Equilibrium Thermodynamic Properties, Structure and Dynamics of the Lithium Helium (Lihe) Van Der Waals Molecule

Home , LiHe

University of Nevada, Reno

Equilibrium Thermodynamic Properties, Structure and Dynamics of the Lithium Helium (LiHe) van der Waals Molecule

A thesis submitted in partial fulﬁllment of the

requirements for the degree of Master of Science in Physics

Naima Tariq

Dr. Jonathan D. Weinstein/Thesis Advisor

May 2015

THE GRADUATE SCHOOL

We recommend that the thesis prepared under our supervision by

NAIMA TARIQ

Entitled

Equilibrium Thermodynamic Properties, Structure and Dynamics Of the Lithium Helium (LiHe) van der Waals Molecule

be accepted in partial fulfillment of the requirements for the degree of

MASTER OF SCIENCE

Jonathan D. Weinstein, Advisor

Andrew Geraci, Committee Member

David M. Leitner, Graduate School Representative

David W. Zeh, Ph.D., Dean, Graduate School

May, 2015

i Abstract

Lithium helium (LiHe) is an interesting van der Waals molecule due to theoretical

interest in its molecular structure and properties. We use cryogenic helium buﬀer gas

cooling to produce high densities of atomic lithium at temperatures ranging from 1–5

Kelvin. LiHe molecules are formed by three body recombination:

Li + He + He ↔ LiHe + He. (1)

The Li density is continuously monitored via laser absorption spectroscopy. LiHe is

detected spectroscopically using both laser induced ﬂuorescence and laser absorption

spectroscopy. The LiHe spectrum shows good agreement with a theoretical model of

the molecular structure, with only a single bound rovibrational state.

In thermal equilibrium, the expected density of LiHe is given by

h2 3/2 /kB T nLiHe = nLi · nHe · e , (2) 2πµkBT where n is the density of the given species and is the binding energy of the single

LiHe bound state. Our data shows good agreement with this model, and we use it to determine the binding energy of the LiHe ground state. The measured binding energy is consistent with the calculated value. We also made attempts to measure the rate coeﬃcients for the reactions of Eq.1. ii

Dedicated to My loving parents Tariq and Amila, my husband Sheraz, my sisters, Shafaq and Zarmish & my brother Farhan, and my daughter Ajwa and son Hsaan. iii Citations to Previously Published Work

Portion of this thesis has appeared previously in the following paper:

“Spectroscopic Detection of the LiHe Molecule”, Naima Tariq, Nada Al

Taisan, Vijay Singh, and Jonathan D. Weinstein, Physical Review Letters

110, 153201 (2013) iv Acknowledgements

All appreciation to almighty Allah without His will and consent we cant proceed a single step. Millions of thanks to Prophet Muhammad peace be upon him who is like a beacon in every aspect whether it is the purpose of education or how to work in a group of people.

I express my gratitude and obligation to my advisor Prof. Jonathan Weinstein for his valuable discussion, kind supervision and encouragement throughout my research work. He also looked closely at the ﬁnal version of the thesis and oﬀering suggestions for improvement.

This gives me pleasure to thank my lab fellows for their collaboration with me in this thesis. I thank Nada Al Taisan and Vijay Singh for their work on spectroscopic search for LiHe molecule; Nancy Quiros for her work on the measurement of formation rates for LiHe molecule; Andrew Kanagin for his moral support.

My sincere regards go to my committee member for serving in my committee and their excellent questions. Furthermore, I would like to thank my friends, teachers and people in the society of physics who have taught and helped me.

Last but not least, I wish to express my deeply felt gratitude towards my Ammi and Abbu who are always supportive in whatever I do. It gives me satisfaction to think that they are always praying for my success. I also want to acknowledge my supportive brother and sisters for their prayers and love over the years. Finally, I would like to thank my husband Sheraz Ahmed for his love, cooperation and encouragements. v

Contents

Abstract...... i Dedication...... ii Citations to Previously Published Work...... iii Acknowledgements...... iv Contents...... v List of Tables...... vii List of Figures...... viii

1 Introduction1 1.1 van der Waals molecules...... 1 1.1.1 Nature of van der Waals Forces...... 2 1.2 Past work on vdW molecules...... 4 1.3 Helium-containing vdW molecules...... 4 1.4 Motivation...... 5 1.5 Three-body recombination...... 6

2 Experiment8 2.1 Lithium production and spectroscopy...... 8 2.2 Measurements of temperature, pressure and helium density...... 10 2.2.1 Temperature measurements...... 10 2.2.2 Pressure measurements...... 14 2.2.3 Helium density measurements...... 14 2.3 Detection setup for LiHe...... 15 2.4 Improvements in LiHe detection setup...... 17 2.5 Search for 6Li4He molecule...... 24 2.5.1 Calibrating the transition frequency for 6Li4He...... 25 2.6 LIF calibration...... 31 2.6.1 Calibration curve for ﬂuorescence with old optics setup.... 32 vi 2.6.2 Calibration curve for ﬂuorescence with new optics setup... 34

3 Equilibrium Thermodynamics Properties of 7Li4He Molecules 38

3.1 Dependence of nLiHe/nLi on helium density and temperature..... 39 3.2 Measurement of the ground-state binding energy of 7Li4He molecule. 48

4 Dynamics of LiHe 57

4.1 Prior theoretical work for the K3 ...... 57

4.2 Upper limit on three body recombination rate coeﬃcient (K3).... 57

4.3 Techniques used for the measurement of K3 ...... 58

Bibliography 68 vii

List of Tables

2.1 Center frequencies, widths and heights of iodine signal from ﬁt func- −5 −3 tion. with y0=0.97±1.8 × 10 (V) , a = 0.02±1 × 10 (V) and −1 x0=14903 (cm )...... 30 2.2 Center frequencies, widths and heights of 6Li4He spectrum from ﬁt −5 −3 function with y0=0.7±3.4 × 10 (V) , a = -0.12±3 × 10 (V) and x0 = 14903 (cm−1)...... 30 2.3 Center frequencies and transitions of Li isotopes...... 32

3.1 The fit parameter c obtained from figures 3.5 to 3.9 and its conversion to ODs ...... 48 3.2 Fit parameters for different helium densities for the measurements of ground state binding energy (B.E.) of 7Li4He molecule...... 56

4.1 The ﬁt parameters obtained from ﬁgures 4.3 to 4.6...... 66 viii

List of Figures

2.1 The optics setup for the experiment. See the text for a description of the setup...... 9 2.2 6Li spectrum after simulations. The vertical lines in black color are the relative line strengths; plotted against the left axis. The lines, from right to left, are 6Li D1 transition from F = 1/2 to F 0 = 3/2, from F = 1/2 to F 0 = 1/2, from F = 3/2 to F 0 = 3/2 and from F = 3/2 to F 0 = 1/2. The continuous curves, plotted against the right axis, are the relative OD. The curves are colored for different temperature... 12 2.3 The plot of ratio of the height of F = 1/2 peak to the valley formed by F = 3/2 and F = 1/2 peaks of 6Li spectrum versus the temperature of the atoms. The fit in black color is discussed in the text...... 13 2.4 The 6Li spectrum in blue color is recorded as a function of time at a helium density of 9 × 1017 cm−3. The fit (in black) is discussed in the text...... 14 2.5 The optics setup for the spectroscopic detection of LiHe molecules. Fluorescence is collected by a lens and steered on photo diode. The PD signal is monitored on the computer (PC) with the help of DAQ. 16 2.6 The LiHe LIF signal (in blue) and the 6Li OD (in green) as a function of time. Both are plotted against the left axis. The red curve, plotted against the right axis, is the synchronized laser scan for both lasers, but scanned over a different frequency range. The Li target is ablated at time t = 0 s. The ablation pulse causes the noise which appear as sharp lines from 0 to ∼ 10 ms. The recorded spectra were taken with the ablation energy of 40 mJ at a helium density of 7 × 1017 cm−3.. 17 ix 2.7 The top view of one of the setup for testing different combinations of lenses with cylindrical lens to get more light on phtodiode (PD). For this face to face lens combination, the lens on the right is LA1002-A and left one is LA1740-B...... 18 2.8 This is a plot for testing different lenses with cylindrical lens to improve collection efficiency for fluorescence signal on phtodiode (PD). Different colors in the graph are for different lenses and their combinations as labeled...... 19 2.9 New improved optics setup for the spectroscopic detection of LiHe molecules. Cylindrical lens is introduced in vacuum (between the cell and 4 K window) while the back to back lens combination is at 300 K window. Fluorescence is collected by the combination of lenses and steered on PD. The PD signal is monitored on the computer with the help of DAQ...... 20 2.10 The bottom view of cryogenic cell before introducing cylindrical lens. 21 2.11 The bottom view of cryogenic cell with cylindrical lens...... 22 2.12 Side view of cryogenic cell with cylindrical lens...... 23 2.13 The 7Li4He LIF (in black) and the 6Li OD (in red) as a function of time after improving detection setup. Both are plotted against the left axis. The blue curve, plotted against the right axis, is the synchronized laser scan for both lasers ,but scanned over a different frequency range. The Li target is ablated at time t = 0 s. The ablation pulse causes the noise which appear as sharp lines from 0 to ∼ 25 ms. The recorded spectra were taken with the ablation energy of 26 mJ at a helium density of 6 × 1017 cm−3...... 24 2.14 The 6Li OD (in red) and the 6Li4He LIF signal (in black) as a function of time after improving detection setup. Both are plotted against the left axis. The blue curve, plotted against the right axis, is the synchronized laser scan for both lasers ,but scanned over a different frequency range. The Li target is ablated at time t = 0 s. The ablation pulse causes the noise which appear as sharp lines from from 0 to ∼ 10 ms in 6Li OD and from 0 to ∼ 50 ms in fluorescence signal. The recorded spectra were taken with the ablation energy of 26 mJ at a helium density of 6 × 1017 cm−3...... 26 x

