# High-Precision Measurements of the Bound Electron’s Magnetic Moment

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## Abstract

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## 1. Introduction

## 2. Sources of Highly Charged Ions

## 3. $\mathit{g}$-Factor Measurements with Highly Charged Ions

#### 3.1. Mainz g-Factor Experiment

#### 3.2. ALPHATRAP

#### 3.3. ARTEMIS

## 4. Conclusions and Outlook

## Author Contributions

## Conflicts of Interest

## References

- Dirac, P.A.M. The Quantum Theory of the Electron. Proc. R. Soc. A
**1928**, 117, 610–624. [Google Scholar] [CrossRef] - Kusch, P.; Foley, H.M. Precision Measurement of the Ratio of the Atomic g Values in the
^{2}P_{3/2}and^{2}P_{1/2}States of Gallium. Phys. Rev.**1947**, 72, 1256. [Google Scholar] [CrossRef] - Kusch, P.; Foley, H.M. On the Intrinsic Moment of the Electron. Phys. Rev.
**1948**, 74, 250. [Google Scholar] [CrossRef] - Schwinger, J. On Quantum-Electrodynamics and the Magnetic Moment of the Electron. Phys. Rev.
**1948**, 73, 416. [Google Scholar] [CrossRef] - Vogel, M.; Quint, W. Magnetic Moment of the bound electron. In Fundamental Physics in Particle Traps; Springer Tracts in Modern Physics; Springer: Berlin/Heidelberg, Germany, 2014; Volume 256. [Google Scholar]
- Vogel, M. The anomalous magnetic moment of the electron. Contemp. Phys.
**2009**, 50, 437–452. [Google Scholar] [CrossRef] - Van Dyck, R.S.; Schwinberg, P.B.; Dehmelt, H.G. New high-precision comparison of electron and positron g factors. Phys. Rev. Lett.
**1987**, 59, 26. [Google Scholar] [CrossRef] [PubMed] - Hanneke, D.; Fogwell, S.; Gabrielse, G. New measurement of the electron magnetic moment and the fine structure constant. Phys. Rev. Lett.
**2008**, 100, 120801. [Google Scholar] [CrossRef] [PubMed] - Kinoshita, T.; Nio, M. Tenth-order QED contribution to the lepton g − 2: Evaluation of dominant α
^{5}terms of muon g − 2. Phys. Rev. D**2006**, 73, 053007. [Google Scholar] [CrossRef] - Bouchendira, R.; Cladé, P.; Guellati-Khélifa, S.; Nez, F.; Biraben, F. New Determination of the Fine Structure Constant and Test of the Quantum Electrodynamics. Phys. Rev. Lett.
**2011**, 106, 080801. [Google Scholar] [CrossRef] [PubMed] - Breit, G. The Magnetic Moment of the Electron. Nature
**1928**, 122, 649. [Google Scholar] [CrossRef] - Beier, T. The g
_{j}factor of a bound electron and the hyperfine structure splitting in hydrogen-like ions. Phys. Rep.**2000**, 339, 79–213. [Google Scholar] [CrossRef] - Yerokhin, V.A.; Pachucki, K.; Harman, Z.; Keitel, C.H. QED Theory of the Nuclear Magnetic Shielding in Hydrogenlike Ions. Phys. Rev. Lett.
