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

Hyperf**in**e Interactions 146/147: 47–52, 2003.

© 2003 Kluwer Academic Publishers. Pr**in**ted **in** **the** Ne**the**rlands.

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 Ma**in**z, D-55099 Ma**in**z, 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** **the**ory **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 s**in**gle 16 O 7+ ion is stored **in** a Penn**in**g trap. Quantum jumps between **the** two sp**in** states (sp**in** up

and sp**in** down) are **in**duced by a microwave field at **the** sp**in** precession frequency **of** **the** bound

electron. The g factor **of** **the** bound electron is obta**in**ed by vary**in**g **the** microwave frequency and

count**in**g **the** number **of** sp**in** flips. Our experimental value for **the** g factor **of** **the** bound electron

is g exp ( 16 O 7+ ) = 2.000 047 026(4). The **the**oretical 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 **the**oretical program to **in**vestigate

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

We constructed a Penn**in**g-trap quantum jump spectrometer for highly charged ions

and, for **the** first time, applied **the** cont**in**uous Stern–Gerlach effect [2] to an atomic

ion [3]. With a novel double-trap technique we determ**in**ed **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 **the**oretical progress made it possible to

determ**in**e **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. Penn**in**g-trap quantum jump spectrometer

In a Penn**in**g trap a charged particle is stored **in** a comb**in**ation **of** a homogeneous

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

conf**in**es **the** particle **in** **the** plane perpendicular to **the** magnetic field l**in**es, and **the**

electrostatic potential **in** **the** direction parallel to **the** magnetic field l**in**es. 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 l**in**es, and **the** axial motion (parallel to **the** magnetic

field l**in**es, 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 determ**in**ed from a

comb**in**ation **of** **the** trapped ion’s three eigenfrequencies w + , w z ,andw − with **the**

formula w 2 c = w2 + + w2 z + w2 − .

The pr**in**ciple **of** **the** cont**in**uous Stern–Gerlach effect is based on a coupl**in**g

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

**the** Penn**in**g trap. This coupl**in**g is achieved by a quadratic magnetic field component

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

**of** **the** Penn**in**g trap, B(z) = B 0 + β 2 z 2 (Figure 1). Due to **the** **in**teraction **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 trapp**in**g force is modified by **the** magnetic **in**teraction, and **the** axial

frequency **of** **the** trapped ion is shifted upwards or downwards, depend**in**g on **the**

Figure 1. Sketch **of** **the** electrode structure and potential distribution **of** **the** double trap. Sp**in** flips are

**in**duced **in** **the** precision trap by a microwave field and detected **in** **the** analysis trap via **the** cont**in**uous

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** scal**in**g with 1/M, **the** frequency

shift becomes smaller for heavier hydrogen-**like** ions.

In our Penn**in**g trap apparatus, which has been described **in** [11], **the**re are two

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

In **the** precision trap, a s**in**gle 16 O 7+ ion is prepared, and **the** sp**in**-flip transition **of**

**the** bound electron is excited by apply**in**g a microwave field. The s**in**gle 16 O 7+ ion

is **the**n transported along **the** magnetic field l**in**es to **the** analysis trap. The r**in**g

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** cont**in**uous Stern–Gerlach effect. The axial oscillation frequency w z ,

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

electronic detection method through **the** image currents which are **in**duced **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 determ**in**ed **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 determ**in**ed non-destructively **in** **the**

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

between **the** two sp**in** states m s = ±1/2 are **in**duced 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 sp**in**-flip transitions are observed as discrete changes **of** **the** ion’s

axial frequency. Figure 2 shows such a quantum jump observed via **the** cont**in**uous

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** cont**in**uous Stern–Gerlach effect on **the** hydrogen-**like**

carbon ion 16 O 7+ makes it possible to determ**in**e its electronic g factor to high

accuracy. Us**in**g **the** cyclotron frequency w c **of** **the** ion for **the** calibration **of** **the**

50 J. VERDÚ ETAL.

Figure 2. Axial frequencies **of** a s**in**gle 16 O 7+ ion for different sp**in** directions. The averag**in**g time

for each resonance curve was 1 m**in** (FFT: fast Fourier transform).

Figure 3. Quantum non-demolition measurement **of** **the** sp**in** state: **the** sp**in**-flip transitions are

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

magnetic field B,**the**g 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 obta**in**ed **in** **the** follow**in**g way. First, we determ**in**e **the** sp**in** direction **in**

**the** analysis trap via **the** cont**in**uous Stern–Gerlach effect. The hydrogen-**like** ion is

**the**n transferred to **the** precision trap where sp**in** flips are **in**duced 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** sp**in** state is analyzed aga**in**. Now **the** ion is moved back to **the**

precision trap, and **the** measurement cycle is started aga**in**. The number **of** sp**in**-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 **in**duced **in** one **of** **the** electrodes from **the** cyclotron motion

**of** a s**in**gle 16 O 7+ ion stored precision trap. The fractional l**in**e width **of** **the** cyclotron resonance is a

few ppb.

Figure 5. Larmor resonance spectrum measured **in** **the** precision trap. Sp**in**-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 sp**in**-flip probability is **the** ratio **of** observed sp**in**

flips to **the** number **of** attempted excitations.

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

frequencies.

F**in**ally, **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 **in**duced **in** **the** trap electrodes

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

obta**in**ed from raw measurement data taken dur**in**g our measurement campaign.

A number **of** small corrections **of** **the** order **of** a few ppb aris**in**g ma**in**ly from **the**

f**in**ite cyclotron energy **of** **the** ion have to be applied **in** order to obta**in** **the** f**in**al

result.

52 J. VERDÚ ETAL.

The g factor **of** **the** bound electron **in** 16 O 7+ is determ**in**ed from **the** measured

Larmor resonance by **in**sert**in**g **the** mass ratio **of** **the** ion and **the** electron **in**to

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 **the**oretical prediction from nonperturbative

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

With**in** **the** error bars **the** measurement confirms **the** validity **of** **the** **the**ory on **the**

ppb-level.

Future measurements **of** **the** g factor **of** **the** bound electron **in** heavier hydrogen**like**

ions will provide even more str**in**gent 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 excit**in**g novel applications [7].

Note added **in** pro**of**

Additional measurements after submission **of** **the** manuscript led to a f**in**al 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 enlighten**in**g and stimulat**in**g discussions. F**in**ancial support

was obta**in**ed from **the** European Union under **the** contract numbers ERB

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

network).

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