CONTINUOUS STERN–GERLACH EFFECT ON ATOMIC IONS

ADVANCES IN ATOMIC, MOLECULAR, AND OPTICAL PHYSICS, VOL. 48

CONTINUOUS STERN–GERLACHEFFECT ON ATOMIC IONS

GUNTHER WERTH 1, HARTMUT HAFFNER 1 and WOLFGANG QUINT 2

1Johannes Gutenberg University, Department of Physics, 55099 Mainz, Germany;2Gesellschaft fur Schwerionenforschung, 64291 Darmstadt, Germany

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191II. A Single Ion in a Penning Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195III. Continuous Stern–Gerlach Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206IV. Double-Trap Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209V. Corrections and Systematic Line Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212VI. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213VII. Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214VIII. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216IX. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

I. Introduction

In 1922 Stern and Gerlach succeeded in spatially separating a beam of silveratoms into two beams, utilizing the force exerted in an inhomogeneous magneticfield on the magnetic moment of the unpaired electron in silver. This so-calledStern–Gerlach effect was the first observation of the directional quantization ofthe quantum-mechanical angular momentum, and represents a cornerstone ofquantum physics (Stern and Gerlach, 1922).Apart from its historical role the effect has been used in numerous atomic

beam experiments to determine the magnetic moments of electrons bound inatomic systems: An atomic beam enters a first inhomogeneous magnetic fieldwhere one spin direction is separated from the other. The polarized atoms thenenter a region with a homogeneous magnetic field where they are subjected toa radio-frequency (rf) field which changes the spin direction. Finally, a secondinhomogeneous magnetic field region analyzes the spin direction. Variation ofthe frequency w of the rf field and recording the spin-flip probability as afunction of w yields a resonance curve. Together with a measurement of thefield strength in the homogeneous magnetic field, the magnetic moment can bederived. Usually the size of the magnetic moment m is given in units of the Bohrmagneton mB = eà/ (2m) and expressed by the g-factor:

m = gsmB, (1)

191 Copyright © 2002 Elsevier Science (USA)All rights reserved

ISBN 0-12-003848-X/ISSN 1049-250X/02 $35.00

192 G. Werth et al. [I

where s is the spin quantum number. Accurate g-factor measurements representa critical test of atomic physics calculations for complex systems (Veseth, 1980,1983; Lindroth and Ynnerman, 1993).There has been an extensive discussion that the Stern–Gerlach effect could

be employed for neutral atoms only. For charged particles the magnetic forceis overshadowed by the Lorentz force acting on a moving charged particle ina magnetic field. Proposals to use the acceleration of electrons moving alongthe field lines of an inhomogeneous B-field to separate the spin directions(Brillouin, 1928) have been discarded by Bohr (1928) and Pauli (1958) on thebasis of Heisenberg’s uncertainty principle. Nevertheless attempts have beenmade, however unsuccessful, to separate the two spin states in an electron beamby using the different signs of the force on the spin exerted by a longitudinalinhomogeneous magnetic field which results in a deceleleration or accelerationof the electrons (Bloch, 1953). Recently, new proposals have come up to performStern–Gerlach experiments on electron beams which would – in contrast tothe analysis by Bohr and Pauli – result in a high degree of spin separationunder carefully chosen initial conditions (Batelaan et al., 1997; Garraway andStenholm, 1999).Dehmelt has pointed out that confining a charged particle by electromagnetic

fields provides a way to circumvent Bohr’s and Pauli’s reasoning (Dehmelt,1988): the force of an inhomogeneous magnetic field on the spin of a particlewhich oscillates in a parabolic potential well leads to a small but measurabledifference of its oscillation frequency for different orientations of the spin. Thusa precise measurement of the oscillation frequency gives information on thespin direction. Dehmelt et al. used this effect for monitoring induced changesof the spin direction of an electron by observing the corresponding changesin the electron’s oscillation frequency. Since the sensitive electronics monitorsthe trapped particle’s spin direction continuously, Dehmelt coined the term“continuous Stern–Gerlach effect” for this technique. In a series of experimentshe and his coworkers have applied this method to measure the magnetic momentof the electron and the positron, which culminated in the most precise values ofa property of any elementary particle (Van Dyck et al., 1987). The experimentalvalue

gexp = 2.0023193043766 (87) (2)

agrees to 10 significant digits with the result of calculations based on the theoryof quantum electrodynamics (QED) for free particles (Hughes and Kinoshita,1999),

gth = 2.0023193044320 (687) (3)

and provides one of the most stringent tests of QED. The theoretical result forthe free-electron g-factor can be expressed in a perturbation series as

gfree = 2

(A0 + A1

(ap

)+ A2

(ap

)2+ A3

(ap

)3+ A4

(ap

)4+ · · ·

), (4)

I] CONTINUOUS STERN–GERLACH EFFECT ON ATOMIC IONS 193

Fig. 1. Calculated expectation values for the electric field strength for hydrogen-like ions ofdifferent nuclear charge Z . (Courtesy Thomas Beier.)

