Download Electronic g Factor of Hydrogenlike Oxygen 16O7+

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Electronic g Factor of Hydrogenlike Oxygen 16 O7
J. Verdú, S. Djekić, S. Stahl, T. Valenzuela, M. Vogel, and G. Werth
Institut für Physik, Johannes-Gutenberg-Universität, D-55099 Mainz, Germany
T. Beier, H.-J. Kluge, and W. Quint
Gesellschaft für Schwerionenforschung, D-64291 Darmstadt, Germany
(Received 11 September 2003; published 5 March 2004)
We present an experimental value for the g factor of the electron bound in hydrogenlike oxygen,
which is found to be gexpt 2:000 047 025 4 1544. The experiment was performed on a single 16 O7
ion stored in a Penning trap. For the first time, the expected line shape of the g-factor resonance is
calculated which is essential for minimizing the systematic uncertainties. The measurement agrees
within 1:1 with the predicted theoretical value gtheory 2:000 047 020 2 6. It represents a stringent
test of bound-state quantum electrodynamics to a 0.25% level. Assuming the validity of the underlying
theory, a value for the electron mass is obtained: me 0:000 548 579 909 6 4 u. This value agrees with
our earlier determination on 12 C5 and allows a combination of both values which is about 4 times more
precise than the currently accepted one.
DOI: 10.1103/PhysRevLett.92.093002
Quantum electrodynamics is an established theory
which describes the interaction of particles with the
quantized radiation field to extremely high precision.
This is manifested particularly in the agreement between
the calculated and measured values of the magnetic moment of the free electron to 12 digits [1].
The situation is more complex when dealing with particles bound to a nucleus of charge Z mainly because the
standard method of calculation by use of a perturbation
expansion is not applicable anymore: while for a free
particle the expansion parameter is the fine structure
constant and is thus small as compared to one, this is
no longer valid for a bound particle. The additional expansion parameter Z may not be small as compared
to one, especially for high nuclear charges Z. For this
case, nonperturbative methods of calculation have been
developed [2].
Recently, the magnetic moment of the electron bound
to a carbon nucleus has been determined with great
precision [3]. This has stimulated theoretical calculations,
and by now values of the magnetic moment of the electron bound in different hydrogenlike ions have been calculated to a high accuracy [4]. In this contribution we
report on an extension of the previous experiment to
hydrogenlike oxygen 16 O7 , using a similar experimental
method, however, with significant improvements. The
result, as discussed below, represents a first step to confirm the Z dependence of the bound-state QED calculations. It furthermore confirms the result for the electron
mass extracted from the measurement on hydrogenlike
carbon and allows the increase of the accuracy of that
value [5,6].
The g factor of a charged particle is defined as the
proportionality constant between its angular momentum
J~ and the corresponding induced magnetic moment ~.
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0031-9007=04=92(9)=093002(4)$22.50
PACS numbers: 32.10.Fn, 06.20.Jr, 12.20.–m, 31.30.Jv
For the electron in the ground state of a hydrogenlike ion
this relation reads
e
~ g
J~;
(1)
2me
where e represents the (positive) elementary charge and
me is the mass of the electron. In the presence of
an external magnetic field B~ B u~ z the Zeeman splitting of the electronic ground state is given by E1s h
~ B~ i and is proportional to the Larmor precession
frequency of the electron E1s h!
L . The magnetic
field can be calibrated by using the cyclotron frequency
of the ion !c q=mi B, with q representing its charge
and mi its mass. By use of this relation and Eq. (1) the g
factor can be obtained in terms of measurable quantities:
q me !L
g2
:
(2)
e mi !c
The static magnetic field necessary for generating the
Zeeman splitting under investigation is also responsible
for the radial confinement of the ions. Additionally, an
electrostatic force along the magnetic field lines is used
for confinement in the axial direction. This kind of arrangement is known as a Penning trap [7]. The motion of
an ion in a Penning trap is a superposition of three
harmonic oscillations. The rotation around the magnetic
field axes is characterized by the reduced cyclotron frequency ! . The oscillation around the electric potential
minimum is given by the axial frequency !z and the drift
of the ion’s orbit on the radial plane by the so-called
magnetron frequency ! . In order to measure the
strength of the magnetic field B we make use of the wellknown invariance theorem [8] which relates the three
trapping frequencies to the cyclotron frequency !2c !2 !2z !2 .
