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The g-Factor of highly charged ions

precision trap with hydrogen-like ion

Motivation and Introduction

Quantum electrodynamics (QED) is one of the most important fundamental theories of the standard model. It describes the interaction between charged particles and electromagnetic fields at all energies and field strengths. QED is able to provide extremely accurate predictions for physical observables and until now all measurements have confirmed the QED predictions. Nevertheless, it is anticipated that QED might fail under extreme conditions and transitions into a superior theory.

The experiments in our group aim to test QED with very high precision under conditions as extreme as possible, especially extremely high field strengths. Since effects of bound-state QED (BS-QED) and hypothetical nonlinearities in theory scale with the field strength this gives an outlook on the limits of validity of fundamental theory. In order to do this, the bound electron in a highly charged ion, which is exposed to the extreme field strengths of the nucleus (up to 1016V/cm), is used as test object. The (spin) g-factor of this electron (the strength of the magnetic interaction of the spin) can be predicted very precisely and also can be experimental measured with comparable accuracy. The comparison of values resulting from theory and experiment can thus be used as an accurate test of BS-QED. Moreover, fundamental constants such as the electron mass me and the fine structure constant α enter the theoretical calculations. In the converse argument, experimental results can therefore be used for precise determinations of these fundamental quantities.

The g-factors of hydrogen-like carbon (H. Häffner et al. [4]) and oxygen (J. Verdú et al. [5]) and recently, after a major upgrade of the apparatus, hydrogen-like silicon have already been measured. These measurements are confirming the validity of the theoretical predictions ([2], [3]), yielding the most stringent test of QED in strong fields to date. The experiment, which is located at the university of Mainz, has been completely renewed and improved in recent years and allows now to measure the g-factors of significantly heavier systems up to hydrogenl-like calcium with drastically increased precision. At the MPIK in Heidelberg, a next-generation experiment, called ALPHATRAP, is currently developed, opening the way for a highly precise determination of the g-factor in extremely heavy systems up to hydrogenlike Pb.

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The g-factor

The g-factor - also called Landé factor - describes the ratio of the magnetic moment μ of a particle and the total angular momentum J: μ = - gj (e/2me) J. J is the result of the vectorial composition of orbital angular momentum and spin: J = L + S (Figure 1).

Coupling of the vectors of spin S and orbital angular momentum L to the total angular momentum J according to the vector model

Figure 1: Coupling of the vectors of spin S and orbital angular momentum L to the total angular momentum J according to the vector model. The vectors S and L show a precessional motion around vector J.

In our experiment we analyse the spin motion. In the case of an electron we have |S| = ½ and the corresponding magnetic moment is denoted by μS. In an external magnetic field B the spin can take on but two discrete orientations out of quantum mechanical reasons, namely parallel or antiparallel to the direction of B (Figure 2).

The spin of a spin ½ particle and thus the magnetic moment have two potential orientations in an external magnetic field.

Figure 2: The spin of a spin ½ particle and thus the magnetic moment have two potential orientations in an external magnetic field. The external magnetic field has the strength B0 in z-direction. Here, gs is the g-factor and μB denotes the Bohr magneton.

These two states correspond to a Zeeman splitting with an energy L, where νL is just the classical Larmor precession frequency of a magnetic dipole. Determining this frequency the g-factor gS can be extracted from the relation above.

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Measurement Principle

Electrically charged particles can be stored by a combination of a weak electrostatic and a strong homogeneous magnetic field. To this end we use a Penning trap system (see figure 3).

Schematic illustration of the Penning trap system used in the Mainz setup
Figure 3: Schematic illustration of the Penning trap system used in the Mainz setup (top). The gold-plated electrodes and the sapphire rings for electrical insulation (bottom).

By means of suitable electronic detection methods both the detection of the trapped particles and the reduction of their kinetic energy (cooling) down to values less than one meV are possible.

The measurements are performed on a single ions stored in a cryogenic Penning trap - the precision trap (PT). The ion exhibits an orbital motion because of the presence of the magnetic field, the so called cyclotron motion. This frequency of motion νc can be detected non-destructively by means of induced image charges and highly sensitive, partly superconductive detection. In a following step the Larmor frequency is determined. To this end the ion is adiabatically transported to the analysis trap (AT), where an inhomogeneity is purposely superimposed to the magnetic field to force the coupling of the spin direction to the motional frequencies of the ion. There, the spin of the ion has to be flipped by a suitable microwave irradiation. The spin flip can be observed in the form of quantum leaps exploiting the extremely accurate determination of the motional frequencies of the stored ion.

Quantum leaps of electron spin
Figure 4: Quantum leaps of electron spin, non-destructively detected by the continuous Stern-Gerlach effect as tiny change of the axial eigenfrequency of the stored ion.

