most cases, hundreds of the ion of interest can be loaded in only 10 gas/laser pulses. This islargely attributed to the high density of each gas pulse.
The UHV region between the skimmer and the PZT valve is pumped separately from themain ion trap chamber by a turbopump (Pfeiffer HiPace 80). The small aperture of the skimmer(2 mm) creates differential pumping during a gas pulse. The short pulse durations and differential

48
pumping keep the ion trap chamber pressure low for reaction experiments that occur after ion
loading. These conditions also minimize chemistry between ions produced by the beam and theirneutral precursors. Pulsed molecular beam loading enables us to isolate highly reactive species in
our ion trap for reaction studies (e.g., C2 H2 + and CCl+ ).
Liquids with reasonable vapor pressures at room temperature can also be used for molecularion precursors. The method is identical to loading from the pulsed leak valve. Since a dilute mixture
is ideal for producing a good supersonic beam, many liquid molecular sources are conveniently
accessible.

Chapter 4
Quantum controlled reaction with Ca+

The following sections are two ion-molecule reactions we studied between laser-cooled Ca+
and oxygen (O2 ) and nitric oxide (NO). Both papers take advantage of the quantum control wehave over Ca+ excited state populations. Presented here are the data and write-ups publishedin the literature with edits to the bibliographical references and formatting to match that of thethesis. The abstracts and acknowledgments have been omitted. A short addendum has also been
added.

Ca+ + NO

4.1

This paper [2] was the first reaction studied in the LIT-TOFMS apparatus. In addition to
helping build the apparatus, control software, and analysis code, I took most of the data and wrote
most of the text.

4.1.1

Introduction

Ions and free radicals play important roles in the chemistry of the interstellar medium [5, 75],
in many combustion processes [9, 10], and in planetary atmospheres [7, 8]. Therefore, studying
reactions between these species in the laboratory is useful for developing models of the chemical
dynamics in these environments. In addition to the importance to interstellar medium chemistry,
radical - ion chemistry is interesting to explore, as reactions may proceed in a fundamentallydifferent manner from stable molecule - ion reactions. For example, radical species such as OH,

50
CH, NH may abstract an H atom or other neutral entities from the ion. Alternatively, they mightreadily add to multiple bonds in the ion acting as a classical electrophile, or insert into single
bonds to form 3-centered 3-electron intermediates. Perhaps most significant, reactions between
radicals and ions are predicted to be barrierless [76–79]; therefore rapid and appreciable reactions
can occur at the low temperatures of important astronomical environments, and these likely playa role in the foundations of the chemical complexity of the universe. Consequently, it is imperative
to study these reactions in low-temperature environments, where an exquisite level of detail can beextracted about these fully quantum mechanical systems. Finally, the same level of detail that canbe extracted from low-temperature experiments offers the chance to anchor advanced theoreticaltreatments of molecular phenomena [80–82].
Even with the considerable potential for increasing our understanding of gas-phase chemical
reactions, experimental studies of ion-radical reactions are extremely rare[83–85]. This is because
laboratory measurements of ion-radical reactions have been notoriously challenging to make, due
to the difficulty of creating dense, pure samples of these reactants. Ion densities are limited by
the Coulomb repulsion between molecules. Radicals, in most cases, must be created in situ bybreaking apart stable species, which leads to low density, contaminated samples. Because both ofthese species are created at low densities, the interaction times for experiments that rely on molecular beams or flow tubes are often too short to have a significant fraction of the molecules react.
Acquiring clean samples of radicals in the gas phase and reacting them with ions at low temperatures poses significant difficulties. However, in the last decade, new experimental techniques havebecome available[22, 23, 78, 86–88], which have the promise to overcome some of the experimental
challenges.
Recently, these techniques were used to study ion-neutral (non-radical) reactions at low
temperatures using laser-cooled ions stored in linear traps. Reactions between laser-cooled atomic
ions and neutral species have been demonstrated in several systems[20, 34, 48, 89–92]. Additionally,
reactions using sympathetically cooled molecular ions stored in ion traps have been carried out[25,31, 33, 36, 42, 93]. Recent experiments also investigated the influence of the internal state of the ions51
on the reaction kinetics. In that work, the quantum state of the laser-cooled atomic ion was varied[28, 32, 34, 54, 94–96], while in other measurements, the reactivity of the molecular ro-vibrational
state[34, 48, 52, 53] or the conformer was tested[50].

4.1.2

Ca+ + NO reactions

In this work, we present results from reactions between neutral radical molecules and lasercooled, electrodynamically trapped ions, where the reaction rates are controlled by changing thequantum-state population of the ions. In particular, we study the reactions of laser-cooled 40 Ca+
at millikelvin temperatures with nitric oxide radicals at room temperature. Since the ions arenearly at rest compared to the neutral molecules, the average collision energy is calculated to beĒcoll /kB T∼ 180 K. There are two possible charged reaction products in this system, CaO+ andNO+ (see Eq. 4.1), where kN O and kCaO are the rate constants for producing NO+ and CaO+
respectively.

Ca+ + NO

k

O
−−N−→
NO+ + Ca

(4.1)

k

CaO
−−
−→ CaO+ + N

Subsequent reactions between CaO+ product ions and NO are energetically forbidden and thus notconsidered.
The thermochemistry associated with the reactions between Ca+ and NO was computedwith high-level coupled-cluster theory and the HEAT thermochemical protocol [3]. Using theseenergies, we can determine the exothermic reaction pathways for our system. Laser cooling of
Ca+ simultaneously populates three electronic states: S1/2 , D3/2 , and P1/2 (Fig. 4.1). At zerotemperature, no reactions can occur when Ca+ is in either the ground (S1/2 ) or next lowest-lying
excited state (D3/2 ). However, reactions can proceed when Ca+ is in the P1/2 state. Here, theoxygen abstraction process is slightly exothermic (∆E = −45(10) meV), while the charge-exchange
process is still energetically forbidden by ∆E = +34(10) meV. For reactions at 180 K, Ēcoll ∼ 16

52

Ca+ + NO

NO+ + Ca
+34meV

P1/2

m

6n

397 nm-45meV

86

Δ

CaO+ + N

D3 / 2

S1/2
Figure 4.1: Comparison of reactant and product energies for the Ca+ + NO system. The exothermicity of the reaction depends on the quantum state of Ca+ . Excited states of Ca+ can be populated
by excitation on the two cooling laser transitions at 397 nm and 866 nm. To achieve cooling, the397 nm laser is red-detuned from resonance by an amount ∆. Only Ca+ in the P1/2 state is energetically allowed to react. The charge exchange product channel barrier is overcome by the thermalenergy of neutral NO at 180 K.(Energies not to scale.)meV, this channel will also be possible, though only with molecules whose thermal energy exceeds
the barrier. Control over both reaction rates is possible by changing the time Ca+ spends in theP1/2 state. In our experiment, this control is accomplished by changing the detuning of the primarycooling laser (λ ∼ 397 nm) from resonance (See Fig. 4.1).

4.1.3

Ion trap apparatus

The experimental apparatus used to study ion-radical reactions uses a combination of a linear
Paul ion trap and a time-of-flight mass spectrometer (TOFMS)[38], similar to Refs. [35–37]. The
trap allows for accumulation of charged products over the course of a reaction (i.e., several minutes)
due to the long lifetime afforded by the large trap depth (2 – 4 eV). Accumulation of products over

53
such a long timescale makes measurements of ion-radical reactions possible even for reaction rates
as slow as one per minute. Additionally, this allows for measurements of accumulated product ionsand not just measurements of the depletion of the reactant. This provides the opportunity to map
out simultaneous, competing reaction pathways. Once a reaction is complete, the products canbe identified by time-of-flight mass spectrometry. Using measurements of the number of ions at
each mass-to-charge ratio, the reaction rates and products of the reaction can be determined. The
combination of an ion trap and TOFMS makes studies of very low density samples of ions andradicals possible.
An experimental run begins by non-resonantly photoionizing (355 nm) a neutral calcium beamand loading the resultant ions into the trap, located in an ultra-high vacuum chamber (< 5 × 10−10torr). 40 Ca+ ions are then laser cooled on the S1/2 → P1/2 transition at 397 nm, where the laser is
red-detuned by ∆ = 30 – 100 MHz (Fig. 4.1). Additionally, a repump laser tuned near resonacewith the D3/2 → P1/2 transition at 866 nm is used to pump ions that have fallen into the D3/2 stateback into the cooling cycle. The ions are cooled to a low enough temperature to form a Coulomb
crystal[97]. The fluorescence from the Coulomb crystal is imaged onto an intensified CCD camera,so that the initial, absolute number of calcium ions can be determined. Next, NO gas is introduced
into the vacuum chamber via a precision leak valve at typical concentrations of 107 to 108 cm−3 .
The NO (98.5% purity, Sigma-Aldrich) is passed through an ascarite cell (NaOH:SiO2 mixture)
to remove any residual water and a copper coil filled with molecular sieve (3Å pore diameter)
submerged in a cold bath at -115◦ C to freeze out any N2 O and NO2 . The pressure of NO ismeasured by a Bayard-Alpert style ion gauge and Agilent controller(XGS-600). The large ion trap
depth ensures that all charged products will remain trapped and have time to sympathetically coolvia interactions with the laser-cooled atomic ions. Once the desired reaction time has elapsed, thetrapping potentials are rapidly turned off (< 500 ns) and high potentials (∼ 2 kV) are pulsed onto
the trap rods to eject ions radially into the TOFMS [38]. The ion current from the microchannel
plate detector is converted into a voltage and recorded by a fast oscilloscope (1 GHz). Arrival times
and integrated peak areas in the TOFMS spectra are then used to determine the absolute number

54

Normalized Ion Number

0.15

NO+
CaO+
0.10

0.05

0

50

100

150

200

250

Reaction Time (s)

300

350

Figure 4.2: NO+ and CaO+ measured ion numbers as a function of time. The ion numbers arenormalized by the initial number of Ca+ determined from the image captured at t = 0. The error
bars represent the standard error of the mean of measured ion numbers from five experimental
runs at each time point. The solid lines indicate fits to the data using a pseudo first-order reaction
rate model. Also shown are three false-color fluorescence images of the Coulomb crystals taken
immediately before ejecting the ions into the TOFMS. The images were taken at t = 50 s, 200 s,
and 320 s. The dark bands in the center and along the outer edges of the crystals indicate thepresence of ions that are lighter and heavier than Ca+ respectively. These ions are the reaction
products.

of ions at each mass-to-charge ratio[38].

4.1.4Reaction Rate Measurements

Reaction rates are determined by repeating experimental runs at different reaction times,
while holding the neutral radical concentration and Ca+ P1/2 state population fixed. An example
reaction curve is shown in Fig. 4.2. Here, the ion numbers are normalized to the initial numberof Ca+ , which is determined from the fluorescence image taken at time t = 0. Example images of
the crystals after reactions with NO are shown at three different times along the reaction curve.

55

Reaction Rate (10-3 s-1 )

1.5

NO+
CaO+
1.0

0.5

0

5

10

15

20

P1/2 State Population (%)

25

Figure 4.3: Measured reaction rates for the two product channels, NO+ and CaO+ , as a functionof Ca+ P1/2 state population. The NO concentration was 2.3(1)×108 cm−3 for all of these data.
The initial number of Ca+ ions in the trap was around 600. The slopes of the fit lines are used todetermine the 180 K reaction rate constants. The statistical uncertainty in the P1/2 state populationis smaller than the points. There is a an offset for the reaction rate at zero excited state population.
However, it corresponds to only a 1 – 2% offset in the P1/2 state population, which is likely due toa small systematic in determining the absolute state population from the florescence.

The dark bands in the center and along the outer edges of the crystals indicate the presence of
non-fluorescing product ions. The solid lines are fits to the data using a pseudo first-order reaction
rate model for each reaction. Explicitly, the model is given by
NO+ (t)kN O=
1 − e−kN O [N O]fP t+
kN O + kCaO
Ca (0)
+CaO (t)kCaO−kCaO [N O]fP t
=
1
−
e
,
kN O + kCaO
Ca+ (0)

(4.2)
(4.3)

where the number of NO+ and CaO+ product ions at a given time, t, are normalized by the numberof calcium ions at t = 0. [N O] is the concentration of neutral NO molecules, and fP is the fractionof Ca+ in the P1/2 state.

56
To demonstrate control of the reaction rates by changing the quantum-state populations of
the Ca+ , we adjust the 397 nm laser detuning between measurements. The detuning controls thefraction of time an ion spends in the excited P1/2 state, and thus its reactivity. The fraction of timeis determined experimentally with the help of the optical Bloch equations [54]. First, we integrate
the intensity of the florescence of the entire pure Ca+ crystal image from the CCD camera as afunction of detuning. Next, we fit these data to the three-level optical Bloch equations. Using thisfit, we can calculate the fraction of time the atomic ions spend in the P1/2 state based on a single
image of the crystal at a known laser detuning. Since the lifetime of the excited state is on theorder of a nanosceond, one can model this excitation as a steady-state system, where the fractionof time in the excited state is equal to the fraction of ions that can react.
The results of the quantum-state controlled reaction can be seen in Fig. 4.3. Here, we keep
the NO concentration fixed at 2.3(1)×108 cm−3 and vary the fraction of ions in the excited P1/2
state. We can vary the reaction rate by up to a factor of five. In our experiment, we were limitedin the range of accessible P1/2 state populations on the lower end by the decreased cooling rate,
which leads to loss of ion crystallization, and on the upper end by the maximum fraction of ionsthat can be in the P1/2 state of the three-level system, assuming the repump laser is on resonance.
In total, we are able to vary the population between approximately 5 and 25%.
We were also able to extract branching ratios and reaction rate constants for the charge
exchange channel, kN O , and for the oxygen abstraction channel, kCaO , using the slopes of thelinear fits and the fixed NO concentration (see Table 4.1.) Note that the NO concentration was
estimated from an ion gauge reading with the gas correction factor for NO. To explore a possible
systematic effect of the pressure measurement, we completed an additional experiment to determine
the rate constants where any systematic effects likely contribute in different ways.
The second experiment measured the reaction rates for producing NO+ and CaO+ at a fixedlaser detuning, while varying the NO concentration. The results of these measurements can be seenin Fig. 4.4. Here, we varied the pressure between 3.4×10−9 and 1×10−8 torr with a fixed fractionof Ca+ in the P1/2 state of 19.1(1.8)%. The lower end of the range was chosen to be well above the57

Reaction Rate (10-3 s-1 )

1.5

NO+
CaO+

1.0

0.5

0.0

0.5

1.0

1.5

2.0

8

2.5
-3

3.0

NO Concentration (10 cm )Figure 4.4: Measured reaction rates for product channels NO+ and CaO+ as a function of neutral
NO concentration in the chamber. The NO pressure in the chamber was varied between 3.4×10−9
and 1×10−8 torr to change the NO concentration. For all of these data, the P1/2 state population
was 19.1(1.8)%. The slopes of the fit lines correspond to the 180 K reaction rate constants.

background pressure before introduction of NO, and the upper end of the range was chosen suchthat the ions still formed a well-resolved Coulomb crystal. Once again, the rate constants fromthese measurements were determined by slopes from linear fits to the data and the excited state
population calculated from the fluorescence images (Table 4.1). The rate constants and branchingratios from the two experiments agree to within statistical uncertainties. This agreement gives us
confidence in the limited impact of any uncontrolled systematic effects.

4.1.5

Discussion

Typical predictions of rate constants using the Langevin capture model do not apply in thissituation. This is because one of the channels is endothermic and involves a charge exchange process,neither of which are well characterized by the simple model. Instead, we present a mechanistic

58

Figure 4.5: Generalized valence-bond (GVB) diagrams that illustrate the processes whereby 2 P
calcium atoms react with NO molecules to form CaO+ and nitrogen atom in their ground states(top); and the charge exchange process that yields Ca atoms and ground state NO+ (bottom).

model for the reactions. Likely mechanisms for the slightly exothermic chemical reaction and thecharge exchange process are shown in Fig. 4.5. The calcium ions and the NO molecules can
combine in four distinct ways, involving A0 and A00 electronic states of either singlet or triplet
multiplicity. Interaction along the 3 A00 pathway correlates with the production of a ground statenitrogen atom and the CaO+ molecule, while interaction along a 1 A0 is consistent with the observed
charge exchange process. A detailed theoretical characterization of the 3 A00 and 1 A0 intermediates
is desirable and is the focus of extensive future work. However, the proposed mechanism is plausible
and consistent with the observed products.

59
Table 4.1: Table of results for the two independent determinations of reaction rate constants andbranching ratios. The listed uncertainties represent the statistical uncertainty.
NO concentration

P1/2 fraction

2.3(2) × 10−11

2.7(3) × 10−11

4.4(7) × 10−127(2) × 10−12

BRN O

0.8(2)

0.8(1)

BRCaO

0.21(5)0.16(3)

cm3
)
s
3
cm
)
kCaO (skN O (

4.1.6

Conclusion

In conclusion, we have demonstrated quantum-state controlled reactions between laser-cooled
Ca+ ions and neutral NO radicals. We utilized an ion trap to accumulate the charged reaction
products over the course of the reaction. By coupling a time-of-flight mass spectrometer to theion trap, we were able to make simultaneous measurements of the product channels. Using thistechnique, we made two independent measurements of the reaction rate constants and branchingratios for NO+ and CaO+ products at 180 K. A realistic mechanistic model for the two productchannels was also presented. In the future, reactants of other important radicals, such as OH andCH can be introduced to the trapped ions while controlling both the internal ro-vibrational states,
as well as the collision energy through the use of a Stark decelerator. These studies will allow forcomplete control over the reactants, and thus lead to a new level of understanding of radical-ion
reactions.

4.2

Ca+ + O2This paper [39] was the second reaction studied in the LIT-TOFMS apparatus. In addition

to helping build the apparatus, control software, and analysis code, I had to retake all of the data
and redo the analysis upon realizing we needed to use an 8-level OBE model. After this, I helped
rewrite the relevant sections of the paper, which ultimately did not have an impact on the overall

60
story.

4.2.1

Introduction

The study of cold, controlled molecular reactions and collisions in the gas phase is a steadily
growing field, offering many opportunities for detailed studies of interactions [98–101]. Several
different experimental pathways are currently used to explore cold molecular interactions withneutral molecules. These include, but are not limited to, crossed-beam experiments using a Stark
decelerator [102, 103], merged-beam experiments [104, 105], co-trapped atoms and moleucles [106,107], buffer-gas cooled molecules inside magnetic traps [108], trapped Rydberg molecules [13],laser-cooled molecules [109–113] and ultracold bi-alkali atom molecules [114, 115]. Another set ofexperiments uses laser-cooled atomic ions confined in radio-frequency (rf) traps as reactants, or asa cold bath to sympathetically cool molecular ions for reaction studies [23, 116].
While most of these experiments benefit from reducing the translational motion via coolingto temperatures below T≤1 K, controlling the internal states of the reactants can offer additional
insights into understanding the reactions. Preparing the reactants in a single quantum state [117–120] could elucidate the state’s influence on reaction pathways. There have been some experimentsthat explore cold reactions where the internal states of the atomic reactant have been controlled
including, but not limited to, Ca+ + O2 [121], Ca+ + H2 [94], Ca+ + H2 O [34], Ca+ + Rb [91],Yb+ + Rb [28], Ba+ + 87 Rb [96], or Ca+ + (CH3 F, CH2 F2 , or CH3 Cl) [54], Ca+ + 6 Li [122], Yb++ 6 Li [123], and Be+ + H2 O [45]. These experiments take advantage of the control of electronic
states of the atomic ion using cooling lasers. However, preparation of pure state-selective molecularsamples is significantly more challenging [13, 50, 52].
Here, we explore the reaction between laser-cooled Ca+ ions and neutral O2 , where thereaction can proceed only when Ca+ is in one of two excited electronic states (2 P1/2 or2 D3/2 ). We
control the relative populations in the different quantum states by the wavelengths of the coolinglasers. Although the reaction of Ca+ + O2 has been observed previously [121], no details on thereaction kinetics or the influence of the quantum states of Ca+ have been reported. In this work, we

61present a detailed study on the reaction Ca+ + O2 → CaO+ + O, including the absolute reaction
rate constants for reactions starting with Ca+ in either the 2 P1/2 -state or 2 D3/2 -state.
In this paper, we begin by discussing the energetics of the reaction (Sec. 4.2.2) before discussing the experimental setup (Sec. 4.2.3). Next, we present results of the reaction measurements
and the dependence of the rates on the quantum state of Ca+ (Sec. 4.2.4). We then discuss theimplications of our results (Sec. 4.2.5), and conclude by giving an outlook for future experiments(Sec. 4.2.6).

4.2.2

Ca+ + O2 Reactions

In the experiments presented here, we show measurements of the reactions between Ca+ ions,which have been laser-cooled to millikelvin temperatures, and O2 molecules at room temperature.
This leads to an average collision energy of ∼ 160 K. This estimation neglects the effect of the
micromotion on the collision energy. Based on the images of our ion crystals and previously
published values for the temperature of similar sized crystals [49], we estimate an upper limit
to the effective kinetic temperature from micromotion to be <Eef f >/kB < 10 K. Although theinclusion of micromotion slightly increases the effective kinetic temperature of the ionic reactants,
its value is still negligible compared to the energy of the O2 molecules.
The reaction of Ca+ with O2 is highly endothermic (+1.7 eV) when Ca+ is in the ground2S
1/2 -state [124], but becomes feasible when the excited states of the atomic cation are involved.

With the 2 P1/2 state, the reaction is exothermic, and for 2 D3/2 , it is thermoneutral to within theaccuracy of a high-level calculation. The numbers here are based on a thermochemical cycle thatinvolves the bond energy (4.202 eV) and ionization potential (6.895 eV) of CaO, the bond energyof O2 , and the ionization potential and relative cation energies of the Ca atom. The numbers
associated with CaO are from high-level theory, and are taken from a previous study [38], while
the others are precise and well-established numbers from the literature.

Ca+ ( 2P1/2 ) + O2 −−→ CaO+ + O∆E = −1.43(2) eV

(4.4)

62

Figure 4.6: Diagram of the Ca+ energy structure and product channel energy (energies are notto scale). The main laser cooling transition for Ca+ at 397 nm excites the ion to the 2 P1/2 -state.
The ion can decay to the ground 2 S1/2 - or the excited 2 D3/2 -state. A second laser at 866 nm
repumps population that accumulates in the dark 2 D3/2 -state back into the cooling cycle. Here,∆397 and ∆866 represent the detuning of the cooling lasers with respect to the two cooling transition
frequencies. The reaction with O2 is energetically possible in either of the two excited states, but
not in the ground state.

