g-factor of a bound electron
g-factor of a bound electron
g-factor of a bound electron
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g-<strong>factor</strong> <strong>of</strong> a <strong>bound</strong> <strong>electron</strong><br />
Ion Stroescu<br />
University <strong>of</strong> Heidelberg<br />
January 11, 2008<br />
Ion Stroescu (Uni Heidelberg) g-<strong>factor</strong> <strong>of</strong> a <strong>bound</strong> <strong>electron</strong> January 11, 2008 1 / 32
Outline<br />
1 History and motivation<br />
2 Theory <strong>of</strong> the free <strong>electron</strong><br />
g-<strong>factor</strong> <strong>of</strong> the free <strong>electron</strong><br />
Penning trap and particle motion<br />
QED contributions<br />
3 Theory <strong>of</strong> the <strong>bound</strong> <strong>electron</strong><br />
g-<strong>factor</strong> <strong>of</strong> the <strong>bound</strong> <strong>electron</strong> in hydrogen-like ion<br />
QED contributions for the <strong>bound</strong> state<br />
4 Experiment<br />
Results and precision<br />
Mass <strong>of</strong> the <strong>electron</strong><br />
5 Outlook: New experiments and future facilities<br />
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Outline<br />
History and motivation<br />
1 History and motivation<br />
2 Theory <strong>of</strong> the free <strong>electron</strong><br />
g-<strong>factor</strong> <strong>of</strong> the free <strong>electron</strong><br />
Penning trap and particle motion<br />
QED contributions<br />
3 Theory <strong>of</strong> the <strong>bound</strong> <strong>electron</strong><br />
g-<strong>factor</strong> <strong>of</strong> the <strong>bound</strong> <strong>electron</strong> in hydrogen-like ion<br />
QED contributions for the <strong>bound</strong> state<br />
4 Experiment<br />
Results and precision<br />
Mass <strong>of</strong> the <strong>electron</strong><br />
5 Outlook: New experiments and future facilities<br />
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History and motivation<br />
The Nobel Prize in Physics 1965<br />
The Nobel Prize was awarded to Richard P. Feynman (USA), Julian<br />
Schwinger (USA) and Sin-Itiro Tomonaga (Japan).<br />
“for their fundamental work in quantum electrodynamics, with<br />
deep-ploughing consequences for the physics <strong>of</strong> elementary particles”<br />
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History and motivation<br />
Quantum electrodynamics<br />
Electron is characterised by mass m and charge e<br />
Both m and e have no theoretical prediction<br />
Excellent describtion <strong>of</strong> the magnetic moment µ by QED, the most<br />
successful theory in physics<br />
Ion Stroescu (Uni Heidelberg) g-<strong>factor</strong> <strong>of</strong> a <strong>bound</strong> <strong>electron</strong> January 11, 2008 5 / 32
History and motivation<br />
Quantum electrodynamics<br />
Electron is characterised by mass m and charge e<br />
Both m and e have no theoretical prediction<br />
Excellent describtion <strong>of</strong> the magnetic moment µ by QED, the most<br />
successful theory in physics<br />
Richard P. Feynman, 1983<br />
“The theory <strong>of</strong> quantum electrodynamics is, I would say, the jewel <strong>of</strong><br />
physics - our proudest possession.”<br />
Ion Stroescu (Uni Heidelberg) g-<strong>factor</strong> <strong>of</strong> a <strong>bound</strong> <strong>electron</strong> January 11, 2008 5 / 32
Outline<br />
Theory <strong>of</strong> the free <strong>electron</strong><br />
1 History and motivation<br />
2 Theory <strong>of</strong> the free <strong>electron</strong><br />
g-<strong>factor</strong> <strong>of</strong> the free <strong>electron</strong><br />
Penning trap and particle motion<br />
QED contributions<br />
3 Theory <strong>of</strong> the <strong>bound</strong> <strong>electron</strong><br />
g-<strong>factor</strong> <strong>of</strong> the <strong>bound</strong> <strong>electron</strong> in hydrogen-like ion<br />