**2011**, 107, 043004. [Google Scholar] [CrossRef] [PubMed] - Zatorski, J.; Oreshkina, N.S.; Keitel, C.H.; Harman, Z. Nuclear Shape Effect on the g Factor of Hydrogenlike Ions. Phys. Rev. Lett.
**2012**, 108, 063005. [Google Scholar] [CrossRef] [PubMed] - Shabaev, V.M. QED theory of the nuclear recoil effect on the atomic g-factor. Phys. Rev. A
**2001**, 64, 052104. [Google Scholar] [CrossRef] - Shabaev, V.M.; Yerokhin, V.A. Recoil Correction to the Bound-Electron g-Factor in H-Like Atoms to All Orders in αZ. Phys. Rev. Lett.
**2002**, 88, 091801. [Google Scholar] [CrossRef] [PubMed] - Yerokhin, V.A.; Indelicato, P.; Shabaev, V.M. Self-energy correction to the bound-electron g factor in H-like ions. Phys. Rev. Lett.
**2002**, 89, 143001. [Google Scholar] [CrossRef] [PubMed] - Yerokhin, V.A.; Indelicato, P.; Shabaev, V.M. Evaluation of the self-energy correction to the g factor of S states in H-like ions. Phys. Rev. A
**2004**, 69, 052503. [Google Scholar] [CrossRef] - Glazov, D.A.; Shabaev, V.M.; Tupitsyn, I.I.; Volotka, A.V.; Yerokhin, V.A.; Plunien, G.; Soff, G. Relativistic and QED corrections to the g factor of Li-like ions. Phys. Rev. A
**2004**, 70, 062104. [Google Scholar] [CrossRef] - Volotka, A.V.; Glazov, D.A.; Shabaev, V.M.; Tupitsyn, I.I.; Plunien, G. Screened QED corrections in lithiumlike heavy ions in the presence of magnetic fields. Phys. Rev. Lett.
**2009**, 103, 033005. [Google Scholar] [CrossRef] [PubMed] - Glazov, D.A.; Volotka, A.V.; Shabaev, V.M.; Tupitsyn, I.I.; Plunien, G. Evaluation of the screened QED corrections to the g factor and the hyperfine splitting of lithiumlike ions. Phys. Rev. A
**2010**, 81, 062112. [Google Scholar] [CrossRef] - Shabaev, V.M.; Andreev, O.V.; Bondarev, A.I.; Glazov, D.A.; Kozhedub, Y.S.; Maiorova, A.V.; Plunien, G.; Tupitsyn, I.I.; Volotka, A.V. Quantum Electrodynamics Effects in Heavy Ions and Atoms. AIP Conf. Proc.
**2011**, 1344, 60–69. [Google Scholar] - Shabaev, V.M.; Glazov, D.A.; Plunien, G.; Volotka, A.V. Theory of Bound-Electron g Factor in Highly Charged Ions. J. Phys. Chem. Ref. Data
**2015**, 44, 031205. [Google Scholar] [CrossRef] - Häffner, H.; Beier, T.; Hermanspahn, N.; Kluge, H.J.; Quint, W.; Stahl, S.; Verdú, J.; Werth, G. High-Accuracy Measurement of the Magnetic Moment Anomaly of the Electron Bound in Hydrogen-like Carbon. Phys. Rev. Lett.
**2000**, 85, 5308. [Google Scholar] [CrossRef] [PubMed] - Verdú, J.; Djekić, S.; Stahl, S.; Valenzuela, T.; Vogel, M.; Werth, G.; Beier, T.; Kluge, H.J.; Quint, W. Electronic g Factor of Hydrogen-like Oxygen
^{16}O^{7+}. Phys. Rev. Lett.**2004**, 92, 093002. [Google Scholar] [CrossRef] [PubMed] - Sturm, S.; Wagner, A.; Schabinger, B.; Zatorski, J.; Harman, Z.; Quint, W.; Werth, G.; Keitel, C.H.; Blaum, K. g-factor of hydrogen-like
^{28}Si^{13+}. Phys. Rev. Lett.**2011**, 107, 023002. [Google Scholar] [CrossRef] [PubMed] - Wagner, A.; Sturm, S.; Köhler, F.; Glazov, D.A.; Volotka, A.V.; Plunien, G.; Quint, W.; Werth, G.; Shabaev, V.M.; Blaum, K. g-Factor of Lithium-like Silicon