where a ≈ 1137 is the fine-structure constant. The coefficients An in Eq. (4) have

been calculated by evaluating the Feynman diagrams of different orders using aplane-wave basis set; they are of the order of unity.In contrast to a free electron, an electron bound to an atomic nucleus

experiences an extremely strong electric field. The expectation value of thefield strength ranges from 109 V/cm in the helium ion (Z = 2) to 1015 V/cmin hydrogen-like uranium (Z = 92) (Fig. 1), and gives rise to a variety of neweffects. The largest change of the bound electron’s g-factor was analyticallyderived by Breit (1928) from the Dirac equation:

gBreit = 23

(1 + 2

√1 − (Za)2

)≈ 2 · (

1 − 13 (Za)

2). (5)

The conditions of extreme electric fields also necessitate changes to be made inthe methods of calculations for the QED contributions to the electron’s magneticmoment. In a perturbative treatment a series expansion in (Za) is made inaddition to that in a. The expansion parameter Za, however, is – at least forlarge Z – no longer small compared to 1. In addition, the expansion coefficientscan be large. For small values of the nuclear charge Z the perturbation expansionmay give reliable results, and calculations were performed which include termsup to order (Za)2 (Grotch, 1970a; Close and Osborn, 1971; Karshenboim et al.,2001). For more accurate theoretical predictions, non-perturbative methods havebeen developed where the solutions of the Dirac equation for the hydrogen-likeion rather than those for the free case are used as a basis set (Beier et al.,2000). The most recent summary of the status of QED for bound systems has

194 G. Werth et al. [I

Fig. 2. Contributions to the g-factor of a bound electron in hydrogen-like ions for different nuclearcharges Z . (Courtesy Thomas Beier.)

Table 1Theoretical contributions to the g-factor in 12C5+

Contribution Size Reference

Dirac Theory 1.9987213544 Breit (1928)

QED, free (all orders) +0.0023193044 Hughes and Kinoshita (1999)

QED, bound, order (a/p ) +0.0000008442(9) Beier et al. (2000)

QED, bound, order (a/p )2, (Za)2 term −0.0000000011(4) Grotch (1970a)

Recoil (in Za expansion) +0.0000000876(1) Yelkovsky (2001)

Finite size correction +0.0000000004 Beier et al. (2000)

been published by Beier (2000). A graphical representation of the bound-statecontributions to the electron g-factor is shown in Fig. 2.

In this contribution we describe an experiment which for the first time appliesthe “continuous Stern–Gerlach effect” to an atomic ion (Hermanspahn et al.,2000). We measured the magnetic moment of the electron bound to a nucleuswith zero nuclear magnetic moment, hydrogen-like carbon (12C5+). For thebound-state contributions of order a(Za)n the existing calculations (Grotch,1970a) deviate already for Z = 6 by as much as 10% from the non-perturbativecalculations to all orders in (Za). The numerical results of the calculations forC5+ are summarized in Table 1. The leading term comes from the solution

II] CONTINUOUS STERN–GERLACH EFFECT ON ATOMIC IONS 195

of the Dirac equation and deviates from the value g = 2 for the free electron(Breit, 1928). The next-largest part is the well-known QED contribution for thefree electron (Hughes and Kinoshita, 1999). The bound-state contributions oforder a (calculated to all orders in Za) are given with error bars which representthe numerical uncertainty of the calculations. The quoted uncertainty of thea2(Za)2 term is an estimate of the contribution from non-calculated higher-order terms. Finally, nuclear recoil contributions have been calculated to lowestorder in Za by Grotch (1970b), Faustov (1970) and Close and Osborn (1971).Recently, Shabaev (2001) presented formulas for a non-perturbative calculation(in Za), and Yelkovsky (2001) presented further numerical results. The nuclear-shape correction was considered numerically by Beier et al. (2000), and recently(for low Z) also analytically by Glazov and Shabaev (2001). The sum of thedifferent contributions leads to a theoretical value for the g-factor in hydrogen-like carbon of

gtheor(12C5+) = 2.0010415899 (10). (6)

II. A Single Ion in a Penning Trap

The experiment is carried out on a single C5+ ion confined in two Penningtraps. In a Penning trap a charged particle is stored in a combination of ahomogeneous magnetic field B0 and an electrostatic quadrupole potential. Themagnetic field confines the particle in the plane perpendicular to the magneticfield lines, and the electrostatic potential confines it in the direction parallel tothe magnetic field lines. In our experiment we use two nearly identical trapsplaced 2.7 cm apart in the magnetic field direction. They consist of a stack of 13cylindrical electrodes of 2r0 = 7mm inner diameter. The difference between thetraps is that in one trap the center electrode is made of ferromagnetic nickel whileall others are machined from OFHC copper. Figure 3 shows a sketch of this setup.The nickel ring distorts the homogeneity of the superimposed magnetic field inthe corresponding trap while the field remains homogeneous in the other trap(see Fig. 3). As will become evident below, the inhomogeneity of the field is thekey element to analyze the direction of the electron spin through the continuousStern–Gerlach effect. Therefore we call the corresponding potential minimum“analysis trap” while we call the one in the homogeneous magnetic field“precision trap.”Each trap uses five of these electrodes to create a potential well, which serves

for axial confinement. We apply a negative voltage U0 to the center electrodewhile we hold the two endcap electrodes at a distance z0 from the center at

