Equilibrium and Dynamic Studies of the Van Der Waals Molecule Titanium-Helium: Tihe

University of Nevada, Reno

Equilibrium and dynamic studies of the van der Waals molecule titanium-helium: TiHe

A dissertation submitted in partial fulﬁllment of the

requirements for the degree of Doctor of Philosophy in Physics

Nancy P. Quir´osAguilar

Dr. Jonathan D. Weinstein/Dissertation Advisor

August 2016

THE GRADUATE SCHOOL

We recommend that the dissertation prepared under our supervision by

NANCY P. QUIRÓS AGUILAR

Entitled

Equilibrium and dynamic studies of the van der Waals molecule titanium-helium: TiHe

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

DOCTOR OF PHILOSOPHY

Jonathan D. Weinstein, Ph.D., Advisor

Timur Tscherbul, Ph.D., Committee Member

Robert Guyer, Ph.D., Committee Member

Roberto Mancini, Ph.D., Committee Member

David Leitner, Ph.D., Graduate School Representative

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

August, 2016

i Abstract

Atoms that are weakly bound by the van der Waals (vdW) interaction are called van der Waals molecules. Conditions for the existence and formation of vdW molecules are favorable at low temperatures due to their weak binding energy. In the experiment described in this thesis, we used laser ablation and He buﬀer gas cooling to form the exotic vdW molecule titanium-helium: TiHe.

A spectroscopic search for the TiHe molecule was initially conducted near the

3 3 −1 a F2 → y F2 atomic Ti transition at 25107.410 cm . Several laser-induced-ﬂuorescence signals were found and studied. However, it was not possible to conclude that these peaks corresponded to a bound TiHe molecule.

We continued the search for TiHe and multiple transitions were found closely blue

3 3 −1 detuned from the a F2 → y F3 atomic Ti transition at 25227.220 cm . Measure- ments of the TiHe binding energy were obtained from the study of its equilibrium thermodynamic properties and good agreement with theoretical predictions for the molecule’s binding energy was obtained. The creation of the TiHe molecule is of great signiﬁcance, since TiHe has interesting properties: TiHe possesses an anisotropic interaction due to its non zero angular momentum and it is the ﬁrst neutral molecule characterized by the Hund’s case e).

We believe that TiHe was formed from the constituent cold atoms through three body recombination. Progress towards measuring the three body recombination rate coeﬃcient K3 will be discussed and preliminary values of this coeﬃcient will be shown. ii

Dedicada a mi mam´aLucy y a mi pap´aBernardo. iii Acknowledgements

I would like to express my deepest gratitude to my advisor Prof. Jonathan D. We- instein for his patience, support and for sharing his knowledge. Without his guidance, this dissertation would not have been possible.

I also want to thank the members of my committee: Prof. Timur Tscherbul, Prof.

Robert Guyer, Prof. Roberto Mancini and Prof. David Leitner for their invaluable advice. Special thanks to Prof. Timur Tscherbul for his theoretical work on TiHe. I want to thank Prof. David Leitner for being an excellent instructor and motivating me to pursue teaching in the future.

I want to acknowledge the lab members that collaborated in this experiment:

Naima Tariq for her work on the spectroscopic search of TiHe. My friend Jocelyn

Sanchez and Andrew Kanagin for their contribution in the measurement of the three body recombination coeﬃcient. Also, thanks to Sunil Upadhyay for his input on the experiment during the Monday meetings.

Finally, I want to give special thanks for the love, support, and friendship to God, my big loving family: my parents (Lucy and Bernardo), my brothers (Jonathan and

Cristopher), my grand parents (Bernardo, Claudia and Lourdes), my aunts and uncles

(especially Vane, Carlos A-B, Joyce and Freddy), my ﬁanc´eMartin and my friends: old and new ones. iv

Contents

Abstract...... i Dedication...... ii Acknowledgements...... iii Contents...... iv List of Tables...... vii List of Figures...... ix

1 Introduction1 1.1 Types of molecular bond...... 1 1.1.1 Intramolecular bonds...... 2 1.1.2 Intermolecular bonds...... 2 1.2 Cold molecules...... 4 1.2.1 Cold molecules applications...... 5 1.2.2 Making cold molecules...... 7 1.3 Past work on cold vdW molecules...... 8 1.3.1 Molecules with He attached...... 9 1.4 TiHe formation...... 11 1.4.1 Three body recombination process...... 11 1.5 Summary of upcoming chapters...... 13

2 TiHe Binding Energy Estimate 15 2.1 Basics on Molecular Spectroscopy...... 15 2.2 TiHe ground state potential...... 17 2.3 TiHe ground state vibrational levels...... 19 2.4 TiHe rotational levels of the ground state...... 21 2.5 Excited state of the TiHe molecule...... 22 v

3 TiHe Experiment 33 3.1 Cryogenic System...... 33 3.2 Detection of Titanium (Ti)...... 35 3.3 Detection of TiHe...... 42 3.3.1 Fluorescence detection improvement...... 44 3.4 Measurement of He density...... 46 3.5 Measurement of Ti density...... 47 3.6 Measurement of temperature...... 48

3 3 4 TiHe signals at a F2 → y F2 Ti transition 65 3 3 4.1 Spectroscopic search of TiHe at a F2 → y F2 Ti transition...... 65 4.2 Equilibrium thermodynamic studies...... 67 4.2.1 Blue detuned spectroscopic TiHe signal...... 68 4.2.1.1 Ti dependence...... 69 4.2.1.2 Temperature dependence...... 69 4.2.1.3 He dependence...... 73 4.2.2 Red detuned spectroscopic TiHe signal...... 75 4.2.2.1 Ti dependence...... 75 4.2.2.2 Temperature dependence...... 76 4.2.2.3 He dependence...... 77

3 3 5 TiHe signals at a F2 → y F3 Ti transition 91 3 3 5.1 Spectroscopic search of TiHe at a F2 → y F3 Ti transition...... 92 5.2 Equilibrium thermodynamic studies...... 96 5.2.1 TiHe signal at 25228.15 cm−1 ...... 96 5.2.1.1 Ti dependence...... 96 5.2.1.2 Temperature dependence...... 96 5.2.1.3 He dependence...... 98 5.2.2 TiHe signal at 25228.03 cm−1 ...... 98 5.2.2.1 Ti dependence...... 98 5.2.2.2 Temperature dependence...... 98 5.2.2.3 He dependence...... 99 5.2.3 TiHe signal at 25228.10 cm−1 ...... 100 5.2.3.1 Ti dependence...... 100 5.2.3.2 Temperature dependence...... 100 5.2.3.3 He dependence...... 101 vi 5.2.4 TiHe signal at 25227.99 cm−1 ...... 101 5.2.4.1 Ti dependence...... 101 5.2.4.2 Temperature dependence...... 101 5.2.4.3 He dependence...... 101

5.2.5 Variations in nHe measurements...... 102 5.2.6 Identiﬁcation of the TiHe peaks...... 104 5.2.6.1 Thermodynamic dependence...... 104 5.2.6.2 Isotope shifts...... 105 5.2.6.3 Comparison between experimental results and theoretical predictions...... 107

6 Measurement of the three body recombination coeﬃcient K3 135 6.1 Photodissociation experiments...... 137

6.1.1 TiHe cross section (σTiHe)...... 141 6.2 Experimental set up...... 143

6.3 Upper limit of K2 ...... 144

6.4 Beam waist (wx, wy)=(0.8, 0.5) mm...... 145 6.4.1 Saturation curves...... 145 6.4.1.1 Peak at 25228.15 cm−1 ...... 145 6.4.1.2 Peak at 25228.03 cm−1 ...... 146

6.4.2 K2 and K3 ...... 147

6.5 Beam waist (wx, wy)=(0.4, 0.3) mm...... 148 6.5.1 Saturation curves...... 148 6.5.1.1 Peak at 25228.15 cm−1 ...... 148 6.5.1.2 Peak at 25228.03 cm−1 ...... 148

Bibliography 169 vii

List of Tables

2.1 Calculated TiHe ground state binding energies for the rotational levels l = 0,1,2 in units of J, cm−1, K and eV using the Lennard-Jones potential...... 21 2.2 Calculated TiHe ground state binding energies for the rotational levels l = 0,1,2 in units of J, cm−1, K and eV using the Buckingham potential. 21

4.1 Binding energies and their relative uncertainty obtained at studying the temperature dependence of the TiHe interaction found at 25108.27 cm−1 at diﬀerent He densities...... 75 4.2 Binding energies and their relative uncertainty obtained at studying the temperature dependence of the TiHe interaction found at 25103.67 cm−1 at diﬀerent He densities...... 76

5.1 Binding energies and their relative uncertainty obtained at studying the temperature dependence of the TiHe interaction found at 25228.15 cm−1 at different He densities...... 97 5.2 Binding energies and their relative uncertainty obtained at studying the temperature dependence of the TiHe interaction found at 25228.03 cm−1 at different He densities...... 99 5.3 Binding energies and their relative uncertainty obtained at studying the temperature dependence of the TiHe interaction found at 25228.10 cm−1 at different He densities...... 100 5.4 Binding energies and their relative uncertainty obtained at studying the temperature dependence of the TiHe interaction found at 25227.99 cm−1 at different He densities...... 102 viii 5.5 Binding energy of the four largest transitions found at the frequency range 25227.95 cm−1 to 25228.20 cm−1. ±0.1 K and ±0.07 cm−1 correspond to the relative error; ±0.25 K and ±0.18 cm−1 to the overall error...... 104 5.6 Measured isotope shift for the TiHe molecules and the atomic isotope shifts from [1]...... 106 5.7 Identification of the TiHe peaks with their corresponding Ti isotope. 107 5.8 Predicted ground state binding energies of TiHe...... 109 5.9 Identification of the spectroscopic signals at 25227.95 cm−1 to 25228.20 cm−1 with the theoretical calculations. ±0.1 K and ±0.25 K are the relative and overall error respectively...... 110

6.1 Saturation ﬁt parameters and calculated values of K2 and K3 at the −1 TiHe frequency = 25228.15 cm and beam waist (wx, wy)=(0.8, 0.5) mm...... 146

6.2 Saturation ﬁt parameters and calculated values of K2 and K3 at the −1 TiHe frequency = 25228.03 cm and beam waist (wx, wy)=(0.8, 0.5) mm...... 146 6.3 Saturation ﬁt parameters at the TiHe frequency = 25228.15 cm−1 and

beam waist (wx, wy)=(0.4, 0.3) mm...... 148 6.4 Saturation ﬁt parameters at the TiHe frequency = 25228.03 cm−1 and

beam waist (wx, wy)=(0.4, 0.3) mm...... 149 ix

List of Figures

2.1 Lennard-Jones potential fit to the TiHe ground state potential from [2]. 18 2.2 Buckingham potential fit to the TiHe ground state potential from [2]. 19 2.3 Ground state TiHe potentials and rotational energy levels for l = 0, 1, 2 using the Lennard-Jones potential...... 22 2.4 Ground state TiHe potentials and rotational energy levels for l = 0, 1, 2 using the Buckingham potential...... 23 2.5 Wavefunctions for the rotational energy levels l = 0, 1, 2 using the Lennard-Jones potential. (The wavefunctions are not normalized).. 24 2.6 Wave functions for the rotational energy levels l = 0, 1, 2 using the Buckingham potential. (The wavefunctions are not normalized)... 25 2.7 The electronic transition from the ground to the excited state of Ti has a frequency of ∼ 25107 cm−1. For this example, it is assumed that the highest or last bound vibrational level of the excited TiHe lies at 7 cm−1 from the excited electronic Ti energy level. The ground vibrational TiHe energy level is supposed to have a very weak binding energy. A molecular transition from the last excited vibrational level to the ground vibrational level would have a frequency of 25100 cm−1. Therefore, we need to scan 7 cm−1 to the red of the Ti atomic transition to be able to find the molecular TiHe transition. According to our calculations, the ground state vibrational level is expected to have a binding energy of around 1 cm−1. Hence, the molecular transition would be found at a frequency of 25101 cm−1...... 27 2.8 Highest last bound vibrational energy level of TiHe in the excited state −78 −136 for a potential with C6 = 2.7×10 and varying C12 from 3.35×10 Jm12 to 2.5 × 10−134 Jm12. The last vibrational energy level lies within ∼ 8 × 10−23 J or ∼ 5 cm−1 from the electronic excited energy level of Ti...... 28 x

2.9 Wavefunction obtained for the Lennard-Jones potential coefficients: −78 6 −135 12 C6= 2.7 × 10 Jm and C12= 4 × 10 Jm ...... 29 2.10 Highest last bound vibrational energy level of TiHe in the excited state −77 6 for a potential with C6 = 2.7 × 10 Jm and varying C12. C12 was changed from 1.25×10−135 Jm12 to 3×10−135 Jm12. The last vibrational energy level lies within ∼ 3×10−23 J or ∼ 1.5 cm−1 from the electronic excited energy level of Ti...... 30 2.11 Wavefunction obtained for the Lennard-Jones potential coefficients: −77 6 −135 12 C6= 2.7 × 10 Jm and C12= 2.5 × 10 Jm ...... 31 −135 2.12 Lennard-Jones potentials for the constant coefficient C12= 3 × 10 12 Jm . The C6 coefficient was varied. The dashed line corresponds −77 6 −78 6 to C6= 2.7 × 10 Jm , C6= 2.7 × 10 Jm for the solid line and −79 6 −77 6 the dotted line has a C6= 2.7 × 10 Jm . C6= 2.7 × 10 Jm is the coefficient with more attractive interaction and that implies the deepest well among the potentials in the figure...... 32

3.1 Simplified schematic of the closed-cycle cryostat used in this experiment: the second stage precools He which is sent through a high impedance line to the “1 K pot” that collects liquid He. The evaporated He is re-condensed and re-circulated into the system...... 34 3.2 Photo of the cell with the Al, Ga, In and Ti targets inside...... 35 3.3 Ti level structure. The atomic Ti transitions used in this experiment are represented by the arrows. Many states that lie between a3F and y3F are omitted for simplicity...... 37 3.4 Pulse energy (mJ) vs Q switch delay (µs) of the Nd:YAG pulse laser used to ablate the Ti target in the cell...... 38 3.5 Ablation laser optics setup...... 39 3.6 Ti optics setup...... 40 xi 3.7 Ti OD spectrum at 25107.410 cm−1, ablation power of 30 mJ, probe power of 0.6 µW and a cell temperature of around 1.2 K is plotted against time after ablation. The curves corresponding to the Ti probe laser scan voltage and the FP peaks are also shown. The Ti signal is given in OD and plotted against time after ablation. The scan voltage, a periodic square wave, is shown on the upper side of the figure. The Ti probe laser is scanning back and forth a frequency range that is 3 3 −1 centered at the transition a F2 → y F2 at 25107.410 cm ...... 51 3.8 Fit of the 48Ti peak using Eq. 3.7. The fit parameters obtained are: −1 y0 = −0.10034 ± 0.000809, y1 = (−15.413 ± 0.51) s , A = 1.0895 ± −6 −6 0.00231, w = (0.0043724 ± 1.13 × 10 ) s, x0 = 0.66428 ± 7.45 × 10 s. 52 3.9 Ti OD spectrum at 25107.410 cm−1, ablation power of 30 mJ, probe power of 0.6 µW and a cell temperature of around 1.2 K vs frequency in wavenumbers...... 53 3.10 TiHe optics setup...... 54 3.11 Voltage divider used to protect the PMT from high voltages. The diode prevents negative voltages to reach the PMT...... 54 3.12 PMT signal vs time. This test signal was taken at a control voltage of 1 V, a 0.5 mW of power for the TiHe probe going through the cell. The ablation laser was off. The curve on top is the laser scan voltage. 55 3.13 PMT signal vs control voltage on the DAQ. These test signals were taken a 0.5 mW of power for the TiHe probe going through the cell, different Q switches...... 56 3.14 Optimal position of the PMT in the optics setup...... 57 3.15 Fluorescense spectrum vs time after ablation for the following conditions: frequency range from 25103.60 cm−1 to 25103.75 cm−1, ablation power 60 mJ, probe power of 3.4 mW, density of He of 3 × 1017 cm−3 and a cell temperature of 1.2 K. The curves corresponding to the TiHe probe laser scan voltage and the FP peaks are also shown. At the t=0.19 s, a pair of very small TiHe peaks appear in the fluorescence spectrum. Fluorescent light was collected using the photodiode detector...... 58 xii 3.16 Fluorescense spectrum vs time after ablation for the following conditions: frequency range from 25103.60 cm−1 to 25103.75 cm−1, ablation power 60 mJ, probe power of 3.0 mW, density of He of 3 × 1017 cm−3 and a cell temperature of 1.2 K. The curves corresponding to the TiHe probe laser scan voltage and the FP peaks are also shown. Several peaks were resolved after adding the PMT in our optics setup.... 59 3.17 Calibration curve at 25107.410 cm−1: 48Ti LIF/ 48Ti OD is plotted against the probe power. The data is fit to a line bx, with b = (1.99 ± V 0.09) µW . The Q switch is 220 µs, the PMT control voltage at the DAQ is 5V and the He density is 3 × 1017 cm−3...... 60 3.18 Calibration curve at 25227.220 cm−1: 48Ti LIF/ 48Ti OD is plotted against the probe power. The data is fit to a line bx, with b = (2.02 ± V 0.06) µW . The Q switch is 220 µs, the PMT control voltage at the DAQ is 5V and the He density is 7 × 1017 cm−3...... 61 3.19 Temperature measured by Ti spectrum vs temperature measured by cell diode. The temperature measured through the Ti spectrum is greater than the temperature measured using the cell diode. He density = 6 × 1017 cm−3...... 62 3.20 48Ti FWHM vs He density. The data represented in squares is the FWHM of Ti obtained from the Voigt profile of the Ti spectrum. Pressure broadening is affecting the Ti peaks. The data is fit to the 8 Eq. 3.22. The fit shows aD = (1.01 ± 0.0075) × 10 Hz and P = (1.07 ± 0.022) × 10−10 Hz cm3. The data shown in circles was obtained

from the relation Eq 3.23 used to deconvolve aD from the Voigt profile of Ti spectrum. The fit is to a constant c = (1.02 ± 0.0040) × 108 Hz. 63 3.21 Temperature measured using the calculated Doppler width of 48Ti and Temperature measured using the 48Ti Voigt profile plotted against the temperature measured from the cell diode. T from the calculated Doppler width shows similar values to the temperature measure from the cell diode. The blue line is: f(T ) = T . Data taken at a He density = 8 × 1017 cm−3...... 64

4.1 Blue detuned spectroscopic signals at the frequency range 25107.8 cm−1 and 25108.4 cm−1. The temperature of the cell is ∼ 1.2 K and the He density is 7.5 × 1017 cm−3...... 66 xiii 4.2 Red detuned spectroscopic signals at the frequency range 25103.35 cm−1 and 25103.75 cm−1. The temperature of the cell is ∼ 1.2 K and the He density is 8 × 1017 cm−3...... 67 3 5 −1 4.3 Spin forbidden Ti transition a F2 → z S2 at 25102.874 cm and He density 8 × 1017 cm−3...... 68 4.4 TiHe OD as a function of Ti OD at frequency 25108.27 cm−1. The data was taken at the temperature range from 1.55 K to 1.9 K and a He density of 5.4 × 1017 cm−3. The data is fit using y = bx; b = (5.6 ± 0.27) × 10−5. The color scale indicates the time after ablation. The TiHe density increases linearly with respect to the Ti density.. 70 4.5 TiHe OD/48Ti OD as a function of time after ablation. The data was 17 −3 taken at a He density of nHe = 5.4 × 10 cm . The data is fit using y = bx; b = (6.6 ± 0.43) × 10−5. The color scale indicates temperature. The ratio TiHe OD/Ti OD remains constant when plotted with respect to time after ablation...... 71 4.6 Fit equation for TiHe density dependence with temperature. E = -2K indicates a bound state. E = 2 K an unbound state. E = 0 a bound (or scattering) level at threshold. The axis are given in logarithmic scale. 72 4.7 TiHe OD/48Ti OD as a function of temperature. The data was taken 17 −3 at a He density of nHe = 1.1 × 10 cm and at a frequency 25108.27 cm−1. From the fit is obtained E = (−0.60 ± 0.3) K and c =(1.01 ± 0.14) × 10−5 K3/2. The color scale indicates the Q switch setting of the ablation laser. By varying the Q switch, the ablation energy is changed. 73 4.8 TiHe OD/48Ti OD as a function of temperature. The data was taken 17 −3 at a He density of nHe = 1.1 × 10 cm and at a frequency 25108.27 cm−1. From the fit is obtained E = (−0.49 ± 0.4) K and c =(1.11 ± 0.18) × 10−5 K3/2...... 74 4.9 TiHe OD/48Ti OD as a function of temperature at the frequency 25108.27 cm−1. E and c are obtained from the fit as discussed in the text. Notice that an increase in He density causes an increase of the TiHe signal...... 79 4.10 TiHe OD T3/2/48Ti as a function of temperature at the frequency 25108.27 cm−1...... 80 xiv 4.11 Binding energy obtained from the study of the temperature dependence of the TiHe signal at 25108.27 cm−1. The fit is to a constant line. E = (0.19±0.14) K...... 81 4.12 c coefficient vs He density. TiHe signal = 25108.27 cm−1. The data is fit using y = bx. b = (2.59 ± 0.27) × 10−22 K3/2cm3. χ2 = 18.01... 82 4.13 c coefficient vs He density. TiHe signal = 25108.27 cm−1. The data is fit using y = Ax2. A = (8.44 ± 1.51) × 10−40 K3/2cm3. χ2 = 11.59.. 83 4.14 Ratio of the height of the peak at 25108.18 cm−1 (Peak A) to the height of the peak at 25108.27 cm−1 (Peak B) versus temperature measured with the cell diode. The fit is to y = a where a=(0.81±0.029). At 17 −3 nHe = 3.1 × 10 cm ...... 84 4.15 TiHe OD vs 48Ti OD at the frequency 25103.67 cm−1 and temperatures from 1.7 K to 1.9 K. The data is fit using y = bx, b = (1.85 ± 0.11) × 10−5. The color scale represents the time after ablation. TiHe OD is linearly proportional to 48Ti OD...... 85 4.16 TiHe OD/48Ti OD as a function of temperature at the frequency 25103.67 cm−1. E and c are obtained from the fit as discussed in the text. Notice that an increase in He density causes an increase of the TIHe signal...... 86 4.17 Binding energy obtained from the study of the temperature dependence of the TiHe signal at 25103.67 cm−1. The fit is to a constant line. E=(0.28±0.14) K...... 87 4.18 Ratio of the height of the peak at 25103.65 cm−1 (Peak A) to the height of the peak at 25103.67 cm−1 (Peak B) versus temperature measured with the cell diode. The fit is to y = a where a=(0.073±0.0066). At 17 −3 nHe = 2.95 × 10 cm ...... 88 4.19 Ratio of the height of the peak at 25103.75 cm−1 (Peak C) to the height of the peak at 25103.67 cm−1 (Peak B) versus temperature measured with the cell diode. The fit is to y = a where a=(0.059±0.0016). At 17 −3 nHe = 2.95 × 10 cm ...... 89 4.20 c coefficient vs He density. TiHe signal = 25103.67 cm−1. The data is fit using y = bx. b = (6.44 ± 0.92) × 10−23 K3/2cm3...... 90 xv

