Measurement of the g Factor of the Bound Electron in Hydrogen-like ...

physik.uni.mainz.de

Measurement of the g Factor of the Bound Electron in Hydrogen-like ...

Hyperfine Interactions 146/147: 47–52, 2003.

© 2003 Kluwer Academic Publishers. Printed in the Netherlands.

47

Measurement of the g Factor of the Bound Electron

in Hydrogen-like Oxygen 16 O 7+

J. VERDÚ 1,2 , T. BEIER 2 ,S.DJEKIC 1 , H. HÄFFNER 2 , H.-J. KLUGE 2 ,

W. QUINT 2 , T. VALENZUELA 1 and G. WERTH 1

1 Institut für Physik, Universität Mainz, D-55099 Mainz, Germany

2 GSI, Planckstr. 1, D-64291 Darmstadt, Germany

Abstract. The measurement of the g factor of the electron bound in a hydrogen-like ion is a highaccuracy

test of the theory of Quantum Electrodynamics (QED) in strong fields. Here we report on the

measurement of the g factor of the bound electron in hydrogen-like oxygen 16 O 7+ . In our experiment

a single 16 O 7+ ion is stored in a Penning trap. Quantum jumps between the two spin states (spin up

and spin down) are induced by a microwave field at the spin precession frequency of the bound

electron. The g factor of the bound electron is obtained by varying the microwave frequency and

counting the number of spin flips. Our experimental value for the g factor of the bound electron

is g exp ( 16 O 7+ ) = 2.000 047 026(4). The theoretical prediction from non-perturbative bound-state

QED calculations is g th ( 16 O 7+ ) = 2.000 047 0202(6).

Key words: magnetic moment, g factor, quantum electrodynamics, fundamental constants.

1. Introduction

A few years ago we started an experimental and theoretical program to investigate

the magnetic moment anomaly of the bound electron in highly charged ions [1].

We constructed a Penning-trap quantum jump spectrometer for highly charged ions

and, for the first time, applied the continuous Stern–Gerlach effect [2] to an atomic

ion [3]. With a novel double-trap technique we determined the g factor of the bound

electron in hydrogen-like carbon 12 C 5+ with an accuracy on the ppb level [4]. We

have performed bound-state QED calculations of the g factor of the bound electron

in hydrogen-like ions with different nuclear charge Z up to uranium 238 U 91+ in a

non-perturbative treatment [5, 6]. Recent theoretical progress made it possible to

determine the electron’s mass in atomic units with unprecedented accuracy from

the measured g factor of the electron in carbon ( 12 C 5+ ) [7, 8].

In this contribution we report on our measurement of the g factor of the bound

electron in hydrogen-like oxygen 16 O 7+ which is part of our efforts to extend the

g-factor measurements to heavy hydrogen-like ions [9].


48 J. VERDÚ ETAL.

2. Penning-trap quantum jump spectrometer

In a Penning trap a charged particle is stored in a combination of a homogeneous

magnetic field B 0 and an electrostatic quadrupole potential [10]. The magnetic field

confines the particle in the plane perpendicular to the magnetic field lines, and the

electrostatic potential in the direction parallel to the magnetic field lines. The three

eigenmotions that result are the trap-modified cyclotron motion (frequency w + ),

the magnetron motion (frequency w − ), which is a circular E × B drift motion perpendicular

to the magnetic field lines, and the axial motion (parallel to the magnetic

field lines, frequency w z ). The free-space cyclotron frequency w c = (Q/M)B 0 of

an ion with charge Q and mass M in a magnetic field B 0 can be determined from a

combination of the trapped ion’s three eigenfrequencies w + , w z ,andw − with the

formula w 2 c = w2 + + w2 z + w2 − .

The principle of the continuous Stern–Gerlach effect is based on a coupling

of the magnetic moment µ of the particle to its axial oscillation frequency w z in

the Penning trap. This coupling is achieved by a quadratic magnetic field component

(“magnetic bottle”) superimposed on the homogeneous magnetic field B 0

of the Penning trap, B(z) = B 0 + β 2 z 2 (Figure 1). Due to the interaction of the

z-component µ z of the magnetic moment with the “magnetic bottle” term the

trapped ion possesses a position-dependent potential energy V m =−µ z (B 0 +β 2 z 2 ),

which adds to the potential energy V el of the ion in the electrostatic well. Therefore,

the effective trapping force is modified by the magnetic interaction, and the axial

frequency of the trapped ion is shifted upwards or downwards, depending on the

Figure 1. Sketch of the electrode structure and potential distribution of the double trap. Spin flips are

induced in the precision trap by a microwave field and detected in the analysis trap via the continuous

Stern–Gerlach effect.