2.15 The 6Li4He LIF (lower axis)and iodine ratio signal (upper axis) versus frequency. The recorded spectra shown here were taken between 100 and 120 ms after the ablation pulse...... 27 2.16 The averaged 6Li4He spectrum (in black) plotted against the lower axis, and the averaged iodine spectrum (in blue) plotted against the upper axis. The curves in red color are fit functions to the spectra described in the text...... 28 2.17 The iodine signal in black color, plotted against the left axis. The red stars * plotted against the right axis, represent the iodine reference as taken from [1]...... 29 2.18 The fluorescence signal of Li and LiHe , plotted against the lower axis, and the iodine signal, plotted against the middle axis, and the iodine references are plotted against the upper right and left axis. The fluorescence lines of atomic Li are described in the table...... 31 2.19 The plot of ratio (6Li LIF/6Li OD) versus the Li probe power. The points are fit to a linear function (in black)...... 33 2.20 The plot of ratio (7Li4He LIF/6Li OD) versus the LiHe probe power. The Li probe power is fixed for this data. The black line represents a linear fit...... 34 2.21 The plot of ratio (6Li LIF/6Li OD) versus the Li probe power after improved detection setup for fluorescence. The black line corresponds to a linear fit...... 35 2.22 With the improved collecting efficiency on fluorescence detector, the ratio 7Li4He LIF/6Li OD as a function of the LiHe probe power is taken. The probe power of Li laser is kept fix. The curve in black is fit as described in the text...... 36 2.23 LiHe probe beam image...... 37

3.1 7Li4He OD plotted as a function of 7Li OD at a constant helium density of 2.3 × 1017 cm−3 and a temperature of 2.45 ± 0.2 K...... 39 3.2 The raw data for LiHe LIF in black, 6Li OD in red (both are plotted against the left axis) and laser scan in blue(plotted against the right axis) with respect to time with the ablation energy of 16 mJ at a constant helium density of 9 × 1017 cm−3...... 40 xi 3.3 The black circles are plot of pulse energy of the Nd:YAG laser versus Q-switch delay time, against the left axis. The blue circles, against the right axis, are the corresponding signal monitored by a photodiode.. 41 3.4 The ratio of 7Li4He LIF to 6Li OD as a function of temperature at a constant helium density of 6 × 1017 cm−3 .The data points at low temperatures are obtained while cell is cold and high temperature data points correspond to warm cell when helium circulation is being stopped in the cryostat...... 42 3.5 The ratio of 7Li4He LIF to 6Li OD as a function of temperature at a constant helium density of 1.2 × 1017 cm−3 and different colors correspond to different ablation energies for Li atoms. The fit is as discussed in the text...... 43 3.6 The ratio of 7Li4He LIF to 6Li OD as a function of temperature at a constant helium density of 6×1017 cm−3 and different colors correspond to different ablation energies for Li atoms. The fit is as discussed in the text...... 44 3.7 The ratio of 7Li4He LIF to 6Li OD as a function of temperature at a constant helium density of 7×1017 cm−3 and different colors correspond to different ablation energies for Li atoms. The fit is as discussed in the text...... 45 3.8 The ratio of 7Li4He LIF to 6Li OD as a function of temperature at a constant helium density of 9 × 1017 cm−3 and different colors represent different ablation energies for Li atoms. The fit is as discussed in the text...... 46 3.9 The ratio of 7Li4He LIF to 6Li OD as a function of temperature at a constant helium density of 1.2 × 1018 cm−3 and different colors correspond to different ablation energies for Li atoms. The fit is as discussed in the text...... 47 3.10 The ratio of 7Li4He OD to 7Li OD as a function of helium density at temperatures from 2.5 to 6 K. The ratio of 7Li4He OD to 7Li OD is linear with the helium density as expected for this chemical reaction. The vertical error bars are from 7Li OD and from the fit for 7Li4He OD to 7Li OD as function of temperature. The horizontal error bars are from helium fill line diameter, ablation pulse and temperature.. 49 xii 3.11 Raw data for ratio of 7Li4He LIF and 6Li OD as a function of temperatures at a constant helium density of 6 × 1017 cm−3 and different colors correspond to different ablation energies for Li atoms...... 50 3.12 The ratio of 7Li4He OD to 7Li OD as a function of temperature at a constant helium density of 1.2 × 1017 cm−3. The fit and error bars are as discussed in the text...... 51 3.13 The ratio of 7Li4He OD to 7Li OD as a function of temperature at a constant helium density of 6 × 1017 cm−3. The fit and error bars are as discussed in the text...... 52 3.14 The ratio of 7Li4He OD to 7Li OD as a function of temperature at a constant helium density of 7 × 1017 cm−3. The fit and error bars are as discussed in the text...... 53 3.15 The ratio of 7Li4He OD to 7Li OD as a function of temperature at a constant helium density of 9 × 1017 cm−3. The fit and error bars are as discussed in the text...... 54 3.16 The ratio of 7Li4He OD to 7Li OD as a function of temperature at a constant helium density of 1.2 × 1018 cm−3. The fit and error bars are as discussed in the text...... 55

4.1 The raw signal of 7Li4He LIF as a function of frequency at a constant helium density of 7 × 1017 cm−3 and ablation energy of 26 mJ.... 59 4.2 The plot of ratio of 7Li4He LIF to 6Li OD versus LiHe probe beam power at a constant helium density of 4 × 1016 cm−3. The Li target is ablated with the energy of 26 mJ. The data, in red , is taken for the LiHe beam waist of 0.15 mm while the one in purple is for 1.5 mm beam waist. The fit is as in the text...... 61 4.3 The ratio of 7Li4He LIF to 6Li OD as a function of pulse period at a constant helium density of 1.48 × 1016 cm−3 and ablation energy of 26 mJ . The fit is as in the text...... 62 4.4 The ratio of 7Li4He LIF to 6Li OD as a function of pulse period at a constant helium density of 2 × 1016 cm−3 and ablation energy of 26 mJ . The fit is as in the text...... 63 4.5 The ratio of 7Li4He LIF to 6Li OD as a function of pulse period at a constant helium density of 3 × 1016 cm−3 and ablation energy of 26 mJ . The fit is as in the text...... 64 xiii 4.6 The ratio of 7Li4He LIF to 6Li OD as a function of pulse period at a constant helium density of 4 × 1016 cm−3 and ablation energy of 26 mJ . The fit is as in the text...... 65 4.7 The plot of τ −1 as a function of helium density. The error bars are from the fit in Fig. 4.3 to 4.6. The fit is a constant...... 67 1

Chapter 1

Introduction

This thesis deals with the production of the LiHe van der Waals molecules in ground state by three-body recombination of Li atoms with helium buﬀer-gas at cryogenic temperatures, their detection spectroscopically and measurement of their binding energy by equilibrium thermodynamic properties. Also, some attempts are made to measure the formation rates for this molecule.

1.1 van der Waals molecules

Molecules can be divided into two groups on the basis of chemical bond strength. If the electron exchange interaction due to overlapping of the outer electron shells of the individual atoms in the molecule is attractive then the molecule is said to have a chemical bond. The chemical bond may be either covalent or ionic, depending on the distribution of electron density in the molecule [2]. A classic example for a covalent bond is the bond between the two H atoms to form H2 molecule. The bond in KCl is an ionic bond. Both of these are intramolecular interactions [3] and possess the bond energies on the order of electron volt(eV) [4].

The intermolecular bonds, containing hydrogen bonds and the van der Waals 2 forces, are weaker than the intramolecular bonds. When a hydrogen atom bound to an electronegative atom then the attraction between them is known as hydrogen bond, such as the bonds between the water molecules H2O[5]. The intramolecular interactions are stronger, by an order of magnitude or two, than the hydrogen bond

[4]. If, on the other hand, the electron exchange interaction is repulsive, then the bond in the molecule is created by the long range or van der Waals interaction; such molecules are known as van der Waals molecules.

1.1.1 Nature of van der Waals Forces

The weakest forces between atoms and molecules among intramolecular and intermolecular forces are van der Waals (vdW) forces [3]. We can divide the vdW forces into three categories based on their strength as explained below:

Dipole-Dipole Forces:

The non uniform distributions of electrons form the partial charges within the

molecules. The molecules become polar, and carry a dipole moment. For exam-

ple, the water molecule (H2O); the hydrogen atom has a slight positive charge, and the oxygen atoms carry a slight negative charge. These slight charges

attract another H2O molecule. The attractive forces between adjacent H2O molecules are known as dipole-dipole forces [3].