 2004 The American Physical Society
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The details of the experimental setup have been
described in [9,10]. In short, it is a stack of two fivepole cylindrical compensated orthogonal Penning traps
[10,11]. The central electrode of each trap, called the ring,
is set to a negative voltage with respect to the two outer
end caps, which are grounded. The voltage applied to the
correction electrodes, Uc , placed between the end caps
and the ring, is carefully selected for achieving maximum harmonicity of the trapping potential [10]. The ring
of one trap, the so-called analysis trap, is made of a
ferromagnetic material which locally deforms the otherwise homogeneous magnetic field, thus creating a magnetic bottle. The curvature of the magnetic field, B2 1 @2 B
2
2! @z2 , has been measured to be B2 10 1 mT=mm .
This allows the determination of the electron’s spin state
by the continuous Stern-Gerlach effect [10,12] as explained below. In the other trap, the so-called precision
trap, the magnetic field inhomogeneity is 3 orders of
magnitude smaller: B2 8:2 0:9 T=mm2 . This permits fractional uncertainties below the ppb (parts per
billion) range in the measurement of !c . Between both
traps the ion can be transported without affecting the spin
state of the electron [9,13].
The three frequencies of the trapped ion, ! , !z , and
! , are measured electronically: the induced mirror
currents upon the trap electrodes are picked up by tuned
resonance circuits, amplified and by fast Fourier transformation (FFT) observed in the frequency domain.
Figure 1 shows the spectrum of the induced voltage by
the reduced cyclotron motion of a single 16 O7 ion. The
line is a least squares fit to a Lorentzian resonance curve.
The center frequency, i.e., the reduced cyclotron frequency, is obtained with a relative uncertainty of 2 1010 . Similarly, the axial frequencies both in the precision and in the analysis trap can be measured [9,10]. The
magnetron frequency is determined indirectly via mode
coupling to the axial degree of freedom [14] with an
uncertainty of 100 mHz which suffices to determine the
magnetic field with a precision of 1010 .
The Zeeman splitting amounts to roughly 105 GHz in
both traps. By irradiation of microwaves of this frequency, spin flips can be induced. They are detected in
the analysis trap, where the ion’s total axial potential
energy is given by Uz q ~ B~ , being the electric
quadrupole potential. The axial frequency is determined
by the second derivative of Uz and is therefore a function
of the size and orientation of ~ . Thus, depending on
whether the spin is up or down, the frequency is different
g
B
by !z up !z down m!
B2 2 0:48 Hz in the
z
16 7
case of O . The spin flips are detected through the
observation of this small jump during the continuous
measurement of !z . However, this is possible only if the
frequency remains constant over the typical measurement time of several minutes. The main anharmonicity
of the axial potential is caused by its octupole term, c4 4
4
@4 1 @ Uz
1
~ @@zB4 , and is proportional to ~.
4! @z4 4! q @z4 Thus, any variation of the ion’s cyclotron energy, i.e., of
its associated magnetic dipole moment, distorts the harmonicity of Uz , making !z unstable. This is compensated
by automatically adjusting the applied correction voltage
necessary to achieve c4 0 at the given measured value
of the cyclotron energy E . The correction voltage Uc
obeys a linear dependence on the cyclotron energy as
shown in Fig. 2. The strong magnetic bottle makes it
very sensitive to fluctuations in E . The actual measurement of the g factor takes place in the precision trap.