MEASUREMENT OF THE g FACTOR OF THE BOUND ELECTRON 49

sign of the z-component µ z of the magnetic moment. This axial frequency shift is

given by

δw z = β 2µ z

, (1)

Mw z

where w z is the unshifted axial frequency. Due to the scaling with 1/M, the frequency

shift becomes smaller for heavier hydrogen-like ions.

In our Penning trap apparatus, which has been described in [11], there are two

positions in the stack of cylindrical electrodes where ions can be trapped (Figure 1).

In the precision trap, a single 16 O 7+ ion is prepared, and the spin-flip transition of

the bound electron is excited by applying a microwave field. The single 16 O 7+ ion

is then transported along the magnetic field lines to the analysis trap. The ring

electrode of this trap is made out of ferromagnetic material (nickel) to produce the

quadratic component of the magnetic field (β 2 = 0.01 T/mm 2 ) which is necessary

to observe the continuous Stern–Gerlach effect. The axial oscillation frequency w z ,

of the single 16 O 7+ ion in the analysis trap is measured non-destructively with an

electronic detection method through the image currents which are induced in the

trap electrodes by the particle motion [12]. A LCR circuit resonant at w z , = 2π ×

369 kHz (with quality factor Q = 2400) is attached to one of the trap electrodes to

optimize the detection sensitivity. The ion’s axial frequency is determined in a fast

Fourier transform (FFT) of the signal across the resonance circuit Figure 2.

3. g-Factor measurement

The quantum state of the 16 O 7+ ion, i.e. the magnetic quantum number m s of the

bound electron in the 1s 1/2 ground state, can be determined non-destructively in the

analysis trap in measurements of the axial frequency of the trapped ion. Transitions

between the two spin states m s = ±1/2 are induced by a microwave field (at

104 GHz) resonant with the Larmor precession frequency w L of the bound electron

¯hw L = g e¯h B = gµ B B. (2)

2m e

Here, g is the g factor of the bound electron and µ B = e¯h/2m e is the Bohr

magneton. The spin-flip transitions are observed as discrete changes of the ion’s

axial frequency. Figure 2 shows such a quantum jump observed via the continuous

Stern–Gerlach effect. In the case of hydrogen-like oxygen 16 O 7+ , the measured

axial frequency shift for a transition between the two quantum levels is w z (↑) −

w z (↓) = 2π × 0.45 Hz. We have plotted the axial frequency versus time when we

irradiate the ion with a microwave field at the Larmor precession frequency of the

bound electron. Frequency jumps of 0.45 Hz can be clearly observed (Figure 3).

The observation of the continuous Stern–Gerlach effect on the hydrogen-like

carbon ion 16 O 7+ makes it possible to determine its electronic g factor to high

accuracy. Using the cyclotron frequency w c of the ion for the calibration of the


50 J. VERDÚ ETAL.

Figure 2. Axial frequencies of a single 16 O 7+ ion for different spin directions. The averaging time

for each resonance curve was 1 min (FFT: fast Fourier transform).

Figure 3. Quantum non-demolition measurement of the spin state: the spin-flip transitions are

observed as small discrete changes of the axial frequency of the stored 16 O 7+ ion.

magnetic field B,theg factor of the bound electron can be calculated from the ratio

of the Larmor precession frequency w L of the electron and the cyclotron frequency

w c of the 16 O 7+ ion when the mass ratio of the ion and the electron is known

g = 2 · wL · Q/M . (3)

w c e/m e

A resonance spectrum of the Larmor precession frequency w L of the bound

electron is obtained in the following way. First, we determine the spin direction in

the analysis trap via the continuous Stern–Gerlach effect. The hydrogen-like ion is

then transferred to the precision trap where spin flips are induced by a microwave

field at frequency w mw ≈ w L . Then the 16 O 7+ ion is transferred back to the analysis

trap where the spin state is analyzed again. Now the ion is moved back to the

precision trap, and the measurement cycle is started again. The number of spin-flip

transitions which occurred in the precision trap is counted. Then the microwave


MEASUREMENT OF THE g FACTOR OF THE BOUND ELECTRON 51

Figure 4. Fourier transform of the voltage induced in one of the electrodes from the cyclotron motion

of a single 16 O 7+ ion stored precision trap. The fractional line width of the cyclotron resonance is a

few ppb.