QED contributions for the <strong>bound</strong> state<br />
4 Experiment<br />
Results and precision<br />
Mass <strong>of</strong> the <strong>electron</strong><br />
5 Outlook: New experiments and future facilities<br />
Ion Stroescu (Uni Heidelberg) g-<strong>factor</strong> <strong>of</strong> a <strong>bound</strong> <strong>electron</strong> January 11, 2008 6 / 32
Theory <strong>of</strong> the free <strong>electron</strong><br />
g-<strong>factor</strong> <strong>of</strong> the free <strong>electron</strong><br />
The magnetic moment <strong>of</strong> the <strong>electron</strong><br />
An electrical current I creates a<br />
magnetic dipole field with the<br />
moment µ<br />
The angular momentum is given<br />
by L = mrv<br />
This results into the relation<br />
⃗µ = − e<br />
2m ⃗ L<br />
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Theory <strong>of</strong> the free <strong>electron</strong><br />
Definition <strong>of</strong> the magneton<br />
g-<strong>factor</strong> <strong>of</strong> the free <strong>electron</strong><br />
The magnetic moment can be expressed in terms <strong>of</strong> the magneton.<br />
⃗µ = − e<br />
2m ⃗ ⃗L<br />
L = −µ M<br />
<br />
, µ M = e<br />
2m<br />
Two common definitions are Bohr magneton and nuclear magneton.<br />
µ B = e<br />
2m e<br />
= 0.579 × 10 −4 eV T<br />
µ N = e<br />
2m p<br />
= 3.152 × 10 −8 eV T<br />
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Theory <strong>of</strong> the free <strong>electron</strong><br />
g-<strong>factor</strong> <strong>of</strong> the free <strong>electron</strong><br />
The anomalous moment <strong>of</strong> the <strong>electron</strong><br />
The g-<strong>factor</strong> <strong>of</strong> a charged particle is defined as<br />
⃗µ = −g e<br />
2m ⃗ S<br />
A particle with pure angular momentum ⃗L has g = 1<br />
If the theory takes into account the spin <strong>of</strong> the <strong>electron</strong>, g = 2 results<br />
directly from Dirac’s equation<br />
QED vacuum fluctuations and polarization slightly increase this value<br />
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Theory <strong>of</strong> the free <strong>electron</strong><br />
Larmor and cyclotron frequency<br />
g-<strong>factor</strong> <strong>of</strong> the free <strong>electron</strong><br />
The Zeeman splitting <strong>of</strong> the <strong>electron</strong> ground state in a magnetic field<br />
⃗B = B · ⃗e z is given by<br />
∆E 1s = −⃗µ · ⃗B<br />
This is proportional to the Larmor precession frequency <strong>of</strong> the <strong>electron</strong><br />
∆E 1s = ω L<br />
The magnetic field is characterised by the cyclotron frequency<br />
ω c = e m B<br />
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Theory <strong>of</strong> the free <strong>electron</strong><br />
g-<strong>factor</strong> <strong>of</strong> the free <strong>electron</strong><br />
How to measure the anomalous moment<br />
Larmor frequency<br />
ω L = g e<br />
2m B<br />
Cyclotron frequency<br />
ω c = e m B<br />
Comparing these two frequencies enables the measurement <strong>of</strong><br />
g = 2 · ωL<br />
ω c<br />
The special configuration needed for this experiment is realised by the<br />
Penning trap<br />
Ion Stroescu (Uni Heidelberg) g-<strong>factor</strong> <strong>of</strong> a <strong>bound</strong> <strong>electron</strong> January 11, 2008 11 / 32
Theory <strong>of</strong> the free <strong>electron</strong><br />
Penning trap and particle motion<br />
Principle <strong>of</strong> the Penning trap<br />
The Penning trap is a<br />
hyperbolic trap<br />
Confinement in axial<br />
direction by electrostatic<br />
field<br />
Confinement in radial<br />
direction by strong<br />
magnetic field<br />
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Theory <strong>of</strong> the free <strong>electron</strong><br />
Penning trap and particle motion<br />
Motion <strong>of</strong> the particle<br />
Axial motion: oscillation in<br />
electrostatic potential with<br />
ω z = √ qV 0 /md 2<br />
