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The lamb shift experiment in muonic hydrogen [Elektronische Ressource] / by Aldo Sady Antognini

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The Lamb Shift Experimentin Muonic HydrogenDissertationsubmitted to the Physics Facultyof the Ludwig{Maximilians{University MunichbyAldo Sady Antogninifrom Bellinzona, SwitzerlandMunich, November 2005st1 Referee : Prof. Dr. Theodor W. H anschnd2 Referee : Prof. Dr. Dietrich HabsDate of the Oral Examination : December 21, 2005Even ifI don’t think,I am.Itsuo TsudaJe suis ou je ne pense pas,je pense ou je ne suis pas.Jacques LacanA mia mamma e mio papacon tanto amoreAbstractThe subject of this thesis is the muonic hydrogen ( p) Lamb shift experiment beingperformed at the Paul Scherrer Institute, Switzerland. Its goal is to measure the 2S 2Penergy di erence in p atoms by laser spectroscopy and to deduce the proton root{mean{3square (rms) charge radius r with 10 precision, an order of magnitude better thanppresently known. This would make it possible to test bound{state quantum electrody-7namics (QED) in hydrogen at the relative accuracy level of 10 , and will lead to animprovement in the determination of the Rydberg constant by more than a factor ofseven. Moreover it will represent a benchmark for QCD theories.The experiment is based on the measurement of the energy di erence between theF=1 F=22S and 2P levels in p atoms to a precision of 30 ppm, using a pulsed laser tunable1=2 3=2at wavelengths around 6 m. Negative muons from a unique low{energy muon beam are1stopped at a rate of 70 s in 0.6 hPa of H gas.

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The Lamb Shift Experiment
in Muonic Hydrogen
Dissertation
submitted to the Physics Faculty
of the Ludwig{Maximilians{University Munich
by
Aldo Sady Antognini
from Bellinzona, Switzerland
Munich, November 2005st1 Referee : Prof. Dr. Theodor W. H ansch
nd2 Referee : Prof. Dr. Dietrich Habs
Date of the Oral Examination : December 21, 2005Even if
I don’t think,
I am.
Itsuo Tsuda
Je suis ou je ne pense pas,
je pense ou je ne suis pas.
Jacques Lacan
A mia mamma e mio papa
con tanto amoreAbstract
The subject of this thesis is the muonic hydrogen ( p) Lamb shift experiment being
performed at the Paul Scherrer Institute, Switzerland. Its goal is to measure the 2S 2P
energy di erence in p atoms by laser spectroscopy and to deduce the proton root{mean{
3square (rms) charge radius r with 10 precision, an order of magnitude better thanp
presently known. This would make it possible to test bound{state quantum electrody-
7namics (QED) in hydrogen at the relative accuracy level of 10 , and will lead to an
improvement in the determination of the Rydberg constant by more than a factor of
seven. Moreover it will represent a benchmark for QCD theories.
The experiment is based on the measurement of the energy di erence between the
F=1 F=22S and 2P levels in p atoms to a precision of 30 ppm, using a pulsed laser tunable1=2 3=2
at wavelengths around 6 m. Negative muons from a unique low{energy muon beam are
1stopped at a rate of 70 s in 0.6 hPa of H gas. Highly excited p atoms are formed, and2
most of them promptly deexcite to the ground state within100 ns. However, there is a
roughly 1% probability that long{lived p atoms with a lifetime of 1:3 s are formed.2S
An incoming muon triggers a pulsed, multi{stage laser system which delivers 0:2 mJ
1per pulse at ’ 6 m with 55 s repetition rate. It consists of two XeCl excimer lasers
followed by dye lasers which pump an oscillator{ampli er frequency{controlled Ti:Sa laser.
Its 6 ns long pulse at 708 nm is then frequency shifted to 6 m via third Stokes production
in a Raman cell lled with hydrogen. The laser pulse has a delay of about 1:5 s with
respect to the prompt muon cascade.
If the laser is on resonance, it induces 2S 2P transitions. The subsequent deexcitation
to the 1S state emits a 1.9 keV K x ray which is detected by large area avalanche
photodiodes. The resonance frequency, and hence the Lamb shift and r , are determinedp
by measuring the intensity of these x rays as a function of the laser wavelength.
A search for the 2S 2P resonance line was performed in November 2003 when a broad
range of laser frequencies was scanned (49:7409 49:8757 THz), corresponding to proton
radii between 0.844 and 0.905 fm. The result of the data analysis is that no signi can t
2S 2P resonance was observed. The negative result is with high probability due to the
low statistics and not to an incorrect search region.
The rst part of this thesis reports on the present status of the Lamb shift theory in
p. Following, there is a detailed description of the apparatus and analysis of the data.
An estimate of the present and future laser{induced event rates are given, together with a
study of the present and future background. In the Appendices are discussed: the energy
levels in H, the proton radius de nition, the relevance of this experiment, the 2S state
population and lifetime, and the spectroscopic properties of the 2S 2P transition.
