Download - Measurement of the Hyper ne Structure of the 7P state and ... · Abstract We report a nal value of the hyper ne splitting of the 7P 1=2 state of 205Tl and 203Tl made using a two-step

Measurement of the Hyperfine Structure of the 7P1/2 state and

8P1/2 state in 205Tl and 203Tl

byGabrielle Vukasin

Professor Protik Majumder, Advisor

A thesis submitted in partial fulfillmentof the requirements for

the Degree of Bachelor of Artswith Honors in Astrophysics

WILLIAMS COLLEGEWilliamstown, Massachusetts

May 18, 2014

Abstract

We report a final value of the hyperfine splitting of the 7P1/2 state of 205Tl and 203Tl made

using a two-step excitation. Our final values are 2173.3(8) MHz and 2153.2(7) MHz respec-

tively. We also measured the isotope shift of the 7S1/2 → 7P1/2 transition to be 534.4(9)

MHz. These experimental hyperfine splitting values are ≈ 20 MHz larger than those mea-

sured by another group in 1988 [1]. Our values bring the experimental values closer to

the theoretical values published in 2001 [2]. Our data consists of spectra taken by scan-

ning the second-step laser 6 GHz. For precise measurement of these spectra, we stabilize the

first-step excitation using a method called laser locking. Using the same experimental layout,

we are now working to measure the the hyperfine splitting of the 8P1/2 state of both isotopes.

Acknowledgments

I would first like to thank my advisor, Tiku Majumder, for his guidance and generous

assistance throughout every step of this thesis. I would like to thank Gambhir Ranjit for

his diligent work in the lab and for acquainting me with the experimental processes of this

lab; Nathan Bricault, for his help in the lab and general cheerfulness; Michael Taylor, for

his expertise; Ward Lopes, for his helpful comments and advice; Sarah Peters, for her help

constructing the external cavity diode laser over the summer; Ben Augenbraun for his work

during Winter Study and the spring; and my friends and family, for all of the support they

provided.

Executive SummaryThis thesis describes the final steps of the precise measurement of the F = 0−F = 1 hyper-

fine splitting of the 7P1/2 state in the two naturally occurring isotopes of thallium: 205Tl and203Tl. Additionally, this thesis describes the preliminary steps of measuring the analogous

hyperfine splitting of the 8P1/2 state in the same two isotopes. We also precisely measure

the hyperfine anomaly and isotope shift between the two isotopes. Hyperfine anomaly and

isotope shift illuminate the isotopic differences in the nuclear structure between 205Tl and203Tl.

Ultimately we would like to use our precise measurements of hyperfine structure to test

the accuracy and guide the refinement of atomic theory surrounding modeling short-range

wavefunctions of the valence electron. Due to thallium’s very heavy nature (Z = 81), accu-

rate atomic theory, coupled with existing and future precision measurements can result in

important atomic-physics-based tests of the Standard Model and physics beyond it. Atomic

theory models benefit from the single valence p electron structure in thallium, allowing a

“semi-hydrogenic” starting point for calculation.

We study hyperfine structure of thallium through a double-excitation of thallium vapor.

The Majumder group has used this method to measure the hyperfine structure through hy-

perfine splitting and Stark shift in both thallium and a similar element, indium, with great

precision [3] [4] [5]. When studying the 7P1/2 state, we use a UV (378 nm) external cavity

diode laser (ECDL) to excite the first transition from 6P1/2 → 7S1/2 and an IR (1301 nm)

ECDL to excite the second transition from 7S1/2 → 7P1/2. Figure 1 shows this double-

excitation scheme. In the future, to study the hyperfine structure of the 8P1/2 state, we will

use the same UV laser to excite the first transition, but with the substitution of a 671 nm

ECDL to excite the second transition of 7S1/2 → 8P1/2.

ii

Figure 1: 7P1/2 Energy Level Diagram

An energy level diagram of the double excitation of the 7S1/2 → 7P1/2 transition. The hyperfine splittings(HFS) of each isotope and the isotope shift are labeled for the 7P1/2 state.

7S1/2

6P1/2

378 nm

Thallium 205Thallium 203F = 1

F' = 1

F'' = 1

F'' = 0

F'' = 1

F'' = 07P1/2 HFS

7P1/2 HFS

7P1/2

Isotope Shift

7P1/2 {

1301 nm

The experimental setup (see figure 2) needed to execute a double excitation of thallium

atoms requires the two lasers to spacially overlap in a heated cell of thallium vapor. In

the 7P1/2 experiment, we lock the UV laser to the 6P1/2(F = 1) → 7P1/2(F ′ = 1) hyper-

fine transition and sweep the IR laser ∼5 GHz across the 7P1/2 resonant frequency. The

locking technique that was developed in this lab previously is based on the thallium atoms

themselves. We then look at the absorption spectra of the IR laser (see figure 3) using a

photodiode and lock-in amplifier because the absorption is very small. Using the same vapor

cell, we can excite either a single isotope or both isotopes by locking the frequency of the

first-step UV laser to a particular frequency point in the UV transition spectrum. Before

the IR laser beam reaches the thallium vapor cell, it is split and half is sent through an

electro-optic modulator (EOM) to the vapor cell and the other half if sent to a Fabry-Perot

(FP) cavity to monitor the frequency scan. Both the EOM and the FP cavity are important

to linearize and calibrate the frequency axis of the absorption spectra.

In our analysis of the resulting spectra, the first step is to fit the FP transmission spectra

to remove the the scan non-linearity caused by the piezoelectric transducer (PZT) used to

tune the laser frequency during the scan. Having removed the non-linearity, we use the

iii

frequency modulated (FM) sidebands produced by the EOM to perform absolute frequency

calibration. As seen in figure 3b, these FM sidebands produce copies of the hyperfine peaks

at a precisely known (600.00 MHz) frequency separation.

378 nm laser

1301 nm laser

Isolator

AOM

Locking Setup

OpticalChopper

Lock-InAmplifier

50/50 BeamSplitter

Dichroic

PD

EOMPD

Half-Wave Plates

FP CavityPBS

Thallium Cell in Oven

To Computer

600 MHz Synthesizer

Shutters

Figure 2: Experimental Setup

As mentioned before, we can choose to excite either one or both isotopes by choosing a

certain frequency with which to lock the UV laser. We can determine the hyperfine splitting

of the 7P1/2 state from the single isotope spectra of each isotopes. To determine the isotope

shift, however, we need to have the spectra of both isotopes on the same frequency axis,

which means we must excite both isotopes at the same time. We can excite Doppler shifted

atoms of both isotopes by locking our UV laser at a frequency between the resonance fre-

quencies of the two isotopes. By studying spectra for both the co and counter-propagating

laser beam configurations, we can remove the relative Doppler shift and reveal the true iso-

tope shift.

The measurement of the hyperfine splitting and isotope shift for the 7P1/2 state of both

isotopes have been completed. We analyzed both single and dual isotope spectra accounting

for many sources of systematic and statistical error. The resulting HFS value of 205Tl is

2173.3(8) MHz and of 203Tl is 2153.2(7) MHz. We found the isotope shift for the 7S1/2 →7P1/2 transition to be 534.4(9) MHz. The error associated with each of these three values

is less than 1 MHz, allowing a careful comparison of our results to previous work. These

results and the corresponding data analysis were published in January 2014 [10]. Our results

show distinct disagreement with the experimental results of the Grexa et al. group [1] and

show improved agreement with theoretical calculations [2].

The rest of this thesis focuses on working towards the measurement of the hyperfine struc-

iv

ture of the 8P1/2 state. The laser controlling the second-step excitation is a 671 nm ECDL

that we have created in our lab. We are working to get the optical setup ready to take

spectra for this state.

Figure 3: 7P1/2 Dual and Single Isotope Spectra

0 1000 2000 3000 4000 5000 6000−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Frequency (MHz)

Nor

mal

ized

Inte

nsity

(a) Dual Isotope Spectrum

0 1000 2000 3000 4000 5000 6000 7000−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Frequency (MHz)

Nor

mal

ized

Inte

nsity

(b) Single Isotope Spectrum

Contents

1 Introduction 1

1.1 The Standard Model and Parity Non-Conservation . . . . . . . . . . . . . . 1

1.2 Purpose for Hyperfine Structure Study of Thallium . . . . . . . . . . . . . . 1

1.3 Measurement of the 7P1/2 Hyperfine Structure . . . . . . . . . . . . . . . . . 3

1.4 Measurement of the 8P1/2 Hyperfine Structure . . . . . . . . . . . . . . . . . 4

2 Atomic Structure Details 5

2.1 Fine Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Hyperfine Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Hyperfine Difference and Hyperfine Anomaly . . . . . . . . . . . . . . 8

2.3 Thallium Atomic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4 Isotope Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.5 Hole Burning and Doppler Broadening . . . . . . . . . . . . . . . . . . . . . 12

2.6 Approximating Atomic Transition Lineshapes in Spectra . . . . . . . . . . . 14

3 Experimental Setup 18

3.1 External Cavity Diode Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.1.1 378 nm Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.1.2 671 nm Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Laser Locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2.1 Importance of Laser Locking . . . . . . . . . . . . . . . . . . . . . . . 22

3.2.2 Laser Locking Set-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3 378 nm Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.4 Experimental Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.5 Detection of the Second Step Signal . . . . . . . . . . . . . . . . . . . . . . . 27

4 Data Analysis and Results for the 7P1/2 Experiment 30

4.1 Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.2 Linearization and Calibration of the Frequency Axis . . . . . . . . . . . . . . 31

v

CONTENTS vi

4.2.1 Fabry-Perot Calibration and Linearization . . . . . . . . . . . . . . . 31

4.2.2 EOM Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.3 Interpreting Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.4 Single Isotope Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.4.1 Single Isotope Error Analysis . . . . . . . . . . . . . . . . . . . . . . 38

4.4.2 Systematic Error Search . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.4.3 Subdividing Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.5 Dual Isotope Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.5.1 Eliminating Doppler Shift with Dual Beam Configuration . . . . . . . 42

4.5.2 Interpreting Dual Isotope Spectra . . . . . . . . . . . . . . . . . . . . 44

4.5.3 Dual Isotope Error Analysis . . . . . . . . . . . . . . . . . . . . . . . 46

4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5 The 8P1/2 Experiment 50

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.2 8P1/2 Simulated Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.2.1 Peak Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.2.2 Single Isotope Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.2.3 Dual Isotope Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.3 Current State of Experiment and Future Work . . . . . . . . . . . . . . . . . 53

A Matlab Code 55

A.1 ThalliumFitting.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

A.1.1 getdataThallium.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

A.1.2 downsampleAndNormalizeThallium.m . . . . . . . . . . . . . . . . . 56

A.1.3 FabryPerotFittingThallium.m . . . . . . . . . . . . . . . . . . . . . . 56

A.1.4 FrequencyLinearizationThallium.m . . . . . . . . . . . . . . . . . . . 58

A.1.5 VoigtFitThallium.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

A.1.6 VoigtFitThallium235.m . . . . . . . . . . . . . . . . . . . . . . . . . . 58

A.2 DataAnalysisThallium.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

B 7P1/2 Data Tables 60

C Detailed Description and Usage of the 671 nm Laser 64

List of Figures

1 7P1/2 Energy Level Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

3 7P1/2 Dual and Single Isotope Spectra . . . . . . . . . . . . . . . . . . . . . iv

2.1 Hyperfine Differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 7P1/2 Energy Level Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 8P1/2 Energy Level Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Hole Burning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.5 Comparison of Three Distributions for Comparable Widths . . . . . . . . . . 14

2.6 The 6P1/2 → 7S1/2 Transition . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.1 Internal Diode Laser Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 671 nm External Cavity Diode Laser . . . . . . . . . . . . . . . . . . . . . . 19

3.3 Laser Locking Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.4 Stability of a Locked Laser versus an Unlocked Laser [6] . . . . . . . . . . . 23

3.5 Simulated Absorption Signal of the 6P1/2(F=1)→ 7S1/2(F’=1) Transition for

Both Isotopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.6 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.7 Red Laser Transmission Spectrum due to Optical Chopper . . . . . . . . . . 28

4.1 Simultaneous Lockin Amplifier and Fabry-Perot Data . . . . . . . . . . . . . 32

4.2 Polynomial of Fabry-Perot Peak Positions . . . . . . . . . . . . . . . . . . . 33

4.3 Second Step Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.4 Single-Isotope Data Run with Residuals . . . . . . . . . . . . . . . . . . . . 36

4.5 Comparison of HFS Values by Data Subsets . . . . . . . . . . . . . . . . . . 41

4.6 Dual Beam Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.7 7P1/2 Dual Isotope Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.8 Beam Misalignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.9 Correlation of Transition Isotope Shift and Doppler Shift . . . . . . . . . . . 48

5.1 Simulated 205 Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

vii

LIST OF FIGURES viii

5.2 Simulated Dual Isotope Spectrum . . . . . . . . . . . . . . . . . . . . . . . . 52

C.1 External Laser Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

C.2 Mount for the Laser Diode and Collimating Lens . . . . . . . . . . . . . . . 72

C.3 Sliding Mount for the Mirror Mount . . . . . . . . . . . . . . . . . . . . . . . 73

C.4 Wedge Diffraction Grating Mount . . . . . . . . . . . . . . . . . . . . . . . . 74

C.5 PZT Tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

C.6 PZT Cap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Chapter 1

Introduction

1.1 The Standard Model and Parity Non-Conservation

The study of atomic structure is of great importance, for the knowledge of subatomic inter-

actions remains incomplete. Experiments of all varieties are essential in testing the Standard

Model of Elementary Particles and physics beyond what we already know. The Standard

Model describes the way particles interact based on the electromagnetic force, the strong

force, and the weak force. The focus of this introduction is on the electroweak interaction,

a force normally studied in accelerator-based experiments. The weak interaction presents

itself in the decay of charged particles, mediated by the W+ and W- bosons. We now know

that the weak interaction is also manifested in the exchange of the neutral Z0 boson by neu-

tral particles. In 1957, an unusual property of the weak force was discovered called parity

non-conservation (PNC), or violation of parity symmetry [7]. Of the three forces that govern

interactions of elementary particles, only the weak force has been proven to violate parity.

PNC of the weak force means that weak force interactions of elementary particles, including

their exchange of W+, W-, and Z0 bosons, show a fundamental “handedness” in nature.

1.2 Purpose for Hyperfine Structure Study of Thallium

The exchange of W+ and W- bosons cause beta-decay of atoms. We are interested in

stable atoms, so we study weak interactions involving the Z0 boson. The Z0 bosons are

exchanged between electrons and nucleons just as photons are exchanged between electrons

and protons in the electromagnetic interaction. Unlike the photon, however, the Z0 boson

has a very short range of influence. The short lifespan of the Z0 boson is about 10−10 seconds

due to its very large mass [6]. This means the maximum distance traveled during its lifetime

is 10−10sec/108m/s ∼= 10−18 meters, assuming its speed is on the order of the speed of light.

1

CHAPTER 1. INTRODUCTION 2

Because this is a very short distance, only electrons traveling through the nucleus, with

an average diameter of 10−15 meters, will have an effect on the nucleus via the Z0 boson

exchange [6]. Although the probability seems low, Z0 boson exchange has been observed

because of the characteristic parity violating signature of the weak force.

In an atom, the observed PNC effect, εPNC , is governed by the equation [8]:

εPNC = Q× C(Z) (1.1)

where Q is the “weak charge” of the weak interaction, C(Z) is a quantity determined by the

wavefunction of the valence electron, and Z is the atomic number of the atom. Q generally

increases as Z increases. It is also the quantity that we wish to measure, which can only be

measured indirectly if we have C(Z) and εPNC . Furthermore, we know that C(Z) ∝ Z3f(Z),

where f(Z) is an increasing first order function [6]. Thus, an atom with a large atomic number

will will show a greater effect of the PNC force from both C(Z) and Q. Because we cannot

calculate the exact wavefunctions for the orbiting electrons for Z ≥ 1, we must approximate

C(Z). The type of study that incorporates such approximations is called ab initio atomic

theory. As a consequence, the accuracy of Q is limited by the accuracy of C(Z) in heavy

complicated atoms. Due to the constraints of C(Z), we chose thallium to study because both

of its natural isotopes, 203Tl and 205Tl, are heavy “hydrogenic” atoms, meaning they each

have essentially one valence electron in the 6P orbital at the ground state. There are actually

three valence electrons, but the first two fill the 6S orbital, so we can treat the 6P electron as

the singular valence electron. Because thallium is a “hydrogenic” atom, the wavefunction of

the 6P electron looks like the wavefunction of the electron of the hydrogen atom to at least

first order. This is a starting point for theoretical approximations of C(Z) and other atomic

structure calculations. Although the atomic number of thallium (Z = 81) is large, εPNC is 9

orders of magnitude smaller than the electromagnetic force. However, it has been measured

before.