196 G. Werth et al. [II

Fig. 3. Sketch of the electrode structure and potential distribution of the double trap.

ground potential. The potential F inside this configuration can be described incylindrical coordinates r, z,ö by an expansion in Legendre polynomials Pi:

F(r,ö) =U0

2

∞∑i = 0

Ci

( r

d

)iPi(cosö), (7)

where d2 =(z20 + r20 /2

)/2 is a characteristic dimension of the trap (Gabrielse

et al., 1989). Two correction electrodes are placed between the center ringand the endcaps. The coefficient C4 which is the dominant contribution to thetrap anharmonicity can be made small by proper tuning of the voltages appliedto the correction electrodes. Essentially then the potential depends on the squareof the coordinates, and is a harmonic quadrupole potential

F =U0

2z2 − r2/2

d2. (8)

We optimize the trap by changing the voltages on the correction electrodes untilthe ion oscillation frequency is independent of the ion’s oscillation amplitudeas characteristic for a harmonic oscillator. With this method we can reduce the


Fig. 4. Ion oscillation in a Penning trap.

dominant high-order term C4, the octupole contribution, to less than 10−5. Theion’s frequency in the harmonic approximation is then given by

wz =

√qU0

Md2. (9)

Radial confinement is achieved by the homogeneous magnetic field directedalong the trap axis. This results in three independent oscillations (axial,cyclotron, and magnetron oscillation) for the ion motion, as depicted in Fig. 4.The fast radial oscillation frequency of the ion in the Penning trap is a perturbedcyclotron frequency w′

c. It differs from the cyclotron frequency

wc =q

MB (10)

of a free particle with charge q and mass M because of the presence of theelectric trapping field, and is given by

w′c =

wc

2+

√w2

c

4−w2

z

2. (11)

It can be expressed also asw′

c = wc − wm, (12)

where wm is the magnetron frequency, a slow drift of the cyclotron orbit aroundthe trap center, given by

wm =wc

2−

√w2

c

4−w2

z

2. (13)


For calibration of the magnetic field we use the cyclotron frequency of thetrapped ion. It can be derived either from Eq. (12) or more reliably from therelation

w2c = w2 ′

c + w2z + w

2m, (14)

since this equation is independent of trap misalignments to first order (Brown andGabrielse, 1986). In this case the measurement of the magnetron frequency wm

is required in addition to a measurement of w′c and wz.

The traps are enclosed in a vacuum chamber placed at the bottom of a heliumcryostat at a temperature of 4K and located at the center of a superconductingNMR solenoid. The helium cryostat provides efficient cryopumping. As an upperlimit we estimate the vacuum in the container to be below 10−16 mbar. Theestimation was derived from the measurements on a cloud of highly chargedions, whose storage time would be limited by charge exchange in collisions withneutral background particles. We observed no ion loss in a cloud of 30 hydrogen-like carbon ions stored for 4 weeks. Together with the known cross section forcharge exchange with helium as the most likely background gas at 4K we obtainan upper limit of 10−16 mbar for the background gas pressure. The magneticfield of the superconducting magnet is chosen to be 3.8 T. At this field strengththe precession frequency of the electron spin is 104GHz. Microwave sources ofsufficient power and spectral purity are commercially available at this frequency.We load the trap by bombarding a carbon-covered surface with an electron

beam. This process releases ions and neutrals of the element under investigationas well as of other elements present on the surface. Higher charge states areobtained by consecutive ionisation by the electron beam. We detect the ionsby picking up the current induced by the ion motions in the trap electrodes.For this purpose superconducting resonant circuits and amplifiers are attachedto the electrodes. Upon sweeping the voltage of the trap, and thus the ions’axial frequencies, the ions get in resonance with the circuit and their signal isdetected. Figure 5 shows such a spectrum, where we identified different elementsand charge states. We eliminate unwanted ion species by exciting their axialoscillation amplitude with an rf field until the ions are driven out of the trap.Ions of the same species have different perturbed cyclotron frequencies in

the slightly inhomogeneous magnetic field of the precision trap, because theyhave different orbits. Therefore, for small ion numbers, single ions can bedistinguished by their different cyclotron frequencies. Figure 6 shows a Fouriertransform of the induced current from 6 stored ions. We excite the ions’ cyclotronmotion individually and thus eliminate them from the trap until a single ion isleft. Typical cyclotron energies for signals as shown in Fig. 6 are of the order ofseveral eV.In order to reduce the ion’s kinetic energy we use the method of “resistive

cooling” which was first applied by Dehmelt and collaborators (Wineland andDehmelt, 1975; Dehmelt 1986). The ion’s oscillation is brought into resonance


Fig. 5. Mass spectrum of trapped ions after electron bombardment of a carbon surface showingdifferent charge states of carbon ions as well as impurity ions (a) before and (b) after removal ofunwanted species.

with the circuits attached to the electrodes. The induced current through theimpedance of the circuit leads to heating of the resonance circuit, and the ion’skinetic energy is dissipated to the surrounding liquid helium bath (Fig. 7). Thisleads to an exponentially decreasing energy with a time constant t given by

t−1 =q2

Md2R, (15)

where R is the resonance impedance of the circuit.For the axial motion we use superconducting high-quality circuits (Q = 1000 at

1MHz in the precision trap and Q = 2500 at 365 kHz in the analysis trap).With the resonance impedances of R = 23MW (analysis trap) and 10MW(precision trap) the cooling time constants are 80ms and 235ms, respectively.For cooling the cyclotron motion we employ a normal-conducting circuit at24MHz (Q = 400) with a resonance impedance of 80 kW. Here we reach coolingtime constants of a few minutes. Figure 8 shows the exponential decrease of theinduced currents from the ion oscillations as the result of axial cooling.