Dipole-Induced Dipole Forces:

These weak interactions occur between the molecules, with the even charge dis-

tribution and the polar molecules. For example if an argon(Ar) atom interact

with a polar HCl molecule then a small instantaneous dipole moment is pro-

duced on Ar atom due to the motion of electrons on one side of the nucleus.

This results weak dipole-induced dipole interactions between Ar atom and HCl

molecules [6]. 3

Induced Dipole-Induced Dipole Forces:

These interactions are also known as dispersion or London forces. These forces

occur when a momentary imbalance in the electron density on an atom or

molecule causes a dipole to be established for an instant. A helium (He) atom

is an example for this type of interaction. A He atom has symmetry in structure.

However, the electron movements around the nuclei of a pair of neighboring He

atoms can become synchronized which induces a small dipole moment on each

atom, resulting induced dipole-induced dipole forces of attraction between the

pairs of atoms. [7]. These weak interactions between He atoms have enough

energy to make He2 and due to extremely weak binding energy, it supports a single bound state [4].

These forces are also known as van der Waals forces. In liquid nitrogen, atoms

are held together by these weak interactions. The dimers and clusters of noble

gas are formed due to London dispersion forces [4].

These forces depends on the internuclear distance (r) ar r−6. They vary directly

with the polarizability and size of the atoms or molecules [5]. As He atoms

are the least polarizable atoms, so London dispersion forces are weak between

them.

Due to the long range interactions between vdW molecules, the dissociation energy is on the order of thermal energy [2]. The existence of these type of molecules is expected whenever two gas phase atoms or molecules are trapped due to their intermolecular attraction at low temperature [8]. 4 1.2 Past work on vdW molecules van der Waals molecules are produced eﬀectively at low temperatures because of the weak binding energies. The popular experimental technique used to cool down atoms and molecules and invetigate the vdW molecules is the supersonic expansion of gases

[7]. The vdW molecules containing argon and neon are produced with this technique and are studied spectroscopically [9].

Luo et. al. [10] and Schöllkopf et. al. [11] confirmed the existence of the first diatomic helium vdW molecule, He2, in mass spectrometric measurements and using a technique for the production of beam containing helium cluster. Luo et. al. used electron impact ionization of a supersonic expansion of helium with translational temperature near 1 mK to observe the helium dimer ion while Schöllkopf et. al. used a transmission grating through which the helium beam was diffracted and they were able to identify it as He2. These measurements are made at source temperatures of 150 and 300 K and source pressures of 170 and 150 bar respectively. The bond length and binding energy of He2 were calculated by the same group later and found to be hri = 52 ± 4 A,˚ and = 1.1 + 0.3/ − 0.2 mK respectively [12].

As listed in ref [13]:

”Excited-state helium vdW molecules have been observed in liquid helium [14], in dense helium gas [15, 16, 17], and in superﬂuid helium nanodroplets [18, 19].”

1.3 Helium-containing vdW molecules

As the helium is the most chemically ”inert” of the noble gases, so the vdW molecules containing He have the least binding energy. For studying these molecules, there is much interest and progress theoretically for last two decades, including questions

4 over their existence [20, 21, 22]. To date, the helium dimer He2 in the gas phase is 5 the the only ground-state helium diatomic molecule that has been directly detected

[10, 12, 11]; also it is found that the other combinations of He atoms, like 3He4He and

3 He2, do not have bound states [23]. There are weak interactions between alkali metals and helium and they posses a

−1 4 −1 well depth of 0.5 − 1.5 cm . On the other hand, for He2, the well depth is 7.6 cm but it has only a single bound state with a binding energy of about 10−3 cm−1.

One might expect that there are no bound states in the alkali-helium dimers [21].

However, Kleinekath¨ofer et. al. [21] predict that there exists a single bound rovibrational state in the X2Σ ground electronic state for all 4He–Alkali diatomic molecules.

They predicted the bond length and binding energy for 6Li4He as hri ∼ 48.5 A˚ and

0.001 cm−1 respectively while for 7Li4He, the values are predicted to be hri ∼ 28 A˚ and

−1 4 0.0039 cm . The binding energies are comparable to He2 [12] while the well depth of LiHe is signiﬁcantly smaller [23]. It is also predicted that Li3He is an unbound vdW molecule [21].

1.4 Motivation

Ag3He is the ﬁrst He containing vdW molecule which has indirect evidence of formation in buﬀer-gas-cooled magnetic traps by Brahms et. al. [24, 25] from Harvard

University. They magnetically trapped silver (Ag) atoms in the presence of dense

3He buffer gas at cryogenic temperatures and observed the Ag–3He spin-change rate coefficient in their experiment. The spin-change coefficient was observed to have a temperature dependence of T −6. They used different models for the formation and calculation of spin relaxation of Ag3He. They calculated the rate constant for the atom-atom collisions as well as for the molecular collisions. The first one has only a weak temperature dependence and does not agree with the experimental data while the molecular spin-change rate constant has good quantitative agreement with exper- 6 imental observations.

The Harvard team could not observe the absorption spectrum of Ag3He molecule.

They propose the following explanations for this result: ”First, the population of

Ag3He clusters in the experiment may have been at least two orders of magnitude smaller than our theoretically predicted value. Second, the molecular transition energies may lie outside our predictions, or the line strengths might be signiﬁcantly smaller than predicted. Third, the pair formation rate may be below 10−35 cm6 s−11, such that thermal equilibrium was not achieved within the experimental diﬀu- sion timescale. Finally, the photodissociation probability per absorbed photon may be close to unity, so that the spectroscopy beam depleted the molecular population below the experimental detection sensitivity.”

The general model provided by Brahms et. al. is based on the observations that the presence of helium buﬀer gas cooling makes the formation rates of helium vdW molecules large enough to easily achieve thermal equilibrium. This model provided us a guidance to look for vdW molecule through three body recombination.

1.5 Three-body recombination

As explained by Esry et. al. [26]:

“Three-body recombination is a three-particle process in which two parti-

cles form a bound state and the third one carries away the binding energy..

Schematically, for a X with two He atoms”,

X + He + He → XHe(v, l) + He + Evl .

where Evl is the binding energy with v and l the rovibrational states.

In our experiment, two He atoms and one Li atom perform three body recombination process in which one He atom combine with Li atom to form a LiHe molecule while 7

the other one carries away energy. This reaction takes place in a cell which contains

cryogenic helium buﬀer gas, described in Ref. [27]. As mentioned above, the general

model used in our experiment is given by Brahms et. al. [25],

K3 Li + He + He LiHe + He , K2

where K3 and K2 are the rate constants of three-body recombination and the collision- induced dissociation respectively.

From the detailed balance principle,

2 n˙ LiHe = −n˙ Li = K3 nHenLi − K2 nHenLiHe , (1.1)

where n denotes the density of species. The density of LiHe will reach thermal

equilibrium with the free Li and He densities if Li lifetime is slower than the formation

and dissociation of LiHe molecules. In thermal equilibrium,n ˙ LiHe = 0, implies

nLiHe = k(T )nLi nHe , (1.2)

where k(T ) = K3/K2 is the chemical equilibrium constant, given by the equation

nLiHe X k = = λ3 g ei/KB T . (1.3) n n dB i Li He i

Here KB is Boltzmann’s constant, T is the temperature, the sum is over all the bound states of the molecule, gi and i are the degeneracies and the (positive) binding energies of these bound states, and λdB is the thermal de Broglie wavelength of the LiHe of reduced mass µ, given by

p 2 λdB = h /2πµKBT.

It is clear from the Eq. 1.3 that the LiHe density increases exponentially with the

low temperature and the assumption T . . 8

Chapter 2

Experiment

This chapter describes the experimental set up to form LiHe molecules through three

body recombination and detect them through laser induced ﬂuorescence (LIF).

From the spectroscopy of LiHe, we know it has only one one bound state [28] , so

7 4 the expected desity of Li He with KBT ,

−3/2 −21 −3 n7Li nHe T n7 4 ≈ 10 cm · · · . (2.1) Li He 1cm−3 1cm−3 1K where n is the density of the given species and T is the temperature. [24]. It is clear from the equation that the detectable quantities of LiHe are formed with high lithium and helium densities and low temperatures. We can achieve three of them with the production of atomic lithium through laser ablation in cryogenic helium buﬀer-gas

[29].

2.1 Lithium production and spectroscopy

We use a frequency-doubled Nd:YAG laser to ablate a solid target of 99.9% pure

Li (of natural isotopic abundances) to produce gas phase lithium and use the laser absorption spectroscopy to detect the Li atoms at 671 nm [30, 31]. The detail of our 9

experimental set up is in [28]. We use photodetector signal to determine the optical

density (OD) of the atomic Li as described in Ref. [32]. We observed optical pumping

of the Li atoms at high power so we used power of ( 1µW) to probe the Li atoms

for the accurate measurements with the beam size of few mm.