In our lab, we study a wide variety of atomic properties in thallium and indium. Some

of these experiments involve long-range and short-range atomic interactions. Short-range

atomic interactions, such as the hyperfine interaction, are useful to study because they give

us insight on the true nature of the electrons’ wave functions close to the nucleus, something

essential for the calculation of parity violation. These atomic structure phenomena are cal-

culated using equations analogous to equation 1.1 but for physical properties other than the

weak charge. For example, the approximated valence electron wavefunction associated with

another physical property would be referred to as “D(Z).” Since we can approximate this

D(Z) in “hydrogenic” atoms using hydrogen as an initial model and we observe the effect

of the phenomena, ε, we can then test and adjust the models for this “D(Z).” The atomic

theory used to approximate the wavefunctions in order to approximate “D(Z)” can then be


applied to C(Z) to make it more accurate, thus making Q more accurate. It is important

to test these theoretical ab initio atomic calculations with experimental results. If atomic

ab initio theory is well understood, then atomic experiments can probe physics of a more

fundamental nature, thus testing the Standard Model.

The specific atomic property that is the focus of this thesis is called the hyperfine struc-

ture. Hyperfine structure is caused by the interaction between the nuclear magnetic moment

and the magnetic field created by the moving electrons. This creates a splitting in the energy

level structure in absence of spin-orbit coupling. This splitting is called hyperfine splitting

(HFS). The purpose of these experiments is to accurately and precisely measure the hyperfine

structure of both thallium 205Tl and 203Tl using methods developed in this lab previously.

More specifically, we would like to test the theoretical calculation of hyperfine structure of

thallium done by the group Kozlov et al. [2] to test and improve their approximations of

quantities determined by wavefunctions of the electrons, which is just a “D(Z)” relating to

HFS. Our results will test the accuracy of and guide refinement of the atomic theory used

to approximate this “D(Z),” which is the ultimate goal of this lab.

A secondary reason for the specific experiment to be explained in the rest of this docu-

ment is to find an accurate value for the measurement of the 7P1/2 hyperfine splitting. The

purpose of this is to get a precise and accurate value to compare to the experimental value of

the group Grexa et al. [1]. In 1988, Grexa et al. published a paper with experimental results

for the HFS and isotope shift, to be explained later, of many hyperfine levels in thallium

[1]. In 1993, this same group published another paper with corrected values for some of the

hyperfine splitting values they published in their previous paper [9]. The 7P1/2 values were

not among those values corrected in the second paper. Therefore, our goal is to remeasure

the 7P1/2 HFS. To ensure calibration accuracy, we employ two complementary methods of

calibration. With the knowledge of success of the measurement of the hyperfine splitting in

the 7P1/2 state, we will use this method for higher levels.

1.3 Measurement of the 7P1/2 Hyperfine Structure

In this first experiment, we measure the 7P1/2 hyperfine structure and transition isotope

shift in both isotopes of thallium. In order to excite the valence electron to the 7P1/2 state,

we use a double excitation. Because of selection rules, we cannot excite the thallium atoms

from 6P1/2 to 7P1/2 with one single wavelength. Electric dipole selection rules of transitions

do not allow for a transition between two levels with the same angular momentum quantum

number, L. Because 6P1/2 and 7P1/2 have the same L = 1, this transition is not possible.

Thus the need for a double excitation of the atoms arises. We use a 378 nm laser to excite


the atoms from their ground state, 6P1/2 (L = 1), to 7S1/2 (L = 0). Then we use a 1301 nm

laser to excite these atoms from 7S1/2 to 7P1/2 (L = 1). Both of these transitions are allowed

by the selection rules.

The work done by David Kealhofer [8] of setting up the experiment and taking the initial

data was finished this year with more data taken and a complete data analysis. The result

of David Kealhofer’s work was an initial measurement of the hyperfine splitting of 205Tl of

2177 ± 1 MHz [8]. With additional trials and data analysis, we find a hyperfine splitting

value for 205Tl and 203Tl, and we find a value for the transition isotope shift between the two

isotopes. This data analysis and error analysis that will be discussed in chapter 4 expands

upon what is in our manuscript published in January of 2014 [10].

1.4 Measurement of the 8P1/2 Hyperfine Structure

Our new experiment focuses on measuring the 8P1/2 hyperfine splitting between the F”=0

and F”=1 hyperfine energy levels. This also requires a double-excitation of 205Tl and 203Tl

vapor using two diode lasers for the same reason as for the 7P1/2. Except for section 4, the

rest of this research focuses on the 8P1/2 hyperfine structure. Section 3 describes the setup of

the entire experiment as well as individual pieces of equipment such as the 671 nm external

cavity diode laser and the locking procedure of the 378 nm laser. Section 5.2 includes the

simulated spectra that we expect to see in the future for this experiment.

Chapter 2

Atomic Structure Details

2.1 Fine Structure

Fine structure is the first deviation from the energy level structure in absence of angular

momentum coupling. In our research we measure the effects of a further deviation, hyperfine

structure, but a certain understanding of fine structure is needed to research additional

deviations. Fine structure is caused by the interactions between the spin and orbital moments

of an electron; it is called spin-orbit coupling [11, p. 187]. Fine structure can be described

by three quantum numbers: J , S, and L. The total angular momentum, J , is the sum of the

angular momentum due to the spin moment of the electron, S, and the angular momentum

due to the orbital moment of the electron, L, which are related by the vector addition

of J = S + L. S equals ±12

for all electrons. L has a range of 0 to n − 1, where n is

the principle quantum number. The following equation relates these terms to produce the

spin-orbit coupling energy, vL,S [11, p. 190]:

v`,S =af2

[J(J + 1)− `(`+ 1)− S(S + 1)] (2.1)

where af is the spin-orbit coupling constant generated by a radial integral. We can also

express how the spin-orbit coupling energy affects the wavefunction of the valence electron

by appending the following Hamiltonian to the Hamiltonian of the total wavefunction [12,

p. 125]:

Hfine = afL · S (2.2)

The quantum numbers S, L and J are used to describe the fine states created by spin-

orbit coupling as follows. Every orbital has two fine structure state; one with J = L + 12

and one with J = L − 12. Each level is denoted, by n, J , and the letter corresponding to

L, producing the notation nLJ . For example, the fine states 6P1/2, 7S1/2, and 8P1/2 are

5

CHAPTER 2. ATOMIC STRUCTURE DETAILS 6

due the spin-orbit coupling of the s = −12

6P electron, s = +12

7S, and s = −12

8P electron

respectively. We will now explore how these states are further split into finer quantized levels.

2.2 Hyperfine Structure

The magnetic field created by the moving valence electron interacts with the magnetic dipole

moment of the nucleus. The result is a coupling between the angular momentum of the

electron, J , and the angular momentum of the nucleus, I. This coupling is called the

hyperfine interaction. It creates splittings of the fine energy states in wavelength on the

order of 10−4 to 10−7 eV [12, p. 168]. Hyperfine splitting can only occur if the spin of the

nucleus is not zero and hence has a nonzero magnetic moment [13, p. 9]. The total angular

moment, F , due to the coupling of the angular momenta of the nucleus and electron is

summed by the vector addition of [11, p. 342]:

F = J + I (2.3)

where F ranges from J − I, J − I + 1, ..., J + I. Therefore, the number of hyperfine

energy states (possible values of F ) is:

# of HF states =

{2I + 1 if I < J

2J + 1 if I > J(2.4)

The reason our lab studies the hyperfine structure in Thallium is because it gives us

insight on the wavefunction of the valence electron. Just as with fine structure and the

spin-orbit coupling addition to the Hamiltonian of the wavefunction of the valence electron,

the addition to the Hamiltonian due to the hyperfine splitting is [12, p. 170] :

Hhf = aI · J (2.5)

a is the magnetic dipole hyperfine coupling constant, which is slightly different for each

isotope due to the slight change in mass and charge density. a also depends on the details

of the wavefunction of the electron in the vicinity of the nucleus. There can be other

multipole terms in the multipole expansion of the gradient of an electric or magnetic field,

due to interactions such as the electrostatic interaction between the electrons and protons

of the nucleus, for nuclei with I > 12. The interactions will contribute more terms to the

Hamiltonian in addition to the two dipole terms we have described thus far: Hfine and

Hhf . However, thallium’s nucleus has a property that I = 12, so we do not have to take

multipole terms beyond dipoles into consideration [13, p. 10]. The other element we study


in our lab, indium, has a nucleus with I = 92

and consequently has dipole, quadrupole, and

octupole terms in the multipole expansion of the magnetic interaction between its electrons

and nucleus.

Now we must find the energy, E, corresponding to the hyperfine Hamiltonian through

Hhfψ = Eψ, which must be a true statement for eigenfunction ψ because Hhf is additive

to the original Hamiltonian. The first step is to find I · J . A simple trick is to take the dot

product of equation 2.3 with itself. This results in:

F · F = I · I + 2I · J + J · J (2.6)

F 2 = I2 + 2I · J + J2 (2.7)

I · J =1

2(F 2 − I2 − J2) (2.8)

Because I · J is actually the dot product of two operators, we should include the eigen-

functions in equations 2.6 through 2.8. J2ψ is the operator of the electronic angular moment

which can be described by [14, p. 32-34]:

J2ψ = h̄2J(J + 1)ψ (2.9)

Similarly, the angular momentum operators F 2 and I2 have energy eigenvalues of h̄2F (F+

1) and h̄2I(I + 1) respectively. Very closely related to the fine structure spin-orbit coupling

energy, the resulting eigenvalue equation, equivalent to equation 2.8, for each hyperfine

energy level is:

∆EHFS =a

2[F (F + 1)− I(I + 1)− J(J + 1)] (2.10)

where a is again the hyperfine constant. To find the energy between two consecutive

hyperfine energy levels, since I and J are the same for both hyperfine levels, we subtract

equation 2.10 for F + 1 from that for F and end up with:

∆EF+1 −∆EF = a(F + 1) (2.11)

This is the energy that we want to find in our experiments, for the 7P1/2 and 8P1/2 states,

between the hyperfine energy levels F = 0 and F = 1, so a is exactly the hyperfine splitting.

We call this energy the hyperfine splitting of a fine structure state. Since the goal of our

experiment is to measure the hyperfine splitting, we will be experimentally finding a value

for a. We will be measuring the effect a has on the addition to the Hamiltonian of equation

2.5.


2.2.1 Hyperfine Difference and Hyperfine Anomaly

Figure 2.1: Hyperfine Differences

This is an exaggerated comparison of the hyperfine splittings of both isotopes of thallium. The level isotopeshift and hyperfine difference, which leads to the calculation of hyperfine anomaly, are both illustrated for a

generic fine structure energy level.

FS Level

F = 1

F = 0205Tl

203Tl

{Isotope Shift

a205

a203

a205/4

3a205/4a203/4

3a203/4

Hyperfine Difference = a205 - a203

The spectra of both isotopes differ in two ways. The first is that the relative sizes of the

hyperfine splittings are different. The second is that the relative sizes of the second transition

in the double excitation differ as well. The former effect is called the hyperfine anomaly and

the latter is called the transition isotope shift, which will be discussed in section 2.4. Both

affects are exaggerated in figure 2.1. It shows the relative sizes of the spacing between F = 0

and F = 1 hyperfine states of this arbitrary fine structure level for both isotopes. Hyperfine

anomaly can be calculated by:

∆ = [H7P,205

H7P,203

g203

g205

− 1] (2.12)

where H7P,205 and H7P,203 are the hyperfine splittings of 7P1/2 of 205Tl and 203Tl respec-

tively. g203 and g205 are the nuclear g-factors of 203Tl and 205Tl. Hyperfine anomaly is caused

by the change in charge distribution due to the additional neutrons because the nuclear

magnetic dipole moment is an average over the total volume of the nucleus and thus so must

be the hyperfine splitting phenomenon [12, p. 178].


2.3 Thallium Atomic Structure

As mentioned in the introduction, of the ground state configuration, [Xe] 4f 145d106s26p1,

we can treat the singular 6P electron as a valence electron because the two 6S electrons are

paired. We are interested in the multiple excited states of this valence electron. In both

figures 2.2 and 2.3, the fine and hyperfine structure of the excited states in which we are

interested in our experiments is expanded. One thing to notice about these diagrams is that

each fine structure state only has two corresponding hyperfine states. Thallium’s nuclear

angular moment is 12. This means that there can only be two hyperfine levels at any given

fine structure level, using equation 2.4.

The spacing between the fine states of the 7S1/2 → 8P1/2 transition in figure 2.3 is larger

than that of the 7S1/2 → 7P1/2 transition in figure 2.2 as an exaggeration of the actual

relative sizes of these two transitions in energy. The electron in the 8P1/2 state has higher

energy than in the 7P1/2 state, so the transition wavelength from 7S1/2 is shorter.

Another point of interest is the spacing of the F = 0 and F = 1 levels relative to the fine

energy levels. Substituting our values for I = 12, J = 12, and F = 0, 1, we find the hyperfine

levels have a shift relative to the fine structure of:

F = 0 : ∆EHFS =a

2[0(0 + 1)− 1

2(1

2+ 1)− 1

2(1

2+ 1)] =

a

2[0− 3

4− 3

4] = −3a

4(2.13)

F = 1 : ∆EHFS =a

2[1(1 + 1)− 1

2(1

2+ 1)− 1

2(1

2+ 1)] =

a

2[2− 3

4− 3

4] = +

a

4(2.14)

These HFS shifts are indicated in figure 2.1. Note that this insures that the weighted

average of the two levels is 0 because F = 1 has three times the number of sublevels as

F = 0.


7S1/2

6P1/2

378 nm

Thallium 205 Thallium 203

F = 1

F' = 1

F'' = 1

F = 0

F' = 0

F'' = 0

F'' = 1

F'' = 0

F' = 1

F' = 0

F = 1

F = 0

21.3 GHz21.1 GHz

12.1 GHz12.3 GHz

7P1/2 HFS7P1/2 HFS

7P1/2 Isotope Shift

7P1/2 {

{

1301 nm

Figure 2.2: 7P1/2 Energy Level DiagramA partial diagram of the hyperfine energy structure of both isotopes for the 6P valence electron. It is not toscale. Each ”nL” label represents the fine structure energy levels and the levels labeled with various degrees

of F represent the hyperfine energy levels. The double excitation experiment from 6P1/2 to 7P1/2 isrepresented. The first excitation from 6P1/2 to 7S1/2 is depicted by the blue arrow at 378 nm. In reality,

the second laser will be scanning a few GHz to reach both of the 7P1/2 F”=0,1 hyperfine levels. Theabsorption peaks on the individual IR scans will be the transitions indicated in red. The isotope shift is

highlighted in green.


8P1/2

7S1/2

6P1/2

378 nm


F = 1

F' = 1

F'' = 1

F = 0

F' = 0

F'' = 0

F'' = 1

F'' = 0

F' = 1

F' = 0

F = 1

F = 0

21.3 GHz21.1 GHz

12.1 GHz12.3 GHz

8P1/2 HFS8P1/2 HFS

8P1/2 Isotope Shift

671 nm

{

{

Figure 2.3: 8P1/2 Energy Level DiagramA partial diagram of the hyperfine energy structure of both isotopes of the 6P valence electron, much like

figure 2.2, which is again not to scale. The double excitation experiment from 6P1/2 to 8P1/2 is representedhere. The 6P1/2 to 7S1/2 transition is again depicted by the blue arrow. The 7S1/2 to 8P1/2 transitions

from F’ = 1 to F” = 0,1 are highlighted in red, at a frequencies near 1301 nm. The red arrows indicate thetwo transitions that will be visible on the red laser spectra. The isotope shift is highlighted in green.


