ARTICLE
Received 29 May 2015 | Accepted 19 Nov 2015 | Published 18 Jan 2016
DOI: 10.1038/ncomms10246 OPEN
Isotope dependence of the Zeeman effect in lithium-like calcium
Florian Khler1,2, Klaus Blaum2, Michael Block1,3,4, Stanislav Chenmarev2,5, Sergey Eliseev2,Dmitry A. Glazov5,6,7, Mikhail Goncharov2, Jiamin Hou2, Anke Kracke2, Dmitri A. Nesterenko8,Yuri N. Novikov2,5,8, Wolfgang Quint1, Enrique Minaya Ramirez2, Vladimir M. Shabaev5, Sven Sturm2, Andrey V. Volotka5,6 & Gnter Werth9
The magnetic moment m of a bound electron, generally expressed by the g-factor m g mB s : 1 with mB the Bohr magneton and s the electrons spin, can be calculated by bound-state quantum electrodynamics (BS-QED) to very high precision. The recent ultra-precise experiment on hydrogen-like silicon determined this value to eleven signicant digits, and thus allowed to rigorously probe the validity of BS-QED. Yet, the investigation of one of the most interesting contribution to the g-factor, the relativistic interaction between electron and nucleus, is limited by our knowledge of BS-QED effects. By comparing the g-factors of two isotopes, it is possible to cancel most of these contributions and sensitively probe nuclear effects. Here, we present calculations and experiments on the isotope dependence of the Zeeman effect in lithium-like calcium ions. The good agreement between the theoretical predicted recoil contribution and the high-precision g-factor measurements paves the way for a new generation of BS-QED tests.
1 Atomic Physics Division and Superheavy Element Physics Division, GSI Helmholtzzentrum fr Schwerionenforschung, Planckstra e 1, 64291 Darmstadt, Germany. 2 Stored and Cooled Ions Division, Max-Planck-Institut fr Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany. 3 Superheavy Element Physics Division, Helmholtz-Institut Mainz, Johann-Joachim Becherweg 36, 55128 Mainz, Germany. 4 Institut fr Kernchemie, Johannes Gutenberg-Universitat, Fritz Strassmann Weg 2, 55128 Mainz, Germany. 5 Department of Physics, St Petersburg State University, Ulianovskaya 1, Petrodvorets,St Petersburg 198504, Russia. 6 Department of Physics, Institut fr Theoretische Physik, Technische Universitat Dresden, Mommsenstra e 13, 01062
Dresden, Germany. 7 Institute for Theoretical and Experimental Physics, Kurchatov Institute, B. Cheremushkinskaya street 25, Moscow 117218, Russia.
8 Petersburg Nuclear Physics Institute, Gatchina, 188300 St Petersburg, Russia. 9 Institut fr Physik, Johannes Gutenberg-Universitat, Staudingerweg 7, 55128 Mainz, Germany. Correspondence and requests for materials should be addressed to F.K. (email: mailto:[email protected]
Web End [email protected] ) or to S.S. (email: mailto:[email protected]
Web End [email protected] ).
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ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms10246
Besides hyperne splitting, isotope shifts of atomic electronic energy levels provide the most common access to nuclear properties1. Typically, the dominating nuclear effects
contributing to isotope shifts are generated by differences in nuclear masses, also denoted as nuclear recoil shifts (mass shifts), and by differences in nuclear sizes due to different spatial distributions of the nuclear charge (eld shift). In absence of the magnetic eld, isotope shifts in highly charged ions were rst measured in refs 2,3. In particular, relativistic nuclear recoil shifts have been previously probed in experiments on the isotope shifts in the binding energy of boron-like argon4 and lithium-like neodymium5.
As already proposed for different magnesium isotopes6, in this paper, we focus on the isotope dependence of the Zeeman effect by studying g-factors of lithium-like calcium isotopes 40Ca17 and 48Ca17 . Featuring on the one hand a 20% mass difference and on the other hand almost identical nuclear charge radii7, these isotopes provide a unique system across the entire nuclear chart to test the relativistic nuclear recoil shift in presence of a magnetic eld.
Most physical effects contributing to g-factors of highly charged ions, for example, the relativistic, radiative, nuclear size or interelectronic-interaction corrections, are calculated using bound-state quantum electrodynamics (QED) in the innite-nuclear-mass approximation. Here, the nucleus is considered as an external Coulomb potential xed in space. This approach is usually denominated as the Furry picture of QED (ref. 8). However, bound-state QED contributions of the studied nuclear recoil shift require calculations beyond the Furry picture, which are presented in the rst part of this paper.
The experimental determination of the tiny g-factor difference Dg g(40Ca17) g(48Ca17 ), which is in the order of
1 10 8, requires four independent high-precision measure
ments: the Larmor-to-cyclotron frequency ratios of both calcium ion species as well as their atomic masses. The frequency ratios have been measured successively with a relative uncertainty of about 7 10 11. For this purpose, we studied single ions
conned in a dedicated Penning-trap set-up9,10. Aiming for atomic masses with relative uncertainties of about 4 10 10, we
also improved the atomic mass of 48Ca by a factor of seven. Here, we used the ofine conguration of the Penning-trap mass spectrometer SHIPTRAP (ref. 11) in combination with the novel phase-imaging ion-cyclotron resonance technique (PI-ICR)12,13. The nally obtained 1.0s agreement between the predicted and measured g-factor difference decisively conrms relativistic recoil corrections in the presence of strong elds. The reinforced understanding of the interaction between the bound electrons and the nucleus provides the opportunity to extract fundamental constants, namely the ne structure constant a, and nuclear properties via g-factor measurements in heavy atomic systems14.