Fig. 6. Fourier transform of the voltage induced in one of the trap electrodes from the cyclotronmotion of 6 stored 12C5+ ions. The inhomogeneous magnetic field of the trap causes ions at differentpositions to have slightly different cyclotron frequencies.

Fig. 7. Principle of resistive cooling.

We do not cool the magnetron motion in a similar way because it ismetastable: in the radial plane the ion experiences an electrostatic force towardsthe negatively biased center electrode. Ion loss is prevented by the presence ofthe magnetic field. Thus the potential energy is an inverted parabola. Thereforereduction of the ion’s magnetron energy results in an increase in the magnetronradius. It is, however, essential to reduce the magnetron radius because ofthe magnetic field inhomogeneities. This is achieved in a well-defined wayby coupling the magnetron motion to the axial motion by a radio-frequencyfield at the sum frequency of both oscillations (Brown and Gabrielse, 1986;Cornell et al., 1990). In the quantum-mechanical picture for the ion motion, theabsorption of a photon from this field increases the quantum number of the axialoscillation by 1 while that of the magnetron oscillation is decreased by 1. Ananalysis of the absorption probabilities in the framework of a harmonic oscillator


Fig. 8. Exponential energy loss of the axial motion of trapped C5+ ions by resistive cooling.

yields that the quantum numbers tend to equalize. This leads to the expectationvalue 〈Em〉 for the magnetron energy,

|〈Em〉| = àwm

(〈km〉 + 12

)= àwm

(〈kz〉 + 12

)=wm

wzàwz

(〈kz〉 + 12

)=wm

wz〈Ez〉.

(16)

The axial oscillation is continuously kept in equilibrium with the cooling circuitand we thus reduce the magnetron orbit to about 10 mm.The mean kinetic energy of a single ion is often expressed in terms of

temperature. This is justified by the statistical equilibrium of the ion and theresonant circuit. The statistical motion of the electrons in the resonance circuitcauses Johnson noise in the trap-electrode voltages, which in turn leads tovarying energies of the ion as a function of time (Fig. 9). Extracting a histogramof the cyclotron energies results in a Boltzmann distribution (Fig. 10) with atemperature of 4.9K close to the temperature of the environment. Calculatingthe temporal autocorrelation function of the energy gives, as expected, anexponential (Fig. 11) with a time constant well in agreement with the measuredcooling time constant.In order to calibrate the magnetic field at the ion’s position with high precision

from Eq. (14) the three oscillation frequencies have to be measured. Becauseof their different orders of magnitude (w′

c/2p = 24MHz, wz/2p = 1MHz,wm/2p = 18 kHz) the required precision is different. w′

c is determined from theFourier transform of the current induced in a split electrode. Figure 12 showsthat the relative linewidth of the resonance, well described by a Lorentzian, is ofthe order of 10−9 and the center frequency can be determined with an accuracyof 10−10. In order to obtain sufficient signal strength the energy of the cyclotronoscillation has to be raised to about 1 eV. Due to the inhomogeneity of the


Fig. 9. Noise power of the induced voltage in a trap electrode from the cyclotron oscillation ofa single trapped ion while its frequency is continuously kept in resonance with an attached tankcircuit.

Fig. 10. Histogram of the probabilities for cyclotron energies. The curve can be well fitted to aBoltzmann distribution, giving a temperature of 4.9(1) K.


Fig. 11. The time-correlation function of the noise in fig. 10 shows an exponential decrease. Thetime constant of 5.40(7)min corresponds to the time constant for resistive cooling of the cyclotronmotion.

Fig. 12. High-resolution Fourier transform of the induced noise at the perturbed cyclotronfrequency. A Lorentzian fit gives a fractional width of 1.4×10−9.

magnetic field this changes the mean field strength along the cyclotron orbit.This has to be considered in the final evaluation of the measurements.The axial frequency wz is determined while the ion is in thermal equilibrium

with the resonance circuit. At a given temperature the thermal noise voltage inthe impedance Z(w) of the axial circuit, given by

Unoise =√4kTRe[Z(w)] dn , (17)

excites the ion motion within the frequency range dn of the ion’s axial resonance.This motion in turn induces a voltage in the endcap electrodes, however at aphase difference of 180º, as can been shown by modeling the system as a drivenharmonic oscillator. Consequently the sum of the thermal noise voltage and


Fig. 13. Axial resonance of a single trapped C5+ ion. The noise voltage across a tank circuitshows a minimum at the ion’s oscillation frequency.

