ADVANCES IN ATOMIC, MOLECULAR, AND OPTICAL PHYSICS, VOL. 48
CONTINUOUS STERN–GERLACH
EFFECT ON ATOMIC IONS
GÜNTHER WERTH 1 , HARTMUT HÄFFNER 1 and WOLFGANG QUINT 2
1 Johannes
Gutenberg University, Department of Physics, 55099 Mainz, Germany;
für Schwerionenforschung, 64291 Darmstadt, Germany
2 Gesellschaft
I.
II.
III.
IV.
V.
VI.
VII.
VIII.
IX.
Introduction . . . . . . . . . . . . . . . . . . .
A Single Ion in a Penning Trap . . . . .
Continuous Stern–Gerlach Effect . . . .
Double-Trap Technique . . . . . . . . . . .
Corrections and Systematic Line Shifts
Conclusions . . . . . . . . . . . . . . . . . . .
Outlook . . . . . . . . . . . . . . . . . . . . .
Acknowledgements . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . .
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191
195
206
209
212
213
214
216
216
I. Introduction
In 1922 Stern and Gerlach succeeded in spatially separating a beam of silver
atoms into two beams, utilizing the force exerted in an inhomogeneous magnetic
field on the magnetic moment of the unpaired electron in silver. This so-called
Stern–Gerlach effect was the first observation of the directional quantization of
the quantum-mechanical angular momentum, and represents a cornerstone of
quantum 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 in
atomic systems: An atomic beam enters a first inhomogeneous magnetic field
where one spin direction is separated from the other. The polarized atoms then
enter a region with a homogeneous magnetic field where they are subjected to
a radio-frequency (rf) field which changes the spin direction. Finally, a second
inhomogeneous magnetic field region analyzes the spin direction. Variation of
the frequency w of the rf field and recording the spin-flip probability as a
function of w yields a resonance curve. Together with a measurement of the
field strength in the homogeneous magnetic field, the magnetic moment can be
derived. Usually the size of the magnetic moment m is given in units of the Bohr
magneton mB = eà/ (2m) and expressed by the g-factor:
m = gsmB ,
191
(1)
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 represent
a 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 force
is overshadowed by the Lorentz force acting on a moving charged particle in
a magnetic field. Proposals to use the acceleration of electrons moving along
the field lines of an inhomogeneous B-field to separate the spin directions
(Brillouin, 1928) have been discarded by Bohr (1928) and Pauli (1958) on the
basis of Heisenberg’s uncertainty principle. Nevertheless attempts have been
made, however unsuccessful, to separate the two spin states in an electron beam
by using the different signs of the force on the spin exerted by a longitudinal
inhomogeneous magnetic field which results in a deceleleration or acceleration
of the electrons (Bloch, 1953). Recently, new proposals have come up to perform
Stern–Gerlach experiments on electron beams which would – in contrast to
the analysis by Bohr and Pauli – result in a high degree of spin separation
under carefully chosen initial conditions (Batelaan et al., 1997; Garraway and
Stenholm, 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 particle
which oscillates in a parabolic potential well leads to a small but measurable
difference of its oscillation frequency for different orientations of the spin. Thus
a precise measurement of the oscillation frequency gives information on the
spin direction. Dehmelt et al. used this effect for monitoring induced changes
of the spin direction of an electron by observing the corresponding changes
in the electron’s oscillation frequency. Since the sensitive electronics monitors
the trapped particle’s spin direction continuously, Dehmelt coined the term
“continuous Stern–Gerlach effect” for this technique. In a series of experiments
he and his coworkers have applied this method to measure the magnetic moment
of the electron and the positron, which culminated in the most precise values of
a property of any elementary particle (Van Dyck et al., 1987). The experimental
value
(2)
gexp = 2.002 319 304 376 6 (87)
agrees to 10 significant digits with the result of calculations based on the theory
of quantum electrodynamics (QED) for free particles (Hughes and Kinoshita,
1999),
gth = 2.002 319 304 432 0 (687)
(3)
and provides one of the most stringent tests of QED. The theoretical result for
the free-electron g-factor can be expressed in a perturbation series as
a a 2
a 3
a 4
+ A3
+ A4
+ ··· ,
(4)
+ A2
gfree = 2 A0 + A1
p
p
p
p
I]
CONTINUOUS STERN–GERLACH EFFECT ON ATOMIC IONS 193
Fig. 1. Calculated expectation values for the electric field strength for hydrogen-like ions of
different nuclear charge Z. (Courtesy Thomas Beier.)