ResultsCalculation of the g-factor difference. The theoretical value of the isotope shift in the atomic g-factors is mainly given by a sum of the nuclear recoil and nuclear size contributions. Considering s-states of highly charged ions, the leading order terms scale with Dgrec mem 1nuclZa2n 2 and
Dgsize 83n3 1m2ec2 2Za4r2nucl (ref. 15), where n
represents the principle quantum number of the valence electron. Further nuclear contributions, for example, nuclear deformation16 and nuclear polarization17 are orders of magnitude smaller and at the current level of experimental as well as theoretical precision extraneous to the g-factor difference. For Z 20 the isotope shift is essentially determined by the mass
shift, which in the case of s-states is of pure relativistic origin.
Considering the two double magic isotopes 40Ca and 48Ca, the nuclear charge radii rnucl(40Ca) 3.4776 (19) fm and
rnucl(48Ca) 3.4771 (20) fm (ref. 7) are surprisingly similar and
by itself subject of present research. In this way, the nuclear recoil shift dominates the g-factor difference of the lithium-like electron conguration to 99.96%.
The lowest order recoil correction, which is non-QED but relativistic, can be derived from Breit equation1821. The full relativistic theory of the nuclear recoil effect on the atomic g-factor has to be formulated in the framework of QED. So far, a systematic approach has been developed to rst order in the electron to nucleus mass ratio me mnucl 1 and to all orders in
Za (ref. 22). As a result, the complete Za-dependence formula for the recoil effect on the g-factor of a hydrogen-like ion has been derived. To zeroth order in Z 1, this formula describes also the recoil effect in a few-electron ion with one electron over closed shells, provided the electron propagators are dened for the vacuum with the closed shells included15. Generally, this leads to the appearance of two-electron nuclear recoil contributions. However, for the (1s)22s-state of a lithium-like ion, the two-electron contributions vanish, and, therefore, to zeroth order in Z 1, one has to evaluate the one-electron contribution only. In the present paper, we evaluate this contribution to all orders in Za for the 2s-state at Z 20 using the corresponding
formula22. This result is combined with the radiative and second order in me mnucl 1 recoil corrections19,21,23,24 to get the total one-
electron contribution. To evaluate the interelectronic-interaction contribution to the recoil effect of the rst and higher orders in Z 1, we extrapolated the related results obtained to the lowest relativistic order25 (Methods section). The uncertainty of this contribution is mainly due to uncalculated higher order relativistic and QED corrections.
To get the total value of the isotope shift, one has also to account for the nuclear size effect. This contribution, being rather small, can be calculated in the one-electron approximation by solving the Dirac equation numerically. Moreover, it can be evaluated using an analytical formula26. The root-mean-square nuclear charge radii and their uncertainties are taken from ref. 7. The uncertainty of the nuclear size contribution includes both the nuclear radius and shape variation effects.
The individual contributions of the calculated isotope difference Dg g(40Ca17 ) g(48Ca17 ) are presented in Table 1. It
is seen that the QED recoil effect, whose calculation requires using QED beyond the Breit approximation and beyond the Furry picture, is about ve times bigger than the total theoretical uncertainty.
Table 1 | Individual contributions of the calculated isotope difference Dg g(40Ca17 ) g(48Ca17 ).
Effects Contributions to Dg/1 10 9
Nuclear recoil: one-electron non-QEDBme/mnucl
12.246
Nuclear recoil: one-electron non-QEDB(me/mnucl)2
0.006
0.123
Nuclear recoil: one-electron QEDBme/mnucl
Nuclear recoil: one-electronQEDBa (me/mnucl)
0.009 (1)
2.051 (25)
Finite nuclear size 0.004 (10)
Total theory 10.305 (27)
For details, see Methods section.
Nuclear recoil: interelectronic-interaction
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NATURE COMMUNICATIONS | DOI: 10.1038/ncomms10246 ARTICLE
Measurement concept. For the experimental determination of the g-factor difference, we measured successively the Zeeman splitting of the respective lithium-like ion in a homogeneous magnetic eld B using single ions conned in a Penning trap. The Larmor frequency nL, which quanties the energy difference between the spin-up and the spin-down state of the valence electron, is given by: vL 12p g2 emeB. We determine the magnetic eld by measuring the
cyclotron frequency vc 12p qionmionB of the ion with electric charge qion and mass mion. In the concluding equation for the g-factor:
g 2
47:943204044 19
u
! dm
48Ca=m48Ca 0:40 p:p:b:
2
The resulting atomic mass agrees within its uncertainty with the previous less accurate measurements32,33.
Measurement of the Larmor-to-cyclotron frequency ratios. Using a triple Penning trap set-up located at the University of Mainz, and described in detail in refs 34,35, we measured the Larmor-to-cyclotron frequency ratio G of both calcium isotopes.
Within a cryogenic (T 4.2 K) ultra-high vacuum chamber
(Po10 16 mbar) a miniature electron beam ion source enables the production of highly charged ions. By means of various cleaning routines35 we remove all unwanted ion species and nally conne a single ion in a ve electrode cylindrical Penning trap with an inner radius of r 3.5 mm. The oscillating ion induces image
charges on the electrode surfaces, which we measure to obtain the axial oscillation frequency. In the attached superconducting, tuned axial resonator the induced oscillating currents generate a measureable voltage signal in the order of a few 10 nV. We detect the signal of the thermalized axial motion (TzB5 K) as a minimum (dip-signal) in the Fourier transform of the thermal noise spectrum of the tank circuit (Fig. 2a). Both radial modes of the ion are thermalized and detected via rf-sideband coupling to the axial resonator generating double dip-signals in the axial frequency spectra. We determine the cyclotron frequency via the BrownGabrielse invariance theorem, where eigenfrequency shifts due to trap misalignment and ellipticity cancel36.