^{28}Si^{11+}. Phys. Rev. Lett.**2013**, 110, 033003. [Google Scholar] [CrossRef] [PubMed] - Sturm, S.; Werth, G.; Blaum, K. Electron g-factor determinations in Penning traps. Ann. Phys.
**2013**, 525, 620–635. [Google Scholar] [CrossRef] - Köhler, F.; Blaum, K.; Block, M.; Chenmarev, S.; Eliseev, S.; Glazov, D.A.; Goncharov, M.; Hou, J.; Kracke, A.; Nesterenko, D.A.; et al. Isotope dependence of the Zeeman effect in lithium-like calcium. Nat. Commun.
**2016**, 7, 10246. [Google Scholar] [CrossRef] [PubMed] - Vogel, M.; Quint, W. Aspects of fundamental physics in precision spectroscopy of highly charged ions in Penning traps. Ann. Phys.
**2013**, 525, 505–513. [Google Scholar] [CrossRef] - Sturm, S.; Köhler, F.; Zatorski, J.; Wagner, A.; Harman, Z.; Werth, G.; Quint, W.; Keitel, C.H.; Blaum, K. High-precision measurement of the atomic mass of the electron. Nature
**2014**, 506, 467–470. [Google Scholar] [CrossRef] [PubMed] - Beier, T.; Häffner, H.; Hermanspahn, N.; Karshenboim, S.G.; Kluge, H.J.; Quint, W.; Stahl, S.; Verdú, J.; Werth, G. New Determination of the Electron’s Mass. Phys. Rev. Lett.
**2002**, 88, 011603. [Google Scholar] [CrossRef] [PubMed] - Shabaev, V.M.; Glazov, D.A.; Oreshkina, N.S.; Volotka, A.V.; Plunien, G.; Kluge, H.J.; Quint, W. g-factor of heavy ions: A new access to the fine structure constant. Phys. Rev. Lett.
**2006**, 96, 253002. [Google Scholar] [CrossRef] [PubMed] - Yerokhin, V.A.; Berseneva, E.; Harman, Z.; Tupitsyn, I.I.; Keitel, C.H. g-Factor of Light Ions for an Improved Determination of the Fine-Structure Constant. Phys. Rev. Lett.
**2016**, 116, 100801. [Google Scholar] [CrossRef] [PubMed] - Sturm, S.; Wagner, A.; Kretzschmar, M.; Quint, W.; Werth, G.; Blaum, K. g-factor measurement of hydrogen-like
^{28}Si^{13+}as a challenge to QED calculations. Phys. Rev. A**2013**, 87, 030501. [Google Scholar] [CrossRef] - Dehmelt, H. Continuous Stern Gerlach effect: Principle and idealized apparatus. Proc. Natl. Acad. Sci. USA
**1986**, 83, 2291–2294. [Google Scholar] [CrossRef] [PubMed] - Hermanspahn, N.; Häffner, H.; Kluge, H.J.; Quint, W.; Stahl, S.; Verdú, J.; Werth, G. Observation of the Continuous Stern–Gerlach Effect on an Electron Bound in an Atomic Ion. Phys. Rev. Lett.
**2000**, 84, 427. [Google Scholar] [CrossRef] [PubMed] - Quint, W.; Moskovkin, D.; Shabaev, V.M.; Vogel, M. Laser-microwave double-resonance technique for g-factor measurements in highly charged ions. Phys. Rev. A
**2008**, 78, 032517. [Google Scholar] [CrossRef] - Von Lindenfels, D.; Wiesel, M.; Glazov, D.A.; Volotka, A.V.; Sokolov, M.M.; Shabaev, V.M.; Plunien, G.; Quint, W.; Birkl, G.; Martin, A.; et al. Experimental access to higher-order Zeeman effects by precision spectroscopy of highly charged ions in a Penning trap. Phys. Rev. A
**2013**, 87, 023412. [Google Scholar] [CrossRef] - Kluge, H.