1
where a ≈ 137
is the fine-structure constant. The coefficients An in Eq. (4) have
been calculated by evaluating the Feynman diagrams of different orders using a
plane-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 the
field strength ranges from 109 V/cm in the helium ion (Z = 2) to 1015 V/cm
in hydrogen-like uranium (Z = 92) (Fig. 1), and gives rise to a variety of new
effects. The largest change of the bound electron’s g-factor was analytically
derived by Breit (1928) from the Dirac equation:
gBreit =
2
3
1+2
1 − (Za)2 ≈ 2 · 1 − 13 (Za)2 .
(5)
The conditions of extreme electric fields also necessitate changes to be made in
the methods of calculations for the QED contributions to the electron’s magnetic
moment. In a perturbative treatment a series expansion in (Za) is made in
addition to that in a. The expansion parameter Za, however, is – at least for
large Z – no longer small compared to 1. In addition, the expansion coefficients
can be large. For small values of the nuclear charge Z the perturbation expansion
may give reliable results, and calculations were performed which include terms
up to order (Za)2 (Grotch, 1970a; Close and Osborn, 1971; Karshenboim et al.,
2001). For more accurate theoretical predictions, non-perturbative methods have
been developed where the solutions of the Dirac equation for the hydrogen-like
ion 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 nuclear
charges Z. (Courtesy Thomas Beier.)
Table 1
Theoretical contributions to the g-factor in 12 C5+
Contribution
Size
Dirac Theory
1.998 721 354 4
Reference
Breit (1928)
QED, free (all orders)
+0.002 319 304 4
Hughes and Kinoshita (1999)
QED, bound, order (a/ p )
+0.000 000 844 2(9)
Beier et al. (2000)
(a/ p )2 ,
(Za)2
−0.000 000 001 1(4)
Grotch (1970a)
Recoil (in Za expansion)
+0.000 000 087 6(1)
Yelkovsky (2001)
Finite size correction
+0.000 000 000 4
Beier et al. (2000)
QED, bound, order
term
been published by Beier (2000). A graphical representation of the bound-state
contributions to the electron g-factor is shown in Fig. 2.
In this contribution we describe an experiment which for the first time applies
the “continuous Stern–Gerlach effect” to an atomic ion (Hermanspahn et al.,
2000). We measured the magnetic moment of the electron bound to a nucleus
with zero nuclear magnetic moment, hydrogen-like carbon (12 C5+ ). For the
bound-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-perturbative
calculations to all orders in (Za). The numerical results of the calculations for
C5+ 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 the
free electron (Hughes and Kinoshita, 1999). The bound-state contributions of
order a (calculated to all orders in Za) are given with error bars which represent
the numerical uncertainty of the calculations. The quoted uncertainty of the
a 2 (Za)2 term is an estimate of the contribution from non-calculated higherorder terms. Finally, nuclear recoil contributions have been calculated to lowest
order 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 nuclearshape correction was considered numerically by Beier et al. (2000), and recently
(for low Z) also analytically by Glazov and Shabaev (2001). The sum of the
different contributions leads to a theoretical value for the g-factor in hydrogenlike carbon of
gtheor (12 C5+ ) = 2.001 041 589 9 (10).