2.4 Isotope Shift

The study of the transition isotope shift is the study of two phenomena called the “field ef-

fect,” also known as the volume effect, and the “mass effect.” The mass effect is proportional

to a change of 1M2 for atoms of large Z [12, p. 194]. Therefore, the mass effect is so small in

thallium that it is negligible compared to the volume effect and we do not need to account

for it in our experiment. The volume effect gets its name because the increased number of

neutrons in different isotopic nuclei increases the volume of the nucleus. This effect mani-

fests in changing the energy of the atom because the charge density of the nucleus changes

when the number of neutrons changes, which will ultimately alter the spectra of the different

isotopes in our experiment [11, p. 339]. Thus, this effect is an important measurement to our

experiment because it illuminates differences in atomic level phenomena due to additional

neutrons.

Unlike the hyperfine splitting, the isotope shift does not rely on a nonzero nuclear spin.

Therefore, we see an isotope shift between isotopes of any kind of nucleus. The transition

isotope shift presents itself as the difference of the frequency of a certain transition for one

isotope and the frequency of the same transition for another isotope. In the first experiment

we find the isotope shift for the 7S1/2 → 7P1/2 transition and in the second experiment we

will find an isotope shift for the 7S1/2 → 8P1/2 transition.

Measuring the transition isotope shift cannot be done using the single isotope data we use

to measure the hyperfine splitting. In order to measure differences between the spectra of

the two isotopes, we need to have their spectra on the same frequency axis. The frequency

axis of each spectrum we take represents relative and not absolute frequency. Therefore, we

take dual isotope data with both isotopes’ F ′′ = 0 and F ′′ = 1 hyperfine energy states on

one spectrum. Dual isotope spectra will be discussed in section 5.2.

2.5 Hole Burning and Doppler Broadening

A gradient of velocity classes of thallium atoms in the ground state arises because atoms in

the gas state within the quartz cell move at different velocities. The only velocity component

of the atoms that creates different velocity classes is the velocity component with respect to

the axis of propagation of the laser. The different velocity classes see the frequency of the

laser light as:

f =fo

1 + vc

(2.15)

according to their respective velocity components, v, and the frequency seen by atoms with

a velocity of zero, fo. In double-excitation laser spectroscopy, the second laser excites a


single velocity class of atoms. By selecting the frequency of the first laser, we excite only one

velocity class of atoms to the second step, in which each doubly-excited atom has relatively

the same Doppler shift. The method of exciting a single velocity class in the first transition

in a two-step excitation is called “hole burning.” Hole burning, or saturated absorption, gets

its name because by exciting a very specific velocity class of atoms, the atoms left at the

ground state now have a Maxwell velocity distribution with a “hole” cut out due to the

missing velocity class, as seen in figure 2.4 [11, p. 381].

Figure 2.4: Hole Burning

The bottom plot represents the absorption profile of the ground state atoms in the presence of hole burning.The top plot represents the absorption profile of the excited atoms due to hole burning.

Frequency

Doppler Width

Hole Burning

Natural Line Width

ExcitedState

GroundState

The line width of the absorption profile of the excited atoms is approximately the nat-

ural line width, with a small contribution from Doppler broadening, if the laser beam is

diverging, for example. This line width is much smaller than the line width of the ground

state absorption profile, which is due to Doppler broadening. Therefore, it is clear that hole

burning reduces the effects of Doppler broadening, although there will always be a small

component of the line width of the absorption profile of the excited atoms due to Doppler


broadening. Additionally, the excited state absorption profile may be further broadened by

“power broadening” if the power is high enough. However, compared to Doppler broadening,

power broadening is negligible for the most part.

2.6 Approximating Atomic Transition Lineshapes in Spectra

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

Frequency

Inte

nsity

Voigt

Lorentzian

Gaussian

Figure 2.5: Comparison of Three Distributions for Comparable Widths

We can model the absorption curve using a Voigt profile, which is a convolution of the

Gaussian and Lorentzian profiles. The Gaussian, Lorentzian, and Voigt distributions are

compared in figure 2.5. These distributions are scaled to have the same full width at half

maximum (FWHM). The biggest difference between the three profiles occurs in the tails

of the distributions where they begin to deviate from each other noticeably. The Gaussian

nature of the peak is due to the inhomogeneous Doppler broadening caused by the Maxwell-

Bolzmann distribution of the kinetic energy of the thallium atoms at T = 400°C to 450°C. The

Doppler effect causes different velocity (kinetic energy) classes to arise and thus is an inho-

mogeneous form of broadening because it affects each atom or groups of atoms differently [6].

Table 2.1: Lifetimes of Excited Thallium Atoms

Fine State Lifetime (ns) [15] Natural Line Width (MHz)7S1/2 7.43 21.47P1/2 61.8 2.588P1/2 177.6 0.896


The Lorentzian nature of the absorption peaks is due to the homogeneous lifetime broad-

ening of the transitions. The term “lifetime” broadening means that the finite lifetime, τ ,

of an electron in an excited state produces uncertainty in energy [8]. This is explained by

the Heisenberg Uncertainty Principle. Uncertainty in lifetime presents itself as the FWHM,

or γ, of the Lorentzian. We call this the natural line width of the transition. Table 2.1

contains the average lifetimes an electron has in the 7S1/2, 7P1/2, and 8P1/2 states and their

corresponding natural line widths from the equation:

γ =1

2πτ(2.16)

Because both the 7P1/2 and 8P1/2 experiments have electrons excited from the 7S1/2 state to

their respective final states, the γ of both experiments must include the total γ of the 7S1/2

state. Therefore, the actual γ’s for the 6P1/2 → 7S1/2, 7S1/2 → 7P1/2 and 7S1/2 → 8P1/2

transitions are:

6P1/2 → 7S1/2 : γ = γ6P + γ7S = 21.4MHz (2.17)

7S1/2 → 7P1/2 : γ = γ7S + γ7P = 24.0MHz (2.18)

7S1/2 → 8P1/2 : γ = γ7S + γ8P = 22.3MHz (2.19)

(2.20)

In practice, given the laser power levels we use, power broadening also occurs. Power

broadening decreases the lifetimes of the atoms in the excited state because stimulated

emission begins to occur at higher laser powers. The true Lorentzian width is about 2 times

greater as a result, giving us actual γ’s of ∼ 50 MHz. Now that we know the causes of the

shape of the transition peaks we should theoretically be getting, we must compare the width

contribution of each. The Doppler width (FWHM) of the Gaussian is calculated using:

∆fFWHM =

√8kT ln 2

mc2fo (2.21)

where fo is the frequency of the transition. The estimate of the relative sizes of the Lorentzian

width and the Doppler width as a measure of the relative number of atoms excited by the

second-step laser in the 7P1/2 experiment is:

γ

ΓDoppler≈ 50MHz

Γ7P,Doppler

≈ 5% (2.22)


Figure 2.6: The 6P1/2 → 7S1/2 Transition

This is the spectrum of the 6P1/2 → 7S1/2 transition including the hyperfine transitions of both isotopes.

0 5 10 15 20 25 30 35 40 45 50

Frequency (GHz)

Tra

nsm

issi

on S

igna

l

F = 1 to F’= 0 F = 1 to F’= 1 F = 0 to F’=1

205Tl

203Tl

Figure 2.6 shows the total spectrum of the hyperfine transitions in the 6P1/2 → 7S1/2

fine structure transition. Using equation 2.21, the Doppler widths for 203Tl and 205Tl of the

F=1→ F’=1 transition are 1072.8 MHz and 1067.5 MHz respectively. For this transition,

the Doppler width is about 50 times greater than the natural line width. Therefore, we only

need to worry about the Gaussian contribution of the Voigt profile for this transition. On

the other hand, the hole burning of the second step excitation largely gets rid of Doppler

broadening. Thus, in the second excitation, we can approximate the transitions in the spec-

tra by Lorentzians. The natural line width dominates, as seen in figure 2.4. This will be

more important when we fit the spectra from the red laser in chapter 4.

Consider the absorption spectrum of the 6P1/2 → 7S1/2 transition for both 203Tl and 205Tl

in figure 2.6. Of the three hyperfine transitions for both isotopes, we are interested in the

F=1→ F’=1 transition, the middle two peaks. These two peaks span only a few GHz of the

entire hyperfine structure of the 6P1/2 → 7S1/2 transition spanning about 35 GHz. We know

that the wavelength of this transition for 205Tl is 377.6800 nm and for 203Tl is 377.6808 nm.


If we scanned the UV laser over only 3 GHz above and below the 377.6800 nm 6P1/2(F=1)

→ 7S1/2(F’=1) transition, we would see the entire F=1 → F’=1 transition, as seen in the

spectrum in figure 3.5. We know that we will be using a Gaussian line shape to approximate

this transition because Doppler broadening is a factor in this transition.

To simulate the locking spectrum with the two modulated acousto-optic modulator (AOM)

frequencies, to be described in the next chapter, we first take the Gaussian equation with

the Doppler widths just calculated. Each Gaussian will have some scaling coefficient that

relates to the height of the absorption peaks. The total absorption spectrum in the sum

(ignoring isotopic differences in line width):

A(f) = A203e−(f−fo,203)2/2Γ2

+ A205e−(f−fo,205)2/2Γ2

(2.23)

where A203 is the relative abundance of 203Tl and A205 is that of 205Tl. Since the relative

abundances of 203Tl and 205Tl are 30% and 70%, we can normalize the signal using the values37

and 1 for A203 and A205 respectively. Finally, the transmission signal is given by:

T (f) = Ioe−αA(f) (2.24)

where α is the optical depth and Io is the intensity at 100% transmission. Since A(f) is nor-

malized to be unity on resonance of 205Tl, α is the optical depth on resonance. We choose

a vapor cell temperature which produces α ≈ 1 because this ensures that the transmission

“dip,” due to absorption of both isotopes, is easily measurable and not saturated.

Chapter 3

Experimental Setup

Both the 7P1/2 and 8P1/2 experiments utilize the same setup except the second laser in the

7P1/2 experiment has a 1301 nm laser in place of the 671 nm laser of the 8P1/2 experiment.

Other minimal differences include the exact Fabry-Perot (FP) cavity length and numerous

focusing and beam-shaping optics. However, the concepts are the same, so I will only de-

scribe the set up of the 8P1/2 experiment in this chapter.

3.1 External Cavity Diode Lasers

Both lasers that we use are external cavity diode lasers (ECDL). This means that the laser

diode is surrounded by a cavity formed by reflective optical elements. A diode laser uses

the p-n junction of a diode to produce a coherent beam of light. Figure 3.1 depicts a simple

version of an internal diode laser cavity. By sending current from the “+Electrode” to the

“-Electrode”, electrons and holes are brought from the valence band to the conduction band.

Holes and electrons then annihilate in the “active region” of the p-n junction. When a hole

from the conduction band of the p-type semiconductor and an electron from the conduc-

tion band of the n-type semiconductor annihilate, a photon is emitted. The photons travel

through the active region and reflect off of the flat faces, which act like mirrors. They travel

back through the active region and if they collide with a free hole and electron, they induce

stimulated emission. The amplified light is emitted through the flat faces of the active region.

An external cavity diode laser works differently than a cavity of a laser diode. One of the

flat faces of the diode is coated with an anti-reflection coating so that no stimulated emission

persists, but spontaneous emission still occurs (becoming a light-emitting diode). The light

then travels to a diffraction grating and the first order diffracted light is reflected back to

the diode, as seen in figure 3.2. The grating selects one wavelength from the spontaneously

18

CHAPTER 3. EXPERIMENTAL SETUP 19

+ Electrode

- Electrode

n-substrate

ActiveRegion

n- type semiconductor

p-typesemiconductor

Current

pn Junction

Flat Faces

Figure 3.1: Internal Diode Laser Cavity

Laser Output

671 nmDiode

DiffractionGrating

Spontaneous Emission fromDiode

First Order Diffraction

1200 lines/mm

Figure 3.2: 671 nm External Cavity Diode Laser

emitted photons to send back to the diode as the first order diffraction beam because the

grating makes the external cavity a Fabry-Perot cavity with discrete wavelengths with which

the cavity will lase. The first order diffraction beam induces stimulated emission in the


diode at its specific wavelength. The amplified light reflects off of the back face of the diode

and returns to the diffraction grating. To change the wavelength of the laser, we apply a

voltage to the PZT that changes the angle of diffraction grating ever so slightly resulting

in a large change in wavelength. This works because the stimulated emission amplifies the

first-order diffracted beam from the grating. So, changing the angle of the grating changes

the wavelength of the first-order diffracted beam thereby changing the wavelength of the

amplified light (diode output). Using the expression,

nλ = dSin(θ) (3.1)

we see that λ and Sin(θ) are directly proportional, so we can change λ accordingly. In this

experiment we want to scan the laser about 7 GHz, which corresponds to a tiny change in

λ of about ≈ 7×109

1014, so we drive the PZT by a ramping the voltage of sufficient size to move

the PZT by the required small distance. The result is continuous up and down frequency

scans that can be used to acquire spectra.

There are three benefits of the ECDL over the laser diode for our experiment. Laser

diodes emit a large spread of frequencies above and below the center frequency. Therefore,

the first benefit of the ECDL is to decrease the spread of frequencies around the center

frequency emitted. Because the first-order diffracted beam is the frequency amplified, there

is less of a range of other frequencies amplified at the same time. The frequency that the

diode laser outputs itself can be changed by changing the temperature and current of the

diode. However, we cannot take a spectrum by changing the temperature and current of the

laser diode because these changes are not smooth, quick enough, nor directly proportional to

the output frequency of the laser diode. Thus the second benefit of the external cavity is to

provide a way to rapidly and proportionally change the frequency of the laser. The ECDL

also allows us to induce lasing everywhere in the gain profile of the laser.

One unfortunate consequence of using the PZT to scan the red laser is that the response

of the PZT to a linear increase in voltage is nonlinear. This creates an issue when we look

at spectra from the scanning red laser because the frequency axis will not be linear. This is

one of the main reasons that we must calibrate the frequency axis.

3.1.1 378 nm Laser

The gain medium of the 378 nm laser is a GaN semiconductor diode. However, it is different

from the red laser because it does not have an anti-reflective coated diode. The wavelength

of 378 nm is short enough that any coating on the diode wears rapidly over time. This is not

the case for the red laser. Therefore, the UV laser is much more susceptible to mode hops


and is not as easily controlled by changing the angle of the grating as is the red laser. There

is a constant battle between the internal laser cavity and the external cavity. If the right

temperature, cavity length, current, etc. are chosen, the influence of the external cavity will

prevail. Because we do not need to take spectra with the UV laser as we do with the red

laser, it is adequate for our experiment that the tuning ability of the UV laser is low. It has

a scanning range of about 2 GHz [8]. Despite its limited tuning ability, we can tune and lock

the UV laser as necessary for hours at a time. It emits between 2− 6 mW depending on its

frequency. This laser system is manufactured by the German company Sacher Lasertechnik.

3.1.2 671 nm Laser

In appendix C, we included an overhead view of the current 671 nm ECDL laser system that

we built ourselves. Initially, we used our own 670 nm laser diode and coated it with anti-

reflection coating ourselves using an evaporation chamber. Unfortunately, the evaporation

chamber broke during the coating process and we did not get to coat the diode fully. When

we tested the capabilities of the semi-AR-coated diode, we found that it was able to lase

in the external cavity, but not at a stable enough frequency. Currently, the 671 nm ECDL

has an AR-coated diode, confocal lens, and laser mount from Toptica Photonics. The laser

diode is an AlGaInP semiconductor that outputs up to 30 mW of power. The specifications

of the laser diode and parts we built can be found in appendix C. We use a Stanford

Research Systems (SRS) LDC501 to control the laser diode’s current and temperature. It

sends current to the laser diode controller and controls the external cavity temperature via a

thermoelectric device mounted below the laser. This ECDL has exactly the same functions

as the 378 nm ECDL except that we can change the laser cavity length. Although, this is

not an extra tuning feature, we have chosen a length of 2 cm and fixed the cavity there.

One piece of this laser that is important to note is a wedge to tilt the diffraction grating

closer to the θ we want in figure 3.2. From the equation 3.1, the angle that will provide us

with a first order diffraction of 671 nm is:

Sin−1(6.71× 10−7m

8× 10−7m) ≈ 54° (3.2)

where 8 × 10−7 m is the inverse of the 1200 lines per millimeter, which is characteristic of

our grating. We cut the actual wedge to be 45°, so we have tilted the diffraction grating an

extra 9° to get to the desired 54°.