Radial motion in an electrostatic<br />
field has two solutions<br />
Magnetron motion: ⃗E × ⃗B drift,<br />
ω −<br />
Cyclotron motion: ω +<br />
Invariance theorem<br />
( q<br />
) 2<br />
ωc 2 = ω+ 2 + ωz 2 + ω− 2 =<br />
m B<br />
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Theory <strong>of</strong> the free <strong>electron</strong><br />
QED contributions<br />
Self-energy and polarisation<br />
In addition to Dirac’s theory,<br />
there are QED effects which<br />
change the energy <strong>of</strong> the ground<br />
state<br />
(a) self-energy<br />
(b) vacuum polarisation<br />
This affects the anomalous moment <strong>of</strong> the <strong>electron</strong><br />
g e /2 = 1 + C 2 (α/π) + C 4 (α/π) 2 + C 6 (α/π) 3 + C 8 (α/π) 4 + ...<br />
First order Schwinger term C 2 = 1/2<br />
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Outline<br />
Theory <strong>of</strong> the <strong>bound</strong> <strong>electron</strong><br />
1 History and motivation<br />
2 Theory <strong>of</strong> the free <strong>electron</strong><br />
g-<strong>factor</strong> <strong>of</strong> the free <strong>electron</strong><br />
Penning trap and particle motion<br />
QED contributions<br />
3 Theory <strong>of</strong> the <strong>bound</strong> <strong>electron</strong><br />
g-<strong>factor</strong> <strong>of</strong> the <strong>bound</strong> <strong>electron</strong> in hydrogen-like ion<br />
QED contributions for the <strong>bound</strong> state<br />
4 Experiment<br />
Results and precision<br />
Mass <strong>of</strong> the <strong>electron</strong><br />
5 Outlook: New experiments and future facilities<br />
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Theory <strong>of</strong> the <strong>bound</strong> <strong>electron</strong><br />
g-<strong>factor</strong> <strong>of</strong> the <strong>bound</strong> <strong>electron</strong> in hydrogen-like ion<br />
The anomalous moment <strong>of</strong> the <strong>bound</strong> <strong>electron</strong><br />
Larmor frequency <strong>of</strong> the <strong>bound</strong><br />
<strong>electron</strong><br />
ω L = g<br />
e B<br />
2m e<br />
Ion cyclotron frequency<br />
ω c =<br />
Q<br />
m ion<br />
B<br />
Anomalous moment<br />
g = 2 · ωL m e Q<br />
ω c m ion e<br />
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Theory <strong>of</strong> the <strong>bound</strong> <strong>electron</strong><br />
QED contributions for the <strong>bound</strong> state<br />
Bound-state QED <strong>of</strong> the <strong>electron</strong> g-<strong>factor</strong><br />
g <strong>bound</strong> (Z α)2 α(Z α)2<br />
= 1 − + + ...<br />
g free 3 4π<br />
Corrections come from Dirac<br />
theory and <strong>bound</strong>-state QED<br />
High Z prohibit the use <strong>of</strong><br />
perturbation theory, since<br />
Z α ≈ 1<br />
QED corrections ∆E ∼ Z 4 /n 3<br />
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Outline<br />
Experiment<br />
1 History and motivation<br />
2 Theory <strong>of</strong> the free <strong>electron</strong><br />
g-<strong>factor</strong> <strong>of</strong> the free <strong>electron</strong><br />
Penning trap and particle motion<br />
QED contributions<br />
3 Theory <strong>of</strong> the <strong>bound</strong> <strong>electron</strong><br />
g-<strong>factor</strong> <strong>of</strong> the <strong>bound</strong> <strong>electron</strong> in hydrogen-like ion<br />
QED contributions for the <strong>bound</strong> state<br />
4 Experiment<br />
Results and precision<br />
Mass <strong>of</strong> the <strong>electron</strong><br />
5 Outlook: New experiments and future facilities<br />
Ion Stroescu (Uni Heidelberg) g-<strong>factor</strong> <strong>of</strong> a <strong>bound</strong> <strong>electron</strong> January 11, 2008 18 / 32
Experiment<br />
Results and precision<br />
GSI experimental setup<br />
Electron beam produces<br />
hydrogen-like C 5+ ions via<br />
collisional ionisation<br />
All charged particles are then<br />
