Nuclear Instruments and Methods in Physics Research B 205 (2003) 15–19
www.elsevier.com/locate/nimb
Determination of the electronÕs mass from
g-factor experiments on 12C5þ and 16O7þ
Thomas Beier a,*, Slobodan Djekic b, Hartmut H€
affner b, Paul Indelicato c,
H.-J€
urgen Kluge a, Wolfgang Quint a, Vladimir M. Shabaev d, Jose Verd
u b,
Tristan Valenzuela b, G€
unther Werth b, Vladimir A. Yerokhin c,d
a
d
Gesellschaft f€ur Schwerionenforschung (GSI), Atomic Physics, Planckstrasse 1, 64291 Darmstadt, Germany
b
Institut f€ur Physik, Universit€at Mainz, 55099 Mainz, Germany
c
Laboratoire Kastler-Brossel, UPMC, case 74, 4 place Jussieu, F-75252 Paris Cedex 05, France
Department of Physics, St. Petersburg State University, Oulianovskaya 1, Petrodvorets, St. Petersburg 198504, Russia
Abstract
We present a derivation of the electronÕs mass from our experiment on the electronic g factor in 12 C5þ and 16 O7þ
together with the most recent quantum electrodynamical predictions. The value obtained from 12 C5þ is
me ¼ 0:0005485799093ð3Þ u, that from oxygen is me ¼ 0:0005485799092ð5Þ u. Both values agree with the currently
accepted one within 1.5 standard deviations but are four respectively two-and-a-half times more precise. The contributions to the uncertainties of our values and perspectives for the determination of the fine-structure constant a by an
experiment on the bound-electron g factor are discussed.
Ó 2002 Elsevier Science B.V. All rights reserved.
PACS: 14.60.Cd; 06.20.Jr; 31.30.Jv; 32.10.Fn
Keywords: ElectronÕs mass; g factor; Highly charged ions; Quantum electrodynamics; Fine-structure constant
1. Introduction
The mass of the electron is a fundamental
physical constant. In atomic mass units, its currently accepted value is given by [1]
me ¼ 0:0005485799110ð12Þ u:
ð1Þ
This value is mainly based on the measurement of
Farnham and co-workers [2,3] who subsequently
*
Corresponding author. Tel.: +49-6159-712116; fax: +496159-712901.
E-mail address: [email protected] (T. Beier).
observed single 12 C6þ ions and clouds of several
electrons in a Penning trap. By comparing the
cyclotron frequencies for both the ion and the
electrons, the ratio of the electron mass to that of
12 6þ
C
is obtained. Earlier determinations were
performed using mainly the proton–electron mass
ratio [4–9] or the antiproton–electron mass ratio
[8]. Wineland et al. [10] measured the ground-state
g factor of 9 Beþ in a Penning trap and compared
their experimental result with the value predicted
in [11] and derived the mass ratio mð9 Beþ Þ=me .
We have developed a setup to measure the g
factor of a single highly charged ion with a precision of about 109 using a Penning trap [12,13]. In
0168-583X/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved.
doi:10.1016/S0168-583X(02)01968-7
16
T. Beier et al. / Nucl. Instr. and Meth. in Phys. Res. B 205 (2003) 15–19
our evaluations of the electronic g factors in 12 C5þ
[13] and 16 O7þ [14], the published uncertainty of
the electron mass proved to be the limiting factor.
It thus seemed reasonable to compare experiment
and the precise theoretical predictions ([15] and
references therein) to obtain an independent value
for the mass of the electron. For carbon, this was
already performed in [16], with a result of
me ¼ 0:0005485799092ð4Þ u. Here we are going to
focus on the progress made since the publication
of [16].
2. Derivation of the electrons mass
The g factor of the electron bound in a hydrogenlike ion is measured by determining the
corresponding Larmor frequency xL , given by
e
B:
ð2Þ
xL ¼ g
2me
The Larmor frequency is the frequency related to a
spin-flip transition between the different magnetic
angular-momentum substates in the given magnetic field B. In order to determine the actual
magnetic field strength, the cyclotron frequency of
the ion is measured,
q
xC ¼
B;
ð3Þ
mion
where q denotes the charge of the ion. From the
value xL =xC , the g factor is obtained as
xL q me
g¼2
:
ð4Þ
xC e mion
Details of this procedure are given in [12–14,16].
The comparison of theoretically evaluated and
experimentally determined g factors for hydro-
genlike carbon and oxygen is given in Table 1. The
experimental uncertainties resulting from the
xL =xC measurement and from the electron mass
given in [1] are presented separately. Experiment
and theory agree reasonably within their error
margins thus forming one of the most stringent
confirmations of the bound-state quantum electrodynamical (QED) calculations carried out by
Yerokhin et al. [17,18]. We stress that the new
calculations for the QED corrections of order
ða=pÞ, presented in [17,18], are in agreement with
the older ones for the same contribution ([19] and
references therein) but about one order of magnitude more precise for the self-energy correction.