Figure 3.3: Laser Locking Setup

PBS

PBS


Half-Wave Plates

378 nm Laser A

OM

Experiment

PID Servo Controller

Differential Photodiodeand Amplifier

3.2 Laser Locking

3.2.1 Importance of Laser Locking

Laser locking is very important to minimize the the drift and instability of the resonance

frequency of the second step transition. By locking the first-step frequency, this fixes the

resonance frequency of the second transition. Without a fixed frequency of the first laser, the

velocity class excited would change over time. Consequently, the resonance frequency of the

second laser would drift many MHz because of the changing Doppler shift due to the change

in which velocity class is excited. This would compromise the reliability of the second-step

spectra which require ≈ 10 seconds to complete.

3.2.2 Laser Locking Set-Up

Figure 3.3 is a diagram of the locking setup. To choose the correct frequency for the first

excitation of both isotopes of thallium, we employ a supplementary smaller oven containing

a vapor cell of thallium. We direct a 378 nm beam through an acousto-optic modulator

(AOM), a device which uses a crystal to create two first-order, diffracted beams 260 MHz

above and below the laser’s frequency. The undiffracted beam proceeds to the main ex-

periment as described later. The two diffracted beams are combined in a polarizing beam

splitter (PBS). They then travel through the supplementary vapor cell and oven setup and

go through another polarizing beam splitter to separate the beams before entering a special

differential photodiode and amplifier system that outputs their difference signal, as seen in


figure 3.5. We use this difference signal as the basis of our locking scheme.

This signal is sent to a commercial servo proportional-integral-differential (PID) con-

troller, which is designed to construct an appropriate correction signal that can be fed back

to the laser. The servo PID controller works by taking the photodiode signal and translating

it to [16]:

Pε(t) + I

∫ t

t−τε(t)dt+D

dε

dt+N (3.3)

where P, I, and D are the respective proportional gain, integral gain, and derivative gain.

ε(t) is the input signal to the PID. τ is a characteristic time constant and N is the offset

adjustable voltage. We use a method called the Ziegler-Nichols tuning method to optimize

the output signal from the PID [8]. After the signal goes through the PID, it applies a

correcting voltage to the piezoelectric transducer (PZT) of the UV laser, which steers the

UV laser frequency to keep it locked. If the values for P and I are too large or too small,

the PID will constantly be overshooting the lock point or reach it too slowly [6]. Therefore,

it is important that these two parameters are chosen carefully.

Figure 3.4: Stability of a Locked Laser versus an Unlocked Laser [6]

A comparison of the frequency of the UV laser when it is and is not locked. The output of the locked UVlaser is in red and the output of the unlocked UV laser

is in black. We convert voltage into frequency using the slope of the linear portion of the difference signal near.

0.0

1.0

2.0

3.0

0 10 20 30 40 50 60 70 80time (minutes)

Freq

uenc

y (u

ncal

libra

ted

units

)

unlocked signal

locked signal

~10 MHz

RMS ~ 0.7 MHz


Figure 3.4 is a comparison of the behavior of the UV laser when it is locked (black) and

when it is unlocked (red). Over an 80 minute period, the unlocked UV laser both jumps and

slowly drifts in frequency by 10’s of MHz from the original frequency. This is not adequate

for our experiment. However, using this method of locking, we see that a noisy drifting laser

output can be made stable at the < 1 MHz level. We calibrate the plot of figure 3.4 so that

we can determine the residual RMS frequency of the locked laser. Initially, the amplitude of

the residual RMS frequency is measured in voltage because it is a measure of the change in

voltage of the difference signal. We can approximate the portion of the difference signal, of

figure 3.5, relatively near the locking point as having a linear slope. The linear slope around

the locking point is ∆v∆f

, so we can use this ratio to convert voltage into frequency. However,

the initial x-axis of the difference signal is voltage scaled by the PZT ramp [6]. To convert

the x-axis of the differential signal into frequency, we use simultaneous FP scans that have

the same voltage x-axis. Since we know the free spectral range (FSR) of the FP cavity, we

can use this to calibrate the x-axis of the difference signal into frequency and then calibrate

the residual RMS voltage into RMS frequency.

3.3 378 nm Signal

We use the 6S1/2(F=1) → 7S1/2(F’=1) transition spectrum to lock the laser. The simulated

effects of the AOM are shown on figure 3.5 by shifting eq. 2.23 by +260 MHz and −260

MHz. The dashed line indicates the unshifted frequency component which is sent to the

experiment. It is then straightforward to take the difference of the two diffracted signals

and plot the result. In figure 3.5, the difference signal below has three corresponding areas

where there is a linear portion of the signal. To take single isotope data, we want to lock

the UV laser at the frequency of the center of the resonance peaks (dashed line) for either

of the two isotopes. As mentioned before, the job of the servo PID is to send a correction

signal to the laser to bring it back to the desired frequency. In the absence of the difference

signal (that is looking at the absorption directly), at the center of the resonant frequency

peaks of either isotope of the 6S1/2(F=1)→ 7S1/2(F’=1) transition, a decrease in absorption

will accompany any change in the frequency of the laser. Consequently, the PID will not be

able to send a correction signal because it does not know in which direction the frequency

of the laser has drifted. The difference signal allows us to easily tell if the laser is drifting

above or below the desired resonance frequency. This is because the slope of the difference

signal at the resonant frequencies of both isotopes is linear.

Besides the resonant frequency of each isotope, there is a third linear location of the

difference signal to which we can lock. Fortunately, this point is directly in between the iso-


−3000 −2000 −1000 0 1000 2000 3000−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Frequency (MHz)

Nor

mal

ized

Tra

nsm

issi

on S

igna

l

Difference Signal

Upshifted SignalDownshifted Signal

Dua

l−Is

otop

e Lo

ck

203

Lock

205

Lock

Figure 3.5: Simulated Absorption Signal of the 6P1/2(F=1) → 7S1/2(F’=1) Transition for Both IsotopesThe dotted blue line is the original signal of that would be transmitted if the UV laser scanned across the

6P1/2(F=1) → 7S1/2(F’=1) transition of both isotopes. The red and green lines represent the up and downshifted versions of the original signal. The solid blue line represents the difference of the up and down

shifted versions of the original signal.

topic resonances. Some atoms of each isotope will be excited because the absorption of each

isotope is not zero at this frequency. To determine the transitional isotope shift, we need to

excite both isotopes simultaneously, so this lock point is crucial. Because the dual isotope

lock point is not at the frequency of either isotopic resonance, we are exciting a blue shifted

velocity class of 203Tl atoms and a red shifted velocity class of 205Tl atoms. Luckily, the

Doppler widths of the isotopic resonances are sufficient that we are able to excite substantial

amounts of both isotopes.

3.4 Experimental Layout

A sketch of the experimental setup is shown in figure 3.4. The double excitation of thallium

requires first and foremost two lasers. The first is a 378 nm external cavity diode laser

(ECDL) and the second is a 671 nm ECDL that we built ourselves. In our earlier 6S1/2

→ 7P1/2 experiment, this laser was an infrared laser tuned to 1301 nm. There is an oven


378 nm laser

671 nm laser

Isolator

AOM

Locking Setup

OpticalChopper

Lock-InAmplifier

50/50 BeamSplitter

Dichroic

PD

EOMPD

Half-Wave Plates

FP CavityPBS


To Computer

200 MHz Synthesizer

Shutters

Figure 3.6: Experimental Setup

containing a quartz cell of naturally occurring thallium (70% 205Tl and 30% 203Tl).

The undiffracted component of the now-stabilized 378 nm laser beam passes through a

50/50 beam splitter. The two outputs send light through our thallium interaction region in

opposite directions. After the beam splitter, there is one shutter in front of each UV beam

that blocks and unblocks each beam before it reaches the interaction region.

The 671 nm laser beam path is more involved. First, the beam is separated in a polarizing

beam splitter. Half of the beam goes into a confocal Fabry-Perot cavity and the other half of

the beam goes through an electro-optic modulator (EOM). The EOM is a device that uses

electro-optic elements and a radio frequency signal (200 MHz) to create frequency modulated

(FM) sidebands above and below the center frequency of the 671 nm laser beam. This is

used for one of the frequency calibration methods, which is described in chapter 4.

The frequency modulated 671 nm beam is then combined via a dichroic mirror with one

of the UV beam components and sent to the interaction region. There are now three laser

beams intersecting in the interaction region: two UV and one red. The shutters can block

and unblock the two UV beams separately before they reach the interaction region. Using

these shutters we can arrange the the two beams copropagating (CO) through the cell or

counterpropagating (CTR) through the cell, as seen in figure 3.6. We align the three beams

so that they spacially overlap in the interaction region. It is important that the two beams,

in both configurations, maximally overlap in the interaction region so that the most UV

excited atoms are excited by the red laser. The decay time for a UV excited atom is 7.43

ns [15]. In this time, the atom can travel a distance of 250m/s× 7.43× 10−9s ≈ 2µm given

the approximate velocity of the hot atoms. Therefore, to have the greatest absorption signal

possible from the red laser, the UV and red beams can be misaligned no more than 2µm, so


that the least amount of the UV excited atoms escape the red laser. After passing through

the interaction region, the red beam then travels to a photodiode, which is connected to a

lock-in amplifier using the signal from the optical chopper as its reference signal. The lock-in

amplifier output is sent to a computer to collect and analyze the final spectra.

3.5 Detection of the Second Step Signal

The first issue that we must deal with is the small scale of the resulting absorption signal.

The diameter of both beams going through the cell is about 2 mm. The second laser can

only excite the already excited 7S1/2 atoms in this small volume constrained by the diameter

of the beam and length of the cell. Therefore, as discussed, it must be very well aligned

with the first laser beam to maximize the amount of excited atoms. On top of this, even

if the beams are perfectly aligned, there will still be a very small absorption signal because

the hole burning, as discussed in chaper 2, only excites a small portion of the UV excited

atoms. We estimated the fraction of ground state atoms excited by the UV laser was ≈ 5%.

So direct detection of the red beam transmission would show a tiny absorption dip with

large background as suggested in figure 3.7a. To decrease the noise and thereby increase the

signal-to-noise ratio, we use an optical chopper and a lock-in amplifier. We set the motor

of the optical chopper to some speed and place it in front of the 378 nm laser so that the

beam is modulated at ≈ 1000Hz. It provides a signal that looks like a Sine function. When

the wheel blocks the UV beam, there is 100% transmission of the red laser beam as seen in

figure 3.7b. This is because the atoms are not excited to the 7S1/2 level enabling them to

absorb the red laser beam. Therefore, the frequency of the modulation of the UV laser beam

is the frequency with which an absorption signal of the red laser is present.

The lock-in amplifier takes the reference signal from the chopper wheel controller and

multiplies it by the input signal. The input signal is the output of the photodiode reading

the power of the scanning red laser. Multiplying the input signal by the reference signal

after some uninteresting math will allow the lock-in amplifier to record the differences be-

tween the red spectra when the UV beam is blocked and unblocked. The lock-in performs

S1(f) − S2(f) for S1(f) as the spectrum of figure 3.7b and S2(f) as the spectrum of fig-

ure 3.7a. The lock-in amplifier will reduce the noise in the S2(f) spectrum because it will

only be sensitive to noise at a frequency near 1000 Hz. In essence, the lock-in will take

the Fourier component of the difference signal at 1000 Hz. The noise from lower frequen-

cies will be ignored by the lock-in amplifier. The second job of the lock-in amplifier is to

amplify the signal as well. The output signal of the lock-in amplifier will look like figure 3.7c.


Figure 3.7: Red Laser Transmission Spectrum due to Optical Chopper

−3000 −2000 −1000 0 1000 2000 30000%

50%

100%

Frequency (MHz)

Tra

nsm

issi

on

S2(f)

(a) UV Beam Unblocked by Optical Chopper

−3000 −2000 −1000 0 1000 2000 30000%

50%

100%

Frequency (MHz)

Tra

nsm

issi

on

S1(f)

(b) UV Laser Beam Blocked by Optical Chopper


−3000 −2000 −1000 0 1000 2000 3000Frequency (MHz)

Arb

itrar

y U

nits

S1(f) − S

2(f)

(c) Output of the Lock-in Amplifier of the Spectrum fromfigure a

Chapter 4

Data Analysis and Results for the

7P1/2 Experiment

This chapter contains the data analysis of the 7P1/2 experiment. Once we begin acquiring

spectra for the 8P1/2 experiment, our analysis procedure will be identical. The discussion

below expands upon the results and error analysis in our Physical Review manuscript [10].

4.1 Data Acquisition

As a first step in data-taking, we lock the UV laser to the appropriate lock point making

sure that both configurations of laser beams through the cell are spacially overlapped to a

maximum by visual alignment of the beam via peak asymmetry minimization. A Labview

program, created by David Kealhofer [8], programs the DAQ to control the shutters, the

FP photodiode, the output of the lock-in amplifier, and the sweep of the piezo. Depending

on the type of spectrum we wish to take, single or dual isotope, the program sets the UV

shutters to the desired configuration. Manually, we turn the EOM on only if we desire single

isotope spectra. The program then sends a voltage signal to the PZT to sweep the IR laser

5-7 GHz up and then back down to the starting frequency of the IR laser. As the sweep

occurs, we sample the output signal of the FP photodiode and lock-in amplifier several hun-

dred times during both the upward and downward sweep of the IR laser frequency. The two

sweeps represent a pair of data runs; one for each sweeping direction. We store each data

run in a separate text file including the point number, along with the sampled FP photo-

diode and lock-in amplifier signals. If we are taking single isotope data, the configuration

of the UV shutters switch to set the other propagation configuration. If we are taking dual

isotope data, the program continues to take pairs of data runs with the same UV shutter

configuration. This process is repeated a few hundred times before we stop acquisition. We

30

CHAPTER 4. DATA ANALYSIS AND RESULTS FOR THE 7P1/2 EXPERIMENT 31

call the collection of the hundreds of pairs (up and down scan) of data runs a “data set.”

Throughout the data acquisition of a data set, the configuration of the beams in both the

CO and CTR configurations remains constant. Other conditions of the experiment remain

constant during the acquisition of a data set. Each data set takes of order one hour. Between

data sets, we tweak the beam alignment of both configurations and change other parameters

of the set up such as oven temperature, IR laser power, IR and UV laser polarization, etc.

We take between 2 and 5 data sets in one day.

4.2 Linearization and Calibration of the Frequency Axis

4.2.1 Fabry-Perot Calibration and Linearization

The Fabry-Perot cavity we use is a parallel plate FP cavity with a finesse of ≈ 50 and an

FSR of approximately 500 MHz. To maintain thermal stability within the cavity, it is made

out of low-expansion material and we cover it with a thermally-insulated box. The purpose

of the parallel confocal Fabry-Perot cavity is to calibrate and linearize the frequency axis

of the final spectra from the 671 nm laser. As mentioned before, when we sweep the laser

by applying a voltage to the PZT, its response is hysteretic and nonlinear. Because we

simultaneously extract the signal data and the FP data, while the IR laser sweeps, both are

subject to the same nonlinearities in the frequency axis. To illustrate this, figure 4.1 contains

the signal spectrum of a 205Tl data run with its corresponding nonlinearized FP data fit with

an Airy function. We convert the raw point number, i, of our scans to a normalized x-axis

scale using the equation:

xi =i−N/2N/2

(4.1)


0

0.2

0.4

0.6

0.8

1

Nor

mal

ized

Am

plitu

de

rmse =0.0061021

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

Normalized Point Number, xi

Nor

mal

ized

Am

pitu

de

rmse =0.018261Airy Function

Figure 4.1: Simultaneous Lockin Amplifier and Fabry-Perot DataA plot of the simultaneously taken samples of FP and lock-in amplifier signal output for 205Tl. The red line

on the FP data represents the Airy function fit to this data. The x-axis is the normalized point numberassociated with each FP and lock-in sample taken. The point number is proportional to the ramp-step of the

voltage applied to the PZT, so we can treat the x-axis as normalized applied PZT voltage.

where N is the total number of points. We see that −1 ≤ xi ≤ 1. When we fit the FP

data to our Airy function, we express the frequency argument as a polynomial function of

xi, and find the best fit coefficients.