Figure 5. Larmor resonance spectrum measured in the precision trap. Spin-flip transitions are driven

by a microwave field at frequency w mw . The cyclotron frequency wc

ion of the 16 O 7+ ion is measured

simultaneously for magnetic-field calibration. The spin-flip probability is the ratio of observed spin

flips to the number of attempted excitations.

frequency w mw is varied and the measurement is repeated at different excitation

frequencies.

Finally, the plot of the quantum jump rate versus excitation frequency yields

the resonance spectrum. For magnetic field calibration, the Larmor frequency is

divided by the cyclotron frequency of the 16 O 7+ ion which is simultaneously measured

in a fast Fourier transform of the image currents induced in the trap electrodes

by the ion motion (Figure 4). The Larmor resonance shown in Figure 5 was

obtained from raw measurement data taken during our measurement campaign.

A number of small corrections of the order of a few ppb arising mainly from the

finite cyclotron energy of the ion have to be applied in order to obtain the final

result.


52 J. VERDÚ ETAL.

The g factor of the bound electron in 16 O 7+ is determined from the measured

Larmor resonance by inserting the mass ratio of the ion and the electron into

Equation (3) [13]. Our value for the g factor of the bound electron in 16 O 7+ is

g exp ( 16 O 7+ ) = 2.000 047 026(4). The most recent theoretical prediction from nonperturbative

bound-state QED calculations is g th ( 16 O 7+ ) = 2.000 047 0202(6) [8].

Within the error bars the measurement confirms the validity of the theory on the

ppb-level.

Future measurements of the g factor of the bound electron in heavier hydrogenlike

ions will provide even more stringent tests of bound-state quantum electrodynamics.

If QED is proven to predict the measured g-factor values, such measurements

enable, in addition, a number of quite exciting novel applications [7].

Note added in proof

Additional measurements after submission of the manuscript led to a final value

for the g-factor of g exp ( 16 O 7+ ) = 2.000 047 024 6(46).

Acknowledgements

We are grateful to N. Hermanspahn, S. G. Karshenboim, V. M. Shabaev, S. Stahl,

and A. S. Yelkhovsky for enlightening and stimulating discussions. Financial support

was obtained from the European Union under the contract numbers ERB

FMRX CT 97-0144 (EUROTRAPS network) and HPRI-CT-2001-50036 (HITRAP

network).

References

1. Quint, W., Phys. Scr. T59 (1995), 203.

2. Dehmelt, H., Proc. Natl. Acad. Sci. U.S.A. 83 (1986), 2291.

3. Hermanspahn, N., Häffner, H., Kluge, H.-J., Quint, W., Stahl, S., Verdú, J. and Werth, G., Phys.

Rev. Lett. 84 (2000), 427.

4. Häffner, H., Beier, T., Hermanspahn, N., Kluge, H.-J., Quint, W., Stahl, S., Verdú, J. and Werth,

G., Phys. Rev. Lett. 85 (2000), 5308.

5. Beier, T., Phys. Rep. 339 (2000), 79.

6. Beier, T., Lindgren, I., Persson, H., Salomonson, S., Sunnergren, P., Häffner, H. and Hermanspahn,

N., Phys.Rev.A62 (2000), 032510.

7. Beier, T., Häffner, H., Hermanspahn, N., Karshenboim, S. G., Kluge, H.-J., Quint, W., Stahl,

S., Verdú, J. and Werth, G., Phys. Rev. Lett. 88 (2002), 011603.

8. Yerokhin, V. A., Indeficato, P., Shabaev, V. M., Phys. Rev. Lett. 89 (2002), 143001.

9. Quint, W., Dilling, J., Djekic, S., Häffner, H., Hermanspahn, N., Kluge, H.-J., Marx, G., Moore,

R., Rodriguez, D., Schönfelder, J., Sikler, G., Valenzuela, T., Verdu, J., Weber, C., Werth, G.,

Hyp. Interact. 132 (2001), 453.

10. Dehmelt, H., Rev. Mod. Phys. 62 (1990), 525.

11. Diederich, M., Häffner, H., Hermanspahn, N., Immel, M., Quint, W., Stahl, S., Kluge, H.-J.,

Ley, R., Mann, R. and Werth, G., Hyp. Interact. 115 (1998), 185.

12. Winefand, D. J. and Dehmelt, H. G., J. Appl. Ph. 46 (1975), 919.

13. Mohr, P. J. and Taylor, B. N., Rev. Mod. Phys. 72 (2000), 351.

14. Shabaev, V. M., E-print archive, physics/0110056 (2001) (http://www.lanl.gov).

More magazines by this user
Similar magazines