The major limitation to the experimental precision is the error margin of the published value for
the mass of the electron [1]. If we employ the
theoretically determined value gtheo for the g factor
and solve Eq. (4) for me instead, we obtain the
values for the electron mass presented in Table 1.
The good agreement between our two determinations is visible. We are not going to perform any
averaging, however, since both in theory and experiment too many correlated effects might be
present that are not yet fully investigated. Not
considering them might lead to a too small uncertainty of the averaged value if employing standard error propagation.
Care has to be taken about the masses of the
ions since the mass of the electron enters also in
their values. The mass of a positive ion of charge
state N is given by
mðA X N þ Þ ¼ mðA X Þ N me þ EB =c2 ;
ð5Þ
where EB refers to the total binding energy of the
missing electrons and c is the speed of light. Taking this into consideration,
Table 1
Experimental and theoretical values for the g factors of 12 C5þ and 16 O7þ , the derived values for the mass of the electron, and the mass
ratio of proton and electron, based on a proton mass of mp ¼ 1:00727646688ð13Þ u [1]
12
g factor from experiment
g factor from theory
Derived me
mp =me
C5þ
2.0010415963(10)(44)
2.0010415901(3)
0.0005485799093(3) u
1836.1526731(10)
16
O7þ
2.0000470268(16)(44)
2.0000470202(6)
0.0005485799092(5) u
1836.1526735(16)
For comparison: in [1] a value of mp =me ¼ 1836:1526675ð39Þ is given. The experimental uncertainties resulting from the xL =xC
measurement and from the electron mass given in [1] are given separately.
T. Beier et al. / Nucl. Instr. and Meth. in Phys. Res. B 205 (2003) 15–19
e
me ¼ ðmðA X Þ þ EB =c2 Þ
q
2 xL
þ1 :
gtheo xC
ð6Þ
In both the numerator and denominator the second term is small compared to the first one. In
particular, binding energies which are hardly better known than to a precision of 106 affect the
relative precision of the final result only on a level
DEB =ðmðA X Þc2 Þ. A detailed discussion on the
binding energies is presented in [20].
3. Uncertainties and the value of a
In the determination of the electron mass, our
experimental error margins are thought to include
all experimental sources of error. A comprehensive
discussion of these uncertainties is provided elsewhere [12,14].
On the theoretical side, the uncertainties quoted
in [17,18] comprise all error margins currently
known for the theory. The value given for the total
uncertainty is largely dominated by an estimate.
For the g factor of an electron bound in the 1s1=2
state of a hydrogenlike ion, the QED contributions
2
of orders ða=pÞ and higher are known only up to
the first term in a (Za) expansion to be
DgðQED; order ða=pÞn Þ
!
2
a n
ðZaÞ
þ ;
1þ
¼ 2 Að2nÞ
p
6
ð7Þ
where 2 Að2nÞ refers to the QED contribution of
the same order to the g factor of the free electron,
i.e. Að2Þ ¼ 1=2, Að4Þ ¼ 0:328 . . . [21,22], etc. An explicit derivation of Eq. (7) can be found in [23].
For precise bound-state QED calculations, an
expansion in powers of ðZaÞ is hardly sufficient (cf.
the reviews [24] and [25] and references therein).
For the order ða=pÞ, a comparison between formula (7) and exact calculations is provided in [19].
For the order ða=pÞ2 and higher orders, however,
calculations non-perturbatively in ðZaÞ were not
yet performed and to get any estimate for these
contributions, Eq. (7) has to be employed. In order
to estimate the error due to this approximation,
the difference of the non-perturbative calculation
and formula (7) for the order ða=pÞ is taken,
17
multiplied by ðAð4Þ =Að2Þ Þ ða=pÞ. To be on the safe
side, in [17,18] these values were multiplied by 1.5
to obtain a reasonable value for the error which at
present also defines the total theoretical error of
the g-factor prediction. The choice of this factor of
1.5 is somewhat arbitrary and a less optimistic
value of 2 was employed for the same factor in
[16]. Increasing the theoretical error in this way
does not influence the total precision of our values
for the electronÕs mass which at present are dominated by the experimental error margins. For the
numerical uncertainties caused by the calculation
of the terms of order ða=pÞ, we refer to [17,18].