The first step to calibrate the frequency axis consists of fitting the FP data versus the

normalized point to an Airy function. Because the ramp-step of the voltage applied to the

PZT is constant, the point numbers are proportional to the applied PZT voltage. Therefore,

we can treat the x-axis as of the plot in figure 4.1 as normalized PZT voltage. We took the

positions of the FP peaks from the Airy function fit. To begin with, we take the frequency

spacing between each successive FP peak to be 500 MHz. We then plotted these frequency

spacings of the FP peaks by their xi positions, see figure 4.2. We fit a fourth-degree polyno-


mial to these data points. We found that the fourth degree polynomial was suitable because

higher-order polynomials did not have fits with any better statistical quality. The position

of the peaks do not fit on a straight line as they would if the PZT scanned linearly. This is

illuminated by the nonzero polynomial coefficients of the fitted line in figure 4.2. It is also

evident in the straight red line that joins the first two peak positions and extends linearly.

We can now use this fourth-order polynomial as a function of normalized point number to

convert the x-axis of figure 4.1 to frequency. In effect, this “linearizes” the x-axis of the

spectra. We are now in a position to fit the atomic spectra with a linearized frequency scale.

Note absolute calibration requires precise knowledge of the true FP FSR. In fact, the true

FSR is slightly larger than 500.0 MHz.

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

6000

Normalized Point Number, xi

Fre

quen

cy (

MH

z)

−4.4438,−29.0075,145.2767,2546.8637,2618.497

Figure 4.2: Polynomial of Fabry-Perot Peak PositionsThe blue line through all of the FP peak positions represents the forth-degree polynomial from the Airyfunction. The red line represents the line fitted to the FP peak positions if the fourth-degree polynomial

were a linear fit instead. This is a plot for FP peak positions for a data run with an upsweeping PZT. Thedown sweeping plots have forth-degree polynomials that have opposite concavity.

The true FSR is above the 500.0 MHz is because the FSR of the FP cavity is determined

by the radius of curvature of each of the confocal mirrors. Thus, the precision of the FSR is

determined by the precision of their radii. However, the radius of curvature of the mirrors

typically differs from the nominal quoted value. After determining the FSR precisely as


described below, it is a simple matter to scale all frequency intervals in our fitted spectra by:

CFP =True FSR

500MHz(4.2)

To find the actual FSR of the FP cavity, we sent a portion of the IR laser beam to a

Burleigh WA-1500 wavemeter, which reads absolute frequencies to a precision of ≈ 30 MHz,

while the FP cavity transmission was sent to an oscilloscope to help us determine where the

transmission peaks of the FP cavity are. We tuned the IR laser to a transmission peak of

the FP and recorded the frequency from the wavemeter. Then we would tune the laser to

a different transmission peak and record that frequency, repeating this process until we had

at least 100 frequencies of FP transmission peaks measured out of a range of ≈ 100 GHz.

To find the FSR, we took pairs of transmission peaks and subtracted their frequencies. The

frequency difference of any pair of these 100 transmission peaks is an integer multiple, n, of

the true FSR. To find the FSR of two distant transmission peaks, we can guess an n such

that the distance between the two frequencies is n times our nominal estimate of the FSR,

“FSRo.” We chose n to minimize the residual between δf and n × FSRo. In this way, we

can find an FSR of improved precision. We took the FSR between every possible pair of

frequencies we recorded using this method, which increases the precision of the average FSR

value. The average FSR we found after several iterations of this process was 501.2(3) MHz.

The corresponding correction factor is therefore 501.2MHz500MHz

= 1.0024(5). This is in agree-

ment with the FP calibration factor of 1.002(1) that David Kealhofer came up with using

the same IR laser and FP cavity [8]. We can thus calibrate the HFS and IS raw data that

we get from our spectra by simply multiplying the raw frequency interval values by 1.0024(5).

4.2.2 EOM Calibration

The second method of calibration of the linearized frequency axis is to use an electro-optic

modulation device (EOM), which provides FM sidebands about the center frequency of the

IR laser at intervals of exactly 600.0 MHz. In our experiment, only the first-order sidebands

are visible in the IR spectra. The resulting IR spectra contain one spectral peak for each

of the two transition frequencies, each surrounded by two sideband copies, see the top of

figure 4.1. We measure the sideband splitting as the difference in frequency between the

resonant spectral peak and its two surrounding sidebands. All linearized spectra start with

the nominal assumption that the FP FSR is 500 MHz. Because we now know that the

FP cavity does not have an FSR of exactly 500.0 MHz, we expect the apparent sideband

splitting to be slightly below 600.0 MHz. We create a correction factor for the linearized

frequency axis by simply dividing the average experimental value of the sideband size into


7S1/2


F' = 1

F'' = 1

F' = 0

F'' = 0

F'' = 1

F'' = 0

F' = 1

F' = 0

7P1/2{ {

7S1/2

7P1/2

Figure 4.3: Second Step TransitionThe red arrows represent the IR laser scanning 6-7 MHz across the 7P1/2-state hyperfine structure

600.0 MHz. The average correction factor, CEOM , from this method was 1.0022(2) for all203Tl data and 1.0018(2) for all 205Tl data [10]. We found that both methods of calibration

were reasonably close and in good statistical agreement with CFP . Furthermore, when we

looked at the spread of calibration factors from each individual data set, they did not vary

more than 0.0002 from their respective average correction factors [10]. Again, once we have

the raw HFS and IS values, we can simply multiply them by the CEOM for the appropriate

isotope.

4.3 Interpreting Spectra

Figure 4.3 represents the hyperfine transitions, 7S1/2(F ′ = 1) → 7P1/2(F” = 0, 1), that the

IR laser scans across. To extract information from these scans, we use the “ThalliumFit-

ting.m” code summarized in appendix A. For simplicity, we will call the data taken from the

lock-in amplifier the signal data. The “ThalliumFitting.m” code first linearizes and converts

the xi axis to a frequency axis of the signal data with the FP data for each data run, using

the FP linearization method previously explained. This code then fits a sum of Lorentzians,

six for single isotope spectra and eight for dual isotope spectra, to each linearized data

run. As mentioned in section 2.6, we approximate the shape of the IR spectral peaks with

Lorentzians because the Gaussian contribution caused by Doppler broadening is negligible


because of our hole-burning technique. The result of this fit is the solid red line in the spectra

in figures 4.4 and 4.7. Each of the blue points represents a data point. Each peak’s Lorentzian

includes a variable for position, common width, and amplitude. We are interested in the

the position variable for each peak. Along with these spectral figures, “ThalliumFitting.m”

also produces four text files for every data set containing these peak positions and their

uncertainty. There is one text file for each combination of beam configuration and sweeping

direction: copropagating upscan, counterpropagating upscan, copropagating down scan, and

counterpropagating downscan. HFS value and four sideband splittings are recorded for each

data run. A second program, “DataAnalysisThallim.m” (appendix A), takes these four files

of peak positions and produces tables with measurements of HFS values, EOM sideband

values, IS values and their corresponding variances. The “DataAnalysisThallium.m” also

creates histograms and comparison figures to assess the quality of the data of a particular

set, which are useful to seek out potential sources of error.

4.4 Single Isotope Spectra

Figure 4.4: Single-Isotope Data Run with Residuals

An upscan 203Tl spectrum [10]. The fitted Lorentzian and the data points from the data run are visibly inagreement, as seen in corresponding residual plot below.

Hyperfine Splitting

Sideband

A

B

C

D

E

F


From the single isotope spectra such as figure 4.4 for each data run, we extract HFS value

and four sideband splittings from each plot. The frequencies of each of the labeled peaks in

figure 4.4 are denoted by νN for peak label N . For the 203Tl isotope, each of the six peaks

in the single isotope spectrum represent:

νA = νB − fAOM (4.3)

νB = fo −3

4H7p,203 −

1

4H7s,203 (4.4)

νC = νB + fAOM (4.5)

νD = νE − fAOM (4.6)

νE = fo +1

4H7p,203 −

1

4H7s,203 (4.7)

νF = νE + fAOM (4.8)

where fo is the transition of 7S1/2 → 7P1/2 in absence of hyperfine structure, fAOM is

the 600 MHz sideband frequency, H7p,203 is the hyperfine splitting of 7P1/2 for 203Tl, and

H7s,203 is the hyperfine splitting of the 7S1/2 level for 203Tl, which subtracts out in our HFS

determination of νE− νB. Because we use fo with no HFS as our reference point, we use the

prefactors −34

and +14

for the F”=0 and F”=1 hyperfine levels, as calculated in equations

2.13 and 2.14 respectively. Figure 4.4 is the resulting spectrum from an upscan of the PZT.

A downscan of the PZT results in a mirror image of the upscan, with A through F labeled

right to left. Experimental spectra for single isotope scans of 205Tl look exactly the same

with roughly 20 MHz greater hyperfine splittings.

The measured splittings of the sidebands going from peak A to peak F are: S1, S2, S3, and

S4. We can calculate the hyperfine splitting values and sideband sizes using the following

equations:

HFS1 = νE − νB (4.9)

HFS2 = νD − νC + 2× fAOM (4.10)

S1 = νB − νA (4.11)

S2 = νC − νB (4.12)

S3 = νE − νD (4.13)

S4 = νF − νE (4.14)

Later we will discuss the importance of the comparison of HFS1 and HFS2 as a mea-

sure of how well we linearized and calibrated the frequency axis. It is important to note

that if the lock point is not quite at the center of isotopic resonance, for the desired isotope,

the entire spectra is slightly Doppler-shifted. However, the hyperfine splitting is not affected.


4.4.1 Single Isotope Error Analysis

From the initial spread of raw hyperfine splitting values, we want to get rid of the data with

bad frequency linearization. One way to do this is to assess the quality of the fitted data.

Bad Lorentzian fits of the scans lead to more error in the position of the peaks and thus more

error in the HFS values. Bad fits occur because the frequency linearization of the scan did

not sufficiently correct for the nonlinearity in the frequency axis. This results in asymmetric

fits. Another cause of poor fits is if the beams are not properly spacially overlapped, leading

to asymmetry in the spectral peaks. Because we change optimize the overlap of the UV

and IR beams between each data set, we effectively randomize the error due to bad spacial

overlap of the beams and does not affect the final average values of our hyperfine splitting.

The only cause of asymmetry that remains as a problem is therefore due to frequency axes

with bad linearization. We can find inconsistent asymmetric fits by measuring the residual

between the data points and fitted Lorentzian of each scan. Figure 4.4 is a scan with a good

residual value. The residual is magnified so that we can better study it. A small fraction of

our fits were rejected based on significantly larger residuals, as measured by the statistical

quantity χ2 (the sum of squares of the deviations of the fit data).

We ultimately want to get a spread of HFS values for both isotopes that resembles a

Gaussian because this will mean we have a normal distribution of our experimental results.

A normal distribution of our results is important because it means we have a resulting HFS

value that is not perturbed by any additional random variables or error. Tables of the raw

results of each data set, by day, exist for each isotope in appendix B. Initially the spread

of this raw data was not in a sufficient normal distribution. Since the calibration of the

frequency axis is one of the main factors that sets our experiment apart from that of Grexa

et al. [1], it should be a major factor used to cut down the raw single isotope data. This

is done by restricting the range of fitted sideband values. Because we expect the average

sideband value to be slightly below 600.0 MHz, the maximum and minimum sideband values

are not centered around 600.0 MHz, but are centered around the estimated average sideband

value. For each isotope, we restrict our data to those scans whose sideband values fall within

the specified range. For 205Tl this range was between 591 MHz and 605 MHz and for 203Tl

this range was between 592 MHz and 604 MHz. It is important to note that adjusting the

exact cutoff range did not change our final mean value for the HFS. Therefore, the resulting

average HFS value is only from data that we have some confirmation of a good frequency

calibration. We apply this approach to all set of data and arrive at a final statistical error

of 0.20 MHz for the 205Tl HFS and 0.25 MHz for the 203Tl HFS.


4.4.2 Systematic Error Search

Equally important to the statistical error, the search for systematic errors is crucial to our

data analysis. First, we considered a number of experimental parameters that ought not

affect our final results. We used five different powers of the second laser in the double excita-

tion [10]. There was no statistically significant difference in the peak positions in the spectra

and thus the HFS values for both isotopes. Second, the relative polarization was changed,

resulting in a relative change in the relative heights of the peaks of the spectra, but we saw

no change in the splitting at the level of statistical error.

The effects of Doppler broadening due to different temperatures of the thallium atoms

should not be significant. We varied the temperature of the main oven that contained the

thallium cell for the final spectra. This also alters the thallium vapor density. Simultane-

ously, these temperatures are not high enough for Doppler broadening effects to change the

profile of the absorption peak shape because we found that only temperatures above 550 °Cwould create Doppler broadened effects in the spectra1 [10]. We saw no effects of this at

lower temperatures between 400°C and 450°C.

We also explored the effect of the speed and width of the laser sweep. The laser sweep is

caused by sending voltage to the piezoelectric device to sweep the laser frequency 6-7 GHz

up and down the frequency scale. There was no difference found from changing the laser

sweep between 5-8 GHz.

4.4.3 Subdividing Data

Table 4.1 shows our results for the 7P1/2 hyperfine splitting for both isotopes. For several

important potential sources of error, we subdivided the data according to categories: the

scanning direction of the PZT, the configuration of the UV and IR lasers, and the 4-3 vs.

5-2 HFS measurement.

Differences in the sweep direction could lead be a result of two issues. The first issue is

that the response of the PZT is different between increasing and decreasing applied voltages,

meaning the PZT’s hysteresis is not the same in both scanning directions. The second issue

could be that the time constant of the lock-in amplifier creates a lag in response time of the

lock-in. In figure 4.5 the hyperfine splitting values for each scan direction differ by a little

bit less than 0.5 MHz in both isotopes. The upscan HFS value is in reasonable statistical

agreement with the downscan HFS value. Since the two scanning directions produce differ-

ences in different parts of the spectra due to the two issues mentioned, the agreement of the

HFS values from both directions shows the validity our linearization method. Any residual

1This effect is called “radiation trapping,” which is due to the relatively immediate absorption of a photon after it has beenemitted by another atom. It happens in high densities. This effect presents itself as Doppler broadened pedestals.


discrepancy was at the 0.3 MHz level, which was included in the final value.

The CO and CTR beam configurations of the UV and IR laser beams excite different

velocity classes of the thallium vapor depending on the locking point of the UV laser. When

we lock to the dual isotope locking point, we excite certain velocity classes of both isotopes.

It must follow that when we lock the UV laser to a single isotope resonant frequency, we

excite not only the zero velocity class of the desired isotope, but also a small amount of

highly Doppler-shifted atoms of the other isotope. The non-resonant isotope could poten-

tially contribute to the IR spectra. This is a problem because this could cause error in

fitting the resonant isotope’s data, and finding the correct frequency splittings of that iso-

tope. Because the Doppler shift has an opposite sign in the CO configuration versus the

CTR configuration, the contribution of the other isotope to the IR spectra will be different

in both configurations. Therefore, if we see no difference in frequency splittings of CO and

CTR single isotope spectra, then we know that the non-resonant isotope’s contribution to

the spectra is negligible. We switch the beam configuration hundreds of times automatically

during the experiment with two shutters hooked up to the DAQ. Looking at figure 4.5, the

CO and CTR HFS values for both isotopes are in statistical agreement.

Lastly, we compared the HFS values resulting from subtracting the peak positions of

the third peak from the fourth peak (δνD − δνC) and the second peak from the fifth peak

(δνE − δνB). The HFS resulting from subtracting the third peak position from the fourth

peak position and adding 1200 MHz, two sideband lengths, is less affected by a potential

residual scan nonlinearity and calibration error in the frequency scale. Comparing the 4-3

and 5-2 HFS values is therefore a good measure of how well-calibrated our frequency scale

is. In figure 4.5, of both isotopes the greatest discrepancy in comparison parameters is be-

tween the 4-3 and 5-2 HFS values. However, the two values are still within their combines

standard deviation of ≈1.5 MHz. In our final error budget, we add a small systematic error

component based on these comparisons.

Finally, we want to add a systematic error from the overall level of disagreement between

the correction factors of the two methods of calibration. Due to the 0.0002 discrepancy be-

tween the frequency modulation correction factors and the Fabry-Perot correction factor, we

associate an error of 0.0002× 2170MHz ≈ 0.4MHz. The error of each source of systematic

error is added in quadrature in conjunction with the statistical error to get our final resulting

error, listed in parentheses of the first line of table 4.1.