Simultaneously to the high-precision phase-sensitive measurement of the modied cyclotron frequency37, lasting about 5 s, we inject microwaves (MW) at the assumed Zeeman transition frequency (nMWE105 GHz) into the apparatus to induce spin-ips. To assess the success of a spin-ip attempt in our Precision trap (PT), we analyse the electron spin-state before and after the probing in a spatially separated Penning trap, the Analysis trap (AT). Here, a large magnetic bottle (B2,z 10(1) 103 T m 2) couples the magnetic moment to the
axial motion, resulting in frequency jumps of the axial oscillation
me mion
qion
e
qion
vLvc 2
me mion
e G 1
the magnetic eld cancels, if in the ratio G nL nc 1 both fre
quencies are probed simultaneously. To obtain the g-factor from the measured frequency ratios G, used in equation (1), the atomic masses of the ions are required. While the masses of 40Ca m(40Ca17) 39.953272233 (22) u with a relative mass
uncertainty of dmion mion 1 0.6 parts per billion (p.p.b.;
refs 27,28) and also of the electron with dme me 1 0.03 p.p.b.
(ref. 9) are known with sufcient accuracy, the tabulated value of the mass of 48Ca is not adequately precise. In the following, we report on high-precision measurements of (i) the 48Ca mass and (ii) the frequency ratios G(40Ca17 ) and G(48Ca17 ).
Determination of the atomic mass of 48Ca17 . With the Penning-trap mass spectrometer SHIPTRAP11, located at GSI
Helmholtzzentrum fr Schwerionenforschung Darmstadt, the atomic mass of 48Ca is directly determined by the measurement of the cyclotron-frequency ratio R of the mass doublet of singly charged 48Ca ions and 12C4 carbon cluster ions:
R nc(48Ca )/nc(12C4) m(12C4)/m(48Ca ). Instead of using
the BrownGabrielse invariance theorem nc2 n
2 nz2 n
2
(ref. 29), both cyclotron frequencies have been determined as the sum of the ions two radial eigenfrequencies nc n n ,
where n is the modied cyclotron frequency and n the
magnetron frequency. Considering a mass difference of Dm m(12C4 ) m(48Ca )E4.8 10 2 u we derive a
systematic shift of the mass ratio DR o1 10 11 caused by
possible misalignments and ellipticity of our trap. At the current level of precision, this effect is negligible.
In each measurement cycle, we produce alternately small clouds (r5 ions) of 48Ca and 12C4 with a laser-ablation ion source30 and separately transfer them into a preparation trap for cooling and centring via mass-selective buffer-gas cooling31 (Fig. 1). Then, the particular cyclotron frequency is measured
in the measurement trap with the novel PI-ICR (refs 12,13; Methods section). Combining the measured cyclotron-frequency ratio R 1.00099010175 (35)stat (17)syst (dR R 1 0.39 p.p.b.)
with the known carbon cluster mass m(12C4) and correcting for the missing electrons and their corresponding binding energies, we obtain the following value for the mass of lithium-like
48Ca (Methods section):
m 48Ca17
Preparation trap
Laser-ablation ion source
Measurement trap
Position-sensitive MCP detector
Image of the ion trap radial motions on a position-sensitive MCP detector
x axis
y axis
Trap-center image
Magnetron-motion
phase image
Cyclotron-motion
phase image
48 Production of of 48Ca+ and12C4+ions
Cooling and centering
Ca+ and 12C4+ ions
Excitation of radial ion motions
Imaging of radial ion motions to determine c
Nd:YAG laser beam
Superconducting magnet
Figure 1 | Ofine conguration of the Penning-trap mass spectrometer SHIPTRAP. The set-up contains a laser-ablation ion source, two Penning traps, one for the preparation of the ion (cooling and centring), the other for the frequency measurement process and a position-sensitive multi-channel plate (MCP) detector for a radial resolution of the ion position. The novel PI-ICR is alternately applied to small clouds of 48Ca and 12C4 ions, determining their respective cyclotron frequencies.
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MW-horn
7 mm
Single ion
a
Precision trap (PT)- Eigenfrequencies measurements
[afii9840]c =[afii9840]c
z
|B| = 3.76 T
[afii9840]2 + [afii9840]2 + [afii9840]2
Probing [afii9840]MW
[afii9796] =
[afii9840]MW
([afii9840] /Hz) 670341
[afii9796]
[afii9796]-resonance
Transport section
Analysis trap (AT)
Spin-flip
~14 cm
- Spin-state b c
detection
[afii9840][afii9840](mHz)
Spin-flip probability (%)
25 20 15 10
5
3 2 1 0 1 2 3
Reflector
Spin-flip
info
Creation trap (CT)
mEBIS Target
- Ion creation
Acceleration electrode
FEP
([afii9796]*/[afii9796]mean1) /1109
Figure 2 | The g-factor experiment for highly charged ions. The triple Penning-trap set-up (gold) comprises: (i) The PT with a homogeneous magnetic eld to determine the frequency ratios G* by measuring the three motional eigenfrequencies and probing the Larmor frequency. (ii) The AT to detect the spin-state of the valence electron. (iii) The Creation trap (CT) for ion creation within a miniature electron beam ion source (mEBIS). To enhance the production rate of 48Ca ions, an enriched calcium target is used with the following isotope composition: 40Ca: 78.77%, 42Ca: 3.02%, 43Ca: 0.62%, 44Ca:9.55%, 46Ca: 0.02% and 48Ca: 8.02%. The set-up is placed in a cryogenic (T 4.2 K) ultra-high vacuum chamber (Po1 10 16 mbar). In a the axial
resonator noise spectrum is shown including the dip-signal of a thermalized single 48Ca17 ion. In b the spin-state of the 48Ca17 ion is detected as an axial frequency jump at an absolute axial frequency of nz,off 412.4 kHz. In c the spin-ip probability is shown in dependence of the measured G*-values,
scaled by the nal central G value Gmean 5138.837 974 37 (58). The black points represent binned data to guide the eye. This data binning is not relevant
for the Gaussian maximum-likelihood (ML) t, shown in red. The dark grey-shaded area illustrates the uncertainty of Gmean and the bright grey area
represents the binomial errors considering the amount of cycles of binned data and the probability of the ML t. Error bars represent the uncertainties of each single axial frequency measurement point is related to the 1 sigma standard deviation.
