Determination of the g-Factor of the bound Electron in highly-charged ions

Motivation

The 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. The QED is able to provide extremely accurate predictions for physical measurands and until now all measurements have confirmed the QED predictions. Nevertheless, it seems imaginable and possible that the QED fails under extreme conditions and transitions into a superior theory.

This experiment aims to test the QED with very high precision under conditions as extreme as possible, especially extremely high field strengths, and thus to give an outlook on the limits of validity of this 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 10¹⁶ V/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 exactly by QED and also can be experimental measured with comparable accuracy. The comparison of those two values represents to date the most accurate test of bound-state QED (BS-QED) [6].

In the past, the g-factors of hydrogen-like carbon (H. Häffner et al. [4]) and oxygen (J. Verdú et al. [5]) have already been measured. These measurements are confirming the validity of the theoretical predictions ([2], [3]). This experiment has been completely renewed and improved in recent years and allows now to measure the g-factors of significantly heavier systems with drastically increased precision. These improvements finally allowed the measurement of the g-factor of ²⁸Si¹³⁺, which provides a much more sensitive test of QED due to the higher charge of the ion and the simultaneous improvement of the precision of the experimental value and the theoretical prediction.

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Introduction

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 (top). The gold-plated electrodes and the sapphire rings for electrical insulation (bottom). - click for bigger version

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 measurement will be performed on a single hydrogen-like silicon ion 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

As shown in figure 6 below the double Penning trap system is inserted into the bore of a superconducting magnet and cooled to 4K 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. - click for bigger version

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Control System

The experiment is fully automated and runs computer-controlled. For this purpose a flexible control system based on Labview and MATLAB has been developed. It allows to control the purchased devices like signal- and pulse generators and -analysers as well as the self-constructed devices like voltage sources for the trap electrodes and the cryogenic electronics. This enables a fast and flexible realization of novel measurement methods.

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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. The development and implementation of novel detection techniques results in a drastic improvement of the achievable precision [7].

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Outlook

With the improved setup many fascinating measurements are possible. On the one hand it is planned to extend the measurement program towards even higher charged systems, e.g. calcium, in order to test the BS-QED in even stronger fields. Furthermore, it is possible to test the correlation between the electrons in many-particle systems, especially in lithium-like ions. The measurement of the g-factor of different isotopes of the same element allows an explicit test of the corresponding contributions of the BS-QED.

In addition, the improved precision provides access to fundamental physical quantities , e.g. the electron mass as well as the fine structure constant α. For example, the re-measurement of the g-factor of carbon by using the new developed phase-sensitive measurement method PnA [7] promises the determination of the electron mass with a precision of about 30 ppt, which improves the literature value by more than one order of magnitude.

In order to realize the numerous measurement tasks and to achieve further improvements, we currently intend to strengthen our team with diploma and/or PhD students.

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References

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	Phys. Rev. 72, 241 (1947)
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	T. Beier, H. Häffner, N. Hermanspahn, I. Lindgren, H. Persson, S. Salomonson and P. Sunnergren
	Phys. Rev. A 62, 032510 (2000)
[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)
[4]	High-accuracy measurements of the magnetic moment anomaly of the electron bound in hydrogenlike carbon.
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	Phys. Rev. Lett. 85, 5308 (2000)
[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)
[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)
[7]	Phase-Sensitive Cyclotron Frequency Measurements at Ultralow Energies
	Sven Sturm, Anke Wagner, Birgit Schabinger, and Klaus Blaum
	Phys. Rev. Lett. 107, 143003 (2011)