YAG Wave meter

Li Cell He LiHe diode laser

Li target Li diode laser

Color filter Fabry Perot Iodine cell Glass wedge Beam blocker Mirror Neutral density Photo diode Iris Lens

Figure 2.1: The optics setup for the experiment. See the text for a description of the setup.

In our experiment, we use the range for the helium buﬀer-gas density from 1.2 ×

1017 to 1.2 × 1018 cm−3 and with the ablation energies of tens of mJ, we can get

Li densities up to 1011 cm−3. We observe the absorption spectrum of 6Li, a less

abundant isotope, for the measurement of atomic densities. We use relation between

density of the atoms and optical density to calculate Li density as explained in [28].

When the Li target is ablated, the cell temperature rises momentarily [33], so we can

measure the translational temperature with the help of absorption spectrum of 6Li as explained in section 2.2.1 because the observed dominant broadening mechanism 10 in our experiment is Doppler broadening. Timescale of 1 ms is enough to thermalize lithium with the buﬀer-gas.

We measure Li and LiHe simultaneously because the time behavior of lithium density is not explained by simple exponential decay at high He densities [34]. The diﬀusion of Li atoms through the He gas makes Li atoms lifetimes approximately

10−1 s [28].

2.2 Measurements of temperature, pressure and

helium density

2.2.1 Temperature measurements

For the cell temperature measurements, we use a silicon diode [35] and ruthenium oxide resistor. The silicon diodes are calibrated relative to a calibrated diode [36].

As mentioned before, another way to determine the cell temperature is by using Li spectrum to measure the translational temperature of atoms [28]. To get simulations at diﬀerent temperatures for the 6Li D1 Doppler broadened spectrum, we follow the steps: First, the relative line strengths of the D1 transitions is plotted as a function of their relative frequencies [37, 31]. Then, Doppler widths of the lines are calculated in hertz as given below: r 1 2K T Γ = B , (2.2) D λ m

6 where λ is Li D1 transition wave length, T is the temperature of the atoms , KB is Boltzmann’s constant and m is 6Li mass [38]. Next, Gaussian proﬁles are applied on each line center and make a sum on them:

x − x 2 A exp − 0 , ΓD 11

where A is the line strength and x0 is the relative frequency in unit of Hz. We also want to include the natural line width so we convolve the Gaussian proﬁles with

Lorentzian proﬁles: (Γ /2)2 Li , 2 2 x + (ΓLi/2) where ΓLi = 5.92 MHz is the natural linewidth of Li [38, 37]. From that, we are able to obtain the optical density versus frequency for 6Li D1 Doppler broadened

spectrum at diﬀerent temperatures as shown in Fig. 2.2. Note that we are neglecting

pressure broadening as we did not observe any evidence of it. We compare the ratio

of the height of F = 1/2 peak to the valley formed between F = 3/2 and F = 1/2

peaks of the measured 6Li spectrum with that in the simulated spectrum at diﬀerent

temperatures ≤ 2.5 K to determine the temperature of atoms and molecules in the

cell. We plot the peak to valley ratio (PV) versus the temperature (T) as shown in

Fig.2.3. We ﬁt the data points to a power function with the form:

PV = a + bT c , (2.3)

this gives a = 0.82584 ± 0.00677, b = 2.0026 ± 0.00965 and c = −2.6355 ± 0.00703.

The expression used to determine the temperature is: r PV − 0.82584 T = −2.6355 . (2.4) 2.0026

It is clear from the ﬁg. 2.2 that it is hard to distinguish the peaks and valleys at

temperatures > 2.5 K. In such cases, we ﬁnd the Doppler width of the lines by

ﬁtting four Gaussian functions to the spectrum as shown in Fig. 2.4 and measure the

temperature using Eq. 2.2. 12

3 4x10 6 Li spectrum F=3/2 4K 40 3.2K 3K 2.8K 2.6K 2.4K 3 2.2K 2K 30 1.8K 1.6K Relative OD F=1/2 1.4K 1.2K 1K 0.8K 2 0.6K 20 Line strength Line strength

1 10

0 0 -400 -200 0 200 400 Relative Frequency [MHz]

Figure 2.2: 6Li spectrum after simulations. The vertical lines in black color are the relative line strengths; plotted against the left axis. The lines, from right to left, are 6Li D1 transition from F = 1/2 to F 0 = 3/2, from F = 1/2 to F 0 = 1/2, from F = 3/2 to F 0 = 3/2 and from F = 3/2 to F 0 = 1/2. The continuous curves, plotted against the right axis, are the relative OD. The curves are colored for diﬀerent temperature. 13

12 Peak to valley ratio Fit

6 Li peak / valley 6

0.5 1.0 1.5 2.0 2.5 T (K)

Figure 2.3: The plot of ratio of the height of F = 1/2 peak to the valley formed by F = 3/2 and F = 1/2 peaks of 6Li spectrum versus the temperature of the atoms. The ﬁt in black color is discussed in the text. 14

0.15 6 Li spectrum Fit

0.10 LiOD 6 0.05

0.00

0.106 0.107 0.108 0.109 0.110 0.111 Time [s]

Figure 2.4: The 6Li spectrum in blue color is recorded as a function of time at a helium density of 9 × 1017 cm−3. The ﬁt (in black) is discussed in the text.

2.2.2 Pressure measurements

For the measurement of the cell pressure, a Pirani gauge [39] at 300 K and the cold

cell are linked with a stainless-steel tube of 0.18 cm inner diameter. We correct

for the thermomolecular pressure ratio with the Weber-Schmidt equation [40, 32] as

discussed in [41].

2.2.3 Helium density measurements

We determine the helium density in the cell, from a pressure guage at room tempera-

ture and from the cell temperature, using Weber-Schmidt equation [41]. The helium 15

density measurement has a uncertainty of ∼ ±20%. The main contributions for this

uncertainty are from momentarily rise in the cell temperature while the Li target is

ablated and the uncertainty in the tube diameter. We expect that this uncertainty

in measuring the density will be even more at low densities.

2.3 Detection setup for LiHe

A schematic optical setup for our experiment is shown in ﬁgures 2.1 and 2.5. We

use a glass wedge to split the LiHe laser beam into reﬂected low power beams and a

transmitted high power beam. We sent one of the low power beams to the wavemeter

to conﬁrm that the tuned laser is at the desired frequency. We use second low power

beam to check the laser mode status by sending it through a Fabry-Perot etalon and

also we linearize the scan by using the fringes of Fabry-Perot etalon. The last low

power beam sent through an iodine cell for the calibration of the LiHe laser.

We probe LiHe molecules with a few mW high power beam by sending it through

the cell. We use a lens of 2-inch diameter to collect the laser induced ﬂuorescence

(LIF) of LiHe from the radiative decay to the ground state. We use a Si photodiode

S2387 − 130R to observe the LIF and cover it with a single-band bandpass ﬁlter which has the average transmission higher than 90% for our spectral range (661.5 to

690.5 nm). The beam size of LiHe laser and the calculation for photodiode position are explained in reference [28]. We observed the LiHe ﬂuorescence signal as a function of time as shown in Fig. 2.6. The details for the transition frequency of LiHe are in ref [28]. 16

Cell Li laser probe beam LiHe

LiHe laser probe beam

50K 300 K PC Lens

DAQ Mirror PD with filter

Figure 2.5: The optics setup for the spectroscopic detection of LiHe molecules. Fluorescence is collected by a lens and steered on photo diode. The PD signal is monitored on the computer (PC) with the help of DAQ. 17

LIF signal 4 6 Li OD 4 Laser Scan

2 Scan voltage [V] 3

0 Li OD 6 2

LiHe Fluorescence [V] -2 1

-4 0

0.00 0.02 0.04 0.06 0.08 0.10 0.12 Time after ablation [s]

Figure 2.6: The LiHe LIF signal (in blue) and the 6Li OD (in green) as a function of time. Both are plotted against the left axis. The red curve, plotted against the right axis, is the synchronized laser scan for both lasers, but scanned over a diﬀerent frequency range. The Li target is ablated at time t = 0 s. The ablation pulse causes the noise which appear as sharp lines from 0 to ∼ 10 ms. The recorded spectra were taken with the ablation energy of 40 mJ at a helium density of 7 × 1017 cm−3.

2.4 Improvements in LiHe detection setup

The optics setup used to detect LiHe molecules through laser induced ﬂuorescence

(LIF)is described in [28]. In order to increase the collection efficiency of LIF system, we added more optics in the experiment. As the LiHe probe beam passes through the cell as a thin line, we chose to put a cylindrical lens LJ1703L2-B [42] in vacuum, outside the cell, along the beam. We tried different combinations with this lens in 18 different ways to collect more light on a long hamamatsu Si photodiode S2387−130R

[43] as shown in Fig. 2.7. For this test, we used a linear array of LEDs in place of laser beam and put a glass window at two inches in front of this linear array and then tried diﬀerent combinations with cylindrical lens. We also tried diﬀerent distances between cylindrical lens and LEDs and measure corresponding signal on photodiode as shown in Fig. 2.8. We used a combination of two lenses set as front faces with cylindrical lens as shown in Fig. 2.7, we also made measurements with this combination back to back.

