The bound-electron g factor
in light ions
Vladimir A. Yerokhin
Peter the Great St. Petersburg Polytechnic University
Precision Physics and Fundamental Physical Constants FFK-2017, May 19, Warsaw
Outline

Free-electron g factor.

Bound-electron g factor: H-like ions.

Bound-electron g factor: Li-like ions.

Electron mass and fine-structure constant determinations.
g factor of free electron: Classical picture
What is g factor ?
Classical electrodynamics: g factor = 1
Relativistic quantum mechanics: g factor = 2
g factor of free electron: Quantum electrodynamics
g factor of free electron: Quantum electrodynamics
g factor of free electron: Quantum electrodynamics
What do we learn from free-electron g factor?
 Best prediction of the (free-field) Quantum Electrodynamics
 Best determination of the fine-structure constant
Bound-electron g factor
Experiments:
Experimental result for 12C5+:
ωL/ωc = 4376.210 500 89 (11)(7)
[Sturm et al. Nature 506, 467 (2014)]
What can we learn from bound-electron g factor?
 Tests of bound-state Quantum Electrodynamics
Advantages:
high experimental accuracy,
weak nuclear effects,
different Z and charge states
 Electron mass

Fine-structure constant

Nuclear magnetic moments, nuclear masses, nuclear radii
g factor of H-like ions: theory
Relativistic
Free QED
Effects
Bound QED
Nuclear
Nuclear recoil
Nuclear size
Nuclear
polarization &
deformation
Leading orders
Dirac g factor:
n
0
Free QED, α (Zα) :
Bound QED, Zα expansion
n
2
Bound QED, α (Zα) :
[H. Grotch 1970, R.N. Faustov 1970, F.E. Close and H. Osborn 1971,…]
4
Bound QED, 1-loop, α (Zα) :
2
4
Bound QED, 2-loop, α (Zα) :
[K. Pachucki, A. Czarnecki, U.D. Jentschura, VAY, 2005]
[K. Pachucki, A. Czarnecki, U.D. Jentschura, VAY, 2005]
[Czarnecki
and
Szafron,
PRA, 2016]
Bound QED, all orders in Zα
Electron Self-energy
Vacuum-polarization
Electric-loop
[Beier et al. PRA 2000]
Magnetic-loop
[Karshenboim &
Milstein 2002, Lee
et al, 2005]
Two-loop QED
2-loop QED corrections, all orders in Zα:
only partial results up to now
VAY and Z. Harman PRA 2013
work in progress …
The main theoretical uncertainty of the g factor !
So far, an “experimental” determination of two-loop QED effects
Nuclear effects
Nuclear recoil
to leading order in Zα
[Grotch, PRA 1970, Faustov, PLB 1970]
to all orders in Zα
[Shabaev, PRA 2001; Shabaev and VAY PRL
2002]
[Faustov 1970, Grotch and Hegstrom 1971,
Close and Osborn 1971, and others]
Finite nuclear size
[VAY et al. JPB 2003; PRA 2016]
Nuclear polarization and deformation
[Nefiodov et al. PLB 2003, Volotka and Plunien PRL 2014, Zatorski et al. PRL 2012]
g factor of H-like carbon: present status
Theoretical
estimate
Derived from
experiment
on Si13+
[J. Zatorsky et al. 2017]
Determination of the electron mass
Bound-electron g factor:
me = 0.000 548 579 909 067 (14)(9)(2) (stat)(sys)(theory)
Independent determination (free electron versus C6+):
me = 0.000 548 579 911 1 (12) [Farnheim et al. PRL 75, 3598 (1995)]
g factor of Li-like ions
Parameters:
Methods
Z α expansion,
1/Z expansion,
all orders in 1/Z
all orders in Zα
g factor of Li-like ions
Parameters:
Unified
theory
1/Z expansion, all orders in Zα
[Volotka et al. PRL 2014; Glazov et al. PRA
2010, PRA 2004; Shabaev et al. PRA 2002]
g (H-like)
1-photon
exchange
g (Li-like)
Electronelectron
interaction
2-photon
exchange
(>=3)-photon
exchange
QED + 1-photon
exchange
QED + Electronelectron
interaction
QED +(>=2)photon
exchange
Zα expansion, all orders in 1/Z
Nonrelativistic Quantum Electrodynamics (NRQED) expansion:
Effective Hamiltonian, describing the interaction of an atom with the external
magnetic field to orders α2, α3, and m/M [Hegstrom 73]:
Numerical calculations:
[Yan PRL 2001; JPB 2002]
[VAY, Pachucki, Puchalski et al., arXiv:submit/1889331]
Unified theory
[VAY, Pachucki, Puchalski et al., arXiv:submit/1889331]
Electron correlation:
1/Z
expansion
Zα expansion
NRQED,
O(α2)
Binding effects in g factor Li-like silicon, current status
Binding effects in g factor Li-like silicon, current status
g-factor of Li-like calcium. Isotope shift
Theory: only one effect (nuclear recoil) contributes !
Nonrelativistic nuclear recoil vanishes (for s states) !
Test of relativistic theory of the nuclear recoil effect.
What’s next: Fine-structure constant ?
If we get α from the free-electron, why not from the bound-electron?
Problems: theory is much more complicated
nuclear effects
Advantages: we can vary Z and the charge state
Fine-structure constant from bound-electron g factor
Possible ways to go:

H-like ions, Z as small as possible (He+)

Li-like + H-like ions, small Z [Yerokhin et al. PRL 116, 100801 (2016)]

B-like + H-like ions, large Z [Shabaev et al. PRL 96, 253002 (2006)]
Weighted difference of H-like and Li-like ions, Low Z.
[Yerokhin et al. PRL 116, 100801 (2016)]
Li-like ion
Difference
Nuclear effects
are suppressed by
three orders of
magnitude.
They do not
create problems
for α
determination.
Nuclear effects are not a problem !
Outlook: Determination of nuclear magnetic moments
g factor of an ion with a spin-nonzero nucleus (I – nuclear spin, J – electron angular momentum, F –
electron + nucleus angular momentum):
All corrections can be parameterized in terms of the shielding constant σ:
Shielding in H-like ions is calculated up to a very high accuracy.
[Yerokhin et al. PRL 2011]
Conclusion
 Bound-electron g factor: high-precision measurements and accurate
calculations.

Tests of bound-state QED

Determination of the electron mass
 In future: access to the fine-structure constant, nuclear magnetic
moments, nuclear masses, nuclear charge radii
Conclusion
 Bound-electron g factor: high-precision measurements and accurate
calculations.

Tests of bound-state QED

Determination of the electron mass
 In future: access to the fine-structure constant, nuclear magnetic
moments, nuclear masses, nuclear charge radii
THANK YOU
FOR YOUR ATTENTION!
Additional
 Bound-electron g factor: high-precision measurements and accurate
calculations.

Tests of bound-state QED

Determination of the electron mass
 Access to the fine-structure constant, nuclear magnetic moments,
nuclear masses, nuclear charge radii