Magnetic Moment of the Proton

The aim of this experiment is to measure the g-factor of the proton with a fractional accuracy better than 1 part-per-billion (ppb).
To achieve this goal, we trap a single proton in a cryogenic Penning trap, a static ion trap consisting of an electrostatic potential created by a stack of electrodes and a constant magnetic field maintained by a superconducting magnet.The cryogenic environment of the trap at 4 K allows a proton to be trapped for well over a year, and enables low-noise, non-destructive, and high fidelity detection of the proton.


(1) Precise measurements of fundamental quantities such as particle magnetic moments provide stringent tests of theoretical models..

(2) The techniques used in our experiment can be applied to similar measurements with antiprotons. A comparison of the magnetic moments of the proton and antiproton is an important test of the fundamental symmetry (CPT symmetry) of matter and antimatter.



The g-factor of the proton can be determined from the ratio of frequencies measured in a Penning trap: the Larmor frequency and the free-cyclotron frequency.

The Larmor frequency is the precession frequency of the proton spin in the magnetic field of the trap. To measure this frequency, radio-frequency field is applied to the proton. When the rf field is resonant with the Larmor frequency, the proton undergoes spin flips. These spin flips can be detected through subtle modifications to the proton's axial motion in the trap.

The free-cyclotron frequency is the frequency of the fast circular motion of the proton in the external magnetic field. The motion of the proton, which carries charge, induces currents in the trap electrodes. To determine the free cyclotron frequency, these tiny currents are directly detected with superconducting resonators.

Via the free cyclotron frequency measurement, the proton itself is used as a precision magnetometer for the external magnetic field.



Determination of the spin state:

To measure the Larmor frequency one has to determine the direction of the proton’s spin. The spin of the proton is coupled to the axial motion of the proton in the trap, via superposition of an inhomogeneous magnet field, a so-called magnetic bottle, on the homogenous background field of the trap. Depending on the spin state, the axial frequency of the proton shifts up- or downwards. This small frequency change is the signal needed for the determination of the proton spin state.

This technique is the so-called continuous Stern-Gerlach effect, it was developed by Dehmelt and co-workers in the 80s for the measurement of the magnetic moment of the electron.

The magnetic moment of the proton is 660 times smaller than for the electron, and the resulting axial frequency shift due to a spin-flip is significantly smaller in a given magnetic bottle. Thus, a strong magnetic bottle is needed to detect the direction of the proton spin state. The precision of the Larmor frequency measurement, however, is limited in an inhomogeneous magnetic field. This problem can be solved by using a double Penning trap technique: the spin state detection and the precision measurement of the proton eigenfrequencies are spatially separated. The direction of the spin is measured in the analysis trap, where we have a superimposed magnetic bottle. The precision measurement of the free cyclotron frequency and the spin-flip irradiation, however, are conducted in the precision trap, where the magnetic field is homogeneous. The proton can be moved between the two traps with a series of transport electrodes.


The double Penning trap:

The figure below shows a cut through the double Penning trap, which consists of cylindrical trap electrodes stacked on one another, the surfaces of the electrodes are gold-plated. The electrodes are separated by sapphire spacers (blue), allowing for application of different voltages to each electrode. The transport section is situated between the analysis trap and precision trap. The distance between the trap centers is 43.7 mm.



State of the experiment:

We recent successfully measured the g-factor of a single proton stored in a Penning trap to high precision for the first time [1]. This incident marks a decisive breakthrough on the pathway to a highly precise CPT test in the baryonic sector using magnetic moments.




In the figure above, the spin flip probability is plotted against the g-factor, calculated from the Larmor frequencies and cyclotron frequencies, normalized with its previous CODATA-value. The g-factor of the proton was extracted from this resonance using a Maximum-Likelihood-Fit, with a relative precision of 3.3 ppb.

A similar experiment with an antiproton would increase the precision of the antiproton g-factor by a factor of 10 000. This is the aim of the BASE experiment being set up at CERN starting in 2013 (Geneva).

The next important experimental milestone is a further increase in precision. To achieve this goal we are currently upgrading the apparatus with new and improved detection systems as well as a self-shielding coil to stabilize the magnetic field. With these improvements in place, the proton g-factor can be determined with a relative precision of 100 ppt or better.

[1] A. Mooser, S. Ulmer, K. Blaum, K. Franke, H. Kracke, C. Leiteritz, W. Quint, C. C. Rodegheri, C. Smorra and J. Walz. Direct high-precision measurement of the magnetic moment of the proton. Nature 509, 596 (2014).


Collaboration with:


• Prof. Dr. Klaus Blaum / MPI for nuclear physics, Heidelberg
• Dr. Wolfgang Quint / GSI Darmstadt.
• Dr. Stefan Ulmer / BASE Collaboration, CERN (Geneva).