removed except for a single C 5+<br />
ion<br />
The Penning trap is located in<br />
the center <strong>of</strong> the copper cylinder<br />
Cryopumping reduces pressure<br />
below 10 −16 mbar<br />
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Experiment<br />
Results and precision<br />
Double-trap measuring technique<br />
Resonant excitation at 104 GHz<br />
<strong>of</strong> transition between the two<br />
spin states in the precision trap<br />
(4 Telsa homogeneous magnetic<br />
field)<br />
Measurement <strong>of</strong> spin-flip<br />
transitions in the analysis trap<br />
(inhomogeneous magnetic field)<br />
Amplitudes <strong>of</strong> the three motions<br />
are reduced below 50<br />
micrometer by cooling to a<br />
temperature T = 4 K<br />
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Experiment<br />
Continuous Stern-Gerlach effect<br />
Results and precision<br />
Electron with spin up and antiparallel magnetic moment is driven<br />
toward weaker fields and the axial frequency is increased<br />
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Experiment<br />
Quantum jump spectroscopy<br />
Results and precision<br />
A microwave field is used to induce the spin-flips<br />
The spin-flip transitions (quantum jumps) are observed as small<br />
discrete changes <strong>of</strong> the axial frequency <strong>of</strong> the stored ion<br />
Ion Stroescu (Uni Heidelberg) g-<strong>factor</strong> <strong>of</strong> a <strong>bound</strong> <strong>electron</strong> January 11, 2008 22 / 32
Experiment<br />
Results and precision<br />
Measurement <strong>of</strong> the Larmor precession frequency<br />
Ion Stroescu (Uni Heidelberg) g-<strong>factor</strong> <strong>of</strong> a <strong>bound</strong> <strong>electron</strong> January 11, 2008 23 / 32
Experiment<br />
Results and precision<br />
g-<strong>factor</strong> <strong>of</strong> a free <strong>electron</strong><br />
Electron: g = 2 × 1.001 159 652 188 4 (43)<br />
Positron: g = 2 × 1.001 159 652 187 9 (43)<br />
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Experiment<br />
g-<strong>factor</strong> <strong>of</strong> a <strong>bound</strong> <strong>electron</strong><br />
Results and precision<br />
g = 1.998 721 354 4 Dirac theory <strong>of</strong> <strong>bound</strong> <strong>electron</strong><br />
+0.000 000 000 4 nuclear volume effect<br />
+0.000 000 087 6 nuclear recoil effect<br />
+0.002 323 663 7 (9) first order QED correction<br />
- 0.000 003 516 2 (2) second order QED correction<br />
2.001 041 589 9 (9) theoretical value<br />
2.001 041 596 4 (10) (44)GSI measurement on C 5+<br />
T. Beier et al, Phys. Rev. Lett. 88, 011603 (2002)<br />
V. Shabaev et al, Phys. Rev. Lett. 88, 091801 (2002)<br />
V. Yerokhin et al, Phys. Rev. Lett. 89, 143001 (2002)<br />
Ion Stroescu (Uni Heidelberg) g-<strong>factor</strong> <strong>of</strong> a <strong>bound</strong> <strong>electron</strong> January 11, 2008 25 / 32
Experiment<br />
Deducing the <strong>electron</strong>’s mass<br />
Mass <strong>of</strong> the <strong>electron</strong><br />
m e<br />
= g ω c e<br />
m ion 2 ω L Q<br />
<strong>electron</strong>’s mass m e /u proton/<strong>electron</strong> mass ratio<br />
CODATA 1998 0.000 548 579 911 0 (12) 1836.152 667 5 (39)<br />
g-<strong>factor</strong> (C 5+ ) 0.000 548 579 909 2 (4) 1836.152 673 1 (10)<br />
Ion Stroescu (Uni Heidelberg) g-<strong>factor</strong> <strong>of</strong> a <strong>bound</strong> <strong>electron</strong> January 11, 2008 26 / 32
Outline<br />
Outlook: New experiments and future facilities<br />
1 History and motivation<br />
2 Theory <strong>of</strong> the free <strong>electron</strong><br />
g-<strong>factor</strong> <strong>of</strong> the free <strong>electron</strong><br />
Penning trap and particle motion<br />
QED contributions<br />
3 Theory <strong>of</strong> the <strong>bound</strong> <strong>electron</strong><br />
g-<strong>factor</strong> <strong>of</strong> the <strong>bound</strong> <strong>electron</strong> in hydrogen-like ion<br />