(6)
II. A Single Ion in a Penning Trap
The experiment is carried out on a single C5+ ion confined in two Penning
traps. In a Penning trap a charged particle is stored in a combination of a
homogeneous magnetic field B0 and an electrostatic quadrupole potential. The
magnetic field confines the particle in the plane perpendicular to the magnetic
field lines, and the electrostatic potential confines it in the direction parallel to
the magnetic field lines. In our experiment we use two nearly identical traps
placed 2.7 cm apart in the magnetic field direction. They consist of a stack of 13
cylindrical electrodes of 2r0 = 7 mm inner diameter. The difference between the
traps is that in one trap the center electrode is made of ferromagnetic nickel while
all 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 in
the 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 the
key element to analyze the direction of the electron spin through the continuous
Stern–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 electrode
while 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 in
cylindrical coordinates r, z, ö by an expansion in Legendre polynomials Pi :
F(r, ö) =
∞
U0 r i
Ci
Pi (cos ö),
2
d
(7)
i=0
where d 2 = z02 + r02 / 2 / 2 is a characteristic dimension of the trap (Gabrielse
et al., 1989). Two correction electrodes are placed between the center ring
and the endcaps. The coefficient C4 which is the dominant contribution to the
trap anharmonicity can be made small by proper tuning of the voltages applied
to the correction electrodes. Essentially then the potential depends on the square
of the coordinates, and is a harmonic quadrupole potential
F=
U0 z 2 − r 2 / 2
.
2
d2
(8)
We optimize the trap by changing the voltages on the correction electrodes until
the ion oscillation frequency is independent of the ion’s oscillation amplitude
as characteristic for a harmonic oscillator. With this method we can reduce the
II]
CONTINUOUS STERN–GERLACH EFFECT ON ATOMIC IONS 197
Fig. 4. Ion oscillation in a Penning trap.
dominant high-order term C4 , the octupole contribution, to less than 10−5 . The
ion’s frequency in the harmonic approximation is then given by
qU0
wz =
.
(9)
Md 2
Radial confinement is achieved by the homogeneous magnetic field directed
along 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 perturbed
cyclotron frequency wc . It differs from the cyclotron frequency
wc =
q
B
M
(10)
of a free particle with charge q and mass M because of the presence of the
electric trapping field, and is given by
wc2 wz2
wc
wc =
+
−
.
(11)
2
4
2
It can be expressed also as
wc = wc − wm ,
(12)
where wm is the magnetron frequency, a slow drift of the cyclotron orbit around
the trap center, given by
wc2 wz2
wc
wm =
−
−
.
(13)
2
4
2
198
G. Werth et al.
[II
For calibration of the magnetic field we use the cyclotron frequency of the
trapped ion. It can be derived either from Eq. (12) or more reliably from the
relation
wc2 = wc2 + wz2 + wm2 ,
(14)
since this equation is independent of trap misalignments to first order (Brown and
Gabrielse, 1986). In this case the measurement of the magnetron frequency wm
is required in addition to a measurement of wc and wz .
The traps are enclosed in a vacuum chamber placed at the bottom of a helium
cryostat at a temperature of 4 K and located at the center of a superconducting
NMR solenoid. The helium cryostat provides efficient cryopumping. As an upper
limit we estimate the vacuum in the container to be below 10−16 mbar. The
estimation was derived from the measurements on a cloud of highly charged
ions, whose storage time would be limited by charge exchange in collisions with
neutral background particles. We observed no ion loss in a cloud of 30 hydrogenlike carbon ions stored for 4 weeks. Together with the known cross section for
charge exchange with helium as the most likely background gas at 4 K we obtain
an upper limit of 10−16 mbar for the background gas pressure. The magnetic
field of the superconducting magnet is chosen to be 3.8 T. At this field strength
the precession frequency of the electron spin is 104 GHz. Microwave sources of
sufficient 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 investigation
as well as of other elements present on the surface. Higher charge states are
obtained by consecutive ionisation by the electron beam. We detect the ions
by picking up the current induced by the ion motions in the trap electrodes.