5.1 Blue detuned spectroscopic signals due to an interaction between Ti and He at the frequency range 25227.95 cm−1 and 25228.20 cm−1. The temperature in the cell is ∼ 1.2 K and the He density is 6.5×1017 cm−3. The atomic transition is oﬀ scale (to the left) at 25227.220 cm−1... 93 5.2 Blue detuned spectroscopic signal due to an interaction between Ti and He at the frequency range 25227.50 cm−1 and 25227.54 cm−1. The temperature in the cell is ∼ 1.2 K, and the He density is 6.5×1017 cm−3. Thermodynamic studies of this peak could not be done due to poor signal to noise...... 94 5.3 Red detuned spectroscopic signal due to an interaction between Ti and He at the frequency range 25218.17 cm−1 and 25218.20 cm−1. The temperature in the cell is ∼ 1.2 K, and the He density is 6.5×1017 cm−3. Thermodynamic studies of this peak could not be done due to poor signal to noise...... 95 5.4 TiHe OD as a function of Ti OD at 25228.15 cm−1. The color scale indicates temperature. The TiHe density increases linearly with respect to the Ti density. The data was taken at a He density of 3.2×1016 cm−3 and ﬁt using y = bx: b = (9.54±0.43)×10−7 for the temperature range between 3 K and 4.5 K (blue), b = (1.64 ± 0.05) × 10−6 for the temperature range between 2.3 K and 3 K (green), b = (2.73 ± 0.07) × 10−6 for the temperature range between 1.8 K and 2.3 K (yellow) and b = (5.52 ± 0.17) × 10−6 for the temperature range between 1.4 K and 1.8 K (red)...... 111 5.5 TiHe OD as a function of Ti OD. The data was taken at the tem-

perature range from 1.7 K to 2.1 K and at a He density of nHe = 3.2 × 1016 cm−3. The data is fit using y = bx, b = (2.80 ± 0.09) × 10−6. The color scale indicates the time after ablation and no dependence on this time is observed so the system is in thermal equilibrium..... 112 5.6 TiHe OD/48Ti OD as a function of temperature at 25228.15 cm−1 at different He densities. E and c are obtained from the fit as discussed in the text...... 113 5.7 Binding energy obtained from the study of the temperature dependence of the TiHe signal at 25228.15 cm−1. The fit is to a constant line. E = (-1.95±0.1) K...... 114 xvi 5.8 c coefficient vs He density. TiHe signal = 25228.15 cm−1. The data is fit using y = bx. b = (9.33 ± 0.78) × 10−23 K3/2cm3...... 115 5.9 TiHe OD as a function of Ti OD at 25228.03 cm−1. The color scale indicates temperature. The TiHe density increases linearly with respect to the Ti density. The data was taken at a He density of 2.6×1017 cm−3 and fit using y = bx: b = (5.01±0.21)×10−6 for the temperature range between 3.2 K and 4 K (blue), b = (1.06 ± 0.05) × 10−5 for the temperature range between 2.2 K and 3.2 K (green), b = (1.63 ± 0.08) × 10−5 for the temperature range between 1.8 K and 2.2 K (yellow) and b = (2.72 ± 0.2) × 10−5 for the temperature range between 1.4 K and 1.8 K (red)...... 116 5.10 TiHe OD/48Ti OD as a function of temperature at 25228.03 cm−1. The data was taken at different He densities. E and c are obtained from the fit as discussed in the text...... 117 5.11 TiHe OD T3/2/48Ti OD as a function of temperature at 25228.03 cm−1. The data was taken at different He densities...... 118 5.12 Binding energy obtained from the study of the temperature dependence of the TiHe signal at 25228.03 cm−1. The fit is to a constant line. E = (-1.52±0.1) K...... 119 5.13 c coefficient vs He density. TiHe signal = 25228.03 cm−1. The data is fit using y = bx. b = (1.20 ± 0.10) × 10−22 K3/2cm3...... 120 5.14 TiHe OD as a function of Ti OD at 25228.10 cm−1. The color scale indicates temperature. The TiHe density increases linearly with respect to the Ti density. The data was taken at a He density of 4.1×1017 cm−3 and fit using y = bx: b = (3.36±0.14)×10−6 for the temperature range between 3 K and 4.5 K (blue), b = (7.56 ± 0.54) × 10−6 for the temperature range between 2 K and 3 K (green) and b = (1.34 ± 0.07) × 10−5 for the temperature range between 1.45 K and 2 K (red)...... 121 5.15 TiHe OD/48Ti OD as a function of temperature at 25228.10 cm−1. The data was taken at different He densities. E and c are obtained from the fit as discussed in the text...... 122 5.16 Binding energy obtained from the study of the temperature dependence of the TiHe signal at 25228.10 cm−1. The fit is to a constant line. E = (-1.33±0.1) K...... 123 xvii 5.17 c coefficient vs He density. TiHe signal = 25228.10 cm−1. The data is fit using y = bx. b = (4.30 ± 0.46) × 10−23 K3/2cm3...... 124 5.18 TiHe OD as a function of Ti OD at 25227.99 cm−1. The color scale indicates temperature. The TiHe density increases linearly with respect to the Ti density. The data was taken at a He density of 4.1×1017 cm−3 and fit using y = bx: b = (5.41±0.19)×10−6 for the temperature range between 3 K and 4.5 K (blue), b = (1.05 ± 0.06) × 10−5 for the temperature range between 2 K and 3 K (green) and b = (2.15 ± 0.09) × 10−5 for the temperature range between 1.45 K and 2 K (red)...... 125 5.19 TiHe OD/48Ti OD as a function of temperature at 25227.99 cm−1. The data was taken at different He densities. E and c are obtained from the fit as discussed in the text...... 126 5.20 Binding energy obtained from the study of the temperature dependence of the TiHe signal at 25227.99 cm−1. The fit is to a constant line. E = (-1.40±0.1) K...... 127 5.21 c coefficient vs He density. TiHe signal = 25227.99 cm−1. The data is fit using y = bx. b = (6.65 ± 0.73) × 10−23 K3/2cm3...... 128 5.22 He density vs Temperature of the cell for the highest He density used in the experiment. The fit is to a line a+bx where a = (7.50±0.21)×1017 cm−3; b = (−5.78 ± 10.2) × 1015cm−3/K. There is no indication of an increase of He density with respect to temperature for this case.... 129 5.23 He density vs Temperature of the cell for the lowest He used in the experiment. The fit is to a line a + bx where a = (2.83 ± 0.18) × 1016 cm−3; b = (1.64±0.94)×1015cm−3/K. There is indication of an increase of He density with respect to temperature...... 130 17 −3 5.24 Temperature dependence data at nHe = 7.9 × 10 cm . The data TiHe OD points from 48Ti OD vs T are represented by the closed circles. The binding energy E and the c coefficient are extracted from the fit: E = (−1.59±0.3) K and c = (6.69±0.9)×10−5 K3/2. The data points from TiHe OD 48 vs T are represented by the open circles. The He density is Ti ODnHe 17 16 −3 given by nHe = (6.90 × 10 + 1.46 × 10 T ) cm . The binding energy E and the c coefficient are extracted from the fit: E = (−1.70 ± 0.3) K and c = (8.78 ± 1.19) × 10−23 K3/2. The difference in the binding energies is 0.1 K...... 131 xviii

16 −3 5.25 Temperature dependence data at nHe = 3.2 × 10 cm . The data TiHe OD points from 48Ti OD vs T are represented by the closed circles. The binding energy E and the c coefficient are extracted from the fit: E = (−1.84 ± 0.2) K and c =(3.92 ± 0.4) × 10−6 K3/2. The data points from TiHe OD 48 vs T are represented by the open circles. The He density is Ti ODnHe 16 15 −3 given by nHe = (2.83 × 10 + 1.64 × 10 T ) cm . The binding energy E and the c coefficient are extracted from the fit: E = (−2.11 ± 0.2) K and c = (1.08 ± 0.11) × 10−22 K3/2. The difference in the binding energies is 0.27 K...... 132 5.26 The solid line indicates the TiHe fluorescence spectrum. The dotted 3 3 lines correspond to the Ti fluorescence spectrum at a F2 → y F3 atomic Ti transiton at 25227.220 cm−1. The identification of some of the TiHe peaks was made by analyzing the correspondence among the Ti isotopes with TiHe peaks and their isotope shift...... 133 5.27 Hund’s coupling case e)...... 134 5.28 Ground and excited state of TiHe. R00=0 is expected to have one transition and R00=2 is expected to have six transitions according to the selection rules: ∆R = 0 and ∆J = 0, ±1...... 134

6.1 The y axis corresponds to LIF and the x axis to intensity. The black curve shows the case of destruction of molecules by the laser, α = 2 V and β = 2 W/cm2. The green curve shows the case of no destruction of molecules y = x...... 139 6.2 Illustration of the fit equation of TiHe LIF. The y axis represents the TiHe LIF and the x axis the probe power. The green line shows the case of no destruction of the molecules and the black curve shows saturation (the molecules are destroyed by means of an intense non homogeneous laser beam). If there is destruction of molecules, less fluorescence is collected at the same probe power. Line = Ax; A= 0.02. The dotted line corresponds to the saturation power P sat =2 W...... 149 xix 6.3 Illustration of the fit equation of TiHe OD. The y axis represents TiHe OD, which is proportional to the number of molecules and the x axis corresponds to the probe power. If the molecules are not destroyed, then the power does not affect the number of molecules and it remains constant at varying the power, as it is seen in the curve called line. If there is destruction of molecules, then the number of molecules or the ODs decrease at increasing the power, as represented in the graph by the black curve. Line: A = 3×10−6. The dotted line corresponds to the saturation power P sat =2 W...... 150 6.4 B factor vs temperature at 25228.15 cm−1. The fit to a constant line c. c=0.097±0.0027. The B factor vs temperature at 25228.03 cm−1 shows a fit: c=0.12±0.0030...... 151 6.5 A removable mirror was located in front of the cell to redirect the TiHe probe beam path. The CCD camera was located at a position that corresponded to the center of the cell. We placed a removable mirror in front of the cell to change the path of the laser beam. From the camera pictures, the beam waist could be obtained...... 152 48 −1 6.6 TiHe LIF/Ti OD vs TiHe probe power at 25228.15 cm , nHe = 16 −1 3.4×10 cm and beam waist = (wx, wy)=(2,5) mm. The data was fit h i using a line Ax; where A = (0.0028±2.75×10−5) V/mW and AP ln 1 + x . sat Psat −5 A = (0.0028±2.93×10 ) V/mW; Psat = (10368±492) mW. There is no indication of destruction of molecules...... 153

6.7 TiHe beam path including telescope: f1 =200 mm and f2 =50 mm.. 154 6.8 Image of the TiHe laser beam at the center of the cell after adding a

telescope with f1 =200 mm and f1 =50 mm in the TiHe laser path. The x and y axis indicate pixels. The color scale corresponds to intensity

(number of photons per pixel). The beam waist is (wx, wy)=(0.8, 0.5) mm...... 155

6.9 TiHe beam path including telescope: f1 =500 mm and f2 =200 mm. 156 6.10 Image of the TiHe laser beam at the center of the cell after adding a

telescope with f1 =500 mm and f1 =200 mm in the TiHe laser path. The x and y axis indicate pixels. The color scale corresponds to inten-

sity (number of photons per pixel). The beam waist is (wx, wy)=(0.4, 0.3) mm...... 157 xx

6.11 Beam waists as a function of position, obtained by moving the camera. O in corresponds to the center of the cell. The beam enters through one of the cell windows and exits through the opposite window. The distance measured towards the entrance window corresponds to a positive location. The beam waists are ﬁt to constant lines. Open circles: Horizontal beam waist = (794.95± 9.74) µm. Vertical beam waist = (447.21±2.55) µm Closed circles: Horizontal beam waist = (400.45±4.92) µm. Vertical beam waist = (307.23±5.89) µm...... 158 6.12 The data shown in the ﬁgures were taken under the following condi- 16 −3 −1 tions: nHe = 2.1×10 cm , frequency = 25228.15 cm , and beam

waist (wx, wy)=(0.8, 0.5) mm. The figure on top corresponds to TiHe LIF/48Ti OD vs power. The data is fit to a line Ax, where A = h i (0.015±0.00031) V/mW, and is also fit to AP ln 1 + x where sat Psat A = (0.024±0.00094) V/mW; Psat = (1.72±0.24) mW. The figure on the bottom shows TiHe OD/48Ti OD vs power. A is set to 2.81×10−6. h i The data is fit to AP 1 ln 1 + x , where A = (2.95±0.12×10−6) sat x Psat V/mW; Psat = (1.72±0.24) mW. The saturation fit is a good fit for both sets of data, which implies that the molecules are being destroyed by the laser beam...... 159 6.13 The TiHe LIF/48Ti OD vs power at frequency = 25228.15 cm−1 and

beam waist (wx, wy)=(0.8, 0.5) mm and diﬀerent He densities.... 160 −1 6.14 Saturation power vs He density at 25228.15 cm and (wx, wy)=(0.8, 0.5) mm. The data is ﬁt to a line bx, b =(8.51±0.65×10−17) mW cm3. 161 6.15 The TiHe LIF/48Ti OD vs power at frequency = 25228.03 cm−1 and

beam waist (wx, wy)=(0.8, 0.5) mm and different He densities.... 162 −1 6.16 Saturation power vs He density at 25228.03 cm and (wx, wy)=(0.8, 0.5) mm. The data is fit to a line bx, b =(1.08±0.10×10−16)mW cm3. 163 −1 6.17 K2 and K3 at the frequency 25228.15 cm (closed circles) and 25228.03 −1 cm (open circles) and beam waist (wx, wy)=(0.8, 0.5) mm. The data −11 3 is fit using a constant line. K2 = (9.68±0.74)×10 cm /s and K3 = −31 6 −1 −10 (2.48±0.19)×10 cm /s at 25228.15 cm . K2 = (1.23 ±0.11)×10 6 −31 6 −1 cm /s and K3 = (2.22 ±0.20)×10 cm /s at at 25228.03 cm ... 164 6.18 The TiHe LIF/48Ti OD vs power at frequency = 25228.15 cm−1 and

beam waist (wx, wy)=(0.4, 0.3) mm and diﬀerent He densities.... 165 xxi

−1 6.19 Saturation power vs He density at 25228.15 cm and (wx, wy)=(0.4, 0.3) mm. The data is ﬁt to a line a + bx, a=(1.22±0.22) mW; b =(3.26±3.29×10−18) mW cm3...... 166 6.20 The TiHe LIF/48Ti OD vs power at frequency = 25228.03 cm−1 and

beam waist (wx, wy)=(0.4, 0.3) mm and diﬀerent He densities.... 167 −1 6.21 Saturation power vs He density at 25228.03 cm and (wx, wy)=(0.4, 0.3) mm. The data is ﬁt to a line a + bx, a=(1.01±0.3) mW; b =(9.92±7.69×10−18) mW cm3...... 168 1

Chapter 1

Introduction

This thesis describes the experimental creation and detection of the van der Waals molecule titanium-helium (TiHe) at low temperatures using laser ablation of titanium and helium buﬀer gas cooling. This introductory chapter explains why the study and creation of new types of cold molecules is important in cold molecule research.

1.1 Types of molecular bond

In this section, a brief overview of the diﬀerent types of chemical bonds will be discussed.

Atoms are held together through chemical bonds to form molecules. The chemical bonds can be divided into: intramolecular bonds and intermolecular bonds.

The electronic structure of the atoms that make up a molecule determine the type of intramolecular bond that will keep them together. The instantaneous arrangement of the electrons in the molecular orbitals of a molecule will attract and repel the cloud of electrons of a nearby molecule, giving rise to the intermolecular bonds [3]. 2 1.1.1 Intramolecular bonds

The intramolecular bonds are characterized by the gaining, losing, or sharing of electrons. The binding energies of the intramolecular bonds are around a few eV. The intramolecular bonds can be classiﬁed as follows [3]:

Ionic: The valence electron (or electrons) are transfered from one atom to the other.

The atoms are bound due to the attraction between ions of opposite charge.

Some ionic compounds are LiF, NaCl and KBr.

Covalent: When atoms share pairs of electrons then the bond between them is co-

valent. Molecules such as H2, Cl2 and CO2 are covalent compounds.

Metallic: In this type of bonding, the electrons move “freely” through a metal, for

example the electrons in Mg, Na and Li.

To understand how electrons are arranged and the shape and size of a molecule, chemists use diﬀerent models. One of the most successful models is the molecular orbital theory (MOT), which is based on combining the atomic orbitals of the individual atoms into molecular orbitals (MO): electrons are not assigned to individual atoms, they are considered to move about the nuclei of the whole molecule. When the molecular orbitals are lower in energy than the atomic orbitals, the MO is bonding.

But when the molecular orbitals are higher in energy than the atomic orbitals, the

MO is antibonding. This theory also predicts the stability of a bond that is known as the bond order.

1.1.2 Intermolecular bonds

The intermolecular bonds between molecules are much weaker than the intramolecular bonds. There are diﬀerent types of intermolecular attractions that are usually called long range interactions. The intermolecular bonds are classiﬁed as [3]: 3

Hydrogen bonding: This type of bonding arises from the attraction between the

hydrogen atom of a polar molecule and the electron pair of the non hydrogen

atom of another molecule. Examples of hydrogen bonding are the HF molecules.

The F atom of a HF molecule is attracted to the H atom of another HF molecule.

Another example is H2O, in which the O atom of one water molecule experiences attraction to the H atom of another water molecule. The binding energy of

hydrogen bonds is ∼ 10−1 to 10−2 eV.

Dipole-Dipole forces: Dipole moments in polar molecules give rise to dipole-dipole

forces. The dipole-dipole forces arise from the attraction of the negative end

of one molecule with the positive end of another molecule. An example of

dipole-dipole interactions can be found in molecules such as CH3CN and LiF.

Dipole-induced dipole forces: The dipole moment of a polar molecule can induce

a dipole in a non polar molecule.

Dispersion forces: The presence of instantaneous dipole moments in molecules that

do not posses permanent dipole moments can inﬂuence the appearance of in-

stantaneous dipoles on nearby molecules. The existence of induced dipoles de-

pends on the molecule’s polarizability. These attractive interactions are called

Dispersion forces or London dispersion forces.

The dipole-dipole and dispersion forces are called van der Waals forces or vdW forces. Complexes that are weakly bound only by long range electrostatic interactions

(vdW forces) are known as van der Waals molecules (vdW molecules). The binding energy of vdW complexes ranges from 10−3 to 10−6 eV. 1 eV is equal to 1.16×104 K; for this reason, it is more convenient to express the vdW binding energies in terms of kelvin. 4 1.2 Cold molecules

Over the last decade, cold molecule research has had enormous progress [4,5,6], and a wide range of applications in many diﬀerent ﬁelds; which makes the study of cold molecules of interdisciplinary interest. Cold molecule research has incorporated scientists from atomic physics, nuclear physics, chemistry, quantum information science, astrophysics, and others [6]. Cold molecules are slow, and this implies that their interaction times are long, which facilitates precision measurements.

Molecules possess more degrees of freedom than atoms. These degrees of freedom can be electronic, vibrational, and rotational, which makes the study of molecules more interesting and complicated than working with atoms [7]. For example, the complexity of the rotational and vibrational energy level structure in molecules usually precludes the use of the most common cooling technique for atoms, which is laser cooling. In order to apply laser cooling in molecules, a molecular closed-cycling transition is needed and they are rare to ﬁnd. Hence, diﬀerent cooling mechanisms have to be implemented to study cold molecules. However, the laser cooling of the polar molecule SrF was achieved by means of a closed-cycling transition that suppressed vibrational and rotational decay channels [8,9]. Also, CaF molecules were cooled by this method, using a closed rotational and almost closed-vibrational transition [10].

It is common to diﬀerentiate between cold and ultracold temperatures. The temperature regime: 1K ≥ T ≥ 1 mK is often referred to as cold. Another deﬁnition of cold is found in reference [5] and it states that the range of cold temperatures can be accessed when the molecules would exhibit its quantum nature and this happens when the de Broglie wavelength is larger or comparable to the size of the molecules. 5

The thermal de Broglie wavelength is expressed as:

s h2 λdB = (1.1) 2πµKBT

Where λdB is in m; µ is the mass in Kg; KB is the Boltzmann constant in J/K and T is the temperature in K.

The TiHe molecule, shows its quantum behavior at 1 K, its λdB ∼ 2 nm. The size of the TiHe molecule is a few Bohr radius (1 Bohr radius = 0.05 nm). The λdB from the three body collisions that form TiHe is ∼ 9 nm. This indicates that λdB for TiHe is comparable to the size of the molecule, consequently, we will be studying TiHe in the cold temperature regime.

The ultracold limit is often designated as T ≤ 1 mK [7]. Partial wave scattering

(s-wave energy below the p-wave barrier) is also used to deﬁne ultracold temperatures

[11].

1.2.1 Cold molecules applications

The following are some applications of cold molecule research:

Understanding the interstellar medium: Interstellar cores have temperatures of

∼10 K. A very large percentage of the molecules found in the interstellar medium

are “exotic” according to terrestrial standards [5]. Cold molecule research can

provide a better understanding of the conditions and formation of interstellar

molecules [12, 13, 14].

Chemical reaction physics: Chemical reactions have been studied by changing

thermodynamic parameters such as temperature and pressure. In the low tem-

perature regime, chemical reactions can be controlled by manipulating internal

molecular states of ultracold molecules or by means of an external electric ﬁeld 6

[15, 16, 17].

High resolution chemistry: Chemical reactions are often studied at room tem-

perature (high T) by averaging over reactants in a wide range of rovibrational

states and collision energies. At low temperature (or low collisional velocities

between reactants), the reactants are in pure states and resonant structures

are resolved. For example, in reference [18], the reaction rate measurements of

He* and H2, He* and Ar; show at low temperature resonant structures that are not present in the calculations obtained from high temperature measurements.

Also, Singh et.al., reported the measurement of the chemical reaction of Li and

CaH at temperatures of 1 K to 2 K. LiH was produced, the reaction rate was

obtained and the vibrational levels were observed. [19].

Testing fundamental physics: The high-resolution spectroscopy of cold molecules

[20] makes them ideal to test the time invariance of fundamental constants (such

as the proton-electron mass ratio or the ﬁne structure constant) by looking for

frequency shifts of molecular transitions [21, 22, 22]. Also, the polar molecule

ThO has been used to study the electron’s electric dipole moment [23].

Quantum information: The ground state of heteronuclear polar molecules pos-

sesses a large dipole moment [24] that can be employed in quantum computing:

molecules in their ground state could be stored in an optical lattice and their

dipole moments aligned by an electric ﬁeld. Each molecule in the lattice would

represent a qubit. Qubit operations can be obtained by ﬂipping the electric

dipole moment of the molecules [4, 25, 26]. 7 1.2.2 Making cold molecules

As it was mentioned in the last section, cooling molecules is more challenging than cooling atoms due to their complex internal level structure. There are two methods to produce cold molecules: direct and indirect. The direct cooling mechanism is based on slowing molecules. Some direct cooling techniques are: buﬀer-gas cooling and molecular beam deceleration. Another possible way to obtain cold molecules is by cooling atoms and then forming the molecules from the cold atomic species. This is the indirect mechanism to produce cold molecules. Examples of indirect cooling mechanisms: photoassociation and Feshbach resonances [11,7,6, 27, 28]. Two of these methods will be discussed (one direct and the other one indirect):

Photoassociation: Through this indirect cooling technique, molecules are created

from cold or ultracold atoms. The presence of a third body is required for the

molecule to form so energy and momentum can be conserved. In the case of

photoassociation, the third body is a photon.

The interaction of two atoms is driven by a laser. A laser photon will be

absorbed by the system and the dimer is excited to an electronic state; it can

decay releasing another photon into an unbound state or a bound level of the

electronic ground state. A second laser can be used to control the decay of the

excited molecule by stimulated emission into its ground state [7, 29].

Buﬀer-gas cooling: This direct cooling technique was implemented by J. Doyle’s

group [30] and it is used to thermalize atoms or molecules via elastic collisions

with cooled He atoms. A cold cell thermally connected to a cryogenic system

contains He buﬀer gas. He is the ideal buﬀer gas since it has the highest vapor

pressure at low temperatures. The vapor pressure for 4He at 1 K is 0.2 torr and

for 3He at 1 K is 9 torr [31]. 8

The type of injection of atomic or molecular species into the cell that contains

He buﬀer gas depends on the experiment. The atoms or molecules that will be

cooled, can be introduced in the cell using: laser ablation [32], capillary ﬁlling

[33], beam injection [34], and others.

In the experiment that will be described in this thesis, we used the He buﬀer

gas technique to cool Ti atoms by elastic collisions with He atoms and by means

of 3-body collisions of Ti and He atoms, we were expecting to produce the TiHe

molecule. We introduced Ti atoms into the cell through laser ablation of a solid

Ti target (more details on laser ablation of the Ti target will be discussed in

Chapter3). Typical He densities in this experiment are 10 16 cm−3 to 1017 cm−3.