Plotting the probability rate of successful spin flip vs. the frequency of the excitation field, the maximum of the spin flip rate (subject to certain corrections of the curve form) represents the Larmor frequency νL. The experimentally determined Larmor and cyclotron frequency yield the g-factor and thus the magnetic moment according to the simple relation:

g = 2(νLc)·(q/M)ion·(m/e)e- ,

where (q/M)ion and (e/m)e- are the charge-to-mass ratio of the ion and the electron, respectively.

Measured g-factor resonance of hydrogen-like silicon.
Figure 5: Measured g-factor resonance of hydrogen-like silicon 28Si13+. A number of such resonances allows the extraction of the g-factor with a precision of 2.6 · 10-10 [6].

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Experimental Setup in Mainz

As shown in figure 3 the measurements of the g-factor are performed in a double Penning-trap system consisting of cylindrical electrodes. The ions with high charge states are produced internally by a combination of a target and an electron beam ion source (EBIS). In figure 6 the double Penning trap system is inserted into the bore of a superconducting magnet and cooled to 4 K by thermal contact with a liquid helium dewar. The complex electronic system is divided in cryo-electronics (above the vacuum chamber) and the room temperature electronics (attached to the hat).

Drawing of the experimental setup in Mainz.
Figure 6: Drawing of the experimental setup in Mainz.

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The experiment is fully set up and has provided the first data in 2011. The new improved cryogenic detection electronics shows a spectacular improvement of detection sensitivity and allows for the first time the measurement at very low temperatures close and below 4.2 K, which is the temperature of the cooled setup. Thereby, the apparatus is constantly being improved. A recently implemented superconducting self-shielded magnetic coil reduces the unwanted influence of external fluctuations of the magnetic field by almost 3 orders of magnitude. Moreover, the development and implementation of novel phase-sensitive detection technique called PnA [7] lead to a drastic improvement of the achievable precision of the measurements.

Altogether, these improvements finally allowed the measurement of the g-factor of 28Si13+, where the g-factor of the electron is measured with a relative uncertainty of 10-11 [6]. This measurement currently provides the most sensitive test of BS-QED. In another measurements series, the g-factor of the electron in lithiumlike 28Si11+ was measured to a level of 10-9 [8], serving as an extremely precise test of relativistic many-electron calculations in a magnetic field. In addition, the improved precision provides access to fundamental physical quantities as the electron mass. The re-measurement of the g-factor of carbon by use of the new PnA method resulted in the determination of the electron mass with a precision of about 30 parts-per-trillion, which improves the literature value by more than one order of magnitude [9,10]. Recently, the g-factors of the lithiumlike calcium isotopes 40Ca17+ and 48Ca17+ were measured to directly test the relativistic recoil effect [11].

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Pushing g-factor experiments to heavier systems with ionization energies over 100 keV, an ex-situ ion production is required. For this purpose, two experiments are under construction within the scope of the HITRAP initiative. The ALPHATRAP experiment at the Max Planck Institute of Nuclear Physics will be coupled to the existing Heidelberg EBIT (HD-EBIT). In that way, the g-factors of all ion systems delivered from the EBIT can be measured with highest precision via the continuous Stern-Gerlach effect. The ARTEMIS external Link experiment is located at the HITRAP facility, which is constructed at the GSI in Darmstadt. It will be specialized on laser-microwave double-resonance spectroscopy for heaviest highly charged ions.

The g-factor experiment in Mainz is currently upgraded. In an extended Penning trap tower, the atomic proton mass will be measured with a relative uncertainty of better than 1·10-11. Here, the cyclotron frequencies of the proton and a highly charged carbon ion will be measured alternately in a highly compensated Penning trap with unchanged field configuration. The impact of disturbing magnetic field fluctuations is significantly reduced by the simultaneous cyclotron frequency measurement of a highly charged ion located in a neighbouring reference trap. Subsequently, we aim for an improved mass of the neutron by the determination of the atomic mass of deuterium and its nuclear binding energy.

Schematic illustration of the new Penning trap system for the high precision measurement of the atomic mass of the proton.
Figure 7: Schematic illustration of the new Penning trap system for the high precision measurement of the atomic mass of the proton.