For this purpose superconducting resonant circuits and amplifiers are attached
to 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 is
detected. Figure 5 shows such a spectrum, where we identified different elements
and charge states. We eliminate unwanted ion species by exciting their axial
oscillation 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 they
have different orbits. Therefore, for small ion numbers, single ions can be
distinguished by their different cyclotron frequencies. Figure 6 shows a Fourier
transform of the induced current from 6 stored ions. We excite the ions’ cyclotron
motion individually and thus eliminate them from the trap until a single ion is
left. Typical cyclotron energies for signals as shown in Fig. 6 are of the order of
several 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 and
Dehmelt, 1975; Dehmelt 1986). The ion’s oscillation is brought into resonance
II]
CONTINUOUS STERN–GERLACH EFFECT ON ATOMIC IONS 199
Fig. 5. Mass spectrum of trapped ions after electron bombardment of a carbon surface showing
different charge states of carbon ions as well as impurity ions (a) before and (b) after removal of
unwanted species.
with the circuits attached to the electrodes. The induced current through the
impedance of the circuit leads to heating of the resonance circuit, and the ion’s
kinetic energy is dissipated to the surrounding liquid helium bath (Fig. 7). This
leads to an exponentially decreasing energy with a time constant t given by
q2
R,
(15)
Md 2
where R is the resonance impedance of the circuit.
For the axial motion we use superconducting high-quality circuits (Q = 1000 at
1 MHz in the precision trap and Q = 2500 at 365 kHz in the analysis trap).
With the resonance impedances of R = 23 MW (analysis trap) and 10 MW
(precision trap) the cooling time constants are 80 ms and 235 ms, respectively.
For cooling the cyclotron motion we employ a normal-conducting circuit at
24 MHz (Q = 400) with a resonance impedance of 80 kW. Here we reach cooling
time constants of a few minutes. Figure 8 shows the exponential decrease of the
induced currents from the ion oscillations as the result of axial cooling.
t −1 =
200
G. Werth et al.
[II
Fig. 6. Fourier transform of the voltage induced in one of the trap electrodes from the cyclotron
motion of 6 stored 12 C5+ ions. The inhomogeneous magnetic field of the trap causes ions at different
positions 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 is
metastable: in the radial plane the ion experiences an electrostatic force towards
the negatively biased center electrode. Ion loss is prevented by the presence of
the magnetic field. Thus the potential energy is an inverted parabola. Therefore
reduction of the ion’s magnetron energy results in an increase in the magnetron
radius. It is, however, essential to reduce the magnetron radius because of
the magnetic field inhomogeneities. This is achieved in a well-defined way
by coupling the magnetron motion to the axial motion by a radio-frequency
field at the sum frequency of both oscillations (Brown and Gabrielse, 1986;
Cornell et al., 1990). In the quantum-mechanical picture for the ion motion, the
absorption of a photon from this field increases the quantum number of the axial
oscillation by 1 while that of the magnetron oscillation is decreased by 1. An
analysis of the absorption probabilities in the framework of a harmonic oscillator
II]
CONTINUOUS STERN–GERLACH EFFECT ON ATOMIC IONS 201
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 expectation
value Em for the magnetron energy,
|Em | = àwm km + 12 = àwm kz + 12
wm
wm
=
àwz kz + 12 =
Ez .
wz
wz
(16)
The axial oscillation is continuously kept in equilibrium with the cooling circuit
and 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 the
resonant circuit. The statistical motion of the electrons in the resonance circuit
causes Johnson noise in the trap-electrode voltages, which in turn leads to
varying energies of the ion as a function of time (Fig. 9). Extracting a histogram
of the cyclotron energies results in a Boltzmann distribution (Fig. 10) with a
temperature of 4.9 K close to the temperature of the environment. Calculating
the temporal autocorrelation function of the energy gives, as expected, an
exponential (Fig. 11) with a time constant well in agreement with the measured
cooling 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. Because
of their different orders of magnitude (wc / 2p = 24 MHz, wz / 2p = 1 MHz,
wm / 2p = 18 kHz) the required precision is different. wc is determined from the
Fourier transform of the current induced in a split electrode. Figure 12 shows
that the relative linewidth of the resonance, well described by a Lorentzian, is of
the order of 10−9 and the center frequency can be determined with an accuracy
of 10−10 . In order to obtain sufficient signal strength the energy of the cyclotron
oscillation has to be raised to about 1 eV. Due to the inhomogeneity of the
202
G. Werth et al.
[II
Fig. 9. Noise power of the induced voltage in a trap electrode from the cyclotron oscillation of
a single trapped ion while its frequency is continuously kept in resonance with an attached tank
circuit.