1.3 Past work on cold vdW molecules

At the end of Section 1.1.2, it was mentioned that atoms that are held together only by vdW forces are called vdW molecules. Conditions for the existence and formation of vdW molecules are favorable at low temperatures due to their weak binding energy.

Experimental and theoretical studies have been performed on vdW molecules in the past decades: the molecular ground state of the noble gas argon (Ar2) was formed and the ground state showed a low binding energy of 76.9 cm−1 (as expected for noble gases) [35]. The alkaline earth metal Mg also showed a low binding energy in its

−1 molecular form as Mg2, the dissociation energy was 399 cm [36]. Studies on alkali metals and rare gas molecules have been done to obtain the excited state molecular potential terms [37]. The molecule NaAr was created using a supersonic expansion of Na and Ar and it had a binding energy of 41.7(8) cm−1 [38]. 9 1.3.1 Molecules with He attached

Helium is the most inert of the noble gases and it has a very small polarizability.

Thus, compounds with He should be the weakest vdW molecules. For this reason, we are interested in studying He complexes.

Polyatomic He molecules: vdW complexes have been studied in prior experiments:

benzene-helium [39], carbonyl sulﬁde helium clusters (HeN -OCS) [40], carbon

monoxide isoptologues in small helium-4 clusters (HeN CO) [41]. These studies oﬀer the possibility of understanding and characterizing new chemical species

[40].

He nanodroplets: He nanodroplets have been used as cryogenic refrigerators for

molecules and clusters, and to improve the resolution of molecular spectroscopy

[11]. He nanodroplet spectroscopy provides a better understanding of long range

forces since He droplets can pick up atoms and molecules to form complexes

when passing by a vapor cell [42]. For example, propyne, gold and silver clusters

in He droplets have been grown [43]. The molecules NaK [44], LiCs and NaCs

[45] in He nanodroplets were observed. Also, superﬂuid He atoms around the

OCS molecule were formed and the binary vdW He-OCS has been characterized

[46].

Diatomic He molecules: According to the molecular orbital theory (MOT) men-

tioned in Section 1.1.1, the diatomic molecule He2 is predicted to not exist: two electrons would be located in a bonding orbital and two of the electrons in

the antibonding orbital, which lead to a zero bond order. This means that no

bonding is expected to be created between two He atoms to form a molecule.

More sophisticated models predict the existence of He2. At low temperatures

this molecule was formed. Its binding was through vdW interactions. He2 was 10 the ﬁrst diatomic He vdW molecule detected. It was seen by Luo et. al. [47] through electron impact ionization of a supersonic expansion He beam and by

Schöllkopf et. al. in diffraction experiments. He atoms from a nozzle at 4.5 K cooled to 1 mK during their flight towards a diffraction grating. Some of the atoms formed He dimers [48]. When the atoms and molecules passed through the small slits of the grating, large and small diffraction peaks were found corresponding to He atoms and He2 molecules respectively. The binding energy found was about 1 mK (∼ 0.7 cm−1)[48, 49].

Brahms, Tscherbul et. al. trapped Ag atoms in the presence of 3He buﬀer gas and proposed the formation of the vdW Ag3He molecule as an explanation for the loss of trapped atoms. A strong temperature dependence of the ratio of the

Ag spin-change to the elastic Ag-3He collision was reported. This spin change rate could not be the result of inelastic Ag-3He collisions according to their collision theory analysis. This spin rate change should be due to the formation of the Ag3He molecule [50, 51]. In reference [51] it has been explained that the formation and dynamics of XHe vdW molecules is favored in He buﬀer gas experiments (where atomic or molecular species is represented by X).

The indirect evidence of Ag3He existence motivated our group to study the vdW molecule LiHe using He buﬀer gas and laser ablation of a solid Li target

[52]. This experiment was successful and the ground state binding energy was measured to be 0.024 ± 0.025 cm−1, which was consistent with the theoretical prediction of 0.039 cm−1 [53]. However, this reaction occurred so fast that the three body recombination coeﬃcient of LiHe could not be measured [54]. For this reason, we decided to study the formation of another He compound, a vdW molecule made of Ti and He. The reason why we chose Ti is due to the prior work at Weinstein lab with Ti [55], TiHe was predicted to have a larger bind- 11

ing energy according to its ground state potential calculations [2] and a bigger

binding energy will allow us to try to measure the three body recombination

coeﬃcient for this reaction. Also, the TiHe molecule has interesting proper-

ties, such as non zero electronic orbital angular momentum that leads to an

anisotropic interaction and it would be the ﬁrst neutral molecule characterized

by the Hund’s case e).

The creation of new vdW complexes provides understanding of long range forces since they represent a way of testing the theory of long range interactions [56,2, 57,

58, 51, 59]. The formation of vdW molecules is important in buﬀer gas experiments since buﬀer gas cooling supplies favorable conditions for the creation of weakly bound compounds [50, 51, 52]. Also, vdW molecules are relevant in supersonic beams, which is another technique used for their creation [60]. Moreover, the vdW molecule formation and dissociation play a role in the spin exchange of the atoms that make up the molecule [61].

1.4 TiHe formation

1.4.1 Three body recombination process

The TiHe vdW molecule is expected to be formed by the three body recombination

process in which one Ti atom interacts with two He atoms to produce a TiHe molecule

and a free He atom. The third body, in this case the atomic He, is needed to stabilize

the bound system by taking away the energy released in the binding process [59].

Since TiHe is weakly bound, the molecule can break due to collisions with He atoms.

The TiHe 3-body reaction is as follows:

K3 Ti + He + He TiHe + He (1.2) K2 12

6 The three body recombination coeﬃcient is K3 in units of cm /s. K2 is the collision- induced dissociation coeﬃcient in units of cm3/s.

The change of TiHe density with respect of time is:

2 n˙ TiHe = −n˙ Ti = K3 nTinHe − K2 nTiHenHe (1.3)

The TiHe density in thermal equilibrium for a single bound state of degeneracy g

is given by [50]:

3 −E/KB T nTiHe = gnTinHeλdB · e (1.4)

Where E is the binding energy. We have taken a sign convention where E is

negative for bound systems, and zero at threshold.

Eq. 1.4 is obtained by calculating the probability for a Ti atom to form a molecule

with a He atom and a binding energy E. Consider a box of volume V ﬁlled with He

atoms and a Ti atom in one of the corners of the box. The probability P for this Ti

atom to form a molecule with a He atom is expressed as:

−E1/KB T g1e PTiHe = · (number of He atoms) (1.5) P −Ei/KB T i gie

Where g represents the degeneracy and the number of He atoms is given by nHeV . Then, the probability can be rewritten as:

−E1/KB T g1e PTiHe = · nHeV (1.6) P −Ei/KB T i gie

P −Ei/KB T The expression i gie corresponds to the partition function Z of the Ti atom in the box [62]:

Z = Zgas + Zbound (1.7) 13

P −Eb/KB T Zbound = b gbe represents partition function of the TiHe molecules formed.

For the conditions of our experiment, Zbound is small compared to Zgas. The partition function is expressed as:

V Z ≈ Zgas = 3 (1.8) λdB Substituting the partition function into Eq. 1.6:

3 −E/KB T PTiHe = g1nHeλdB · e (1.9)

If there are many Ti atoms in the cell, the density of TiHe nTiHe is given by:

nTiHe = nTi · PTiHe (1.10)

From Eq. 1.10, the density of TiHe in thermal equilibrium can be obtained and it matches the expression shown in Eq. 1.4 that will be used throughout this manuscript to conﬁrm that TiHe is being produced and to obtain the TiHe ground state binding energy.

1.5 Summary of upcoming chapters

In this introductory chapter, it has been shown that the cold molecule research is a rapidly growing ﬁeld that has many interesting and interdisciplinary applications.

This thesis will present an examination of cold molecules in the weakly bound limit with the creation of the exotic vdW TiHe.

Chapter2 will present the TiHe ground state binding energy predictions. In

Chapter3 the experimental set-up used to create the TiHe molecule will be discussed.

Chapters4 and5 will show the spectroscopic signals found due to an interaction of

Ti and He and which of these signals correspond to molecular transitions of TiHe. 14

Also, the agreement of the measured binding energy with new theoretical calculations will be shown. Chapter6 will explain how to obtain the K2 and K3 coeﬃcients of the TiHe chemical reaction and the attempts made to measure them. 15

Chapter 2

TiHe Binding Energy Estimate

This chapter will focus on the calculations done to estimate the predicted ground state binding energy of the TiHe molecule based on the TiHe molecular potential of

Klos, Rode, et al. 2004 [2]. Discrepancies between these binding energies and the ones measured in our experiment motivated the recalculation of the ground state potential of TiHe.

2.1 Basics on Molecular Spectroscopy

Electrons in atoms (as in molecules) are described by their electronic orbitals.A molecule possesses more degrees of freedom than atoms. These degrees of freedom are given by the rotational and vibrational motion between the nuclei, and they give rise to rotational and vibrational energy levels.

The quantum mechanical treatment of a rotating diatomic molecule system which consists of two nuclei of mass m1 and m2 about the center of mass (rigid rotor), shows that the molecule can only rotate with discrete energies EJ . In the rigid rotor 16

approximation [63]:

EJ = BJ(J + 1) (2.1)

Where J represents the rotational quantum number:

J = 0, 1, 2, 3.... (2.2)

EJ is the energy of the Jth rotational level and it is often expressed in units of wavenumber [cm−1]. B is the rotational constant deﬁned as:

2 B ≡ ~ (2.3) 2I

Where ~ is the Planck constant and I is the moment of inertia for the diatomic molecule expressed by:

I = µr2 (2.4)

µ is the reduced mass:

m m µ = 1 2 (2.5) m1 + m2 and r is the distance between the two nuclei.

The vibrational energy levels of a diatomic molecule can be approximated by assuming that the bond between the two nuclei acts like a spring that is governed by the Hooke’s Law. The energy levels of the quantum mechanical harmonic oscillator are the following:

Eν = (ν + 1/2)hω (2.6) 17

where ν is the vibrational quantum number:

ν = 0, 1, 2, 3.... (2.7)

2.2 TiHe ground state potential

Before beginning the experimental process of ﬁnding the TiHe molecule, we calculated

the ground state binding energy (or energies depending if there are several vibrational

and rotational levels) to see if a bound molecule was expected to exist. Also, it is

convenient to estimate the rotational and vibrational energy levels in the ground and

excited state to guide our spectroscopic search for the molecule.

To calculate the vibrational and rotational energy levels of the TiHe molecule in

the ground state, we ﬁt the predicted potential curve of TiHe found in reference [2]

to the Lennard-Jones potential (VLJ ) which describes the interaction of two neutral particles (atoms, molecules):

−12 −6 VLJ(r) = C12r − C6r (2.8)

For our case, r is the separation distance between the nuclei of Ti and He. The

−6 6 term C6r [Jm ] accounts for the attraction of the particles. The repulsion between

−12 12 the particles is described by C12r [Jm ]. From the ﬁt of the TiHe ground state interaction to the Lennard-Jones potential, the C12 and C6 coeﬃcients are obtained:

−134 12 C12 = 5.578 × 10 Jm (2.9)

−78 6 C6 = 4.3124 × 10 Jm (2.10) 18

The Lennard-Jones potential ﬁt is shown in Fig. 2.1.

Figure 2.1: Lennard-Jones potential ﬁt to the TiHe ground state potential from [2].

We also used the Buckingham potential to ﬁt the points from the TiHe ground state interaction. The Buckingham potential is expressed as:

−r C −6 VB(r) = C2e 3 − C1r (2.11)

The coeﬃcients found from the Buckingham potential are:

−78 6 C1 = 3.663 × 10 Jm (2.12)

−16 C2 = 1.6998 × 10 J (2.13) 19

−11 C3 = 3.6526 × 10 m (2.14)

The Buckingham ﬁt is shown in Fig. 2.2.

Figure 2.2: Buckingham potential ﬁt to the TiHe ground state potential from [2].

2.3 TiHe ground state vibrational levels

In order to estimate the vibrational energy levels of the TiHe molecule in the ground state, we followed a simple method that allowed us to solve the radial Schrödringer equation to find a wavefunction ψ(r) for the effective potential Veff (r) that accounts for the centrifugal interaction and the Lennard-Jones (or the Buckingham potential) 20 described above (Eq. 2.15). The radial component of the effective potential interaction

Veﬀ (r) of the TiHe molecule is as follows:

~2 1 Veﬀ (r) = 2 l(l + 1) + V (r) (2.15) 2µTiHe r The ﬁrst term is the centrifugal interaction that depends on the angular momentum quantum number and the term V (r) corresponds to the Lennard-Jones or the

Buckingham potential.

The value of the constant in Eq. 2.15 is:

2 ~ = 9.09 × 10−43(Js)2/Kg (2.16) 2µTiHe The method used to calculate the vibrational and rotational energy levels is popu- larly known as “wag the dog method” which consists of finding a normalizable solution of the Schrödringerequation ψ(r) by guessing a value for the energy E. If the solutions blow up exponentially at large r, then, ψ(r) is not normalizable. When the tail of the wavefunction ψ(r) at large r “wags” for different guessed values of E, it means that we passed over an allowed energy value [64]. This method neglects the angular momentum coupling in the molecule and the θ, φ dependence of the anisotropic TiHe potential.

When solving the Schr¨odringerequation for Veﬀ (r) as given in Eq. 2.15 for the lowest angular momentum quantum number: l = 0, only one energy level was found

(as shown in the next section). We concluded that the ground state potential of TiHe has only one bound vibrational energy level. 21

l ×10−23 J cm−1 K ×10−4 eV

0 1.72 0.87 1.22 1.1

1 1.32 0.66 0.92 0.82

2 0.56 0.28 0.39 0.35

Table 2.1: Calculated TiHe ground state binding energies for the rotational levels l = 0,1,2 in units of J, cm−1, K and eV using the Lennard-Jones potential.

l ×10−23 J cm−1 K ×10−4 eV

0 1.64 0.83 1.16 1.02

1 1.24 0.62 0.87 0.77

2 0.47 0.24 0.34 0.29

Table 2.2: Calculated TiHe ground state binding energies for the rotational levels l = 0,1,2 in units of J, cm−1, K and eV using the Buckingham potential.

2.4 TiHe rotational levels of the ground state

Diﬀerent values of the angular momentum l were used to ﬁnd the rotational energy levels. Table 2.1 and Table 2.2 present a summary of the predicted TiHe rotational ground state binding energies (BE) using the Lennard-Jones potential and the Buck- ingham potential as V .

The plots in Fig. 2.3 and Fig. 2.4, show the Veﬀ (r) ground state TiHe potentials with their respective energy levels using for V (r) in Eq. 2.15 the Lennard-Jones and the Buckingham potential for TiHe respectively. Fig. 2.5 and Fig. 2.6 illustrate the wave function corresponding to the rotational levels l = 0, 1 and 2 corresponding to the Lennard-Jones and Buckingham potentials.

These estimations of the TiHe ground state binding energies do not correspond to the ones found experimentally. A new ground state TiHe potential had to be theoretically calculated, as will be explained in Chapter5. 22

Figure 2.3: Ground state TiHe potentials and rotational energy levels for l = 0, 1, 2 using the Lennard-Jones potential.

2.5 Excited state of the TiHe molecule

In this experiment, the laser-induced-ﬂuorescence detection technique was used to

ﬁnd a spectroscopic signal that can be attributed to the TiHe molecule (this will be explained in Chapter3). The ﬂuorescent signal would correspond to a transition from the excited state of the TiHe molecule to its ground state. It was essential to know at which frequency ranges a transition or transitions can occur in order to have an idea of where to look for them. This is why a prediction of the molecular excited vibrational and rotational levels for TiHe is necessary. Unfortunately, there are no predicted excited states potentials for the interaction of Ti and He that can be used in our calculations.

The binding energy of the vdW molecules is very small, hence, their potentials 23

Figure 2.4: Ground state TiHe potentials and rotational energy levels for l = 0, 1, 2 using the Buckingham potential. are very shallow [56]. For this reason, a molecular TiHe transition is expected to be found near an atomic Ti transition. We wanted to estimate how far apart in frequency a molecular TiHe transition would be located from the atomic Ti transition.

With this purpose, we roughly estimated how far red detuned a transition from the highest vibrational energy level of TiHe in the excited state is located from the atomic transition. We were interested in the transition from the last vibrational level of the excited state of TiHe to the ground state because this might be the ﬁrst transition that we would detect, also it is expected to be the strongest [65]. This estimate would indicate to us how far in frequency we needed to scan with the probe laser to ﬁnd a transition from TiHe.

For example, if the highest vibrational level of the excited TiHe molecule lies at

−1 3 3 −1 7 cm from the excited electronic Ti transition a F2 → y F2 at 25107 cm , then 24

53 3.5x10 l =0 3.0 l =1 2.5 l =2 ] 1/2 2.0

1.5

Wavefunction [m 1.0

0.5

0.0

-9 0.4 0.6 0.8 1.0 1.2 1.4 1.6x10 r[m]

Figure 2.5: Wavefunctions for the rotational energy levels l = 0, 1, 2 using the Lennard-Jones potential. (The wavefunctions are not normalized). the molecular TiHe transition to the ground state will be found at the frequency

25100 cm−1: (25107 cm−1 - 7cm−1). This means that we need to scan 7cm−1 in the direction of smaller frequencies from the atomic Ti transition (red detuned), as it is shown in Fig. 2.7

To obtain the wavefunction of the last bound vibrational energy level (l = 0), we applied the “wag the dog method” explained earlier. For the potential V (r) (Eq. 2.15), the Lennard-Jones potential was used and we guessed a value for the C6 coeﬃcient.

The choice of C6 is based on the previous experiment performed in Weinstein Lab: the formation and thermodynamic studies of the vdW molecule LiHe. The value 25

24 4x10

l =0 l =1 3 l =2 ] 1/2

2 Wavefunction [m 1

0 -9 0.4 0.6 0.8 1.0 1.2 1.4 1.6x10 r[m]

Figure 2.6: Wave functions for the rotational energy levels l = 0, 1, 2 using the Buckingham potential. (The wavefunctions are not normalized).

−78 6 of the C6 coeﬃcient is 2.7 × 10 Jm for the excited LiHe and it comes from the calculations found in reference [66].

We started our calculations by fixing C6 and varying the C12 coefficient to obtain possible excited potentials of TiHe and find its last bound vibrational energy level.

−78 6 In Fig. 2.8, the value of the C6 coeﬃcient was set to 2.7×10 Jm . From the ﬁgure, the range at which the last vibrational level could be found lies within ∼ 4 cm−1 from the electronic excited energy level of Ti. This means that we should scan ∼ 3 cm−1

(∼ 25107 cm−1 - 4 cm−1 + 1 cm−1 (estimated binding energy of the ground state)), if the excited state of TiHe behaves as described. The graph in Fig. 2.9 shows the 26

−78 6 −135 wavefunction of the last vibrational level for C6= 2.7×10 Jm and C12= 4×10 Jm12.

In Fig. 2.10, the process of maintaining the C6 coeﬃcient constant and varying

−77 6 the C12 coeﬃcient was repeated. The value chosen for C6 is 2.7 × 10 Jm . The last vibrational energy level lies within ∼ 1.5 cm−1 from the electronic excited energy level of Ti. To ﬁnd a transition from the excited state of TiHe from a potential with the previous characteristics, we need to scan ∼ 0.5 cm−1.

Because we did not know the value of C6, we repeated the process for a range of C6 values. Fig. 2.11 shows the highest vibrational state wavefunction for the Lennard-

−77 6 −135 Jones potential with the coeﬃcients C6= 2.7 × 10 Jm and C12= 2.5 × 10

12 Jm . This wavefunction has more nodes than the wavefunction in Fig. 2.9.A C6=

−77 6 −78 6 2.7 × 10 Jm allows to have more vibrational states than a C6= 2.7 × 10 Jm .

A decrease of the C6 coeﬃcient implies that there is a stronger repulsion between

Ti and He; this is why a smaller C6 can hold fewer vibrational levels, as shown in Fig. 2.12.

These calculations give us an idea of how far red detuned from the atomic Ti transition we need scan to ﬁnd a spectroscopic signal due to TiHe. In the experiment, we decided to scan about ∼ 10 cm−1 red detuned and ∼ 5 cm−1 blue detuned from the atomic Ti transition. 27

25107 cm-1 7 cm-1

(25107-7) cm-1 = 25100 cm-1

0 cm-1

Figure 2.7: The electronic transition from the ground to the excited state of Ti has a frequency of ∼ 25107 cm−1. For this example, it is assumed that the highest or last bound vibrational level of the excited TiHe lies at 7 cm−1 from the excited electronic Ti energy level. The ground vibrational TiHe energy level is supposed to have a very weak binding energy. A molecular transition from the last excited vibrational level to the ground vibrational level would have a frequency of 25100 cm−1. Therefore, we need to scan 7 cm−1 to the red of the Ti atomic transition to be able to ﬁnd the molecular TiHe transition. According to our calculations, the ground state vibrational level is expected to have a binding energy of around 1 cm−1. Hence, the molecular transition would be found at a frequency of 25101 cm−1. 28

-2

-4

-6

-23 -8x10 Energy of the last vibrational state [J]

-134 0.5 1.0 1.5 2.0 2.5x10 12 C12 [Jm ]

Figure 2.8: Highest last bound vibrational energy level of TiHe in the excited state −78 −136 12 for a potential with C6 = 2.7 × 10 and varying C12 from 3.35 × 10 Jm to 2.5 × 10−134 Jm12. The last vibrational energy level lies within ∼ 8 × 10−23 J or ∼ 5 cm−1 from the electronic excited energy level of Ti. 29

0.5 ] 1/2 0.0

-0.5 Wavefunction [m

15 -1.0x10

-9 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8x10 r[m]

Figure 2.9: Wavefunction obtained for the Lennard-Jones potential coeﬃcients: C6= −78 6 −135 12 2.7 × 10 Jm and C12= 4 × 10 Jm . 30

-1

-2

-3 Energy of the last vibrational state [J]

-23 -4x10 -135 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0x10 12 C12 [Jm ]

Figure 2.10: Highest last bound vibrational energy level of TiHe in the excited state −77 6 for a potential with C6 = 2.7 × 10 Jm and varying C12. C12 was changed from 1.25 × 10−135 Jm12 to 3 × 10−135 Jm12. The last vibrational energy level lies within ∼ 3 × 10−23 J or ∼ 1.5 cm−1 from the electronic excited energy level of Ti. 31

200 ]

1/2 0

-200

Wavefunction [m -400

-600

-9 0.5 1.0 1.5 2.0x10 r[m]

Figure 2.11: Wavefunction obtained for the Lennard-Jones potential coeﬃcients: −77 6 −135 12 C6= 2.7 × 10 Jm and C12= 2.5 × 10 Jm . 32

0 E[J]

-2

-4

-20 -6x10 -10 1 2 3 4 5 6 7x10 r[m]

−135 Figure 2.12: Lennard-Jones potentials for the constant coefficient C12= 3 × 10 12 −77 Jm . The C6 coefficient was varied. The dashed line corresponds to C6= 2.7 × 10 6 −78 6 −79 Jm , C6= 2.7 × 10 Jm for the solid line and the dotted line has a C6= 2.7 × 10 6 −77 6 Jm . C6= 2.7 × 10 Jm is the coefficient with more attractive interaction and that implies the deepest well among the potentials in the figure. 33

Chapter 3

TiHe Experiment

In this chapter, the apparatus and the experimental efforts made towards the creation and detection of the van der Waals (vdW) TiHe molecule will be discussed. We used laser ablation of a solid Ti target and He buffer gas to cool the Ti atoms. TiHe molecules form through collisions of Ti and He atoms. Through laser absorption and laser-induced-fluorescence spectroscopy we detected Ti atoms and TiHe molecules.

Also, it is discussed how the temperature, the He and Ti densities were determined.

3.1 Cryogenic System

We needed to be able to reach low temperatures to create the TiHe molecule since the bond of the vdW molecules is very weak. The cryogenic system employed in this experiment provided temperatures of around 1 K. The refrigerator used is a closed- cycle 4He fridge of the evaporation type. The main cooling power of the cryostat was given by a commercial pulse-tube cooler (temperatures from ambient to 3 K were reached) and the homebuilt recirculating 4He fridge allowed the system to achieve even lower temperatures.