The present value of the fine-structure constant
a is given by a ¼ 1=137:03599976ð50Þ [1]. In the
calculation of the g factor, this leads to an uncertainty of 2 1011 for 12 C5þ and of 3 1011 for
16 7þ
O , respectively. This insensitivity on the finestructure constant is a feature of the particular Z
region investigated so far, as the bound-state correction to the Dirac value
for theffig-factor, given by
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
gDirac ¼ ð2=3Þ 1 þ 2
1 ðZaÞ
2
and the leading
QED correction, given by ða=pÞ, almost cancel
here (and thus the value of the g factor is rather
close to 2). This is different, however, for even
moderately higher Z. If we consider only the two
mentioned contributions as crucial for the determination of a, it is quite possible to obtain also a
value for a from the g-factor measurements from
aðfrom gÞ
2
3
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
3 41
1
8 2
¼ 2
Z ðgexpt 2 sÞ5;
4Z p
p
3
ð8Þ
where the ‘‘’’ in front of the square root has to be
taken for Z 6 5. Here, s is a small number, given by
the theoretical value of g with the crucial leading
terms (in a) subtracted:
a 2
2
s ¼ gtheo 2 þ ðZaÞ :
p 3
ð9Þ
The remaining dependence sðaÞ is negligible in the
region Z 20 and below if a ‘‘reasonable’’ a (e.g.
the value from [1]) is employed in the bound-state
QED calculations. However, care should be taken
when going to higher Z. In order not to perform
18
T. Beier et al. / Nucl. Instr. and Meth. in Phys. Res. B 205 (2003) 15–19
systematical errors, it has to be checked whether
more terms have to be subtracted separately.
The uncertainty da ðfrom gÞ from the determination Eq. (8) is given by
" #1=2
2
1
8 2
da ðfrom gÞ ¼
Z gexpt 2 s
p
3
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
:
ð10Þ
dgexpt
þ dgtheo
perspectives, we therefore include also a curve for
da=a with a theoretical precision for the g factor of
109 over the whole range. At about Z ¼ 70, the
precision in determining a from a g-factor experiment would match that of the value currently
published in [1]. Ions of these charge numbers are
planned to be investigated in the future HITRAP
facility at GSI [28].
If we take 40 Ca19þ as an example and consider an
experimental precision of 1 109 and the current
theoretical one of 1:4 108 [18] for this ion, we
thus obtain an error margin of 5:4 107 for a
which is only slightly less precise than those values
obtained by several other methods as listed in [1].
In Fig. 1, we display the current total theoretical
uncertainties for gtheo [18] and the resulting uncertainties for a according to Eq. (10), assuming an
experimental precision of 1 109 , over a large
range of Z. The theoretical uncertainty increases
2
due to the ða=pÞ -bound-state contributions (cf.
the discussion above) and also because of nuclear
size and structure effects [26,27]. These latter are
thought to be partly overcome by investigating
also g factors of lithiumlike ions [26]. For future
4. Summary
fractional uncertainty
10
10
10
10
10
We have presented our way of determining the
mass of the electron from a high-precision g-factor
experiment on hydrogenlike carbon and oxygen
together with precise theoretical QED predictions.
The obtained value for the mass of the electron
from carbon is me ¼ 0:0005485799093ð3Þ u which
is four times more precise than that of the best
experiment up to now [2]. The value obtained from
oxygen (me ¼ 0:0005485799092ð5Þ u) still is twoand-a-half times more precise than that from [2].
Both values match each other and agree also with
the older one within 1.5 standard deviations. The
precision of our experiment is such that even a
determination of a competitive with other methods
seems possible if we investigate heavier ions.
-5
-6
(δα/α) from g, δgexp = 10
−9
(δgtheo/gtheo )
-7
(δα/α) if δgtheo = 10
-8
-9
Acknowledgements
−9
(δα/α) from CODATA 1998
20
30
40
50
60
70
nuclear charge Z
80
90
Fig. 1. Current uncertainty of the theoretical predictions for g,
based on the values for 32 S15þ , 40 Ar17þ , 40 Ca19þ , 52 Cr23þ ,
74
Ge31þ , 132 Xe53þ , 208 Pb81þ and 238 U91þ published in [18], and
derived values for da=a, according to Eq. (10) and assuming an
experimental uncertainty of 1 109 . In addition the calculated
values for da=a are shown, assuming also a theoretical uncertainty of 1 109 over the whole range of Z. For Z P 70, this
uncertainty becomes smaller than that given in the recent
compilation of fundamental constants [1]. Wiggles in the curves
are due to spline interpolation between the calculated values.
We thank R.N. Faustov, A.V. Nefiodov, A.S.
Yelkhovsky and S.G. Karshenboim for stimulating and helpful discussions. Our work was supported by the European Community through the
TMR network ‘‘EUROTRAPS’’ (contract number
ERB FMRX CT 97-0144) and through the RTD
network ‘‘HITRAP’’ (contract number HPRI-CT2001-50036), by the Russian Foundation for Basic
Research (project no. 01-02-17248), and by the
programme ‘‘Russian Universities’’ (project no.
UR.01.01.072).
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