Cylindrical lens LEDs

Lenses Glass window

Figure 2.7: The top view of one of the setup for testing diﬀerent combinations of lenses with cylindrical lens to get more light on phtodiode (PD). For this face to face lens combination, the lens on the right is LA1002-A and left one is LA1740-B.

We also tried an aspheric lens KPA055-C, a biconvex lens [44] and a plano convex spherical lens [45] one by one with the cylindrical lens. We found that the best 19 combination is two back to back lenses as shown in Fig. 2.8. We put the cylindrical lens between cell and 4K window, and used back to back combination of lenses at

300K window as shown in Fig. 2.9. The upper lens is LA1002-A, a plano convex lens

[46] and the lower one is LA1740-B, a plano convex spherical lens [45].

10 Back to Back Front Faces Aspheric lens 9 Plano convex spherical lens Biconvex lens

7 PD signal [v] 6

4 2.7 2.8 2.9 3.0 3.1 3.2 Cylindrical lens position from linear LED array [inch]

Figure 2.8: This is a plot for testing different lenses with cylindrical lens to improve collection efficiency for fluorescence signal on phtodiode (PD). Different colors in the graph are for different lenses and their combinations as labeled. 20

Cell Li laser probe beam LiHe

LiHe laser probe beam

Cylindrical Lens 4K

50K 300K Lenses

PD with filter DAQ PC

Figure 2.9: New improved optics setup for the spectroscopic detection of LiHe molecules. Cylindrical lens is introduced in vacuum (between the cell and 4 K window) while the back to back lens combination is at 300 K window. Fluorescence is collected by the combination of lenses and steered on PD. The PD signal is monitored on the computer with the help of DAQ. 21

Figure 2.10: The bottom view of cryogenic cell before introducing cylindrical lens. 22

Figure 2.11: The bottom view of cryogenic cell with cylindrical lens. 23

He line

Cylindrical lens

Figure 2.12: Side view of cryogenic cell with cylindrical lens.

There are two amplification stages for the detector. The first one changes voltage to current with a feedback resistor of 100 MΩ and the second one amplifies voltage with two gains , 1 and 11. We set the gain on 1 for the experiment. We found that the LiHe fluorescence signal is improved by 10 times after new detection setup and with low gain on detector as shown in fig. 2.13. 24

LIF signal 4 1.5 Laser Scan 6 LiOD

2 Scan voltage [V]

1.0 0 LiOD 6 He Fluorescence [V] 4 0.5 -2 Li 7

-4 0.0

0.00 0.05 0.10 0.15 Time after ablation [s]

Figure 2.13: The 7Li4He LIF (in black) and the 6Li OD (in red) as a function of time after improving detection setup. Both are plotted against the left axis. The blue curve, plotted against the right axis, is the synchronized laser scan for both lasers ,but scanned over a diﬀerent frequency range. The Li target is ablated at time t = 0 s. The ablation pulse causes the noise which appear as sharp lines from 0 to ∼ 25 ms. The recorded spectra were taken with the ablation energy of 26 mJ at a helium density of 6 × 1017 cm−3.

As explained in ref. [28], we identify the molecule as 7Li4He based on the excited state structure calculations and ground-state hyperﬁne splitting.

2.5 Search for 6Li4He molecule

With the improvement in detection setup, we expected to observe 6Li4He spectrum.

We started searching for its transition from atomic Li transition frequency as de- 25

scribed for 7Li4He in [28]. We use an iodine cell with the dimensions of 9.5 mm radius ×100 mm length at 300 K for the calibration of the transition frequency of the

LiHe molecule. The molecular iodine transmission spectrum as a frequency reference

[47,1] is utilized. We observed one pair of 6Li4He lines as shown in ﬁg. 2.14.

2.5.1 Calibrating the transition frequency for 6Li4He

To calibrate the transition frequency of 6Li4He we compare our iodine peaks with the

reference iodine spectrum.

We follow the same steps, for linearizing the scan to get 6Li4He, as described for

7Li4He in [28]. After that, we append the iodine ratio as a function of frequency over

the reference iodine spectrum as shown in Fig. 2.17. By shifting the frequency, we are

able to match one line of our iodine signal to the reference line at (∼ 14902.894 cm−1)

and another line at (∼ 14903.0177 cm−1) frequency by eye, which shows that the

Fabry-Perot eatlon has a free spectral range of 1.5 GHz. We take more spectra for

iodine signal, along with 6Li4He ﬂuorescence, and match them with this spectrum at

frequency of (∼ 14903.0177 cm−1) and (∼ 14903.0479 cm−1). After these calculations,

we plot the 6Li4He LIF along with the iodine ratio signal versus frequency as shown

in Fig. 2.15. The ﬁt function for the averaged iodine spectrum is a linear background

with three Gaussians as below:

2 2 2 −((x−C1)/w1) −((x−C2)/w2) −((x−C3)/w3) f(x) = y0 + a ∗ (x − x0) + h1 ∗ e + h2 ∗ e + h3 ∗ e

Similarly, we ﬁt the same function with two Gaussians to the averaged 6Li4He

spectrum as shown in ﬁg. 2.16. The ﬁt parameters are given in the table 2.2. The

error from the iodine calibration and from the ﬁt of an individual LiHe line cause

the uncertainty in frequency measurements as discussed in [28] for 7Li4He. Here are

uncertainty in the frequency measurements for 6Li4He.

From iodine lines we have, 26

LIF signal Laser Scan 4 6 LiOD 1.5

2 Scan voltage [V]

1.0 0 Li OD 6 He fluorescence [V] 4

Li -2 6 0.5

-4 0.0 0.00 0.05 0.10 0.15 0.20 Time after ablation [s]

Figure 2.14: The 6Li OD (in red) and the 6Li4He LIF signal (in black) as a function of time after improving detection setup. Both are plotted against the left axis. The blue curve, plotted against the right axis, is the synchronized laser scan for both lasers ,but scanned over a diﬀerent frequency range. The Li target is ablated at time t = 0 s. The ablation pulse causes the noise which appear as sharp lines from from 0 to ∼ 10 ms in 6Li OD and from 0 to ∼ 50 ms in ﬂuorescence signal. The recorded spectra were taken with the ablation energy of 26 mJ at a helium density of 6 × 1017 cm−3. 27

0.982 0.980 0.978 0.976 Iodine signal 0.20

0.18

0.16

0.14 He fluorescence [V] 4

Li 0.12 6

0.10 14903.00 14903.05 14903.10 14903.15 -1 Frequency [cm ]

Figure 2.15: The 6Li4He LIF (lower axis)and iodine ratio signal (upper axis) versus frequency. The recorded spectra shown here were taken between 100 and 120 ms after the ablation pulse. 28

0.982 0.980 0.978 Iodine signal 0.26 Fit

0.25

0.24

0.23

0.22 He fluorescence [V] 4 Li

6 0.21

0.20

14903.00 14903.05 14903.10 14903.15 14903.20 -1 Wavenumber [cm ]

Figure 2.16: The averaged 6Li4He spectrum (in black) plotted against the lower axis, and the averaged iodine spectrum (in blue) plotted against the upper axis. The curves in red color are ﬁt functions to the spectra described in the text. 29

12 0.980 10

0.978 Iodine reference 8

0.976 6 Iodine signal 4 0.974

2 0.972 0

3 14.9027 14.9028 14.9029 14.9030 14.9031x10 -1 Frequency [cm ]

Figure 2.17: The iodine signal in black color, plotted against the left axis. The red stars * plotted against the right axis, represent the iodine reference as taken from [1]. 30

Table 2.1: Center frequencies, widths and heights of iodine signal from ﬁt function. −5 −3 −1 with y0=0.97±1.8 × 10 (V) , a = 0.02±1 × 10 (V) and x0=14903 (cm ) center frequency (cm−1) heights (V) width (cm−1)

14903.0169 -0.0015±1.3 × 10−5 0.0106±2 × 10−4

14903.0316 -0.0011±3 × 10−5 0.0073±2 × 10−4

14903.0526 -0.002±8 × 10−6 0.0140±1 × 10−4

Table 2.2: Center frequencies, widths and heights of 6Li4He spectrum from ﬁt func- −5 −3 −1 tion with y0=0.7±3.4 × 10 (V) , a = -0.12±3 × 10 (V) and x0 = 14903 (cm ) . center frequency (cm−1) heights (V) width (cm−1)

14903.09956 ±3.6 × 10−5 0.050581 ±6 × 10−4 0.00355±3.6 × 10−5

14903.10664 ±8.3 × 10−5 0.03756 ±2 × 10−4 0.00536 ±8 × 10−5

14903.0526 - 14903.0169 = 0.0357 cm−1

while from the ref. [1],

14903.0479 - 14903.0177 = 0.0302 cm−1

which gives the estimated errors in absolute wavenumbers ∼ 0.006 cm−1.