He gas at 300 K is pumped by a rough pump at a pressure of 1 atm; the high 34 pressure of the 4He makes it easier to condense but there is a lot of energy that needs to be removed in the process. For this reason, the system has a series of pre-coolers and once the He is cooled, it is sent to a ≈ 3 K condenser at a few hundred Torr.

The liquid He is sent through a high impedance line to a pot at low pressure known as the “1 K pot”. The evaporated He is sent back into the cryostat to re-condense. A simpliﬁed schematic of the cryogenic system employed in this experiment is illustrated in Fig. 3.1. Details on the cryostat and its design can be found in reference [67].

He in He out

2nd High Stage impedance

1 K Pot

Figure 3.1: Simpliﬁed schematic of the closed-cycle cryostat used in this experiment: the second stage precools He which is sent through a high impedance line to the “1 K pot” that collects liquid He. The evaporated He is re-condensed and re-circulated into the system.

The creation of the TiHe molecule takes place in the cell which is thermally anchored to the “1 K pot”. The cell is made of copper, its length is 10 cm and it is ﬁlled with He buﬀer gas. Inside of the cell there are solid targets: Ti, Al, Ga and In. For this experiment we focused our attention on the Ti target that is 1.0 inch diameter and 0.125 inch thick and 99.995% pure. Fig. 3.2 shows the cell and the solid 35 targets inside.

The temperature of the cell was measured by a ruthenium oxide resistor and a silicon diode. A room temperature pressure gauge was used to determine the He density in the cell. More about temperatures and density will be discussed in later sections.

Indium

Titanium

Aluminum

Gallium

Figure 3.2: Photo of the cell with the Al, Ga, In and Ti targets inside.

3.2 Detection of Titanium (Ti)

From the metal group, the element titanium (Ti) with atomic number of 22 was used in our cold molecule experiment. Ti has several naturally occurring isotopes: 36

46Ti, 47Ti, 48Ti, 49Ti and 50Ti. The isotope 48Ti is 73.8% abundant. The other

isotopes show abundances of 8.0%, 7.4%, 5.5% and 5.4% for 46Ti, 47Ti, 49Ti and 50Ti

respectively. There are three isotopes with even mass: 46Ti, 48Ti, 50Ti and I = 0 (I is the nuclear spin), and two odd mass isotopes: 47Ti with I = 5/2 and I = 7/2 for

49Ti.

The electronic ground state conﬁguration of Ti is [Ar]4s23d2. The spin-orbit ~ ~ ~ interaction, HSO = AL·S, couples the electronic (L) and the spin angular momentum ~ ~ ~ ~ (S). The total angular momentum is given by Ja=L+S. For Ti, the coupling of the two electrons in the d level, results in S = 1 and L = 3. The ﬁne structure

splitting of the energy levels of Ti is given by the quantum number Ja, where Ja

3 = 2,3,4. Consequently, the ground state of Ti is a F2,3,4. A simpliﬁed schematic of the Ti structure level is shown in Fig. 3.3. The Ti transitions used in this work are:

3 3 −1 3 3 −1 a F2 → y F2 at 25107.410 cm and a F2 → y F3 at 25227.220 cm [68]. We were interested in detecting Ti atoms because we wanted to assure that Ti was present in the cell to produce TiHe molecules. Also, the spectrum of Ti provided information of the translational temperature of the Ti atoms; this will be explained in detail in Section 3.6.

Ti atoms were ejected into the cell by laser ablation of the solid target using a frequency-doubled Nd:YAG laser with a pulse width of ∼ 5 ns and wavelength of 532

nm. The pulsed output beam energies could be controlled by changing the Q switch

delay, as it is seen on Fig. 3.4. On this research, we used Q switch values from 160 µs to 240 µs that corresponded to energies of ∼34 mJ to 8 mJ (we have a 50:50 beam splitter in the YAG path, only half the energy reaches the target). The ablation laser path is shown in Fig. 3.5. The production and the translational temperature of

Ti were aﬀected by the amount of energy used to ablate the solid target [69]. The ablation laser caused the release of hot Ti atoms and ions into the cell. The Ti atoms 37

Figure 3.3: Ti level structure. The atomic Ti transitions used in this experiment are represented by the arrows. Many states that lie between a3F and y3F are omitted for simplicity. thermalized on a timescale of a few 100 µs [70]; we began our study of TiHe at times from ∼ 0.2 s to 1.5 s after the ablation pulse. The physical processes related to the dynamics of the ablation plume before these times were not considered in this work.

A grating stabilized diode laser was used as the Ti probe laser [71]. The optics setup for the Ti laser through the cell is illustrated in Fig 3.6. The Ti probe laser beam was split off into three beams. One beam passed through the cell and was used for laser absorption spectroscopy: a photodiode with a BG3 color filter [72] (to block light at other frequencies) was used to measure the power of the transmitted beam. Another beam was sent to a Fabry-Perot (FP) cavity to confirm that the laser 38

Pulse energy [mJ] energy Pulse 20

0 140 160 180 200 220 240 260 Q switch delay [s]

Figure 3.4: Pulse energy (mJ) vs Q switch delay (µs) of the Nd:YAG pulse laser used to ablate the Ti target in the cell. was single mode. The third beam went to a wavemeter to ensure that the laser was operating at the desired frequency. The error on the measurement of the frequency from the wavemeter is 0.1 cm−1. The mirrors used in this experiment were broadband

E02 mirrors [73] that reﬂected 98% blue laser light.

To detect Ti atoms we used laser absorption spectroscopy. This detection technique is based on the idea that the intensity of the laser light will experience attenu- ation due to absorption when it passes through some medium according to:

−OD I = I0e (3.1) Fabry-Perot Glass Waveplate cavity wedge

Ti target

Neutral Mirror densities Ti He

Prism Detector TiHe 39

Ablation laser

Beam splitter Beam blocker

Detector Ti target

He Lens Ti

Glass wedge TiHe

Figure 3.5: Ablation laser optics setup.

Where I0 is the intensity of the laser light before it passes through the medium. I is the transmitted intensity and OD is the optical density. For the case of a uniform

density sample of constant cross sections, the optical density is given by [63]:

OD = nσL (3.2)

Where n is the density of the atoms or molecules usually given in units of cm−3;

σ is the cross section of the absorber in cm2 and L is the light’s path length through

the absorber in units of cm. For this experiment L=10 cm, which is the length of the

cell.

We determined the Ti ODs by means of a photodetector that measured the changes

in voltages caused by the absorption of light when it passed through the atoms. The

background voltage was measured when the laser light passed through the cell and

there were no Ti atoms in it (no ablation of the Ti target). When the Ti target was 40

Fabry-Perot Glass Waveplate Wavemeter cavity wedge

Ti target

Neutral Mirror densities Ti He

Prism Detector TiHe

Ti probe laser

Figure 3.6: Ti optics setup. ablated, some of the laser light was absorbed by the Ti atoms and this absorption caused a decrease in the voltage measured by the photodetector. Transmission of the laser light is given by the following expressions:

I T = = e−OD (3.3) I0

V T = (3.4) V0

Where V is the voltage on the photodetector with atoms in the cell and and V0 is the voltage on photodetector with no atoms in the cell (background voltage).

By relating Eq. 3.3 and Eq. 3.4 it is obtained,

V OD = −ln (3.5) V0 41

For small ODs, the Taylor expansion of Eq. 3.3 and the fact that A + T = 1 (A is

absorption), it is concluded that:

OD ≈ A (3.6)

The OD signal from the Ti atoms in our experiment was monitored on the com- puter (PC) via the Data Acquisition System (DAQ) that was connected to the detec-

3 3 tor used for laser absorption. The spectrum of Ti for the transition a F2 → y F2 at 25107.410 cm−1 is shown in Fig. 3.7. There are two spectra of Ti in the ﬁgure. These spectra correspond to the laser scanning up or down in frequency: the Ti spectrum at the time range from 0.64 s to 0.68 s is the mirror image of the Ti spectrum from

0.68 s to 0.72 s.

The background of the Ti spectrum (Fig. 3.7) was caused by fluctuations in the power of the laser and etalon effects on the windows of the cell: most of the laser light was transmitted and part of the light was reflected. The background seems to have a slope. To analyze the Ti peaks (this process was also applied for the TiHe signal analysis), we fit the Ti spectrum to a combination of a Gaussian and a linear function that accounts for the slope of the background signal. Fig. 3.8 shows a Ti spectrum and the fit for the 48Ti peak. The fit equation is the following:

2 −(x−x0) w2 f(x) = y0 + y1 · (x − x0) + A · e (3.7)

Where y0 is the intersection with the y-axis, y1 is the slope of the background signal, A is amplitude of the peak, w is the width (this quantity was also used to obtain the temperature) and x0 is the center of the peak which is the time after ablation.

As follows from Eq. 3.2, the optical density (OD) is proportional to the density of

Ti atoms. The data taken in this experiment was normalized with respect to Ti OD 42

(the reason for this normalization will be made clear in the next chapter). At high He

densities, we saw smaller values for the Ti OD, so we measured the Ti OD of the most

abundant isotope: 48Ti. At low He densities, we used the 46Ti isotope to determine

the OD because more Ti atoms were present and the 48Ti OD signal saturated under

these conditions. The data is presented in terms of 48Ti OD (we converted from 46Ti

OD to 48Ti OD using their isotopic abundance).

The sharp peaks in Fig. 3.7 are the Fabry-Perot fringes used to determine if the

laser is working in single mode. The spacing of the FP peaks is 1.5 GHz (30 GHz

= 1 cm−1). The Fabry-Perot cavity allowed us to determine the relative frequencies with an estimated error of 2%, however, we normally used the separation in frequency of the Ti isotopes for greater accuracy. In Fig. 3.9, the Ti OD spectrum is plotted against frequency (in wavenumbers) and its isotopes are identiﬁed. Typical probe powers for the Ti laser are in the order of µW to prevent optical pumping.

3.3 Detection of TiHe

Because the OD of the TiHe signal was too low to observe via absorption spectroscopy, we used laser-induced ﬂuorescence (LIF) to detect TiHe molecules via scattered photons. The idea of LIF is as follows: a laser will excite the TiHe molecules from their ground electronic state to their excited electronic state. Then, light will be emitted by the molecule when decaying back to the ground state.

In this detection process, we used as the TiHe probe laser a doubled Ti-sapphire laser with a power of a few mW. The optics setup for the detection of TiHe was as shown in Fig. 3.10. The TiHe probe beam was also divided into three beams (similarly to the Ti probe beam): the beam that went through the cell, the Fabry-Perot cavity and the wavemeter. To collect ﬂuorescent light, we had an in-vacuum cylindrical lens placed outside the cell, a back to back plano convex and plano convex spherical lens 43

combination located outside the vacuum chamber [54] and a photomultiplier tube

(PMT) below the lens combination. A FF01-400/40-25 color ﬁlter [74] was placed

before the photomultiplier to block scattered light at undesired frequencies.

When the TiHe probe laser was sent through the cell, we noticed that the back-

ground level on the absorption detector (∼V) increased a few mV. We placed waveplates (to change the polarization state of the laser light) in the path of Ti and TiHe before they entered the cryogenic cell to reduce the leak of the TiHe probe laser into the detector used to measure the absorption of Ti (we have a beam splitter in front of the Ti detector). By rotating the waveplates, we found conditions for which the contribution of the TiHe probe laser was minimum and the contribution of Ti probe laser was maximum into the absorption detector.

The Ti and TiHe probe laser beams entered on the east face of the cryostat, horizontally separated (on the north-south plane) by 10 mm. Inside of the cell, the beams overlapped in order to ensure that they were mostly interacting with the same atoms and molecules due to the non-uniformity of the Ti density after the ablation of the solid target. At the west face of the cryostat, the beams were 10 mm apart.

After analyzing the PMT signal, we noticed that fluorescent light coming from the atoms was interfering with the fluorescent light from the molecules; to reduce this effect, the Ti probe beam was moved 7 mm upwards (on the vertical axis) from the

TiHe beam.

We expected that a TiHe transition would be located close in frequency to a Ti atomic transition as explained in Chapter2. To ﬁnd the transition frequency of the

TiHe molecule, we tuned the TiHe probe laser to an atomic Ti transition and then we changed the TiHe probe beam by small frequency steps until we saw a signal on the PMT. The signals observed are discussed in Chapter4 and Chapter5. 44 3.3.1 Fluorescence detection improvement

At the beginning of this experiment, a Hamamatsu Si photodiode S2387 [75] was used to collect the ﬂuorescent light emitted by the TiHe molecules. To improve our signal-to-noise, we replaced the photodiode detector by a photomultipler of the type

Hamamatsu 9305 [76] with the goal of increasing the ﬂuorescent signal.

We built a protection circuit, a voltage divider with a diode to make sure that the photomultiplier was not damaged by very high input voltages. We added a diode to prevent negative input voltages to reach the photosensor. The circuit is illustrated in Fig. 3.11. We used R1 = 10 kΩ and R2 = 1 kΩ. The incoming voltage Vin was reduced in the following way:

R2 Vout = Vin (3.8) R1 + R2

The protection circuit was tested for a Vin = 15 V and a Vout of 1.37 V was measured.

We examined the photosensor at different conditions to find the most appropriate for our experiment and tried not to exceed 10 µA of output current as required by the specifications of the product. The time it takes for the photosensor to turn on is about 100 ms and the time the time it takes to turn off is about 400 ms. The photomultiplier is very sensitive when it turns on, as seen in Fig. 3.12. We turned on the photosensor right after the ablation pulse so the light from the ablation process was not collected because it could damage the device.

We also studied the signal measured by the photomultiplier when the TiHe probe laser beam was going through the cell with a power of 0.5 mW and varied the ablation energies and the control voltages (Vin). We changed the control voltage through the DAQ system and we found that 6 V was suitable for our experiment, since we collected a decent signal from the PMT, and the output current was a lot less than 10 µA. 45

The signal on the PMT with respect to the control voltage on the DAQ is shown in

Fig. 3.13.

We aligned the long axis of the PMT with the direction of the probe beam. Then, we moved the device transversally to ﬁnd a location in which we maximized the light collected, as in Fig. 3.14. The TiHe probe laser was tuned to the Ti transition

3 3 −1 a F2 → y F2 at 25107.410 cm and a probe power of 2.3 mW. The amount of fluorescent light depended on the Ti present in the cell, this is why the vertical axis was divided by the Ti OD. We fit the data points to a Gaussian function and the center of the Gaussian curve given by the fit indicated the best location for the PMT. To reduce noise at frequencies higher than 3 kHz, we added a low frequency preamplifier in the path of the signal from the PMT to the DAQ.

The spectrum of TiHe for the frequency range 25103.60 cm−1 to 25103.75 cm−1 is shown before and after the PMT (Fig. 3.15 and Fig 3.16). The ﬂuorescence signal collected by the photodiode, is shown in Fig. 3.15. The peaks could not be studied further since the signal-to-noise was very poor. In Fig 3.16, the spectrum was taken at similar conditions to the spectrum shown before the PMT. The pair of peaks are easily distinguished from the background and other peaks nearby are resolved.

TiHe OD is proportional to the molecular density in the cell. We can convert the

LIF signal into OD by means of a calibration. We tuned the TiHe probe laser to the

3 3 3 3 a F2 → y F2 or the a F2 → y F3 atomic Ti transitions using probe powers on the order of µW. We simultaneously measured Ti via LIF and absorption spectroscopy.

Then, varied the probe powers and measured the absorption and ﬂuorescence signal; this way we could calibrate the OD values for greater powers. The graphs on Fig 3.17

3 3 3 3 and Fig 3.18 show the calibration curves at a F2 → y F2 and a F2 → y F3 atomic Ti transitions respectively.

TiHe OD can be found by means of the slope of the calibration curve in the 46

following way:

TiHe LIF (V) TiHe OD = V (3.9) slope( µW ) · probe beam power (µW) Most of the data that will be presented in the following chapters was taken using a control voltage on the DAQ = 6 V. The calibration curves were taken at 5 V control voltage on the DAQ, hence, we used the information collected in Fig. 3.13 to make the appropriate conversion.

3.4 Measurement of He density

The pressure of He inside the cell was read by a room temperature pressure gauge that is connected through a thin line (0.09 cm inner radius) to the cell. The He density in the cell was determined via the Weber-Schmidt equation [77] that presents a correction for the thermomolecular pressure ratio. Details on the calculations of the

He density in the cell are shown in reference [78]. The uncertainty associated with the He density measurements is ±20%.

There were two other factors that could alter the He density in the cell. One of

them is the increase of the cell walls temperatures. Helium that was adsorbed at cold

temperatures may have evaporated at higher temperatures. This situation may cause

a systematic error in the binding energy measurements of the TiHe molecule (this

will be discussed in Chapter5, Section 5.2.5). The other element that could aﬀect

the He density in the cell is the ablation process that produces a momentary heating

of the cell when the Ti target is ablated [30]. These eﬀects are not easy to account

for since the pressure gauge was not inside of the cell and cannot read instantaneous

changes in the cell pressure.

The He densities in this experiment vary from 3 × 1016 cm−3 to 8 × 1017 cm−3. 47 3.5 Measurement of Ti density

The quantity OD provides information related to the Ti density in the cell (see

Eq. 3.2). The density of Ti can be calculated from the measured OD:

TiOD nTi = (3.10) σDTiL

Where σDTi is the Doppler cross section for Ti deﬁned as [79]:

γ σDTi = 0.89 σ0 (3.11) ΓDTi

3 3 −1 γ is the decay rate for the a F2 → y F2 Ti transition in units of s (the value for the desired Ti transition is found in [68]), ΓDTi is the Doppler width in rad/s and

−2 σ0 is the resonance cross section for Ti in cm . These quantities are mathematically described as:

1 1 γ = = (3.12) τ 23 ns

λ2 2J 0 + 1 σ = Ti (3.13) 0 2π 2J + 1 Doppler width in units of rad/s: r ω0 2KBT ΓDT i = (3.14) c MTi

Doppler width in units of Hz: r 1 2KBT ΓDTi = (3.15) λ MTi

Where 2J + 1 ground state statistical weight and 2J 0 + 1 is the excited state sta-

tistical weight. KB the Boltzmann constant. For Ti, the wavelength at the transition 25107.410 cm−1 is λ = 398.3 nm, ω = 2π ν = 4.73 × 1015 rad/s. The mass of Ti is 48

48 amu =7.98 × 10−26 Kg. The cell length is L = 10 cm. Then, the Ti Doppler cross section is:

−11 1 2 1/2 σDTi ≈ 3 × 10 √ [cm K ] (3.16) T And the Ti density becomes:

√ −3 9 cm nTi ≈ 3 × 10 TiOD T √ (3.17) K

3.6 Measurement of temperature

Throughout this experiment, a precise measurement of the temperature played an

important role in determining the binding energy of the TiHe molecule, as will be

explained in the next chapters. The temperature of the cell was measured by a

ruthenium oxide resistor and a silicon diode. The translational temperature of the Ti

atoms was obtained by the Doppler width of the Ti spectrum (Eq. 3.15).

As described in Section 3.2, the spectrum of Ti in the DAQ appears as Ti OD

versus time after ablation, and time can be converted into frequency in Hz using the

Ti isotope shifts. At high He densities, the temperature measured using the isotope

shifts is much higher than the temperatures from the cell diode, as shown in Fig 3.19.

This indicates that the Ti width is aﬀected by Doppler and pressure broadening.

To accurately measure temperatures, we had to correct for pressure broadening.

The Ti linewidth is made of the natural linewidth, Doppler broadening and pressure

broadening. The goal is to deconvolve the Voigt proﬁle of the Ti peaks into its

Gaussian and Lorentzian contributions, ﬁnd a mathematical expression of the Doppler

width and from this relation, determine the translational temperature of the atoms. 49

Reference [80] states that:

q 2 2 aV ≈ 0.5 1.0692 aL + 0.86639 aL + 4 aD (3.18)

Where aV is the Voigt FWHM, aL is the Lorentzian FWHM and aD is the Doppler FWHM.

The Ti and TiHe peaks are ﬁt to a combination of a linear and a Gaussian function, as shown in Section 3.2. The FWHM can be easily obtained from the width w in the following way:

√ FWHM = 2 ln2 w (3.19)

From the ﬁt of the Ti peaks and their FWHM, we obtained their Voigt proﬁle:

√ aV = 2 ln2 w (3.20)

We wanted to ﬁnd the FWHM caused by pressure broadening, aL, such as:

aL = P nHe (3.21)

Where P is the pressure broadening coeﬃcient in units of Hz cm3.

The 48Ti FWHM which includes pressure, Doppler broadening and the natural

linewidth is presented as a function of He density in Fig. 3.20. The 48Ti FWHM

increases with He density due to pressure broadening. The pressure broadening coef-

ﬁcient in this experiment is ∼ 1 × 10−10 Hz cm3. This value was obtained after ﬁtting

the data to the expression given in Eq. 3.18 with aL as in Eq. 3.21:

q 2 2 f(x) = 0.5 1.0692 P x + 0.86639 (P x) + 4 aD (3.22) 50

The Doppler FWHM (aD) was calculated using the expression given in Eq. 3.23. This quantity was used in Eq. 3.15 to obtain the translational temperature of Ti.

q 2 2 aD = 0.5 (2 aV − 1.0692 P nHe) − 0.86639 (P nHe) (3.23)

The pressure broadening of the Ti spectrum causes inconsistent measurements of the translational temperatures of the Ti atoms with respect to the cell wall temperatures obtained from the cell diode. The deconvolution of the Voigt proﬁle into its

Lorenztian and Gaussian contributions is a reasonable correction for the temperature extracted from the Ti spectrum.

The translational temperature of the Ti atoms from their Doppler widths is higher than the temperature read by the cell diode and the resistor. The reason for this situation is due to the heating of the gas by ablation [69]. The Ti gas temperature is usually 0.5 K to 1.5 K higher than the cell wall temperature, as shown in Fig. 3.21.

The diﬀerence in temperature depends on the ablation power and the time after ablation studied. Thermalization happens on a ∼ 1 s timescale. When the temperature is obtained from the Ti spectrum, instantaneous changes in temperature are being considered, but the cell diode and resistors are measuring an averaged temperature.

In subsequent chapters, the temperature used in the study of the thermodynamic properties to obtain the binding energy of the TiHe molecule, is the translational temperature of the Ti atoms. 51

4 3 2 1

Scan [V] 0 -1

1.2 0.24 1.0 0.22 0.8 0.20

0.6 0.18 FP [V] 0.4 0.16 Ti OD 0.2 0.14 0.0 0.12 -0.2 0.10

0.64 0.66 0.68 0.70 0.72 Time [s]

Figure 3.7: Ti OD spectrum at 25107.410 cm−1, ablation power of 30 mJ, probe power of 0.6 µW and a cell temperature of around 1.2 K is plotted against time after ablation. The curves corresponding to the Ti probe laser scan voltage and the FP peaks are also shown. The Ti signal is given in OD and plotted against time after ablation. The scan voltage, a periodic square wave, is shown on the upper side of the ﬁgure. The Ti probe laser is scanning back and forth a frequency range that is 3 3 −1 centered at the transition a F2 → y F2 at 25107.410 cm . 52

1.0

0.8

0.6 Ti spectrum Fit 0.4 Ti OD 0.2

0.0

-0.2

0.655 0.660 0.665 0.670 0.675 Time [s]

Figure 3.8: Fit of the 48Ti peak using Eq. 3.7. The ﬁt parameters obtained are: −1 y0 = −0.10034 ± 0.000809, y1 = (−15.413 ± 0.51) s , A = 1.0895 ± 0.00231, w = −6 −6 (0.0043724 ± 1.13 × 10 ) s, x0 = 0.66428 ± 7.45 × 10 s. 53

2.5

48 Ti 2.0

1.5

Ti OD 1.0

0.5 50 Ti 46 47 49 Ti Ti Ti 0.0

25107.36 25107.40 25107.44 25107.48 -1 Frequency [cm ]

Figure 3.9: Ti OD spectrum at 25107.410 cm−1, ablation power of 30 mJ, probe power of 0.6 µW and a cell temperature of around 1.2 K vs frequency in wavenumbers. Fabry-Perot Glass Waveplate cavity wedge

Ti target

Neutral Mirror densities Ti He 54 Prism TiHe

Fabry-Perot Top view: cavity

Wavemeter TiHe probe Detector laser

Waveplate

Glass wedge

Mirror

Ti target

Side view: Ti He

TiHe

Figure 3.10: TiHe optics setup.

