Table Of Content

Pis’ma v ZhETF Double–Resonance g Factor Measurements by Quantum Jump Spectroscopy Wolfgang Quint+, Behnam Nikoobakht∗, Ulrich D. Jentschura∗ +Gesellschaftfu¨rSchwerionenforschung (GSI), Planckstraße1,64291Darmstadt,Germany 8 ∗Max–Planck–Institut fu¨rKernphysik,Postfach103980, 69029Heidelberg,Germany 0 andInstitutfu¨rTheoretischePhysik,Universita¨tHeidelberg,Philosophenweg16,69120Heidelberg,Germany 0 Submitted12November2007 2 Resubmitted∗) n a J Withtheadventofhigh-precisionfrequencycombsthatcanbridgelargefrequencyintervals,newpossibil- 2 ities have opened up for the laser spectroscopy of atomic transitions. Here, we show that laser spectroscopic techniquescanalsobeusedtodeterminetheground-stateg factorofaboundelectron: Ourproposalisbased ] h on a double-resonance experiment, where the spin state of a ground-state electron is constantly being read p out by laser excitation to the atomic L shell, while the spin flip transitions are being induced simultaneously - by a resonant microwave field, leading to a detection of the quantum jumps between the ground-state Zee- m man sublevels. The magnetic moments of electrons in light hydrogen-like ions could thus be measured with o advanced laser technology. Corresponding theoretical predictions are also presented. t a . sPACS:12.20.Ds, 14.60.Cd, 31.30.Jv, 13.40.Em, 06.20.Jr, 31.15.-p c i s y Recently, there has been a dramatic progress in hyperfine transitions as well as electronic and nuclear g hthe precision laser spectroscopy of atomic transitions, factors [11, 12, 13]. pwith uncertaintiesonthe orderof10−14 fortwo-photon Thebound-electron(Landé)g factorforanelectron j [transitions in hydrogen [1] and even 10−17 for ultra- bound in an ion with a spinless nucleus is the propor- 2violet (UV) electric-quadrupole transitions in the mer- tionality constant relating the Zeeman energy ∆E in v cury ion[2] (statisticaleffects led to a limitation onthe the magneticfieldB (directedalongthez axis)andthe 71order of 10−16 for the evaluation of the latter measure- Larmor precession frequency ωL to the magnetic spin 0ment). By contrast, microwave measurements of the projection mj =−21,12 onto that same z axis. In natu- 1bound-electron g factor in hydrogen-like ions [3, 4, 5] ral units (~=c=ǫ0 =1), we have . 1have been restricted to a comparatively low level of ac- 1curacy, namely in the range of 10−10, where the most ∆E =mjωL =mjgjµBB, (1) 7 accurate values have been obtained for bound electrons :0in hydrogen-like carbon 12C5+ and oxygen 16O7+ (for whereµB =−e/(2me)istheBohrmagneton,expressed v in terms of the electron charge e and the electron mass anintroductoryreviewsonbound-electrongfactorsand i m . Deviations from the Dirac–Breit [14] prediction Xvariousrelatedexperimentalas wellas theoreticaltech- e g (1S) = 2(1+2p1 (Zα)2)/3 are due to quantum rniques,see[6,7,8]). Itistemptingtoaskiftheaccuracy j − a electrodynamic (QED), nuclear and other effects. gap between the two categories of measurements might The purpose of this note is to answer the following leave room for improvement of the g factor determina- question: “Is it possible to apply ultra-high precision tion,asbothmeasurementsinvestigatethepropertiesof atomic laser spectroscopy to bound-electron g factor bound electrons. More specifically, the question arises measurements?” Our answer will be affirmative. iftheadditional“channels”providedbylaserexcitation In contrast to the continuous Stern–Gerlach ef- amongthe discrete states of the boundsystem, andthe fect [15], and complementing a recent proposal for a additional possibilites for the laser cooling of ions (fol- high-precision measurement of the g factor in a highly lowing the original ideas formulated in Refs. [9, 10]), chargedion[16],thecurrentproposalisbasedonaPen- can be used as auxiliary devices to improve the accu- ning trap and will be studied here in conjunction with racy of the g factor determination via quantum jump the hydrogen-like helium ion 4He+, which seems to be spectroscopy. We also note that double-resonancetech- well suited for an experimental realization in the near niques for stored ions have already been shownto open future. The magnetic field strength B in the Penning up attractive experimental possibilities with respect to trap can be calibrated via a measurement of the cyclotron frequency ω of the trapped ion, and the Landé c 1 2 W.Quint W., B.Nikoobakht B., U.D.Jentschura U.D. g-factorfor the bound electronis determined by the re- tic term of order (Zα)2. Suppose now that one sin- lation gle 4He+ ion in the Penning trap is prepared in the g =2(Z 1) me ωL , (2) Zeeman sublevel m = +1 of the electronic ground j − mion ωc state 1S . Narrowj-band u2ltraviolet (UV) electromag- 