Dnsfz g4p2 mBmion B2nz, which are caused by changes of the electrons
spin direction. This so-called continuous SternGerlach effect38 enables the spin-state detection. In case of the 48Ca17 ion Dnsfz amounts to only 140 mHz at an absolute frequency of nz
gmeas 48Ca17
412.4 kHz, which represents a signicant experimental challenge. Figure 2b illustrates the distinct detection of a spin-ip in the AT. Considering the limiting axial frequency resolution in the AT, we implement a proper cycle weighting to reduce the statistical uncertainty (Methods section).
During the automated measurement process, we probe the Zeeman transition several 100 times at different MW frequencies nMW. Combining the corresponding measured frequency ratios G* nMW nc 1 with the binary information of the spin-ip, we
obtain a G-resonance (Fig. 2c), which depicts the spin-ip probability in the PT versus the measured frequency ratios. With a weighted Gaussian maximum-likelihood t, we extract the mean value Gmean. This value has to be corrected for several
systematic shifts (Methods section and ref. 39).
DiscussionCombining the calcium masses with the measured frequency ratios G(40Ca17 ) 4,282.42953545 (30) and G(48Ca17 )
5,138.83795612 (42), we derive the most precise g-factor values for lithium-like ions from equation (1):
gmeas 40Ca17
11:70 0:16
Gstat 0:03
Gsyst 1:38
mion 10 9; 5
where the uncertainties of the frequency ratios and the mass measurements are listed separately. Obviously, the uncertainties in the masses of the isotopes dominate the total uncertainty. Since the dominant systematic shifts of the frequency ratios, the image charge shift (Table 3 and Methods section), scales with the mass of the ion, it cancels in the g-factor difference. Consequently, the denoted systematic uncertainty of the frequency ratios is smaller than the quadratically summed statistical uncertainties of the G-ratios given in equations (3) and (4). The comparison of the measured value of the g-factor difference with the theoretical prediction of this work:
Dgtheo gtheo 40Ca17
gtheo 48Ca17
Gsyst 110
mion
! dg=g 5:6 10 10; 3
1:99920204055 10
Gstat 12
10:305 27
10 9 6 allows for the rst time a direct test of the relativistic interaction of the electron spin with the motile nucleus. Although at present the experiment conrms the calculation only at the 10% level, the uncertainty of the measured frequency ratios is on the level of the QED recoil contribution.
Assuming QED calculations are correct within the given error bar, one may use the small uncertainty of the theoretically predicted g-factor difference in combination with the measured frequency ratios and the mass of 48Ca17 to determine the isotopic mass difference: Dm m(48Ca)
m(40Ca) 7.9899317834 (54) u. The uncertainty of this indirectly
obtained mass difference is a factor 5.7 smaller than the directly measured mass difference.
The combination of high-precision measurements of Larmorto-cyclotron frequency ratios, atomic masses of the lithium-like
gmeas 48Ca17
1:99920202885 12
Gstat 13
Gsyst 80
mion
! dg=g 4:1 10 10; 4
The statistical, systematic and ion mass uncertainties are given separately. The absolute values for the g-factors (Table 2) provide
a stringent test of many-electron QED calculations in a magnetic eld40,41. The g-factor difference nally yields the sought-after isotope difference:
Dgmeas gmeas 40Ca17
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Table 2 | Theoretical g-factor contributions for the lithium-like calcium ions 40Ca17 and 48Ca17 .
Effects g(40Ca17 ) g(48Ca17 )
Dirac value (point nucleus) 1.99642601090 QED,Ba 0.002325555 (5)
QED,Ba2 0.000003520 (2)
Interelectronic interaction 0.000454290 (9) Screened QED 0.000000370 (7)
Finite nuclear size 0.00000001441 (2) 0.00000001441 (2) Nuclear recoil 0.00000006185 (15) 0.00000005154 (12)
Total theory 1.999202042 (13) 1.999202032 (13)
Measured g-factor 1.9992020405 (11) 1.99920202885 (82)
The Dirac value, as well as the QED, interelectronic-interaction and screened QED corrections cancel in the g-factor difference. The two predicted g-factors agree with the measured values.
Table 3 | Systematic shifts and uncertainties of the G measurements.