Protection diode

Vin

Vout

Figure 3.11: Voltage divider used to protect the PMT from high voltages. The diode prevents negative voltages to reach the PMT. 55

4 2 0 -2 -4 Scan voltage [V] -0.10

-0.15

-0.20

-0.25

-0.30 Signal from photomultiplier [V]

-0.35 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Time [s]

Figure 3.12: PMT signal vs time. This test signal was taken at a control voltage of 1 V, a 0.5 mW of power for the TiHe probe going through the cell. The ablation laser was oﬀ. The curve on top is the laser scan voltage. 56

260 4 Q switch [µs] 250

240 Data 3 230 220

2 Signal on PMT [V]

0 0 2 4 6 8 Control voltage on DAQ [V]

Figure 3.13: PMT signal vs control voltage on the DAQ. These test signals were taken a 0.5 mW of power for the TiHe probe going through the cell, diﬀerent Q switches. 57

12 Data Fit

8 Ti OD [V/OD] 48

6 Ti LIF/ 48

12 14 16 18 20 PMT position [mm]

Figure 3.14: Optimal position of the PMT in the optics setup. 58

4 2 0 -2 Scan [V] -4 0.18 0.6 0.16

0.4 FP [V] 0.14

0.2 0.12

Laser Induced Fluorescence [V] 0.0 0.10

0.15 0.20 0.25 0.30 0.35 0.40 Time [s]

Figure 3.15: Fluorescense spectrum vs time after ablation for the following conditions: frequency range from 25103.60 cm−1 to 25103.75 cm−1, ablation power 60 mJ, probe power of 3.4 mW, density of He of 3 × 1017 cm−3 and a cell temperature of 1.2 K. The curves corresponding to the TiHe probe laser scan voltage and the FP peaks are also shown. At the t=0.19 s, a pair of very small TiHe peaks appear in the ﬂuorescence spectrum. Fluorescent light was collected using the photodiode detector. 59

4 2 0 -2 Scan [V] -4 1.0 0.16

0.9 0.14

0.8 FP [V]

0.12 0.7

0.6 0.10 Laser Induced Fluorescence [V] 0.5 0.20 0.25 0.30 0.35 0.40 0.45 0.50 Time [s]

Figure 3.16: Fluorescense spectrum vs time after ablation for the following conditions: frequency range from 25103.60 cm−1 to 25103.75 cm−1, ablation power 60 mJ, probe power of 3.0 mW, density of He of 3 × 1017 cm−3 and a cell temperature of 1.2 K. The curves corresponding to the TiHe probe laser scan voltage and the FP peaks are also shown. Several peaks were resolved after adding the PMT in our optics setup. 60

100

80 Data Fit

60 Ti OD [V/OD] 48 40 Ti LIF/ 48

0 0 10 20 30 40 Power [µW]

Figure 3.17: Calibration curve at 25107.410 cm−1: 48Ti LIF/ 48Ti OD is plotted V against the probe power. The data is ﬁt to a line bx, with b = (1.99 ± 0.09) µW . The Q switch is 220 µs, the PMT control voltage at the DAQ is 5V and the He density is 3 × 1017 cm−3. 61

60 Data Fit

40 Ti OD [V/OD] 48 Ti LIF/ 48 20

0 0 5 10 15 20 25 Power [µW]

Figure 3.18: Calibration curve at 25227.220 cm−1: 48Ti LIF/ 48Ti OD is plotted V against the probe power. The data is ﬁt to a line bx, with b = (2.02 ± 0.06) µW . The Q switch is 220 µs, the PMT control voltage at the DAQ is 5V and the He density is 7 × 1017 cm−3. 62

3.5

3.0

2.5

Ti spectrum [K] spectrum Ti 2.0 46

1.5

1.0

T measured T by 0.5

0.0 1.4 1.6 1.8 2.0 2.2 T measured by cell diode [K]

Figure 3.19: Temperature measured by Ti spectrum vs temperature measured by cell diode. The temperature measured through the Ti spectrum is greater than the temperature measured using the cell diode. He density = 6 × 1017 cm−3. 63

8 1.4x10

1.2

1.0

0.8

Ti FWHM [Hz] 0.6 Data aD 48 Fit Fit 0.4

0.2

0.0 17 0 2 4 6x10 -3 He density [cm ]

Figure 3.20: 48Ti FWHM vs He density. The data represented in squares is the FWHM of Ti obtained from the Voigt profile of the Ti spectrum. Pressure broadening is affecting the Ti peaks. The data is fit to the Eq. 3.22. The fit shows aD = (1.01 ± 0.0075) × 108 Hz and P = (1.07 ± 0.022) × 10−10 Hz cm3. The data shown in circles was obtained from the relation Eq 3.23 used to deconvolve aD from the Voigt profile of Ti spectrum. The fit is to a constant c = (1.02 ± 0.0040) × 108 Hz. 64

5 Ti spectrum [K] Ti spectrum

48 4

2 Voigt 1 Doppler T measured T using

0 1.5 2.0 2.5 3.0 T measured using the cell diode [K]

Figure 3.21: Temperature measured using the calculated Doppler width of 48Ti and Temperature measured using the 48Ti Voigt proﬁle plotted against the temperature measured from the cell diode. T from the calculated Doppler width shows similar values to the temperature measure from the cell diode. The blue line is: f(T ) = T . Data taken at a He density = 8 × 1017 cm−3. 65

Chapter 4

3 3 TiHe signals at a F2 → y F2 Ti transition

This chapter presents the search and the equilibrium thermodynamic studies of the

ﬂuorescent spectroscopic signals attributed to a Ti and He interaction that were found

3 3 −1 red and blue detuned from the atomic Ti transition a F2 → y F2 at 25107.410 cm . From these particular transitions, it was not possible to conclude the existence of a bound TiHe molecule.

3 3 4.1 Spectroscopic search of TiHe at a F2 → y F2

Ti transition

We expected to produce the TiHe molecules in the ground state by the three body

recombination process, and detect them via LIF by means of a probe laser tuned to

the molecular transition frequency. Details on the experiment and the LIF detection

were explained in Chapter3.

3 3 The TiHe probe laser was tuned at the a F2 → y F2 Ti transition at 25107.410 66 cm−1, and from this atomic transition we changed the frequency of the TiHe probe laser until we saw a ﬂuorescence signal. We found several peaks blue and red detuned from the atomic Ti transition. In Fig. 4.1, the blue detuned spectroscopic signals are shown, and in Fig. 4.2 the red detuned peaks are presented.

0.55

0.50

0.45 Fluorescence[V] 3 0.40 x10 25.1078 25.1080 25.1082 25.1084

0.35 25107.8 25108.0 25108.2 25108.4 -1 Frequency [cm ]

Figure 4.1: Blue detuned spectroscopic signals at the frequency range 25107.8 cm−1 and 25108.4 cm−1. The temperature of the cell is ∼ 1.2 K and the He density is 7.5 × 1017 cm−3.

The peaks centered at the frequencies 25107.95 cm−1 and 25103.42 cm−1 are very broad and information extracted from them would be very noisy. This is why these peaks are not included in the study of the thermodynamic properties presented in the next sections.

It is interesting to point out that while we were scanning the frequency range in 67

2.05

2.00

1.95

1.90

1.85 Fluorescence[V]

1.80

1.75

25103.4 25103.5 25103.6 25103.7 -1 Frequency [cm ]

Figure 4.2: Red detuned spectroscopic signals at the frequency range 25103.35 cm−1 and 25103.75 cm−1. The temperature of the cell is ∼ 1.2 K and the He density is 8 × 1017 cm−3. order to ﬁnd a spectroscopic signal or signals from TiHe, we came across the spin

−1 3 5 forbidden Ti transition at 25102.874 cm [68]: a F2 → z S2. The spectrum of this transition is shown in Fig. 4.3.

4.2 Equilibrium thermodynamic studies

This section discusses the studies of the thermodynamic properties of the blue (frequency range from 25107.80 cm−1 to 25108.4 cm−1) and red (frequency range from

−1 −1 3 25103.35 cm to 25103.70 cm ) detuned spectroscopic signals found near the a F2 → 68

2.0

1.5

1.0 Fluorescence[V]

0.5

25102.82 25102.84 25102.86 25102.88 25102.90 -1 Frequency [cm ]

3 5 −1 Figure 4.3: Spin forbidden Ti transition a F2 → z S2 at 25102.874 cm and He density 8 × 1017 cm−3.

3 −1 y F2 Ti atomic transition at 25107.410 cm . For convenience, we discuss these signals as if they originated from bound TiHe molecules. By examining their thermodynamic properties, we can determine if this is the case.

4.2.1 Blue detuned spectroscopic TiHe signal

The dependence of the TiHe density (nTiHe) on the temperature (T ), He density (nHe)

−1 and Ti density (nTi) is analyzed for the peak 25108.27 cm . 69

4.2.1.1 Ti dependence

We measured the TiHe density with respect to the Ti density at a constant He density

and a narrow temperature range, as in Fig. 4.4. Notice that the axes of the graph

correspond to Ti OD and TiHe OD. The quantity OD is proportional its density. We

measured the Ti OD from laser absorption spectroscopy and converted the TiHe LIF

to TiHe OD, as explained in Chapter3. From the ﬁgure, we see that the dependence

of TiHe OD on the Ti OD is linear, as expected in thermal equilibrium (Eq. 1.4), and

independent of the time after ablation. Thus, the TiHe molecules were forming on a

time scale faster than the time at which we observed them, see Fig. 4.5. Because the

TiHe OD is linear in the TiOD, we used the ratio: TiHe OD/Ti OD in subsequent

analysis.

4.2.1.2 Temperature dependence

The TiHe density in thermal equilibrium is given by Eq. 1.4. To study the temperature

dependence of TiHe OD/48Ti OD at constant He density, we are going to focus on

the expression:

n h2 3/2 TiHe −E/KB T = nHe ge (4.1) nTi 2πµKBT Rewriting Eq. 4.1 in terms of T (temperature) and the fit parameters c (c coefficient) and E (binding energy), we obtain the fit equation for the study of the ratio of TiHe density with Ti density as a function of temperature:

f(T ) = cT −3/2e−(E/T ) (4.2)

The illustration in Fig. 4.6 shows how the TiHe density and the Ti density ratio should behave as a function of temperature for a bound or unbound state of Ti and

He. A negative binding energy indicates that the system is bound and the production 70

-4 3.0x10 Data 2.5 Fit

2.0

1.5 Time after ablation [s] after ablation Time TiHeOD 0.3 1.0

0.2 0.5 0.1

0.0 0 1 2 3 4 5 48 Ti OD

Figure 4.4: TiHe OD as a function of Ti OD at frequency 25108.27 cm−1. The data was taken at the temperature range from 1.55 K to 1.9 K and a He density of 5.4 × 1017 cm−3. The data is ﬁt using y = bx; b = (5.6 ± 0.27) × 10−5. The color scale indicates the time after ablation. The TiHe density increases linearly with respect to the Ti density. of TiHe increases exponentially at low temperatures, as expected from Eq. 4.1.A positive binding energy would correspond to an unbound state, such as a scattering resonance.

The temperature in the cell was varied by changing the temperature of the cell walls. The gas temperature was measured through Ti spectroscopy, as explained in

17 −3 Chapter3. At a constant He density nHe = 1.1 × 10 cm , we present in Fig. 4.7 TiHe OD/48Ti OD as a function of temperature. The ﬁt values E and c correspond 71

Data -4 1.2x10 Fit

1.0

Ti OD Ti 0.8 48

0.6 Temperature[K]

TiHeOD/ 0.4 1.90 1.80 0.2 1.70 1.60

0.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30

Time after ablation [s]

Figure 4.5: TiHe OD/48Ti OD as a function of time after ablation. The data was 17 −3 taken at a He density of nHe = 5.4 × 10 cm . The data is ﬁt using y = bx; b = (6.6 ± 0.43) × 10−5. The color scale indicates temperature. The ratio TiHe OD/Ti OD remains constant when plotted with respect to time after ablation.

to: (0.6 ± 0.3) K and (1.01 ± 0.14) × 10−5 K3/2 respectively. The same data was averaged and is shown in Fig. 4.8. The averaged version of the data shows the ﬁt values: E = (−0.49 ± 0.4) K and c =(1.11 ± 0.18) × 10−5 K3/2.

Fig. 4.9 shows the temperature dependence of TiHe OD/48Ti OD at diﬀerent

He densities and Fig. 4.10 presents the dependence of TiHe OD T3/2/48Ti OD with respect to temperature at diﬀerent densities of He. It is not possible to conclude that at low temperature, the production of TiHe is increasing, which is expected if the signal studied corresponds to a molecular transition of TiHe. 72

E = -2 K E = 0 K 1 E = 2 K

0.1 f(T)

0.01

0.001

2 3 4 5 6 7 8 2 3 4 5 6 7 8 1 10 100 T

Figure 4.6: Fit equation for TiHe density dependence with temperature. E = -2K indicates a bound state. E = 2 K an unbound state. E = 0 a bound (or scattering) level at threshold. The axis are given in logarithmic scale.

A summary of the binding energies at the peak frequency 25108.27 cm−1 is shown in Table 4.1.

The graph in Fig. 4.11 shows the averaged binding energy E obtained for the

transition 25108.27 cm−1. The value of E corresponds to (0.19±0.14) K and the

error indicates the relative statistical uncertainty. The binding energy value (close to

zero) does not allow us to conclude whether the signal found at 25108.27 cm−1 is due

to a bound molecular transition or a scattering resonance. 73

5 10 9 Data 8 Fit 7 6

4 Ti OD 48 3

240 [s] switch Q

TiHe OD/ TiHe 2 220 200 180 160

6 10 9 2.0 2.5 3.0 Temperature [K]

Figure 4.7: TiHe OD/48Ti OD as a function of temperature. The data was taken 17 −3 −1 at a He density of nHe = 1.1 × 10 cm and at a frequency 25108.27 cm . From the ﬁt is obtained E = (−0.60 ± 0.3) K and c =(1.01 ± 0.14) × 10−5 K3/2. The color scale indicates the Q switch setting of the ablation laser. By varying the Q switch, the ablation energy is changed.

4.2.1.3 He dependence

The different sets of data for TiHe OD/ 48Ti OD vs temperature at constant He density are fit using Eq. 4.2. The c coefficient is extracted from the fit and it provides information of the He density dependence of TiHe.

The illustration in Fig. 4.12 shows the c coeﬃcient with respect to He density.

The linear ﬁt of the data has a slope b = (2.59 ± 0.27) × 10−22 K3/2cm3 and a

2 2 χ = 18.01. Fig. 4.13 shows the same data (c vs nHe) using a quadratic ﬁt: y = Ax , 74

6 Data Fit

5 Ti OD

48 4

TiHe OD/ TiHe -6 3x10

2.0 2.5 3.0 Temperature [K]

Figure 4.8: TiHe OD/48Ti OD as a function of temperature. The data was taken at 17 −3 −1 a He density of nHe = 1.1 × 10 cm and at a frequency 25108.27 cm . From the ﬁt is obtained E = (−0.49 ± 0.4) K and c =(1.11 ± 0.18) × 10−5 K3/2.

A = (8.44 ± 1.51) × 10−40 K3/2cm3 and χ2 = 11.59. The χ2 of the quadratic ﬁt seems to indicate that this is a better ﬁt. Consequently, instead of the TiHe molecule, we could be forming the molecule TiHe2:

2 3 2 −E/T nTiHe2 = nTinHeg λdB · e (4.3)

Summary of the blue detuned signals: It is inconclusive from the He density

−1 dependence whether the peak at 25108.27 cm corresponds to TiHe or TiHe2, or whether it corresponds to a bound molecule or a scattering resonance. 75

He density [cm−3] E [K]

6.2 × 1016 0.43±0.5

1.1 × 1017 -0.49±0.4

2.9 × 1017 -0.59±0.5

3.1 × 1017 0.30±0.4

5.3 × 1017 0.93±0.4

5.4 × 1017 0.21±0.3

7.6 × 1017 0.24±0.3

Table 4.1: Binding energies and their relative uncertainty obtained at studying the temperature dependence of the TiHe interaction found at 25108.27 cm−1 at diﬀerent He densities.

The behavior of the peak at 25108.18 cm−1 is expected to be similar to the studied

signal at 25108.27 cm−1, since the ratio of the height of these two peaks was studied

17 −3 with respect to temperature at nHe = 3.1 × 10 cm and it remains constant, as shown in Fig. 4.14.

4.2.2 Red detuned spectroscopic TiHe signal

The TiHe density at the frequency 25103.67 cm−1 is studied as a function of the

thermodynamic properties: Ti density (nTi), He density (nHe) and temperature (T ).

4.2.2.1 Ti dependence

−1 The nTiHe dependence on nTi at the frequency 25103.67 cm and the temperature range from 1.7 K to 1.9 K is shown in Fig. 4.15. The data is ﬁt to y = bx with b = (1.85 ± 0.11) × 10−5. The TiHe density is proportional to the Ti density and it is independent of the observation time. 76

He density [cm−3] E [K]

2.5 × 1017 1.73±0.5

2.9 × 1017 -0.08±0.25

4.8 × 1017 0.065±0.4

4.8 × 1017 -9.07±2.13

8.9 × 1017 -0.040±0.3

8.9 × 1017 0.68±0.26

Table 4.2: Binding energies and their relative uncertainty obtained at studying the temperature dependence of the TiHe interaction found at 25103.67 cm−1 at diﬀerent He densities.

4.2.2.2 Temperature dependence

The TiHe OD/48Ti OD at 25103.67 cm−1 is analyzed as a function of temperature.

Fig. 4.16 shows the temperature dependence of the red detuned spectroscopic signal

at diﬀerent He densities.

The binding energies from the ﬁt of the data TiHe OD/ 48Ti OD vs temperature

at the TiHe frequency 25103.67 cm−1 is presented in Table 4.2.

The averaged binding energy E for the TiHe interaction seen at the frequency

25103.67 cm−1 is shown in Fig. 4.17. According to the linear ﬁt, E=(0.28±0.14) K, and as explained before, the error corresponds to the relative statistical uncertainty.

These peaks do not allow us to conclude that the red detuned spectroscopic signals would correspond to the TiHe molecule, for the same reason discussed previously for the blue detuned peaks.

It is expected that the binding energy of the red detuned peaks seen at 25103.65 cm−1 and 25103.67 cm−1 does not diﬀer signiﬁcantly from the binding energy obtained from the peak at 25103.75 cm−1, since the measured ratio of these peaks with 77

respect to the studied signal at 25103.67 cm−1 remains constant when plotted versus temperature, as shown in Fig. 4.18 and Fig. 4.19.

4.2.2.3 He dependence

Fig. 4.20 shows the c coeﬃcient extracted from the ﬁt of the data TiHe OD/ 48Ti

OD vs temperature at constant He density for the transition 25103.67 cm−1. The c

coeﬃcient is plotted against the diﬀerent He densities for which the data was taken.

The c coeﬃcient vs He density is ﬁt using a linear function y = bx with b = (6.44 ±

0.92) × 10−23 K3/2cm3. A quadratic ﬁt is not a good ﬁt for this transition.

Summary of the red detuned signals: The spectroscopic signal found at

25103.67 cm−1 is linearly proportional to the density of Ti and He. However, after studying the temperature dependence of this signal, we were not able to conclude that this interaction would correspond to a bound TiHe molecule, since its binding energy is E =(0.28±0.14) K.

The signals found at the frequency range from 25107.8 cm−1 to 25108.4 cm−1 could be due to a bound-bound transition from the ground state potential of TiHe

3 3 to an excited state potential lying close to the atomic Ti transition a F2 → y F2 at 25107.410 cm−1. Likewise, the signals found at the frequency range 25103.35 cm−1 and 25103.75 cm−1 can be the result of a bound-bound transition from the ground state potential of TiHe to an excited state potential lying close to the forbidden atomic

3 5 3 3 −1 Ti transition a F2 → z S2 a F2 → y F2 at 25102.874 cm . Unfortunately, there is no indication from the equilibrium thermodynamic studies that these transitions are bound, since the averaged binding energy is very close to 0 K. Another possibility that

3 3 could explain the E ∼ 0 K is that the excited state at the a F2 → y F2 transition does not support bound states, but this idea would not account for the appearance 78

of the red detuned signals. One last explanation for these spectroscopic signals is

that they are due to resonance-enhanced photoassociation [29,4]. If the signals are

a consequence of photoassociation only, then the peaks would be ∼ 1cm−1 wide. But they are not. It is possible that there is a resonance shape on the TiHe ground state potential that is producing quasibound states above threshold and these states are the ones that are excited giving rise to the peaks that were discussed in this chapter.

