The g-Factor of highly charged ions

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 m_e 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 = - g_j (e/2m_e) J. J is the result of the vectorial composition of orbital angular momentum and spin: J = L + S (Figure 1).

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 μ_S. 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).

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, g_s is the g-factor and μ_B denotes the Bohr magneton.

These two states correspond to a Zeeman splitting with an energy hν_L, where ν_L is just the classical Larmor precession frequency of a magnetic dipole. Determining this frequency the g-factor g_S 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).

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 ν_c 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.

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 ν_L. The experimentally determined Larmor and cyclotron frequency yield the g-factor and thus the magnetic moment according to the simple relation:

g = 2(ν_L/ν_c)·(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.

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).

Figure 6: Drawing of the experimental setup in Mainz.

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Status

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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Outlook

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  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.

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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ALPHATRAP

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.

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References

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