Ca+ ( 2D3/2 ) + O2 −−→ CaO+ + O∆E = +0.01(2) eV(4.5)

We note that the endothermicity of the 2 D3/2 -state reaction is comparable to the uncertainty inthe calculated energy. However, the collision energy is observed to be sufficient to counter thisendothermicity, if indeed it is present. Thus, the reaction rate for Ca+ + O2 will depend on the
population of both excited states of Ca+ .
We control the relative populations of Ca+ in its two excited states by adjusting the detuningof both the main cooling laser at 397 nm and the repump laser at 866 nm (Fig. 4.6). To determine
the fraction of the time the Ca+ spends in each of its three electronic states, we use the 8-level
optical Bloch equations (OBE) calibrated to the magnitude of the fluorescence from the Coulomb

63

100
2

S 1/2

2

D3/2

2

P 1/2

Quantum state population (%)

90
80
70
60
50
40

V

IV

III

30
20
10
0
-80

-70

-60

-50

-40

397

-30

-20

-10

0

(MHz)

Figure 4.7: Predictions of the state distribution of the Ca+ ion from the 8-level OBE with a fixed
866 nm detuning (+42 MHz). The state populations for the 2 S1/2 , 2 P1/2 , and 2 D3/2 states areshown. The vertical lines represent the parameters used for three measurements used to determine
the rate constants presented in Section 4.2.4 (see Table 4.2). To obtain the other state distributions
shown in Table 4.2, we also changed the 886 nm laser detuning.

crystal following the method described in [54]. Once calibrated, the 8-level OBE are used to
predict the populations in each state based on the detunings of both lasers used for laser cooling.
An example set of curves for the three states is shown in Fig. 4.7. Here, the repump laser detuning
is fixed, while the main cooling laser detuning is varied.

64
4.2.3

Ion Trap Apparatus

Details of the experimental setup for measuring reactions between cations and neutrals havebeen described recently [2, 38]. In the following, we present a short description of the apparatus
with an emphasis on aspects relevant for the present experiment.
Ca+ ions are created from non-resonant photo-ionization of calcium atoms at 355 nm (7mJ/pulse, 10 Hz, beam diameter ≈ 1 mm2 ) and trapped using a segmented, linear quadrupole ion
trap [38]. The trapped Ca+ ions are laser cooled [125] to secular temperatures close to the Doppler
cooling limit resulting in the formation of Coulomb crystals [97]. Laser cooling is performed on the
2S2
1/2 - P1/2 transition, driven at 397 nm by a fiber-coupled diode laser (NewFocus, 3.5 mW, 600µm beam diameter). A second fiber-coupled diode laser at 866 nm (NewFocus, 9 mW, 2 mm beam
diameter) is used to repump the ions back into the cooling cycle on the 2 P1/2 - 2 D3/2 transition.1
Both laser frequencies are measured and locked using a wavemeter (HighFinesse/ Ångstrom WSU30) with a precision of ± 2.5 MHz around the center frequency. The wavemeter is calibrated daily
to a 780 nm laser, locked to a transition in 87 Rb by saturation absorption spectroscopy.
Detection of the trapped ions is done by two different methods. We can image the fluorescencefrom the Coulomb crystal onto an intensified CCD camera or eject the ions into a time-of-flight mass
spectrometer (TOF-MS), which is radially coupled to the linear ion trap [38]. Optical detection
has the advantage of being non-destructive, but is sensitive to only Ca+ . Non-fluorescing ions canbe inferred only by a change in the shape of the Coulomb crystal in combination with results frommolecular dynamics simulations [126]. In the present experiment, we use the optical detection ofpure Ca+ crystals to determine the initial number of trapped atomic ions by fitting the area of thecrystal and knowing the density of ions in the trap [38]. To accurately determine the number of all
trapped reactant and product ions, we eject all ions in the trap into the TOF-MS. This allows us
to determine the absolute number of ions at each mass in a single shot [38]. Thus, we can study
the reactions by the increase of products, as well as by the loss of reactants.
1 40
Ca+ has a second possible cooling scheme, which uses the 2 S1/2 - 2 P3/2 transition for cooling and the 2 P3/2
- 2 D5/2 transition for repumping. The two transition wavelengths — 393 nm and 854 nm — are not within the
bandwidth of our laser systems. We therefore neglect these electronic states for the scope of this work.

65
An experimental sequence starts by loading a Ca+ Coulomb crystal into the ion trap andremoving any impurity ions by varying the trapping fields. Next, an image of the crystal is taken to
determine the initial number of Ca+ ions. At this point, O2 is introduced into the chamber througha leak valve for a predetermined length of time. After the set reaction time has been reached, alltrapped ions are ejected into the TOF-MS and a mass spectrum is produced. The total number ofions (product + reactant) is also determined from the full mass spectrum [38] at each time point
to ensure the number of ions remains constant during a reaction measurement and ions are notlost from the trap. To test the conservation of ions in the trap, we sum the number of ions inevery mass channel and compare this sum at each time step in the reaction to the initial number of
Ca+ ions. This check is especially important when measuring the highly exothermic, excited-state
reaction of Ca+ + O2 , where the charged products may have > 1 eV of kinetic energy.
There is one notable improvement to the apparatus compared to our previous work [38],which concerns how we introduce the neutral reactant. Instead of having the O2 present during thetrap loading sequence, we now implement a pulsed-leak valve [127]. The valve consists of an allmetal leak valve (Kurt Lesker LVM series) with a 3-way solenoid valve (Parker Hannifin) connectedto the input port. The 3-way valve connects the input port of the leak valve to either a roughingpump or a bottle of O2 diluted to 60% in N2 . In this configuration, the leak valve is held open
at a desired leak rate at all times and the 3-way valve controls whether or not O2 flows into thechamber. Using the 3-way valve to initiate the introduction of O2 into the trapping chamber, weachieve a stable background O2 pressure in around 1−−2 seconds. This time sets the uncertainty
in the initial time for the reactions.We note that the pressure of O2 was measured using an ion
gauge, which can give readings that are accurate to within only a factor of 2-3. This systematic
will not affect the relative rates between the two states, however.
Although care is taken to introduce pure O2 gas into the vacuum chamber, there is always thepossibility that contaminants can be entrained in the gas flow. Since all trapped ions are detectedin the TOF-MS spectrum simultaneously, we can measure any background reactions between Ca+
and H2 O (the dominant contaminant), which lead to the production of CaOH+ . Therefore, we can

66
account for parasitic background reactions in our modeling of the reaction rates.

4.2.4

Reaction Rate Constant Measurements

We model the reactions in our experiment using pseudo-first-order kinetics, where we assume
that the density of O2 does not change during a reaction measurement. We model the system byincluding reactions that start with the Ca+ in the two excited states. Equation (4.6) describes thegrowth of the charged product, CaO+ .
−keff [O2 ]t[CaO+ ] = [Ca+)0 ](1-e(4.6)

Here, [CaO+ ] is the number of CaO+ ions as a function of reaction time, t, [Ca+0 ] is the initial
number of trapped Ca+ ions, kef f is the effective reaction rate constant, and [O2 ] is the oxygen
concentration in the chamber. The effective reaction rate constant, kef f , is given by
kef f = kP fP + kP fD ,(4.7)

where kP (kD ) is the rate constant for reactions with Ca+ in the P (D)-state, and fP (fD ) is thefraction of the Ca+ ions in the P (D)-state.
To extract the quantum-state dependent reaction rate constants, we first measure kef f atseveral different state populations. We accomplish this by varying the detunings of the coolinglasers, while maintaining a constant O2 concentration. An example of one such rate measurement
is shown in Fig. 4.8. With our experimental parameters, the 2 P1/2 -state population can be varied
between 4% and 16%, while the 2 D3/2 -state range is between 8% and 50%. We are not able
to work outside this range, as either the heating rates become too large such that we can not
maintain a stable Coulomb crystal due to the energy imparted by collisions with the backgroundgas or the population is not possible due to the coupling in an eight-level system. A summaryof the determined kef f for the different state populations is given in Table 4.2. We perform a
two-dimensional fit to these data using Eqn. 4.7 as the model to extract two fit parameters, kP
and kD . The data and the fit can be seen Fig. 4.9. The measurements are shown as diamond

67

Normalized CaO+ Number

0.15

0.1

0.05

0
0

10

20

30

40

50

60

Time(s)Figure 4.8: Example reaction measurement showing the growth of CaO+ product ions, normalized
by the initial number of Ca+ ions (∼ 800). The fraction in the 2 P1/2 -state is 0.16 and 2 D3/2 -stateis 0.22, with an O2 density of 8.9×107 cm−3 . The solid line is a fit using Eqn. (4.6), where kef f
is the only free parameter. The error bars correspond to the standard error of the mean of 7
measurements at each time point.

points, where the color represents kef f . The fit is shown as contour lines of constant kef f . The
rate constants from the fit are given in Table 4.3. The rate constant for Ca+ in its 2 D3/2 -state isabout three times lower than 2 P1/2 -state reactions.

68
Table 4.2:

Measured reaction rate constants for pairs of Ca+ excited state populations shown in

Fig. 4.9. The fractional populations in the 2 P1/2 -state (fP ) and 2 D3/2 -state (fD ) are shown withthe corresponding measured effective rate constant (kef f ). The data point numbers correspond tomeasurements shown in Fig. 4.9.
Data pointfP

fD

kef f (10−11 cm3 s−1 )

I

0.13

0.32

4.1(7)

II

0.16

0.22

3.5(4)

III

0.15

0.28

4.1(6)

IV

0.08

0.15

2.3(1)

V

0.04

0.08

1.4(1)

VI

0.07

0.50

3.6(4)

Table 4.3: The resulting rate constants kP (kD ) for the three quantum states of Ca+ . Uncertainties
are from the 90% confidence interval for the fit parameters, kP (kD ).
Ca+ statek (10−10 cm3 s−1 )

2S
1/2

0

2P

4.2.5

1/2

1.8(4)

2D
3/2

0.5(2)

Discussion

There are two possible explanations as to why the rate constants for the different quantum
states are not the same. The first, and most likely, of these is related to the energy of the 2 D3/2 state. The calculated energy of the Ca+ (2 D3/2 ) + O2 entrance channel is coincident, to within
the estimated uncertainty, to that of the CaO+ + O outgoing channel. If the reaction is indeed

10
4

VI

4.

2e

-1

0.45
0.4

4.

1

3e
-1
1

2.
4e
-1

69

8e

-1
1

3.
6e
-1
1

1

3.5

0.35

I

4.

2e

1.
8e

-1
1

0.25

3e
-1
1

2.

III

3.

6e

4e
-1
1

-1

1

II

1.
2e

0.2

3

-1
1

0.3

2.5

keff cm3/s

Ca+ 2D3/2-state fractional pop.

0.5

-11

-1
1

0.15

IV
1.

0.1

4e

-1

1

2

0.05

2

-1
1

2.

1

V

6e
-1

3e

8e
-1

1.5
0
0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Ca+ 2P1/2-state fractional pop.
Figure 4.9: Dependence of the effective rate constant, kef f , on the 2 P1/2 - and 2 D3/2 -state populations of Ca+ . The experimental measurements are shown as diamonds, where the color of thepoint represents the value of kef f . For all data shown, the O2 concentration was kept constant at
8.9×107 cm−3 . We perform a two-dimensional fit to the data using Eqn. 4.7, which allows us to
extract kP and kD . The contour lines represent constant kef f values of the fit.

endothermic by an amount comparable to the collision energy, then only a portion of the Boltzmann
distribution of the O2 molecules have enough energy to react. Assuming the ions are at rest andthe O2 is at 300 K, the estimated collision energy is 160 K. To have a reduction of the rate constantby a factor of three (i.e., only one-third of the collisions have enough energy to react), an energy
barrier of ca. 15 meV would have to exist. Given that this is in quantitative agreement withthe calculated energy, this is an entirely plausible explanation for the observations. However, theuncertainty in the calculation includes the possibility that the reaction is actually exothermic, so

70
one needs to be conservative in making such an assertion.
If the possible endothermicity of the reaction involving Ca+ (2 D3/2 ) is not the explanation
for the slower rate, then a mechanistic rationalization is needed. The association of the calcium
ion and O2 will involve some sort of intermediate complex, and thus it is necessary to consider thefull triatomic potential energy surface. Given that the reaction involves an excited state calcium
atom, the lowest electronic states of “CaO+
2 ” lay well below the energy of the entrance channel,
and a reaction pathway involving many highly excited electronic states is likely. While a detailed
theoretical study of such a process is well beyond the scope of this paper, some indication of thestructures and electronic states involved can be gleaned from Fig. 4.10. This displays calculated
energetics obtained with coupled-cluster theory based on single, double, and perturbative triple
excitations (CCSD(T)) [128, 129] together with the cc-pCVTZ basis set [130]. This level of theory
is somewhat less rigorous than that used to calculate the reaction energetics documented earlier,differences are likely small (i.e., +0.01 eV for the effectively thermoneutral channel v. -0.02 with
CCSD(T)/cc-pCVTZ) and the results are expected to be accurate to within ca. 0.02−−0.04 eV forthe states shown in the figure. Only the lowest doublet and quartet states of various symmetries
are shown, all of which are both bona fide minima on the potential energy surface and qualitatively
described by a single Slater determinant (thus treated well by CCSD(T)).
Figure 4.10 shows that the association of Ca+ and O2 can potentially lead to states belonging
to several manifolds of spin multiplicity and spatial symmetry, the lowest of which lies very far
below the association asymptote. Favorable association of the 2 P1/2 and 2 D3/2 states of the atomic
calcium ion with ground state O2 correlates with 2 A2 and 2 B2 states of the C2v intermediate,
where the former arises from pairing of in-plane electrons (forming a σ bond) and the latter froma π interaction. The lowest states of these symmetries will correlate adiabatically with Ca+ (2 S)+ O2 (3 Σ), with higher-lying 2 A2 and 2 B2 states correlating adiabatically with the Ca+ (2 P1/2 ) or2D32
3/2 + O2 ( Σ) reactants. The minimum-energy path for the approach of the P1/2 state will be in

Cs symmetry on excited 2 A’ and 2 A” surfaces that correlates with 2 B2 and 2 A2 states of the favoredcyclic structure, respectively; the approach of the 2 D3/2 -state atomic cation is symmetry allowed in71

Figure 4.10: An energy-level diagram featuring electronic states of reactants, intermediates, andthe observed CaO+ + O(3 P) product channel. All energies calculated at the CCSD(T) level, apartfrom those levels involving excited states of molecular oxygen, that use experimental excitationenergies.

C2v . Although the excited states that adiabatically correlate to reactants have not been calculated
in this cursory exploration, it is likely that rapid internal conversion will produce vibrationally hotcomplexes associated with the lowest two electronic states (Fig. 4.10) that can then dissociate
to observed products. Dissociation of complexes associated with these cyclic geometries would
certainly encounter a barrier, but the complex will form with roughly 4−−5 eV of internal energy,
which should be sufficient to surmount it. However, the possibility remains that the barriers forthese processes are sufficiently great that they are above the entrance channel associated with
Ca+ (2 D3/2 ). If so, then another state of the complex that might come into play is the 2 B1 state

72of the open form, which likely dissociates without a barrier. In such a scenario, the lower-energy
entrance channel would react by a completely different pathway, and perhaps at the slower rate
observed experimentally.
Note that the possibility discussed above is based on a very simple thermodynamic observation (the fact that the 2 B1 intermediate lies below products) and the assumption that its dissociation
to CaO+ and atomic oxygen is barrierless. Taking a theoretical investigation of the reaction process
beyond this would involve a significant amount of effort; it is perhaps more productive to focus on
reducing the uncertainty for the calculated reaction energy for Ca+ (2 D3/2 ) + O2 , the establishedendothermicity of which could successfully account for the difference in rate.

4.2.6

Summary

We measured the ionic products from the photo-induced reaction of Ca+ + O2 → CaO+
+ O using laser-cooled and trapped samples of Ca+ ions. The product ions were trapped andsympathetically cooled by the atomic ions and detected using a TOF-MS. The reaction is energetically allowed only when the Ca+ is in one of two electronically excited states (2 P1/2 or 2 D3/2 ). By
adjusting the detunings of the cooling lasers, the quantum-state population of the Ca+ ions canbe controlled. Using this control, we were able to determine reaction rate coefficients for reactions
starting in either the 2 P1/2 - or 2 D3/2 -state. The 2 P1/2 -state rate constant is a factor of three larger
than for the 2 D3/2 -state. We propose two possible explanations for this difference. The 2 D3/2 -state
reaction might be endothermic by a very small amount or, perhaps, proceeds through a different
mechanism involving the 2 B1 state of the charged CaO2 intermediate.
In future work, we plan to extend these types of measurements to more complex systems,
such as reactions with organic molecules, and with additional control over the neutral reactant
using a Stark decelerator. We will attach a traveling-wave Stark decelerator [131, 132] to the ion
trap apparatus, similar to the experiments of the Oxford group [133] and the recent proposal by
the Willitsch group [134]. This will allow us to tune the velocity of the neutral reactant down
to 10 ms−1 , enabling studies of ion-molecule reactions below 10 K. A Stark decelerator can also

73
produce molecules in a single quantum state, thus enabling studies of the influence of the internal
molecular states on the reaction. Ultimately, by introducing state-selective molecular ions into
the trap [135, 136], fully quantum-state resolved molecular reactions at cold temperatures will bepossible.

4.2.7

Ca+ + H2 O

In order to accurately measure the state-dependent rate constants for Ca+ + O2 , we neededto account for a parallel reaction occuring: Ca+ + H2 O −−→ CaOH+ + H. Water is the largest
contaminant in our vacuum chamber and this reaction can be observed even when the chamberpressure drops below 5 × 10−10 torr. As an added bonus to the published reaction, we measured thestate-dependent rates of Ca+ + H2 O. We cannot report the rate constants because we do not know
the absolute concentration of water in the chamber. Instead, we measure the associated reaction
rate and its dependence on Ca+ quantum-state populations. In doing so, we verified the curious
result reported in [34], where the reaction rate is dramatically faster for Ca+ in the d-state than
the rate is for Ca+ in the p-state. This is shown in Fig. 4.11, which is a state-dependent rate fit for
Ca+ + H2 O. The rates themselves are summarized in Tab. 4.4. Their relatively large uncertainties
come from the fact that tiny numbers of CaOH+ ions were created from the parallel reaction. The
CaOH+ signal was typically 10 times smaller than the corresponding CaO+ signal.
Table 4.4: The resulting reaction rates for the three quantum states of Ca+ . Uncertainties are fromthe 90% confidence interval for the fit parameters.
Ca+ state

rate (10−3 s−1 )

2S
1/2

0

2P
1/2

1(4)

2D
3/2

2(2)

74
10-4
0.001214

5VI0.0012

5

0.45

0.001

12

0.001

0.4

0.001

0.35
0.0007

10

I

5

0.3

0.0007

5

III

0.25

0.00078

5II0.00050.2

rate s-1

Ca+ 2D3/2-state fractional pop.

0.5

0.0005

0.15

6

IV

0.00050.0002

0.1

5

V

4

0.000250.05

0.0002

5

0
0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Ca+ 2P1/2-state fractional pop.
Figure 4.11: Dependence of the reaction rate on the 2 P1/2 - and 2 D3/2 -state populations of Ca+ forthe reaction Ca+ + H2 O. The experimental measurements are shown as diamonds, where the colorof the point represents the value of the rate. We perform a two-dimensional fit to the data, whichallows us to estimate the state dependent rates. The contour lines represent constant reaction rate
values of the fit.

4.2.8

Addendum

During the publication process of this paper, we were alerted to the fact that a 3-level OBE
model does not accurately reflect the state-populations of calcium. In the presence of a magnetic
field, the energy levels of Ca+ split into 8 distinct Zeeman levels and thus an 8-level OBE must
be used. This introduces two new variables that were present but not measured in the original
experiment: the magnetic field experienced by the crystal, and the 866nm repump laser polarization

75

Figure 4.12: Calibration data used to predict calcium p-state populations by coulomb crystal
fluorescence. The red line is the output of the 3-level OBE model used in [2]. Retroactive use of
the 8-level OBE is not possible and thus the y-axis scaling of the p-state is too large by as muchas a factor of 2.

angle with respect to the magnetic field determined quantization axis. These were measured andapplied correctly for the Ca+ + O2 study [39], but not for the Ca+ + NO study [2]. Unfortunately,
we cannot go back in time to measure the two variables required and retroactively apply the 8-level
OBE. Fortunately, it will not have much of an impact on the results reported in [2]. The 3-level
OBE still qualitatively predicts the excited state population of calcium as a function of coulomb
crystal fluorescence intensity. This can be seen in the callibration data taken for Ca+ + NO shownin Fig. 4.12. What would change, however, is the quantitative predictions of the state-populations.
Specifically, the the state-populations are too large by as much as a factor of 2. This would also
affect the calculations of the absolute rate-constants reported. They should be increased by thecorresponding factor.

Chapter 5
Isomer specific chemistry and isotopologue reactions of C2 H2 + + C3 H4

The following is a pre-print draft of a paper yet to be submitted. This study investigates theisomer specific chemistry between sympathetically cooled C2 H2 + cations and two isomers of C3 H4 :allene and propyne. Paramount to this investigation was the use of isotope substitution of both theionic and neutral reactants. Presented here are the data and analysis drafted for future publication.
In addition to helping build the apparatus, control software, and analysis code, I acquired mostof the data presented and wrote the majority of the text. Electronic structure calculations were
provided by our fantastic theory collaborators John Stanton, James Thorpe, and Lam Nguyen at
the University of Florida Gainesville.