Effects 40Ca17 (p.p.t.) 48Ca17 (p.p.t.) Image charge shift 941 (47) 1130 (57)
Image current shift 11 (12) 0.6 (10)
Magnetic eld imperfections 0.46 (31) 0.45 (37) Line-shape model of the dip-signal 0 (14) 0 (12) Electric eld imperfections 0.00 (39) 0.00 (51)
n measurement 0.0 (30) 0.0 (26)
Drift of axial potential 0.0 (12) 0.0 (12) Relativistic shift 0.010 (1) 0.010 (1)
Line-shape model G resonance 0.0 (6) 0.0 (6)
Gstat from lin. extrapol. to zero E 4,282.42953943 (21) 5,138.83796192 (30)
G (corrected for syst. shifts) 4,282.42953545 (21)stat (22)syst 5,138.83795612 (30)stat (30)syst
In the upper part the relative systematic shifts and their uncertainties are listed, which have to be added to the G measurements to derive the nal G values.
isotopes 40Ca17 and 48Ca17 and corresponding g-factor calculations, presented in this paper, enables a variety of fundamental studies. Besides the test of many-electron QED calculations in a magnetic eld by considering the absolute values of the g-factors or the indirect determination of the isotopic mass difference, the analysis of the measured and predicted g-factor difference between the calcium isotopes deepens the understanding of the interaction between the bound electrons and the nucleus. A further reduction of the mass uncertainties will enable an even more stringent test of the relativistic recoil predictions in the future. The validation of QED calculations is a prerequisite for further fundamental measurements in atomic physics, for example, the determination of the ne structure constant a via g-factor measurements of heavy, highly charged ions14.
Methods
Calculation of the isotope shift. The main contribution to the isotope shift
Dg g(40Ca17 )g(48Ca17 ) results from the nuclear recoil effect that must
be calculated including the relativistic, QED and interelectronic-interaction contributions. As the nuclear size effect is rather small, it can be evaluated in one-electron approximation by solving the Dirac equation.
Consider rst the nuclear recoil effect on the atomic g-factor to zeroth order in 1/Z. In this approximation, the me mnucl 1 nuclear recoil contribution to the g factor
of an ion with one electron over closed shells is given in refs 22,23.
Dg
1 mBma
expi o
j jr
r dil rirl
j jr
1
o2r
8
is the transverse part of the photon propagator in the Coulomb gauge. The tilde sign indicates that the related quantity (the wave function, the energy and the Coulomb-Green function o
) must be calculated in presence of the
homogeneous magnetic eld B directed along the z axis. As we consider an ion with one valence electron over the closed shells, the Coulomb-Green function is dened aso P
exp i o
whereF is the Fermi energy and Z-0. In equation (7), the summation over the repeated indices(k 1,2,3), which enumerate components of the three-dimensional vectors, is
implicit. Formula (7) incorporates both one- and two-electron nuclear recoil contributions to zeroth order in 1/Z. For the (1s)22s-state of a lithium-like ion, the (1/Z)0 two-electron contribution is zero and, therefore, we restrict our consideration to the one-electron contribution only. For the practical calculations, the one-electron contribution is conveniently represented by a sum of low-order (non-QED) and higher order (QED) term, Dg DgL DgH:
DgL
1 mBma
j i
h j o
F
1,
n iZ
n
1
2mnucl
a i
B0
@@Baj p2
aZr a p
a n
n p
9
1 ma
me2mnucl haj r p
aZ
2r r a
z
jai;
z
DgH
1 mBma
i 2pmnucl
Z1
1
do @
ji
B0
@Baj Dk o
pk; V
o i0
o
a
Dk o
pk; V
o i0
;
10
where V(r) (aZ)/(r) is the Coulomb potential induced by the nucleus and
n r r 1. The low-order term can be derived from the relativistic Breit equation,
while the derivation of the higher-order term requires using QED beyond the Breit
i 2pmnucl
Z1
1
do @
@Baj pk Dk o
pk Dk o
eAkcl j~ai
B0
:
7
eAkcl ~G
o
a
Here, c 1, eo0, mB is the Bohr magneton, ma is the angular momentum
projection of the state a, pk irk is the momentum operator, Acl [B r]/2 is
the vector potential of the homogeneous magnetic eld B directed along the z axis, Dk(o) 4paZalDlk(o),
Dil o; r
1 4p
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approximation. For this reason, we call them the non-QED and QED contributions, respectively.
The low-order term DgL can be evaluated analytically42:
DgL
8
me mnucl
2k2E2 kmeE m2e2m2ej j 1
; 11
9
4
where E is the Dirac energy and k 1jl
1
2 j 12
.
To the two lowest orders
in aZ, we have
(R-R mean) / 110
0
: 12
As follows from this formula, for an s-state (k 1) the non-relativistic
contribution to DgL vanishes and the low-order term comes from pure relativistic (B(aZ)2) origin.
The calculation of the higher order term, DgH, is a much more difcult task. For the 1s-state it is calculated in ref. 42. In the present paper we performed the corresponding calculation for the 2s-state. Details of this calculation and the corresponding results for other ions will be published elsewhere.
In addition to the main one-electron nuclear recoil contribution, we have to consider the radiative (Ba) nuclear recoil correction and the (me/mnucl)2 nuclear
recoil correction. To the lowest order in aZ, these corrections were evaluated in refs 19,21,23,24. We need also to account for the interelectronic-interaction effects of the rst and higher orders in 1/Z. To evaluate these effects we extrapolatethe lowest order relativistic results from ref. 25. The uncertainty of the interelectronic-interaction contribution is mainly due to uncalculated higher-order relativistic and QED corrections.
To get the total value of the isotope shift, we also evaluate the nuclear size correction. The root-mean-square nuclear charge radii and their uncertainties are taken from ref. 7. The uncertainty of the nuclear size contribution includes both the nuclear radius and shape variation effects. The individual contributions to the isotope shift of the g-factor for 40Ca17 and 48Ca17 are presented in Table 1.