Fig. 10. Histogram of the probabilities for cyclotron energies. The curve can be well fitted to a
Boltzmann distribution, giving a temperature of 4.9(1) K.
II]
CONTINUOUS STERN–GERLACH EFFECT ON ATOMIC IONS 203
Fig. 11. The time-correlation function of the noise in fig. 10 shows an exponential decrease. The
time constant of 5.40(7) min corresponds to the time constant for resistive cooling of the cyclotron
motion.
Fig. 12. High-resolution Fourier transform of the induced noise at the perturbed cyclotron
frequency. 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 in
the impedance Z(w) of the axial circuit, given by
Unoise =
4kT Re[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 a
phase difference of 180º, as can been shown by modeling the system as a driven
harmonic oscillator. Consequently the sum of the thermal noise voltage and
204
G. Werth et al.
[II
Fig. 13. Axial resonance of a single trapped C5+ ion. The noise voltage across a tank circuit
shows a minimum at the ion’s oscillation frequency.
Fig. 14. High-resolution Fourier transform of the axial noise near the center of the resonance
frequency of the axial detection circuit.
the induced voltage leads to a reduced total power around the axial frequency
of the ion. This appears as a minimum in the Fourier transform of the axial
noise as shown in Fig. 13. A spectrum with a resolution of 10 mHz (Fig. 14)
shows that the center frequency can be determined to about 24 mHz. A different
approach to explain the appearance of a minimum in the axial noise spectrum
was taken by Wineland and Dehmelt (1975) considering the equivalent electric
circuit of an oscillating ion in the trap. The magnetron frequency wm is
measured by sideband coupling to the axial motion. If the ion is excited at
the difference between the axial and magnetron frequencies, the ion’s axial
II]
CONTINUOUS STERN–GERLACH EFFECT ON ATOMIC IONS 205
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 this
frequency determination is below 100 mHz.
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). Considering
only an octupole contribution to the trap potential, characterized by a coefficient C4 in the potential expansion (5), these shifts amount to
2
w
1
3
Dwz
= C4
Ec + 6Em ,
(18)
Ez − 3
wz
qU0 2
wc
2 2
Dwc
wz
1
3 wz
= C4
Ec − 6Em ,
−3Ez +
(19)
wc
qU0 wc
2 wc
2
Dwm
wz
1
=
Ec + 6Em .
6Ez − 6
(20)
wm
qU0
wc
Ez , Ec and Em are the energies in the axial, cyclotron and magnetron degrees
of 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 the
corresponding frequency shifts are below 10−10 and need not be considered here.
The coefficient C6 of the dodekapole contribution to the trapping potential has
been calculated to an accuracy of 10−3 for our trap geometry. The frequency
shifts 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 value
of the oscillation frequencies of an ion with finite kinetic energy as compared
to the ion at rest. A series expansion of the B-field in axial direction,
Bz = B0 + B1 z + B2 z 2 + · · ·
(21)
gives frequency shifts
2
wz
Dwc
1 B2 1
=
Ec + Ez + 2Em ,
−
wc
mwz2 B0 àwc
wc
Dwz
1 B2 1
[Ec − Em ] ,
=
wz
mwz2 B0 àwc
Dwm
1 B2 1
[2Ec − Ez − 2Em ] .
=
wm
mwz2 B0 àwc
(22)
(23)
(24)
The size of the inhomogeneity term B2 is measured by application of a bias
voltage 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 each
position. We obtain B2 = 8.2(9) mT/mm2 . The shift in the perturbed cyclotron
frequency which is of most interest here is dominated by the axial energy Ez ,
and amounts to Dwc / wc = 7×10−9 for an axial temperature of 100 K.
III. Continuous Stern–Gerlach Effect
The g-factor of the bound electron as defined by Eq. (1) can be determined by
a measurement of the energy difference between the two spin directions in a
magnetic field B:
(25)
DE = hnL = gmB B,
where nL is the Larmor precession frequency. We induce spin flips by applying
magnetic dipole radiation which is blown into the trap structure by a microwave
horn.