1/2 wheretheelectron-ionmassratiome/mionisanexternal netic radiation with σ+ polarization and angular fre- input parameter and Z is the nuclear charge number. quency ωUV = 2π 9.87 1015Hz drives the Lyman- stroInngahPomenongienngetoruasp,maasginnegtliec4fiHeeld+Bioninisthcoenpfilnaendebpyera- αFigtr.a1n.sitTiohnis1iSs1a/2c×(cid:0)lomsejd=c+×y21cl(cid:1)e⇔bec2aPu3s/e2 d(cid:0)mecja=y +by23(cid:1)e,msiese- pendiculartothemagneticfieldlinesandbyaharmonic sion of a fluorescence photon is only possible to the ini- electrostatic potential in the direction parallel to the tial state 1S (m = +1) (if one ignores one-photon 1/2 j 2 field lines [17]. The three eigenmotions of a stored ion ionization into the continuum). Due to the short life- arethe trap-modifiedcyclotronmotion(frequency ω+), time of the upper state τ(2P ) 99.7ps, the fluores- 3/2 theaxialmotion(frequencyωz),andthemagnetronmo- cenceintensityof[2 τ(2p3/2)]−1 ≈5.01 109 photons/s tion(frequencyω−). Thefree-spacecyclotronfrequency under saturation conditions ma≈kes it×possible to de- ωc =qB/mionofanionwithchargeqcanbedetermined tect a single trapped ion with high sensitivity [23]. from the three eigenfrequencies by [18] The Rabi frequency of the UV transition is given by ωc2 =ω+2 +ωz2+ω−2. (3) ΩunRiatbsio=f W1.3/0c8m×2.107Hz√IUV,whereIUV ismeasuredin Experimentally, the eigenfrequencies of the stored ion Building a continuous-wave(cw) laserthat operates canbemeasuredbynon-destructivedetectionoftheim- at the Lyman-α transition of 4He+, with a wavelength agecurrentswhichareinducedinthetrapelectrodesby of 30.37nm, is certainly not a trivial task. However, a the ionmotion. Measurementsonthe levelofδω /ω = cw laser operating at the corresponding Lyman-α tran- c c 7 10−12 are achieved [19, 20, 21] by careful anhar- sition for atomic hydrogen, with λ = 121.56nm, has × monicitycompensationoftheelectrostatictrappingpo- already been demonstrated [24]. A possible pathway is tential, optimizing the homogeneity and temporal sta- higher-harmonicgenerationwhichhasrecentlybeende- bility of the magnetic field close to the Penning trap’s scribed in Ref. [25] andleads to a pulsed UV excitation center,andcoolingthemotionalamplitudesofthesingle with a high repetition rate and a potentially discon- trapped ion to low temperatures. Further optimization tinuous probing of the Zeeman ground-state sublevel. of experimental techniques should make it possible to Note that a discontinuous probing of the ground-state reach an accuracy of (better than) δω /ω = 10−12. In sublevels does not inhibit the quantum jump detection c c ourproposedg factormeasurement,advantagecouldbe scheme as outlined below. Groundwork for a detailed taken of the fact that two frequencies of the same par- analysisofthedynamicsofapulsedexcitationschemein ticle are measured simultaneously (cyclotron vs. spin- averymuchanalogousatomicsystemhasrecentlybeen flip), whereas in a mass measurement, the cyclotron laid in Ref. [26]; in principle, one only has to ensure frequencies of two different particles have to be deter- that the light intensity of the Lyman-α source during a mined. single laser pulse is sufficient to discern the presence or In the following, we concentrate on the 4He+ sys- absence of fluorescence. tem, where the total angular momentum is equal to During excitation of the transition the total electron angular momentum J. In the pres- 1S1/2 (cid:0)mj=+21(cid:1) ⇔ 2P3/2 (cid:0)mj=+23(cid:1) and detection of ence of the magnetic field B in the Penning trap, the the corresponding fluorescence photons, a microwave Zeeman splitting of the electronic ground state 1S1/2 fieldwithfrequencyωMW inresonancewiththe spinflip of the 4He+ ion is given by Eq. (1). Correspondingly, transition 1S1/2 (cid:0)mj=+21(cid:1) ⇔ 1S1/2 (cid:0)mj =−12(cid:1) in the excited state 2P is split into four Zeeman sub- the electronic ground state is irradiated on the single 3/2 levels ∆E = mjgj(2P3/2)µBB, with mj = ±21,±32. trapped 4He+ ion (Fig. 1). Successful excitation The Landé g factor can be obtained easily according of the spinflip transition results in an instanta- to a modified Dirac equation which forms a basis for neous stop of the fluorescence intensity, because the bound-state analysis [22] lower Zeeman level 1S1/2(cid:0)mj =−21(cid:1) is not excited by the narrow-band Lyman-α radiation. A quan- 4 α 2 g (2P )= + (Zα)2, (4) tum jump is thus directly observed with essentially j 3/2 3 3π − 15 100% detection efficiency [27]. A second spinflip where we take into account the leading QED and rel- 1S1/2 (cid:0)mj =−12(cid:1) → 1S1/2 (cid:0)mj =+21(cid:1) restores the ativistic contributions and use Z = 2 for the relativis- fluorescenceintensity. Aplotofthequantumjumprate Double–Resonance g Factor Measurements... 