Figure 4.5: Comparison of HFS Values by Data Subsets

The upper graph is a comparison of 203Tl HFS values, and the lower graph is a comparison of the 205TlHFS values. Both graphs are representations of the hyperfine splitting values based on six parameters:

copropagating (CO) laser configuration, counterpropagating (CTR) laser configuration, upscan (UP) of thepiezo, downscan (DN) of the piezo, 3rd peak position subtracted from the 4th peak position, and 2nd peak

position subtracted from the 5th peak position. All of these comparisons come from the same set of data, soCO and CTR, UP and DN, and 4-3 and 5-2 each represent the entire final data set. Error bars represent

onestandard deviation of the HFS mean value determined from the uncertainty associated with the peak positions.

2152

2152.5

2153

2153.5

2154

2154.5Hyperfine Splitting Values by Data Subsets

203T

l Hyp

erfin

e S

plitt

ing

(MH

z)

CO CTR UP DN 4−3 5−2

2172

2172.5

2173

2173.5

2174

2174.5

205T

l Hyp

erfin

e S

plitt

ing

(MH

z)

CO CTR UP DN 4−3 5−2

Table 4.1: 7P1/2 Results

7P1/2 HFS 205Tl 7P1/2 HFS 203Tl 7S1/2 - 7P1/2 Isotope ShiftFinal Result (MHz) 2173.3(8) 2153.2(7) 534.4(9)

Statistical Error (MHz) 0.20 0.25 0.50

Systematic Error (MHz)Laser Sweep 0.30 0.30 0.20Beam Propagation (co vs. counter) 0.3 0.2 N/AFrequency Calibration 0.55 0.45 0.50Scan Linearization 0.20 0.20 0.20Thallium Cell Temp. 0.20 0.20 0.20Alignment of Beams N/A N/A 0.25Correl. with HFS and Doppler Shift N/A N/A 0.30


4.5 Dual Isotope Spectra

4.5.1 Eliminating Doppler Shift with Dual Beam Configuration

In the single isotope trials, we purposefully choose to excite the zero velocity class atoms by

choosing the locking point to be on resonance of the 6P1/2 → 7S1/2 transition. Even if the

class of atoms excited by locking is just off resonance, and they do have nonzero velocity, we

do not have to worry about the Doppler shift caused by the nonzero velocity because both

hyperfine transitions will be affected by the same Doppler shift. Since we only care about

the relative difference in the hyperfine transitions, we can ignore the Doppler shift. Thus,

there is no issue with Doppler shift in the single isotope spectra.

However, we do have to account for the Doppler shift in the dual isotope spectra. The

locking point we choose to take the red dual isotope spectra is in the middle of the resonances

of the 6P1/2 → 7S1/2 transition of both isotopes at 377.6804 nm. A non-zero velocity class

of atoms of each isotopes will then be excited. This produces a problem because the veloc-

ity classes of atoms of both isotopes are different. Therefore, we cannot ignore the relative

Doppler shift if we want to measure a quantity such as the isotope shift. Since we must take

differences in of transitions from both isotopes to find the isotope shift in the dual isotope

spectra, we need to find a way to eliminate the Doppler shift of both isotopes completely in

order to calculate the isotope shift. This is still not a problem for calculating HFS of each

isotope because each isotope’s peak is Doppler shifted by the same amount.

The solution emerges in the simultaneous use of the CO and CTR configurations. Both

of these configurations and the velocity classes of atoms they excite are depicted in figure

4.6. The velocity class of 205Tl atoms excited by the UV laser will be moving towards the

direction of propagation of the UV beam because with a resonance of 377.6808 nm, these

atoms will be blue shifted. The velocity class of 203Tl atoms excited by the UV laser will be

moving away from the UV beam because they are redshifted atoms of the resonant wave-

length 377.6800.


Figure 4.6: Dual Beam Configurations

The UV beam excites moving velocity classes of thallium atoms along the direction of propagation asshown. In the CO configuration, the 205Tl atoms are blue shifted by the UV and IR beams. The 203Tl

atoms are traveling in the opposite direction, so they are red shifted by both beams. In the CTRconfiguration, the 205Tl atoms are blue shifted by the UV beam and red shifted by the IR beam. The 203Tl

atoms are red shifted by the UV beam and blue shifted by the IR beam.

Counterpropagating Configuration

UV Beam

203Tl Atoms205Tl Atoms

Copropagating Configuration

UV Beam

IR Beam

IR Beam

203Tl Atoms 205Tl Atoms

In the CO configuration, the 205Tl atoms are blue shifted and the 203Tl atoms are red-

shifted in the final spectra because the red beam propagates in the same direction as the UV

beam. In the CTR configuration, however, the 205Tl atoms will be moving away from the IR

beam (redshifted) and the 203Tl atoms will be moving towards the red beam (blue shifted).

We can calculate the velocity and Doppler shift of the two isotopes in both configurations

using the equations:

∆f =v

cfo (4.15)

∆f2 =fo,2fo,1

∆f1 (4.16)

Equation 4.16 is found by setting the velocity found from equation 4.15 for f1 equal to

the velocity of equation 4.15 for f2, another frequency. 205Tl is found to have a Doppler shift

of about 880 MHz and 203Tl is found to have a Doppler shift of about 755 MHz with the UV

laser [8]. Regardless of the exact Doppler shifts of the isotopes, their sum is related to the


isotopic peak splitting in the UV spectrum. Using equation 4.15 the velocities of the excited205Tl and 203Tl atoms are approximately 330 m/s and 285 m/s respectively. The Doppler

shifts of the 205Tl and 203Tl atoms in the IR beam are approximately 219 MHz and 256 MHz

respectively, where fo,1 = c378nm

and fo,2 = c1301nm

.

The only difference in the frequencies of one isotope’s Doppler shifted transition peaks

in the CO configuration and the CTR configuration is the sign of the Doppler shift. If

we average the CO and CTR frequencies for a specific hyperfine transition of a single iso-

tope, we will get the non-Doppler shifted value of this transition. Therefore, we eliminate

the IR Doppler shift using the average of the CO and CTR frequency of each transition peak.

4.5.2 Interpreting Dual Isotope Spectra

0 1000 2000 3000 4000 5000 6000−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Frequency (MHz)

Nor

mal

ized

Inte

nsity

A

CB

D

E

F G

H

Figure 4.7: 7P1/2 Dual Isotope Spectrum

The dual isotope spectra have eight peaks. Four peaks correspond to the two hyperfine

transitions of each isotope in the CO configuration and four peaks correspond to the two


hyperfine transitions of each isotope in the CTR configuration. As discussed, these sets of

peaks are Doppler shifted form one another. The frequency parametrization of each peak in

figure 4.7 is described by the following equations:

7S1/2 → 7P1/2

Peak Equation Prop. Config. Hyperfine Transition IsotopeνA = fo − 3

4H7p,205 − 14H7s,205 − δf205 CO F ′ = 1→ F” = 0 205

νB = fo − 34H7p,205 − 1

4H7s,205 + δf205 CTR F ′ = 1→ F” = 0 205νC = fo + 1

4H7p,203 − 14H7s,203 + I7s−8p − δf203 CTR F ′ = 1→ F” = 0 203

νD = fo − 14H7p,203 − 1

4H7s,203 + I7s−8p + δf203 CO F ′ = 1→ F” = 0 203νE = fo + 1

4H7p,205 − 14H7s,205 − δf205 CO F ′ = 1→ F” = 1 205

νF = fo − 34H7p,205 − 1

4H7s,205 + δf205 CTR F ′ = 1→ F” = 1 205νG = fo + 1

4H7p,203 − 14H7s,203 + I7s−8p − δf203 CTR F ′ = 1→ F” = 1 203

νH = fo + 14H7p,203 − 1

4H7s,203 + I7s−8p + δf203 CO F ′ = 1→ F” = 1 203

where fo is the frequency of the 205Tl 7S1/2 → 7P1/2 transition in the absence of HFS.

I7s−7p is the isotope shift between the two isotopes for the transition 7S1/2 → 7P1/2. δf203

and δf205 are the Doppler shifts due to the IR excitation only. From these 8 peaks, labeling

their frequency splittings by δνij, we can isolate the HFS values for each isotope as well as

the transition isotope shift:

HFS205,1 = νE − νA = νEA (4.17)

HFS205,2 = νF − νB = νFB (4.18)

HFS203,1 = νG − νC = νGC (4.19)

HFS203,2 = νH − νD = νHD (4.20)

I7S−7P =3 |

νG + νH

2−νE + νF

2| + |

νC + νD

2−νA + νB

2|

4+HFS7S,203 −HFS7S,205

4(4.21)

The transitional isotope shift, equation 4.21, is calculated by taking averages of the CO

and CTR peaks to remove the Doppler shift leaving the frequency due to the transitional

isotope shift and the frequency due to the hyperfine anomaly. The second term in this

equation removes the affects of hyperfine anomaly in the 7S1/2 level. The previous 7S1/2

HFS work done in this lab provides precise values for these terms. This second term is

important because isotope shift is the difference between the the 7P1/2 states in the absence

of hyperfine states. Using this equation, the value of the transition isotope shift for the 7S1/2

→ 7P1/2 that we found is 534.4(9) MHz. We can take this transition isotope shift and convert

it into a 7P1/2 level isotope shift because this transition isotope shift is the difference of the

7S1/2 and 7P1/2 level isotope shifts. The 7S1/2 level isotope shift was previously determined

to be +409.0(3.8) MHz [4]. We simply take the difference between the 7S1/2 level isotope


shift and the 7S1/2 → 7P1/2 transition isotope shift, 409.0(3.8) MHz − 534.4(9) MHz, to get

the 7P1/2 level isotope shift of -125.4(4.0) MHz.

The total Doppler shift is the sum of the Doppler shift due to the IR excitation of the 205Tl

atoms and the Doppler shift due to the IR excitation of the 205Tl atoms. We can estimate

the total Doppler shift by taking the difference between the four corresponding peaks in the

CO and CTR configurations:

∆fDtotal =| δf203 | + | δf205 |=1

4[δνBA + δνDC + δνFE + δνHG] (4.22)

Each difference accounts for twice the Doppler shift by which both peaks are shifted. The

expected value of equation 4.22 is 475 MHz, by estimating the effects of the Doppler shift

due to the two velocity classes excited by the CO and CTR configurations. This is estimated

using the isotopic separation in figure 3.5 of 1636 MHz:

∆fDtotal =fIRfUV× 1636MHz =

378 nm

1301nm× 1639MHz ≈ 475MHz (4.23)

4.5.3 Dual Isotope Error Analysis

Many of the systematic errors that plague the single isotope data also affects the dual isotope

data. Therefore, similar systematic error values are associated with the dual isotope data.

Because we do not use the frequency modulation (FM) calibration method with the dual

isotope scans, we must estimate a frequency-scale correction factor based on the results of

single isotope scans. We chose to the average of the single isotope correction factors from

the FM method of 1.0020.

There are three additional sources of systematic error to consider in the dual isotope scans.

The first is the geometric alignment of the beams. We rely on the exact counterpropagation

since our isotope shift values come from taking the average of CO and CTR. Because there

are simultaneous CO and CTR configurations of the UV and IR beams, three beams must

be perfectly aligned inside of the thallium cell. We noticed that there was a greater scatter

and more asymmetries in isotope shift data from the dual isotope spectra than there was of

HFS values from the single isotope spectra. Due to the increase in asymmetries present, we

attributed the scatter to the greater difficulty of aligning three laser beams in the dual iso-

tope experiments than of aligning the two laser beams in single isotope experiments. Again,

realigning the beams between sets of data taken effectively randomizes this error and we can

associate an error for misalignment of all of the sets. We take as our final error as the scatter

between data sets. However, we should consider any biases misalignment could produce.


Figure 4.8: Beam Misalignment

Unaligned UV Beam

Aligned UVBeam

Red Beam

Using both the CO and CTR configurations to eradicate the Doppler shift can fail if

the Doppler shift of a certain transition peak is not the same in both configurations. This

can happen if the geometrical alignment of the beams is not perfectly overlapping in both

configurations. For example, if the red beam and the CO configuration UV beam overlap

perfectly, but the CTR configuration UV beam comes into the cell at an angle, θ, with the

red beam, different velocity classes of atoms will be excited and thus the Doppler shifts will

be different in each configuration. The Doppler shift is at a maximum when the beams are

aligned perfectly, so the Doppler shift of misaligned beams will result in a smaller Doppler

shift. This reduces the total Doppler shift by:

Sin(θ)× δf(IR) ≈ 0.001m

1m× 475MHz ≈ 0.5MHz (4.24)

where the dimensions of the collimator are 1 m × 1 mm, which allow us to approximate

Sin(θ) ≈ θ.

Using the approximate Doppler shift of the IR excited atoms in both isotopes calculated

earlier, 0.5 MHz. Figure 4.9 is a plot of the isotope shift versus Doppler shift. A fitted

line shows a slight trend between isotope shift and Doppler shift. The mean isotope shift of

approximately 534 MHz correlates to a Doppler shift of about 473.5 MHz, which is slightly

below the expected value. This suggests that the CO and CTR configurations may not have

eradicated the Doppler shift fully due to misalignment of the three beams.

The second additional source of systematic error is the correlation between the HFS val-

ues and the transitional isotope shift (IS). We can calculate IS and HFS values from dual

isotope spectra, we restricted our data to those runs for which the corrected HFS values

agreed with the single isotope scans within ±5 MHz.

The last additional source of error is the correlation of the total Doppler shift and the


Figure 4.9: Correlation of Transition Isotope Shift and Doppler Shift

Scatterplot of transition isotope shift versus the total Doppler shift for one set of data.


transitional isotope shift. The total Doppler shift is the sum of the Doppler shift due to both

excitations. We made scatterplots of IS vs. Doppler shift and used them to look for any

correlation between the IS and the total Doppler shift. From these plots of every data set,

such as figure 4.9, we some times saw that the average value of the total Doppler shift is ≈0.5 MHz lower than expected (475 MHz). This could be due to a small beam misalignment

as we have already discussed. For both the total Doppler shift and HFS correlations with

IS, we considered the extra error associated with the correlation slope and then combined

the two sources of error in the last entry of table 4.1.

4.6 Conclusions

Table 4.2: Comparing Results of 7P1/2 HFS and IS Values

Group 203Tl HFS (MHz) 205Tl HFS (MHz) 7P1/2 ISGrexa et al.(1988) [1] 2134.6 ± 0.8 2155.5 ± 0.6 N/AHermann et al. (1993) [9] N/A N/A -130 ± 5Theoretical Results[2] N/A 2193 N/AOur Results 2153.2 ± 0.7 2173.3 ± 0.8 -125.4 ± 4.0

Combining all sources of statistical and systematic error we see that both of our results

exceed those of Grexa et al. [1] by about 20 MHz. However, both of our experiment and

the experiment of Grexa et al. [1] quote errors below 1 MHz. Therefore, it seems that the

calibration errors that plagued the earlier experiments of Grexa et al. [1] may have also been

present in their quoted HFS values of 7P1/2 for both isotopes. The most recent ab initio

theory calculation of the 205Tl hFS is 2193 MHz, with an error of about 2-3% [2]. Thus,

our results improve agreement with the atomic theory, with a ≈ 20 MHz closer HFS value

than the 1988 results. The isotope shift of the 7P1/2 level is in very good agreement with

the previously determined value of -130(5)MHz [4]. Because the isotope shift is a measure

of the effects of an increased number of neutrons in the nucleus, it gives us insight into the

difference in mean square isotopic charge radius. We can access this using the equation 2.12,

getting a hyperfine anomaly, which is related to the ratio of 203Tl and 205Tl HFS values, of

∆7P1/2 = −5(4) × 10−4 [10]. This barely resolved value is in agreement with the hyperfine

anomaly calculated in a previous experiment in this lab for ∆7S1/2 [4].