-J.; Beier, T.; Blaum, K.; Dahl, L.; Eliseev, S.; Herfurth, F.; Hofmann, B.; Kester, O.; Koszudowski, S.; Kozhuharov, C.; et al. HITRAP: A facility at GSI for highly charged ions. Adv. Quantum Chem.
**2007**, 53, 83–98. [Google Scholar] - Donets, E.D. Electron Beam Ion Sources. In The Physics and Technology of Ion Sources; Brown, I.G., Ed.; John Wiley and Sons: New York, NY, USA, 1989. [Google Scholar]
- Schneider, D.; DeWitt, D.; Clark, M.W.; Schuch, R.; Cocke, C.L.; Schmieder, R.; Reed, K.J.; Chen, M.H.; Marrs, R.E.; Levine, M.; et al. Ion-collision experiments with slow, very highly charged ions extracted from an electron-beam ion trap. Phys. Rev. A
**1990**, 42, 3889–3895. [Google Scholar] [CrossRef] [PubMed] - Gonzalez Martinez, A.J.; López-Urrutia, J.C.; Braun, J.; Brenner, G.; Bruhns, H.; Lapierre, A.; Mironov, V.; Orts, R.S.; Tawara, H.; Trinczek, M.; et al. Benchmarking high-field few-electron correlation and QED contributions in Hg
^{75+}to Hg^{78+}ions. Phys. Rev. A**2006**, 73, 052710. [Google Scholar] [CrossRef] - Gabrielse, G. Why Is Sideband Mass Spectrometry Possible with Ions in a Penning Trap? Phys. Rev. Lett.
**2009**, 102, 172501. [Google Scholar] [CrossRef] [PubMed] - Angeli, I. A consistent set of nuclear rms charge radii: Properties of the radius surface R(N,Z). At. Data Nucl. Data Tables
**2004**, 87, 185–206. [Google Scholar] [CrossRef] - Furry, W.H. On Bound States and Scattering in Positron Theory. Phys. Rev.
**1951**, 81, 115. [Google Scholar] [CrossRef] - Angeli, I.; Marinova, K.P. Table of experimental nuclear ground state charge radii: An update. At. Data Nucl. Data Tables
**2013**, 99, 69–95. [Google Scholar] [CrossRef] - Crespo López-Urrutia, J.R. Progress at the Heidelberg EBIT. J. Phys. Conf. Ser.
**2004**, 2, 52. [Google Scholar] - González Martínez, A.J.; López-Urrutia, J.C.; Fischer, D.; Orts, R.S.; Ullrich, J. The Heidelberg EBIT: Present Results and Future Perspectives. J. Phys. Conf. Ser.
**2005**, 72, 012001. [Google Scholar] [CrossRef] - Alonso, J.; Blaum, K.; Djekic, S.; Kluge, H.J.; Quint, W.; Schabinger, B.; Stahl, S.; Verdú, J.; Vogel, M.; Werth, G. A miniature electron-beam ion source for in-trap creation of highly charged ions. Rev. Sci. Instrum.
**2006**, 77, 03A901. [Google Scholar] [CrossRef] - Buchauer, L. Konstruktion einer kompakten Elektronenstrahl-Ionenfalle mit Permanentmagneten für Fluoreszenzmessungen. Bachelor’s Thesis, Ruprecht-Karls-Universität, Heidelberg, Germany, 2012. [Google Scholar]
- Von Lindenfels, D.; Vogel, M.; Quint, W.; Birkl, G.; Wiesel, M. Half-open Penning trap with efficient light collection for precision laser spectroscopy of highly charged ions. Hyperfine Interact.
**2014**, 227, 197–207. [Google Scholar] [CrossRef]