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 Dyck
et al., 1986): the quadrupole potential of the Penning trap depends on the square
of the coordinates (6) leading to a linear force acting upon the charge of the
stored ion. Considering the force upon the magnetic moment of the bound
electron 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 distortion
which can be described in first approximation by
2
2
ÀB = ÀB0 + 2B2 z − r ê − zÀr .
(27)
2
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 = −2mz B2 z,
(28)
which is linear in the axial coordinate. It adds to the electric force from the
quadrupole trapping field acting on the particle’s charge. Since both forces are
linear in the axial coordinate the ion motion is still described by a harmonic
oscillator (Fig. 15). The axial frequency, however, depends on the direction of
the magnetic moment m with respect to the magnetic field:
wz = wz0 + 12 dwz = wz0 +
mz B2
.
M wz0
(29)
The value of B2 in our set-up was calculated using the known geometry and
magnetic 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 potential
for the spin-magnetic moment in a bottle-like magnetic field. The strength of the potential depends
on 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 time
for each resonance line was 1 min.
experimentally by applying a bias voltage between the endcap electrodes of the
analysis trap and measuring the cyclotron frequency of the ion at different axial
positions. The calculated and experimental values for B2 in our experiment agree
within their uncertainties of 10% and yield B2 = 1 T/cm2 . For hydrogen-like
carbon the frequency difference dwz / 2p between the two spin states amounts to
0.7 Hz at a total frequency of wz0 / 2p = 365 kHz.
As evident from Fig. 16, the axial frequency can be determined to better
than 100 mHz. Fig. 17 demonstrates that after 1 min. averaging the expected
frequency difference between the two spin states becomes obvious. Driving
the spin-flip transition, we can distinguish the two possible axial frequencies,
0.7 Hz apart as calculated from the trap parameters. Varying the frequency of
the microwave field and counting the number of induced spin flips per unit time
yields a resonance curve as shown in Fig. 18. The shape of this resonance is
asymmetric due to the inhomogeneity of the magnetic field. The general shape
of 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 with
microwaves at the Larmor precession frequency showing two distinct values which correspond to
the two spin directions of the bound electron.
Fig. 18. Number of observed spin flips per unit time vs. the frequency of the inducing field. The
solid line is a fit according to Eq. (29).
by Brown (1985) as a complex function of the trap parameters, the ion’s energy
and 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 averaged
over the Boltzmann distribution of the energy:
∞
c (wL ) =
dE d wL − wL0
0
eEz
1+
mwz2
kTe−E/kT
Q(wL − wL0 )
wL − wL0
=
exp −
.
DwL
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 as
wL = w0 (1 + ez 2 ) on the axial coordinate. A least-squares fit of this function
to the data points of Fig. 18 yields the Larmor frequency with a relative
uncertainty of 10−6 (Hermanspahn et al., 2000). This is sufficient to measure
the binding correction to the g-factor in C5+ . The bound-state QED corrections
for 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 the
inhomogeneous magnetic field as required for the analysis of the spin direction
via the continuous Stern–Gerlach effect. In fact the inhomogeneity of the field
was chosen to be as small as possible, but still large enough to be able to
distinguish 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 inducing
spin flips and analyzing the spin direction (Häffner et al., 2000). This is achieved
by transferring the ion after a determination of the spin direction from the
analysis trap to the precision trap. The voltages at the trap electrodes are changed
in such a way that the potential minimum in which the ion is kept is moved
towards 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 therefore
adiabatic. Once in the precision trap, the ion’s motional amplitudes are prepared
by coupling the ion to the resonant circuits. We then apply the microwave field
to induce spin flips. After the interaction time, typically 80 s, and an additional
cooling time, the ion is moved back to the analysis trap. Here the spin direction
is analyzed again. In principle one measurement of the axial frequency would be
sufficient to determine whether it has changed by 0.7 Hz as compared to the value
before transport into the precision trap. If, however, the ion is not brought back
with the same radial motional amplitudes to the analysis trap, the axial frequency
may have changed by as much as 1 Hz. This is because of the magnetic moment
connected 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 precision
trap. A change in axial frequency of about 0.7 Hz 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 doubt
the spin direction after return to the analysis trap. Figure 19 shows several cycles
for a spin analysis. The total time for a complete cycle is about 30 min.