Chapter 5

The 8P1/2 Experiment

5.1 Introduction

Now that we have completed the 7P1/2 experiment and published the results, our focus is

now to find the hyperfine splitting and isotope shift for the 8P1/2 level. The experimental

layout of this experiment, as described in chapter 3, is depicted in figure 3.4. The equipment

changes this experiment requires are the second step laser, the FP cavity, and the EOM

synthesizer. Instead of a 600 MHz synthesizer, the EOM of this experiment would use a 200

MHz synthesizer. The greatest change made between these two experiments is the choice of

the second step laser. Instead of the 1301 nm ECDL laser, we use a 671 nm ECDL laser

built in our lab, see section 3.1. Because the second step laser has been changed, we must

also change some specifications of the FP cavity.

As mentioned before, in 1993 the Hermann et al. group [9] republished the values of HFS

and IS after finding a calibration error in their previous results published in 1988 [1]. Unlike

the 7P1/2 state, this group quotes republished HFS values for the 8P1/2 state. For the 203Tl

isotope they found an HFS of 777.4 ±0.7 MHz and for the 205Tl they found an HFS of 789.2

± 0.6 MHz [9]. Just as the 7P1/2, the 8P1/2 results will also help test the accuracy and

guide the refinement of the atomic theory used to approximate quantities that depend on

the wavefunctions of electrons.

5.2 8P1/2 Simulated Spectra

5.2.1 Peak Approximation

The spectra are simulated using Lorentzians of equal natural line width to approximate the

shape of the peaks corresponding to the absorption transitions for the same reason that we

fit Lorentzians to the IR spectra. These HFS values are much smaller than the HFS of the

50

CHAPTER 5. THE 8P1/2 EXPERIMENT 51

7P1/2 state. For consistent precision with the 7P1/2 experiment, we need to have a FP cavity

with a different FSR and smaller frequency modulated sidebands from the EOM.

5.2.2 Single Isotope Spectra

Figure 5.1: Simulated 205 Spectrum

0 500 1000 1500 2000 2500 30000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency (MHz)

Nor

mal

ized

Pea

k H

eigh

ts

A

B

C

D

E

F

Figure 5.1 is a simulated spectrum of 205Tl taken by scanning the red laser 3 GHz. The

simulated spectrum of 203Tl is exactly the same except the space between peaks B and E

are smaller by roughly a few tens of MHz. Except for the smaller frequency splittings, this

spectrum is identical to the 7P1/2 single isotope spectra. Peak B is the resonance peak of the

7S1/2(F ′ = 1)→ 8P1/2(F” = 0) transition. Peak E is the resonance peak of the 7S1/2(F ′ = 1)

→ 8P1/2(F” = 1) transition. Peaks A, C, D, and F represent the upshifted and downshifted

peaks due to the EOM. The equations for the peak positions are as follows:

νA = νB − fEOM (5.1)

νB = fo −3

4H8p,205 −

1

4H7s,205 (5.2)

νC = νB + fEOM (5.3)

νD = νE − fEOM (5.4)

νE = fo +1

4H8p,205 −

1

4H7s,205 (5.5)

νF = νE + fEOM (5.6)


where fEOM is the sideband frequency of 200 MHz. H8p,205 is the hyperfine splitting for

the 8P1/2 state of 205Tl and H7s,205 is the hyperfine splitting for the 7S1/2 state for 205Tl.

The heights of the frequency modulated (FM) sideband peaks relative to their corresponding

resonant frequency peaks are defined by Bessel functions. Here I have assumed a so-called

“modulation depth” of 1.

5.2.3 Dual Isotope Spectra

Figure 5.2: Simulated Dual Isotope Spectrum

The solid lines represent the part of the spectrum due to the CO configuration of the UV and red laserbeams. The dashed lines represent the part of the spectrum due to the CTR configuration.

0 500 1000 1500 2000 2500 3000 35000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency (MHz)

Nor

mal

ized

Pea

k H

eigh

ts

A

B

C

D

E

F

G

H

The dual isotope spectrum, figure 5.2, was constructed by taking the single isotope equa-

tion for each isotope, without the sidebands, and offsetting each by the doppler shift each

isotope encounters in both CO and CTR configurations. Because the frequency splittings in

the hyperfine structure of the 8P1/2 state are smaller than those of the 7P1/2 state, the order

of the peaks in the dual isotope 8P1/2 spectrum is not the same as the order of the peaks in

the 7P1/2 spectrum. The equations for the frequencies of these peaks are:

Peak Equation Prop. Configuration Hyperfine Transition IsotopeνA = fo − 3

4H8p,205 − 14H7s,205 − δf205 CO F ′ = 1→ F” = 0 205

νB = fo − 34H8p,203 − 1

4H7s,203 − I7s−8p − δf203 CTR F ′ = 1→ F” = 0 203νC = fo + 1

4H8p,205 − 14H7s,205 − δf205 CO F ′ = 1→ F” = 1 205

νD = fo − 34H8p,205 − 1

4H7s,205 + δf205 CTR F ′ = 1→ F” = 0 205νE = fo + 1

4H8p,203 − 14H7s,203 + I7s−8p − δf203 CTR F ′ = 1→ F” = 1 203

νF = fo − 34H8p,203 − 1

4H7s,203 + I7s−8p + δf203 CO F ′ = 1→ F” = 0 203νG = fo + 1

4H8p,205 − 14H7s,205 + δf205 CTR F ′ = 1→ F” = 1 205

νH = fo + 14H8p,203 − 1

4H7s,203 + I7s−8p + δf203 CO F ′ = 1→ F” = 1 203


where fo is the transition frequency of 7S1/2 → 7P1/2 for 205Tl. I7s−8p is the isotope shift

between the two isotopes for the transition 7S1/2 → 8P1/2. When we fit the actual data for

this experiment in the future, there may be some error associated in the determination of

the positions of peaks E and F because they appear very close together in this simulated

spectra. The calculations of the hyperfine splittings, the isotope shift and the Doppler shifts

are also different:

HFS205,1 = νC − νA = νCA (5.7)

HFS205,2 = νG − νD = νGD (5.8)

HFS203,1 = νE − νB = νEB (5.9)

HFS203,2 = νH − νF = νHF (5.10)

I7S−7P =3 |

νE + νH

2−νG + νC

2| + |

νF + νB

2−νA + νD

2|

4+HFS7S,203 −HFS7S,205

4(5.11)

5.3 Current State of Experiment and Future Work

The first thing to do is to get the 671 nm laser lasing and tuning in a single mode fashion.

This will be done by focusing the spontaneous emission that comes from the laser diode so

that the beam does not diverge much at least a meter from the source. The diffraction grat-

ing must be angled ≈ 54° and then carefully aligned so that they first-order diffraction beam

and the output of the laser diode spacially overlap well. Once the 671 nm laser is lasing,

we will check its wavelength using the wavemeter and then tune it to 671 nm. We will also

check that the laser is lasing in single mode using the spare FP cavity that is coated for this

wavelength from the previous indium experiment. One will keep adjusting and tweaking the

temperature and current of the laser until it can scan at least ≈ 5 GHz across the 671 nm

7S1/2 → 8P1/2 resonance.

Next we will have to get the properly sized 200 MHz synthesizer for the EOM for 671nm

laser. Once this is ordered, it will be simple to align the beam through the EOM to get the

maximum power through the device.

The FSR of the FP cavity from the 7P1/2 experiment will not be sufficient for the 8P1/2

experiment. We would like to have an FSR of 300 MHz. The HFS of both isotopes is smaller,

so we need to have a smaller FSR. We have the longer rods cut to increase the size of the FP

cavity. However, we do not have the correct mirrors for this length. Our original plan was to

get plano-concave lens blanks with a focal length of 25 to 30 cm when coated. Because the

evaporator broke, we do not have the capability to coat lens blanks. Therefore, we will need


to purchase two mirrors of this specification. They can have a diameter of either an inch or

half and inch because we currently have mirror mounts for one inch diameter mirrors with

shop-made mounts for half inch diameter mirrors.

Before we can take data for the 8P1/2 experiment, we need a Labview program for the data

acquisition. Our computer crashed and we lost the program used for the data acquisition for

the 7P1/2 experiment. Fortunately, there is a copy of a similar program that can be modified

to fit the needs of our experiment, so we do not have to start from scratch. The Matlab data

analysis code described in appendix A is ready to analyze the 8P1/2 data once it is taken. As

long as the Labview program creates a data text file for each run containing the count num-

ber as the first column, the FP signal voltage as the second column, and the lock-in amplifier

voltage (from photodiode of red laser) as the third column. Each file name must specify the

run number, the propagation configuration and the laser sweep direction in its title, such as

88Copupscan.txt. All of the text files for each data run for the same set should be automati-

cally put in a folder labeled with the date, isotope, and data set number, such as Thu, Jul 18,

2013 205 1. Following these guidelines will ensure easy use of the Matlab data analysis code.

Appendix A

Matlab Code

This code was a collaboration of efforts from Nathan Schine, David Kealhofer, Dr. Gambhir

Ranjit, and myself. The following code was used to take the data from the 7P1/2 experiment

and find the hyperfine splitting for both isotopes, as well as the isotope shift. The results

of this code were used in the analysis of the 7P1/2 experiment described in chapter 4. This

code will be easily modifiable to analyze the 8P1/2 data that will be taken in the future.

A.1 ThalliumFitting.m

All of the code in the following section can be found in the “MATLAB” folder of the mid-

dle computer. This code takes in a matrix of data from the experiment as three columns

of numbers. The first column numbers each data point taken. The second column is the

data from the lock-in amplifier. The third column is the data from the Fabry-Perot of

the 671 nm laser. Lastly, if there is a fourth column, it holds the frequency modulation

signal data. Each data set must have the following filename exactly including the spaces:

Z : \Data Thallium\DayofWeek, Month Day, Y ear TypeofData TrialNumber\. It is a

file that contains all of the scans from a certain trial. DayofWeek is a three letter abbrevia-

tion for the day of the week with the first letter capitalized. Month is the first three letters

of the month also with the first letter capitalized. Day is the number of the day. Year is

the year in four numbers. TypeofData is “203” for 203Tl single isotope data, “205” for 205Tl

single isotope data, and “235” for dual isotope data.

The output of this code is a matrix, Z, with the peak positions and their standard devia-

tions. Z also has a goodness of fit parameter to determine how well the polynomial fits the

data taken. This code can be run for the single isotope 205 or 203 data as well as the dual

isotope data. The only change with the dual isotope data is to use “VoigtFitThallium235.m”

instead of “VoigtFitThallium.m.”

The rest of this file names “FName” corresponding to the configuration of the lasers.

55

APPENDIX A. MATLAB CODE 56

“n1” is the number of data scans in the data trial folder, so you must change the second

number for each data trial file you run through this code. The even “n1’s” mean the data

from a scan in the copropagating laser configuration and the odd ones means the data is

from a scan in the counterpropagating laser configuration. For each “n1” in the data trial

folder, there are two files: the first is for the down scan of the piezo and the second is for

the up scan of the piezo. “n2” differentiates down scan, 1, from up scan, 2. The rest of this

code calls the other files to be mentioned in the following subsections. This code is executed

for each of the subfiles in the main data trial file by looping through n1 and n2.

A.1.1 getdataThallium.m

This code opens each file and reads the three columns of data. If your data has more or less

than three columns you must add of take away “%f” for each column in your data file. If

you have more than 2 headerlines, change the “2” in the ”data” declaration accordingly.

A.1.2 downsampleAndNormalizeThallium.m

This code decreases the number of data points in each data file, because in order to get good

fits later in the “ThalliumFitting.m” code, you must decrease the number of data points. It

also makes the fitting part of the program run faster. If you have to edit this part of the

code, you must be weary of getting rid of too many points for a good fit.

First, the x-axis is normalized, resulting in the column vector “x-Norm” from -1 to 1.

Then we choose a factor by which to down-sample the data . This is manifested in the vari-

able ”fmDS factor” for the frequency modulation (FM) data if it exists, “fpDS factor” for

the Fabry-Perot (FP) data, and “saDS factor” for the spectrum analyzer (SA) data. What

ever number you choose for each down-sample factor, n, the “downsample” method will keep

every nth data point starting from the first point. The down-sampled data is then plotted

versus the normalized x-axis, down-sampled by the corresponding factor for the FM data,

the FP data, and the SA data.

A.1.3 FabryPerotFittingThallium.m

This program uses “sympeaksThallium.m” to get the peak heights and locations (on the

frequency axis) of the down-sampled FP data (x fp DS and fp DS). In lines 12 through 25,

the loop is executed if the first peak is in a position greater than or equal to “nPoints” and

the last peak is in a position less than or equal to a position “nPoints” from the end of the


graph. “ind11” is the position “nPoints” before the first peak, so “xBegin” is just the posi-

tion “nPoints” units before the first peak. Similarly, “xEnd” is defined as the position of the

last peak plus “nPoints.” “indx fp” is then defined as the portion of “x fp DS,” the x-axis

values, between “xBegin” and “xEnd.” Then “fp” is defined as the FP data corresponding

to the x-axis values of “indx fp” and “xfp” is defined as the values of “x fp DS” equivalent

to the positions “indx fp.” Finally, “Uin” is defined as the matrix made of the two column

vectors: peak positions and peak heights (the same as UinFP). Lastly, “N” is defined as

“NFP,” which is the number of Fabry-Perot peaks in the scan.

The if condition of lines 28 through 42 is similar to the last one except it is executed if the

first peak occurs in a position before “nPoints.” “ind11” is defined as the position “nPoints”

before the second peak and “xBegin” is defined as the position “ind11.” “xEnd” is the same

as it was in the previous if statement. Lines 36 through 40 are the same as the previous

if statement. Then “Uin” is defined as matrix of peak position and heights of the second

through last peak.

The third similar if condition is executed if the first peak position is greater than or equal

to “nPoints” and the last peak position is greater than or equal to “nPoints” before the end

of the scan. “ind11” and “xBegin” are the same as in the first if statement. However, this

if statement contains another if statement. The first part of the statement is executed if the

“xfpPeak” value, peak position, for the last peak minus that of the second to last peak is

less than or equal to half of the difference of the x values of the middle peak and the peak

in front of it. “xEnd” is defined as “nPoints” after the third to last peak. “Uin” is defined

as the matrix of peak positions and heights of all of the peaks except the last two. If the

second part of the if statement is executed, it is similar to the first part except “Uin” is the

matrix of peak positions and heights of all the peaks but the last one.

Next, this program plots “xfp” vs. “fp” and then fits the FP data to a sum of Lorentzians

based on how many FP peaks there are, “N.” The bounds for this fit are set in lines 89

thorough 94, where “I” is defined as the vector of ones with length N. Lines 96 through 178

have the actual parameters to fit the FP peaks to sums of Lorentzians and then these fits

are plotted with their corresponding RMSE values. These plots are only saved if the RMSE

value, “gdFP,” is less than “goodnessThreshold” defined at the beginning of ThalliumFit-

ting.m. The “b” coefficients are also saved as vector “FPoints.” If the RMSE value is not

greater than “goodnessThreshold,” then the whole program is called again.

sympeaksThallium.m

This method takes the frequency (x value) and Fabry-Perot height data as well as and av-

eraging number and minimum peak height for each FP peak. It outputs the peak heights


(pks) and their frequency locations (locs).

A.1.4 FrequencyLinearizationThallium.m

This script first plots the “b” Lorentzian coefficients versus a frequency axis spacing the

points 500 MHz apart, which is the FSR of the Fabry-Perot. The code then fits a fourth

order polynomial to the data and plots this, saving it in a certain file.

A.1.5 VoigtFitThallium.m

This program first uses sympeaksThallium.m to get the positions and heights of each of the

six peaks of the single isotope spectra. Lines 8 through 19 run change the “avg” and rerun

sympeaksThallium.m until exactly 6 peaks are found. Then it plots these peaks, where “fre-

qSIG” are the corresponding x values for the down-sampled signal data, “sig DS.” Before

this plot is created, the signal data is fit to a sum of Lorentzians. The output of this fit is

the fit results, “cf SIG,” and the goodness of fit, “gd SIG.” The fit results are then plotted

in the same frame as the down-sampled signal data. The “a” and “b” coefficients from the

Lorentzian model are declared. Finally, an if statement is executed and will print the matrix

Z of position and standard deviation of each of the six peaks if the goodness of fit parameter

is less than “goodnessThreshold” defined in ThalliumFitting.m. The standard deviation of

each peak position is calculated by subtracting the confidence bounds of the coefficients of

the fit from each of the “b” coefficients. If the goodness of fit parameter is not sufficient, the

“goodnessThreshold” is increased by 0.01 and the program is called again to refit the peaks.

