g-factor of a bound electron

g-factor of a bound electron 

Ion Stroescu 

University of Heidelberg 

January 11, 2008 

Ion Stroescu (Uni Heidelberg) g-factor of a bound electron January 11, 2008 1 / 32

Outline 

1 History and motivation 

2 Theory of the free electron 

g-factor of the free electron 

Penning trap and particle motion 

QED contributions 

3 Theory of the bound electron 

g-factor of the bound electron in hydrogen-like ion 

QED contributions for the bound state 

4 Experiment 

Results and precision 

Mass of the electron 

5 Outlook: New experiments and future facilities 


Outline 

History and motivation 









4 Experiment 






The Nobel Prize in Physics 1965 

The Nobel Prize was awarded to Richard P. Feynman (USA), Julian 

Schwinger (USA) and Sin-Itiro Tomonaga (Japan). 

“for their fundamental work in quantum electrodynamics, with 

deep-ploughing consequences for the physics of elementary particles” 



Quantum electrodynamics 

Electron is characterised by mass m and charge e 

Both m and e have no theoretical prediction 

Excellent describtion of the magnetic moment µ by QED, the most 

successful theory in physics 



Quantum electrodynamics 

Electron is characterised by mass m and charge e 

Both m and e have no theoretical prediction 

Excellent describtion of the magnetic moment µ by QED, the most 

successful theory in physics 

Richard P. Feynman, 1983 

“The theory of quantum electrodynamics is, I would say, the jewel of 

physics - our proudest possession.” 


Outline 

Theory of the free electron 









4 Experiment 







The magnetic moment of the electron 

An electrical current I creates a 

magnetic dipole field with the 

moment µ 

The angular momentum is given 

by L = mrv 

This results into the relation 

⃗µ = − e 

2m ⃗ L 



Definition of the magneton 


The magnetic moment can be expressed in terms of the magneton. 

⃗µ = − e 

2m ⃗ ⃗L 

L = −µ M 

 

, µ M = e 

2m 

Two common definitions are Bohr magneton and nuclear magneton. 

µ B = e 

2m e 

= 0.579 × 10 −4 eV T 

µ N = e 

2m p 

= 3.152 × 10 −8 eV T 




The anomalous moment of the electron 

The g-factor of a charged particle is defined as 

⃗µ = −g e 

2m ⃗ S 

A particle with pure angular momentum ⃗L has g = 1 

If the theory takes into account the spin of the electron, g = 2 results 

directly from Dirac’s equation 

QED vacuum fluctuations and polarization slightly increase this value 



Larmor and cyclotron frequency 


The Zeeman splitting of the electron ground state in a magnetic field 

⃗B = B · ⃗e z is given by 

∆E 1s = −⃗µ · ⃗B 

This is proportional to the Larmor precession frequency of the electron 

∆E 1s = ω L 

The magnetic field is characterised by the cyclotron frequency 

ω c = e m B 




How to measure the anomalous moment 

Larmor frequency 

ω L = g e 

2m B 

Cyclotron frequency 

ω c = e m B 

Comparing these two frequencies enables the measurement of 

g = 2 · ωL 

ω c 

The special configuration needed for this experiment is realised by the 

Penning trap 




Principle of the Penning trap 

The Penning trap is a 

hyperbolic trap 

Confinement in axial 

direction by electrostatic 

field 

Confinement in radial 

direction by strong 

magnetic field 




Motion of the particle 

Axial motion: oscillation in 

electrostatic potential with 

ω z = √ qV 0 /md 2 

Radial motion in an electrostatic 

field has two solutions 

Magnetron motion: ⃗E × ⃗B drift, 

ω − 

Cyclotron motion: ω + 

Invariance theorem 

( q 

) 2 

ωc 2 = ω+ 2 + ωz 2 + ω− 2 = 

m B 




Self-energy and polarisation 

In addition to Dirac’s theory, 

there are QED effects which 

change the energy of the ground 

state 

(a) self-energy 

(b) vacuum polarisation 

This affects the anomalous moment of the electron 

g e /2 = 1 + C 2 (α/π) + C 4 (α/π) 2 + C 6 (α/π) 3 + C 8 (α/π) 4 + ... 