QED contributions for the <strong>bound</strong> state<br />
4 Experiment<br />
Results and precision<br />
Mass <strong>of</strong> the <strong>electron</strong><br />
5 Outlook: New experiments and future facilities<br />
Ion Stroescu (Uni Heidelberg) g-<strong>factor</strong> <strong>of</strong> a <strong>bound</strong> <strong>electron</strong> January 11, 2008 27 / 32
Outlook: New experiments and future facilities<br />
Overview <strong>of</strong> the HITRAP facility at GSI<br />
Experiments with highly charged ions<br />
at low energies:<br />
g-<strong>factor</strong> measurements (tests <strong>of</strong><br />
QED)<br />
Hyperfine structure in<br />
hydrogen-like ions<br />
Fundamental constants m e and<br />
α<br />
Fundamental symmetries CPT<br />
and parity<br />
Laser and X-ray spectroscopy<br />
with cold HCI<br />
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Outlook: New experiments and future facilities<br />
g-<strong>factor</strong> <strong>of</strong> a <strong>bound</strong> <strong>electron</strong> for U 91+ at HITRAP<br />
U 73+ coming from SIS is<br />
stripped to U 91+ before the<br />
injection into ESR<br />
10 5 ions/pulse every 10 s<br />
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Outlook: New experiments and future facilities<br />
Facility for Antiproton and Ion Research<br />
Ion Stroescu (Uni Heidelberg) g-<strong>factor</strong> <strong>of</strong> a <strong>bound</strong> <strong>electron</strong> January 11, 2008 30 / 32
Conclusion<br />
Conclusion<br />
The g-<strong>factor</strong> is defined as ⃗µ = −g e<br />
2m ⃗ S<br />
Ion Stroescu (Uni Heidelberg) g-<strong>factor</strong> <strong>of</strong> a <strong>bound</strong> <strong>electron</strong> January 11, 2008 31 / 32
Conclusion<br />
Conclusion<br />
The g-<strong>factor</strong> is defined as ⃗µ = −g e<br />
2m ⃗ S<br />
Zeeman splitting in a magnetic field is ∆E 1s = −⃗µ · ⃗B = ω L<br />
Ion Stroescu (Uni Heidelberg) g-<strong>factor</strong> <strong>of</strong> a <strong>bound</strong> <strong>electron</strong> January 11, 2008 31 / 32
Conclusion<br />
Conclusion<br />
The g-<strong>factor</strong> is defined as ⃗µ = −g e<br />
2m ⃗ S<br />
Zeeman splitting in a magnetic field is ∆E 1s = −⃗µ · ⃗B = ω L<br />
The cyclotron frequency is given by ω c = e m B<br />
Ion Stroescu (Uni Heidelberg) g-<strong>factor</strong> <strong>of</strong> a <strong>bound</strong> <strong>electron</strong> January 11, 2008 31 / 32
Conclusion<br />
Conclusion<br />
The g-<strong>factor</strong> is defined as ⃗µ = −g e<br />
2m ⃗ S<br />
Zeeman splitting in a magnetic field is ∆E 1s = −⃗µ · ⃗B = ω L<br />
The cyclotron frequency is given by ω c = e m B<br />
Measurement <strong>of</strong> the g-<strong>factor</strong><br />
g = 2 · ωL<br />
ω c<br />
Ion Stroescu (Uni Heidelberg) g-<strong>factor</strong> <strong>of</strong> a <strong>bound</strong> <strong>electron</strong> January 11, 2008 31 / 32
Conclusion<br />
Conclusion<br />
The g-<strong>factor</strong> is defined as ⃗µ = −g e<br />
2m ⃗ S<br />
Zeeman splitting in a magnetic field is ∆E 1s = −⃗µ · ⃗B = ω L<br />
The cyclotron frequency is given by ω c = e m B<br />
Measurement <strong>of</strong> the g-<strong>factor</strong><br />
g = 2 · ωL<br />
ω c<br />
The anomalous moment <strong>of</strong> the <strong>bound</strong> <strong>electron</strong> is given by<br />
g = 2 · ωL m e Q<br />
ω c m ion e<br />
Ion Stroescu (Uni Heidelberg) g-<strong>factor</strong> <strong>of</strong> a <strong>bound</strong> <strong>electron</strong> January 11, 2008 31 / 32
References<br />
References<br />
J. Verdú, S. Djekić, S. Stahl, T. Valenzuela, M. Vogel, G. Werth, T.<br />
Beier, H.-J. Kluge, and W. Quint,<br />
Phys. Rev. Lett. 92, 093002 (2004).<br />
N. Hermanspahn, H. Häffner, H.-J. Kluge, W. Quint, S. Stahl, J.<br />
Verdú, and G. Werth,<br />
Phys. Rev. Lett. 84, 427 (2000).<br />
H. Dehmelt,<br />
Proc. Natl. Acad. Sci. USA 83, 2291-2294 (1986).<br />
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