Fig. 14. High-resolution Fourier transform of the axial noise near the center of the resonancefrequency of the axial detection circuit.

the induced voltage leads to a reduced total power around the axial frequencyof the ion. This appears as a minimum in the Fourier transform of the axialnoise as shown in Fig. 13. A spectrum with a resolution of 10mHz (Fig. 14)shows that the center frequency can be determined to about 24mHz. A differentapproach to explain the appearance of a minimum in the axial noise spectrumwas taken by Wineland and Dehmelt (1975) considering the equivalent electriccircuit of an oscillating ion in the trap. The magnetron frequency wm ismeasured by sideband coupling to the axial motion. If the ion is excited atthe difference between the axial and magnetron frequencies, the ion’s axial


energy increases, leading to an increased current at the ion’s axial frequency.We detect this as a peak in the detection circuit signal. The uncertainty in thisfrequency determination is below 100mHz.Imperfections in the trap geometry may change the motional frequencies.

These changes have been calculated by different authors (Brown and Gabrielse,1982; Kretzschmar, 1990; Gerz et al., 1990; Bollen et al., 1990). Consideringonly an octupole contribution to the trap potential, characterized by a coeffi-cient C4 in the potential expansion (5), these shifts amount to

Dwz

wz= C4

1qU0

[32Ez − 3

(wwc

)2

Ec + 6Em,

](18)

Dw′c

w′c

= C41

qU0

(wz

wc

)2[−3Ez +

32

(wz

wc

)2

Ec − 6Em

], (19)

Dwm

wm=

1qU0

[6Ez − 6

(wz

wc

)2

Ec + 6Em

]. (20)

Ez, Ec and Em are the energies in the axial, cyclotron and magnetron degreesof freedom. The tuning of the trap potential results in a coefficient C4

as small as 10−5. For the energies of the motions in thermal equilibrium thecorresponding frequency shifts are below 10−10 and need not be considered here.The coefficient C6 of the dodekapole contribution to the trapping potential hasbeen calculated to an accuracy of 10−3 for our trap geometry. The frequencyshifts arising from this perturbation scale with (E/qU0)2 and are negligible here.The residual inhomogeneity of the magnetic field in the precision trap arising

from the nickel ring electrode in the analysis trap 2.7 cm away changes the valueof the oscillation frequencies of an ion with finite kinetic energy as comparedto the ion at rest. A series expansion of the B-field in axial direction,

Bz = B0 + B1z + B2z2 + · · · (21)

gives frequency shifts

Dw′c

w′c

=1

mw2z

B2

B0

1àwc

[−

(wz

wc

)2

Ec + Ez + 2Em

], (22)

Dwz

wz=

1mw2

z

B2

B0

1àwc

[Ec − Em] , (23)

Dwm

wm=

1mw2

z

B2

B0

1àwc

[2Ec − Ez − 2Em] . (24)

The size of the inhomogeneity term B2 is measured by application of a biasvoltage between the endcap electrodes. This shifts the ion’s position in the axial

206 G. Werth et al. [III

direction by a calculable amount, and the cyclotron frequency is measured at eachposition. We obtain B2 = 8.2(9)mT/mm2. The shift in the perturbed cyclotronfrequency which is of most interest here is dominated by the axial energy Ez,and amounts to Dw′

c/w′c = 7×10−9 for an axial temperature of 100K.

III. Continuous Stern–Gerlach Effect

The g-factor of the bound electron as defined by Eq. (1) can be determined bya measurement of the energy difference between the two spin directions in amagnetic field B:

DE = hnL = gmBB, (25)

where nL is the Larmor precession frequency. We induce spin flips by applyingmagnetic dipole radiation which is blown into the trap structure by a microwavehorn.For the detection of an induced spin flip we follow a route developed in the

determination of the g-factor of the free electron (Dehmelt, 1986; Van Dycket al., 1986): the quadrupole potential of the Penning trap depends on the squareof the coordinates (6) leading to a linear force acting upon the charge of thestored ion. Considering the force upon the magnetic moment of the boundelectron by the inhomogeneous field in the analysis trap we get

F = −∇(m · B). (26)

The nickel ring in the analysis trap creates a bottle-like magnetic field distortionwhich can be described in first approximation by

ÀB = ÀB0 + 2B2

[z2 − r2

2e − zÀr

]. (27)

The odd terms vanish in the expansion because of mirror symmetry of the field.The corresponding force on the magnetic moment in axial direction is

Fz = −2mzB2z, (28)

which is linear in the axial coordinate. It adds to the electric force from thequadrupole trapping field acting on the particle’s charge. Since both forces arelinear in the axial coordinate the ion motion is still described by a harmonicoscillator (Fig. 15). The axial frequency, however, depends on the direction ofthe magnetic moment m with respect to the magnetic field:

wz = wz0 + 12dwz = wz0 +

mzB2

Mwz0. (29)

The value of B2 in our set-up was calculated using the known geometry andmagnetic susceptibility of the nickel ring electrode. We also determined it

III] CONTINUOUS STERN–GERLACH EFFECT ON ATOMIC IONS 207

Fig. 15. Axial parabola potential for an ion in a quadrupole trap including the magnetic potentialfor the spin-magnetic moment in a bottle-like magnetic field. The strength of the potential dependson the spin direction. Upper curve: spin down; lower curve: spin up.