First order Schwinger term C 2 = 1/2 


Outline 

Theory of the bound electron 









4 Experiment 







The anomalous moment of the bound electron 

Larmor frequency of the bound 

electron 

ω L = g 

e B 

2m e 

Ion cyclotron frequency 

ω c = 

Q 

m ion 

B 

Anomalous moment 

g = 2 · ωL m e Q 

ω c m ion e 




Bound-state QED of the electron g-factor 

g bound (Z α)2 α(Z α)2 

= 1 − + + ... 

g free 3 4π 

Corrections come from Dirac 

theory and bound-state QED 

High Z prohibit the use of 

perturbation theory, since 

Z α ≈ 1 

QED corrections ∆E ∼ Z 4 /n 3 


Outline 

Experiment 









4 Experiment 





Experiment 


GSI experimental setup 

Electron beam produces 

hydrogen-like C 5+ ions via 

collisional ionisation 

All charged particles are then 

removed except for a single C 5+ 

ion 

The Penning trap is located in 

the center of the copper cylinder 

Cryopumping reduces pressure 

below 10 −16 mbar 


Experiment 


Double-trap measuring technique 

Resonant excitation at 104 GHz 

of transition between the two 

spin states in the precision trap 

(4 Telsa homogeneous magnetic 

field) 

Measurement of spin-flip 

transitions in the analysis trap 

(inhomogeneous magnetic field) 

Amplitudes of the three motions 

are reduced below 50 

micrometer by cooling to a 

temperature T = 4 K 


Experiment 

Continuous Stern-Gerlach effect 


Electron with spin up and antiparallel magnetic moment is driven 

toward weaker fields and the axial frequency is increased 


Experiment 

Quantum jump spectroscopy 


A microwave field is used to induce the spin-flips 

The spin-flip transitions (quantum jumps) are observed as small 

discrete changes of the axial frequency of the stored ion 


Experiment 


Measurement of the Larmor precession frequency 


Experiment 


g-factor of a free electron 

Electron: g = 2 × 1.001 159 652 188 4 (43) 

Positron: g = 2 × 1.001 159 652 187 9 (43) 


Experiment 

g-factor of a bound electron 


g = 1.998 721 354 4 Dirac theory of bound electron 

+0.000 000 000 4 nuclear volume effect 

+0.000 000 087 6 nuclear recoil effect 

+0.002 323 663 7 (9) first order QED correction 

- 0.000 003 516 2 (2) second order QED correction 

2.001 041 589 9 (9) theoretical value 

2.001 041 596 4 (10) (44)GSI measurement on C 5+ 

T. Beier et al, Phys. Rev. Lett. 88, 011603 (2002) 

V. Shabaev et al, Phys. Rev. Lett. 88, 091801 (2002) 

V. Yerokhin et al, Phys. Rev. Lett. 89, 143001 (2002) 


Experiment 

Deducing the electron’s mass 


m e 

= g ω c e 

m ion 2 ω L Q 

electron’s mass m e /u proton/electron mass ratio 

CODATA 1998 0.000 548 579 911 0 (12) 1836.152 667 5 (39) 

g-factor (C 5+ ) 0.000 548 579 909 2 (4) 1836.152 673 1 (10) 


Outline 

Outlook: New experiments and future facilities 









4 Experiment 






Overview of the HITRAP facility at GSI 

Experiments with highly charged ions 

at low energies: 

g-factor measurements (tests of 

QED) 

Hyperfine structure in 

hydrogen-like ions 

Fundamental constants m e and 

α 

Fundamental symmetries CPT 

and parity 

Laser and X-ray spectroscopy 

with cold HCI 



g-factor of a bound electron for U 91+ at HITRAP 

U 73+ coming from SIS is 

stripped to U 91+ before the 

injection into ESR 

10 5 ions/pulse every 10 s 



Facility for Antiproton and Ion Research 


Conclusion 

Conclusion 

The g-factor is defined as ⃗µ = −g e 

2m ⃗ S 


Conclusion 

Conclusion 


2m ⃗ S 

Zeeman splitting in a magnetic field is ∆E 1s = −⃗µ · ⃗B = ω L 


Conclusion 

Conclusion 


2m ⃗ S 


The cyclotron frequency is given by ω c = e m B 


Conclusion 

Conclusion 


2m ⃗ S 



Measurement of the g-factor 

g = 2 · ωL 

ω c 


Conclusion 

Conclusion 


2m ⃗ S 



Measurement of the g-factor 

g = 2 · ωL 

ω c 

The anomalous moment of the bound electron is given by 

g = 2 · ωL m e Q 

ω c m ion e 


References 

References 

J. Verdú, S. Djekić, S. Stahl, T. Valenzuela, M. Vogel, G. Werth, T. 

Beier, H.-J. Kluge, and W. Quint, 

Phys. Rev. Lett. 92, 093002 (2004). 

N. Hermanspahn, H. Häffner, H.-J. Kluge, W. Quint, S. Stahl, J. 

Verdú, and G. Werth, 

Phys. Rev. Lett. 84, 427 (2000). 

H. Dehmelt, 

Proc. Natl. Acad. Sci. USA 83, 2291-2294 (1986).

g-factor of a bound electron

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