While the ion is in the precision trap its cyclotron frequency wc = (q/M ) B
is measured simultaneously with the interaction with the microwaves. This
ensures that the magnetic field is calibrated at the same time as the possible
spin flip is induced. The field of a superconducting solenoid fluctuates at the
level of 10−8 −10−9 on the time scale of several minutes. Figure 20 shows a
measurement of the cyclotron frequency of the ion in the precision trap over
a time span of several hours. Every 2 min the center frequency of the cyclotron
resonance was determined. The change in cyclotron frequency has approximately
a Gaussian distribution with a full-width-at-half-maximum of 1.2×10−8 . This
may impose a serious limit on the precision of measurements as in the case of
high-precision mass spectrometry using Penning traps (Van Dyck et al., 1993;
Natarajan et al., 1993). However, the simultaneous measurement of cyclotron and
Larmor 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=2
wL m
.
wC M
(31)
The mass ratio of the electron to the ion can be taken from the literature. In our
case of 12 C5+ , Van Dyck and coworkers (Farnham et al., 1995) measured it with
high 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 trapped
ion in a period of several hours. Data were taken every 2 min. The distribution is fitted by a Gaussian
with a full width of 1.2×10−8 .
Fig. 21. Measured spin-flip probability vs. ratio of Larmor and cyclotron frequencies. The data
are least-squares fitted to a Gaussian.
plot the spin flip probability, i.e. the number of successful attempts to change the
spin direction divided by the total number of attempts, we obtain a resonance line
as shown in Fig. 21. The maximum attainable probability is 50% when the
amplitude of the microwave field is high enough. To avoid those saturation effects
we take care to keep the amplitude of the microwave field at a level that the
maximum probability for a spin flip at resonance frequency is below 30%. In
addition 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 more
symmetric. For a constant homogeneous magnetic field in the precision trap the
lineshape would be a Lorentzian with a very narrow linewidth determined by
the coupling constant g to the cooling circuit. However, the observed lineshape
can be well described by a Gaussian. The fractional full width is 1.1×10−8 . This
reflects the variation of the magnetic field during the time the ion spends in the
precision trap which is of the same order of magnitude (see Fig. 20). The line
center 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 from
the fact that the field in the precision trap is not perfectly homogeneous.
As mentioned above, the ferromagnetic nickel ring placed 2.7 cm away in
the analysis trap causes a residual inhomogeneity in the precision trap. The
expansion coefficient from Eq. (21) gives B2 = 8 mT/ mm2 , three orders of
magnitude smaller than in the analysis trap. Therefore we still have to consider
an asymmetry in the line profile. Performing such an analysis gives a maximum
deviation 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 the
ion’s energies. In order to obtain a sufficiently strong signal of the induced
current from the cyclotron motion in the precision trap, the ion’s energy has
to be raised to about 1 eV. This finite cyclotron energy has a large magnetic
moment and thus shifts the axial frequency as compared to vanishing cyclotron
energy even in the precision trap by about 1 Hz. To account for this shift
we grouped our data of the spin flip probabilities according to the different
axial frequency shifts in the precision trap corresponding to different cyclotron
energies, 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 2
Systematic uncertainties (in relative units) in the g-factor determination of 12 C5+
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 . Of
the same order of magnitude are frequency shifts caused by changes of the
trapping potential due to ground loops when the computer controls are activated.
The interaction of the ion with its image charges changes the frequencies
by 3×10−10 , but can be calculated with an accuracy of 10%. Relativistic shifts
are of the order of 10−10 at typical ion energies, but do not contribute to the
uncertainty at the extrapolation to zero energy. A list of uncertainties of these
corrections is given in Table 2. The quadrature sum of all systematic uncertainties
amounts to 3×10−10 . The final experimental value for the frequency ratio wL / wC
in 12 C5+ is
wL
= 4376.210 498 9(19)(13).
