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The Electron Mass and Calcium Isotope Shifts pp 5–32Cite as

The g-Factor - Exploring Atomic Structure and Fundamental Constants

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Abstract

Slightly more than 100 years ago Ernest Rutherford in 1911 and Niels Bohr in 1913 made the first fundamental steps to explain the atomic structure of nature (Rutherford, Philos Mag Ser 6 21(125):669–688, 1911, [1]; Bohr, Philos Mag Ser 6 26(151):1–25, 1913, [2]). Since then, enormous efforts have been undertaken, such that the SM nowadays is able to predict properties of elementary particles up to the thirteens digit (Aoyama et al. Phys Rev Lett 109(11):111807, 2012, [3]; Hanneke, et al., Phys Rev Lett 100(12):120801, 2008, [4]; Bouchendira et al., Phys Rev Lett 106(8):080801, 2011, [5]). In the following chapter I will illuminate the present understanding of the fundamental electromagnetic dynamics in atomic structure. The main focus will be set on the present workhorse of the underlying theory, the so-called bound-state quantum electrodynamics (BS-QED): the bound-electron g-factor.

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Notes

  1. 1.

    Relative uncertainties of the g-factor of the free electron: \((\delta g/g)_{\text {theo}}=0.8\cdot 10^{-12}\) and \((\delta g/g)_{\text {exp}}=0.3\cdot 10^{-12},\) see also Sect. 2.2.

  2. 2.

    In the following a homogeneous magnetic field will always point in z-direction.

  3. 3.

    Even stronger fields can be reached by muonic ions or even more exotic atomic systems. However, the measurement time is strongly limited by their short lifetimes.

  4. 4.

    Here, the relativistic expectation value of \(1/r^2\) has been used from [15]. For nuclear charges larger than 30, such a relativistic calculation is essential, e.g. for uranium \((Z=92)\) the mean electric field derived by the non-relativistic Schrödinger equation is a factor of 2.8 smaller than the relativistic calculation. The mean electric field of a \(1s_{1/2}\) electron is:

    figure a
  5. 5.

    The following explanation roughly outlines the 1 / Z parameter of the perturbative expansion of the interelectronic interaction: In general, the parameter of the perturbative expansion is given by the matrix-element of the perturbation operator over the typical energy difference. The matrix-element of the Coulomb repulsion scales with \(\alpha /<r_{12}>,\) where \(<r_{12}>\) (distance between two electrons) scales inversely with \((\alpha Z)^{-1}.\) The energy difference typically scales with \((\alpha Z)^2.\)

  6. 6.

    In this paragraph I quote the theoretical g-factors, which have been published together with the measurements at that time. Most calculations improved in the last years.

  7. 7.

    In principle, also nuclear polarizations contribute to isotope shifts. However, at the present level of precision this contribution can be neglected.

  8. 8.

    Binding energies from NIST table [70]: \(E_{{\text {bind}}}=18804(4)-11756.4449(80)=6747.5(4.0)\,{{\text {eV}}}\) and \(1{{\text {u}}}=931 494 061 (21){{\text {eV/c}}}^2.\)

  9. 9.

    In general the binding energies of different isotopes varies due to their different masses (mass shift) and their different charge distributions (field shift). For calcium the field shift dominates, which scales with \(Z^5\) to \(Z^6\) [77]. Since the binding energy of a 1s electron in uranium differs by 200 eV between a hypothetical point-charge distribution and the measured charge distribution [94], the field shift for 1s electrons in calcium isotopes should be smaller than \(\varDelta m/m=200\,{\text {eV}}/92^5\cdot 20^5/40\,{\text {GeV}}=2.6\cdot 10^{-12}\). For 2s electrons this effect is even smaller.

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  1. Department of Stored and Cooled Ions, Max-Planck-Institut für Kernphysik, Heidelberg, Germany

    Florian Köhler-Langes

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Köhler-Langes, F. (2017). The g-Factor - Exploring Atomic Structure and Fundamental Constants. In: The Electron Mass and Calcium Isotope Shifts. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-50877-1_2

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