5.1

Introduction
Small carbocations and neutral organic species such as C2 H2 + and C3 H4 form some of thebasic building blocks in organic chemistry and contribute to a diverse field of chemical environments. For example, acetylene has been observed in the interstellar medium (ISM)[137], where it
is likely to play a central role in the formation of complex organic molecules through condensation reactions leading to larger carbocations.[75, 138, 139] Characterization of reaction mechanismsthat contribute to condensation reactions can help to explain the diverse species of complex organic
matter observed in the ISM.[140] Propyne and allene (C3 H4 ) are also expected to play a role insimilar polymerization reactions in the ISM.[141] An additional chemical environment of interest isplanetary atmospheres. C3 H4 was recently detected in the atmosphere of Titan, and is expected to77have a role in the formation of tholins that make up its dense haze layer.[142] All three reactants arealso common constituents in fuels. The chemistry of acetylene flames have been studied in detailto investigate soot formation mechanisms.[143–145] The role of propyne and allene isomerization in
flames have also been characterized, previously.[146] Our work adds valuable insight to the broad
knowledgebase on these fundamental hydrocarbons using new experimental techniques of coolingof molecular ions co-trapped with laser-cooled atomic ions in ion traps.
Linear Paul ion traps (LIT) have proven to be fruitful environments to study gas-phase ion
chemistry of molecular ions. Laser-cooled atomic ions can be used to sympathetically cool cotrapped molecular ions to translational temperatures below 10K. The cold, trapped ion ensemble
forms a Coulomb crystal structure that can be stably trapped and interrogated for minutes to hours.
By exposing the Coulomb crystal to a neutral reactant gas, chemistry can be observed to occur
at low concentrations (106 cm−3 , < 10−8 Torr) and over long timescales.[23] Another advantage of
LITs is the ability to purify reactants using radio frequency (RF) secular excitation, which enables
mass dependent ejection of ions from the trap.[147, 148] We use this technique to isolate acetylene
ions in our trap from other carbocations trapped in the molecular ion loading process.
In addition to the ability to create clean, cold molecular ion samples, the LIT offers a way
to determine the reaction products with high mass resolution and sensitivity. Recently, reaction
studies in the Coulomb crystal environment have been improved by (radially) coupling to a timeof-flight mass spectrometer (TOFMS) to the LIT.[35–38, 149, 150] Low translational temperatures
provide the initial conditions necessary for a high-resolution determination of the mass-to-chargemratio ( ∆m
> 1000) and an accurate count of each constituent mass. The TOFMS also detects all

constituent masses in the ion trap simultaneously. Because reaction product ions are also trappedand sympathetically cooled, one can follow reactions both by the depletion of reactants and by thegrowth of product channels. This enables a more complete description of reaction kinetics.
These detection methods also allow for extensions beyond reaction kinetics into the realmof chemical dynamics, which can be characterized by isotope substitution studies.[43, 44] Evenfor small organic molecules, there can be a large combination of isotope substituted reactants

78
available. Following the isotopic distribution of product channels is a powerful extension to basic
kinetic studies. These data can validate and/or exclude kinetic pathways proposed by potential
energy surface calculations. Overall, this degree of control and measurement sensitivity is criticalto tracking reaction pathways in the experiment. With these experimental techniques, we were able
to study here the previously unexplored reactions between acetylene cations and C3 H4 .
What follows in this article is a detailed investigation into the isomer specific chemical mechanisms observed in the reactions between acetylene cations, allene, and propyne. This begins withan overview of the experimental and computational methods we used to study the reactions. After
that, the paper is divided into two major sections, one for reactions involving propyne and anotherfor reactions with allene. Each reaction section is structured the same way. First is the descriptionof experimental results, which include a model of the observed reaction kinetics, measured reaction
rate constants, branching ratios for primary products, and select branching ratios for secondary
products. Second is a description of computational results, which cover the reaction potential
energy surface and supporting calculations. Last is a discussion of how the experimental results
validate the computational results. The discussion section also includes a comparison of the resultsin this inquiry to those in the literature (where possible), as well as possible sources of uncertainty.
Finally, the paper concludes with a summary of the presented results and an outlook on future
work.

5.2

Methods

Experimental: The details of our LIT-TOFMS apparatus have been described in previous work.[2,38, 39] We will give a short overview of the experimental apparatus, emphasizing the new aspects
and experimental procedures. See Fig. 5.1 for a schematic layout of the apparatus.
A major addition from previous descriptions was a molecular beam source, which was needed
to study reactions with molecular ions. A piezo actuated (PZT) valve backed by acetylene gas
diluted in argon (6%, 30 PSIG) produced a pulsed, supersonic beam in a diferentially pumped
region of the vacuum system. The molecular beam was skimmed using a 2 mm diameter skimmer,

79
to camera

pulsed leak valve

microscope objective

shutter
Ca skimmer

Kimballchamber

Ca ovenion trap

PZT skimmer
coolingbeams

to TOFMS

ionization
beams

PZT valve

Figure 5.1: Schematic render of the LIT-TOFMS apparatus. A cross-sectional view of the sphericaloctagon vacuum chamber reveals the ion trap at the center of our experiment. The source of Ca+
is an effusive beam of neutral calcium, generated by an oven. The source of molecular ions is apulsed, supersonic gas jet of neutral acetylene seeded in a noble carrier gas. These neutral beams
are intersected by pulsed ionization lasers in the center of our trap to load atomic and molecularions. Ca+ ions are addressed by the cooling lasers. To observe the laser induced fluorescence from
Ca+ , a microscope objective (outside the UHV chamber) is implemented. A pulsed leak valve (alsooutside the UHV chamber) is used for the introduction of the neutral reactant.

which allowed the ion trap chamber to recover to a UHV pressure (∼ 10−10 Torr) quickly after
operation of the pulsed valve. The resultant molecular beam passed through the center of the ion
trap along the radial direction, where constituents of the beam could be ionized and loaded into the
trap. In particular, acetylene cations were produced by the intersection of a focused, pulsed dyelaser (LIOP-TEC, 2mJ/pulse, f = 20 cm) and the molecular beam at the center of the ion trap. A(1+1’) resonance enhanced multi-photon ionization (REMPI) scheme was used at ∼216nm to load
C2 H2 + . [151] For C2 D2 + , ∼218nm was used to address the same intermediate excited state.[152]Both the dye laser and PZT valve were run synchronously at a 10 Hz rep rate, which loaded
hundreds of molecular ions into the trap in <1s. Due to the high reactivity of acetylene cations

80

Ion Signal (V)Ca++
240 ions

Ion Mass
C2H2+
Figure 5.2: Left: False-color fluorescence microscope image of a typical two-component Coulomb
crystal and its corresponding TOFMS trace (right). The TOFMS data were produced by a single
trap loading, i.e., not averaged over multiple experimental cycles. The mass spectrum shows ∼240
C2 H2 + ions were co-trapped with ∼600 Ca+ ions.

and the high ionization photon energy, this loading scheme also loaded many contaminant ions. To
remove all contaminant ions, a resonant secular mass excitation scheme was implemented.[147, 148]
Radio-frequency (RF) excitations were used to remove all unwanted masses, thus purifying theacetylene sample for the reaction study. Typically, 150–250 acetylene ions remained in the trap
after purification, and contaminant ions constituted .5% of the acetylene number.
Calcium ions (600–900) were then loaded into the trap and laser cooled to . 1 K.[38] The
cooled calcium ions sympathetically cooled the acetylene ions via the Coulomb-Coulomb interaction to translational temperatures below 10K, similar to previous work.[49] Additional RF secular
excitations were applied to remove from the trap the isotopes of calcium not addressed by the cooling lasers. This point in the experimental procedure marked the initial condition (time, t=0) for
reaction measurements. See Fig. 5.2 for a typical trapped ensemble after loading and purification.
Reactions were then carried out by admitting a neutral reactant gas into the vacuum chamber

81
by a pulsed leak valve scheme.[127] This raised the UHV chamber base pressure of 10−10 Torr toa constant pressure of about ∼ 3 × 10−9 Torr within 2 s. The neutral gas was a 7% mixture ofallene or propyne diluted in a noble gas (argon or helium). After the Coulomb crystal was exposedto the neutral reactant for a variable reaction time of 10–300 s, the pulsed leak valve was closed
and the neutral gas was pumped out of the UHV chamber, which quickly (< 2 s) returned to thebase pressure. Pressures were measured by a hot-cathode ion gauge. Subsequently, the crystal wasejected into the TOFMS to determine the number of ions in the trap, as well as their mass-to-charge
ratio. This measurement scheme was repeated for various reaction times (10-15 times per reactiontime), which allowed us to create reaction curves simultaneously for the reactants and products(e.g., Fig. 5.3).
Because reaction curves were measured at a constant chamber pressure – and thus, constantneutral reactant concentration ([n] ∼ 106 cm−3 ) – reaction rate constants could be determined.
Fitting reaction curves to a reaction model (e.g., Fig. 5.5) gave the effective rate constant, kef f foreach pathway. All reaction pathways were treated as pseudo-first order with the neutral reactant
being in excess. The absolute rate constant, k, was then given by the relationship kef f = k·[n].
To measure the effects of isotope substituted reactants and more carefully map out reaction
pathways, deuterium labelled reactants (CDN isotopes, >98% D) were used. For the rest of thepaper, we will refer to the explicit chemical formulas of the reactants to emphasize which hydrogensites were substituted for deuterium. In addition to unlabelled acetylene (C2 H2 ), allene (H2 C3 H2 ),and propyne (HC3 H3 ), we utilized C2 D2 , DC3 H3 , HC3 D3 , and DC3 D3 gasses diluted in argon (7%).
D2 C3 D2 was also used, but diluted in helium (10%) for unrelated technical reasons. Reaction curves
were mapped out and fit to models in the same way as described above. The product channel
distributions for the isotopologue reactions were then compared to help verify the potential energy
surfaces.
Computational: Potential energy surfaces were assembled via quantum chemical calculations.
We calculated the propyne surface using our hybrid model: CCSD(T)/ANO1//ANO0. This meansthat we calculate molecular geometries and vibrational frequencies using CCSD(T)/ANO0, and

82
1.2

C2D+2
C3D+3

Normalized Ion Number

1

C3D+4
C5D+5

0.8

0.6

0.4

0.2

0
0

20

40

60

80

100

120

140

160

180

200

Time(s)Figure 5.3: Primary products of the reaction C2 D2 + + DC3 D3 as a function of time. Each point
represents the mean and standard error of 12 measurements. Secondary reaction products havebeen omitted for clarity. Ion numbers are normalized to the initial number of acetylene ions in thetrap (∼ 200 ions).

then a single point energy from CCSD(T)/ANO1 using this geometry. We have not performed
rigorous analysis of the errors of this model, but relative errors should around ±2 kcal/mol (90meV). For the allene surface, the mHeat+[3] method was used to improve the accuracy of thecalculations to about ±0.23 kcal/mol (10 meV). Kinetics were performed on the propyne surface
to estimate branching ratios of the products. We calculated rate coefficients from the standard
Rice–Ramsperger–Kassel–Marcus (RRKM) equation.

5.3

Propyne – HC3 H3

5.3.1

Experimental results

Over the course of the reaction time (ranging from 100–300s), all trapped acetylene cations
reacted away into product ions (see Fig. 5.3). The primary product ions for the reaction C2 H2 + +HC3 H3 are C3 H3 + , C3 H4 + , and C5 H5 + . Because C3 H4 + and Ca+ are both mass 40 u, the two83

Normalized Ion Number

1.2
4x C5H+5

C 2H+2

1

0.8
4x C6H+7

0.6

0.4
4x C7H+5

0.2

0
0

50

100

150

200

Time(s)Figure 5.4: Example detection of multiple orders of products from C2 H2 + + HC3 H3 . The productorder is apparent by the delay of onset in appearance and their mass is too large to be a productof C2 H2 + + C3 H4 . C5 H5 + was first order, C6 H7 + second, and C7 H5 + was third. Ion numbers havebeen normalized to initial C2 H2 + number. Product channels have been multiplied by 4 to enhanceclarity of the differences in reaction order. Note that not all measured product channels are shownin this plot.

ion species are not readily distinguished from each other. The product assignment was directly
confirmed by the isotopologue reaction C2 D2 + + DC3 D3 , which shifted the product C3 H4 + toC3 D4 + (mass 44 u) and allowed the product channel to be assigned unambiguously (see Fig. 5.3).
The product C3 H4 + can also be inferred by the detection of heavier-mass product ions, C6 H5 + andC6 H7 + , resulting from the secondary reaction C3 H4 + + HC3 H3 .[141] These products were evidently
second order in comparison to the primary products, because their onset occurs later in time andtheir mass-to-charge ratio was too large to be a product of C2 H2 + + C3 H4 (See Fig. 5.4). Of thethree primary order products, only C3 H4 + was observed to react further with the neutral propyne inexcess in the chamber. Third order reactions also occurred, which resulted in product ions C7 H5 +and C7 H7 + , as shown in Fig. 5.4.

84
From the results of C2 H2 + + HC3 H3 , a reaction model was developed depicting the relationship between reactant and product ions (see Fig. 5.5). The large amount of branching in
the reaction led to very small ion numbers per product channel, especially for higher order products. Detailed relationships between higher order products were ambiguous and are left as such inthe model (represented by curly brackets). Branching becomes even more prevalent in analogous
models for the isotopologue reactions studied, as H/D scrambling is observed within each productorder.

C2 H2++ propyne

C3 H3++ propyne
C3 H4+C5 H5+C6 H5++ propyne

C6 H7+C7 H5+C7 H7+

Figure 5.5: Reaction model for propyne. Each arrow represents a bimolecular gas-phase reaction
between the ion and a neutral propyne. Vertically stacked ions are of the same reaction order. The
exact relationship between higher order products was undetermined and beyond scope of this work.
This ambiguity is represented by the curly brackets.

After developing a reaction model, absolute rate constants for each isotopologue reaction were
determined. Ion numbers were normalized by the initial acetylene ion number for each reaction.
The decay of the normalized reactant ion number vs. time was fit to determine the rate. The rate
constant was then obtained by dividing the rate by the neutral gas concentration. The results aretabulated in Tab. 5.1. The uncertainties reported are derived using the 90% confidence intervalof the rate fit parameter. The uncertainties are purely statistical and do not include any possible
inaccuracy introduced by the method of concentration measurement.
Branching ratios for the primary products C5 Hx Dy + (x + y = 5) and C3 Hx Dy + (x + y = 3)
were determined by a fit of normalized product ion numbers to the reaction model. These results
are reported as percentages in Tab. 5.2 for the C5 Hx Dy + (x + y = 5) channel and Tab. 5.3 for85
the C3 Hx Dy + (x + y = 3) channel. The rate constant for each of these pathways can be calculated
by multiplying the branching ratio with its corresponding isotopologue rate constant in Tab. 5.1.
The branching ratio of the C3 H4 + channel, and its analogous isotopologues, can be inferred bysubtracting the branching ratios of the two other primary products from 100%.
Table 5.1: Reaction rate constants for all measured isotopologue reactions. All rate constants
determined by fit of the loss of charged reactants. Uncertainties are derived from the 90% confidence interval of the fit and do not account for possible inaccuracies in the neutral concentration
measurement.
Reactants3k 10−9 cms

C2 H2 + + HC3 H36(1)C2 H2 + + DC3 H36(1)C2 H2 + + HC3 D37(1)C2 H2 + + DC3 D36(1)C2 D2 + + HC3 H38(2)C2 D2 + + DC3 H38(3)C2 D2 + + HC3 D36(2)C2 D2 + + DC3 D3

7(2)

86
+

+

Table 5.2: Branching ratios for the reaction C2 H2 + HC3 H3 −−→ C5 H5 + H and its isotopologues
that led to C5 Hx Dy + (x+y = 5). Numbers are the branching ratios (in %) for each mass channel as
determined by the fit of the model. Uncertainties are the 90% confidence interval of the branchingratio fit. Blank spaces indicate the mass channel was either not populated or not statisticallydifferent from zero at the 10% significance level.

Products

C 5 H5 +C5 H4 D+C 5 H3 D 2 +C5 H2 D3 +C5 HD4 +C5 D5 +

Mass (U)

65

66

67

68

69

70

C2 H2 + + HC3 H321(3)C2 H2 + + DC3 H3

3(2)

11(2)

7(1)

17(2)

C2 H2 + + HC3 D3
C2 H2 + + DC3 D3
C2 D2 + + HC3 H3C2 D2 + + DC3 H3C2 D2 + + HC3 D3
C2 D2 + + DC3 D3

12(2)
4(2)

6(1)

11(2)
8(5)

14(5)
16(4)

5(5)
13(3)

87
+

+

Table 5.3: Branching ratios for the reaction C2 H2 + HC3 H3 −−→ C3 H3 + C2 H3 and its isotopologues that led to C3 Hx Dy + (x + y = 3). Numbers are the branching ratios (in %) for each mass
channel as determined by the fit of the model. Uncertainties are the 90% confidence interval of
the branching ratio fit. Blank spaces indicate the mass channel was either not populated or not
statistically different from zero at the 10% significance level.
Products

C 3 H3 +C3 H2 D+

C3 HD2 +C3 D3 +

Mass (U)

39

40

41

42

C2 H2 + + HC3 H353(9)C2 H2 + + DC3 H327(4)40(20)C2 H2 + + HC3 D3

5(1)

33(8)

19(3)

6(1)

18(3)

30(20)C2 H2 + + DC3 D3
C2 D2 + + HC3 H3C2 D2 + + DC3 H3

31(5)

24(9)

5(1)

40(20)

11(2)C2 D2 + + HC3 D3

27(3)

25(5)

C2 D2 + + DC3 D3

5(2)

36(4)

It was also possible to estimate the branching ratios for selected second order products that
stemmed from the reaction C3 H4 + + HC3 H3 . This was done by fitting a truncated version of themodel that assumed the products C6 H5 + and C6 H7 + are the final products and do not react further.
As such, the branching ratios tabulated in Tab. 5.4 are likely lower bounds on the actual branchingratios. More product branches were observed from these isotopologue reactions than are reported.
Some of the detected mass-to-charge ratios could not be assigned a unique chemical formula dueto the possibility of H/D swapping. Since the secondary reactions were not the focus of this study,
those products have been omitted.

88+

Table 5.4: Estimated branching ratios for the reaction C3 H4 + HC3 H3 −−→ products and its
selected isotopologues. Numbers are the branching ratios (in %) for each mass channel, as determined by the fit of the truncated model. Uncertainties are the 90% confidence interval of the
branching ratio fit. Blank entries indicate the mass channel was not statistically different from zeroat the 10% significance level. Detected mass-to-charge ratios that could not be assigned a uniquechemical formula have been omitted.
Products

C 6 H5 +C 6 H7 +C6 D5 +

C6 D7 +

Mass (U)

77

79

82

86

C2 H2 + + HC3 H3

2(1)

13(3)
2(1)

6(1)

4(1)

9(3)

C2 H2 + + DC3 D3
C2 D2 + + HC3 H3

2(1)

7(2)

C2 D2 + + DC3 D3

5.3.2

Computational results

The experimentally observed formation of C3 H3 + and C5 H5 + from the reaction of C2 H2 +with propyne is indicative of a reaction pathway featuring a short-range, bound reaction complex.
Figure 5.6 displays a coupled-cluster, CCSD(T), potential energy surface (PES) that comprises
the salient stationary points (minima and transition states) for this system, where the (zero-point
corrected) energies refer to the undeuterated species.
The formation of the acetylene-propyne cation complex is initiated by a charge transfer fromacetylene cation to propyne, culminating in the complex labelled as INT0 in Figure 5.6. This
process is exothermic by about ∼1 eV; the lack of collisional cooling in the ion trap, coupledwith barrier heights below the entrance energy, enables the vibrationally hot complex to rapidlyinterconvert between the many minima displayed in Figure 5.6. This reaction ultimately produces
C5 H5 + , C3 H4 + , and C3 H3 + via the loss of a hydrogen atom (PRD1, PRD3, and PRD5), the lossof a neutral acetylene (PRD4), and the loss of a vinyl radical (PRD2), respectively.
The product yields can be simulated by employing a collisionless treatment of the chemical

89

Figure 5.6: Potential energy surface (PES) connecting the propyne and acetylene cation to experimentally observed primary products. The energy of each stationary point is calculated using
CCSD(T) in the scheme described in the main text. All energies are relative to separated acetylene
cation and propyne. Dashed lines between stationary points indicate reactions that occur via a barrierless process. Note that the initial point of the PES is given for after the first charge exchange
has taken place already.

master equation in conjunction with RRKM theory.[71]We estimate that reactions between propyne and acetylene cations, which proceed throughthe complex displayed in Figure 5.6 result in product yields of roughly 40 % C3 H3 + and 60 %C5 H5 + , in contrast to the experimentally observed product ratios of roughly 70 % C3 H3 + and 30% C5 H5 + . This suggests that another pathway may be in play, namely one where charge transfer
between propyne and acetylene occurs, but formation of a reactive complex is suppressed. This is
supported by the experimentally observed build-up of C3 D4 + in Fig. 5.3. Combinations of long-90
range charge exchange mechanisms and short range, bound, complex formation can be dependent
on the reaction impact parameter. These effects have been studied in other ion-molecule systems
velocity map imaged (VMI) products from crossed-beam experiments.[153] Without a measured
angle-dependent reactive cross-section, it is difficult to predict a priori the percentage of ionizedpropyne molecules that would take this second path.

