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Magnetic Moment of the Proton

This experiment aims at a high precision measurement of the magnetic moment, or g-factor of the proton. The g-factor is a dimensionless proportionality constant relating the magnetic moment of a charged particle to its spin S

μ = g (e/2mp) S,

where e/mp is the charge-to-mass ratio of the proton. A single proton is stored in a Penning trap and investigated by means of radio frequency spectroscopy techniques. The trap is cooled to a temperature of 4K which is near absolute zero temperature. This allows arbitrarily long storage times of the particle in the Penning trap. The detection of the proton and the measurement of its eigenfrequencies is carried out with highly sensitive detection systems [1].


The magnetic moment of the free proton has never been measured before. The currently most precise value, which is known at a level of 10 ppb, is extracted from the magnetic field dependence of the hyperfine structure of atomic hydrogen by means of a maser. Based on the measurement the g-factor of the free proton can be extracted theoretically. This experiment is limited by the wall shift. With our experimental method a precision measurement at the level of 1ppb, at least, should be possible.

The experimental principle which is used in our experiment can be directly applied to the antiproton. Currently the magnetic moment of the antiproton is known at the level of one part in a thousand only. With our experimental method the precision can be improved by a factor of one million and a new high precision comparison between the magnetic moment of the proton and the antiproton becomes possible. This will be the first high precision test of the matter-antimatter symmetry with a baryonic vectorquantity.

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The g-factor determination in a Penning trap reduces to the measurement of two frequencies

g = 2 ωLc

the Larmor-frequency ωL and the free cyclotron frequency ωc.

The Larmor-frequency ωL is the precession frequency of the proton spin in the magnetic field of the Penning trap. The frequency can be measured by means of the continuous Stern-Gerlach effect.

The cyclotron frequency ωc is the frequency of the fast circular motion of the proton in the magnetic field. The oscillating particle induces image currents in the electrodes of the Penning trap, which can be measured with highly sensitive detection systems.

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Measurement of the Proton Spin - Continuous Stern-Gerlach Effect

For the measurement of the Larmor frequency, the direction of the proton spin has to be determined. Therefore a strong magnetic field inhomogeneity is superimposed to the Penning trap, which coupled the spin direction of the particle to its axial oscillation frequency. The inhomogeneity is introduced by a ferromagnetic ring electrode, which is made out of Co/Fe. The axial frequency becomes a function of the spin direction. A spin quantum-jump changes the axial frequency by only three ten millionth parts, 200mHz out of 680 kHz. This tiny frequency shift is the measuring signal.

This measuring principle has has been introduced by the Nobel Prize laureate Hans G. Dehmelt who called it "Continuous Stern-Gerlach Effect". It has been applied with great success to measure the magnetic moment of the electron, the positron and the bound electron. These experiments involved magnetic moments on the level of the electron magneton, the Bohr magneton. The magnetic moment of the proton, the nuclear magneton, is 660 times smaller and spin quantum-jumps are thus much harder to detect. Compared to the experiments where an electron is involved, the experimental sensitivity has to be increased by at least 1000 times to detect the spin direction of the proton. Therefore we developed a Penning trap with the strongest magnetic field inhomogeneity which has ever been superimposed to a particle trap. Within a typical distance of 1.5mm the magnetic field changes already by 1 Tesla.

The strong magnetic bottle is required to detect the spin direction of the particle. On the other hand, the large magnetic field inhomogeneity reduces the experimental precision which can principally be achieved. This problem is solved by application of the double-Penning trap technique. The detection of the proton-spin direction and the precision measurements of the Larmor- and the cyclotron frequency are separated in space. The spin direction is measured in the "analysis trap" where the large magnetic bottle is superimposed. The precession measurements of the frequencies are carried out in the "precision trap" where the magnetic field is by a factor of 100000 more homogeneous Between the single experimental steps the particle has to be transported between the Penning traps.

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Setup of the Double-Penning trap

The figure shows the double-Penning trap tower which consists of stacked cylindrical electrodes. The trap stack is mounted on a sealing flange, which can be seen on the left hand side of the figure. Cryogenic feedthroughs are soldered on the flange. By means of these parts biasing voltages and radio frequency signals are guided to the trap. On the right hand side the cylindrical housing of the cyclotron detection system is shown. The whole construction has a total length of about 20 cm.

Double-Penning trap tower
Figure 1: Double-Penning trap tower - click to enlarge


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Experimental Status

With our cryogenic double-Penning trap system, the detection of the very first proton-spin quantum-jumps has been demonstrated [2]. This is a crucial milestone towards the first direct comparison of the magnetic moment of the proton and the antiproton.

In the analysis trap the spin transition rate was measured as a function of the spin-flip drive frequency The result of that measurement is shown in the figure below.

Spin transition rate
Figure 2: Spin transition rate measured as a function of the spin-flip drive frequency

These data correspond to a Larmor frequency measurement with a precision of a tenth of a thousandth part, already. Combined with a measurment of the cyclotron frequency the precision of the g-factor of the antiproton can be improved by a factor of 10. Currently measurements are performed in our laboratory to improve the precision by another factor of 10 at least.



After the successful detection of first spin quantum-jumps with a single trapped proton, a new apparatus is constructed to apply our method to measure the magnetic moment of a single antiproton. Our former Ph.D. student and current collaborator Stefan Ulmer obtained a highly competitive research grant – a RIKEN Initiative Research Unit externer Link – to set up this experiment.

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[1]   S. Ulmer, K. Blaum, H. Kracke, S. Kreim, A. Mooser, W. Quint, C. C.Rodegheri, und J. Walz, Rev. Sci. Instr. 80, 123302 (2009) externer Link
[2]   S. Ulmer, C. C.Rodegheri, K. Blaum, H. Kracke, A. Mooser, W. Quint und J. Walz, Phys. Rev. Lett. 106, 253001 (2011) externer Link
[3]   S. Ulmer, K. Blaum, H. Kracke, A. Mooser, W. Quint, C. C. Rodegheri, and J. Walz, Phys. Rev. Lett. 107, 103002 (2011) externer Link

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