The splitting between the 6Li4He peaks, by ﬁtting it to the function as discussed above, is 0.007 cm−1 ± 5.6 × 10−4 or 212 MHz ±17 MHz which is consistent with the ground state hyperﬁne structure of atomic lithium, 228.2 MHz. The full spectrum of

6Li4He and 7Li4He with atomic 6Li and 7Li is shown in ﬁg. 2.18 31

Atomic 6Li & 7Li

Atomic 7Li

Atomic 6Li

7Li4He 6Li4He

Figure 2.18: The ﬂuorescence signal of Li and LiHe , plotted against the lower axis, and the iodine signal, plotted against the middle axis, and the iodine references are plotted against the upper right and left axis. The ﬂuorescence lines of atomic Li are described in the table. 2.6 LIF calibration

There is a proportional relation between density and OD of the atom or the molecule

in the cell

OD = n σD l , (2.5)

where n is the average atomic or molecular density, σD is the Doppler broadened absorption cross-section, and l is the cell length.

To study the thermodynamic equilibrium properties of LiHe given by Eq. 2.1, we need to measure the LiHe density. Due to linear relationship between the LiHe OD 32

Table 2.3: Center frequencies and transitions of Li isotopes. isotope of atomic Li Center frequency (cm−1) transition levels

6 2 2 Li 14903.3 2s S1/2 → 2p P1/2

6 2 2 Li 14903.64 2s S1/2 → 2p P3/2

7 2 2 Li 14903.66 2s S1/2 → 2p P1/2

7 2 2 Li 14903.99 2s S1/2 → 2p P3/2

and density, we need to calibrate LiHe LIF for the LiHe OD measurement. As the

LiHe ODs are small, so we could not get the absorption spectra of the LiHe. We use

the absorption spectroscopy of 6Li for the calibration of LIF.

2.6.1 Calibration curve for ﬂuorescence with old optics setup

For the calibration of the LIF, we use diﬀerent probe power for taking the absorption

and LIF spectra of 6Li simultaneously using the optics setup as in Fig. 2.5. We use

the curve in the Fig. 2.19 as a calibrating curve as the ratio 6Li LIF/6Li OD is linearly

V proportional to the probe power. We obtained the slope of 0.094 µW after ﬁtting the data to line function:

6Li LIF V = 0.094 × probe beam power . (2.6) 6Li OD µW We also measure the LiHe spectra with the absorption and LIF spectroscopy simultaneously as a function of LiHe probe powers to check if high powers of LiHe probe beam cause the saturation or optical pumping of the LiHe signal. We kept Li probe power ﬁx. Fig. 2.20 shows that there is no optical pumping or saturation, otherwise, at high powers the LiHe LIF/6Li OD would be ﬂat.

LiHe LIF 6Li LIF = . (2.7) LiHe OD 6Li OD 33

3.0 6 6 Li LIF / Li OD Linear fit

2.5 100 Time after ablation (ms)

80 2.0 Li OD 6 60

1.5 40

20 1.0 Li LIF (V) / 6

0.5

0.0 0 5 10 15 20 25 30 Li probe power (µW)

Figure 2.19: The plot of ratio (6Li LIF/6Li OD) versus the Li probe power. The points are ﬁt to a linear function (in black).

In our experiment, we excite the LiHe molecules with the probe beam of 7.5 mW power, so

LiHe LIF V = 0.094 × 7.5 × 103µW , (2.8) LiHe OD µW

LiHe LIF LiHe OD = . (2.9) 705 V This Equation is used for LiHe OD calculation. 34

6 2.0 LiHe LIF / Li OD Linear fit Time after ablation (s)

-3 1.5 70x10 Li OD

6 60 50 1.0 40 30

LiHe LIF (V) / 0.5

0.0 0 2 4 6 8 10 LiHe probe power (mW)

Figure 2.20: The plot of ratio (7Li4He LIF/6Li OD) versus the LiHe probe power. The Li probe power is ﬁxed for this data. The black line represents a linear ﬁt.

2.6.2 Calibration curve for ﬂuorescence with new optics setup

After increasing collection eﬃciency on ﬂuorescence with the optics setup shown in

Fig. 2.9, we take a calibration curve for ﬂuorescence. The plot for ratio 6Li LIF/6Li

OD vs the Li probe power shows that we have been collecting more light on the detector as shown in Fig. 2.21. 35

Time after ablation [ms] 6 6 Li LIF [V] / Li OD 120 6 Linear Fit 110 100 5 90 80 Li OD

6 4

3 Li LIF [V] / 6 2

0 0 20 40 60 80 Li probe power [µW]

Figure 2.21: The plot of ratio (6Li LIF/6Li OD) versus the Li probe power after improved detection setup for ﬂuorescence. The black line corresponds to a linear ﬁt.

We measured the ratio 7Li4He LIF/6Li OD as a function of the LiHe probe power to see optical pumping or saturation of the LiHe signal for the probe beam with high powers. Fig. 2.22 shows that for high probe beam powers, the ratio is no longer linear. This indicates that the saturation phenomenon occurs at these powers. We

ﬁt this data to the the function:

f(x) = a ∗ x/(b + x)

V where a = 2.8622 mW is the slope and b = 20.11 mW is the saturation power. 36

7 4 6 Li He fluorescence [V] / Li OD Time after ablation [ms] Fit 250

200 1.5 150

100 Li OD 6 50

1.0 He fluorescence [V] / 4 Li

7 0.5

0.0

0 5 10 15 LiHe probe power [mW]

Figure 2.22: With the improved collecting efficiency on fluorescence detector, the ratio 7Li4He LIF/6Li OD as a function of the LiHe probe power is taken. The probe power of Li laser is kept fix. The curve in black is fit as described in the text.

We measure the LiHe beam size by recording the beam images with a CCD camera

[48]. The beam is few mm as shown in Fig. 2.23. 37

0 50 100 150 200 250 300 350 400 450 500 550 600 0 50 100 150 200 250 300 350 400 450

Figure 2.23: LiHe probe beam image. 38

Chapter 3

Equilibrium Thermodynamics

Properties of 7Li4He Molecules

In thermal equilibrium, from Eq. 2.1, the expected density of LiHe is given by

−3/2 −21 −3 n7Li nHe T n7 4 ≈ 10 cm · · · . (3.1) Li He 1cm−3 1cm−3 1K

We measured the LiHe signal as a function of nLi ,nHe and T [13]. We determine the LiHe OD from the LIF signal by calibrating with the absorption spectroscopy of

6Li isotope as described in the section 2.6. The 7Li4He OD as a function of 7Li OD is shown in Fig. 3.1. We choose to measure 6Li OD, less abundant isotope, due to high atomic densities in the experiment and then convert them into 7Li OD using the known isotopic ratio [30].

From Fig. 3.1, it is clear that the LiHe density is linearly dependent on the lithium density, consistent with Eq.2 and thermal equilibrium. Moreover, the relation between the Li and LiHe densities is measured to be independent of the observation time after the laser pulse, indicating that the equilibration of the Li and LiHe populations occurs on a timescale fast compared to diﬀusion.

We measure the ratio nLiHe/nLi from Eq. 3.1 at diﬀerent helium densities and 39

Data -4 4x10 Linear Fit

3 He OD 4 2 Li 7

0 0 5 10 15 20 25 7Li OD

Figure 3.1: 7Li4He OD plotted as a function of 7Li OD at a constant helium density of 2.3 × 1017 cm−3 and a temperature of 2.45 ± 0.2 K. temperatures by establishing a linear relationship between Li and 7Li4He densities.

3.1 Dependence of nLiHe/nLi on helium density and

temperature

6 We measure the LiHe LIF and Li OD simultaneously at ﬁxed nHe as shown in Fig. 3.2, then plot the ratio of LiHe ﬂuorescence to 6Li OD as a function of temperature.

The temperature is measured by the absorption spectroscopy of 6Li spectrum as

described in section 2.2.1. We change the temperature of the experiment by stopping 40

3 2 Laser scan [V]

2 0 Li OD 6 LiHe LIF [V]

-2 1

-4 0 -3 20 40 60 80x10 Time after ablation [s]

Figure 3.2: The raw data for LiHe LIF in black, 6Li OD in red (both are plotted against the left axis) and laser scan in blue(plotted against the right axis) with respect to time with the ablation energy of 16 mJ at a constant helium density of 9×1017 cm−3. the helium circulation in the cryostat [27] and ablate the Li target with different ablation energies. We obtain different ablation energies by changing the Q switch delay as shown in Fig. 3.3. The data in figures 3.5 to Fig.3.9 is plotted for the temperatures above 2.5 K and the time after 50 ms of the ablation pulse. The fit for graphs from Fig. 3.5 to Fig.3.9 ,with the assumption that there is only one ground state bound state and with KBT is,

n7 4 f(T ) = Li He = c ∗ T −3/2 (3.2) n7Li

where c is some coeﬃcient and T is temperature in kelvin. 41

140 140

120 120 YAG photodiode [mV] 100 100

80 80

60 60 Pulse energy [mJ] 40 40

20 20

0 200 250 300 350 400 Q switch delay [µs]

Figure 3.3: The black circles are plot of pulse energy of the Nd:YAG laser versus Q-switch delay time, against the left axis. The blue circles, against the right axis, are the corresponding signal monitored by a photodiode. 42

7 4 6 4 Li He LIF/ Li OD

2 Li OD 6 He LIF/

4 1

Li 9 7 8 7 6

4 0 3 4 5 2x10 Temperature [K]