Fig. 16. Axial frequencies of a single C5+ ion for different spin directions. The averaging timefor each resonance line was 1min.

experimentally by applying a bias voltage between the endcap electrodes of theanalysis trap and measuring the cyclotron frequency of the ion at different axialpositions. The calculated and experimental values for B2 in our experiment agreewithin their uncertainties of 10% and yield B2 = 1 T/cm2. For hydrogen-likecarbon the frequency difference dwz/2p between the two spin states amounts to0.7Hz at a total frequency of wz0/2p = 365 kHz.As evident from Fig. 16, the axial frequency can be determined to better

than 100mHz. Fig. 17 demonstrates that after 1min. averaging the expectedfrequency difference between the two spin states becomes obvious. Drivingthe spin-flip transition, we can distinguish the two possible axial frequencies,0.7Hz apart as calculated from the trap parameters. Varying the frequency ofthe microwave field and counting the number of induced spin flips per unit timeyields a resonance curve as shown in Fig. 18. The shape of this resonance isasymmetric due to the inhomogeneity of the magnetic field. The general shapeof the Larmor resonance in an inhomogeneous magnetic field has been derived

208 G. Werth et al. [III

Fig. 17. Center of the axial frequency for a single C5+ ion when irradiated continuously withmicrowaves at the Larmor precession frequency showing two distinct values which correspond tothe two spin directions of the bound electron.

Fig. 18. Number of observed spin flips per unit time vs. the frequency of the inducing field. Thesolid line is a fit according to Eq. (29).

by Brown (1985) as a complex function of the trap parameters, the ion’s energyand the field inhomogeneity. However, assuming that the ion’s amplitude z(t) is

IV] CONTINUOUS STERN–GERLACH EFFECT ON ATOMIC IONS 209

constant during the time 1/DwL, the line profile is given by a d-function averagedover the Boltzmann distribution of the energy:

c (wL) =

∞∫0

dE d(wL − wL0

(1 +

eEz

mw2z

))kTe−E/kT

=Q(wL − wL0)

DwLexp

{−wL − wL0

DwL

}.

(30)

Here Q(wL −wL0) is the step function, which is 0 for wL <wL0 and 1 for wL >wL0,and e is a linewidth parameter so that the Larmor frequency depends aswL = w0(1 + ez2) on the axial coordinate. A least-squares fit of this functionto the data points of Fig. 18 yields the Larmor frequency with a relativeuncertainty of 10−6 (Hermanspahn et al., 2000). This is sufficient to measurethe binding correction to the g-factor in C5+. The bound-state QED correctionsfor C5+, however, are 4×10−7 and were not observed in this measurement.

IV. Double-Trap Technique

The limitation in accuracy of the experiment described above stems from theinhomogeneous magnetic field as required for the analysis of the spin directionvia the continuous Stern–Gerlach effect. In fact the inhomogeneity of the fieldwas chosen to be as small as possible, but still large enough to be able todistinguish the two spin directions.We obtained an improvement of three orders of magnitude in the accuracy of

the measured magnetic moment by spatially separating the processes of inducingspin flips and analyzing the spin direction (Haffner et al., 2000). This is achievedby transferring the ion after a determination of the spin direction from theanalysis trap to the precision trap. The voltages at the trap electrodes are changedin such a way that the potential minimum in which the ion is kept is movedtowards the precision trap. The transport takes place in a time of the order of 1 s,which is slow compared to any oscillation period of the ion and is thereforeadiabatic. Once in the precision trap, the ion’s motional amplitudes are preparedby coupling the ion to the resonant circuits. We then apply the microwave fieldto induce spin flips. After the interaction time, typically 80 s, and an additionalcooling time, the ion is moved back to the analysis trap. Here the spin directionis analyzed again. In principle one measurement of the axial frequency would besufficient to determine whether it has changed by 0.7Hz as compared to the valuebefore transport into the precision trap. If, however, the ion is not brought backwith the same radial motional amplitudes to the analysis trap, the axial frequencymay have changed by as much as 1Hz. This is because of the magnetic momentconnected with the cyclotron and magnetron motion. To circumvent this problem

210 G. Werth et al. [IV

Fig. 19. Determination of the spin direction in the analysis trap after transport from the precisiontrap. A change in axial frequency of about 0.7Hz indicates that the spin was up (left) or down (right)when the ion left the precision trap.

we induce an additional spin flip in the analysis trap to determine without doubtthe spin direction after return to the analysis trap. Figure 19 shows several cyclesfor a spin analysis. The total time for a complete cycle is about 30min.While the ion is in the precision trap its cyclotron frequency wc = (q/M )B

is measured simultaneously with the interaction with the microwaves. Thisensures that the magnetic field is calibrated at the same time as the possiblespin flip is induced. The field of a superconducting solenoid fluctuates at thelevel of 10−8−10−9 on the time scale of several minutes. Figure 20 shows ameasurement of the cyclotron frequency of the ion in the precision trap overa time span of several hours. Every 2min the center frequency of the cyclotronresonance was determined. The change in cyclotron frequency has approximatelya Gaussian distribution with a full-width-at-half-maximum of 1.2×10−8. Thismay impose a serious limit on the precision of measurements as in the case ofhigh-precision mass spectrometry using Penning traps (Van Dyck et al., 1993;Natarajan et al., 1993). However, the simultaneous measurement of cyclotron andLarmor frequencies eliminates most of this broadening. Using Eqs. (10) and (25)we obtain the g-factor as the ratio of the two measured frequencies

