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The Fine Structure Constant and Electron (g‐2) Factor
Nick Hutzler Physics 135c Spring 2007
Theory Review
• A (spin) magnetic moment μ quantifies how a particle interacts with magnetic fieldsparticle interacts with magnetic fields
• The electron has μ proportional to spin
• The fine structure constant α characterizes the
hh
h Sg
Sm
eg Bμ==μ 2
strength of the EM interaction
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1371
4 0
2
≈πε
=α ceh
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Theory Review
• The (spin) factor g is dimensionless and varies b ti l tby particle type– Classical charged sphere: g = 1– Dirac point particle: g = 2– Neutron: g = ‐3.826 085 45(90)– Proton: g = 5.585 694 713(46)Proton: g 5.585 694 713(46)
• The anomalous magnetic moment is defined as a = (g‐2)/2
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Theory Review
• Why measure g?– Probe of possible internal structure– One of best tests of QFT/QED, new physics– Fine structure constant α determined by g+QED
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Theory Review
• Why isn’t g = 2?T l l F– Tree level Feynman diagram gives g = 2
– Higher order corrections from quantum fluctuations add ~.001
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Theory Review
• g and α are closely related– a has a series expansion in α (remember that α is commonly expanded in perturbation series)
– The Ci are calculated from QED– The aμτ hadronic weak terms are the QED muon/tau
( ) ( ) ( ) weakhadronic aaaCCCa ++++++= μτπαπαπα L33221
μτ,hadronic,weakcontribution, and non‐QED hadronic and weak contributions (respectively)
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Theory Review
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History
• Earlier αmeasurements– Quantum Hall effect
– Atomic spectra
– Josephson effect
αμ
= 20cRK
hcm
R e2
2
∞
α=
h φ∂
– Recoil measurements, numerology, others
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teh
U∂φ∂
= 2
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Philosophy
• Why does α have this value?Anthropic Principle: small changes in the value of α– Anthropic Principle: small changes in the value of αwould dramatically change the universe
– Numerology: Arthur Eddington calculated α=1/136 using numerology and gave the Eddington Number:
N = 2256/ α– Feynman: “one of the greatest damn mysteries of physics: a magic number that comes to us with nophysics: a magic number that comes to us with no understanding by man”
– James Gilson: α has a formula:
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( ) ( )( )( ) 0359997867137129137
29137137
137.//
/tan/cos≈
⋅π⋅ππ
=α
The Experiment
• We shall discuss a recent measurement of gd lti l l ti fand resulting calculation of α
• New Measurement of the Electron Magnetic Moment Using a One‐Electron Quantum Cyclotron– Odom, Hanneke, D’Urso, GabrielseOdom, Hanneke, D Urso, Gabrielse– Harvard Physics Department– PRL 2006
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The Experiment
• Overview:– Put an electron in cyclotron motion– Measure jumps between lowest quantum states– Calculate g– Calculate α
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Penning Traps
• Static fields used to trap charged particlescharged particles– Uniform axial B field– Quadrupole E field
• Earnshaw’s Theorem: static electric fields alone cannot confine aalone cannot confine a charged particle
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Penning Traps
• Four motions:– Spin (νs)Spin (νs)– Cyclotron (νc = 2 νs/g)– Axial (νz)– Magnetron (νm)
• In the present experiment:– νs ≈ νc’ ≈ 149 GHzs c– νz’ ≈ 200 MHz– νm’ ≈ 134 kHz– Prime denotes “actual”
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Particle Motion
• Spin motion is essentially free• Cyclotron motion is damped by synchrotron radiation (~0.1 s in free space)– However, in this trap the fields are tuned so that cyclotron motion is far from a cavity mode
– Lifetime extended to several seconds
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Energy Levels
• Measure jumps between lowest spin (ms) and cyclotron (n) levels
• νa’ and fc’ are anomaly ( ) ( ) ( )2212
1212 s'cscs m/nhh/nmh
gm,nE ++δ−ν++ν=
and spin‐up frequencies (respectively)
• νc is classical cyclotron frequency eB/2πm
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Energy Levels
• νc is obtained using the Brown GabrielseBrown‐Gabrielse Invariance Theorem:
• This allows us to l l /2
( ) ( ) ( ) ( )2222 'm'z'cc ν+ν+ν=ν
calculate g/2:
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( )( )'c'z'c'c'z'a
c
'c'a
f//ff/g
2232
12 22
ν+δ+ν−ν
+≅νν+ν
=
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What is Measured?
• Spin and cyclotron jumps induce a changejumps induce a change in νz’
• A resonant circuit drives the axial motion at νz’(SEO)
• Quantum Jump• Quantum Jump Spectroscopy
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What is Measured?
• Anomaly transitions are created by inducingcreated by inducing spin‐flips with a perpendicular B field
• Cyclotron transitions are induced by microwavesB h d i l• Both drives are always on, but tuned so that only one affects the electron
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Results
• g/2 = 1.001 159 652 180 85(76)• That’s 76 parts per trillion• Certainly of the most accurate measurements ever made
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Sources of Error
• Magnetic field instabilities– After settling for several months the superconductingAfter settling for several months, the superconducting solenoid is constant to 10‐9/hours
– Maintain in 0.3K environment– Perform measurements at night
• Fitting/statistics– ~50 measurements/night
• Cavity radiation modes• Cavity radiation modes• Blackbody Photon Excitation
– Cool sample to
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Fine Structure Constant
• This g‐value translates toα‐1 = 137.035 999 710(90)(33)[0.66][0.24]
= 137.035 999 710(96)[0.70]
• The top line errors are experimental, QED respectively (second line gives total error)
• Square brackets give error in ppb
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Fine Structure Constant
• C4,C6,C8,C10 require the evaluation of 7 72 891 12 672 F di7,72,891,12 672 Feynman diagrams respectively
• C8 is known fairly well, C10 is still being calculated
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Important Conclusions
• QED continues to be a highly accurate d i ti f th h i l lddescription of the physical world
• The electron can still be viewed as having no internal structure
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Electron Internal Structure
• Internal constituents must have m > 130 GeV– Calculations made assume no internal structure– Thus corrections must come from very high energy/short distance scales
– i.e. constituents must be very tightly bound• No QED error: m > 600 GeVNo QED error: m 600 GeV
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References
• New Measurement of the Electron Magnetic Moment Using a One‐Electron Quantum CyclotronUsing a One Electron Quantum CyclotronOdom, Hanneke, D’Urso, Gabrielse. PRL 97, 030801
• New Determination of the Fine Structure Constant from the Electron g Value and QEDGabrielse, Hanneke, Kinoshita, Hio, Odom. PRL 91 030802
• Fully Quantum Measurement of the Electron Magnetic MomentMagnetic MomentBrian Odom, Thesis, Harvard University, 2004
• A Finer Constant, Czarnecki, Nature 443 p. 516• James Gilson, fine‐structure‐constant.org
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