In Table 2 we list the various contributions to the g-factor of 40Ca17 and
48Ca17 . The Dirac value, as well as the QED, interelectronic-interaction, and the screened QED corrections17 cancel out in the isotope difference. The nite nuclear size and nuclear recoil corrections lead inherently to the isotope shift.
The PI-ICR measurement scheme. After the transfer of the ions from the preparation trap into the centre of the measurement trap (Fig. 1), the coherent components of their magnetron and the axial motions are damped via 1 ms dipole rf-pulses at the corresponding motional frequencies to amplitudes of about 0.01 and0.4 mm, respectively. These steps are required to reduce a possible shift in the ratio of the 48Ca and 12C4 ions due to the anharmonicity of the trap potential and inhomogeneity of the magnetic eld to a level well below 10 10 (see ref. 13 for details). After this preparatory step, the radius of the ion cyclotron motion is increased to a radius of 0.5 mm to set the initial phase of the cyclotron motion. Then, two excitation patterns, called in this work magnetron-motion phase and cyclotron-motion phase, are applied alternately to measure the ion cyclotron frequency nc. In the magnetron-motion phase pattern the cyclotron motion is rst converted to the magnetron motion with the same radius. Then, the ions perform the magnetron motion for 100 ms accumulating a certain magnetron phase.
After 100 ms have elapsed, the ions position in the trap is projected onto a position-sensitive detector by ejecting the ions from the trap towards the detector43. In the cyclotron-motion phase pattern the ions rst perform the cyclotron motion for 100 ms accumulating a certain cyclotron phase with a consecutive conversion to the magnetron motion and again projection of the ion position in the trap onto a position-sensitive detector. The angle between the ion-position images corresponding to two patterns with respect to the trap centre image is proportional to the ion cyclotron frequency nc. Pulse patterns are applied for a total measurement time of B5 min. On this measurement scale the magnetron-motion phase and cyclotron-motion phase can be considered to be measured simultaneously. Data with 45 detected ions per cycle are not considered in the analysis to reduce a possible shift in the ratio of the 48Ca and 12C4 ions due to ionion interaction. To eliminate a possible cyclotron-frequency shift, which arises due to incomplete damping of the coherent component of the magnetron motion, the time between the damping of the magnetron and axial motions and the excitation of the ion cyclotron motion is varied over the period of the magnetron motion. The positions of the magnetron motion and cyclotron motion phase spots are chosen such that the angle between the phase spots, calculated with respect to the centre of the measurement trap, do not exceed few degrees. This is required to reduce the shift in the ratio of the
48Ca and 12C4 ion masses due to the possible distortion of the ion-motion projection onto the detector to a level well below 10 10 (ref. 13).
Data sets for the ion cyclotron-frequency ratio R. The cyclotron frequencies nc of the 48Ca and 12C4 ions are measured alternately for several days. The total measurement period is divided in 45 B1-h periods. In addition, each 5 min measurement is divided in 10 30-s periods. For each of the 45 1-h periods the ratio R1 h of the cyclotron frequencies 48Ca and 12C4 ions is obtained along with the inner and outer errors44 by tting to the 12C4 frequency points a polynomial of fth order P2(t) with constant coefcients a0, a1, a2, a3, a4 and a5 and to the 48Ca
frequency points a polynomial P1(t) R1 h P2(t). The nal cyclotron-frequency
ratio Rmean is the weighted mean of the R1 h ratios, where the maximum of the inner and outer errors of the R1 h ratios are taken as the weights to calculate Rmean (Fig. 3). The difference between the inner and outer errors does not exceed 10%. The nal frequency ratio R with its statistical and systematic uncertainties is Rmean 1.00099010175 (35)stat (17)syst. The systematic uncertainty in the
frequency-ratio determination originates from the anharmonicity of the trap potential, the inhomogeneity of the magnetic eld and the distortion of the ion-motion projection onto the detector13.
The atomic mass of 48Ca17 . The mass of a C4 cluster is calculated by considering the dissociation energy: Ediss 18.0(17) eV (ref. 45), the
ionization energy: Eion 11.0(7) eV (ref. 46) and the missing electron:
m C4
4 12u 1 me Dm Ediss
Dm Eion
DgL
me mnucl
1 j j 1
k2
k
2
1
2 k2
k
4
aZ
2
n2
4
8
0 10 20 30 40
Measurement time (hour)
Figure 3 | Data sets of the cyclotron-frequency ratio measurements R1 h
of 48Ca and 12C4 at SHIPTRAP. The red line and the grey-shaded band illustrate the mean ratio Rmean and the s.d. For details on the plotted error
bars see text.
Maximum-likelihood fit:
No spin-flip Spin-flip up Spin-flip down
z in AT (Hz)
12
10
Probability density (Hz1 )
8
6
4
2
0 0.2 0.1 0.0 0.1 0.2
Figure 4 | Probability density of the measured axial frequency differences in the AT. Duz is the axial frequency difference of subsequent measurements in the AT with 30 s spin-ip drives in between. From a maximum-likelihood (ML) t, which combines three Gaussian distributions (red: no spin-ip (sf), green: spin-ip up, blue: spin-ip down), the following parameters are extracted: the spin-ip rate: 26.5%, the frequency jitter: sDu 25 mHz and the axial frequency jump due to a spin-ip:
Dusfz 140 mHz.