**Figure 1.**Five-electrode cylindrical Penning-trap with attached electronics for ion detection and cooling.

**Figure 2.**Fourier transform of the noise voltage across the axial resonance circuit attached to a trap electrode in the presence of a single trapped ${}^{12}$C${}^{5+}$ ion. The ion shortens the noise of the resonator and generates a dip signal with a full width at half maximum (FWHM) of 0.5 Hz. The total oscillation frequency of about 670 kHz can be determined from the fitted lineshape (red line) with an uncertainty of 100 mHz. The insert shows the complete resonator and the corresponding fit (blue line).

**Figure 3.**Detection of three induced spin flips of a single ${}^{12}$C${}^{5+}$ ion by a change of the ion’s axial oscillation frequency. The frequency change amounts to 0.58 Hz in a total frequency of 411 kHz. Every four axial frequency measurement cycle microwaves around the Larmor frequency are induced.

**Figure 4.**Illustration of the trap setup. (

**Left**) Overview of the superconducting magnet setup with the Penning-trap located in the magnetic field center; (

**Right**) Trap chamber with Penning-trap setup.

**Figure 5.**The ALPHATRAP Penning-trap setup. The trap chamber (

**right inset**) is connected to the room-temperature beamline (green); A cryogenically operateable valve (

**left inset**) allows for sealing the cryogenic section during the measurement. This limits the in-flux of rest-gas atoms from the room-temperature section and allows reaching the close-to-perfect vacuum conditions required for the storage of heavy highly-charged ions. To provide optimal mechanical stability, the cryogenic section is cooled by a specially developed liquid helium cryostat.

**Figure 6.**Overview of the ALPHATRAP ion sources. The tt-EBIT provides few-electron ions of low to medium Z. Recently, the extraction and transport of highly charged ions from the tt-EBIT to the end of the room-temperature beamline has been demonstrated. The connection to the HD-EBIT gives access to few-electron ions up to the highest Z.

**Figure 7.**Schematic of the setup inside the superconducting magnet (

**left**) and schematic of the Penning-trap arrangement (

**right**).

**Figure 8.**Charge state spectrum of in-trap created argon ions (

**left**) and fine structure level scheme with Zeeman substructure of boron-like argon (

**right**) with energy level shifts by second and third order contributions indicated (not to scale).

**Table 1.**Calculated contributions to the g-factor of the electron bound in different hydrogen-like ions [31] and comparison to experimental values. The uncertainty in the theory reflects the estimates on the size of uncalculated higher-order terms. For comparison, the experimental results are listed. The uncertainties of the experimental values include the uncertainty of the electron mass at the time of the measurement.

Contribution | ${}^{12}$C${}^{5+}$ | ${}^{16}$O${}^{7+}$ | ${}^{28}$Si${}^{13+}$ |
---|---|---|---|

Dirac value | +1.998 721 354 391 (1) | +1.997 726 003 06 (2) | +1.993 023 571 551(5) |

free QED | +0.002 319 304 358 (1) | +0.002 319 304 358 (1) | +0.002 319 304 358 (1) |

BS-QED | +0.000 000 843 391 (6) | +0.000 001 594 38 (11) | +0.000 005 855 67 (165) |

nuclear size | +0.000 000 000 407 | +0.000 000 001 55 (1) | +0.000 000 020 468 |

nuclear recoil | +0.000 000 087 629 | +0.000 000 116 97 | +0.000 000 205 881 |

theory total | +2.001 041 590 117 (6) | +2.000 047 021 28 (11) | +1.995 348 957 93 (165) |

experiment | +2.001 041 592 44 (232) | +2.000 047 025 4 (46) | +1.995 348 959 04 (81) |

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**MDPI and ACS Style**

Sturm, S.; Vogel, M.; Köhler-Langes, F.; Quint, W.; Blaum, K.; Werth, G. High-Precision Measurements of the Bound Electron’s Magnetic Moment. *Atoms* **2017**, *5*, 4.
https://doi.org/10.3390/atoms5010004

**AMA Style**

Sturm S, Vogel M, Köhler-Langes F, Quint W, Blaum K, Werth G. High-Precision Measurements of the Bound Electron’s Magnetic Moment. *Atoms*. 2017; 5(1):4.
https://doi.org/10.3390/atoms5010004

**Chicago/Turabian Style**

Sturm, Sven, Manuel Vogel, Florian Köhler-Langes, Wolfgang Quint, Klaus Blaum, and Günter Werth. 2017. "High-Precision Measurements of the Bound Electron’s Magnetic Moment" *Atoms* 5, no. 1: 4.
https://doi.org/10.3390/atoms5010004