5.3.3

Discussion

The distribution of products for the C5 Hx Dy + (x+y = 5) product channel exhibited all of theisotopic combinations of hydrogen and deuterium possible (see Tab. 5.2). These H/D scrambled
products suggested the formation of a C5 H6 + complex with a PES conducive to H/D swapping.
Such a surface would need shallow minima along the pathway where hydrogen sites exchange.
This condition is met by the transition states in Fig. 5.6 are all submerged well below the energyof charge exchange. One of many examples of isomerization that leads to swapping can be seen
following steps (INT2) thru (INT4) in Fig. 5.6. Hydrogen-site swapping in these steps is shown
from the methyl group of propyne to the other end of the complex. There is also ambiguity in thesource and destination of the transferred hydrogen atom. Combined with similar pathways on the
PES, any of the hydrogen/deuterium atoms from either acetylene or propyne could end up in the
final product. Because there is only loss of a single hydrogen from the system to form C5 Hx Dy +(x + y = 5), a maximum of two isotopic arrangements can exist for each isotopologue reaction.
We also observed that the loss of a methyl group hydrogen is preferred (largest branching ratio) in
every isotopologue reaction. This may be informative when considering future RRKM calculations.
The C3 Hx Dy + (x+y = 3) product channel in the isotopologue experiments told a very similarstory to that of C5 Hx Dy + (x + y = 5). We observed experimentally of all but three of the allowed
combinations (see Tab. 5.3). The three missing combinations are C2 D2 + + DC3 H3 −−→ 39 u
(C3 H3 + ) and 42 u (C3 D3 + ), and C2 H2 + + DC3 D3 −−→ 40 u (C3 H2 D+ ). These are likely missing
because the low ion numbers in each of the different product channels allow us to differentiate only abranching ratio of ≥5% from zero for the C3 H3 + channel. Additionally, the neutral reactant gasses

91
are specified to >98% isotopic purity only from the factory. We cannot rule out the possibility
H/D swapping occurred in the stainless steel lines delivering the gas to the vacuum chamber.
Accordingly, reactions may result in products with hydrogen atoms that do not originate fromthe assumed reactants. This was notably the case for the reaction of C2 D2 + + DC3 D3 , where
we observe the product C3 HD2 + , which should not be possible. While this was a clear influenceof contamination, the levels were low enough to believe that the main results of the measured
isotopolouge reactions were not appreciably influenced. The combined observations of C5 Hx Dy +(x + y = 5) and C3 Hx Dy + (x + y = 3) product channel isotopic distributions validate the proposed
PES pathways.
The lack of secondary reactions observed in the experiment between propyne and the C3 H3 +and C5 H5 + primary product channels notably indicated stable isomers of the products. The moststable isomers for both carbocations involve resonant ring structures (PRD2 and PRD5 in Fig.
5.6). Thermodynamic calculations verify that the cyclopropenyl cation is the only isomer of C3 H3 +that is exothermic with respect to the reactants. Thus, PRD2 is very likely the only observed
product isomer of C3 H3 + . For C5 H5 + , there are three isomers that thermodynamic calculations
show are exothermic with respect to the reactants (PRD1,PRD3, and PRD5), all of which involve
ring structures. The barriers leading to any of these products are all well below the initial reactionenergy and thus do not rule out any of the isomers. A future study involving a titration reaction or
spectroscopy is needed to identify the C5 H5 + structures produced by the C2 H2 + + HC3 H3 reaction.
Another prediction of the computational results was that both long-range and short-range
charge exchange could lead to the C3 H4 + product. By long-range we mean no bond is formed
between reactants in the process of charge exchange. By short-range, we mean the C5 H6 + boundcomplex is formed (see Fig. 5.6), and the reaction exits the surface via either PRD4 or the unnamed
entrance channel with propyne cation, which ultimately produces a C3 H4 + product. The former,
long-range process, would show no signs of H/D scrambling in the isotopic distribution of charge
exchange products, while the latter, short-range process, could. Unfortunately, our isotopologue
reaction measurements cannot distinguish these two possibilities for two reasons: First, the possible

92
mass channels for charge exchange products overlap considerably with those of C3 H3 + products.
Second, secondary reactions between C3 H4 + and propyne continually deplete the charge exchange
product formed by the primary reaction.[141] This prevented sufficient build-up of detectable primary reaction products in many (but not all) isotopologue reactions. Those two factors prevented
unambiguous determination of C3 H4 + product channels. A possible way around these limitations
would be to monitor the second order products from the resulting C3 H4 + + HC3 H3 reaction and its
isotopologues. That method is also untenable in our experiment due to the immense branching andisotopic scrambling of the second order products leading to branching ratios typically ≤3%. Thereis no way to unambiguously assign chemical formulas to such secondary product mass channels
with the levels of ionic and neutral contaminants present in this study.
Impurities also contribute to the Uncertainty in the branching ratios. As stated in themethods section, a small number of ionic impurities, relative to acetylene ion numbers (.5%),
remain trapped after RF secular excitation cleaning. For C2 H2 + , the largest impurity occurred at
mass 27 u (C2 H3 + ). For C2 D2 + , the largest impurities were masses 27 and 29 (C2 H3 + ,C2 HD+ ,and/or C2 D2 H+ ). Bimolecular reactions between these contaminant cations and neutral propyne
are expected to form products that overlap with the primary product channels of interest.[154]Some of the experimentally observed products from secondary reactions, however, can be
assigned chemical formulas. We can thus compare these results with previous measurements of
the reaction C3 H4 + + HC3 H3 by Anicich et. al.[141]. Both experiments observed the C6 H7 + andC6 H5 + products. Notably, we did not see the C3 H5 + product channel that Anicich et. al. did.
Since the product channel is exothermic, we suggest two explanations why we do not see it. One
possibility is there is a barrier to this process that cannot be overcome by the translationally cold
collision temperatures in our apparatus, ∼150 K, while those in the cited ion cyclotron resonance(ICR) experiment could be on the order of ∼1000 K due to their method of ion production. Anotherpossibility is that our reaction took place in a lower pressure environment, . 10−9 Torr, than thatof the ICR reaction, ≥ 10−6 Torr. Their experiment investigated the effects of pressure on productbranching ratios and found that as the pressure is reduced from 10−6 Torr to 10−8 Torr, the C6 H7 +

93product becomes preferred over the C3 H5 + product. The more than two orders of magnitude extra
gas pressure could kinetically stabilize a pathway on the potential energy surface leading to thisproduct. Building a potential energy surface to investigate secondary reaction pathways would
likely provide insight into this discrepancy.
Lower absolute concentrations in our experiment also caused some challenges. These arereflected in measured overall reaction rate constants of the isotopologue reactions. All of therate constants are considerably larger than predicted by the Langevin model for C2 H2 + + HC3 H3 ,3kL ≈ 2 × 10−9 cms . We attribute this discrepancy to the inaccuracy of our absolute pressure
measurement made at extremely low gas densities. This inaccuracy is ultimately due to systematiceffects inherent to the ion gauge used to measure the introduced neutral gas pressure, which canbe upwards of a factor of two or more in the < 10−8 Torr pressure regime.[155] Despite this fact,
relative differences in the rate constants could have revealed kinetic isotope effects. However, Noneof the rate constants measured appear to show isotope effects within their respective statisticaluncertainties.

5.4

Allene – H2 C3 H2

5.4.1

Experimental Results

The measurements in the system C2 D2 + + H2 CCCH2 followed the same steps as describedfor propyne above. The observed primary product ions for the reaction C2 H2 + + H2 C3 H2 wereC3 H3 + and C3 H4 + (see Fig. 5.7). Compared to the reaction with propyne, C5 H5 + was notably
not observed. Also in contrast to the propyne case, the products of the deuterium substituted
aceyetlene reaction did not show isotopic scrambling. Similar to the propyne case, C3 H4 + andCa+ are both mass 40 u, which were not readily distinguished from each other. However, theassignment of this product was directly confirmed by the isotopologue reaction C2 D2 + + D2 C3 D2 ,which shifted the product C3 H4 + to C3 D4 + (mass 44 u) and allowed the product channel to be
assigned unambiguously. The C3 H4 + product channel can also be inferred by the existence of

94
1.2

C2D+2
C3H+3

Normalized Ion Number

1

0.8

0.6

0.4

0.2

0
0

20

40

60

80

100

120

140

160

180

200

Time(s)Figure 5.7: Primary product of the reaction C2 D2 + + H2 C3 H2 as a function of time. Each point
represents the mean and standard error of 10 measurements. Secondary and higher order products
have been omitted for clarity. Ion numbers are normalized to the initial acetylene in the trap
120 ions.

heavier-mass product ions, C6 H5 + and C6 H7 + . These products were the result of the reaction
C3 H4 + + H2 C3 H2 [141] and were evidently second order in comparison to the primary products andtheir mass-to-charge ratio is too large to be a product of C2 H2 + + C3 H4 (see Fig. 5.8). Only C3 H4 +was observed to react further with the neutral allene in excess in the chamber. We also observed
third order reactions resulting in product ions C7 H5 + and C7 H7 + .
From the results of C2 H2 + + H2 C3 H2 , a reaction model was developed depicting the relationship between reactant and product ions (see Fig. 5.9). Branching in the reaction led to very small
ion numbers per product channel for higher order products. Detailed relationships between higher

95

Normalized Ion Number

1.2
C 2H+21

1 C 3H+3

0.8

0.6
20 C 6H+7

0.4

20 C 7H+5

0.2

0
0

50

100

150

200

250

300

Time(s)Figure 5.8: Example detection of multiple orders of products from C2 H2 + + H2 C3 H2 . Whichproduct order is apparent by delay of onset in appearance and their mass was too large to be a
product of C2 H2 + + C3 H4 . C3 H3 + was first order, C6 H7 + second, and C7 H5 + was third. Ion
numbers have been normalized to initial C2 H2 + number. Second and third order product channels
have been multiplied by 20 to enhance the clarity of the differences in reaction order. Note that
not all measured product channels are shown in this plot.

order products were ambiguous and are left as such in the model (depicted by curly brackets).

C2 H2++ alleneC3 H3++ alleneC3 H4+C6 H5+C6 H7++ allene

C7 H5+C7 H7+

Figure 5.9: Reaction model for allene. Each arrow represents a bimolecular gas-phase reaction
with the ion and a neutral allene. Vertically stacked ions are of the same reaction order. The exact
relationship between higher order products was undetermined and beyond scope of this work. This
ambiguity is represented by the curly brackets.

96
After developing a reaction model with allene, absolute rate constants for each isotopologue
reaction were determined in the same way as for propyne. Ion numbers were normalized by the
initial acetylene ion number for each reaction and the decay of the normalized reactant ion number
vs. time was fit to determine the rate. The rate constant was then obtained by dividing the rateby the neutral gas concentration. The results are tabulated in Tab. 5.5 with the uncertainties
derived from the 90% confidence interval of the rate fit parameter. The uncertainties are purely
statistical and do not include any possible inaccuracy introduced by the method of concentration
measurement.
Branching ratios were determined for the primary product C3 H3 + – in an identical manner
to propyne – by a fit of normalized product ion numbers to the reaction model. These results arereported as percentages in Tab. 5.6. The rate constant for each of these pathways can be calculated
by multiplying the branching ratio with its corresponding isotopologue rate constant in Tab. 5.5.
The branching ratio of the C3 H4 + channel, and its analogous isotopologues, can be inferred bysubtracting the branching ratios in Tab. 5.6 from 100%.
Table 5.5: Reaction rate constants for all measured isotopologue reactions. All rate constants
determined by fit of the loss of charged reactants. Uncertainties are derived from the 90% confidence interval of the fit and do not account for possible inaccuracies in the neutral concentration
measurement.

3
Reactants
k 10−9 cms
C2 H2 + + H2 C3 H2
C2 H2 + + D2 C3 D2
C2 D2 + + H2 C3 H26(1)1.3(2)
6(2)Table 5.6: Branching ratios for the reaction C2 H2 + + H2 C3 H3 −−→ C3 H3 + + C2 H3 and its isotopologues. Numbers are the branching ratios (in%) for each mass channel as determined by the fit of
the model. Uncertainties are the 90% confidence interval of the branching ratio fit. Blank spaces
indicate the mass channel was not statistically different from zero at the 10% significance level.
Products
Mass (U)C 2 H2 + + H 2 C 3 H2C 2 H2 + + D 2 C 3 D 2
C2 D2 + + H2 C3 H2

C3 H3 +
39
100(20)90(10)C 3 H2 D +40

C3 HD2 +41

C3 D3 +
42

40(20)

6(1)

44(5)

97
It was also possible to estimate the branching ratios for some second order products resulting
from the reaction C3 H4 + + H2 C3 H2 . This was done by fitting a truncated version of the modelthat assumed the products C6 H5 + and C6 H7 + are the final products and do not react further. As
such, the branching ratios tabulated in Tab. 5.4 are likely lower bounds on the actual branchingratios. Like for the propyne data, detected mass-to-charge ratios that could not be assigned aunique chemical formula have been omitted.
Table 5.7: Selected isotopologue data for C6 H5 + and C6 H7 + products from allene. Numbers are
the branching ratios (%) for each mass channel as determined by the fit of the truncated model.
Uncertainties are the 90% confidence interval of the branching ratio fit. Blank spaces indicate themass channel was not statistically different from zero at the 10% significance level.
Products

C 6 H5 +C6 H7 +

C6 D5 +

C6 D7 +

Mass (U)

77

79

82

86

C2 H2 + + H2 C3 H2

3(1)

12(2)

C2 H2 + + D2 C3 D2
C2 D2 + + H2 C3 H2

5.4.2

1.0(6)2(1)10(2)

Computational Results

In contrast to propyne, the only product (with appreciable yield) of the reaction C2 D2 + +H2 C3 H2 is C3 H3 + . It is significant to note that only fully hydrogenated C3 H3 + is observed, with noevidence of H/D scrambling noticeable. This suggests that no intimate reaction complex between
the two species is formed, else the reaction would proceed on the identical reaction pathway on
the PES of propyne, starting at PRD4, as shown on in Figure 5.6. This would invariably lead tosome C5 H5 + products and/or isotope scrambling of C3 H3 + products, similar to propyne. Thus,
the allene reaction pathway must be described by a different portion of the full PES than propyne.
We propose the acetylene cation and allene must undergo charge-exchange at long-range
exclusively. A PES of the subsequent reaction of undeuterated allene cation, calculated using the

98

Figure 5.10: Potential energy surface connecting the of allene and acetylene cation to experimentally observed primary products. The energy of each stationary point is calculated using themHEAT+ method.[3] All energies are relative to separated acetylene cation and allene.

mHEAT+ method,[3] is displayed in Figure 5.10. If nearly all of the internal energy released in the
charge-transfer process is converted into C3 H4 + vibrations, the allene cation has sufficient energy
to rapidly isomerize to INT3 in Figure 5.10, and subsequently tunnel through TS3 to release ahydrogen atom and form cyclic C3 H3 + (PRD1). Since the allene cation is a Jahn-Teller systemand distorts significantly from the neutral,[156] this is a plausible hypothesis. It should be notedthat this process was first proposed by Mebel and Bandrauk,[157] but recalculated here with ahigher level of theory. An analogous process has also been suggested in a VMI study of the chargetransfer reaction C+ + NH3 . The authors observed evidence of a resonant energy transfer from theexcess charge-exchange energy into the umbrella bending mode that connects the geometry of NH3

99
to NH3 + .[158]
The PES in Fig. 5.10 was calculated for the non-deuterated case only. While zero-point
energy corrections were not explored for deuterium substitutions on the entire surface, it wasdetermined that the thermochemistry of the c-C3 H3 + (PRD1) product channel reaction remains
exothermic for any choice of the studied isotopologue reactants.

5.4.3

Discussion

Almost all the allene cations formed undergo the proposed mechanism leading to cyclic
C3 H3 + . While it is a somewhat surprising result that all the internal energy available from chargetransfer is converted to internal energy of the allene cation, it is entirely consistent with the appearance potential of C3 H3 + observed in photoionization studies of allene,[159] where C3 H3 + appears atenergies ∼1.79 eV above the threshold for ionization of allene. If the excess energy from the chargetransfer is localized in the allene cation, it would be 1.71 eV above the threshold, and the results
observed here might be expected. Moreover, in the photoionization experiments, samples of allene
were ionized and detected much more quickly than the ions studied in this experiment. Thus, thetunneling process is apparently quite efficient, and appears dominant under the conditions of the
Coulomb crystal experiment.
The results of the C2 H2 + + D2 C3 D2 isotopologue reaction (see Tab. 5.6) bolster the proposedmechanism leading to primarily C3 D3 + products. While tunneling of a D atom should be muchless efficient than an H atom, it is still observed experimentally. This is likely due to the factthat kinetically there is only one other option, which is to vibrationally relax into C3 D4 + . Given
the extremely low background pressure environment, . 10−8 torr, the vibrationally excited C3 D4 +would not kinetically stabilize on a short enough timescale to prohibit tunneling. We can also
conclude that the tunneling time-scale is shorter than the vibrational relaxation time for C3 D4 +due to radiative decay.
The other two observed products from the C2 H2 + + D2 C3 D2 isotopologue reaction, masses
40 u and 41 u, are not predicted by the proposed reaction mechanism. Mass 41 u is very likely

100
the result of ionic contaminants or neutral reactant contaminants in the form of H/D swappingin the gas delivery lines. This is identical to the sources of uncertainty described in the propyne
discussion section. Mass 40 u, however, is observed in such abundance that it must be the resultof a reaction involving C2 H2 + . We assume the neutral reactant to be an unidentified contaminant
in our D2 C3 D2 sample, as the mass 40 u product occurs only when using the fully deuterated
isotopologue of allene and in no other combination (see Tab. 5.6).
We can also compare our results of secondary reactions between C3 H4 + + allene to thoseof Anicich et. al.[141]. For isotopologue reactions involving H2 C3 H2 , we report good quantitative
agreement with the major products and branching ratios. For the deuterium substituted reaction
with D2 C3 D2 , the resulting ion numbers in the different product channels were too few to make aquantitative comparison.
The reaction rate constants tabulated in Tab. 5.5 provide largely the same insights foracetylene cations reacting with allene as they do for reactions with propyne. We attribute the large
difference between reactions with H2 C3 H2 and D2 C3 D2 to be from the inaccuracies of the ion gauge
at low pressures being different for different gasses. H2 C3 H2 was diluted in argon and D2 C3 D2 wasdiluted in helium. It is likely that the systematic inaccuracy of the two gas pressure measurements
are different. In light of this possibility, the rate constants do not appear to be affected by isotopic
substitution of the acetylene cation within the reported statistical uncertainties. Alternatively,
however, the reaction with D2 C3 D2 really could be kinetically slower, due to the longer tunnelingtime of a deuterium atom compared to that of an hydrogen atom.

5.5

Summary and outlook
Isotopic substitution is a powerful and well-tested method for gaining insight into chemical

reaction processes. We applied this methodology to bimolecular reactions of C2 H2 + + C3 H4 in
the gas phase in the extremely isolated and specific environment of a Coulomb crystal. For reactions with different isomers of C3 H4 , propyne and allene, we measured a fundamentally different
reaction mechanism leading to isomer specific reaction products. Together with high-level theory

101
calculations, we gained understanding into the isomer specific reaction mechanism.
The reaction C2 H2 + + HC3 H3 and its isotopologues showed the likely formation of the C5 H6 +reaction complex. This complex produced primary products C3 H3 + and C5 H5 + . Reaction rate constants, branching ratios, and isotopic product distributions were measured for the primary products.
These results were used to validate the calculated PES, which demonstrated pathways leading toextensive H/D swapping. Secondary reactions between C3 H4 + and propyne were measured andcompared to results in the literature.
The reaction C2 H2 + + H2 C3 H2 and its isotopologues exhibited only C3 H3 + as its major
product channel, which suggests no short-range reaction complex was formed. This was further
bolstered by a lack of H/D scrambling in observed products. These results validated a calculated
PES based on the unimolecular decay of C3 H4 + after it converted the energy of the charge-transfer
process almost entirely into vibrational energy. Reaction rate constants, branching ratios, andisotopic product distributions were also measured for the primary products. Secondary reactions
also occurred between C3 H4 + and allene and the results, where possible, were compared to theliterature.
Given the copious added insight from isotopologue reactions in this system, this methodology
will become an indispensable tool in the characterization of future reactions studied in our ion trap.
The ability to unambiguously assign reaction products will aid in modeling reaction pathways
contributing to ISM chemistry and interactions in isolated environments. Further revelations of
unique ion-molecule reaction mechanisms will contribute to the basic understanding of how chemicalbonds are formed and broken.
In the future, we plan to couple a Stark decelerator to slow the neutral reactant and control
the collision energy available for reaction.[132, 160] This will enable detailed studies of reaction
dynamics. Isotopic substitution will be another useful handle in such experiments that allow for
precise changes in reaction energetics through zero-point energy corrections. While such changes
in energy are typically negligible, the highly controlled reaction environment of a combined ion
trap–Stark decelerator apparatus will have the goal to leverage those changes to do precise ther-

102
mochemistry measurements; the resulting experiments may set new benchmarks for theory andquantum chemical calculations.

Chapter 6
Translationally cold trapped CCl+ reactions with acetylene (C2 H2 )The following is a pre-print draft of a paper submitted The Journal of Chemical Physics
for publication. This study investigates the reaction mechanisms of CCl+ + C2 H2 , which revealed
previously unknown pathways to two cations actively studied in the chemistry of the ISM: C3 H+
and C3 H2 + . Presented here are the data and analysis drafted and under review for publication. In
addition to helping build the apparatus, control software, and analysis code, I acquired some of thedata presented and aided in writing the text. Electronic structure calculations were carried out at
JILA by our excellent post-doc Katherine Catani.

6.1

Introduction
Chlorine containing compounds have been found to exist in extraterrestrial environments in-cluding the dense and diffuse clouds of the interstellar medium (ISM). Interest in chlorine chemistry
in these areas was initially fueled by detection of HCl, HCl+ , and H2 Cl+ in these areas. However,these small molecules are acknowledged to account for only a minor fraction of the overall chlorine
abundance.[161–170] More recently, CH3 Cl was detected towards the low mass protostar IRAS
16293-2422 and comet Churyumov-Gerasimento.[171] Organohalogens, including CF+ have been
detected in these remote environments, but their overall chemical roles are not completely understood. A possible reservoir containing both carbon and chlorine is the carbon monochloride cation(CCl+ ). CCl+ has been proposed as an intermediate in the overall chlorine cycle in diffuse clouds,
although its overall abundance is not known. The major predicted pathways to CCl+ formation

104

Figure 6.1: Ground-state structures of CCl+ and C2 H2 .

are through reactions of C+ with HCl and H2 Cl.[161, 162, 172–177] Previous kinetic measurements, mainly using ion cyclotron resonance (ICR) spectrometry and selected ion flow tube (SIFT)
techniques, found CCl+ to be stable and non-reactive.[161, 162, 178] Products were observed forreactions with NH3 , H2 CO, CH3 Cl, CHCl3 , CCl4 , CHCl2 F, and CHClF2 .[161, 162, 178] However,with NO, H2 , CH4 , N2 , O2 , CO, and CO2 , no reactions were observed.[161, 162, 178] Although
only a few reactions with relevant interstellar species have been measured, it is conceivable that
CCl+ could undergo reactive collisions with varying neutrals, such as acetylene (C2 H2 ; previously
detected and thought to be abundant).[142, 179–182] Reactions of CCl+ with neutral carbonaceous
species would likely produce larger carbocation species.
Simple carbocation species like C2 H2 , c-C3 H2 + , and l-C3 H+ (observed cationic reaction products), have long been speculated to be present in the ISM. Recently, l -C3 H+ was detected towards
dark nebula in Orion and confirmed with laboratory spectroscopic measurements.[183–188] The
role of C3 H2 + in these environments is less certain. Both neutral l -C3 H2 and c-C3 H2 have been
detected towards dark clouds in the interstellar medium by their radioemission lines.[189] c-C3 H2 ,in particular, has been detected in diffuse clouds, where ionizing radiation is more prevalent, andthe role of its cation has been speculated as its probable precursor. Insight into additional formation pathways of these fundamental carbocations could prove useful for refining chemical models
that rely on energetic and electronic information.
Understanding the details of these and other ion-neutral chemical reactions benefit from studies at low temperatures and are beginning to be explored using new experimental techniques. In
particular, trapped Coulomb crystals have become increasingly popular as a way of studying cold
ion-neutral chemical reactions.[2, 25, 39, 43–45, 47, 50, 51, 53] Here, the reactant ion is sympa-

105
thetically cooled to low translational temperatures by elastic collisions with co-trapped laser-cooled
calcium ions. This technique allows for controlled interrogation of cold ion-neutral reactions overlong timescales.
To characterize the CCl+ cation and its reactivity in remote environments, we have measured
the reactive pathways of gas-phase CCl+ with C2 H2 in a linear Paul ion trap coupled to a time-offlight mass spectrometer (TOF-MS). The coupled ion trap enables co-trapping of sympatheticallycooled CCl+ with laser cooled Ca+ . This creates a controlled environment to study reactions withC2 H2 that may be present is some areas of the ISM, planetary atmospheres, or comets. The reactionof CCl+ + C2 H2 leads to a possible pathway to the formation of c-C3 H2 + and l -C3 H+ and to ameasurement of the branching ratio of these two products. Complementary electronic structure
calculations are used to aid in the discussion of the experimental results.