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For the measurement of the g-factor of the electron in extremely heavy ions, a novel cryogenic Penning trap, called ALPHATRAP, is currently set up at MPIK. It features an access to the Heidelberg EBIT [12], which can deliver even the heaviest highly charged ions up to 208Pb81+. With these systems it becomes possible to probe QED at the boundary of our knowledge in the most extreme fields. The ions are transported via a beamline including ion-optic elements and diagnostic stations to ALPHATRAP. There, the trap region is surrounded by a liquid-helium cryostat to provide a cryogenic environment. To ensure sufficient vacuum conditions for the storage of highly charged ions despite the connection to the room-temperature vacuum, a cryogenic vacuum valve is developed to cut the beamline off the trap setup. Besides mechanical differences due to the open system most technical elements can in principal be adopted from the Mainz setup. However, adjustments are necessary for several issues. For instance, the higher charge states of the ions lead to a corresponding systematic shift of the ion frequencies due to the image-charge effect and, thus, a larger trap diameter is needed compared to the Mainz-trap. Another example is the even more complex and extensive detection system, which is currently under development.

The new ALPHATRAP setup.
Figure 8: The new ALPHATRAP setup. Ions can be injected from the Heidelberg HD-EBIT via a beam-port from the top. The design is based on the Mainz apparatus, but numerous improvements and new technologies have been implemented.

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Open positions and contact

In order to realize the numerous measurement tasks and to achieve further improvements both for the Mainz setup and for the development of ALPHATRAP, we currently intend to strengthen our team with diploma and/or PhD students. Feel free to contact Dr. Sven Sturm () and Alexander Egl ().

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[1] Fine structure of the hydrogen atom by a microwave method
W.E. Lamb and R.C. Retherford
Phys. Rev. 72, 241 (1947) externer Link
[2] gj factor of an electron bound in a hydrogenlike ion
T. Beier, H. Häffner, N. Hermanspahn, I. Lindgren, H. Persson, S. Salomonson and P. Sunnergren
Phys. Rev. A 62, 032510 (2000) externer Link
[3] Self-energy correction to the bound-electron g factor in H-like ions
V.A. Yerokhin, P. Indelicato and V.M. Shabaev
Phys. Rev. Lett. 89, 143001 (2002) externer Link
[4] High-accuracy measurements of the magnetic moment anomaly of the electron bound in hydrogenlike carbon.
H. Häffner, T. Beier, N. Hermanspahn, H. J. Kluge, W. Quint, S. Stahl, J. Verdú and G. Werth
Phys. Rev. Lett. 85, 5308 (2000) externer Link
[5] Electronic g factor of hydrogenlike Oxygen 16O7+
J. Verdú, T. Beier, S. Djekic, H. J. Kluge,W. Quint, S. Stahl, T. Valenzuela, M. Vogel and G. Werth
Phys. Rev. Lett. 92, 093002 (2004) externer Link
[6] g Factor of Hydrogenlike 28Si13+
S. Sturm, A. Wagner, B. Schabinger, J. Zatorski, Z. Harman, W. Quint, G. Werth, C. H. Keitel, and K. Blaum
Phys. Rev. Lett. 107, 023002 (2011) externer Link
[7] Phase-Sensitive Cyclotron Frequency Measurements at Ultralow Energies
Sven Sturm, Anke Wagner, Birgit Schabinger, and Klaus Blaum
Phys. Rev. Lett. 107, 143003 (2011) externer Link
[8] g Factor of Lithiumlike Silicon 28Si11+
A. Wagner, S. Sturm, F. Köhler, D. A. Glazov, A. V. Volotka, G. Plunien, W. Quint, G. Werth, V. M. Shabaev, and K. Blaum
Phys. Rev. Lett. 110, 033003 (2013) externer Link
[9] High-precision measurement of the atomic mass of the electron
S. Sturm, F. Köhler, J. Zatorski, A. Wagner, Z. Harman, G. Werth, W. Quint, C. H. Keitel, and K. Blaum
Nature 506, 467-470 (2014) externer Link
[10] The electron mass from g-factor measurements on hydrogen-like carbon 12C5+
F. Köhler, S. Sturm, A. Kracke, G. Werth, W. Quint and K. Blaum
J. Phys. B: At. Mol. Opt. Phys. 48, 144032 (2015) externer Link
[11] Isotope dependence of the Zeeman effect in lithium-like calcium
F. Köhler, K. Blaum, M. Block, S. Chenmarev, S. Eliseev, D. A. Glazov, M. Goncharov, J. Hou, A. Kracke, D. A. Nesterenko, Y. N. Novikov, W. Quint, E. Minaya Ramirez, V. M. Shabaev, S. Sturm. A. V. Volotka and G. Werth
Nat. Commun. 7, 10246 (2016) externer Link
[12] Optimization of the charge state distribution of the ion beam extracted from an EBIT by dielectronic recombination
J. R. Crespo López-Urrutia, J. Braun1, G. Brenner, H. Bruhns, A. Lapierre, A. J. González Martı́nez, V. Mironov, R. Soria Orts, H. Tawara, M. Trinczek and J. Ullrich
Rev. Sci. Instrum. 75, 1560 (2004) externer Link