iiiContents
Abstract i
Contents ii
List of Tables vi
List of Figures vii
1 Overview of the muonic hydrogen Lamb shift experiment 1
2 Present status of the 2S 2P Lamb shift in muonic hydrogen 7
2.1 Vacuum polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Finite nuclear size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Relativistic recoil corrections . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Fine and hyper ne structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
F=2 F=12.5 The E(2P 2S ) energy splitting . . . . . . . . . . . . . . . . . . . 143=2 1=2
3 Muon beam, target, and electronics 17
3.1 Low{energy muon beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.1.1 Cyclotron trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.1.2 Muon extraction channel . . . . . . . . . . . . . . . . . . . . . . . . 22
3.1.3 The 5 T solenoid with the muon detector . . . . . . . . . . . . . . . 23
3.2 Gas target . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2.1 Detectors for the 1.9 keV energy x rays: the LAAPDs . . . . . . . . 27
3.2.2 Electron detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2.3 Anti{coincidence detector . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2.4 Intermezzo about foils . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.3 Electronics of the data acquisition system . . . . . . . . . . . . . . . . . . . 33
4 The laser system 35
4.1 Excimer lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 Dye lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.3 Continuous wave Ti:Sa laser . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.3.1 Wavelength control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.3.2 Frequency stabilization . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.4 Pulsed Ti:Sa oscillator and ampli er . . . . . . . . . . . . . . . . . . . . . . 44
iii4.4.1 Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.4.2 Injection seeding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.4.3 Chirp in the Ti:Sa oscillator . . . . . . . . . . . . . . . . . . . . . . . 46
4.4.4 Ti:Sa ampli er . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.5 Raman cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.6 Q (1) Stokes{shift in H . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5501 2
4.7 Water absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.8 Frequency calibration of the 6 m light . . . . . . . . . . . . . . . . . . . . 57
4.9 FP calibration and stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.10 Summary of the frequency control of our laser system . . . . . . . . . . . . 62
4.11 The 6 m multipass cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.12 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5 Measurements 71
5.1 Analysis of the waveform digitizer signals . . . . . . . . . . . . . . . . . . . 71
5.2 Event classi cation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.2.1 Signal versus particle identi cation . . . . . . . . . . . . . . . . . . . 73
5.2.2 Event construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.3 X-ray and electron energy spectra: K energy cut and delayed electron cut 74
5.4 Second{muon cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.5 Electron time spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.6 Time calibration and resolution . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.7 LAAPDs x-ray e ciency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.7.1 Detector e ciency " . . . . . . . . . . . . . . . . . . . . . . . . . . 83x
5.7.2 Total e ciency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85x
6 Search for the 2S 2P resonance 87
6.1 \Laser ON" and \Laser OFF" data . . . . . . . . . . . . . . . . . . . . . . . 87
6.2 2 keV x-ray energy and time spectra . . . . . . . . . . . . . . . . . . . . . . 87
6.3 Resonance line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.4 Problems during 2003{run . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.5 Event and background rate in 2003 beam time . . . . . . . . . . . . . . . . 95
7 Future improvements of the apparatus 97
7.1 Thin{disk laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
7.2 Future event rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.3 Background rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
7.4 Possible background reduction . . . . . . . . . . . . . . . . . . . . . . . . . 103
7.5 Measuring time and search of the resonance . . . . . . . . . . . . . . . . . . 104
7.6 Future extension of the experiment . . . . . . . . . . . . . . . . . . . . . . . 106
ivA Theory of hydrogen energy levels 107
A.1 Bohr energy levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
A.2 Dirac energy levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
A.3 The Lamb shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
A.4 Radiative corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
A.5 Perturbative and all{order approaches to the self{energy . . . . . . . . . . . 113
A.6 One{loop self{energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
A.7 Two{loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
A.8 Finite nuclear size and nuclear structure corrections . . . . . . . . . . . . . 118
A.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
B Bound{state QED test and extraction of Rydberg constant 123
B.1 QED test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
B.2 Rydberg constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
B.3 Lamb shift and R uncertainty related to and m=M . . . . . . . . . . . . 1261
B.3.1 Lamb shiftty caused by . . . . . . . . . . . . . . . . . . 126
B.3.2 Uncertainty of R caused by the uncertainty of m=M . . . . . . . . 1261
C Electron{proton scattering experiments 129
C.1 Elastic scattering cross sections . . . . . . . . . . . . . . . . . . . . . . . . . 129
C.2 Measurements and extraction of the nuclear structure . . . . . . . . . . . . 131
C.3 Structure functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
D Proton radius 135
D.1 Proton radius in scattering experiments . . . . . . . . . . . . . . . . . . . . 135
D.2 radius in atomic hydrogen . . . . . . . . . . . . . . . . . . . . . . . . 136
D.3 Problems related to the de nition of the proton radius . . . . . . . . . . . . 136
E 2S 2P transition probability 137
E.1 2S 2P probabilities and matrix elements . . . . . . . . . . . . . 137
E.2 2S 2P linewidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
E.3 Laser intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
E.4 Two{level system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
E.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
F 2S 2P transition systematics 145
F.1 Doppler broadening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
F.2 Zeeman e ect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
F.2.1 Linear versus quadratic Zeeman e ect . . . . . . . . . . . . . . . . . 146
F.2.2 Orbital and spin magnetic moments . . . . . . . . . . . . . . . . . . 146
F.2.3 Zeeman e ect of the hyper ne levels . . . . . . . . . . . . . . . . . . 147
F.2.4 Anomalous Zeeman e ect versus Breit{Rabi solution . . . . . . . . . 148
F.3 Collisional shift and broadening . . . . . . . . . . . . . . . . . . . . . . . . . 149
F.3.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
vF.3.2 2S 2P energy shift . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
F.3.3 Electric eld and interatomic potential . . . . . . . . . . . . . . . . . 151
F.3.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
G Population and lifetime of the 2S state 155
G.1 Muonic hydrogen formation and cascade processes . . . . . . . . . . . . . . 155
G.2 Population of the 2S state . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
G.3 Long and short{lived 2S components . . . . . . . . . . . . . . . . . . . . . . 159
G.4 Lifetime of the 2S long{lived state . . . . . . . . . . . . . . . . . . . . . . . 161
H Background of the 2S 2P resonance 163
H.1 Background from muon transfer to carbon . . . . . . . . . . . . . . . . . . . 164
H.2 Uncorrelated background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
H.3 Total background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
References 175
Acknowledgements 185
vi