g = 2wL

wC

m

M. (31)

The mass ratio of the electron to the ion can be taken from the literature. In ourcase of 12C5+, Van Dyck and coworkers (Farnham et al., 1995) measured it withhigh accuracy using a Penning trap mass spectrometer.We measure the induced spin flip rate for a given frequency ratio of the

microwave field and the simultaneously measured cyclotron frequency. When we

IV] CONTINUOUS STERN–GERLACH EFFECT ON ATOMIC IONS 211

Fig. 20. Distribution of magnetic field values measured by the cyclotron frequency of a trappedion in a period of several hours. Data were taken every 2min. The distribution is fitted by a Gaussianwith a full width of 1.2×10−8.

Fig. 21. Measured spin-flip probability vs. ratio of Larmor and cyclotron frequencies. The dataare least-squares fitted to a Gaussian.

plot the spin flip probability, i.e. the number of successful attempts to change thespin direction divided by the total number of attempts, we obtain a resonance lineas shown in Fig. 21. The maximum attainable probability is 50% when theamplitude of the microwave field is high enough. To avoid those saturation effectswe take care to keep the amplitude of the microwave field at a level that themaximum probability for a spin flip at resonance frequency is below 30%. Inaddition we can take saturation into account using a simple rate-equation model.

212 G. Werth et al. [V

In contrast to the single-trap experiment the lineshape is now much moresymmetric. For a constant homogeneous magnetic field in the precision trap thelineshape would be a Lorentzian with a very narrow linewidth determined bythe coupling constant g to the cooling circuit. However, the observed lineshapecan be well described by a Gaussian. The fractional full width is 1.1×10−8. Thisreflects the variation of the magnetic field during the time the ion spends in theprecision trap which is of the same order of magnitude (see Fig. 20). The linecenter can be determined from a least squares fit to 1×10−10.

V. Corrections and Systematic Line Shifts

The main systematic shifts of the Larmor and cyclotron resonances arise fromthe fact that the field in the precision trap is not perfectly homogeneous.As mentioned above, the ferromagnetic nickel ring placed 2.7 cm away inthe analysis trap causes a residual inhomogeneity in the precision trap. Theexpansion coefficient from Eq. (21) gives B2 = 8mT/mm2, three orders ofmagnitude smaller than in the analysis trap. Therefore we still have to consideran asymmetry in the line profile. Performing such an analysis gives a maximumdeviation as compared to the symmetric Gaussian fit of 2×10−10. In addition,the inhomogeneity of the magnetic field causes a shift of the line with theion’s energies. In order to obtain a sufficiently strong signal of the inducedcurrent from the cyclotron motion in the precision trap, the ion’s energy hasto be raised to about 1 eV. This finite cyclotron energy has a large magneticmoment and thus shifts the axial frequency as compared to vanishing cyclotronenergy even in the precision trap by about 1Hz. To account for this shiftwe grouped our data of the spin flip probabilities according to the differentaxial frequency shifts in the precision trap corresponding to different cyclotronenergies, and extrapolated the ratios wL/wC to zero cyclotron energy (Fig. 22).We find a slope of D(wL/wC)/EC = −1.09(5)×10−9 eV−1. Other systematic shifts

Fig. 22. Extrapolation of measured frequency ratios to vanishing cyclotron energy.

VI] CONTINUOUS STERN–GERLACH EFFECT ON ATOMIC IONS 213

Table 2Systematic uncertainties (in relative units) in the g-factor determination of 12C5+

Contribution Relative size

Asymmetry of resonance 2×10−10

Electric field imperfections 1×10−10

Ground loops in apparatus 4×10−11

Interact. with image charges 3×10−11

Calibration of cyclotron energy 2×10−11

Sum 2.3×10−10

are less important: From the residual imperfection of the electric trapping field(C4 = 10−5) we calculate a shift of the cyclotron frequency of 1×10−10. Ofthe same order of magnitude are frequency shifts caused by changes of thetrapping potential due to ground loops when the computer controls are activated.The interaction of the ion with its image charges changes the frequenciesby 3×10−10, but can be calculated with an accuracy of 10%. Relativistic shiftsare of the order of 10−10 at typical ion energies, but do not contribute to theuncertainty at the extrapolation to zero energy. A list of uncertainties of thesecorrections is given in Table 2. The quadrature sum of all systematic uncertaintiesamounts to 3×10−10. The final experimental value for the frequency ratio wL/wC

in 12C5+ iswL

wC= 4376.2104989(19)(13). (32)

The first number in parentheses is the statistical uncertainty from the extrap-olation to vanishing cyclotron energy, the second is the quadrature sum of thesystematical uncertainties. Taking the value for the electron mass in atomic units(M (12C) = 12) from the most recent CODATA compilation (Mohr and Taylor,1999) we arrive at a g-factor for the bound electron in 12C5+ of

gexp(12C5+) = 2.0010415963 (10)(44). (33)

Here the first number in parentheses is the total uncertainty of our experiment,and the second reflects the uncertainty in the electron mass.