The binding energy predictions shown in Chapter2 suggest that the bound TiHe molecule is expected to form under the conditions of our experiment. For this reason, we decided to continue with the search for the TiHe molecule at a diﬀerent Ti transition. Suitable Ti transitions to search for the TiHe molecule could be [68]:

3 3 −1 a F2 → y F3 at 25227.220 cm

3 3 −1 a F2 → y D1 at 25317.815 cm

3 3 −1 a F2 → y D2 at 25438.906 cm

3 3 −1 a F2 → y D3 at 25643.699 cm

3 3 −1 The search for the TiHe molecule at a F2 → y F3 at 25227.220 cm and the spectroscopic signals found, are the topic of the next chapter. 79

-3 He density [cm ] 4 17 7.6x10 17 3 5.4x10 17 5.3x10 17 2 3.1x10 17 2.9x10 17 1.1x10 16 -4 6.2x10 10

6 5 4

Ti OD 3 48

-5 TiHe OD/ 10

6 5 4 3

0 3 4 2x10 Temperature [K]

Figure 4.9: TiHe OD/48Ti OD as a function of temperature at the frequency 25108.27 cm−1. E and c are obtained from the ﬁt as discussed in the text. Notice that an increase in He density causes an increase of the TiHe signal. 80

-3 3 He density [cm ] 17 7.6x10 17 2 5.4x10 17 5.3x10 17 3.1x10 17 -3 2.9x10 17 10 1.1x10 17 6.2x10 6 5 4 3 Ti OD 48 / 2 3/2

-4 10

TiHe OD*T 5 4 3

-5 10

0 3 4 2x10 Temperature [K]

Figure 4.10: TiHe OD T3/2/48Ti as a function of temperature at the frequency 25108.27 cm−1. 81

1.0

0.5

0.0

Binding energy [K] energy Binding -0.5 Energy Fit

-1.0

17 0 2 4 6x10 -3 Helium density [cm ]

Figure 4.11: Binding energy obtained from the study of the temperature dependence of the TiHe signal at 25108.27 cm−1. The ﬁt is to a constant line. E = (0.19±0.14) K. 82

-4 3.0x10

c coefficient ] 2.5 Fit 3/2 2.0

1.5

1.0 c coefficient c coefficient [K

0.5

0.0 0 2 4 6 17 -3 8x10 Helium density [cm ]

Figure 4.12: c coeﬃcient vs He density. TiHe signal = 25108.27 cm−1. The data is ﬁt using y = bx. b = (2.59 ± 0.27) × 10−22 K3/2cm3. χ2 = 18.01 83

-4 4x10

c coefficient

] Fit

3/2 3

c coefficient c coefficient [K 1

0 2 4 6 17 -3 8x10 Helium density [cm ]

Figure 4.13: c coeﬃcient vs He density. TiHe signal = 25108.27 cm−1. The data is ﬁt using y = Ax2. A = (8.44 ± 1.51) × 10−40 K3/2cm3. χ2 = 11.59 84

240 Q switch [s] switch Q

1.5 220 Data Fit 200 180

160 1.0 Peak A/B Peak Peak

0.5

0.0 1.6 1.8 2.0 2.2 Temperature [K]

Figure 4.14: Ratio of the height of the peak at 25108.18 cm−1 (Peak A) to the height of the peak at 25108.27 cm−1 (Peak B) versus temperature measured with the cell 17 −3 diode. The ﬁt is to y = a where a=(0.81±0.029). At nHe = 3.1 × 10 cm 85

-5 Data 1.5x10 Fit

1.0 Time after ablation [s] after ablation Time TiHeOD

0.6 0.5 0.4

0.2

0.0 0.0 0.2 0.4 0.6 0.8

48 Ti OD

Figure 4.15: TiHe OD vs 48Ti OD at the frequency 25103.67 cm−1 and temperatures from 1.7 K to 1.9 K. The data is ﬁt using y = bx, b = (1.85 ± 0.11) × 10−5. The color scale represents the time after ablation. TiHe OD is linearly proportional to 48Ti OD. 86

-3 -3 He density [cm ] 17 10 8.9x10 17 8.9x10 17 4.8x10 17 4.8x10 17 2.9x10 17 2.5x10

-4 10 Ti OD 48 -5 10 TiHe OD/

-6 10

0 3 4 2x10 Temperature [K]

Figure 4.16: TiHe OD/48Ti OD as a function of temperature at the frequency 25103.67 cm−1. E and c are obtained from the ﬁt as discussed in the text. Notice that an increase in He density causes an increase of the TIHe signal. 87

-2

-4

-6 Energy Fit

Binding energy [K] energy Binding -8

-10

17 0 2 4 6 8x10 -3 Helium density [cm ]

Figure 4.17: Binding energy obtained from the study of the temperature dependence of the TiHe signal at 25103.67 cm−1. The ﬁt is to a constant line. E=(0.28±0.14) K. 88

0.8

0.6

0.4 Data 240 [s] switch Q Fit 220 200 Peak A/ Peak B Peak A/ Peak 180 0.2 160

0.0 1.6 1.8 2.0 2.2 2.4 Temperature [K]

Figure 4.18: Ratio of the height of the peak at 25103.65 cm−1 (Peak A) to the height of the peak at 25103.67 cm−1 (Peak B) versus temperature measured with the cell 17 −3 diode. The ﬁt is to y = a where a=(0.073±0.0066). At nHe = 2.95 × 10 cm . 89

0.8

0.6

0.4 Q switch [s] Qswitch Data 240 220 Peak C/ Peak B Peak C/ Peak Fit 200 0.2 180 160

1.6 1.8 2.0 2.2 2.4 Temperature [K]

Figure 4.19: Ratio of the height of the peak at 25103.75 cm−1 (Peak C) to the height of the peak at 25103.67 cm−1 (Peak B) versus temperature measured with the cell 17 −3 diode. The ﬁt is to y = a where a=(0.059±0.0016). At nHe = 2.95 × 10 cm . 90

-5 8x10 c coefficient Fit ] 3/2 6

c coefficient c coefficient [K 2

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

Figure 4.20: c coeﬃcient vs He density. TiHe signal = 25103.67 cm−1. The data is ﬁt using y = bx. b = (6.44 ± 0.92) × 10−23 K3/2cm3. 91

Chapter 5

3 3 TiHe signals at a F2 → y F3 Ti transition

A spectroscopic search of the TiHe molecule was conducted and multiple signals were

3 3 −1 found blue detuned from the a F2 → y F3 Ti atomic transition at 25227.220 cm . This chapter shows the equilibrium thermodynamic studies for these TiHe spectroscopic signals. The binding energy of four diﬀerent peaks was measured from the

TiHe density dependence on temperature. These binding energies allowed us to conclude that the TiHe molecule was created. However, the binding energies obtained through the experiment do not match with the theoretical predictions shown in Chap- ter2. For this reason, new calculations were made by theoretical collaborators. A corrected TiHe ground state potential was calculated and good agreement of the experiment with the new theoretical model was obtained. A comparison between the experimental and theoretical results is shown in this chapter. 92 3 3 5.1 Spectroscopic search of TiHe at a F2 → y F3

Ti transition

In the previous chapter, the eﬀorts towards ﬁnding spectroscopic signals that could

3 3 be attributed to a bound interaction between Ti and He at the a F2 → y F2 atomic Ti transition were shown. From this work, binding energies of around 0 Kelvin were obtained by studying the equilibrium thermodynamic properties of the spectroscopic signals. From these binding energy values, it was impossible to conclude that the peaks corresponded to a bound TiHe interaction.

In order to ﬁnd a spectroscopic ﬂuorescence signal (or signals) from TiHe, we

3 decided to start over the search for the molecule at a diﬀerent Ti transition: a F2 →

3 −1 y F3 at 25227.220 cm . From this atomic Ti transition, we changed the frequency of the laser in small steps until we detected a ﬂuorescence signal. Multiple transitions were found blue detuned from the atomic Ti transition, as shown in Fig. 5.1. The equilibrium studies of these signals will be presented in the following sections. Also, the identiﬁcation of these peaks based on the new theoretical calculations will be discussed.

Other peaks were found (Fig. 5.2 and Fig. 5.3), but their signal to noise was very small and further studies on these peaks could not be performed in this experiment.

The peak located at 25218.18 cm−1 could be a transition to a more deeply bound vibrational level in the excited state. However, we do not have enough data to conclude that this assignment is correct. 93

1.6

1.4

1.2

1.0 Fluorescence [V] Fluorescence

0.8 25227.95 25228.00 25228.05 25228.10 25228.15 25228.20

25227.95 25228.00 25228.05 25228.10 25228.15 25228.20 -1 Frequency [cm ]

Figure 5.1: Blue detuned spectroscopic signals due to an interaction between Ti and He at the frequency range 25227.95 cm−1 and 25228.20 cm−1. The temperature in the cell is ∼ 1.2 K and the He density is 6.5 × 1017 cm−3. The atomic transition is oﬀ scale (to the left) at 25227.220 cm−1. 94

1.08

1.06

1.04

1.02

1.00 Fluorescence [V] Fluorescence

0.98

0.96 25227.50 25227.51 25227.52 25227.53 25227.54 -1 Frequency [cm ]

Figure 5.2: Blue detuned spectroscopic signal due to an interaction between Ti and He at the frequency range 25227.50 cm−1 and 25227.54 cm−1. The temperature in the cell is ∼ 1.2 K, and the He density is 6.5 × 1017 cm−3. Thermodynamic studies of this peak could not be done due to poor signal to noise. 95

1.04

1.02

1.00

0.98

0.96

0.94

Fluorescence [V] Fluorescence 0.92

0.90

0.88

25218.170 25218.180 25218.190 25218.200 -1 Frequency [cm ]

Figure 5.3: Red detuned spectroscopic signal due to an interaction between Ti and He at the frequency range 25218.17 cm−1 and 25218.20 cm−1. The temperature in the cell is ∼ 1.2 K, and the He density is 6.5 × 1017 cm−3. Thermodynamic studies of this peak could not be done due to poor signal to noise. 96 5.2 Equilibrium thermodynamic studies

The equilibrium thermodynamics studies of the peaks found at the frequencies 25228.15

cm−1, 25228.03 cm−1, 25228.10 cm−1 and 25227.99 cm−1 are presented in this section.

5.2.1 TiHe signal at 25228.15 cm−1

The TiHe density dependence (nTiHe) on temperature (T ), He density (nHe) and Ti

−1 density (nTi) is studied at the frequency 25228.15 cm .

5.2.1.1 Ti dependence

The TiHe density dependence on Ti density in thermal equilibrium is expected to

be linear (Eq. 1.4). To conﬁrm this linear dependence, the ﬂuorescence signal was

converted into OD (as explained in Chapter3). We plotted the TiHe OD with respect

to 48Ti OD, as shown in Fig. 5.4. The data is colored by temperature, divided by

segments according to their temperature, and then ﬁt to a line. The linear dependence

of TiHe OD on 48Ti OD is conﬁrmed.

The graph in Fig. 5.5 shows the TiHe OD with respect to 48Ti OD, as in the

previous graph and for the same conditions. Data points from a small temperature

range (from 1.97 K to 2.10 K) were extracted to study the dependence at “constant

temperature”; for this reason, fewer data points are presented in the plot. The data

is colored with respect to time after ablation. There is no dependence on time, so it

is concluded that the system had already reached thermal equilibrium. Subsequent

data is presented as the ratio: TiHe OD/Ti OD.

5.2.1.2 Temperature dependence

The dependence of TiHe OD/48Ti OD is studied with respect to temperature at a

constant He density and the data obtained is ﬁt to Eq. 4.2. From the ﬁt, the values 97

He density [cm−3] E [K]

3.2 × 1016 -1.84±0.2

5.8 × 1016 -2.21±0.3

1.3 × 1017 -1.85±0.2

2.6 × 1017 -2.57±0.3

4.1 × 1017 -2.10±0.2

5.4 × 1017 -1.93±0.2

5.5 × 1017 -1.51±0.3

7.9 × 1017 -1.59±0.3

Table 5.1: Binding energies and their relative uncertainty obtained at studying the temperature dependence of the TiHe interaction found at 25228.15 cm−1 at different He densities. of the binding energy (E) and the c coefficients (c) can be extracted. We performed this measurement with He densities from 3.2 × 1016 cm−3 to 7.9 × 1017 cm−3. Fig. 5.6 shows TiHe OD/48Ti OD vs temperature at different He densities.

The TiHe density of the peak at 25228.15 cm−1 follows the expected dependence on temperature in thermal equilibrium: at low temperatures, the TiHe production increases exponentially.

The binding energies extracted from the ﬁt of the peak at the frequency 25228.15 cm−1 are shown in Table 5.1.

Fig. 5.7 presents the binding energy E as a function of He density for the TiHe peak found at 25228.15 cm−1. The averaged binding energy E is equal to (-1.95±0.1)

K and ±0.1 K corresponds to the relative error from the statistical sources. We can conclude that the spectroscopic signal at 25228.15 cm−1 corresponds to a bound

TiHe molecule. The systematic error is ±0.25 K (this error will be discussed in later sections). 98

5.2.1.3 He dependence

As seen in Fig. 5.6, the TiHe density increases with the He density. To study this quantitatively, we examine the dependence of the c coeﬃcient (from Eq. 4.2) on the

He density.

Fig. 5.8 shows the c coeﬃcient vs He density and a linear dependence is observed.

From this, we conclude that this line corresponds to TiHe and not to TiHe2.

Summary of the signal at 25228.15 cm−1: The TiHe density is linearly proportional to the Ti and He densities. At low temperatures, the TiHe production increases exponentially (conﬁrmed by the ﬁt of the data). This transition is consistent with a bound molecular TiHe transition.

5.2.2 TiHe signal at 25228.03 cm−1

The TiHe density is studied for the spectroscopic signal at the frequency 25228.03 cm−1 as a function of the Ti density, He density and temperature.

5.2.2.1 Ti dependence

Fig. 5.9 indicates that the TiHe density dependence of the signal at 25228.03 cm−1 is linear with respect to 48Ti OD.

5.2.2.2 Temperature dependence

At constant He density, the TiHe OD/48Ti OD is studied with respect to temperature at 25228.03 cm−1. The graph in Fig. 5.10 shows TiHe OD/48Ti OD as a function of temperature at diﬀerent He densities. Fig. 5.11 shows the TiHe OD T3/2/48Ti OD temperature dependence at diﬀerent He densities. Notice that at low temperatures, the ratio TiHe OD T3/2/48Ti OD increases. 99

He density [cm−3] E [K]

3.2 × 1016 -1.49±0.2

5.8 × 1016 -1.75±0.3

1.3 × 1017 -1.62±0.3

2.6 × 1017 -1.96±0.3

4.1 × 1017 -1.87±0.2

5.4 × 1017 -1.29±0.3

5.5 × 1017 -1.03±0.2

7.9 × 1017 -1.17±0.3

Table 5.2: Binding energies and their relative uncertainty obtained at studying the temperature dependence of the TiHe interaction found at 25228.03 cm−1 at diﬀerent He densities.

A summary of the binding energies extracted from the ﬁt of the peak at the

frequency 25228.03 cm−1 are shown in Table 5.2.

The averaged binding energy E as a function of He density for the TiHe peak

found at 25228.03 cm−1 is shown in Fig. 5.12. E is equal (-1.52±0.1) K and ±0.1

K is the relative error from the statistical sources. The systematic error is ±0.25 K.

This binding energy value also indicates that we have found a molecular transition of

TiHe.

5.2.2.3 He dependence

The c coeﬃcient vs He density is shown in Fig. 5.13. The TiHe density shows a clear linear dependence on He density.

Summary of signal at 25228.03 cm−1: The study of the signal at 25228.03 cm−1 with respect to Ti density, temperature and He density is as expected. The 100

He density [cm−3] E [K]

3.2 × 1016 -1.12±0.2

5.8 × 1016 -1.70±0.3

2.6 × 1017 -2.22±0.4

4.1 × 1017 -1.00±0.2

5.4 × 1017 -1.57±0.3

Table 5.3: Binding energies and their relative uncertainty obtained at studying the temperature dependence of the TiHe interaction found at 25228.10 cm−1 at diﬀerent He densities. interaction found at this frequency corresponds to a bound TiHe molecule.

5.2.3 TiHe signal at 25228.10 cm−1

5.2.3.1 Ti dependence

The TiHe density dependence on Ti density is linear as shown in Fig. 5.14. The data is colored by temperature.

5.2.3.2 Temperature dependence

The graph in Fig. 5.15 shows TiHe OD/48Ti OD as a function of temperature at diﬀerent He densities.

The binding energies extracted from the ﬁt for the peak at the frequency 25228.10 cm−1 are shown in Table 5.3.

Fig. 5.16 shows the averaged binding energy E as a function of He density at

25228.10 cm−1. E = (-1.33±0.1) K and ±0.1 K is the relative error from the statistical sources. The systematic error is ±0.25 K. This value of E leads us to conclude that this transition is in agreement with a bound TiHe molecule. 101

5.2.3.3 He dependence

Fig. 5.17 shows the the c coeﬃcient vs He density at 25228.10 cm−1. The TiHe den-

sity increases linearly with the He density.

Summary of the signal at 25228.10 cm−1: The spectroscopic signal at

25228.10 cm−1 is consistent with a bound TiHe molecule.

5.2.4 TiHe signal at 25227.99 cm−1

5.2.4.1 Ti dependence

The density of TiHe increases linearly with respect to the Ti density, as in Fig. 5.18.

The data is colored by temperature.

5.2.4.2 Temperature dependence

The temperature dependence of TiHe OD/48Ti OD is shown in Fig. 5.19 at diﬀerent

He densities.

Table 5.4 shows the binding energies at 25228.10 cm−1 extracted from the ﬁt of the TiHe OD/48Ti OD temperature dependence.

The averaged binding energy of the peak found at 25227.99 cm−1 is shown in

Fig. 5.20. E = (-1.33±0.1) K and ±0.1 K is to the relative error from the statistical sources. The transition at this frequency is consistent with a bound transition of the

TiHe molecule.

5.2.4.3 He dependence

The TiHe density dependence at 25227.99 cm−1 is shown in Fig. 5.21. 102

He density [cm−3] E [K]

3.2 × 1016 -1.44±0.2

5.8 × 1016 -2.11±0.5

2.6 × 1017 -2.13±0.4

4.1 × 1017 -0.97±0.2

5.4 × 1017 -1.69±0.3

Table 5.4: Binding energies and their relative uncertainty obtained at studying the temperature dependence of the TiHe interaction found at 25227.99 cm−1 at diﬀerent He densities.

Summary of the signal at 25227.99 cm−1: The signal at 25227.99 cm−1 co- incides with a bound TiHe molecule.

The dependence of the TiHe density obtained from the four largest peaks shown in Fig. 5.1 were studied with respect to Ti density, He density, and temperature. The expected dependences in thermal equilibrium have been conﬁrmed. For this reason, we conclude that these spectroscopic signals correspond to molecular transitions of

TiHe.

5.2.5 Variations in nHe measurements

The He density in the cell was assumed to remain constant during the process of varying the temperature of the cell walls, considering that the cell was closed and He was not supposed to come in or out the cell while we were taking data. However, there might have been He stuck to the cell walls that could evaporate when the temperature of the cell walls was increased. If this was the case, then this change in He density might have aﬀected the values of the binding energy presented in previous sections.

For this reason, we wanted to study the He density dependence with respect to the 103

cell temperature for the highest and the lowest He densities used in this experiment

(∼ 8 × 1017 cm−3 and ∼ 3 × 1016 cm−3).

The Weber-Schmidt equation, mentioned in Chapter3, enabled us to convert the

pressure in the cell read by the pressure gauge (which was at 300 K and connected

through a small tube to the cell) into He density. For this task, we needed to know the

temperature of the cell for a speciﬁc cell pressure and substitute these temperature

and pressure values into the equation. We had a calibrated diode and a resistor

attached to the cell that allowed us to measure the cell temperature.

The temperature was measured using the cell diode and the resistors (we had an-

other resistor at room temperature for calibration). The resistors have high sensitivity

at low temperatures and the cell diode at high temperatures. Fig. 5.22 corresponds

to the highest He density. This ﬁgure shows that there was not a signiﬁcant change in

the He density with respect to temperature measured using the resistors at low tem-

peratures and the cell diode at high temperatures. The graph presented in Fig. 5.23,

for the case of the lowest He density, shows that there was an increase in He density

with respect to temperature. We note that this increase is within the error bars of

the high density ﬁt.

This variation of the He density could be causing a systematic error in our data

and this eﬀect is not easy to account for, since the pressure gauge was not located

inside of the cell, thus it cannot determine instantaneous changes in the He density;

the gauge measured the average pressure in the cell.

In order to determine how the change in He density with respect to temperature

could be aﬀecting our data, we go to Eq. 4.1 and correct for He density. The data so

far has been presented as the ratio of TiHe OD with respect to 48Ti OD. Instead of

TiHe OD TiHe OD considering 48 , we want to study the temperature dependence of 48 . The Ti OD Ti OD nHe quantity nHe is given by a+bx where x is temperature and the parameters a and b are 104

Peak frequency [cm−1] Energy [K] Energy [cm−1]

25228.15 -1.95±0.1 -1.37±0.07

25228.10 -1.33±0.1 -0.93±0.07

25228.03 -1.52±0.1 -1.07±0.07

25227.99 -1.40±0.1 -0.99±0.07

Table 5.5: Binding energy of the four largest transitions found at the frequency range 25227.95 cm−1 to 25228.20 cm−1. ±0.1 K and ±0.07 cm−1 correspond to the relative error; ±0.25 K and ±0.18 cm−1 to the overall error. the fit parameters shown in Fig. 5.22 and Fig. 5.23. In Fig. 5.25, the binding energy extracted from the fit of the density corrected data changed. At low He densities, the change is about 0.27 K and at the high He densities, the change is negligible. After averaging these binding energy changes for all the data at different He densities, we found that the overall correction in the averaged binding energy due to a change of

He density with respect to temperature is ∼ 0.25 K.

In Table 5.5 is presented a summary of the peaks with their respective ground state binding energy in kelvins and wavenumbers.

5.2.6 Identiﬁcation of the TiHe peaks

The TiHe peaks shown in Fig. 5.1 can be identiﬁed from their:

5.2.6.1 Thermodynamic dependence

The spectroscopic signals located at 25228.15 cm−1, 25228.10 cm−1, 25228.03 cm−1

−1 and 25227.99 cm were studied with respect to their thermodynamic properties (nTi, nHe and T ). From this analysis, it was concluded that these signals correspond to molecular transitions, their binding energies were determined and they show diﬀerent 105

values which suggests that these peaks are transitions from diﬀerent energy levels in

the ground state.

From the binding energy measurements, we can conclude that the signal at 25228.15

cm−1 is a transition from one the rotational level and that the transitions at 25228.10

cm−1, 25228.03 cm−1 and 25227.99 cm−1 are transitions from another rotational level.

The measured binding energies of the 25228.10 cm−1, 25228.03 cm−1 and 25227.99 cm−1 signals and their uncertainty do not allow us to infer that there is a splitting in the energy levels for these three transitions. Also, not all the peaks seen in the

ﬂuorescence spectrum in Fig. 5.1 could be analyzed since their signal-to-noise makes them diﬃcult to study with respect to temperature.

5.2.6.2 Isotope shifts

We looked for correlations between the TiHe peaks and the isotopes of Ti. The TiHe

3 3 −1 probe laser was tuned to the atomic Ti transition a F2 → y F3 at 25227.220 cm and a few µW coming from this laser were used to obtain a fluorescence sprectrum of Ti. In Fig. 5.26, the Ti fluorescence spectrum is overlapped with the TiHe fluorescence spectrum; the 48Ti peak is centered at the frequencies 25228.15 cm−1, 25228.03 cm−1 and 25227.99 cm−1, which are the highest peaks seen in the TiHe fluorescence spectrum. We assumed that these peaks are 48Ti4He since 48Ti is the most abundant

isotope. We wanted to ﬁnd patterns in the TiHe spectrum that could be attributed

to the isotopes of Ti.

We compared the equivalence in height of the peaks at 25228.19 cm−1 and 25228.08

cm−1 with 50Ti, and they match. The peaks 25228.19 cm−1 and 25228.08 cm−1 are

50Ti4He. The TiHe peak at 25228.10 cm−1 and the isotope 46Ti show equivalent

distances in frequency from the 48Ti4He and the 48Ti peaks respectively, but, they do

not match in height. We believe that there might be another transition overlapping 106

Isotopes Molecular shift [cm−1] Atomic shift [cm−1]

46 − 48 0.045 0.043

47 − 48 0.019 0.021

50 − 48 0.043 0.040

Table 5.6: Measured isotope shift for the TiHe molecules and the atomic isotope shifts from [1].

with 46Ti4He at 25228.10 cm−1. There is a growing peak at 25227.98 cm−1 that has

also been identiﬁed as 46Ti4He. At 25227.95 cm−1, there is an overlap of a 46Ti4He peak with an unidentiﬁed transition at this frequency. There is a very tiny bump in the TiHe spectrum at 25228.13 cm−1 that could be caused by 47Ti, and another small

peak at 25228.015 cm−1 that also correlates with 47Ti. These peaks are identiﬁed as

47Ti4He. However the peak at 25228.015 cm−1 looks more sharp and deﬁned than

47Ti, this means that this peak could also have some overlap with another molecular

transition.

Table 5.6 shows the TiHe distances in wavenumbers of the peaks seen in the TiHe

3 3 spectrum and the Ti isotope shifts for the a F2 → y F3 Ti atomic transition at 25227.220 cm−1 reported in reference [1]. The molecular shifts diﬀer from the atomic

isotope shifts by about 0.002 cm−1. Table 5.7 shows the identiﬁcation of the TiHe

peak with their corresponding Ti isotope.

It is important to mention that the transitions that were identiﬁed by the isotope

recognition method have been identiﬁed as X Ti4He molecules (where X is the Ti

isotope), because the abundance of 4He is 99.999862% and 3He is only 0.000138%

abundant. We do not expect to have suﬃcient signal to noise to observe X Ti3He

molecules. 107

Peak frequency [cm−1] Ti isotope-He

25227.95 46Ti4He

25227.98 46Ti4He

25228.01 47Ti4He

25228.03 48Ti4He

25228.08 50Ti4He

25228.10 46Ti4He

25228.13 47Ti4He

25228.15 48Ti4He

25228.19 50Ti4He

Table 5.7: Identiﬁcation of the TiHe peaks with their corresponding Ti isotope.

5.2.6.3 Comparison between experimental results and theoretical predic-

tions

Table 2.1 shows the predicted TiHe ground state binding energies using the TiHe ground state potential in reference [2]. The experimentally measured ground state binding energies presented in Table 5.5 do not agree with the predictions from Chap- ter2. For this reason, theoretical collaborators of this research, Dr. Timur Tscherbul and Dr. Jacek Klos, supported the TiHe experiment with theoretical calculations.

Dr. Jacek Klos re-calculated the TiHe molecular potentials and Dr. Timur Tscherbul calculated the bound energy levels from the potential surfaces.