A.1.6 VoigtFitThallium235.m

The same as VoigtFitThallium.m, except it fits the 8 peaks of the dual isotope spectra.

Finally, the data and plots are stored in files in writetofile.m and writetofile235.m, where

the vector Z containing the peak positions and standard deviation of these positions.

A.2 DataAnalysisThallium.m

This next program takes the matrix Z containing the peak positions and the error associated

with each peak and takes the difference of certain peaks to find the hyperfine splitting and


isotope shift values.

Appendix B

7P1/2 Data Tables

This is a collection of data tables for every day data was taken that averages certain param-

eters for each set of data. Every thing is measured in MHz. The length of each data set is

the number of scans averaged in this set. Config. stands for beam configuration and scan

direction. The elements of this consist of copropagation upscan (CO UP), copropagation

downscan (CO DN), counterpropagating upscan (CTR UP), and counterpropagating down-

scan (CTR DN). Avg. Sideband is the average size of the four sidebands in each scan. SB

SD is the standard deviation of this average. CF is the correction factor calculated by the

average sideband size and CF SD is its standard deviation. Raw HFS is the raw average of

HFS values and Raw SD is its standard deviation. Corr. HFS is the corrected HFS values

by the CF and Corr. SD is its standard deviation.

05/13/2013

Config. Avg. Sideband SB SD CF CF SD Raw HFS Raw SD Corr. HFS Corr. SD

Set 1: length = 247

CO UP 599.225069 0.027793 1.001293 0.000046 2150.554786 0.020544 2152.267926 0.088513

CO DN 598.308688 0.033373 1.00282 0.000056 2147.596188 0.040923 2151.723277 0.162924

CTR UP 599.662091 0.021113 1.000563 0.000035 2149.906829 0.016945 2151.859674 0.070216

CTR DN 598.976999 0.019857 1.001708 0.000033 2147.110487 0.025404 2151.121266 0.101192

Set 2: length = 94

CO UP 597.914177 0.022403 1.003488 0.000038 2149.623203 0.018932 2153.098954 0.071570

CO DN 598.882208 0.044142 1.001866 0.000074 2148.234745 0.060526 2150.924963 0.223707

CTR UP 598.274420 0.018616 1.002884 0.000031 2149.098746 0.016811 2152.665342 0.062481

CTR DN 599.414904 0.034663 1.000976 0.000058 2147.126528 0.051246 2150.634060 0.186566

05/14/2013


Set1 has bad difference calculations for CO UP an CO DN

Set 2: length = 292

CO UP 597.668583 0.012846 1.003901 0.000022 2147.994169 0.010715 2152.784888 0.039354

CO DN 597.308918 0.045438 1.004505 0.000076 2140.566269 0.036129 2149.852594 0.156578

CTR UP 598.275656 0.008254 1.002882 0.000014 2148.947074 0.00715 2153.034756 0.026313

60

APPENDIX B. 7P1/2 DATA TABLES 61

CTR DN 597.968929 0.043240 1.003397 0.000073 2141.515923 0.036123 2150.208305 0.154118

07/01/2013


Set 1: length = 64

CO UP 599.423061 0.050822 1.000962 0.000085 2151.762938 0.037879 2151.569539 0.152954

CO DN 598.606943 0.311394 1.002327 0.000521 2153.217411 0.293636 2151.934021 1.176743

CTR UP 599.684768 0.049406 1.000526 0.000082 2151.436736 0.038251 2151.007109 0.149958

CTR DN 599.178493 0.220097 1.001371 0.000368 2153.621526 0.222758 2151.250065 0.857648

Set 2: length = 120

CO UP 599.173903 0.018008 1.001379 0.000030 2152.412765 0.013353 2151.423701 0.055886

CO DN 598.121032 0.177471 1.003141 0.000298 2155.195613 0.161450 2152.161202 0.678184

CTR UP 599.244300 0.024801 1.001261 0.000041 2152.746965 0.019563 2151.806923 0.077985

CTR DN 598.722981 0.190146 1.002133 0.000318 2155.707720 0.187128 2152.662616 0.752259

Set 3: length = 328

CO UP 600.107493 0.022261 0.999821 0.000037 2150.286246 0.013109 2150.229782 0.077870

CO DN 600.091609 0.045469 0.999847 0.000076 2149.932103 0.070281 2149.745355 0.193623

CTR UP 600.851333 0.020837 0.998583 0.000035 2149.499806 0.013350 2149.372054 0.071273

CTR DN 600.800063 0.031840 0.998668 0.000053 2149.392668 0.053384 2149.190490 0.139073

07/2/2013


Set 1: length = 84

CO UP 596.607217 0.034790 1.005687 0.000059 2149.184931 0.032376 2154.088967 0.115989

CO DN 596.876638 0.065814 1.005233 0.000111 2148.583967 0.049704 2153.464050 0.188334

CTR UP 596.603610 0.033319 1.005693 0.000056 2149.882461 0.033197 2154.633041 0.116926

CTR DN 596.963545 0.064307 1.005086 0.000108 2149.206953 0.050241 2153.559623 0.186553

Set 2: length = 202

CO UP 599.546569 0.018125 1.000756 0.000030 2150.958436 0.014640 2151.224647 0.056773

CO DN 600.557618 0.040368 0.999071 0.000067 2149.473709 0.032139 2149.135116 0.121732

CTR UP 599.559218 0.017824 1.000735 0.000030 2150.528298 0.015144 2150.954341 0.058215

CTR DN 600.941263 0.040860 0.998434 0.000068 2149.188763 0.033912 2148.773994 0.126602

07/03/2013


Set 1: length = 30

CO UP 599.383129 0.034596 1.001029 0.000058 2155.317110 0.035989 2154.016160 0.128041

CO DN 600.232584 0.108541 0.999613 0.000181 2149.680665 0.088861 2150.242247 0.386978

CTR UP 599.960962 0.031784 1.000065 0.000053 2156.172632 0.035614 2154.250982 0.124363

CTR DN 600.531930 0.101965 0.999114 0.000170 2150.374150 0.089672 2149.786965 0.376044

Set 2: length = 150

CO UP 598.956705 0.015341 1.001742 0.000026 2153.886953 0.016074 2154.587980 0.056244

CO DN 600.114229 0.028806 0.999810 0.000048 2153.969431 0.026717 2153.430390 0.111268

CTR UP 599.355429 0.010075 1.001075 0.000017 2154.168483 0.011439 2154.592703 0.039092

CTR DN 600.414993 0.040069 0.999309 0.000067 2153.763547 0.039820 2152.962339 0.162665

Set 3: length = 270

CO UP 600.048329 0.015164 0.999919 0.000025 2148.369572 0.013931 2148.558211 0.055215

CO DN 599.915111 0.022433 1.000142 0.000037 2154.800503 0.020976 2152.626423 0.076788


CTR UP 600.246491 0.014557 0.999589 0.000024 2148.086145 0.014345 2148.310202 0.055752

CTR DN 599.886514 0.022091 1.000189 0.000037 2154.791207 0.022069 2152.600667 0.079152

Table B.1: 203Tl Raw Data Sets by Date

03/15/2013

Config. Avg. Sideband CF Raw HFS Raw SD Corr. HFS Corr. SD

Set 1

CTR UP 598.59 1.00235 2171.09 0.65 2177.88 0.58

CO UP 599.10 1.00151 2172.77 0.58 2174.088 0.58

CTR DO 598.72 1.00214 2170.81 0.58 2175.33 0.63

CO DO 598.83 1.00195 2170.68 0.63 2175.57 0.61

03/15/2013

Config. Avg. Sideband CF Raw HFS Raw SD Corr. HFS Corr. SD

Set 1

CTR UP 598.92 1.0018 2170.67 0.15 2174.58 0.15

CO UP 598.72 1.00214 2171.7 0.23 2176.35 0.23

CTR DO 599.05 1.00158 2170.52 0.16 2173.95 0.16

CO DO 598.75 1.00209 2171.83 0.24 2176.37 0.24

Set 2

CTR UP 599 1.00166 2168.79 0.16 2172.39 0.16

CO UP 598.59 1.0024 2167.04 0.17 2172.24 0.17

CTR DO 599.03 1.00163 2169.65 0.18 2173.19 0.18

CO DO 598.61 1.00232 2168.34 0.19 2173.37 0.19

Set 3

CTR UP 599.01 1.00165 2169.00 0.10 2172.58 0.10

CO UP 598.78 1.00204 2167.28 0.11 2171.70 0.11

CTR DO 599.16 1.00141 2168.93 0.10 2171.99 0.10

CO DO 598.87 1.0019 2167.23 0.13 2171.35 0.13

Set 5

CTR UP 597.37 1.0044 2167.89 0.20 2177.43 0.20

CO UP 597.39 1.00437 2166.00 0.18 2175.47 0.18

CTR DO 597.63 1.00397 2168.04 0.43 2176.65 0.43

CO DO 597.64 1.00394 2166.36 0.34 2174.90 0.34

Set 7

CTR UP 600.29 0.99951 2170.26 0.21 2169.20 0.21

CO UP 600.13 0.99979 2168.35 0.24 2167.89 0.24

CTR DO 600.77 0.99873 2169.67 0.2 2166.91 0.20

CO DO 600.46 0.99923 2168.00 0.19 2166.33 0.19

Table B.2: 205Tl Raw Data Sets by Date


Table B.3: Dual Isotope Data

Date Raw Isotope Shift (MHz) Std. Dev. (MHz) Corr. Factor Corrected Isotope Shift (MHz)5/16/13 534.65 0.43 1.002 535.725/17/13 533.23 0.72 1.002 534.305/20/13 534.16 0.73 1.002 535.235/23/13 540.16 0.23 1.002 541.246/27/13 530.25 0.20 1.002 531.316/28/13 530.98 0.19 1.002 532.04

Appendix C

Detailed Description and Usage of the

671 nm Laser

The AR coated laser diode, collimating lens, and diode mount all came from the company

Toptica Photonics. The following papers are some specifications about the AR coated diode.

Following these drawings is a labeled picture of the laser in its current form.

64

APPENDIX C. DETAILED DESCRIPTION AND USAGE OF THE 671 NM LASER 65


Figure C.1: External Laser CavityThis is an overhead picture of the current red laser. The coated diode and surrounding confocal lens (1) lieinside a mount (2) sending the beam at the center of the diffraction grating (3) that has 1200 lines per mm.

We mount the diffraction grating on a 45° metal wedge (4) attached to a mirror mount (5). The metalwedge is near the angle for the first order diffracted beam to leave the laser cavity perpendicular to thecavity. One of the tuning nuts is replaced by a piezoelectric driver (6) on the mirror mount (5). The

piezoelectric transducer (PZT) is hooked up to a signal generator (7). The SRS LDC501 current hookup(9) and temperature control hookup (8) are both female 9 pin connectors. A thermoelectric device

underneath the laser cavity changes the temperature of the laser. The thermometer (10) for this device is inthe laser bed closest to the diode mount (2), because the temperature that we really care about is the

temperature of the diode. We can angle the diffraction grating vertically using the knob(12).


The next five figures are the CAD drawings of the laser pieces. These CAD drawings

were created by Sarah Peters ’14. The first is the original diode and collimating lens mount.

We did not use this in the final laser cavity because we bought a coated laser diode from

Toptica Photonics that came with a collimating lens and mount. If these drawings are used

to build another ECDL laser in the Littrow configuration, make sure you make mount for

the diode and collimating lens such as this. All of the pieces, except as noted, are built out

of aluminum. The following details about each piece of the laser are adopted from the work

of Sarah Peters ’14.

The Base

We used a scrap piece of aluminum bar about 3 wide for the base, or thermal reservoir.

This does not have to have a precise size, it just needs to be wider than the diffraction grat-

ing mount. It needs to have tapped holes to secure the track and to secure it to the table.

The holes that attach it to the optical table will need to be spaced by integer multiples of

an inch, due to the spacing of holes on standard optical tables. Additionally, two tapped

holes are needed to attach the plastic plate that holds the connections to the electric circuitry.

The Track

The track is an aluminum rectangle with a 14” notch cut at one end to allow the laser

mount to attach and slide horizontally. We ended up not even using this notch because we

got a new diode later in the year that had its own laser mount. We attached this laser mount

by cutting a 45°notch to fit the bottom of the mount and tapped two holes for the screws it

came with. We then chamfered the edges to allow for sliding fit of the mirror mount. Holes

to attach the track to the base were drilled and tapped. Lastly, we drilled a small hole in the

side, near the laser mount, to fit the temperature sensor with thermal-electric grease/epoxy.

The Laser Mount: Figure C.2

The central hole for the tube containing the diode and collimating lens was created using

an adjustable reamer to ensure a precise fit of the odd size. In order to secure the tube in

place, we cut a slit at the top and drilled and tapped two holes so that two setscrews can

tighten the drilled hole for the tube. On the bottom leg of the mount, we drilled and tapped

a hole so that we can secure the mount with a setscrew once a suitable position on the track

is chosen. The laser mount is 1 thick, which may be excessive can be cut down to 0.5 and

still maintain stability when attached to the track.

Sliding Mount for the Mirror Mount: Figure C.3

The piece is based on the dimensions of the Newport U100-P2K mirror mount. The


lengths of the arms do not need to be exact. However, the holes drilled to match the clear-

ance holes in the mirror mount must be precise. We used a vertical bandsaw to cut the

L-shape of the mount where the mirror mount will attach. We drilled and tapped two holes,

one on each leg, and put in setscrews to secure the mount on the track.

The Wedge: Figure C.4

The wedge attaches the diffraction grating to the mirror mount. It holds the diffraction

grating at a 45°angle. You can adjust the mirror mount using the set screws and the PZT

to get the diffraction grating at the correct angle depending on the wavelength of the laser

diode. The four holes are drilled based on the holes in the mirror mount.

The PZT System: Figures C.5 and C.6

The barrel and piston system that holds the PZT consists of three pieces: the barrel, the

cap, the sliding cylinder. This system ensures reproducible translation of the mirror mount

by the PZT. The cap remains in place. The PZT pushes the sliding cylinder and then the

sliding cylinder will push on the mirror mount. We put a ball bearing inside the end of the

sliding cylinder to eliminate possible horizontal translation.

The barrel is made of brass and has an inner and outer part of the tapped hole. The

outer part of the tapped hole has a setscrew. The inner part is wider and will contain the

PZT and the cylinder. The tapped portion of the hole needs to be done very carefully with a

0.240 drill and a ?-100 tap. It is important to note that the exterior and interior of the barrel

both have two sections with different diameters. These sections are not the same length in

the exterior versus the interior.

The cap and sliding cylinder are made of steel and hold the PZT in place. Each piece

has a slit made to fit the PZT, so to encapsulate it. We feed the electrical leads of the PZT

out of the barrel. The dimensions are based on the 3.5 × 4.5 × 5mm piezoelectric actuator

from Thorlabs (Part Number AE0203D04F). However, if one decides to reproduce this laser,

one must measure the exact dimensions of the PZT and decide upon a depth for the slits on

the cap and sliding cylinder. We made sure that the pieces had a snug fit, so that no other

axial translation is possible. The other end of the cap is flat. The other end of the sliding

cylinder has a conical hole the ball bearing is placed.

To assemble this system, we drilled holes a bit larger than 0.04 to fit the wires through

the mirror mount and barrel. It is important to choose a place for this hole on the mirror

mount that will not damage it. We chose to put the hole 0.30 from the outermost edge of

the mirror mount. We then had to disassemble the Thorlabs mirror mount by removing one

end of the springs that holds the two faces together. We removed the commercial barrel and

setscrew and replaced them with barrel. The setscrew we used was a 100TPI setscrew. We


placed the flat face of the cap against the setscrew. Then we gently pulled the PZT electric

leads through the barrel holes and fit the PZT head into the slit of the cap. We fit the sliding

cylinders slit side against the other side of the PZT head. Finally, we put a ball bearing on

the other side of the cylinder and reassembled the mirror mount.


Figure C.2: Mount for the Laser Diode and Collimating Lens


Figure C.3: Sliding Mount for the Mirror Mount


Figure C.4: Wedge Diffraction Grating Mount


Figure C.5: PZT Tube


Figure C.6: PZT Cap

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