6.2

Experimental Methods
Kinetic data for the reactions of CCl+ + C2 H2 were recorded at different interaction time

steps by monitoring the masses of all ions present in a custom built linear Paul ion trap radiallycoupled to a time-of-flight mass spectrometer (TOF-MS). Details of the apparatus have been described previously,[2, 38, 39] and only a brief outline is presented here. CCl+ was formed throughnon-resonant multiphoton ionization of tetrachloroethylene (TCE; C2 Cl4 ). Vapor from TCE wasseeded in a pulsed supersonic expansion of helium gas (3 % TCE in ∼1000 Torr He). The skimmed
molecular beam was overlapped with 216 nm photons from a frequency-doubled, 10 Hz pulsed dyelaser (LIOP-TEC LiopStar; 10 ns pulse, 100 µJ/pulse) in the center of the ion trap forming C 35Cl+ ,C 37Cl+ , C2 + , C2 H+ , 35Cl+ , and 37Cl+ . Radio frequency (RF) secular excitation of contaminant ion
masses was used to eject all cation species from the trap except C 35Cl+ cations (subsequently denoted as CCl+ ). Next, calcium was added to the chamber via effusive beam from a resistively heatedoven and non-resonantly photoionized with 355 nm light from the third harmonic of a Nd:YAG laser(Minilite, 10 Hz, 7 mJ/pulse). Trapped calcium ions were Doppler laser cooled with two external
cavity diode lasers (ECDLs) forming a Coulomb crystal composed of Ca+ and a pure sample

106
of CCl+ . Both ECDLs were fiber-coupled with one tuned to the 2 S1/2 −2 P1/2 transition of Ca+(397 nm; New Focus, 3.5 mW, 600 µm beam diameter) and the other to re-pump Ca+ ions back into
the cooling cycle on the 2 P1/2 −2 D3/2 transition (866 nm; New Focus, 9 mW, 600 µm beam diameter). Both laser frequencies were measured and locked using a wavemeter (HighFinesse/Ångstrom
WSU-30).
Upon formation of a mixed-species Coulomb crystal, the RF trapping amplitude was dropped
to allow any ions in metastable higher angular-momentum orbits to escape from the trap. This was
repeated three times, leaving only cold Ca+ and sympathetically cooled CCl+ . During the course ofeach experimental run, trapped Ca+ ions were monitored continuously via their fluorescence using
an intensified CCD camera. CCl+ and other product ions do not fluoresce and were not directly
seen in the images, but rather were inferred due to changes in the shape of the Coulomb crystal.
Non-fluorescing ions were detected and quantified by ejecting the trapped species into a TOF-MS,
which enabled unambiguous determination of chemical formula based on the ion mass-to-charge
ratio (m/z).
In a typical experimental run, ∼ 100 CCl+ ions were loaded and sympathetically cooledby ∼ 800 calcium ions. Once the mixed Coulomb crystal was formed, neutral C2 H2 (∼ 4.7%
in Ar at 298 K) was leaked into the vacuum system containing the trap at a constant pressureof ∼ 4.6(3) × 10−9 Torr via a pulsed leak valve scheme.[127] The gas pulse duration was one ofmultiple neutral gas exposure time steps (0, 5, 10, 20, 40, 80, 160, and 260 s). At the end of eachtime step, a mass spectrum was measured and ion numbers for each mass channel were recorded.
An experimental run for every time step was repeated four times and the number of ions fromeach mass channel of interest was averaged at each time step. Each mass channel of interest
was normalized by the initial number of CCl+ ions, and the average normalized ion numbers andstandard error of the mean are plotted as a function of the different time steps as shown in Figure
6.2.

107

6.3

Computational Methods
Structures, vibrational frequencies, and relaxed potential energy scans for CCl+ + C2 H2

were determined using the M06-2X/aug-cc-pVTZ level of theory. Potential energy surfaces were
scanned along bond lengths, rotations, and angles to identify minima and saddle points. The M062X stationary points were used for CCSD/aug-cc-pVTZ calculations of structures and harmonic
vibrational frequencies. The optimized structures were then used for single point energy calculations
at the CCSD(T) level of theory with extrapolation to the complete basis set (CBS) limit using theaug-cc-pVXZ (X = T,Q) basis sets.[190–192] The energies were corrected for vibrational zero-point
energy from the calculated CCSD/aug-cc-pVTZ level of theory. Energies extrapolated to the CBS
limit should account for possible basis set superposition errors. Structures for CCl+ and C2 H2
are shown in Figure 6.1, with further geometric parameters and calculated harmonic vibrationalfrequencies (CCSD/aug-cc-pVTZ) provided in the Supplemental Information.
Other computational studies on CCl+ itself recommend using multireference methods.[193–196] The computational cost of using multireference methods for the larger potential energy surface
would be prohibitive, therefore we use the CCSD/aug-cc-pVTZ level of theory, which gives reasonable agreement with previously published computational and experimental data for CCl+ andC2 H2 separately.[193–197] Reported CCSD(T)/CBS energies should be accurate to within 0.04 eV.
Computations were compared to previous theoretical work and experimental data where available
for the reactants and products in order to determine the best level of theory to determine accurateenergetics. DFT and CCSD calculations were performed using both the Gaussian 16 and the Psi4
version 1.3.2 computational packages.[198, 199] Relaxed potential energy scans for the complexes
at the M06-2X/aug-cc-pVTZ level of theory were undertaken using Gaussian 16.

108

6.4

Results and Discussion

6.4.1

CCl+ + C2 H2 reaction measurements

Using room temperature (298 K) C2 H2 , the average collision energy with translationally cold(∼ 10 K) CCl+ is 17 meV (∼ 197 K). The excess collision energy should not be a significant source ofenergy to the reactions as the predicted products, c-C3 H2 + + Cl and l -C3 H+ + HCl are -0.15 and 0.07 eV exothermic, respectively (Equations 6.1 and 6.3) based on CCSD(T)/CBS level calculations:

CCl+ + C2 H2 −−→ c-C3 H2 + + Cl
(6.1)
∆E = −0.15 eV

CCl+ + C2 H2 −−→ l -C3 H2 + + Cl
(6.2)
∆E = 0.08 eV

CCl+ + C2 H2 −−→ l -C3 H+ + HCl(6.3)∆E = −0.07 eV
Several other product channels were explored theoretically, however, their endothermicities
were significantly greater than the calculated average collision energy (greater than 1 eV above
the reactant energies), and therefore disqualified from the discussion of products. In addition, thel -C3 H2 + isomer could contribute to the m/z 38 signal, however, the pathway is endothermic by
about 0.08 eV, making it very unlikely to be formed (see Equation 6.2).
As shown in Figure 6.2, the initial ionic products that are formed as CCl+ reacts away areindeed C3 H+ and C3 H2 + (m/z 37 and 38, respectively). At about 40 seconds, the rates into theintermediate C3 H+ and C3 H2 + species are overtaken by the rates of these intermediate species reacting with the excess C2 H2 present. This is illustrated by the turning over of the two corresponding
curves (green and black traces) and growth of the secondary products of larger carbonaceous species(red trace), mainly comprising C5 H2 + and C5 H3 + . The red trace plotted in Figure 6.2 includes

109

normalized ion number

1.2
1
0.8

CCl+
C3 H+

0.6

C3 H+2

0.4+

C5 Hx (x=2,3) & larger

0.2
0

0

50

100

150

200

250

time (s)Figure 6.2: Measured ion numbers of CCl+ (blue ×), l-C3 H+ (green ◦), c-C3 H2 + (black 2), andC5 Hx + (x = 2, 3, and higher; red ∗) as a function of time. Data are normalized by the initial
ion number of CCl+ (∼ 100). Each data point represents the mean and standard error from four
experimental runs per time point. The averaged data are fit using a pseudo-first order reaction
rate model (solid lines).

all mass channels above m/z 61, which should account for any secondary and tertiary reaction
products (or larger). Additions of C2 H2 to C3 Hx + (x = 1, 2) have been characterized previously,
and are not modeled computationally.[200–204]
Experimental reaction rates are determined by fitting the reaction data, shown in Figure 6.2
to a pseudo-first-order reaction rate model; the fits are represented by solid lines. For the formationof intermediate and secondary products, the model is explicitly defined as:
d[CCl+ ]= −(k37 + k38 )[CCl+ ]
dt

(6.4)

110

d[C3 H+ ]= k37 [CCl+ ] − k37 63 [C3 H+ ]
dt

(6.5)

d[C3 H2 + ]= k38 [CCl+ ] − k38 63 [C3 H2 + ]
dt

(6.6)

d[C5 Hx + ]= k37 63 [C3 H+ ] + k38 63 [C3 H2 + ]
dt

(6.7)

where the product ions of C3 H+ (m/z 37), C3 H2 + (m/z 38), and C5 Hx + (x=2,3, and higher; m/z≥ 62) at time t are normalized with respect to the number of CCl+ (∼ 100) ions at t = 0. Rate
constants are estimated using the measured concentration of neutral C2 H2 , which is measured usinga Bayard-Alpert ion gauge ≈ 5(1) × 106 cm3 . It should be noted that the uncertainty quoted here isstatistical, there is larger systematic uncertainty introduced by using an ion gauge in the 10−10 Torr
regime. The resulting reaction rates and rate constants are given in Table 6.1. These calculated
rate constants are on the order of the rate calculated using Langevin collision theory, 1.07 × 10−9
cm3 s−1 .
Table 6.1: Results from the fits to the pseudo-first order reaction rate models for the formationof intermediate and secondary products from the reaction of CCl+ + C2 H2 . Included are the rate
constants for the single pressure measurement of C2 H2 ≈ 5(1) × 106 cm3 .
Speciesm/zk (s−1 )k/[C2 H2 ] (cm3 s−1 )C 3 H+

37

0.5(3) × 10−2

0.9(5) × 10−9

C 3 H2 +38

2.0(3) × 10−2

4(1) × 10−9

C5 Hx +a62 (and larger)

4(1) × 10−2

7(2) × 10−9aformation of C5 Hx + and higher order products from both C3 H2 + and C3 H+

From the calculated rate constants, the branching ratio between the two intermediate products is determined to be about 4:1, with the Cl loss product channel (formation of c-C3 H2 + ) favored
over the HCl loss product channel (formation of l -C3 H+ ).

111
Further confirmation of the intermediate and secondary products was obtained by reactionof CCl+ with C2 D2 , where the product masses shifted to m/z 38 for C3 D+ and m/z 40 for C3 D2 +as expected. Additionally, for both reactions with C2 H2 and C2 D2 , the total ion numbers as afunction of trap time were also collected to ensure conservation of trapped ions. These data are
included in the Supplementary Information.

6.4.2

Modeling the CCl+ + C2 H2 reaction

In order to determine the relative energetics of the different pathways, an abridged potential energy surface of the CCl+ + C2 H2 reaction was explored; the key features that lead to thepreferable intermediate products are shown in Figure 6.3. Starting from the two reactants, atvarying approaches, a stable C3 H2 Cl+ C2v complex forms, which is exothermic by about 4.73 eV[CCSD(T)/CBS]. From this complex, the main experimentally observed products correspond to
Cl and HCl loss respectively, discussed below. Geometric parameters of each stationary point,
harmonic vibrational frequencies, and energies (CCSD/aug-cc-pVTZ), as well as a lower level M062X/aug-cc-pVTZ potential energy surface are given in the Supplementary Information.

6.4.2.1

Cl loss

The favored products, c-C3 H2 + + Cl, are obtained through a barrierless dissociation pathway,
with the products residing about -0.15 eV below the reactants. The c-C3 H2 + isomer is favored overthe l -C3 H2 + by about 0.23 eV (22 kJ/mol) and the barrier to rearrangement from linear to cyclic
has been calculated in previous studies to be between 1.62-1.75 eV.[205] Indeed, it may be possiblethat both isomers are present, however, as noted above the l -C3 H2 + + Cl products are slightly
endothermic, by about 0.08 eV, making it very unlikely to be formed.

6.4.2.2

HCl loss

The other intermediate product channel, forming l -C3 H+ through HCl loss, goes through thesame cyclic intermediate as the Cl loss channel [2]. From the cyclic minima, several rearrangements

112

Figure 6.3: The main features of the potential energy surface for CCl+ + C2 H2 calculated at the
CCSD(T)/CBS level of theory. The favored Cl loss product, dissociates without a barrier, formingc-C3 H2 + + Cl [8]. The HCl loss channel leading to l-C3 H+ + HCl [7] involves a few stationary
points with lower energies than the dissociation limit of -0.1 eV. Here, the bare ‘+’ indicates infinite
separation.

are required before eventually losing neutral HCl from l -C3 H+ . A few shallow minima were foundwhere the HCl moiety rearranges along the conjugated pi-system of l -C3 H+ , the lowest energy
isomer is shown [6]. The structures and energies (CCSD/aug-cc-pVTZ) of these different isomers
are given in the Supplementary Information.
The experimentally observed branching ratio favors c-C3 H2 + over l -C3 H+ by about a factor
of four. In the absence of Rice-Ramsperger-Kassel-Marcus (RRKM) theory/master equation simulations, it is difficult to probe the potential energy surface barriers and their possible contribution
to this observed branching ratio between the two primary product channels. The comparison is113
supported by the increased number of steps required for the HCl loss channel compared to the
Cl loss channel. It should be noted that over several levels of theory the barrier height of the H
abstraction before HCl loss ([3] in Figure 6.3) is variable based on the level of theory employed,with the CCSD(T)/CBS level giving the largest barrier. The uncertainty in the level of theory(0.04 eV) could push this barrier slightly endothermic, which would kinetically favor the Cl loss
channel.

6.5

Conclusions
The reaction of C2 H2 with sympathetically cooled CCl+ was examined using a linear Paulion trap coupled to a TOF-MS. Although CCl+ has been previously predicted to be non-reactive,
it is clear that collisions with C2 H2 result in either Cl or HCl loss. The primary products formed
from these two loss channels are the astrochemically relevant c-C3 H2 + and l -C3 H+ , respectively.
The rate constants measured for the formation of the two primary products are on the order of
those predicted through Langevin modeling. Furthermore, the Cl loss channel, occurring through abarrierless dissociation pathway, is favored by a factor of four compared to HCl loss channel, which
requires several rearrangements before losing HCl.
The theoretical work presented in this study would benefit from further exploration at ahigher level of theory, possibly with multireference methods to characterize CCl+ and its initial
complexation with C2 H2 . In addition, kinetic modeling, possibly using RRKM theory, would be
beneficial for a comprehensive comparison to experimentally measured branching ratios.
In future work, we plan to explore further the reactivity of CCl+ with other astrochemically
relevant neutrals. In addition, a natural extension to this work would be to study similar reactionsof CCl+ with more control over the neutral reactant, using state-selected molecular ions from a
traveling wave Stark decelerator.[132, 160] This will allow further study of ion-neutral reactions in
low temperature regimes, where the internal quantum states and external motion can be controlledwith high precision.

114

6.6

Supplementary information
Data included in this supplementary information consist of reaction curves from reactionsof CCl+ + acetylene (C2 H2 ) and CCl+ + deuterated acetylene (C2 D2 ; used to verify reaction
products). Also included are measured total ion numbers (a sum of all detected ion channels in agiven TOF-MS trace) for each reaction; used to demonstrate conservation of ions in the linear Paulion trap over the entire reaction. The full calculated potential energy surface for the discussion of
the CCl+ + C2 H2 reaction is included at the CCSD/aug-cc-pVTZ level of theory with additional
structural and energetic information provided for each stationary point.

115

total ionslinear fit

total ion number

1200
1100
1000
900
800
700
0

50

100

150

200

250

time (s)Figure 6.4: Measured total ion numbers of the CCl+ + C2 H2 reaction as a function of time. Eachdata point represents the mean and standard error from 4 experimental runs per time point. The
total ion data are fit using a 1st order polynomial giving a slope of -0.04 with an uncertainty of 0.3(the 95% confidence interval of the fit parameter) indicating the number of ions are conserved over
the reaction.

116

1.5

normalized ion number

CCl

+

C D

+

3

+
3 2

C D

+
5 x

C D (x=2,3) & larger

1

0.5

0

0

50

100

150

200

250

300

time (s)Figure 6.5: Measured ion numbers of CCl+ (blue ×), C3 D+ (green ◦), C3 D2 + (black 2), and C5 Dx(x = 2, 3 and higher; red ∗) as a function of time for the reaction of CCL+ with duetrated acetylene
C2 H2 . The product masses shift as expected due to deuterium substitution. Data are normalized
by the initial ion numbers of CCl+ (∼ 100). Each data point represents the mean and standard
error from 4 experimental runs per time point. The average data are fit using a pseudo-first order
reaction rate model (solid lines). C3 D2 + (m/z 40) overlaps with the Ca+ channel. To fit thischannel, the initial average Ca+ ion numbers are subtracted from the m/z 40 channel at each time
step. There is some variability in the loaded numbers of Ca+ , making the ion numbers and rates
extracted from the 40 channel less certain.

117

total ionslinear fit

total ion number

1600
1500
1400
1300
1200
1100
0

50

100

150

200

250

300

time (s)Figure 6.6: Measured total ion numbers of the CCl+ + C2 D2 reaction as a function of time. Eachdata point represents the mean and standard error from 4 experimental runs per time point. The
raw total ion data are fit using a 1st order polynomial giving a slope of -0.09 with an uncertainty of
0.2 (the 95% confidence interval of the fit parameter) indicating the number of ions are conserved
over the reaction.

118
Table 6.2: Results from the fits to the pseudo-first order reaction rate models for the formationof intermediate and secondary products from the reaction of CCl+ with C2 H2 (from Table I in
main text) and C2 D2 . Included are the rate constants for the single pressure measurement ofC2 H2 ≈ 5(1) × 106 cm3 and C2 D2 ≈ 7(1) × 106 cm3 . For the C2 D2 case, rates extracted fromthe data are less certain and the majority of the reported error is due to the calcium ion number
uncertainties, as the C3 D2 + product channel overlaps with the Ca+ channel
CCl+ + C2 H2Speciesm/zk (s−1 )k/[C2 H2 ] (cm3 s−1 )C 3 H+

37

0.5(3) × 10−2

0.9(5) × 10−9

C 3 H2 +38

2.0(3) × 10−2

4(1) × 10−9

C5 Hx +a62 (and larger)

4(1) × 10−2

7(2) × 10−9

CCl+ + C2 D2Speciesm/zk (s−1 )k/[C2 D2 ] (cm3 s−1 )

C3 D+

38

1(1) × 10−2

1(2) × 10−9

C3 D2 +

40

6(2) × 10−2

8(4) × 10−9

C5 Dx +b64 (and larger)

9(24) × 10−2

1(3) × 10−8abformation of C5 Hx + and higher order products from both C3 H2 + and C3 H+

formation of C5 Dx + and higher order products from both C3 D2 + and C3 D+

119Table 6.3: Previously measured and calculated constants of CCl+for comparison with current

computational results.

CCl+re (Å)ωe (cm−1 )De (eV)

Ref

Exp.

1.5378

1178

6.55

[197]

MRCI

1.54211172

M06-2X/aug-cc-pVTZa

1.53411212

CCSD/aug-cc-pVTZa

1.53301169

X+ 1 Σ+ ...(5σ)2 (6σ)2 (2πx )2 (2πy )2 (7σ)2athis work

[195]

6.59

120
Table 6.4: Electronic energies and vibrational zero-point corrections from calculations at the
CCSD/aug-cc-pVTZ level of theory with accompanying CCSD(T)/CBS single point energies, allin Hartrees. Stationary point numbers correspond to those on the CCl+ + C2 H2 potential energy
surface in Figure 6.3. Structures 6 and 10-12 are energetically similar isomers where the HCl moiety rearranges along the l -C3 H+ backbone. The other linear isomer of C3 H2 + is also included forreference. All structures are illustrated in Figure 6.8.
Stationary Point

Electronic Energy

Vibrational ZPE correction

Single Point Energy

CCSD/aug-cc-pVTZ

CCSD/aug-cc-pVTZ

CCSD(T)/CBS[1a] CCl+-497.3093456

0.0027851-497.4439045[1b] C2 H2-77.2104448

0.0271492-77.2986219[2]-574.7086427

0.0375602-574.9238578[3]-574.5202927

0.0294604-574.7423639[4]-574.5572188

0.0326468-574.7740926[5]-574.5254313

0.0277394-574.7466993[6]-574.5569850

0.0310633-574.7779988[7a] HCl-460.3686914

0.0068998-460.4594045[7b] l -C3 H+-114.1521467

0.0201516-114.2829693[8a] Cl-459.7017467

0.0000000-459.7883079[8b] c-C3 H2 +-114.8378178

0.0333737-114.9630686[9] l -C3 H2 +-114.8212372

0.0265948-114.9479331[10]-574.5530070

0.0306167-574.7741961[11]-574.5339041

0.0311430-574.7528099[12]-574.5325391

0.0306695-574.7513854

121

relative energy / eV

0

[1]

[3]

[7]

[5]

[8]

[4]

–1

[6]

–2
–3
–4
[2]

–5

Figure 6.7: Potential energy surface calculated at the CCSD/aug-cc-pVTZ level of theory forthe reaction of CCl+ with C2 H2 . Energies (in eV) are from single point CCSD(T)/aug-cc-pVTZ
calculations extrapolated to the complete basis set limit. Structures, energies, and geometries are
given in Figure 6.8 and Tables 6.4 and 6.5, respectively.