Figure 3.4: The ratio of 7Li4He LIF to 6Li OD as a function of temperature at a constant helium density of 6 × 1017 cm−3 .The data points at low temperatures are obtained while cell is cold and high temperature data points correspond to warm cell when helium circulation is being stopped in the cryostat. 43

6 6 LiHe LIF [V] / Li OD Fit 5

220 Q switch [µs]

4 200 180 3 160 140 Li OD 6 2 LiHe LIF [V] / 0.1 9 8 7 6

0 4 3x10 Temperature [K]

Figure 3.5: The ratio of 7Li4He LIF to 6Li OD as a function of temperature at a constant helium density of 1.2×1017 cm−3 and different colors correspond to different ablation energies for Li atoms. The fit is as discussed in the text. 44

6 LiHe LIF / Li OD 260 Fit 240 Q switch [µs] 220 200 180 1 160 140 9

Li OD 8 6 7

LiHe LIF / 5

0 4 3x10 Temperature [K]

Figure 3.6: The ratio of 7Li4He LIF to 6Li OD as a function of temperature at a constant helium density of 6 × 1017 cm−3 and different colors correspond to different ablation energies for Li atoms. The fit is as discussed in the text. 45

3 6 LiHe LIF [V] / Li OD Fit 240 2 220 Q switch [µs] 200 180 160 Li OD

6 1 9 140 8 7 6 5 LiHe LIF [V] / 4

0 4 5 3x10 Temperature [K]

Figure 3.7: The ratio of 7Li4He LIF to 6Li OD as a function of temperature at a constant helium density of 7 × 1017 cm−3 and different colors correspond to different ablation energies for Li atoms. The fit is as discussed in the text. 46

6 LiHe fluorescence [V] / Li OD 2 Fit 240

220 Q switch [µs] 200 180 160 1

Li OD 140

6 9 8 7 6

5 LiHe LIF [v] / 4

0 4 5 3x10 Temperature [K]

Figure 3.8: The ratio of 7Li4He LIF to 6Li OD as a function of temperature at a constant helium density of 9 × 1017 cm−3 and different colors represent different ablation energies for Li atoms. The fit is as discussed in the text. 47

6 LiHe LIF [V] / Li OD 240 2 Fit 220 Q switch[µs] 200 180 160 140 Li OD 6 1 9 8

6 LiHe LIF [V] /

0 4 5 3x10 Temperature [K]

Figure 3.9: The ratio of 7Li4He LIF to 6Li OD as a function of temperature at a constant helium density of 1.2×1018 cm−3 and different colors correspond to different ablation energies for Li atoms. The fit is as discussed in the text. 48

Table 3.1: The ﬁt parameter c obtained from ﬁgures 3.5 to 3.9 and its conversion to ODs . −3 7 4 6 7 4 7 nHe (cm ) c ( Li He LIF/ Li OD) c ( Li He OD / Li OD) 1.2 ×1017 1.0778±0.042 1.242 ×10−4 ± 4.843 × 10−6

6 ×1017 4.2723±0.149 4.9242 ×10−4 ± 1.72 × 10−5

7 ×1017 3.4389±0.164 3.9555 ×10−4 ± 1.89 × 10−5

9 ×1017 4.3214±0.164 4.98184×10−4 ± 1.89 × 10−5

1.2 ×1018 5.7873±0.134 6.6655×10−4 ± 1.54 × 10−5

This data is shown in Fig. 3.10. As per from Eq. 3.1, we expect the ratio of

7Li4He OD to 7Li OD to increase linearly with the helium density which is consistent with the data as shown in Fig. 3.10.

3.2 Measurement of the ground-state binding en-

ergy of 7Li4He molecule

We use dependence of nLiHe/nLi on temperature to measure the ground state binding energy of 7Li4He. We measure LiHe LIF signal and 6Li OD simultaneously, as shown in Fig 3.2, at diﬀerent helium densities. We plot the ratio of LiHe LIF and 6Li OD with respect to temperature as shown in Fig. 3.11. The lower end of the temperature range is limited by the base temperature of our cryostat and ablation-induced heating.

At higher temperatures, our signal-to-noise ratio suﬀers due to low LiHe densities.

We convert LiHe ﬂuorescence into LiHe OD using calibration curve as described in chapter 2 and 6Li OD to 7Li OD using their isotopic ratio [30]. The vertical error bars in the data is from the measurements of 7Li OD and from binning the data while the horizontal error bars are from temperature measurements. 49

-4 7 4 7 8x10 Li He OD / Li OD Linear Fit

6 Li OD 7

4 He OD / 4 Li 7 2

0 18 0.0 0.2 0.4 0.6 0.8 1.0 1.2x10 -3 Helium Density [cm ]

Figure 3.10: The ratio of 7Li4He OD to 7Li OD as a function of helium density at temperatures from 2.5 to 6 K. The ratio of 7Li4He OD to 7Li OD is linear with the helium density as expected for this chemical reaction. The vertical error bars are from 7Li OD and from the ﬁt for 7Li4He OD to 7Li OD as function of temperature. The horizontal error bars are from helium ﬁll line diameter, ablation pulse and temperature. 50

4 260 240 Q switch [µs] 3 220 200 180 2 160 140 Li OD 6

1 9 8 7 6 He LIF [V] / 4 5 Li 7 4

0 3 4 5 6 2x10 Temperature [K]

Figure 3.11: Raw data for ratio of 7Li4He LIF and 6Li OD as a function of temperatures at a constant helium density of 6 × 1017 cm−3 and diﬀerent colors correspond to diﬀerent ablation energies for Li atoms.

We measure the the ground-state binding energy of 7Li4He molecule by ﬁtting the data as shown in ﬁgures 3.12 to 3.16 to the function

n7 4 f(T ) = Li He = c ∗ T −3/2 ∗ exp(b/1.38 × 10−23 ∗ T ). (3.3) n7Li where c is some coeﬃcient , b is the binding energy in joule and T is temperature in

−23 −1 kelvin and 1.38 × 10 is value for kB in J K . The measured values for the ground state binding energy at diﬀerent helium densities are averaged using the formula

P B.E.i 2 ¯ i σi B.E. = P 1 2 i σi 51

-4 10 9 7 4 7 8 Li He OD / Li OD Fit 7 6

4 Li OD 7 3 He OD/

4 2 Li 7

-5 10 9 8 0 3 4 5 6 2x10 Temperature [K]

Figure 3.12: The ratio of 7Li4He OD to 7Li OD as a function of temperature at a constant helium density of 1.2 × 1017 cm−3. The ﬁt and error bars are as discussed in the text. 52

7 4 7 3 Li He OD / LiOD Fit

2 Li OD 7

-4 10 He OD/ 9 4

Li 8 7 7 6

0 3 4 5 2x10 Temperature [K]

Figure 3.13: The ratio of 7Li4He OD to 7Li OD as a function of temperature at a constant helium density of 6 × 1017 cm−3. The ﬁt and error bars are as discussed in the text. 53

7 4 7 Li He OD / Li OD Fit 2

Li OD -4 7 10 9 8 7

He OD / 6 4 Li

7 5

0 3 4 5 6 2x10 Temperature [K]

Figure 3.14: The ratio of 7Li4He OD to 7Li OD as a function of temperature at a constant helium density of 7 × 1017 cm−3. The ﬁt and error bars are as discussed in the text. 54

3 7 4 7 Li He OD / Li OD Fit

2 Li OD 7

-4 10

He OD / 9 4

Li 8 7 7

0 3 4 5 2x10 Temperature [K]

Figure 3.15: The ratio of 7Li4He OD to 7Li OD as a function of temperature at a constant helium density of 9 × 1017 cm−3. The ﬁt and error bars are as discussed in the text. 55

4 7 4 7 Li HeOD / Li OD Fit

2 Li OD 7 HeOD / 4 Li 7 -4 10 9 8 7

6 0 3 4 5 2x10 Temperature [K]

Figure 3.16: The ratio of 7Li4He OD to 7Li OD as a function of temperature at a constant helium density of 1.2 × 1018 cm−3. The ﬁt and error bars are as discussed in the text. 56

Table 3.2: Fit parameters for diﬀerent helium densities for the measurements of ground state binding energy (B.E.) of 7Li4He molecule.

−3 nHe (cm ) c B.E. (J) 1.2 ×1017 1.0094 ×10−4 ± 1.84 × 10−5 7.8932×10−25±6.22 ×10−25

6×1017 4.11 ×10−4 ± 3.96 × 10−5 9.1721×10−24±3.84 ×10−24

7×1017 7.0358 ×10−4 ± 1.47 × 10−4 -1.7171×10−24±8.12 ×10−25

9×1017 7.6971×10−4 ± 1.91 × 10−4 -1.3541×10−24±1.02 ×10−25

1.2×1018 7.378×10−4 ± 1.38 × 10−4 -2.2894×10−25± 8.51 ×10−25

with the uncertainty by

2 1 σ B.E. = P 1 2 i σi is −0.0153 ± 0.0198 cm−1 which is consistent with the calculated value [49].