(32)
wC
The first number in parentheses is the statistical uncertainty from the extrapolation to vanishing cyclotron energy, the second is the quadrature sum of the
systematical uncertainties. Taking the value for the electron mass in atomic units
(M (12 C) = 12) from the most recent CODATA compilation (Mohr and Taylor,
1999) we arrive at a g-factor for the bound electron in 12 C5+ of
gexp (12 C 5+ ) = 2.001 041 596 3 (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 theoretical
calculations 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 . Bound
QED contributions of order (a/ p )2 are too small to be observed. The nuclear
recoil 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. Taking
this for granted we can use experimental and theoretical numbers to determine
a more accurate value for the atomic mass of the electron, since this represents
by 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 = 0.000 548 579 909 2(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 to
that of a carbon ion in a Penning trap (Farnham et al., 1995):
m = 0.000 548 579 911 0(12).
(35)
VII. Outlook
The continuous Stern–Gerlach effect, using the frequency dependence of the
axial oscillation on the spin direction of an ion confined in a Penning trap
when an inhomogeneous field is superimposed, is a powerful tool to measure
magnetic moments of charged particles with great precision. This accurate
knowledge of magnetic moments is very important for tests of QED calculations.
The g−2 experiment on free electrons by Dehmelt and coworkers (Van Dyck
et al., 1987) was a first example, followed now by the first application to an
atomic ion. The method described above is applicable to any ion having a
magnetic moment on the order of a Bohr magneton, provided it can be loaded
into the trap. For a given axial frequency
and magnetic inhomogeneity B2 , the
√
frequency 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 with
heavier hydrogen-like ions. Currently the stability of the electric trapping field
limits the maximal resolution of the axial frequency measurements: a jitter of
the trapping voltage by 1 mV, typical for state-of-the-art high-precision voltage
sources, induces frequency changes of 100 mHz for trap parameters as in our
case. However, materials with higher magnetic susceptibilities than nickel, such
as Co−Sm alloys, produce a larger magnetic inhomogeneity and therefore a
larger frequency splitting, allowing to proceed to heavier ions. In addition, the
induced magnetic inhomogeneity scales with the cube of the inverse radius of
the 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 for
various hydrogen-like ions. The parameters B0 = 3.8 T, B2 = 1 T/cm2 , and wz / 2p = 365 kHz
are those of our experiment. The frequency difference scales linear with the magnetic field
inhomogeneity B2 .
frequency difference for the two spin states significantly. This would also have
the advantage that it reduces the amount of ferromagnetic material placed in the
analysis trap, and so helps to improve the homogeneity in the precision trap. To
further improve the homogeneity in the precision trap the distance between the
two traps can be increased. Finally, shim coils may be used to make the field
in the precision trap more homogeneous. We believe that we can maintain the
presently achieved precision with other ions as well, and hope to even increase
it when we apply some of the measures for improvement. This would result in a
more significant test of higher-order bound-state QED contributions since they
increase 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 corrections in these systems but also correlation effects with the remaining electrons
which change the g-factor significantly. When applied to hydrogen-like ions with
non-zero nuclear spin the transition frequencies between spin states depend on
the nuclear magnetic moment. Measuring the different transition frequencies
yields the magnetic moment of the nucleus. This would be of special interest,
because all nuclear magnetic moments so far have been determined using neutral
atoms or singly ionized ions. The effective magnetic field seen by the nucleus in
these systems differs from the applied magnetic field by shielding effects of the
electron 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 the
first time, test atomic-physics calculations on electron shielding.
VIII. Acknowledgements
The measurements described above are performed in close collaboration to
GSI /Darmstadt. We gratefully acknowledge financial support from its Atomic
Physics group (Prof. H.-J. Kluge). Several doctoral and diploma students were
and are actively involved in the experiments: Stefan Stahl, Nikolaus Hermanspahn, Jose Verdú, Tristan Valenzuela, Slobodan Djekic, Michael Diederich,
Markus Immel, and Manfred Tönges. We appreciated stimulating discussions
with our colleagues: Thomas Beier, Andrzej Czarnecki, Ingvar Lindgren, Savely
Karshenboim, Vasant Natarajan, Hans Persson, Sten Salomonson, Vladimir
Shabaev, 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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