We determine the microwave frequency !MW which
maximizes the probability of inducing a spin flip. This
frequency would ideally correspond to the Larmor frequency !L of the electron in 16 O7 . However, in order to
obtain the g factor and according to Eq. (2) the value of
the magnetic field also needs to be known. Since B fluctuates with time, a separate measurement of the Larmor
and cyclotron frequencies broadens the expected resonance, thus reducing the accuracy. This is avoided by
measuring the spin flip probability as a function of the
frequency ratio defined through !!MW
. The basic
c
measurement cycle is as follows: (i) The electron’s spin
FIG. 1 (color online). Cyclotron resonance spectrum of a
single hydrogenlike oxygen ion in the precision trap.
FIG. 2 (color online). Uc as a function of the cyclotron
energy in the analysis trap. The error bars are smaller than
the symbols. The zero point of the energy corresponds to 4 K.
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direction is determined in the analysis trap. (ii) The ion is
transported to the precision trap. Simultaneously with the
measurement of !c the ion is irradiated with microwaves
of the frequency !MW thus representing a try of inducing
a spin flip by the frequency ratio !MW =!c . (iii) The
ion is transported back to the analysis trap and a new
determination of the electron’s spin state is performed. By
comparison with the previous spin direction it can be
determined whether for the tried a spin flip was induced
in the precision trap or not.
A scan over the relevant region of the instantaneous
frequency ratio yields a resonance curve of the spin flip
probability as shown in Fig. 3. The uncertainty assigned
to each experimental data point is calculated assuming a
binomial probability distribution [9]. The value of for
which the spin flip probability resonance P reaches its
maximum is denoted by ^ . It is determined from a least
squares fit to a Lorentzian, as is justified below. Its value
is ^ 4164:376 187 8 31, where the given uncertainty is
purely statistical. In a perfectly homogeneous magnetic
field the spin flip probability is given by a simple
Lorentzian resonance curve. However, because of the
nonvanishing B2 term of the magnetic field in the precision trap, the Larmor frequency and, consequently, P
are functions of the ion’s kinetic energies: !L !L E ; Ez ; E [15]. In order to measure !c the cyclotron
motion has to be excited to initial energies of E0 3 eV.
During the try of one and because of the interaction
with the detection electronics, the cyclotron motion of the
ion undergoes resistive cooling [9]. Therefore, its cyclotron energy follows an exponential cooling curve:
E t E0 expt= . The cooling time constant has
been measured to be 298 4 s for a single oxygen
ion. The axial motion stays in thermal equilibrium with
the corresponding detection electronics. Thus, the energy
Ez fluctuates according to a Boltzmann distribution
expEz =kB Tz . This distribution has been observed experimentally [9] and yields an axial temperature of Tz 61 7 K. The magnetron energy is constant and equals
FIG. 3. Experimental spin flip probability. All data are corrected to zero cyclotron energy.
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the magnetron cooling limit E !!z kB Tz 0:10 0:01 meV [15]. Its influence upon the spin flip resonance is
very small. Inserting the known relation !L !L E ; Ez ; E into the ideal Lorentzian curve, the true
spin flip probability is obtained as the sum of the probabilities for the ion to experience one spin flip for any of
the energy pairs E ; Ez possible during the try of a
certain . It is therefore given by
Zt
1 Z1
P dEz eEz =kB Tz dt0 IEz ; t0 : (3)
2tkB Tz 0
0
The integration limit t is the measurement time of 72 s
and identical to the time needed for recording a cyclotron
spectrum as shown in Fig. 1. The integrand I is given by
I 2
2
g0 0
E0 et =
z Ez $ 2
:
(4)
I represents the modified Lorentzian spin flip probability
for a given axial energy Ez and at a time t0 within the
measurement. The parameter $ is defined as $ 2 qe mmei .