It can be expected that we can measure binding energy spectroscopically by com-

paring this data to photoassociation (PA)spectroscopy. Unfortunately, the PA signal

is expected to be small under our experimental conditions, and the PA linewidth

would be expected to be on the order of 1 cm−1 [50]. 57

Chapter 4

Dynamics of LiHe

This chapter explains the attempts we made to measure the formation rates for LiHe

molecule as discussed in section 1.5.

4.1 Prior theoretical work for the K3

Suno et. al. [26] studied three body recombination in helium-helium-alkali metal collisions at cold temperatures. They used adiabatic hyperspherical representation and predicted the three body recombination rate coeﬃcient is ∼ 10−27 cm6s−1 for

7Li4He at a temperature of 10 mK.

4.2 Upper limit on three body recombination rate

coeﬃcient (K3)

The broadening can be of diﬀerent kinds; Doppler broadening, pressure broadening

and broadening due to natural line width. The life time of the ground state also causes

the broadening. We measure the upper limit on K2 using the spectral line width of one of the LiHe peak. The line width of LiHe peak at frequency 14902.74 cm−1 is 58

229 MHz as shown in Fig. 4.1. The collision time for the spectral line width is,

τ −1 = 2 ∗ π ∗ f = 1.25 × 109s−1 which gives,

τ −1 −9 3 −1 K2 2 × 10 cm s . nHe .

(4.1) at the helium density of 7 × 1017 cm−3.

7 4 For Li He, with a single bound state with KBT ,

nLiHe −21 ∼ 10 nHe (4.2) nLi

Comparing equ. 1.3 and 4.2 , we can relate K3 and K2 as,

−21 K3 ∼ 10 K2 (4.3)

−30 6 −1 This shows that upper limit on K3 . 1.4 × 10 cm s .

4.3 Techniques used for the measurement of K3

We use a pulse of resonant light of higher intensity to destroy any ground-state LiHe molecules in our sample, and observe their re-population. First, we must show that we can optically deplete the LiHe population. To measure intensity, we measure the beam power with a power meter and use a CCD camera to record the LiHe beam images [48]. 59

1.0

0.8

0.6 He Fluorescence [V] 4

Li 0.4 7

0.2

14902.72 14902.74 14902.76 14902.78 -1 Wavenumber [cm ]

Figure 4.1: The raw signal of 7Li4He LIF as a function of frequency at a constant helium density of 7 × 1017 cm−3 and ablation energy of 26 mJ. 60

To investigate this, we look at the signal as a function of intensity. We plot the ratio of LiHe ﬂuorescence to Li OD as a function of the beam power (or intensity) and then ﬁt the data to the function with the assumption that the beam intensity and the density of LiHe molecules is inhomogeneous.

f(x) = A ∗ B ∗ ln(1 + x/B)

Where A is oﬀset and B is saturation power in watt.

The Fig. 4.2 shows the average ratio versus probe beam power at diﬀerent beam

waists. The data with higher intensity bend faster than the data taken with low

intensity beam which is indication that we can deplete the LiHe population.

We use an acoustic optical modulator (AOM) to chop the high intensity LiHe

beam with diﬀerent frequencies. It is based on the acousto-optic eﬀect. The sound

waves (usually at radio frequency) are used to diﬀract and shift the frequency of light.

In our experiment, we chop the LiHe beam at diﬀerent frequencies and expect to see

a behavior for the ratio of 7Li4He LIF to 6Li OD vs pulse period as shown in Fig.

4.3 to 4.6 at diﬀerent helium densities. The LiHe beam waist for all the graphs is

∼ 0.55 mm. We normalize the ratio with the temperature using Eq. 3.2 for ruling

out the temperature dependence. The data is ﬁt to the following function, with the

assumption that beam is on and oﬀ with duty cycle of 50%,

N f(x) = norm − (τ ∗ (c − 1)(1 − e(0.5∗x/(−τ))) ∗ (1 − e−x/2∗c∗τ))/(1 − e(c∗0.5∗x+0.5∗x/(−c∗τ))) x

non Where c is ,Nnorm is c*noff and τ is collision time in seconds. From the ﬁt noff parameters , we can measure τ and then by plotting τ −1 vs helium density, we can

ﬁnd K2 which is slope of the graph as in Eq. 4.1. The ﬁt parameters are in the table 61

Avg ratio [V/OD] 0.10 Fit Li OD 6 0.08

0.06

0.04 He flourescence [V] / 4 Li 7 0.02 Avg

0.00 0 2 4 6 8 10 12 14 16 LiHe probe power [mW]

Figure 4.2: The plot of ratio of 7Li4He LIF to 6Li OD versus LiHe probe beam power at a constant helium density of 4 × 1016 cm−3. The Li target is ablated with the energy of 26 mJ. The data, in red , is taken for the LiHe beam waist of 0.15 mm while the one in purple is for 1.5 mm beam waist. The ﬁt is as in the text. 62 Time after ablation[s] 7 4 6 norm Li He LIF [V] / Li OD Fit 0.20 0.15 2 0.10

Li OD 0.05 6

He LIF [V] / 0.01 4

Li 9 7 8

norm 7

2 3 4 5 6 7 2 3 4 5 6 7 -7 -6 -5 10 10 10 Pulse Period [s]

Figure 4.3: The ratio of 7Li4He LIF to 6Li OD as a function of pulse period at a constant helium density of 1.48 × 1016 cm−3 and ablation energy of 26 mJ . The ﬁt is as in the text. 63 Time after ablation [s] 7 4 6 norm Li He LIF [V] / Li OD 0.25 3 Fit 0.20

0.15

0.10 Li OD 6 2 He LIF [V] / 4 Li 7

norm 0.01 9

2 3 4 5 6 7 2 3 4 5 6 7 2 -7 -6 -5 10 10 10 Pulse Period [s]

Figure 4.4: The ratio of 7Li4He LIF to 6Li OD as a function of pulse period at a constant helium density of 2 × 1016 cm−3 and ablation energy of 26 mJ . The ﬁt is as in the text. 64

-3 7 4 6 34x10 norm Li He LIF [V] / Li OD Time after ablation [s] Fit 32 0.25

30 0.20

Li OD 0.15

6 28 0.10 26

24 He LIF [V] / 4 Li 7 22 norm 20

2 3 4 5 6 2 3 4 5 6 2 -7 -6 -5 10 10 10 Pulse Period [s]

Figure 4.5: The ratio of 7Li4He LIF to 6Li OD as a function of pulse period at a constant helium density of 3 × 1016 cm−3 and ablation energy of 26 mJ . The ﬁt is as in the text. 65

7 4 6 Time after ablation [s] 5 norm Li He LIF [V] / Li OD Fit 0.30 0.25 0.20 0.15

Li OD 0.10 6

4 He LIF [V] / 4 Li 7

norm -2 3x10

2 3 4 5 6 2 3 4 5 6 2 -7 -6 -5 10 10 10 Pulse Period [s]

Figure 4.6: The ratio of 7Li4He LIF to 6Li OD as a function of pulse period at a constant helium density of 4 × 1016 cm−3 and ablation energy of 26 mJ . The ﬁt is as in the text. 66

Table 4.1: The ﬁt parameters obtained from ﬁgures 4.3 to 4.6. −3 −1 −1 nHe (cm ) τ (s ) Nnorm c 1.48×1016 1.5458×107 ± 6.58 × 106 0.018112 ± 3.7 × 10−4 0.62639 ± 0.0526

2×1016 1.7053×107 ± 3 × 106 0.022527 ± 2.6 × 10−4 0.47346 ± 0.0339

3×1016 1.4324×107 ± 1.8 × 106 0.043782 ± 3.5 × 10−4 0.46531 ± 0.0197

4×1016 1.287×107 ± 1 × 106 0.064987 ± 5 × 10−4 0.3779 ± 0.0169

4.1. We obtain the values for τ by repeating the experiment and plot τ −1 as a function of helium density as shown in Fig. 4.7 which shows an unexpected ﬁt for the plot. The graph shows that τ −1 is independent of helium density which does not match with the Eq. 4.1. We predict that the collision time is faster than the excited state lifetime due to which we observe diﬀerent ratio at low and high pulse period in

Fig. 4.3 to 4.6.

Due to the weak forces between the LiHe molecules, the destruction and recombination rates are so fast that we can not measure them. We could make reaction slow by lowering the helium density but the signal to noise on the ﬂuorescence detector limits the measurements.

In the future experiment, we will search for TiHe molecule and measure three body recombination rate coeﬃcient for this reaction. We pick this molecule because our group has previous experience working with atomic titanium, and we are able to generate high densities of Ti atoms through laser ablation and the binding energy of

48Ti4He is predicted to be on the order of 0.72 cm−1 [51], which leads to the higher

TiHe densities than expected for LiHe. 67

6 20x10

15 ] -1

10 inv tau [s

-1 inv tau [s ] 5 Fit

0 15 0 10 20 30 40x10 -3 Helium Density [cm ]

Figure 4.7: The plot of τ −1 as a function of helium density. The error bars are from the ﬁt in Fig. 4.3 to 4.6. The ﬁt is a constant. 68

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