VI. Conclusions

A comparison of the experimentally obtained result of Eq. (33) to the theoreticalcalculations presented in Table 1 shows that the bound-state QED effects of

214 G. Werth et al. [VII

order a/p in hydrogen-like carbon are verified at the level of 5×10−3. BoundQED contributions of order (a/p )2 are too small to be observed. The nuclearrecoil part has been verified to about 5%.It is believed that uncalculated terms of higher-order QED contributions do not

change the theoretical value beyond the presently quoted uncertainties. Takingthis for granted we can use experimental and theoretical numbers to determinea more accurate value for the atomic mass of the electron, since this representsby far the largest part in the total error budget (Beier et al., 2001). Using Eqs.(6) and (33) we obtain from Eq. (31) the electron’s atomic mass as

m = 0.0005485799092(4). (34)

This is in agreement with the CODATA electron mass (Mohr and Taylor, 1999)based on a direct determination by the comparison of its cyclotron frequency tothat of a carbon ion in a Penning trap (Farnham et al., 1995):

m = 0.0005485799110(12). (35)

VII. Outlook

The continuous Stern–Gerlach effect, using the frequency dependence of theaxial oscillation on the spin direction of an ion confined in a Penning trapwhen an inhomogeneous field is superimposed, is a powerful tool to measuremagnetic moments of charged particles with great precision. This accurateknowledge of magnetic moments is very important for tests of QED calculations.The g−2 experiment on free electrons by Dehmelt and coworkers (Van Dycket al., 1987) was a first example, followed now by the first application to anatomic ion. The method described above is applicable to any ion having amagnetic moment on the order of a Bohr magneton, provided it can be loadedinto the trap. For a given axial frequency and magnetic inhomogeneity B2, thefrequency splitting depends as 1/

√qM on the mass M of the ion and its charge

state q (Fig. 23). This will impose technical limitations when working withheavier hydrogen-like ions. Currently the stability of the electric trapping fieldlimits the maximal resolution of the axial frequency measurements: a jitter ofthe trapping voltage by 1mV, typical for state-of-the-art high-precision voltagesources, induces frequency changes of 100mHz for trap parameters as in ourcase. However, materials with higher magnetic susceptibilities than nickel, suchas Co−Sm alloys, produce a larger magnetic inhomogeneity and therefore alarger frequency splitting, allowing to proceed to heavier ions. In addition, theinduced magnetic inhomogeneity scales with the cube of the inverse radius ofthe ring electrode. Thus a reduction in size of the analysis trap increases the

VII] CONTINUOUS STERN–GERLACH EFFECT ON ATOMIC IONS 215

Fig. 23. Difference in axial frequency for two spin directions in a bottle-like magnetic field forvarious hydrogen-like ions. The parameters B0 = 3.8 T, B2 = 1 T/cm2, and wz /2p = 365 kHzare those of our experiment. The frequency difference scales linear with the magnetic fieldinhomogeneity B2.

frequency difference for the two spin states significantly. This would also havethe advantage that it reduces the amount of ferromagnetic material placed in theanalysis trap, and so helps to improve the homogeneity in the precision trap. Tofurther improve the homogeneity in the precision trap the distance between thetwo traps can be increased. Finally, shim coils may be used to make the fieldin the precision trap more homogeneous. We believe that we can maintain thepresently achieved precision with other ions as well, and hope to even increaseit when we apply some of the measures for improvement. This would result in amore significant test of higher-order bound-state QED contributions since theyincrease quadratically with the nuclear charge (see Fig. 2).The method can also be applied to more complicated systems: a measurement

of the electronic g-factor in lithium-like ions would test not only the QED cor-rections in these systems but also correlation effects with the remaining electronswhich change the g-factor significantly. When applied to hydrogen-like ions withnon-zero nuclear spin the transition frequencies between spin states depend onthe nuclear magnetic moment. Measuring the different transition frequenciesyields the magnetic moment of the nucleus. This would be of special interest,because all nuclear magnetic moments so far have been determined using neutralatoms or singly ionized ions. The effective magnetic field seen by the nucleus inthese systems differs from the applied magnetic field by shielding effects of theelectron cloud. In a measurement on hydrogen-like ions this shielding is strongly

216 G. Werth et al. [IX

reduced, and comparison with data obtained on neutral systems would, for thefirst time, test atomic-physics calculations on electron shielding.

VIII. Acknowledgements

The measurements described above are performed in close collaboration toGSI/Darmstadt. We gratefully acknowledge financial support from its AtomicPhysics group (Prof. H.-J. Kluge). Several doctoral and diploma students wereand are actively involved in the experiments: Stefan Stahl, Nikolaus Her-manspahn, Jose Verdu, Tristan Valenzuela, Slobodan Djekic, Michael Diederich,Markus Immel, and Manfred Tonges. We appreciated stimulating discussionswith our colleagues: Thomas Beier, Andrzej Czarnecki, Ingvar Lindgren, SavelyKarshenboim, Vasant Natarajan, Hans Persson, Sten Salomonson, VladimirShabaev, Gerhard Soff, and Alexander Yelkovsky.The experiments are part of the TMR network ERB FMRX CT 97-0144

“EUROTRAPS” of the European Community.

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