The constants and z are given by BB2 m g!0 2
i
and z BB2 2mi !g0 ! . For a single 16 O7 ion in the precision trap, the values 1:1 0:1 109 eV1
and z 7:5 0:8 107 eV1 result. The parameter
contains the influence of the microwave
21 2 BMW
B
power upon the resonance, where BMW is the strength of
the magnetic field component of the microwaves. Finally,
g0 represents the pursued g factor, given by the ratio of
the ideal unperturbed Larmor to cyclotron frequencies,
as defined in Eq. (2).
In Fig. 4, Eq. (3) has been used to simulate the spin flip
probability resonance for four different axial temperatures. For this purpose, the theoretical value of g0 [4] is
used. It shows the evolution of ^ towards the correct
frequency ratio with vanishing axial temperature. In general, this is true for all ion energies, since for vanishing
motion amplitudes the influence of the magnetic field
inhomogeneity disappears. Thus, the frequency ratio 0
FIG. 4 (color online). Calculated spin flip resonance curves
according to Eq. (3). The zero point is set at the frequency ratio
corresponding to the theoretical value of g0 .
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TABLE I. Systematic corrections to the frequency ratio ^ of
a single 16 O7 ion. The systematic uncertainty of 0 is the
quadratic sum of all systematic uncertainties.
Description
Value
Syst. uncertainty
0
^ with
Extrap. Tz ! 0
Extrap. E ! 0
Time base FFT
Zero-reference E0
Microwave purity
Electrical instabilities
Elec. trap anharmon.
Relativistic shifts
4164.376 187 8
0:000 005 0
0
0:000 000 3
0:000 000 3
6 107
6 1011
2 107
3 107
2 107
3 108
1 108
4 109
Final result 0
4164.376 183 4
7 107
E0
corresponding to the true g factor is obtained by a multiple extrapolation of ^ to zero motional energies in all
degrees of freedom. The theoretical model contained in
Eq. (3) predicts a linear dependence of ^ with the cyclo@ ^
tron energy of @E
2:0 0:2 106 eV1 . This
linear behavior is confirmed experimentally by a mea@ ^
sured dependence of @E
1:8 0:3 106 eV1 . In
the experimental resonance curve presented in Fig. 3 all obtained for different nonvanishing cyclotron energies
have been corrected to zero E . Because of the linearity
of ^ ^ E the symmetry of the underlying Lorentzian
shape of the spin flip probability is not affected by the
cyclotron energy. In contrast, a finite value of Tz deforms
this symmetry. The deviation caused by the use of a
simple Lorentzian curve for obtaining ^ , as shown in
Fig. 3, is smaller than the statistical uncertainty for the
actual temperature of Tz 61 K. Assuming the validity
of Eq. (3) all necessary remaining corrections are calculated and given in Table I together with the estimated
systematic uncertainties.
Using the accepted values for the electron mass me
[16], the mass of the neutral oxygen atom mO [17] and the
sum of the binding energies of the seven missing electrons of Ebin 1:258 66 9 106 u [18], the mass of
the hydrogenlike oxygen ion is mi mO 7me Ebin 15:991 075 819 2 u. Inserting mi , me , and 0 into Eq. (2)
yields the g factor of the electron bound in hydrogenlike
oxygen:
g0 2:000 047 025 4 15 44:
(5)
The first uncertainty is the quadratic sum of the statistical
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and systematic uncertainties in the measured frequency
ratio 0 . The second one is caused by the uncertainty of
the electron mass given in [16]. The combined experimental uncertainty of 4.6 ppb represents about 0.27% of
the bound-state QED contributions (including recoil) to
the theoretical g factor of 1715 ppb [4]. Since the uncertainty of the accepted electron mass dominates over the
measurement uncertainty, the argument can be turned
around and, under the assumption of the validity of the
calculations presented in [4], a new mass for the electron
can be obtained. It is me 0:000 548 579 909 6 4 u in
agreement with the value from measurements on 12 C5
[4,5]. A combination of both values improves the current
accepted CODATA [16] value by a factor of about 4 [6].
This work was supported by the European Union
(HITRAP HPRI-CT-2001-50036) and the BMBF. We
thank R. Ley for stimulating discussions.
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