122+[5] -0.17[8a][9] +0.08+[2] -4.73[6] -0.93[3] -0.01[7] -0.07[7b][10] -0.84[7a][11] -0.25+[1a][1] 0.00
[1b]

[8a]
+

[4] -0.79[8] -0.15[12] -0.22[8b]Figure 6.8: Structures of stationary points on the potential energy surface in Figure 6.7. In eachcase, energies are given relative to the CCl+ + C2 H2 limit [1]. Geometries were computed at the
CCSD/aug-cc-pVTZ level. Structures 6 and 10-12 are the several minima with different rearrangements of the HCl moiety along the C3 H+ backbone. Also included is the l -C3 H2 + isomer energy.
Geometries for each stationary point are provided in Table 6.4.

Table 6.5: CCSD/aug-cc-pVTZ calculated geometries for stationary points on potential energy
surface represented in Figure 6.5. Structures for each point are depicted in Figure 6.8 and electronic
and zero point energies are provided in Table 6.4

stationary point

atom

x

y

z

[1a]

C

0

0

-1.14136Cl

00

0.39167C

000.599904H001.656109C

00-0.599904

[1b]

123
stationary point

[2]

[3]

[4]

[5]

[6]

atom

x

y

z

H

0

0

-1.656109

C00-0.345000

Cl

0

0

1.286760

C

00.678367-1.527787

H01.600247-2.078554

C0-0.678367-1.527787

H0-1.600247-2.078554

C

0.018354

0.536955

0Cl-0.768290-0.963326

0C

1.394966

0.4583160H0.755356-1.120267

0C

0.703112

1.675055

0H

0.7022562.750014

0C

0.005983

0.4744400Cl-0.085875-1.279367

0C

0.860059

1.562082

0C-0.594289

1.6275230H-1.455953

2.265192

0H

1.199848-1.501703

0C-1.162870

0.048628

0.038439

Cl

1.682021-0.086273-0.050671

C-1.744158

1.243183-0.008853

H

2.149974

0.074659

1.128091

C-1.954728-0.900496

0.022612

H-2.623554-1.740519

0.008556

C

1.223681

0.310537

0124
stationary point

atom

x

y

z

Cl

-1.280714-0.425591

0C

0.019856

0.951579

0H-2.290150

0.383234

0C

2.394211-0.025809

0H

3.413458-0.336866

0

Cl

0

0

0.035656H00-1.237164

C001.197535H002.270605C00-0.027009

C00-1.361224

[8a]

Cl

0

0

0

[8b]

C

0

0.693545

0.333591

C0-0.693545

0.333591C00-0.821531

H

0-1.595173

0.918901

H01.595173

0.918901

[7a]

[7b]

Chapter 7

Conclusion

7.1

Summary
The work presented in this thesis represents a new branch in a growing tree of experiments,

which are working towards cold, controlled, ion-molecule collision studies. Our approach combined
a linear Paul ion trap with a TOFMS. While we were not the first to do so, we developed a simplified
version of LIT-TOFMS electronics without sacrificing any performance. The experiment also did
not rely on molecular dynamics simulations for calibration or ion detection.
The new apparatus was used to first study reactions between laser-cooled Ca+ with the
only radical molecules available in gas bottles, NO and O2 . Reactions with both of these neutral
species observed rates dependent on the electronic quantum state of Ca+ . Via the cooling lasers, thequantum state of Ca+ was manipulated, and state dependent reaction rate constants were measured.
The resulting products of these reactions were also interesting in their own right. There is not muchchemical intuition to draw upon for what bonds will break or form when two electrophiles react.
More reactions between ions and radicals is a very exciting direction for future studies.
We have developed methods of loading clean, isolated samples of highly reactive molecularions. This lead to isomer specific reaction studies of C2 H2 + + C3 H4 that had not been previously
studied. Two stable isomers of C3 H4 , propyne and allene, displayed very different chemistry withC2 H2 + . Such a study was made possible by the highly specified creation and detection of ions in
our apparatus. Through the use of isotope substitution, we completed a thorough investigation of
the potential energy surface used to model the reaction. Such in-depth studies of isomer specific126
reaction models provide insight into the foundational question of how structure determines function.
Another study of molecules relevant to the ISM performed was that of CCl+ + C2 H2 . Benefiting from the methods developed in the previous studies, clean loading of CCl+ was achieved.
This reaction produced two very interesting carbocations, C3 H+ and C3 H2 + , which are centralcomponents of ISM chemistry. This study showed that halogenic compounds outside of flourine
likely contribute to the chemistry of intersteller clouds and are worth further study.

7.2

OutlookThe use of an ion trap and laser-cooled Ca+ produces cold, controlled ions. The next stepin studying cold, controlled molecular-ion collisions is to gain similar levels of control over neutral
molecules. Conveniently, such a technology has been developed and thoroughly investigated in
the Lewandowski lab. Stark decelerators produce cold, velocity-tunable, quantum state selected,
beams of neutral molecules. As an added bonus, neutral radicals are particularly well suited for
Stark deceleration. A large leap forward in cold-controlled radical-ion reaction studies would bemade by connecting the ion trap apparatus to a Stark decelerator in the lab. This is way easier said
than done. As mentioned in the introduction of this thesis, multiple groups have been pursuingthe exact proposal of a combined ion trap-decelerator experiment for years.
We hope to succeed in this endeavor through the use of an in-vacuum shutter and an electrostatic hexapole guide. The main elements of the molecular beam line are shown in Fig. 7.1. The
in-vacuum shutter will separate the decelerated beam of neutral radicals from its non-decelerated
carrier gas. The electrostatic hexapole will then guide the decelerated beam into the ion trap center. The pictured decelerator is a traveling-wave ring decelerator (TWRD) [131, 132]. The end
result would be the ability to study ion-radical reactions with control over the collision energy andquantum state of the neutral molecule.

127

Hexapole guide
TWRD

LIT-TOFMS

In-vacuum shutter

Figure 7.1: Schematic render of the partially constructed coupling between a TWRD and the ion
trap from this thesis. An in-vacuum molecular beam shutter separates the line of site from the
TWRD to the guiding hexapole. The guiding hexapole delivers the molecular beamline to the trap
center.

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Appendix A

Electronics circuit diagrams

Here are the most recently updated drawings of the circuit diagrams used in the experiment.
With the exception of the DC shim box, these circuits are not laid out on printed circuit boards(PCBs).

A.1

Ion trap driver
The heart of the ion trap driver is a resonant step-up transformer coil. The primary side

(three windings) trades high-current for high-voltage into two secondary sides, which are handwound in a bifilar arrangement (26 windings, each) around a toroidal core (Mircometal, T200-2).
The secondary sides are center-tapped so that two, high-amplitude RF voltages are produced 180
degrees out of phase. A drawing of the ion trap driver is shown in Fig. A.1. High-Q, shim inductors(∼100 nH) are hand soldered in line with some of the four, transformer secondary side outputs tobalance the RF amplitude of each phase. 12 SHV cables (∼30 pF/foot) deliver the high voltage
to the trap via vacuum feedthroughs. These constitute almost the entirety of the capacitive trapload.
DC blocking capacitors are in series between the rods and the RF transformer. They prevent
the DC shim voltages from finding their way to the cener-taps (nominally ground). The DC shim
voltages are coupled to the rod output via a 1 MΩ resistor. There is an added buffer cap to createa low-pass filter for RF traveling back to the DC shim box. This is not strictly needed as the 1 MΩ
resistor effectively blocks the low-current RF.

8

7

6

5

4

3

2

1

145

ROD6

Secondary

ROD5

Primary side of Coil

ROD4

To DC Shim Box

13 turns per phase

C13
100nF

2

2

C15

D
P1T1

C2

P1T
6 uH

P1T2

HV switch Top
1C3

P1 Quench

GND

1

1

R3

2
R1
1M
1

C1
2.2nF

D

2

C14

P1T

R2

2-3 turns

P1T3

HV_TOP

To Trap Electrodes

C4
2

P1B1

P1B
6 uH

C5

P1B2

C6

EOM

1
R6

2

2

R4

R5
2

C

1

P1B3
1

1

GND

P1 Quench

C16
100nF

C

C17
ROD1

ROD2

ROD3

P1B

C18

GND

ROD9

ROD8

D1

ROD7

To DC Shim Box
C19
100nF

S

2

2
R9
1

GND

1

2

C21

P2T1

TTL_IN

primary side quench

C8

B

P2T2

2

B

R8

C7
2.2nF

R7

P2T

1

1

Q1
MTP2N50E
G2PRIMARY

C20

D

P2T
6 uH

C9

1

HV switch Bottom
P2 Quench

P2T3

HV_BOTTOM

2

To Trap Electrodes

P2B
6 uH
1

GND

P2 Quench

C10

P2B1

C11

P2B2

C12
1

R12

2

R11
2
1

P2B3
R10
2
1

P2B

A

A

C22
100nF

JILA ELECTRONICS LAB

C23
GND

ROD10

ROD11

ROD12

C24

8

7

6

5

4

BUILT FOR:

DESIGN BY:
BUILT BY:

To DC Shim Box

DN:

3

DRAWN BY:
DATE:

SHEET

2

OF

1

Figure A.1: Main RF driver for the ion trap. Primary and secondary sides of the step-up transformer are separated by the dashed line. The quench boards and high-voltage switches are represented schematically as two and three-way switches, respectively. Connections to the DC shim box
are also labeled.

A.2

HV switches and quench boards
The HV switches and RF quench boards are needed to convert the ion trap electrodes froma trap to the repeller and extractor of a TOFMS. The RF quench shorts the secondary sides of themain transformer to the center-tap, removing the high-voltage RF quickly. The gates drivers (MIC4420) of the main fets that short the coil must float up with the center-tap to HV DC. The way
we accomplish this is by floating a battery up with the center-tap. This provides HV + 12V with

146
extremely low current noise. The batteries we use are recharchable (Tenergy NiMH 12V 2000mAh),
and have enough capacity to last at least 12 hours of continuous experimental runs. We used to use
DC-DC converters to provide the floating power. These are based on switch mode power supplies
which provide continous power and are isolated via a transformer. Unfortunately, the switchingfrequency of almost all comercially available supplies is around 100 kHz. This produces RF noise
capable of exciting secular resonances of our ions, which is unacceptable. In principle, one could
return to a switch-mode power supply solution that was home-made if the frequency is >1 MHz.
The High voltage switches then pulse high voltage DC onto the ion trap rods. This isaccomplished using a push-pull configuration of two N-channel Power mosfets (IXTL2N450). Highcurrent for fast switching is provided by the two series buffer capacitors. There are two copies of
the circuit in Fig. A.2, one for each secondary winding. Each has a separate battery pack and highvoltage power supply to provide different repeller and extractor voltages.

147

Figure A.2:

High voltage switches and quench circuit diagrams. The floating power supply is a

12VDC NIHM battery pack that rests on a glass plate to allow it to float up to 2kV.

A.3

DC shim box
DC shim voltages are required to overlap the geometric center of the trap and the RF node.

DC voltages are also required to produce the axial trapping potential. These voltages are suppliedby the DC shim box to the trap rods through the main RF amplifier. The output voltages forgroupings of rods are adjusted simultaneously by the potentiometers mounted on the front of the
DC shim box. The potentiometer for the endcap voltages (axial trapping potential) has been
replaced by the analog output of a NI-DAQ box. This enables digital, programatic control over the
endcap potential. These control voltages are sent through a buffer circuit, one for each of the eight

148
potentiometers, before the control voltage is sent to the summing matrix.
The heart of the shim box is a big array of resistors leading to 12 summing op-amp circuits,
one for each trap rod segment. The resistors feeding into a summing amp divides the control voltage
from the potentiometer before being added. 100kΩ → 1× and 200kΩ → 0.5× the input voltage.
The full voltage is for pots that provide 0–10V (pots 1, 4, 5, and 6). The halved voltages are forpotentiometers that provide -5–5V (pots 2, 3, 7, and 8). The summing amps add the input control
voltages to create the final output voltage for a rod segment. There is relatively little cross-talk, butjust enough to notice with the ions. Even mV can affect the ion cloud position on the micron-scale.
The summing amps also have an added current buffer on their output to provide quiet, stable DC
voltages. Lots of buffer caps are used in this circuit to minimize AC noise. They are all picked tohave minimal ESR.
All the amps are laid out on a PCB. The circuit diagram is shown in Fig. A.3. There is asolitary mistake on the PCB layout in the box power buffer amp. This is documented in the lab
notebook.

Rods 1 - 6

+12V
+

1uF
C28

+

POT4OUT

R22
1M

C30
10nF

R90
100K

6
8

+
4

2

OP-07CP

4

6

R24
1M

2

C33
10nF

8

+
4

POT8OUT-12V

POT1OUT

OP-07CPPOT3OUT

+

1uF
C34

POT4OUTPOT5OUT

R49
100K
R91
200K

C62
1uF

S

D

D14
200V

CW

R51
1K

D13
12V
S

D
G

D16
200V

R56
1K

D15
12V

CW

2

Q10
BSP89

S
G

D18
200V

R61
1K

D20
200V

ROD11OUT

8

+
4

J14
1
2

D

R73
1K

6

U23
3

C91
470nF

7
-

Q22
BSP89

G

OP-07CP

S

1
R114
50k
+12V

1uF
C93

+

G

C101
100nF
C60
1nF
ROD6OUT
C61
470nF

C64
1nF
6
8

+

R112
50k

2

D

R48
1K

POT1OUTPOT5OUTPOT8OUT

R79
100K
R106
100K

R77
680

R75
100K

C96
1nF
ROD12OUT

+12V
+

R107
200K

C98
1uF
2

Q12
BSP89

G

OP-07CP

6
8

+

Q24
BSP89

G

OP-07CP

R113
50k

2

D

R78
1K

S

1

1uF
C99

J15
1

7
-

4

+

C97
470nF

C100
1nF

U24
3

S

1
CW

4

J9
1

7

1uF
C63

Q23
BSP135-12V

Q11
BSP135

U18
3

+

C90
1nF

C94
1nF

C92
1uF

2

D
G

R47
680

2

R93
100K

R72
680
+12V

+

+12V

R92
100K

R74
200K

S

R111
50k

R45
100K

+

POT8OUT

J8
1

OP-07CP

C65
100nF

1

Q21
BSP135

R70
100K

+12V

C35
1uF

U13

3

1

C55
470nF

1

D
7
2

J23
Pot8

D4
200V

D

6
8

+

1uF
C57

-12V

POT8IN

R115
50k
+12V

C95
100nF

R43
1K

-

1uF
C31
+

S

1

-12V

7

+12V
+

2
Q20
BSP89

G

OP-07CP

8

+

1uF
C87

J13
1

D

R68
1K

6

U22

ROD5OUT

U17
3+

ROD10OUT
C85
470nF

7
-

4

C54
1nF

C58
1nF

1

-12V

2
3

+12V

C56
1uF

POT7OUT

R105
200K

G

R40
100K

+

C84
1nF

C88
1nF

C86
1uF

Q9
BSP135

CW

3

2

R44
200K

2

+

+12V

R42
680

U12

1

POT3OUT
C32
1uF

POT8OUT

R67
680
+12V

R104
100K

+

C59
100nF

7

POT6OUT

S

R110
50k

-12V

+12V

POT1OUT

Q8
BSP89

G

1

4

D

+

6
OP-07CP

8

+

D

R38
1K

U16

3

1uF
C51

+

-12V

G

G

CW
2

R89
100K

POT6OUT

J7
1

7

OP-07CP

1

4

C27
10nF

POT6OUT

R88
100K

C49
470nF

C52
1nF

D22
200V

POT4OUTROD4OUT

+12V

C50
1uF

R65
100K

R69
100K

D24
200V

8

+

R35
100K

+

Q19
BSP135
C89
100nF

C48
1nF

R66
1K

6

U11

R20
1M

C29
1uF

-

3

2

R116
50k
+12V

R71
1K

7
2

2

R87
200K

S

1

4

-12V

R37
680

CW

POT6IN

R39
100K

2
Q18
BSP89

G

OP-07CP

8

+

1uF
C81

D23
12V

POT3OUT

6

U21
3

R103
200K

J12
1

D

R63
1K

-

R76
1K

POT1OUT

+

J22

R102
200K

Q7
BSP135
D

+

1uF
C25

ROD9OUT
C79
470nF

7

2

+

C53
100nF

-12V

POT7IN

POT8OUT

R101
100K

+12V
-12V

POT5OUT

OP-07CP

8
1

4

+12V

1

POT7OUT

C78
1nF

C82
1nF

C80
1uF

D

6

U10
+

C24
10nF

J21

2

S

R109
50k

+

-

3

2
R18
1M

+

POT5OUT

R62
680
+12V

R100
100K

D19
12V

7
2

J20
1

POT4OUT

Q6
BSP89

G

OP-07CP

8
1

4

POT5IN

6

U15
+

D

R33
1K

-

3

1uF
C45

POT1OUT

1

7

2

R85
200K

C26
1uF

J6

S

R86
100K

C43
470nF

C46
1nF

C44
1uF

D21
12V

+

S

POT7OUT+ROD3OUT

+12V

R84
200K

D17
12V

POT5OUT

+12V

R30
100K

R34
100K

S

+

-12V

POT2OUTG

POT1OUT
1uF
C22

G

1

4

R60
100K

R64
100K

D

R15
1M

OP-07CP

8

+

C21
10nF

C83
100nF
C42
1nF

CW

3

2

Q17
BSP135

-12V

R32
680

D

1

POT4OUTCW

6U9

8-32MOUNTINGHOLE

CW

C47
100nF

-

J19

S

1
R117
50k
+12V

Q5
BSP135

C23
1uF

7
2

D

POT4IN

2
Q16
BSP89

G

OP-07CP

8

+
4

D6
200V

X4

6

U20
3

1uF
C75

+

J11
1

D

R58
1K

-

+12V
-12V

+
X3

ROD8OUT
C73
470nF

7

2

CW

+

+12V

-12V

X2

R108
50k

C72
1nF

C76
1nF

C74
1uF

R99
200K

POT8OUT

2

+

S

1

1uF
C39

R57
680
+12V

R98
200K

POT7OUT

Q4
BSP89

G

OP-07CP

8

+
4

X1

6

U14
3

D

R28
1K

-

1uF
C19

POT4OUT

1

7

2
+

-12V

J5

D8
200V

1

C37
470nF

C40
1nF

C38
1uF

1

4

C18
10nF

+

D10
200V

4

2

TL071CPROD2OUT

+12V

R83
200K

POT7OUT

R55
100K

R59
100K

D12
200V

8

+

POT3OUT

OP-07CP

8

R25
100K

R29
200K

POT2OUT
6

U8
+

R31
1K

U5
3

-

3

1

-10REG

R36
1K

6
C13
0.1uF

AD584

6

R41
1K

50k
R1

2

J18

-

R46
1K

VBG

3

7

POT3IN

7
2

C77
100nF
C36
1nF

R26
1K

COM

2.5V

R3
10K

2

Q15
BSP135

-12V

R27
680

S

STB

1

C41
100nF

D7
12V

5V

R118
50k

+12V

S

4

10V

CAP

R13
1M

5

+V

CW

7

C20
1uF

D9
12V

2

U6
8

+

S

1

4

Q3
BSP135
-12V

1

Q14
BSP89

G

OP-07CP

8

+

2

+12V
1uF
C10

+
+12V

J3

+12V

6

U19
3

1uF
C69

+

J10
1

D

R53
1K

-

S

+12V

ROD7OUT
C67
470nF

7

2

R97
200K

POT8OUT

C66
1nF

C70
1nF

C68
1uF

R96
200K

POT7OUT

S

R5
50k

D5
12V

+

-12V

Q2
BSP89

G

OP-07CP

8
1

4
1uF
C15

+10REG
R4
10K

6

U3
+

D

R10
1K

-

3

D2
200V

2

R82
200K

POT7OUT

1

4

+

R95
100K

POT6OUT

2

R52
680
+12V

R94
100K

POT4OUT

1

D11
12V

R11
1M

POT2OUT

OP-07CP

8

+

C14
10nF

J4

7

R8
1K

POT6OUT
6

U7

2

S

D

-

3

1
-12V

C4
2.2uF

2

R81
100K

C8
470nF

C11
1nF

C9
1uF

S

2

J17

IN
+

1

+

R50
100K

R54
100K

POT1OUTROD1OUT

+12V

R80
200K

POT2OUT

C7
1nF

D3
12V

7

POT2IN

C16
1uF

R7
100K

R17
100K

POT1OUT+3

Q13
BSP135
C71
100nF

R9
680

D1
12V

+12V

C12
1uF

2

J2

C17
100nF

S

GND OUT

C6
0.1uF

+

U4
LM7912C
1

+

-12V
+

G

R2
1M
3

GND
2

CW

C5
0.22uF

OUT

Q1
BSP135

D

+

IN

+12V

1uF
C2

G

2

+12V

POT1OUT

1

4

C1
10nF

OP-07CP

8

+

G

1

1

6

U1

2

U2
LM7812CT

J1

-

3

1

CW

2

J16

+12V

G

7

POT1INBox Power

149

Rods 7 - 12

C3
1uF

-12V
-12V

Figure A.3: Circuit diagra for DC shim box PCB. Buffer amps for potentiometer control voltages
feed into the summing matrix via dividing resistors. The summing amps then generate the buffered
DC voltages that are applied to the rods.

A.4

RF secular excitation
RF secular excitations are used to eject specific masses from the ion trap. They are sinu-

soidal frequency sweeps generated by two SRS DS345 function generators. A mini-circuits RF
splitter/combiner combines the two signals into a single BNC which is coupled to one of the secondary center-taps via a HV solid state relay. The relay protects the function generators from theHV pulse used for ion extraction into the TOFMS. A schematic (hand) drawing of the circuit isshown in Fig. A.4. A time profile of the excitation sequence is given in Fig. A.5.