47:9994514126 20 u: The
mass differences between all three possible cluster structureslinear, rhombus and triangular pyramidalare already covered by the uncertainties of the dissociation and ionization energies. For the determination of the mass of lithium-like 48Ca we have to correct the mass of singly charged 48Ca, m(48Ca1 ) m(C4 )/R,
by the 16 missing electron masses and the corresponding ionization energies: Dm(Ebind) 7.2438 (43) 10 6 u, where Ebind 6,747.5 (40) eV (ref. 47) and
1u 931,494,061 (21) eV c 2:
m 48Ca17
m 48Ca1
16 me Dm Ebind
47:943204044 19
u: 13
The atomic mass of neutral 48Ca. For completeness, we also specify the atomic mass of neutral 48Ca. Correcting for the mass of the missing electron and its
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NATURE COMMUNICATIONS | DOI: 10.1038/ncomms10246 ARTICLE
a b
40Ca -1) / 1 109
off
2
48Ca 1) / 1 109
off
0
1
2
([afii9796] mean/[afii9796]
1
00.0
0 0 5 10 15 20 25
0
4
([afii9796] mean/[afii9796]
0.1 0.2 0.3 0 2 4 6 8 10
U2exc(Vpp2) U2exc(Vpp2)
r2 (m2) / 1109 r2+ (m2) / 1109
+
8 12
20 40 60E (eV)
0 20 40 60 80 100E (eV)
Figure 5 | C-resonances at different modied cyclotron energies. We measure for both isotopes various G-resonances at different modied cyclotron energies. Here we show the extracted mean values Gmean, normalized by the constant values: G40Caoff 4; 282:429538772 in a and G48Caoff
5; 138:837973696 in b in dependence of the modied cyclotron energy, which is proportional to the squared amplitude Uexc of the rst excitation pulse. The slope is mainly given by the relativistic mass shift in the cyclotron frequency. The indicated error bars illustrate the statistical uncertainty of the weighted maximum-likelihood t (see the dark-shaded area in Fig. 2c).
unknown:
binding energy Ebind 6.11315520 (25) eV (ref. 47) we obtain:
m 48Ca
m 48Ca1
0:5; if : Dnz4spin-flip cut
1 me Dm Ebind
8 >
<
>
:
G Dn A;Dn ;s
j
2 r Dn A;Dn ;s
j
47:952522652 19
u; 14
which is in good agreement with the literature value of m(48Ca) 47.952522765
(129) (ref. 28) but a factor seven more precise.
Cycle weighting of the C-resonances. In the magnetic bottle of the AT the axial frequency jump caused by an induced spin-ip scales with the inverse of the ions mass. In contrast to our previous measurements, where the axial frequency shifts have been: Dnsfz 560 mHz for 12C5 (ref. 10), Dnsfz 240 mHz for 28Si13 (ref. 9)
and 28Si11 (ref. 40), it is a particular challenge to resolve the spin-states for the calcium isotopes, where Dnsfz 170 mHz for 40Ca17 and only Dnsfz 140 mHz
for 48Ca17 . We measure axial phase differences of subsequent measurements by applying a coherent detection technique, which includes three steps: (i) The axial phase is imprinted by a 10 ms dipolar excitation. (ii) The axial phase evolves for a certain time Tevol. (iii) The phase is measured via the axial detection system. With a phase-evolution time of Tevol 1 s and a readout-time of 552 ms, a spin-ip
corresponds to an axial phase-shift of Djsfz 360 deg Ttot Dnsfz 78 deg
for 40Ca17 and Djsfz 65 deg for 48Ca17 . In Fig. 4 1,790 averaged axial
frequency differences of 48Ca17 are histogrammed. Here, we determine each axial frequency by averaging over four successive phase measurements.
Between these measurement sequences, we try to induce spin-ips for 30 s at maximum MW-power and at a xed MW-frequency. The plotted probability density rAT is modelled by a superposition of three Gaussian distributions:
rAT Dnz A; Dnsfz ; sDn
wAT Dnz
16
where the spin-ip cut is 70 mHz for 48Ca17 . In a normal measurement cycle, we try to induce a spin-ip at least three times in the AT and then proceed with this measurement process, until the cut-criterion |Dnz|4 spin-ip cut is fullled for the rst time. For the rst and the last frequency jump in the AT, which fulls this criterion, the AT-weight is calculated. The spin-ip probability in the PT (wPT) is calculated from the two AT-weights: before entering the PT wbeforeAT
and directly
G Dn A; Dn ;s
2 r Dn A;Dn ;s
j
j
0:5; if : Dnzo spin-flip cut;
:
wPT wbeforeAT 1 wafterAT
after leaving the PT wafterAT
wafterAT 1 wbeforeAT
17
wPT 1 corresponds to a spin-ip in the PT, wPT 0 corresponds to no spin-ip in
the PT and wPT 0.5 corresponds to no spin-ip information in the PT. The
Gaussian line-shape of the G-resonance, which has been analysed in refs 9,34, gets modied by adding a fourth t-parameter (offG), which describes the wrong spin-ip detection rate in the PT:
GPT G ; AG; Gc; sG; offG offG
AG 2ps2G
e : 18 The PT-weight nally has to be included in the maximum-likelihood function:
L AG; Gc; sG; offG
Y
N
i1
wPT i GPT G i; AG; Gc; sG; offG
19
which is used, to extract the nal mean value Gmean. In comparison to the common cut-analysis, we improve the relative uncertainty of Gmean by 20 p.p.t.