The ground state of the TiHe molecule was calculated by Dr. Tscherbul. From his theoretical calculations and our experimental measurements, we concluded that

TiHe is a good Hund’s case e) molecule. Hund’s cases represent diﬀerent scenarios in which angular momenta can be coupled in diatomic molecules. To date, the only 108 compound studied under this case is HeKr+ ion [81]. The fact that TiHe falls into the Hund’s case e), makes TiHe more interesting, since it would be the ﬁrst example of a ground state diatomic molecule described by this coupling case [82].

The coupling of angular momenta of Hund’s case e) is shown in Fig. 5.27, and it is the ideal case to study the TiHe molecule since the magnitude of the spin- orbit coupling A, is comparable to the spin-orbit coupling of Ti (because of the weak interaction of TiHe). The magnitude of A is on the order of 102 cm−1. The rotational constant B of the TiHe molecule is ∼ 0.2 cm−1 (from the experimental results). The coupling of L (or S) to axis, ∆E, is experimentally unresolvable, but we believe that is smaller than 0.1 cm−1. Consequently, A > B > ∆E, this is consistent with the limiting case Hund’s e) [82]. The pattern of rotational levels and the TiHe spectrum were predicted.

3 The ground state of Ti is a F2, this implies that the atomic angular momentum

00 for the TiHe ground state is Ja = 2 (the double prime is used for the ground state and prime for the excited state). According to the theoretical calculations, a molecular transition from TiHe must have a change in the rotational angular momentum equal to zero (∆R = 0). The TiHe ground state is made of one vibrational level and four rotational levels R00 = 0, 1, 2 and 3. Dr. Tscherbul calculated the binding energy for each rotational level with its respective J value (total angular momentum quantum number), where J = |Ja − R|, ..., Ja + R. Cases:

00 00 00 Ja = 2 and R = 0: J = 2.

00 00 00 Ja = 2 and R = 1: J = 1, 2, 3.

00 00 00 Ja = 2 and R = 2: J = 0, 1, 2, 3, 4.

00 00 00 For the Ja = 2 and R = 3 level, only J = 1 shows a bound energy. A summary of the rotational ground state theory is found in Table 5.8. 109

00 00 00 −1 Ja R J E(K) E(cm ) 2 0 2 -1.824 -1.268

2 1 1 -1.529 -1.063

2 1 3 -1.508 -1.048

2 1 2 -1.469 -1.021

2 2 0 -0.910 -0.632

2 2 1 -0.884 -0.614

2 2 4 -0.878 -0.610

2 2 2 -0.860 -0.598

2 2 3 -0.846 -0.588

2 3 1 -0.008 -0.005

Table 5.8: Predicted ground state binding energies of TiHe.

By comparing the theoretical binding energy values with the ones obtained ex-

perimentally through the study of the equilibrium thermodynamic properties, it is

noticed that there is good agreement with the new calculations. Table 5.9 shows the

TiHe signals that were studied through the experiment and their identiﬁcation with

theory.

3 3 The studied TiHe line is closely detuned from the a F2 → y F3 Ti transition.

0 This means that the TiHe excited state has an atomic angular momentum Ja = 3. We have identiﬁed some of the transitions seen experimentally as R00=0 and R00=1. From the theoretical calculations, it was obtained that ∆R = 0. Therefore, the excited state of TiHe should have R0=0 and R0=1 rotational energy levels. The theoretical calculations predict that in the ground state there should be rotational levels R00=2, however, we did not see these transitions. We believe that the excited state does not support the R0=2 energy levels, since the TiHe signals are blue detuned, which means 110

Peak frequency [cm−1] Energy [K] ID]

25228.15 -1.95 R00=0

25228.10 -1.33 R00=1

25228.03 -1.52 R00=1

25227.99 -1.40 R00=1

Table 5.9: Identiﬁcation of the spectroscopic signals at 25227.95 cm−1 to 25228.20 cm−1 with the theoretical calculations. ±0.1 K and ±0.25 K are the relative and overall error respectively.

that the TiHe excited state is more weakly bound. We infer that the excited states

of TiHe should be:

0 0 0 Ja = 3 and R = 0: J = 3.

0 0 0 Ja = 3 and R = 1: J = 2, 3, 4.

Fig. 5.28 shows the allowed transitions from the ground to the excited state of

TiHe. There is only one transition from R00 = 0 to R0 = 0 (∆R = 0). This transition was observed experimentally and it is consistent with the peak seen at 25228.15 cm−1.

00 0 00 0 This peak is identiﬁed as a transition from Ja = 2 to Ja = 3 and J = 2 to J = 3. There are six allowed transitions from R00 = 1 to R0 = 1. Table 5.9 shows that three of the peaks studied experimentally are consistent with R00 = 1 to R0 = 1 transitions. However, we were not able to identify these peaks according to their

J 00 to J 0 transition. We believe that the peaks that were not studied according to their thermodynamic properties (the other peaks found showed a small signal to noise that precluded further analysis): 25228.01 cm−1, 25227.97 cm−1 and 25227.95 cm−1, correspond to the three remaining transitions from R00 = 1 to R0 = 1. 111

-4 1.2x10 4.0

3.5 T[K] 1.0 3.0 Data 2.5 2.0 0.8 1.5

0.6 TiHe OD 0.4

0.2

0.0 0 5 10 15 48Ti OD

Figure 5.4: TiHe OD as a function of Ti OD at 25228.15 cm−1. The color scale indicates temperature. The TiHe density increases linearly with respect to the Ti density. The data was taken at a He density of 3.2 × 1016 cm−3 and ﬁt using y = bx: b = (9.54 ± 0.43) × 10−7 for the temperature range between 3 K and 4.5 K (blue), b = (1.64 ± 0.05) × 10−6 for the temperature range between 2.3 K and 3 K (green), b = (2.73 ± 0.07) × 10−6 for the temperature range between 1.8 K and 2.3 K (yellow) and b = (5.52 ± 0.17) × 10−6 for the temperature range between 1.4 K and 1.8 K (red). 112

1.0 [s] Time 0.8 -6 20x10 0.6 0.4 0.2

10 TiHe OD TiHe

5 Data Fit

0 0 2 4 6 8 48 Ti OD

Figure 5.5: TiHe OD as a function of Ti OD. The data was taken at the temperature 16 −3 range from 1.7 K to 2.1 K and at a He density of nHe = 3.2 × 10 cm . The data is ﬁt using y = bx, b = (2.80 ± 0.09) × 10−6. The color scale indicates the time after ablation and no dependence on this time is observed so the system is in thermal equilibrium. 113

-3 4 He density [cm ] 17 7.9 x10 3 17 5.5x10 17 5.4x10 2 17 4.1x10 17 2.6x10 17 -4 1.3x10 16 10 5.8x10 16 3.2x10 6 5 4 3 Ti OD

48 2

-5 10 TiHe OD/ 6 5 4 3

-6 10

0 3 4 2x10 Temperature [K]

Figure 5.6: TiHe OD/48Ti OD as a function of temperature at 25228.15 cm−1 at diﬀerent He densities. E and c are obtained from the ﬁt as discussed in the text. 114

-1.5

-2.0

-2.5 Binding energy Binding [K]

Data -3.0 Fit

17 0 2 4 6 8x10 -3 Helium density [cm ]

Figure 5.7: Binding energy obtained from the study of the temperature dependence of the TiHe signal at 25228.15 cm−1. The ﬁt is to a constant line. E = (-1.95±0.1) K. 115

-5 8x10 Data Fit 6 ] 3/2 4 c [K c

0 0 2 4 6 17 -3 8x10 Helium density [cm ]

Figure 5.8: c coeﬃcient vs He density. TiHe signal = 25228.15 cm−1. The data is ﬁt using y = bx. b = (9.33 ± 0.78) × 10−23 K3/2cm3. 116

4.0 3.5 -4 3.0 T[K] Data 1.5x10 2.5 2.0

1.0 TiHe OD TiHe

0.5

0.0 0 1 2 3 4 5 6 48Ti OD

Figure 5.9: TiHe OD as a function of Ti OD at 25228.03 cm−1. The color scale indicates temperature. The TiHe density increases linearly with respect to the Ti density. The data was taken at a He density of 2.6 × 1017 cm−3 and ﬁt using y = bx: b = (5.01 ± 0.21) × 10−6 for the temperature range between 3.2 K and 4 K (blue), b = (1.06 ± 0.05) × 10−5 for the temperature range between 2.2 K and 3.2 K (green), b = (1.63 ± 0.08) × 10−5 for the temperature range between 1.8 K and 2.2 K (yellow) and b = (2.72 ± 0.2) × 10−5 for the temperature range between 1.4 K and 1.8 K (red). 117

-3 4 He density [cm ] 17 3 7.9 x10 17 5.5x10 17 2 5.4x10 17 4.1x10 17 2.6x10 17 -4 1.3x10 10 16 5.8x10 16 3.2x10 6 5 4 3 Ti OD

48 2

-5 10 TiHe OD/ 6 5 4 3

-6 10

0 3 2x10 Temperature [K]

Figure 5.10: TiHe OD/48Ti OD as a function of temperature at 25228.03 cm−1. The data was taken at diﬀerent He densities. E and c are obtained from the ﬁt as discussed in the text. 118

-3 -3 10 He density [cm ] 17 8 7.9 x10 7 17 5.5 x10 6 17 5.4 x10 5 17 4.1 x10 17 4 2.6 x10 17 1.3 x10 16 3 5.8 x10 16 3.2 x10 2 Ti OD 48 / -4

3/2 10 8 7 6 5 4 TiHe OD*T 3

-5 10 8

0 3 2x10 Temperature [K]

Figure 5.11: TiHe OD T3/2/48Ti OD as a function of temperature at 25228.03 cm−1. The data was taken at diﬀerent He densities. 119

-0.8

-1.0

-1.2

-1.4

-1.6

-1.8 Energy -2.0 Binding energy [K] Fit -2.2

-2.4

17 0 2 4 6 8x10 -3 Helium density [cm ]

Figure 5.12: Binding energy obtained from the study of the temperature dependence of the TiHe signal at 25228.03 cm−1. The ﬁt is to a constant line. E = (-1.52±0.1) K. 120

-4 1.0x10 Data Fit

0.8 ]

3/2 0.6 c [K c 0.4

0.2

0.0 0 2 4 6 17 -3 8x10 Helium density [cm ]

Figure 5.13: c coeﬃcient vs He density. TiHe signal = 25228.03 cm−1. The data is ﬁt using y = bx. b = (1.20 ± 0.10) × 10−22 K3/2cm3. 121

-5 5x10 Data 3.5 3.0 T[K] 4 2.5 2.0 1.5

3 TiHe OD 2

0 0 1 2 3 48 Ti OD

Figure 5.14: TiHe OD as a function of Ti OD at 25228.10 cm−1. The color scale indicates temperature. The TiHe density increases linearly with respect to the Ti density. The data was taken at a He density of 4.1 × 1017 cm−3 and ﬁt using y = bx: b = (3.36 ± 0.14) × 10−6 for the temperature range between 3 K and 4.5 K (blue), b = (7.56 ± 0.54) × 10−6 for the temperature range between 2 K and 3 K (green) and b = (1.34 ± 0.07) × 10−5 for the temperature range between 1.45 K and 2 K (red). 122

-3 He density [cm ] 3 17 5.4x10 17 4.1x10 17 2 2.6x10 16 5.8x10 16 3.2x10

-5 10 8 7 6 5 Ti OD

48 4

2 TiHe OD/

-6 10 8 7 6 5 4

0 3 2x10 Temperature [K]

Figure 5.15: TiHe OD/48Ti OD as a function of temperature at 25228.10 cm−1. The data was taken at diﬀerent He densities. E and c are obtained from the ﬁt as discussed in the text. 123

-1.0

-1.5

-2.0

-2.5 Binding energy Binding [K] Energy Fit -3.0

17 0 2 4 6 8x10 -3 Helium density [cm ]

Figure 5.16: Binding energy obtained from the study of the temperature dependence of the TiHe signal at 25228.10 cm−1. The ﬁt is to a constant line. E = (-1.33±0.1) K. 124

-5 3.0x10 Data 2.5 Fit

2.0 ] 3/2 1.5 c [K c

1.0

0.5

0.0 1 2 3 4 5 6 17 -3 7x10 Helium density [cm ]

Figure 5.17: c coeﬃcient vs He density. TiHe signal = 25228.10 cm−1. The data is ﬁt using y = bx. b = (4.30 ± 0.46) × 10−23 K3/2cm3. 125

-5 8x10 Data 3.5 3.0 T[k] 2.5 6 2.0 1.5

4 TiHe OD

0 0 1 2 3 48 Ti OD

Figure 5.18: TiHe OD as a function of Ti OD at 25227.99 cm−1. The color scale indicates temperature. The TiHe density increases linearly with respect to the Ti density. The data was taken at a He density of 4.1 × 1017 cm−3 and ﬁt using y = bx: b = (5.41 ± 0.19) × 10−6 for the temperature range between 3 K and 4.5 K (blue), b = (1.05 ± 0.06) × 10−5 for the temperature range between 2 K and 3 K (green) and b = (2.15 ± 0.09) × 10−5 for the temperature range between 1.45 K and 2 K (red). 126

6 -3 He density [cm ] 17 5 5.4x10 17 4 4.1x10 17 2.6x10 16 3 5.8x10 16 3.2x10 2

-5 10 8

Ti OD 7

48 6 5 4

3 TiHe OD/

-6 10 8 7 6 5 0 3 2x10 Temperature [K]

Figure 5.19: TiHe OD/48Ti OD as a function of temperature at 25227.99 cm−1. The data was taken at diﬀerent He densities. E and c are obtained from the ﬁt as discussed in the text. 127

-1.0

-1.5

-2.0

-2.5 Binding energy Binding [K] Energy Fit -3.0

17 0 2 4 6 8x10 -3 Helium density [cm ]

Figure 5.20: Binding energy obtained from the study of the temperature dependence of the TiHe signal at 25227.99 cm−1. The ﬁt is to a constant line. E = (-1.40±0.1) K. 128

-5 6x10

Data 5 Fit

4 ] 3/2 3 c [K c

0 1 2 3 4 5 6 17 -3 7x10 Helium density [cm ]

Figure 5.21: c coeﬃcient vs He density. TiHe signal = 25227.99 cm−1. The data is ﬁt using y = bx. b = (6.65 ± 0.73) × 10−23 K3/2cm3. 129

17 7.5x10 ]

-3 7.0

6.5 Fit

6.0 Hellium density [cm 5.5

5.0 1.5 2.0 2.5 3.0 Temperature [K]

Figure 5.22: He density vs Temperature of the cell for the highest He density used in the experiment. The ﬁt is to a line a + bx where a = (7.50 ± 0.21) × 1017 cm−3; b = (−5.78 ± 10.2) × 1015cm−3/K. There is no indication of an increase of He density with respect to temperature for this case. 130

16 ] 3.0x10 -3

2.5 Fit

Helium density [cm 2.0

1.5 1.5 2.0 2.5 3.0 Temperature [K]

Figure 5.23: He density vs Temperature of the cell for the lowest He used in the experiment. The ﬁt is to a line a + bx where a = (2.83 ± 0.18) × 1016 cm−3; b = (1.64 ± 0.94) × 1015cm−3/K. There is indication of an increase of He density with respect to temperature. 131

2 Data with no density variations included Fit -4 Data with density variations included 10 9 Fit 8 TiHeOD/( -22 7 910 6 8

TiOD 5 7 48 48 6 4 *n TiOD 5

3 4 TiHeOD/ He 3 2 )

0 3 4 2x10 Temperature [K]

17 −3 Figure 5.24: Temperature dependence data at nHe = 7.9 × 10 cm . The data TiHe OD points from 48Ti OD vs T are represented by the closed circles. The binding energy E and the c coefficient are extracted from the fit: E = (−1.59 ± 0.3) K and c = −5 3/2 TiHe OD (6.69 ± 0.9) × 10 K . The data points from 48 vs T are represented by the Ti ODnHe 17 16 −3 open circles. The He density is given by nHe = (6.90×10 +1.46×10 T ) cm . The binding energy E and the c coefficient are extracted from the fit: E = (−1.70 ± 0.3) K and c = (8.78 ± 1.19) × 10−23 K3/2. The difference in the binding energies is 0.1 K. 132

7 6 Data with no density variations included 2 Fit Data with density variations included

5 Fit TiHeOD/( 4

-22 3 10 9 TiOD 8 48 48 7 *n TiOD 2 6 5 He TiHeOD/

4 )

-6 10 3 9 8

0 3 4 2x10 Temperature [K]

16 −3 Figure 5.25: Temperature dependence data at nHe = 3.2 × 10 cm . The data TiHe OD points from 48Ti OD vs T are represented by the closed circles. The binding energy E and the c coefficient are extracted from the fit: E = (−1.84 ± 0.2) K and c −6 3/2 TiHe OD =(3.92±0.4)×10 K . The data points from 48 vs T are represented by the Ti ODnHe 16 15 −3 open circles. The He density is given by nHe = (2.83×10 +1.64×10 T ) cm . The binding energy E and the c coefficient are extracted from the fit: E = (−2.11 ± 0.2) K and c = (1.08 ± 0.11) × 10−22 K3/2. The difference in the binding energies is 0.27 K. 133

-1 Ti frequency [cm ] 25227.15 25227.20 25227.25 25227.30 25227.35 -1 Ti frequency [cm ] 25227.05 25227.10 25227.15 25227.20 25227.25 10 1.8 6 -1 Ti frequency [cm ] 8 1.6 25227.20 25227.25 25227.30 25227.35 25227.40 5 [V] fluorescence Ti

6 4 1.4 TiHe spectrum Ti spectrum Ti spectrum 3 1.2 Ti spectrum 4

TiHe fluorescence [V] 2 1.0 2 25227.95 25228.00 25228.05 25228.10 25228.15 25228.201 0.8 0

25227.95 25228.00 25228.05 25228.10 25228.15 25228.20 -1 TiHe frequency [cm ]

Figure 5.26: The solid line indicates the TiHe fluorescence spectrum. The dotted 3 3 lines correspond to the Ti fluorescence spectrum at a F2 → y F3 atomic Ti transiton at 25227.220 cm−1. The identification of some of the TiHe peaks was made by analyzing the correspondence among the Ti isotopes with TiHe peaks and their isotope shift. 134

Hund’s case (e)

J: Total angular momentum

Ja Ja: L and S coupling

R: Rotational angular J R momentum of the nuclei

L: Electronic orbital angular momentum S: Electronic spin angular momentum

Figure 5.27: Hund’s coupling case e).

3 R’=1 4 J’ =3 2 a

R’=0 3

2 R’’=1 3

1 J’’a=2

R’’=0 2

Figure 5.28: Ground and excited state of TiHe. R00=0 is expected to have one transition and R00=2 is expected to have six transitions according to the selection rules: ∆R = 0 and ∆J = 0, ±1. 135

Chapter 6

Measurement of the three body recombination coeﬃcient K3

The purpose of this chapter is to present our experimental efforts towards measuring the two body collisional dissociation coefficient K2 and the three body recombination coefficient K3 of the TiHe molecule, using a technique that we call the Saturation Model. The TiHe peaks studied are 25228.15 cm−1 and 25228.03 cm−1 due to their good signal-to-noise.

We attempted to measure K2 and K3 for the vdW LiHe molecule. Nevertheless, the experimental approaches to accomplish this task were unsuccessful. We think that the formation of LiHe was so fast that were not able to measure it [54]. This is precisely one of the motivations for our group to work with the TiHe molecule, since its binding energy was expected to be ∼ 1 cm −1 and in fact, according to our measurements, is twice this value. A bigger binding energy would be expected to slow

K2 because not every collision with a He atom would break the TiHe molecule, and consequently we might be able to experimentally obtain K2 and K3 for TiHe. The TiHe molecule was expected to form through the three body recombination 136 process, in which one Ti atom collides with two He atoms and TiHe is created. The inverse process also occurs, in which a TiHe molecule collides with a He atom and the molecule is destroyed through these collisions. The 3-body reaction for TiHe is as follows:

K3 Ti + He + He TiHe + He (6.1) K2 The TiHe density formation rate is given by:

2 n˙ TiHe = −n˙ Ti = K3 nTinHe − K2 nTiHenHe (6.2)

Where:

6 K3: Three body recombination coeﬃcient [cm /s].

3 K2: Two body collisional dissociation coeﬃcient [cm /s].

−3 nTi: Density of Ti [cm ].

−3 nHe: Density of He [cm ].

−3 nTiHe: Density of TiHe [cm ] that we measure. For our initial model, we

consider nTiHe for a single state and ignore state-changing collisions.

2 K3 nHe: Three body recombination rate [1/s].

K2 nHe: Two body collisional dissociation rate [1/s].

τ= 1 : How long it takes before TiHe is destroyed by a collision [s]. K2nHe

1 τ= 2 : How long it takes before Ti collides with He to form TiHe [s]. K3nHe 137

In thermal equilibriumn ˙ TiHe = 0. This leads to:

K n 3 = TiHe (6.3) K2 nHenTi Also, it has been shown that in thermal equilibrium the density of TiHe for a

single bound state follows the mathematical expression:

h2 3/2 −E/kB T nTiHe = nTinHe · ge (6.4) 2πµKBT Combining these last two expressions:

2 3/2 K3 h = · ge−E/kB T (6.5) K2 2πµKBT

This indicates that if we know the value of K3 we can calculate K2 and vice versa. The other quantities on the right hand side of Eq. 6.5 are known in our experiment.

6.1 Photodissociation experiments

We wanted to destroy the TiHe molecules by means of an intense laser and measure the ﬂuorescence of the molecules that were left in the cell as a function of power to obtain the quantity Psat (saturation power, which will be explained later). From Psat we expect to calculate K2. We called the model that would allows us to extract Psat as the Saturation Model.

The TiHe density formation rate when a laser light is interacting with the molecules is given by:

2 n˙ TiHe = K3 nTinHe − K2 nTiHenHe − Kl nTiHeI (6.6)

2 Where Kl [cm /J] is the photodissociation coeﬃcient and I is the intensity of the TiHe probe laser beam [Watts/cm2]. We are assuming that the excited state 138 population is much smaller than the ground state population (nexc << ng). The formation rate of TiHe will decrease for large values of I. The quantity KlI [1/s] is the photodissociation rate.

If the laser is on for a long time, then a “new” equilibrium will be reached by the system:

2 0 = K3 nTinHe − K2 nTiHenHe − Kl nTiHeI (6.7)

The laser-induced ﬂuorescence (LIF) of the molecules is proportional to the intensity of the laser and the density of TiHe present in the cell.

TiHe LIF = SNTiHeI = SnTiHeIV (6.8)

Where NTiHe = nTiHeV is the number of molecules and V is a volume element. S is a constant that takes in account the detector eﬃciency and the solid angle factor of the PMT (since only a fraction of the scattered photons is collected).

Solving Eq. 6.7 for the density of TiHe:

2 K3nTinHe nTiHe = (6.9) K2nHe + KlI After substituting Eq. 6.9 into Eq. 6.8, the following relation is obtained:

αI TiHe LIF = (6.10) β + I AK n n2 Where α = 3 Ti He [V] and β = K2nHe [W/cm2]. Kl Kl Eq. 6.10 shows the expression of the TiHe LIF for a homogeneous beam and

Fig. 6.1 presents the LIF with respect to intensity for the case of destruction and no destruction of molecules. 139

10 Saturation Line 8

y 6

0 0 2 4 6 8 10 12 x

Figure 6.1: The y axis corresponds to LIF and the x axis to intensity. The black curve shows the case of destruction of molecules by the laser, α = 2 V and β = 2 W/cm2. The green curve shows the case of no destruction of molecules y = x.

However, the case of a uniform beam does not apply in our experiment. The laser beam is a Gaussian beam, which means that its intensity I is not homogeneous in the radial direction r:

h −2r2 i 2 I(r) = I0e w (6.11)

Where I0 is the intensity of the beam at the center, r=0. 140

Rewriting Eq. 6.7 in terms of an inhomogeneous Gaussian beam:

σ I(r) 0 = K n n2 − K n n − TiHe n (6.12) 3 Ti He 2 TiHe He hν TiHe

σTiHeI(r) In Eq. 6.12, the photodissociation coeﬃcient Kl is represented by hν . The TiHe LIF is expressed as:

Z Z ∞ TiHe LIF = S nTiHe(r)I(r)σTiHedV = S 2πrLnTiHe(r)I(r)σTiHedr (6.13) V 0

Where L is length of the cell imaged on the PMT and S is a proportionality constant that accounts for the solid angle and the detector eﬃciency rate.

