QED test envisaged
Measurements of the bound electron g-factor in highly charged ions
By Georg Wolschin
The precise calculation of the anomaly of the magnetic moment of the electron is one of the great successes of quantum electrodynamics (QED). According to Diracs theory, the free electron g- factor, which determines the magnetic moment in multiples of Bohrs magneton e/2m_e, is exactly equal to 2. In the context of quantum electrodynamics, however, one can calculate the deviation from 2 due to the cloud of virtual photons surrounding the electron in a series expansion in powers of the fine-structure constant a. Here, the probability for emission of a virtual photon is proportional to a. Schwinger had calculated the first-order term in 1949. Today, the expansion coefficients up to the eigth order are known, and quantum electrodynamics has strengthened its role as the most precise physical theory.
Hans Dehmelt and his colleagues succeeded in 1976 to measure the magnetic moment of a single electron caught in a Penning trap. At that time, the theoretical value (with 4 expansion coefficients) was confirmed with an accuracy of about 2 in 10^7. Meanwhile, the accuracy of both theory and experiment has been substantially improved.
Of particular interest is the measurement of the g-factor of an electron bound in the groundstate of a highly charged hydrogen-like ion, which has successfully been performed at GSI. Compared to the magnetic moment of the free electron with g_free = 2.002 319 304 377(4), an additional relativistic correction arises accounting for the relativistic electron motion in the 1s-state. G. Breit had calculated this term in a solution of the Dirac equation for bound states already in 1928. In addition, there are quantum-electrodynamic corrections due to the binding. Both correction terms are proportional to the square of the nuclear charge and hence, measurements with very heavy ions are desirable. Since the determination of the g-factor of the bound electron is not very sensitive to nuclear structure effects (in particular, in case of spherical nuclei with highlying excited states), it is - at least in principle - possible to envisage QED-tests for the bound electron.
g-measurements in traps...
Experimental data on the bound-electron g-factor in light hydrogenic systems are available for hydrogen, deuterium, and the helium^+ - ion. In the present experiment the first measurement of the bound-electron g-factor in a multiply charged hydrogen-like ion (C^5+) stored in a Penning trap was performed. In addition, the g-factor in hydrogen-like lead- and bismuth-ions could be determined through laser spectroscopy of the 1s-hyperfine state at the storage ring ESR. Here, the analysis of the fluorescence signal and hence, the accuracy of the bismuth measurement has been improved such that the relativistic correction to the g-factor can be tested.
With the carbon experiment in a Penning trap, the relativistic correction accounting for the 1s-binding of the electron could already be tested with an accuracy of 3*10^-3. Current and future experiments with the trap should allow for additional tests of the QED-terms for the bound state with sufficient precision.
In a Penning trap (figure 1) the C^5+ - ion is stored in a combination of a homogenous magnetic filed and an electrostatic quadrupole field. The amplitudes of the three eigenmotions of the particle (axial motion, circular drift perpendicular to the magnetic field, and cyclotron motion) are reduced below 50 micrometer by cooling to a temperature of T = 4 Kelvin. The bound-electron g-factor is then determined from the Larmor-precession frequency of its magnetic moment in the magnetic field. The precession frequency is measured by resonant excitation at 104 Gigahertz of the transition between the two spin states (spin up and down) of the bound electron in the 4 Tesla-magnetic field of the Penning trap. A microwave field is used to induce the spin-flips. This method had been pioneered by Dehmelt and collaborators in 1987 in experiments with a single electron in a trap, with a very good accuracy of the resulting free-electron g-factor. The spin-flip transitions (quantum jumps) are observed as small discrete changes of the axial frequency of the stored ion (figure 2).
The experimental g-factor of the bound electron in hydrogen-like carbon deducted from this measurement is g_e(C^5+) = 2.001 040(4). The significant reduction as compared to the value for the free electron is due to the relativistic correction in the 1s- state. Its theoretical value of -0.001 278 646 is confirmed by the measurement with an accuracy of 3*10^-3. Then the quantum-electrodynamic correction has to be added; considering all terms that correspond to the emission and reabsorption of a single virtual photon, it amounts to + 0.84*10^-6. Since the experiment has an accuracy of 2*10^-6, the QED-terms for the bound electron cannot yet be tested. However, through improvements of the measurement of the spin-flip transition, it should be possible to increase the accuracy to about 10^-8, such that a test of quantum electrodynamics for the bound electron in hydrogen-like systems can be envisaged. The best results are expected for heavy ions with spherical nuclei such as Pb^81+.
..and in the storage ring ESR
A complimentary experiment was performed at the storage ring ESR of GSI. Already in 1994, the transition between the hyperfine-levels of the 1s-ground state of heavy hydrogen-like ions such as bismuth^82+ had been induced using a laser beam. Through a measurement of both transition probability (which is proportional to the third power of the transition energy) and transition energy, the g-factor of the bound electron could be determined. However, lifetime-measurements in lead and bismuth performed in 1994 and 1998 were not accurate enough and hence, the bismuth-experiment was repeated.
The ions revolve at an energy of 3oo million electronvolt per particle in the storage ring. Every 0.02 seconds, 5-nanosecond-pulses of a frequency-doubled neodymium-YAG-laser in the straight cooler section of the ESR (figure 3) induce hyperfine-transitions with about 5.1 electronvolt transition energy. The ensuing fluorescence signal received by a photomultiplier is recorded as function of time after laser excitation (figure 4) by two independent methods. From the measured lieftime (decay probability) of the excited 1s-hyperfine level and the transition energy, together with the information about the nuclear spin and the nuclear g-factor, the experimental bound-electron g-factor is determined.
The accuracy of this experiment is now sufficient to successfully test the relativistic correction to the g-factor of the bound electron. For a QED-test the experiment is not yet precise enough. However, taking into account the QED-terms on the one-photon level, one calculates a theoretical bound-electron g-factor of g_theo(Bi^82+) = 1.7310, which agrees with the experimental value in the first three digits, whereas one obtains 1.7281 without the QED-terms.
For the proposed tests of quantum-electrodynamical corrections to the bound-electron g-factor in highly charged heavy hydrogen-like ions, the accuracy has to be considerably improved. Both the experiments with single ions in traps and the ones at the storage ring are of great physical interest, although the precision obtainable in the trap will remain superior.
Fig.1 : Penning trap used for the first measurement of the magnetic moment of the bound electron in hydrogen-like C^5+.
Fig. 2: Measurement of the Larmor precession frequency of the magnetic moment of the bound electron in the magnetic field of the trap through resonant excitation of the transition between spin-up and spin-down states. Small discrete changes of the axial frequency of the stored carbon-ions are observed.
Fig.3: Neodymium-YAG-laser in the ESR used to excite the 1s hyperfine state of bismuth^82+ - ions.
Fig.4:Fluorescence signal of hydrogen-like bismuth^82+ as function of time after laser excitation of the 1s hyperfine state in the laboratory frame. From the resulting half-live of 524,9±2,0 microseconds, the g-factor of the bound electron is determined.
cf. GSI-Nachrichten 3/99 for full article return