Data sets of the C-resonances. Various G-resonances are recorded at different modied cyclotron energies during the phase-evolution time of the modied cyclotron mode and the simultaneous probing of the Larmor frequency nMW in the
PT. In Fig. 5 the mean values from the maximum-likelihood t, see equation (19), are plotted for 40Ca17 (a) and 48Ca17 (b). The slope is given mainly by the relativistic mass shift in the cyclotron frequency. The Larmor frequency is far less susceptible to relativistic shifts owing to the slow Thomas precession of the electron, which is bound to the heavy ion, leading to a suppression by a factor nL=nc. From linear extrapolations to zero modied cyclotron energy we derive our statistical G-values:
Gstat 40Ca17
4; 282:42953943 21
1 wPT i
1 GPT G i; AG; Gc; sG; offG
;
Gsf up Dnz A=2; Dnsfz
Gno sf Dnz 1 A; 0; sDn
j
Gsf down Dnz A=2; Dnsfz
; sDn
15
where Gno sf is the Gaussian distribution of the axial frequency differences without spin-ips with an amplitude (1-A), a mean value of zero and a s.d. of sDn . Gsf up
and Gsf down are the Gaussian distributions with spin-ip up (mean value: Dnsfz )
and spin-ip down (mean value: Dnsfz ). From a maximum-likelihood t, the
following three parameters are extracted: (I) the spin-ip rate: 26.5%, (II) the frequency jitter: sDn 25 mHz and (III) the axial frequency jump due to a spin-
ip: Dnsfz 140 mHz. For the different data sets of 48Ca17 , we determine a
frequency jitter of sDn def std Dnz
= 2
;
; sDn
p 305 mHz. More precisely, we started
with a tiny frequency jitter of 25 mHz and ended with a larger jitter of 35 mHz, although we optimized the trap harmonicity and checked the ion temperature. The reason of the declined frequency stability is unclear, but probably related to varying radiofrequency noise from external sources. As the largest measured jitter is only 22.9 times smaller than the cut-frequency difference of Dnsfz=2
; 20
Gstat 48Ca17
5; 138:83796192 30; 21 which have to be corrected by systematic shifts.
Systematic shifts and uncertainties of C(40Ca17 ) and C(48Ca17 ). The systematic shifts of the Larmor-to-cyclotron frequency ratios and the corresponding uncertainties are listed in Table 3. The dominant systematic shift and uncertainty is given by the image charge shift. Here, the induced image charges at
70 mHz the
probability of error of 0.54.5% is not negligible and has to be considered.
Instead of using a data analysis based on simple quality cuts, to decrease the probability of error and in that way losing statistics, we introduce the following AT-weight wAT for each spin-ip, in a way that wAT 0, if the electron is in spin-
down, wAT 1, if the electron is in spin-up and wAT 0.5, if the spin-state is
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the Penning trap electrode surfaces generate an additional effective electric potential, which shifts the radial eigenfrequencies of the ion. In ref. 39 the shift of the cyclotron frequency is analytically calculated:
Dnc nc
1:92
mion
8pe0r3B20 ; 22
where r is the inner radius of the Penning trap. Due to the r 3-scaling, this shift can be reduced in future experiments by increasing the size of the Penning trap. All other systematic shifts, which are at least one order of magnitude smaller than the image charge shift, are explained in refs 10,35.
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Acknowledgements
This work was supported by the Max Planck Society, the EU (ERC grant no 290870; MEFUCO), International Max Planck Research School for Quantum Dynamics in Physics, Chemistry and Biology (IMPRS-QD), GSI Helmholtzzentrum fr Schwerionenforschung, the Helmholtz Alliance HA216/EMMI, Russian Foundation forBasic Research (RFBR) (grants no. 13-02-00630 and No. 14-02-31316), the St Petersburg State University (grant nos. 11.38.269.2014 and 11.38.237.2015), Deutsche Forschungsgemeinschaft (DFG) (grant no. VO1707/1-2), Bundesministerium fr Bildung und Forschung (BMBF) (grant no. 01DJ14002) and the FAIR-Russia Research Center.
Author contributions
F.K., S.S., A.K. and J.H. performed the measurements on the Larmor-to-cyclotron frequency ratios. V.M.S., D.A.G. and A.V.V. carried out the QED calculations. S.E., M.G. and E.M.R. performed the measurement of the mass of 48Ca. F.K., S.S., V.M.S., S.E., G.W. and K.B. prepared the manuscript. All authors discussed the results and contributed to the manuscript at all stages.
Additional information
Competing nancial interests: The authors declare no competing nancial interests.
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How to cite this article: Khler, F. et al. Isotope dependence of the Zeeman effect in lithium-like calcium. Nat. Commun. 7:10246 doi: 10.1038/ncomms10246 (2016).
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Copyright Nature Publishing Group Jan 2016
Abstract
The magnetic moment μ of a bound electron, generally expressed by the g-factor μ=-g μB s h-1 with μB the Bohr magneton and s the electron's spin, can be calculated by bound-state quantum electrodynamics (BS-QED) to very high precision. The recent ultra-precise experiment on hydrogen-like silicon determined this value to eleven significant digits, and thus allowed to rigorously probe the validity of BS-QED. Yet, the investigation of one of the most interesting contribution to the g-factor, the relativistic interaction between electron and nucleus, is limited by our knowledge of BS-QED effects. By comparing the g-factors of two isotopes, it is possible to cancel most of these contributions and sensitively probe nuclear effects. Here, we present calculations and experiments on the isotope dependence of the Zeeman effect in lithium-like calcium ions. The good agreement between the theoretical predicted recoil contribution and the high-precision g-factor measurements paves the way for a new generation of BS-QED tests.
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