The quantity nTiHe(r) is found by solving Eq. 6.12 and using Eq. 6.11:

C n (r) = 1 (6.14) TiHe h −2r2 i σTiHe w2 C2 + hν I0e 2 With C1 = K3 nTinHe and C2 = K2 nHe. After substituting Eq. 6.14 into Eq. 6.13 and solving the integral, the expression of the TiHe LIF is given by:

P TiHe LIF = APsat ln 1 + (6.15) Psat The TiHe OD is expressed by:

1 P TiHe OD = TiHe ODinitial Psat ln 1 + (6.16) P Psat P is the power, A, and P sat are:

A = SnTiHeσTiHeL (6.17) 141

nTiHe as in Eq. 6.3

2 hνπw K2nHe Psat = (6.18) 2σTiHe

Psat is the power at which the photodissociation rate and the collisional dissociation rates are equal.

The ﬁt equations for TiHe LIF vs power and TiHe OD vs power have been found

and are shown in Eq. 6.15 and Eq. 6.16. These mathematical expressions are given in

2P terms of power (I = πω2 ) since in this experiment we can easily measure this quantity. These relations have been plotted in Fig. 6.2 and Fig. 6.3.

6.1.1 TiHe cross section (σTiHe)

The two body collisional dissociation coeﬃcient K2 can be calculated from Eq. 6.18:

2σTiHePsat K2 = 2 (6.19) hνπw nHe To complete this part of the experiment, the He density and the beam waist were

48 changed. The quantity Psat was extracted from the ﬁt of the data TiHe LIF/ Ti OD vs power.

The TiHe resonant light scattering cross section σTiHe is expressed as:

σTiHe = σTi · FC · (angular momentum factors) = B · σTi (6.20)

Where σTi is the Ti cross section, and it is calculated using the Doppler width. FC is the TiHe Franck Condon factor that indicates how much an electronic transition

gets split up by vibrational transitions.

The B factor indicates how much the σTiHe is reduced. B was found in the 142 following way:

Ti OD = nTi σTi L (6.21)

The Ti OD was measured and TiHe OD was calculated as explained in Chapter3.

TiHe OD = nTiHe σTiHe L = nTiHe (σTi B) L (6.22)

Then, the B factor is expressed as:

TiHe OD n B = Ti (6.23) Ti OD nTiHe Using Eq. 6.4, B is expressed as:

1 TiHe OD T 3/2 21 E/kB T B = 1.33 × 10 −3 3/2 e (6.24) cm K gTi OD nHe Using Eq. 6.24, the B factor was obtained in our experiment. Fig. 6.4 shows B as a function of temperature and it is around 0.1.

We are using the relations shown in Section. 3.5 to calculate the Ti cross section at T = 1.25 K (temperature that was used to take the saturation data). For the Ti

3 3 0 transition a F2 → y F3: J=2 and J = 3, γ = 22 ns [68]. At these conditions, the Ti cross section is:

−11 2 σTi ≈ 4 × 10 cm (6.25)

The TiHe cross section σTiHe is given by:

−12 2 σTiHe ≈ 4 × 10 cm (6.26) 143

Then, the K2 coeﬃcient is:

2 6 cm Psat K2 ≈ 5 × 10 2 (6.27) J w nHe The value of hν in Eq. 6.27 is ∼ 5 × 10−19 J.

The beam waist and the He density were varied during the experiment. The saturation power is found from the ﬁt of the data.

6.2 Experimental set up

As it was explained before, we wanted to destroy the molecules by means of an intense laser. The fluorescent light emitted by the molecules that were not destroyed will be studied at different probe laser powers to obtain the saturation power Psat. The coefficients K2 and K3 will be calculated from this measurement. In order to achieve this goal, we decreased the TiHe probe beam waist to generate higher intensities. To measure the laser beam waist, images of the TiHe laser beam were recorded on a CCD camera, as shown in Fig. 6.5. The picture taken of the laser beam was fit to a two dimensional Gaussian function. From the fit parameters, the horizontal and vertical waists were extracted in pixels. Then, we converted the beam waist from pixels to micrometers (1 pixel = 5.86 µm). This is how we measured the beam waist of the TiHe probe laser beam.

Fig. 6.6 shows the TiHe LIF/ Ti 48OD vs power for the beam waist used during our spectroscopic search. According to the ﬁgure, no indication of destruction of molecules is seen. The beam waist at the center of the cell was: (wx, wy)=(2, 5) mm. Similarly at the frequency 25228.03 cm−1, there is no evidence of destruction of molecules.

We reduced the beam waist from (wx, wy)=(2, 5) mm to (wx, wy)=(0.8, 0.5) mm 144

by adding a telescope in the TiHe probe laser path. The focal distances of the lenses

used in the telescope were: f1 =200 mm and f2 =50 mm. This reduction in the beam

2P waist implied an increase of the intensity by a factor of 30 (I = πw2 ). Fig. 6.7 shows the path of the TiHe laser beam after adding the telescope and Fig. 6.8 presents a

picture of the laser beam.

In the next section, the data collected at (wx, wy)=(0.8, 0.5) and the frequencies 25228.15 cm−1 and 25228.03 cm−1 is presented.

The beam waist was made even smaller to see a bigger saturation eﬀect. The

lenses of the telescope were changed to f1 =500 mm and f2 =200 mm. The new

beam waist was (wx, wy)=(0.4, 0.3) mm. The TiHe probe laser intensity increased by a factor of 3. In Fig. 6.9, the optics set up is shown and Fig. 6.10 presents the beam image at this smaller beam waist.

Fig. 6.11 shows the beam waists at diﬀerent positions in the cell and it is concluded that the beam waists remained uniform through the cell, which was the desired condition.

6.3 Upper limit of K2

The lifetime (t) of the ground state of TiHe can cause a broadening of the spectral lines. We call this type of broadening existence broadening, which is similar to the natural linewidth and the transit-time broadening [83]. The transit-time broadening is related to the broadening that atoms and molecules experience when they cross a laser beam. Because of the ﬁnite interaction time with the beam, there is a ∆t. The broadening of a spectral line can be obtained by measuring its FWHM (since FWHM

∼ 1/t). The shorter the ground state lifetime, the greater the broadening. 145

The existence broadening is expressed as:

1 ∆ω ∼ (6.28) existence ∆t The FWHM of TiHe in angular frequency at the highest He density that we used in

this experiment (8 × 1017 cm−1) is ∼ 9.40×108 rad/s (for both frequencies: 25228.15

cm−1 and 25228.03 cm−1). This FWHM of the TiHe spectral lines, also includes

1 Doppler and pressure broadening. For this reason an upper limit for K2 (τ ∼ ) K2nHe is given by:

FWHM −9 3 K2 ≥ = 1.2 × 10 [cm /s] (6.29) nHe

6.4 Beam waist (wx, wy)=(0.8, 0.5) mm

6.4.1 Saturation curves

From the saturation and the linear ﬁt of the data, we can determine if the TiHe

molecules are destroyed by the TiHe probe laser beam. In Fig. 6.12, an example of

the data collected for TiHe LIF/48Ti OD vs power and TiHe OD/48Ti OD vs power

at 25228.15 cm−1 are shown. These sets of data are ﬁt to a line and to the saturation

function. From the ﬁts, it is concluded that the molecules are being destroyed, the

data is clearly not part of the linear regime. A similar behavior is seen at the frequency

25228.03 cm−1.

6.4.1.1 Peak at 25228.15 cm−1

TiHe LIF/48Ti OD vs power taken at diﬀerent He densities is shown in Fig. 6.13. The

saturation power extracted from these sets of data and the coeﬃcients K2 and K3 are summarized in Table 6.1. Fig. 6.14 shows Psat vs density. As shown in the ﬁgures and 146

−3 3 6 nHe [cm ] A [V/mW] Psat [mW] K2 [cm /s] K3 [cm /s] 1.4 × 1017 0.10±0.0027 16.13± 3.37 (1.31±0.27) ×10−10 (3.35±0.70) ×10−31

7.3 × 1016 0.050±0.00093 5.79±0.63 (9.02±0.98) ×10−11 (2.31±0.25) ×10−31

2.1 × 1016 0.024±0.00094 1.72 ±0.24 (9.31±0.13) ×10−11 (2.38±0.33) ×10−31

1.1 × 1016 0.0072±0.00064 1.97 ±0.47 (2.03±0.48) ×10−10 (5.21±1.24) ×10−31

Table 6.1: Saturation ﬁt parameters and calculated values of K2 and K3 at the TiHe −1 frequency = 25228.15 cm and beam waist (wx, wy)=(0.8, 0.5) mm.

−3 3 6 nHe [cm ] A [V/mW] Psat [mW] K2 [cm /s] K3 [cm /s] 1.4 × 1017 0.092±0.0023 23.05± 7.1 (1.88±0.57) ×10−10 (3.40±1.04) ×10−31

7.3 × 1016 0.058±0.00132 7.36±1.01 (1.15±0.16) ×10−10 (2.07±0.28) ×10−31

2.1 × 1016 0.021±0.00094 2.17 ±0.31 (1.17±0.17) ×10−10 (2.13±0.30) ×10−31

1.1 × 1016 0.0075±0.00076 2.38 ±0.65 (2.46±0.66) ×10−10 (4.46±1.19) ×10−31

Table 6.2: Saturation ﬁt parameters and calculated values of K2 and K3 at the TiHe −1 frequency = 25228.03 cm and beam waist (wx, wy)=(0.8, 0.5) mm.

tables, the saturation power increases with respect to He density (Eq. 6.18). At higher

He densities, there are more TiHe molecules and they are formed faster through the

three body recombination process.

6.4.1.2 Peak at 25228.03 cm−1

Fig. 6.15 presents The TiHe LIF/48Ti OD vs power taken at diﬀerent He densities.

Table 6.2 shows the coeﬃcients extracted from the ﬁts and the calculated K2 and K3.

Fig. 6.16 shows Psat vs density and a linear relation is seen. 147

6.4.2 K2 and K3

48 The quantity Psat extracted from the TiHe LIF/ Ti OD vs power data (or TiHe

48 OD/ Ti OD vs power), is used to calculate K2. From K2 we obtained K3 (Eq. 6.5).

Fig. 6.17 shows a plot of K2 and K3 against He density. The values obtained for

K2 and K3 are: At 25228.15 cm−1:

−11 3 K2 = (9.68±0.74)×10 cm /s.

−31 6 K3 = (2.48±0.19)×10 cm /s.

At 25228.03 cm−1:

−10 6 K2 = (1.23 ±0.11)×10 cm /s.

−31 6 K3 = (2.22 ±0.20)×10 cm /s.

The peak at 25228.03 cm−1 was identiﬁed as R=1 and the binding energy measured was (-1.52±0.1) K. The peak at 25228.15 cm−1 is R=0 with a binding energy of (-

1.95±0.1) K. The calculated rate at which the R=1 molecules at 25228.03 cm−1 dissociate by collisions with He atoms is faster than the rate at R=0. This behavior is consistent with the fact that the R=0 molecules are more bound and it is more diﬃcult for them to be destroyed by collisions.

These TiHe three body recombination coeﬃcient values are comparable with the ones found for other types of molecules. For the case of LiHe [59] at temperatures

−1 −27 6 −27 ranges of 10 mK to 1 mK, the predicted K3 is around 1×10 cm /s to 2×10 cm6/s. However, the trend might indicate that at higher temperatures the expected

K3 would be smaller. This is also the case for systems such as H+H+K, H+H+Cs and H+H+Li [84] for which at temperatures of ∼ 0.1 K the predicted K3 is about

−29 6 ×10 cm /s and the trend looks like K3 decreases at higher temperatures. 148

−3 nHe [cm ] A [V/mW] Psat [mW] 1.8 × 1017 0.13±0.0052 2.04± 0.46

5.9 × 1016 0.056±0.0018 1.30±0.17

2.7 × 1016 0.026±0.0011 1.45 ±0.22

1.1 × 1016 0.0053±0.0012 1.59 ±0.99

Table 6.3: Saturation ﬁt parameters at the TiHe frequency = 25228.15 cm−1 and beam waist (wx, wy)=(0.4, 0.3) mm.

6.5 Beam waist (wx, wy)=(0.4, 0.3) mm

6.5.1 Saturation curves

6.5.1.1 Peak at 25228.15 cm−1

Fig. 6.18 presents The TiHe LIF/48Ti OD vs power taken at diﬀerent He densities.

Table 6.3 shows the coefficients extracted from the fits. Fig. 6.19 shows Psat vs density. From the figures and the table, we notice that the saturation power does not follow the expected trend. We believe that at this smaller beam waist, the intensity is so large, we may violate our assumption that the ground state population is much greater than the excited state population (nexc << ng). The K2 and K3 values are not reliable at these conditions.

6.5.1.2 Peak at 25228.03 cm−1

Fig. 6.20 presents The TiHe LIF/48Ti OD vs power taken at diﬀerent He densities.

Table 6.4 shows the coeﬃcients extracted from the ﬁts. Fig. 6.21 shows Psat vs density. As seen in the previous section, we believe that the transition has been saturated and

we cannot obtain the K2 and K3 values. 149

−3 nHe [cm ] A [V/mW] Psat [mW] 1.8 × 1017 0.13±0.0039 5.96± 2.44

5.9 × 1016 0.062±0.0022 1.43±0.26

2.7 × 1016 0.024±0.00092 1.32 ±0.17

1.1 × 1016 0.0053±0.00073 1.18 ±0.46

Table 6.4: Saturation ﬁt parameters at the TiHe frequency = 25228.03 cm−1 and beam waist (wx, wy)=(0.4, 0.3) mm.

Figure 6.2: Illustration of the ﬁt equation of TiHe LIF. The y axis represents the TiHe LIF and the x axis the probe power. The green line shows the case of no destruction of the molecules and the black curve shows saturation (the molecules are destroyed by means of an intense non homogeneous laser beam). If there is destruction of molecules, less ﬂuorescence is collected at the same probe power. Line = Ax; A= 0.02. The dotted line corresponds to the saturation power P sat =2 W. 150

Figure 6.3: Illustration of the ﬁt equation of TiHe OD. The y axis represents TiHe OD, which is proportional to the number of molecules and the x axis corresponds to the probe power. If the molecules are not destroyed, then the power does not aﬀect the number of molecules and it remains constant at varying the power, as it is seen in the curve called line. If there is destruction of molecules, then the number of molecules or the ODs decrease at increasing the power, as represented in the graph by the black curve. Line: A = 3×10−6. The dotted line corresponds to the saturation power P sat =2 W. 151

0.20 Data Fit

0.15

0.10 B B factor

0.05

0.00 1.5 2.0 2.5 3.0 3.5 Temperature [K]

Figure 6.4: B factor vs temperature at 25228.15 cm−1. The ﬁt to a constant line c. c=0.097±0.0027. The B factor vs temperature at 25228.03 cm−1 shows a ﬁt: c=0.12±0.0030. 152

CCD camera

TiHe probe beam

Figure 6.5: A removable mirror was located in front of the cell to redirect the TiHe probe beam path. The CCD camera was located at a position that corresponded to the center of the cell. We placed a removable mirror in front of the cell to change the path of the laser beam. From the camera pictures, the beam waist could be obtained. 153

-2 1.2x10

1.0 Data Linear Fit 0.8 Saturation Fit

Ti OD [V/OD] OD Ti 0.6 48

0.4 TiHe LIF/ LIF/ TiHe 0.2

0.0 0 1 2 3 4 Power [mW]

48 −1 Figure 6.6: TiHe LIF/Ti OD vs TiHe probe power at 25228.15 cm , nHe = 16 −1 3.4×10 cm and beam waist = (wx, wy)=(2,5) mm. The data was ﬁt using a h i line Ax; where A = (0.0028±2.75×10−5) V/mW and AP ln 1 + x . A = sat Psat −5 (0.0028±2.93×10 ) V/mW; Psat = (10368±492) mW. There is no indication of destruction of molecules. 154

Fabry-Perot Glass Waveplate cavity wedge

Ti target

Neutral Mirror densities Ti He

Prism TiHe

Beam from the TiHe probe laser

Waveplate

Glass wedge

Up and to the cell Mirror

f1 Lens

Figure 6.7: TiHe beam path including telescope: f1 =200 mm and f2 =50 mm. 155

550 600 650 700 750 800 850 900 950 1000 1050 1100 1150 1200 1250 1300 250

300 100 350 80 400

60 450

500 40 550

20 600 650 700

Figure 6.8: Image of the TiHe laser beam at the center of the cell after adding a telescope with f1 =200 mm and f1 =50 mm in the TiHe laser path. The x and y axis indicate pixels. The color scale corresponds to intensity (number of photons per pixel). The beam waist is (wx, wy)=(0.8, 0.5) mm. 156

Fabry-Perot Glass Waveplate cavity wedge

Ti target

Neutral Mirror densities Ti He

Prism TiHe

Beam from the TiHe probe laser

Waveplate f1 Glass wedge

Up and to the cell Mirror

Lens

AOM

Figure 6.9: TiHe beam path including telescope: f1 =500 mm and f2 =200 mm. 157

720 760 800 840 880 920 960 1000 1040 1080 1120 1160 400 100 450

80 500

60 550 600 40 650

20 700 750

Figure 6.10: Image of the TiHe laser beam at the center of the cell after adding a telescope with f1 =500 mm and f1 =200 mm in the TiHe laser path. The x and y axis indicate pixels. The color scale corresponds to intensity (number of photons per pixel). The beam waist is (wx, wy)=(0.4, 0.3) mm. 158

800

f1=200 mm f2=50 mm Horizontal beam waist 700 Vertical beam waist Horizontal fit Vertical fit

f1=500 mm f2=200 mm 600 Horizontal beam waist Vertical beam waist Horizontal fit Vertical fit 500 Beam Beam waist [m]

400

300

2 1 0 1 2 Cell location [in]

Figure 6.11: Beam waists as a function of position, obtained by moving the camera. O in corresponds to the center of the cell. The beam enters through one of the cell windows and exits through the opposite window. The distance measured towards the entrance window corresponds to a positive location. The beam waists are ﬁt to constant lines. Open circles: Horizontal beam waist = (794.95± 9.74) µm. Vertical beam waist = (447.21±2.55) µm Closed circles: Horizontal beam waist = (400.45±4.92) µm. Vertical beam waist = (307.23±5.89) µm. 159

0.10 Data Linear Fit Saturation Fit 0.08

0.06 Ti OD [V/OD] OD Ti 48 0.04

TiHe LIF/ LIF/ TiHe 0.02

0.00 0 1 2 3 4 5 6 7 Power [mW]

-6 2.5x10

2.0 Ti OD Ti 48 1.5

Data 1.0

TiHe OD/ OD/ TiHe Linear Fit Saturation Fit 0.5

0.0 0 1 2 3 4 5 6 7 Power [mW]

Figure 6.12: The data shown in the figures were taken under the following conditions: 16 −3 −1 nHe = 2.1×10 cm , frequency = 25228.15 cm , and beam waist (wx, wy)=(0.8, 0.5) mm. The figure on top corresponds to TiHe LIF/48Ti OD vs power. The data is fit to h i a line Ax, where A = (0.015±0.00031) V/mW, and is also fit to AP ln 1 + x sat Psat where A = (0.024±0.00094) V/mW; Psat = (1.72±0.24) mW. The figure on the bottom shows TiHe OD/48Ti OD vs power. A is set to 2.81×10−6. The data is fit to h i AP 1 ln 1 + x , where A = (2.95±0.12×10−6) V/mW; P = (1.72±0.24) mW. sat x Psat sat The saturation fit is a good fit for both sets of data, which implies that the molecules are being destroyed by the laser beam. 160

0.7 -3 He density [cm ] 17 1.4x10 16 0.6 7.3x10 16 2.1x10 16 0.5 1.1x10 Ti OD

48 0.4

0.3

TiHe LIF/ 0.2

0.1

0.0 0 2 4 6 8 Power [mW]

Figure 6.13: The TiHe LIF/48Ti OD vs power at frequency = 25228.15 cm−1 and beam waist (wx, wy)=(0.8, 0.5) mm and diﬀerent He densities. 161

15 Fit

5 Saturation power [mW]

0 17 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4x10 -3 nHe [cm ]

−1 Figure 6.14: Saturation power vs He density at 25228.15 cm and (wx, wy)=(0.8, 0.5) mm. The data is ﬁt to a line bx, b =(8.51±0.65×10−17) mW cm3. 162

0.7 -3 He density [cm ] 17 1.4x10 0.6 16 7.3x10 16 2.1x10 16 0.5 1.1x10

Ti OD 0.4 48

0.3

TiHe LIF/ 0.2

0.1

0.0 0 2 4 6 8 Power [mW]

Figure 6.15: The TiHe LIF/48Ti OD vs power at frequency = 25228.03 cm−1 and beam waist (wx, wy)=(0.8, 0.5) mm and diﬀerent He densities. 163

20 Fit

10 Saturation power [mW] 5

0 17 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4x10 -3 nHe [cm ]

−1 Figure 6.16: Saturation power vs He density at 25228.03 cm and (wx, wy)=(0.8, 0.5) mm. The data is ﬁt to a line bx, b =(1.08±0.10×10−16)mW cm3. 164

3.0x10 -10 (w x,w y)=(0.8,0.5) mm 2.5 25228.15 cm -1 Fit 2.0 25228.03 cm -1

/s] Fit 3 1.5 [cm 2 K 1.0

0.5

0.0 0.4 0.8 1.2 1.6x10 17 nHe [cm -3 ]

-31 6x10 (w x,w y)=(0.8,0.5) mm 25228.15 cm -1 5 Fit 25228.03 cm -1 Fit

/s] 4 6

[cm 3 3 K 2 1 0 0.4 0.8 1.2 1.6x10 17 nHe [cm -3 ]

−1 Figure 6.17: K2 and K3 at the frequency 25228.15 cm (closed circles) and 25228.03 −1 cm (open circles) and beam waist (wx, wy)=(0.8, 0.5) mm. The data is ﬁt using a −11 3 −31 6 constant line. K2 = (9.68±0.74)×10 cm /s and K3 = (2.48±0.19)×10 cm /s −1 −10 6 −31 at 25228.15 cm . K2 = (1.23 ±0.11)×10 cm /s and K3 = (2.22 ±0.20)×10 cm6/s at at 25228.03 cm−1. 165

-3 He density [cm ] 17 0.25 1.8x10 16 5.9x10 16 2.7x10 16 0.20 1.1x10

Ti OD 48 0.15

0.10 TiHe LIF/

0.05

0.00 0 1 2 3 4 Power [mW]

Figure 6.18: The TiHe LIF/48Ti OD vs power at frequency = 25228.15 cm−1 and beam waist (wx, wy)=(0.4, 0.3) mm and diﬀerent He densities. 166

2.5

2.0 Fit

1.5

1.0

Saturation power [mW] 0.5

0.0 17 0.0 0.5 1.0 1.5x10 -3 nHe [cm ]

−1 Figure 6.19: Saturation power vs He density at 25228.15 cm and (wx, wy)=(0.4, 0.3) mm. The data is ﬁt to a line a + bx, a=(1.22±0.22) mW; b =(3.26±3.29×10−18) mW cm3. 167

0.4

-3 He density [cm ] 17 1.8x10 16 0.3 5.9x10 16 2.7x10 16 1.1x10 Ti OD 48 0.2 TiHe LIF/ 0.1

0.0 0 1 2 3 4 Power [mW]

Figure 6.20: The TiHe LIF/48Ti OD vs power at frequency = 25228.03 cm−1 and beam waist (wx, wy)=(0.4, 0.3) mm and diﬀerent He densities. 168

6 Fit

2 Saturation power [mW]

0 17 0.0 0.5 1.0 1.5x10 -3 nHe [cm ]

−1 Figure 6.21: Saturation power vs He density at 25228.03 cm and (wx, wy)=(0.4, 0.3) mm. The data is ﬁt to a line a + bx, a=(1.01±0.3) mW; b =(9.92±7.69×10−18) mW cm3. 169

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