150

Figure A.4: Secular excitation relay circuit schematic.

Actual sweep duration = 1ms, not to scale

Excitation

Relay trigger

Sweep time (s)Time

Figure A.5: Time sequence of the secular excitation and control trigger.

Appendix B

Labview control and automation

B.1

Automation
Automation is essential to the success of any experiment. People can’t be trusted to reli-

ably reproduce experiments from run to run and we rely on the help of electronics and computer
controlled hardware to pull of insanely complicated things like trapping and laser cooling of ionsin vacuum. To prove this point, Fig. B.1 is a block diagram of most of the actively controlled
components

152
Andor
CameraBertan HV
supplyHV pulser

PZT Valve

Stepper motor
arduinoOven shutter
stepper motor

Minilite shutter
arduinoMinilite shutter

PLV arduino

Parker 3-way
valve

Control PC
DIO Breakout
board

Ion Gauge
controller

Dye laser
Flipper mirror

Powerlite Yag
Power Supply
x2

Big YAGshutterarduinoTrigger selectClock divider

SRS Fn. Gen.

x4

DC Batteriesx2

RFSwitch

NI-6009 DAQ

HV Switches

Cooling
Fan

Big YAGshutter
3.5 MHzx2

RF Main Amp Box

Op-amp

x12

x12

x2

x2

Diode laser
shutter arduino

x2

Diode
SRS
HVLaser
supplycontrollers

SRS HV supply

Jordan TOF
voltage div.

Kepco DC
supplyB-field
coil

Kepco DC
supplyCa Oven vacuum
feedthrough

Diode
SRS
HVLaser
supplyshutters

SRS
SRSFn.
Fn.Gen.
Gen.

MCP

RF
splitterx2

SRSSRS
HVHV
supplysupplies

Trap Rod vacuum
feedthroughs

Powered by 120AC, 208AC,
USB, or high voltage/current
supplyDC Shim Box
NI-6002 DAQ

Wavelength
meter

x2

RF PreAmp

VCA

Fiber
SwitchPowered by JILA power supply+/- 15V or +5V
x2

RF
Mixer

x2

GHz
Scope

Master trigger
from DIO
breakout board

Sec exc

Powered by dedicatedDC supply
Low Voltage DC

RF monHigh Voltage DC
RF pickup
coil

MHz
Scope

High Current DC
Trigger/Logic

Low Voltage RF
High Voltage RF/DC

Figure B.1: Block diagram of all the control electronics in the experiment.

Automation of the experiment has other benefits as well. For example the experiment runs
faster. An experiment has to run faster than any uncontrolled long term-drifts in conditions.
Automation also allows for active feedback (servo) on long term-drifts that are too quick for experimental timescales. There is also, of course, the practical benefit of being less work. But let’s
be real, you’ll spend that time doing something else and ultimately just doing more, and that’s agood thing!

B.2

Labview control code
I want to say a few words about the philosophy behind the control code and some bestpractices that will serve you into the future when working with it. Great care was taken to153minimize the amount of work and headaches that you, dear reader, will have to experience in the
future.
There is no way to future proof experimental control code. As new people and ideas come inand out of the lab, the experiment will change and the control code will have to adapt accordingly.
This can easily produce a problem with version control. As edits are made to the code, one will
have to carefully track which version of the code went with which state of the experiment. This is
impossible. I have watched many try and fail to pull-off good version control but this system will,
without fail, break down. So if you can’t predict the future perfectly or keep good enough track ofwhat has happened in the past, what recourse do you have?
The solution I have implemented relies on a philosophy of additive coding. Change nothing,
only add new features. There is only one version of the code that grows ever larger as the demandsof the experiment change. When slight changes are needed, resist the urge to make them onfunctioning code. Copy and paste it with a new name and add it to the library of other functions.
If you can adhere to this original vision, I believe you will be rewarded.

B.2.1

Producer consumer architecture

To help make the additive coding philosophy a reality, the code had to be built on an architecture. This is something that all real coders do for every project. For some reason, doing thisin labview is viewed as noteworthy. The architecture I have chosen is called a producer consumer
architecture.
Since labview is a graphical programming language, the producer consumer architecture isrepresented graphically by a vertical stack of parallel (not nested!!) while loops. The general flowof data is from top to bottom. Data is generated in the top loop, to be handled by the next loop
down, which can further down another loop on for more handling ad infinitum. In practice, I neverneeded to use more than three loops. The first handles user input. This passes instructions tohardware which are carried out by the second loop. If the hardware generated data that needs tobe saved, it is passed to a third loop that does the actual saving.

154
All data/instuctions are passed from one loop to the next in the form of messages on queues.
A message consists of a one word instruction (string) and can optionally carry with it any amountof data (cluster). A queue is an object that stores messages. It functions like a first-in, first-out(FIFO) buffer. This allows the loops to handle tasks asynchronously. That’s a nice feature for a
program that is supposed to control hardware. If you need to interrupt what the program is doing,
you can add a command to the front of the queue (priority message) which is then handled next.
Make sure the task associated with each message is small enough that the program doesn’t ever
get hung up. Never include a while loop in your tasks. This will ensure that the control code isalways ready to respond to new user inputs.
Ultimately, to run the experiment, one wants to generate a list of tasks and have it carryout these tasks in-order and potentially repetitively. This is exactly what scripting languages
are designed to do. Since labview is not a scripting language (runs functions simultaneously andasynchronously) I had to do my best to implement a script-like interface. The way it works is there
is a table which all the details of your script are written in. Each pair of lines in the table is atask, which I call a step. Each task has a name and can have an arbitrary number of parametersnames with it that are the first line. The second line are the values that go along with the named
parameters. A series of these steps build an experimental script, which I call a sequence. The
sequence table can be saved, loaded, and edited, in both labview and in any text editor (evenin excel). Running the sequence repetitively is called a scan. You can have the program change
sequence values automatically as it runs the scan. The parameters of the scan are stored in a
separate scan table. All of this functionality is in the main brain module, and hopefully will neverneed to be written again.

B.2.2

Code hierarchy

I wanted to be able to run each piece of hardware (or small groups of hardware) used in theexperiment, individually. I also wanted to be able to stop a VI and add functionality to it whileother pieces of hardware were still running. This mean I had to build a system of modular VIs

155that can talk to one another, but don’t rely on one another. This is exactly what I did. It isroughly structured like this: Each piece of hardware has its own module (VI). Each module is anindependent producer-consumer system with a queue to pass messages from the front panel controls
to the hardware it is supposed to run. To allow the modules to talk to one another, the queues
are made to be functional global variables (also called an action engine). This allows any module
to load messages onto any other modules queue, and thus control them programatically. The only
module that utilizes the ability to talk to other modules is the main brain module aside from one
or two exceptions.
Fig. B.2 is a flowchart of each module and the hardware they control. This was done
from memory so please excuse errors or omissions. When something goes wrong or you need newfunctionality, it should point you to the correct VI.

156

Software modules (VIs)

Main Brain

Hardware

CameraAndor camera (USB)

OscopeTextronix Scope (GPIB→USB)

IGs

Ion Gauge controller (RS232→USB)

DS345

2x Function generators (GPIB→USB)

Laser Lock

NI-6002 (USB)High-Finesse Wavelength Meter (USB)
Laser Fiber Switch (RS232→USB)

2x Laser PZT control voltage (analog)
2x Laser shutter Arduino trigs (digital)

Trap RF/DC

NI-6009 (USB)RF control voltage (analog)
DC control voltage (analog)

DIO Board

NI-6009 (USB)2x trigger select box (digital)

DIO board (PCI)

PLV Relay Arduino trigger
Quench triggers
HV pulse triggers
RF switch trigger
Oscope trigger

Minilite shutter trigger
Dye Laser shutter trigger
Dye laser flipper mirror trigger
PZT valve trigger
Ca oven Arduino trigger

Flow of control

Figure B.2: Block diagram of all VI modules and what hardware they each control.

You may also find in the project file some extra, legacy modules that have been retired. The
save and launcher modules had nasty bugs associated with them. Now main brain handles all savefeatures and you can launch the module VIs by hand. Speaking of saving, everything is written
to an HDF5 file. That stands for hierarchical data format, and is exactly that. It is a clever way
to store multiple data types in the same data file with common, named headings that make themeasy to access later. Gone are the days of counting header lines. Now you just call your dataset or
attributes by name. As an added bonus, they are stored in binary and can make use of compression.
This is perfect for our experiment which typically generates two images and 4 scope traces worthof data (∼ 7 MB) per shot. The compressed binary HDF5 files usually are only 1.5 MB.
The same additive philosophy should be applied to the HDF5 file structure. If you are

157
generating data from a new source, or want to organize data from an old source in a different way,
make a new folder/dataset for it! This will allow old anylsis code to be compatible with your new
file structure.

B.2.3

How to add a step

To add functionality to a module, you need to make a new button on the front panel. Thisbutton will then need to generate an event in the producer loop. This event will then add a new,unique message to the consumer loop queue. The consumer loop queue will then need a new casewith a name identical to the message added to the queue. The functionality is then added to the
new case with the new message name. If any data needs to be included in the message –like thevalue of a parameter– a control must be added to the front panel with a name corresponding tothat of the parameter. The parameter must then be added to the VIs type definition. The valueof the parameter is then added to the message generation event. The value can then be pulledfrom the message using the variant-to-data VI with the type definition being fed into its data-type
input. The output cluster can be un-bundled by name with the name of the control corresponding
to the parameter. The value is then available to the case where the new message is being handled.
To call this new step from main brain, one has to edit two .cfg files that are in the same
directory as main brain. The “main brain steps.cfg” and the “main brain step timings.cfg” areloaded by main brain when the VI is run. The first .cfg file is how the table auto-populates withparameters and default values when you add a step. The second is how main brain knows how longa step takes when running a sequence.

B.2.4

How to add a module

Use the module template found in the labview project file. Open communication to the new
hardware to the left of the producer and consumer loops, and close it to the right. Make newfunctional global variables for any queues needed (probably only one). Follow the instructions
above to add steps and also make them available to main brain.

158
One additional step is needed to add the new module to main brain. A call to the new queuemust be added by hand along with the type-def for the new module in the “run step” sub-VI. This
only needs to be done once. After the run step knows about your new global queue, it can send it
any message you decide to generate in the future.

Appendix C

MATLAB analysis code

C.1

OBE global fit and model
Described in Ch. 3 is the procedure to calibrate the OBE model to fluorescence data measured

by the experiment. The hierarchy of code is as follows: A script, OBEalldatafit.m, is used to format
all of the data. It then calls the global fit function,pop8fit.m, to produce the calibration parameters.
The parameters are then fed into pop8model.m, to predict the state populations. These three files
are reproduced below.
OBEalldatafit.m% close all
clear all% xvar = [p397 p866 d397 d866 alpha]% x0,k = [c397 c866 L397 L866 B_mag]% note: laser detuning in this analysis defined as delta = w_laser - w_0% this is a holdover from the Gingell thesis% These functions pull and format raw data from their respective% measurement files[det397x,det397y] = detuningscan397();[det866x,det866y] = detuningscan866();[pwr866x,pwr866y] = powerscan866();[pwr397x,pwr397y] = powerscan397();[alphax,alphay] = alphascan();% vectorize and normalize data for global fit
xdata = [det397x;det866x;pwr866x;pwr397x;alphax];ydata = [det397y;det866y;pwr866y;pwr397y;alphay];norm = max(ydata);
ydata = ydata/norm;
x0 = [1,1,6,12,3];[k,~,resid,~,~,~,J] = lsqcurvefit(@pop8fit,x0,xdata,ydata);
kfit = k';ci = nlparci(kfit,resid,'jacobian',J,'alpha',0.10);
kunc = abs(ci(:,1)-ci(:,2))./2;

160% build table to report fit params
param = {'c397';'c866';'L397';'L866';'B_mag'};
unit = {'none';'none';'MHz';'MHz';'Gauss'};
T=table(param,unit,kfit,kunc)
out = pop8model(k,xdata);
maxp = max(out(:,2));% plot s,d,p-states vs blue detuning
figureplot(det397x(:,3),pop8model(k,det397x))
legend('S','P','D');
title('397 detuning scan')ylabel('State Populations (%)')
xlabel('\Delta_{397} (MHz)')% plot all data and fits
figurehold onplot(det397x(:,3),maxp*det397y/norm,'ko')
det397model = pop8model(k,det397x);
plot(det397x(:,3),det397model(:,2),'r-')title('397 detuning scan')ylabel('P-State Population (%)')
xlabel('\Delta_{397} (MHz)')figurehold onplot(det866x(:,4),maxp*det866y/norm,'ko')
det866model = pop8model(k,det866x);
plot(det866x(:,4),det866model(:,2),'r-')title('866 detuning scan')ylabel('P-State Population (%)')
xlabel('\Delta_{866} (MHz)')figurehold onplot(pwr866x(:,2),maxp*pwr866y/norm,'ko')
pwr866model = pop8model(k,pwr866x);
plot(pwr866x(:,2),pwr866model(:,2),'r-')title('866 power scan')ylabel('P-State Population (%)')
xlabel('866 nm power before chamber (mW)')
figurehold onplot(pwr397x(:,1),maxp*pwr397y/norm,'ko')
pwr397model = pop8model(k,pwr397x);
plot(pwr397x(:,1),pwr397model(:,2),'r-')title('397 power scan')ylabel('P-State Population (%)')
xlabel('397 nm power in front beam before chamber (mW)')
figurehold onplot(alphax(:,5),maxp*alphay/norm,'ko')alphamodel = pop8model(k,alphax);plot(alphax(:,5),alphamodel(:,2),'r-')title('\alpha scan')

161
ylabel('P-State Population (%)')
xlabel('\alpha (rad)')% plot fluorescence vs p-state
figurelb = [0,0];
ub = [Inf,0];
pvsffit = fit(ydata*norm,out(:,2),'poly1','Lower',lb,'Upper',ub);plot(pvsffit,ydata*norm,out(:,2),'k.')ylabel('P-state population (%)')
xlabel('BG subtracted fluorescence')

pop8fit.mfunction [ppop] = pop8fit(params,vars)% note: laser detuning in this analysis defined as delta = w_laser - w_0% this is a holdover from the Gingell thesis code at the core of the% analysis. I.e.: delta_analysis = -1*delta_literature% Program to calculate the steady state populations of a trapped% Ca+ ion in interaction with two coherent light fields, assuming% 8 sub-levels. Calculations are made with the assumption that% the magnetic field is perpendicular to the propagation% direction of the lasers at angle alpha to their polarizations.
%% Laser Waists% 1/e^2 area of blue/red in mm^2
A397 = 0.34^2*pi*1e-6; %mm^2 -> m^2 area @ 397 nm
A866 = 1.17^2*pi*1e-6; %mm^2 -> m^2 area @ 866 nm%% Variable inputs and outputs (measured from experiment)
ppop = zeros(size(vars,1),1);
for jj = 1:size(vars,1)
P397 = vars(jj,1); % 397nm power in mW
P866 = vars(jj,2); % 868nm power in mW
detb = vars(jj,3); % 397 detuning in MHz (red detune is positive)
detr = vars(jj,4); % 866 detuning in MHz (red detune is positive)
alpha = vars(jj,5);% angle between B-field and 866 polarization axes in,→radians%% Model parameters (to be fit)
c1 = params(1); % Scale factor for effective 397nm intensity
c2 = params(2); % Scale factor for effective 866nm intensity
l1 = params(3); % Linewidth of 397nm Diode Laser (MHz)l2 = params(4); % Linewidth of 866nm Diode Laser (MHz)B = params(5); % B-field magnitude in gauss%% Constants
h = 6.626068e-34; hbar = h/(2*pi); % J*s
ub = 9.74e-24; % bohr magneton in J/T
g1 = 1e-6/7.7e-9;%Ca+ P->S Decay Constant MHz
g2 = 1e-6/94.3e-9;%Ca+ P->D Decay Constant MHz%% Generation of Rabi Frequencies from Laser Power Data
I12 = c1*1e-3*P397/A397; % Effective intensity in Wm^-2
I32 = c2*1e-3*P866/A866; % Effective intensity in Wm^-2
O12 = rabi8(I12,1);
O23 = rabi8(I32,2);%% Conversion/Consolidation of Data with units of Frequency%Detunings converted to angular Mhz
D1 = 2*pi*detb;
D2 = 2*pi*detr;%Laser Linewidths

162
l1 = 2*pi*l1; %Laser linewidths input in USER DATA in khz
l2 = 2*pi*l2; %and converted to angular Mhz%B-field splittingu = 1e-6*1e-4*B*ub/hbar;%% MAIN STEADY STATE CALCULATION%In the steady state, L*rho = 0%To enforce the correct normalisation we replace%one equation with the normalisation condition
L = M8(O12,O23,D1,D2,l1,l2,g1,g2,u,alpha);
L(1,:) = 0;
L(1,1:9:64) = 1;
x = zeros(64,1); x(1) = 1;
rhoSS = L\x;
popSS = diag(real(reshape(rhoSS,8,8)));popSS = 100.*popSS./sum(popSS);% N1 = (popSS(1) + popSS(2));% S-state pop
N2 = (popSS(3) + popSS(4));% P-state pop% N3 = (popSS(5) + popSS(6) + popSS(7) + popSS(8)); % D-state pop% Espop = N2+N3;% P + D-state pop%% Designate output
ppop(jj) = N2; % normalize to P-max so intensity data can be fit
end
pmax = max(ppop);
ppop = ppop/pmax;
ppop(vars(jj,3)<=0) = 1e-6;
end%% Function to calculate Rabi frequency% function omega = rabi(I,which)%3-level
function omega = rabi8(I,which)%8-level%I = Intensity in Wm^-2%which = 1 for 397 nm, 2 for 866 nm
c = 299792458; % M/s
h = 6.626068e-34; hbar = h/(2*pi);
switch whichcase 1 %397 nm laser
wl = 1/(25191.51e2);
lt = 7.7e-9; %natural lifetime
case 2 %866 nm laser
wl = 1/(25191.52e2 - 13650.19e2);
lt = 94.3e-9; %natural lifetimeend
g = 1/lt; %Spontaneous decay rate in Hz
omega = sqrt((3 * I * wl^3 * g)/(16 * pi^2 * c * hbar));%8-level
omega = omega/(1e6); % Angular Mhz
end%% Formation of Liouvillian Matrix% Density matrix treatment of 8 level atom in a magnetic field
function L=M8(O1,O2,D1,D2,l1,l2,g1,g2,u,alpha)gS=2; gP=2/3; gD=4/5; % Lande factors
c=cos(alpha); s=sin(alpha); w3=3^0.5; E=eye(8);ii=(1:8)'*ones(1,8); i1=reshape(ii',1,64); % first index
i2=reshape(ii,1,64); % second index
H=diag([D1,D1,0,0,D2,D2,D2,D2]+0.5*u*[-gS,gS,-gP,gP,-3*gD,-gD,gD,3*gD]);
HSP=-O1/w3*[ -c s ; s c ];HDP=-O2/2/w3*[ w3*s 0 ; 2*c s ; -s 2*c ; 0 -w3*s ];H(1:2,3:4)=HSP; H(3:4,1:2)=HSP'; H(5:8,3:4)=HDP; H(3:4,5:8)=HDP';% Relaxation

163
C1=(2/3*g1)^0.5*E(:,1)*E(4,:);C2=(2/3*g1)^0.5*E(:,2)*E(3,:);C3=(1/3*g1)^0.5*(E(:,1)*E(3,:)-E(:,2)*E(4,:));
C4=(1/2*g2)^0.5*E(:,5)*E(3,:)+(1/6*g2)^0.5*E(:,6)*E(4,:);C5=(1/6*g2)^0.5*E(:,7)*E(3,:)+(1/2*g2)^0.5*E(:,8)*E(4,:);
C6=(1/3*g2)^0.5*(E(:,6)*E(3,:)+E(:,7)*E(4,:));
C7=(2*l1)^0.5*(E(:,1)*E(1,:)+E(:,2)*E(2,:));
C8=(2*l2)^0.5*(E(:,5)*E(5,:)+E(:,6)*E(6,:)+...
E(:,7)*E(7,:)+E(:,8)*E(8,:));
CC=(C1'*C1+C2'*C2+C3'*C3+C4'*C4+C5'*C5+C6'*C6+C7'*C7+C8'*C8);% Form effective Hamiltonian and Liouvillian
H=H-1i/2*CC; L = -1i*(kron(H,E) - kron(E,H'));% Add feeding terms
L=L+C1(i1,i1).*C1(i2,i2); L=L+C2(i1,i1).*C2(i2,i2);
L=L+C3(i1,i1).*C3(i2,i2); L=L+C4(i1,i1).*C4(i2,i2);
L=L+C5(i1,i1).*C5(i2,i2); L=L+C6(i1,i1).*C6(i2,i2);
L=L+C7(i1,i1).*C7(i2,i2); L=L+C8(i1,i1).*C8(i2,i2);
end

pop8model.mfunction [out] = pop8model(params,vars)% Program to calculate the steady state populations of a trapped% Ca+ ion in interaction with two coherent light fields, assuming% 8 sub-levels. Calculations are made with the assumption that% the magnetic field is perpendicular to the propagation% direction of the lasers at angle alpha to their polarizations.
%% Laser Waists% 1/e^2 area of blue/red in mm^2
A397 = 0.34^2*pi*1e-6; %mm^2 -> m^2 area @ 397 nm
A866 = 1.17^2*pi*1e-6; %mm^2 -> m^2 area @ 866 nm%% Variable inputs and outputs (measured from experiment)
spop = zeros(size(vars,1),1);
ppop = zeros(size(vars,1),1);
dpop = zeros(size(vars,1),1);
for jj = 1:size(vars,1)
P397 = vars(jj,1); % 397nm power in mW
P866 = vars(jj,2); % 868nm power in mW
detb = vars(jj,3); % 397 detuning in MHz (red detune is positive)
detr = vars(jj,4); % 866 detuning in MHz (red detune is positive)
alpha = vars(jj,5);% angle between B-field and 866 polarization axes in,→radians%% Model parameters (to be fit)
c1 = params(1); % Scale factor for effective 397nm intensity
c2 = params(2); % Scale factor for effective 866nm intensity
l1 = params(3); % Linewidth of 397nm Diode Laser (MHz)l2 = params(4); % Linewidth of 866nm Diode Laser (MHz)B = params(5); % B-field magnitude in gauss%% Constants
h = 6.626068e-34; hbar = h/(2*pi); % J*s
ub = 9.74e-24; % bohr magneton in J/T
g1 = 1e-6/7.7e-9;%Ca+ P->S Decay Constant MHz
g2 = 1e-6/94.3e-9;%Ca+ P->D Decay Constant MHz%% Generation of Rabi Frequencies from Laser Power Data
I12 = c1*1e-3*P397/A397; % Effective intensity in Wm^-2
I32 = c2*1e-3*P866/A866; % Effective intensity in Wm^-2
O12 = rabi8(I12,1);

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O23 = rabi8(I32,2);%% Conversion/Consolidation of Data with units of Frequency%Detunings converted to angular Mhz
D1 = 2*pi*detb;
D2 = 2*pi*detr;%Laser Linewidths
l1 = 2*pi*l1; %Laser linewidths input in USER DATA in khz
l2 = 2*pi*l2; %and converted to angular Mhz%B-field splittingu = 1e-6*1e-4*B*ub/hbar;%% MAIN STEADY STATE CALCULATION%In the steady state, L*rho = 0%To enforce the correct normalisation we replace%one equation with the normalisation condition
L = M8(O12,O23,D1,D2,l1,l2,g1,g2,u,alpha);
L(1,:) = 0;
L(1,1:9:64) = 1;
x = zeros(64,1); x(1) = 1;
rhoSS = L\x;
popSS = diag(real(reshape(rhoSS,8,8)));popSS = 100.*popSS./sum(popSS);
N1 = (popSS(1) + popSS(2));% S-state pop
N2 = (popSS(3) + popSS(4));% P-state pop
N3 = (popSS(5) + popSS(6) + popSS(7) + popSS(8)); % D-state pop% Espop = N2+N3;% P + D-state pop%% Designate output
spop(jj) = N1;
ppop(jj) = N2;
dpop(jj) = N3;
endout = [spop,ppop,dpop];end%% Function to calculate Rabi frequency% function omega = rabi(I,which)%3-level
function omega = rabi8(I,which)%8-level%I = Intensity in Wm^-2%which = 1 for 397 nm, 2 for 866 nm
c = 299792458; % M/s
h = 6.626068e-34; hbar = h/(2*pi);
switch whichcase 1 %397 nm laser
wl = 1/(25191.51e2);
lt = 7.7e-9; %natural lifetime
case 2 %866 nm laser
wl = 1/(25191.52e2 - 13650.19e2);
lt = 94.3e-9; %natural lifetimeend
g = 1/lt; %Spontaneous decay rate in Hz
omega = sqrt((3 * I * wl^3 * g)/(16 * pi^2 * c * hbar));%8-level
omega = omega/(1e6); % Angular Mhz
end%% Formation of Liouvillian Matrix% Density matrix treatment of 8 level atom in a magnetic field
function L=M8(O1,O2,D1,D2,l1,l2,g1,g2,u,alpha)gS=2; gP=2/3; gD=4/5; % Lande factors
c=cos(alpha); s=sin(alpha); w3=3^0.5; E=eye(8);ii=(1:8)'*ones(1,8); i1=reshape(ii',1,64); % first index

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i2=reshape(ii,1,64); % second index
H=diag([D1,D1,0,0,D2,D2,D2,D2]+0.5*u*[-gS,gS,-gP,gP,-3*gD,-gD,gD,3*gD]);
HSP=-O1/w3*[ -c s ; s c ];HDP=-O2/2/w3*[ w3*s 0 ; 2*c s ; -s 2*c ; 0 -w3*s ];H(1:2,3:4)=HSP; H(3:4,1:2)=HSP'; H(5:8,3:4)=HDP; H(3:4,5:8)=HDP';% Relaxation
C1=(2/3*g1)^0.5*E(:,1)*E(4,:);C2=(2/3*g1)^0.5*E(:,2)*E(3,:);C3=(1/3*g1)^0.5*(E(:,1)*E(3,:)-E(:,2)*E(4,:));
C4=(1/2*g2)^0.5*E(:,5)*E(3,:)+(1/6*g2)^0.5*E(:,6)*E(4,:);C5=(1/6*g2)^0.5*E(:,7)*E(3,:)+(1/2*g2)^0.5*E(:,8)*E(4,:);
C6=(1/3*g2)^0.5*(E(:,6)*E(3,:)+E(:,7)*E(4,:));
C7=(2*l1)^0.5*(E(:,1)*E(1,:)+E(:,2)*E(2,:));
C8=(2*l2)^0.5*(E(:,5)*E(5,:)+E(:,6)*E(6,:)+...
E(:,7)*E(7,:)+E(:,8)*E(8,:));
CC=(C1'*C1+C2'*C2+C3'*C3+C4'*C4+C5'*C5+C6'*C6+C7'*C7+C8'*C8);% Form effective Hamiltonian and Liouvillian
H=H-1i/2*CC; L = -1i*(kron(H,E) - kron(E,H'));% Add feeding terms
L=L+C1(i1,i1).*C1(i2,i2); L=L+C2(i1,i1).*C2(i2,i2);
L=L+C3(i1,i1).*C3(i2,i2); L=L+C4(i1,i1).*C4(i2,i2);
L=L+C5(i1,i1).*C5(i2,i2); L=L+C6(i1,i1).*C6(i2,i2);
L=L+C7(i1,i1).*C7(i2,i2); L=L+C8(i1,i1).*C8(i2,i2);
end

C.2

Reaction model fits
Described in Ch. 2 are the kinetic rate equations for pseudo-first order reactions. In theexperiment, we often see a nasty mess of reaction pathways that do not have easy to work withanalytic solutions. This can make fitting your data to extract rates a pain. So we rely on MATLAB’s
ode45 integrator to do the heavy lifting. If the reaction pathways are known, you can write down
the set of coupled differential equations to fit your data to. Then it must be properly vectorized tomake use of the ode45 integrator. After doing so, the set of differential equations can be passed toa non-linear, least squares, curve-fitting routine to extract rates and confidence intervals. Below is
an example of this procedure used to analyze the Ca+ + O2 reaction presented in Ch. 4 and [39].
The hierarchy of code is as follows: A script, O2analysis8level.m, is used to format all of
the data. It then calls the model fit function,O2fit.m. The model fit function then calls the set ofdifferential equations defined in O2model.m. These three files are reproduced below.
O2analysis8level.m
clear allclose all

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% basepath = 'S:\Data\2019\04 April\26\Batman\analyzed o2 data\';
det397 = [20,20,20,40,60,20,20,20,20,20,20,40,60,30];
det866 = [20,0,-20,-20,-20,20,0,-20,20,0,-20,-20,-20,60];% obe parameters:
p397 = 1.45*ones(length(det397),1);
p866 = 12.85*ones(length(det397),1);alpha = mod(260-25,90)*2*pi/180*ones(length(det397),1);% k = [0.6,1.039,3,13,4.28];
k = [0.86,1.8,3,3,5.3];
xdata = [p397,p866,det397',(det866-22)',alpha];model=pop8model(k,xdata);
ppop = model(:,2)/100;
dpop = model(:,3)/100;% imageca = [];% tofca = [];for ii=1:length(det397)
if ii>=14
basepath = 'S:\Data\2019\04 April\30\Batman\analyzed o2 data\';scan = [basepath,'blu',num2str(det397(ii)),'red',num2str(det866(ii)),'h c,→
ip_scan.h5'];elseif ii>=9basepath = 'S:\Data\2019\04 April\29\Batman\analyzed o2 data\';scan = [basepath,'blu',num2str(det397(ii)),'red',num2str(det866(ii)),'h c,→
ip_scan.h5'];elseif ii>=6
basepath = 'S:\Data\2019\04 April\25\Batman\analyzed o2 data\';scan = [basepath,'blu',num2str(det397(ii)),'red',num2str(det866(ii)),'_ c,→scan.h5'];else
basepath = 'S:\Data\2019\04 April\26\Batman\analyzed o2 data\';scan = [basepath,'blu',num2str(det397(ii)),'red',num2str(det866(ii)),'t c,→estrxn_scan.h5'];end[ratefit,ratefitunc,conc(ii),concunc(ii)] = O2fit(scan,0);%conc(ii) = conc(1);%concunc(ii) = conc(1);% for O2%rate(ii) = ratefit(1);%rateunc(ii) = ratefitunc(1);% for H2O
rate(ii) = ratefit(2);rateunc(ii) = ratefitunc(2);end% % for O2% kfit = rate./conc;% kfitunc = kfit.*sqrt((rateunc./rate).^2+(concunc./conc).^2);% for H2O
kfit = rate;
kfitunc = rateunc;% 2D fit rate constants and propogate uncertainties[fit2d,gof,output] = fit([ppop,dpop],kfit','poly11',...
'Weights',1./(kfitunc').^2,'Upper',[0,+Inf,+Inf],'Lower',[0,-Inf,-Inf]);
ci = confint(fit2d,.90);
kunc = (abs(ci(1,:) - ci(2,:))/2)';
k = [fit2d.p00,fit2d.p10,fit2d.p01]';
names = {'S','P','D'};

167
T = table(k,kunc,'RowNames',names)% produce reduced data set% eqns from wiki on: Reduced chi-squared statistic
ppopred = ppop(1:5);
dpopred = dpop(1:5);
det397red = det397([1:5,14]);
det866red = det866([1:5,14]);for ii=1:length(ppopred)
xi = kfit(ppop==ppopred(ii) & dpop==dpopred(ii));
wi = 1./(kfitunc(ppop==ppopred(ii)&dpop==dpopred(ii))).^2;
kfitred(ii) = sum(xi.*wi)/sum(wi);kfituncred(ii) = sqrt((sum(wi.*xi.^2)*sum(wi)-(sum(xi.*wi))^2)/(sum(wi))^2);
end
ppopred(6) = ppop(14);dpopred(6) = dpop(14);kfitred(1,6) = kfit(14);kfituncred(1,6) = kfitunc(14);% 2D fit plot
figure()hold oncolormap('jet')scatter(ppopred,dpopred,72,kfitred,'filled')
axis([0 1.15*max(ppopred) 0 1.15*max(dpopred)]);ph = plot(fit2d, 'Style', 'Contour');set(ph, 'Fill', 'off', 'LineColor', 'auto','LineWidth',2);grid off;
C = get(ph, 'ContourMatrix');
clabel(C, ph,'FontSize',10);h = colorbar;
ylabel(h, 'k_{eff} cm^3/s','FontSize',12);
xlabel('P-State Population','FontSize',12);ylabel('D-state Population','FontSize',12);O2fit.m% O2fit.mfunction [kfit,kunc,conc,concunc] = O2fit(scan,zeichen)% calculate concentration
mix = 0.6;
mixunc = mix*1/55;[po2,po2unc] = pressuredata(scan);% po2=1.75e-9;% po2unc = 1.75e-12;
conc = po2*mix*3.3e16;
concunc = conc.*sqrt((po2unc./po2).^2 +(mixunc./mix).^2);% read in data from reaction time scanions = h5read(scan,'/analysis/ions');
totalions = h5read(scan,'/analysis/totalions');
traptime = h5read(scan,'/analysis/scanvar');% prepare data for fit
masses = [40,56,57];
Ndata = ions(:,masses);
Tdata = traptime;

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uTdata = unique(Tdata);% reduce data for plotting
for jj = 1:length(uTdata)
sub = Ndata(Tdata == uTdata(jj),:);
Ndatamean(jj,:) = mean(sub);Ndatasem(jj,:) = std(sub)/sqrt(length(sub));end% Time dither and sort for ODE solver
Tdata = Tdata + 0.0001*rand(numel(Tdata),1);[Tdata,index] = sort(Tdata);
Ndata = Ndata(index,:);% fit parameters
k0 = [1e-3*ones(1,length(masses)-1),mean(totalions)];
lb = [2*zeros(1,length(masses)-1),mean(totalions)-std(totalions)];
ub = [1*ones(1,length(masses)-1),mean(totalions)+std(totalions)];
opts = optimset('Display','off');[kfit,~,resid,~,~,~,J] = lsqcurvefit(@O2model,k0,Tdata,Ndata,lb,ub,opts);
kfit = kfit';
ci = nlparci(kfit,resid,'jacobian',J,'alpha',0.10);
kunc = abs(ci(:,1)-ci(:,2))./2;% rate = kfit(1);% rateunc = kunc(1);% build fit lines for plotting
Tfit = min(Tdata):max(Tdata);
Nfit = O2model(kfit,Tfit);% plot reduced data and fits
if (zeichen)figure('Position',[5 5 800 900])
for ii = 1:length(masses)
subplot(4,2,ii)
errorbar(uTdata,Ndatamean(:,ii),Ndatasem(:,ii),'bo')hold onplot(Tfit,Nfit(:,ii),'r-')ylabel('ions')
title(num2str(masses(ii)))
hold offend
xlabel('time (s)')
suptitle(scan)
end
end

O2model.mfunction Nout = O2model(k,t)% kinetics model that can be passed as a function argument for lsq-fitting% of data%
Model:%
n40(t=0) = N0%n40dot = -(k56 + k57)*n40%
n56dot = k56*n40%
n57dot = k57*n40
%
%
With:
%
Variables: N(1) = n40%
N(2) = n56

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%
%
%
%

N(3) = n57
Parameters k(1) = k56
k(2) = k57
k(3) = N0

N0 = [k(3),0,0];[~,Nout] = ode45(@DifEq,t,N0);function dN = DifEq(t,N)
Ndot = zeros(length(N0),1);
Ndot(1) = -(k(1)+k(2))*N(1);
Ndot(2) = k(1)*N(1);
Ndot(3) = k(2)*N(1);dN = Ndot;
end
end

C.3

TOFMS integrator
When we acquire a raw TOFMS trace, the time axis is rescaled to mass and we can read

off the mass-to-charge ratios that were in the trap. To determine the number of ions in each mass
channel, the peaks must be binned and integrated. Here is the function that accomplishes this.
h5analysisintegrator.m% h5analysis_integrator.m% by James Greenberg% 2019/01/15% read in scope traces from a data file, integrate ion numbers, and write% results back to the data file.
function []=h5analysis_integrator(file)% read in analysis attributes (integrator settings)
p1 = h5readatt(file,'/analysis','p1');
p2 = h5readatt(file,'/analysis','p2');
masswindow = h5readatt(file,'/analysis','masswindow');
zoomtracethresh = h5readatt(file,'/analysis','zoomtracethresh');minpeakheight = h5readatt(file,'/analysis','minpeakheight');
nvsperion = h5readatt(file,'/analysis','nvsperion');dt = h5readatt(file,'/Oscope/Traces','dt');
maxmass = h5readatt(file,'/analysis','maxmass');% read in traces
toftrace = h5read(file,'/analysis/toftrace');tofzoomtrace = h5read(file,'/analysis/tofzoomtrace');toftime = h5read(file,'/analysis/toftime');msgid = 'signal:findpeaks:largeMinPeakHeight';
warning('OFF',msgid);
masses = 1:maxmass; % index of array corresponds to AMU. only integrate up,→
to mass 140

170

ions = zeros(size(toftime,1),length(masses));
fwhm = zeros(size(toftime,1),length(masses));peaktime = zeros(size(toftime,1),length(masses));zoom = zeros(size(toftime,1),length(masses));
val_min = 6.5e-6;
val_max = 7.5e-6;
for jj = 1:size(toftime,1)
trace = toftrace(jj,:)';
zoomtrace = tofzoomtrace(jj,:)';
time = toftime(jj,:)';[~,idx_min] = min(abs(time-val_min));[~,idx_max] = min(abs(time-val_max));[peak,loc] = max(trace(idx_min:idx_max));
if peak > 0.03
p2 = sqrt(40)-(time(loc+idx_min)*1e6)*p1;
end
for ii = 1:length(masses)[~,masswinl] = min(abs(time*1e6 - (sqrt(masses(ii)-masswindow) ,→p2)/p1));[~,masswinr] = min(abs(time*1e6 - (sqrt(masses(ii)+masswindow) ,→p2)/p1));[ions(jj,ii),fwhm(jj,ii),peaktime(jj,ii),zoom(jj,ii)] =h5analysis_traceintegrate(trace(masswinl:masswinr),zoomtrace(ma c,→,→sswinl:masswinr),dt,zoomtracethresh,minpeakheight,nvsperion);peaktime(jj,ii) = peaktime(jj,ii)*dt*1e6 + time(masswinl)*1e6; %,→converts from relative points to absolute time in micro seconds
end
end
tofmass = (toftime*1e6*p1 + p2).^2;
h5write(file,'/analysis/tofmass',tofmass);
h5write(file,'/analysis/ions',ions);
h5write(file,'/analysis/fwhm',fwhm);
h5write(file,'/analysis/peaktime',peaktime);
h5write(file,'/analysis/zoom',zoom);h5writeatt(file,'/analysis','analyzed',1);h5writeatt(file,'/analysis','p2',p2);end
function [ions,fwhm,peaktime,zoom] = h5analysis_traceintegrate(trace,zoomtrace, c,→dt,zoomtracethresh,minpeakheight,nvsperion)interpoints = 10;
peakheight = max(zoomtrace);zoom = 1;
if peakheight < minpeakheightions = 0;
fwhm = 0;
peaktime = 0;

171
returnend
if peakheight > zoomtracethreshzoom = 0;
peaktrace = trace;
trint = interp1(1:length(peaktrace),peaktrace,1:1/interpoints:l c,→ength(peaktrace));[peak,center] = max(trint);
for ii = center:-1:1
left = ii;
if trint(ii) < 0.004
break
end
end
for jj = center:1:length(trint)
right = jj;
if trint(jj) < 0.004
break
end
end
ions = simps(trint(left:right))*dt*1e9/nvsperion/interpoints;
leftside = trint(1:center);rightside = trint(center:length(trint));[~,left_FWHM] = min(abs(leftside-peak/2));[~,right_FWHM] = min(abs(rightside-peak/2));
fwhm = (center+right_FWHM-left_FWHM)*dt*1e6/interpoints;
peaktime = center/interpoints;else
peaktrace = zoomtrace;
smztr = smooth(peaktrace,3);[peak,center]=findpeaks(smztr,'MinPeakHeight',minpeakheight);
if isempty(center)[peak,center]=findpeaks(zoomtrace,'MinPeakHeight',minpeakhe c,→ight);
end
center = center*interpoints;
if isempty(center) %|| length(center)>=3
ions = 0;
fwhm = 0;
peaktime = 0;
returnend
trint = interp1(1:length(peaktrace),peaktrace,1:1/interpoints:l c,→ength(peaktrace));for ii = center(1):-1:1
left = ii;
if trint(ii) < 1e-4
break
end
end
for jj = center(length(center)):1:length(trint)
right = jj;
if trint(jj) < 1e-4
break
end
end
if left == rightions = 0;
fwhm = 0;
peaktime = 0;

172
returnend
ions = simps(trint(left:right))*dt*1e9/nvsperion/interpoints;[peak,kk] = max(peak);center = center(kk);leftside = trint(1:center);rightside = trint(center:length(trint));[~,left_FWHM] = min(abs(leftside-peak/2));[~,right_FWHM] = min(abs(rightside-peak/2));
fwhm = (center+right_FWHM-left_FWHM)*dt*1e6/interpoints;
peaktime = center/interpoints;
end
end

C.4

Image fitting code
Since the TOFMS detection is destructive, we cannot simultaneously measure the initial

number of ions for a given coulomb crystal and the final ions after exposure to a reactant gas. One
way around this is to infer the initial number of ions from an image of the crystal taken before
exosure to the reactant gas. This procedure is described in Ch. 3 and [38]. Here is the functionthat turns images into ion numbers.
h5analysisimagefit.m% h5analysis_integrator.m% by James Greenberg% 2019/01/15% read in images from a data file, calculate ion numbers, and write% results back to the data file.
%modified to subtract dark ions by O. A. Krohn 02/19/2020
function []=h5analysis_imagefit(file)% read in analysis attributes (image fit settings)
theta = h5readatt(file,'/imageanalysis','theta');countsthreshold = h5readatt(file,'/imageanalysis','countsthreshold');
iondensity = h5readatt(file,'/imageanalysis','iondensity');% read in image(s)imageset = h5info(file,'/Images');
numimages = numel(imageset.Datasets);
for ii = 1:numimages
imagename = ['/Images/',num2str(ii)];
pic = h5read(file,imagename)';pic(pic<countsthreshold) = 0;
pic = imrotate(pic,theta,'crop');vtol = 0.05;
vert = sum(pic,2);173
y = (1:512)';
vertfit = fit(y,vert,'smoothingspline','SmoothingParam',0.001);
y = 1:0.01:512;
vthresh = vtol*max(vert);
vert = abs(vertfit(y)-vthresh);[~,vindex] = sort(vert,'ascend');majoraxis(ii) = abs(y(vindex(1))-y(vindex(2)))/2;y0(ii) = min([y(vindex(1)),y(vindex(2))]) + majoraxis(ii);htol = 0.1;
horz = sum(pic,1)';x = (1:512)';horzfit = fit(x,horz,'smoothingspline','SmoothingParam',0.001);
x = 1:0.01:512;
hthresh = htol*max(horz);horz = abs(horzfit(x)-hthresh);[~,hindex] = sort(horz,'ascend');minoraxis(ii) = abs(x(hindex(1))-x(hindex(2)))/2;x0(ii) = min([x(hindex(1)),x(hindex(2))]) + minoraxis(ii);[darkpks, locs] = findpeaks(horz);[~,locsind] = sort(darkpks,'descend');darkradius(ii) = abs(x(locs(locsind(1)))-x(locs(locsind(2))))/2;% ellipse fit with no cylinder cut out%
imageions(ii) =,→iondensity*(4*pi*minoraxis(ii).^2.*majoraxis(ii)/3)*(1.6*10^-4)^3;%ellipse with a cylinder cut out. Added 02/19/2020
imageions(ii) = iondensity*(4*pi*minoraxis(ii).^2.*majoraxis(ii)/3-pi*d c,→
arkradius(ii).^2*2*majoraxis(ii))*(1.6*10^-4)^3;
darkions(ii) =,→iondensity*(pi*darkradius(ii).^2*2*majoraxis(ii))*(1.6*10^-4)^3;
end
end