3 mj states(seeFig.2). Forthisprocess,weobtainanioniza- 12SS11//22 22PP13//............................................22............................................................................................................ ..............................................................................................................................................................................................................................................................................................ωMWωU..................................................................V..............................................................................................................................................................................[σ+]...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... −−−−++++11311311////////22222222 witriustosieiesoninnsettncpihshzetioectahanytrtelh,doitaoinsewismnnsoehigrteosreae-antirdctnotieitezetoaiapaeonosntendfaitnsohleγiodntpiecfyeol=rRne1namtt.avi26toebeee.3nx4ani1i9pstsfouir5×oofepr×nnqter1ehsduo10nep0eo−tin−ofni22acr23PRltyusm3ieen−o/ifxs22in.1pt.ao×[s(cid:0)(2lnT−m6oIlthU]fyγoj.iViWsp=Itt,nrh)l/woe.e+pcaphHmldro23aeiesr(cid:1)nr2srtee,ctesiroiot,IcpanUoiwalntareVnee--,l Laser-microwavedouble-resonanceexcitationscheme to its square root, it might be preferable to work at re- using circularly polarized UV light for excitation of duced laser intensities in order to increase the lifetime the 1S1/2 ⇔ 2P3/2 transition and a microwave field ofthehydrogen-likechargestateoftheion. However,at fordrivingthespinfliptransition1S1/2 `mj=+21´⇔ anincidenttypicallaserintensityof100W/cm2,thelife- 1S1/2 `mj =−21´. time of 4He+ is 0.401s against ionization, and this has to be compared to a Rabi frequency of 1.308 108Hz. The 4He+ ion has about 108 Rabi cycles be×fore it is versus excitation microwave frequency at ωMW ωL ionized, and so the ionization channel does not limit ≈ yields the resonancespectrum ofthe Larmorprecession the feasibility of the measurement at all. frequency. The cyclotron frequency ω , which also c Finally, we take notice of the ac Stark shift of the enters Eq. (2), is measured simultaneously by non- 1S–2P transition due to non-resonant levels, which is destructive electronic detection of the image currents 0.0968Hz IUV, with IUV given in W/cm2. The ac induced in the trap electrodes [5]. × Stark shift of the UV transitionaffects the two ground- state Zeeman levels slightly differently, but the relative Absolutetransitionfrequenciesof4He+ relevanttothe shift of the spinflip transition frequency between them excitationschemegiveninFig.1. TheZeemansplitting is a fourth-order effect and is suppressed with respect is excluded. to the ac Stark shift by a factor of ωL/ωUV < 10−4 Nuclear charge radius 1S 2P frequency and thus negligible on the level 10−12 in units of the 3/2 ⇔ microwave frequency, at a typical laser intensity of r2 1/2 =1.673(1)fm 9868722559.240(237)MHz h i 100W/cm2. Also,experimentalproceduresforprevious r2 1/2 =1.680(5)fm 9868722558.650(477)MHz g factor measurements [4, 5] have included an extrapo- h i lation to zero intensity of the microwavefields, and the same canbe done with the drivingUV laserfield in the proposed measurement scheme. Alternatively, one can ...1ε..SS...11././.22........................................................................................................................................................................................................................................................................................................................................2..............................................................P.......................................m........................................3.........................................../.............j...............................2...................................=..............................................ω..............................................................1....U................................................/.........V.............................................2..........................................[.............................σ..................................................................................................................+...........].......ε.ε.................DD.....................................53............//.........m.22.....................ω.j.....................U.=.................V....................3...........[........./σ......................2..............................+.......]...........m...j...=....5../..2.. flsprtpftaihyeireocplseeranttdfItoteognierromrcrda.mfotoaniuArtottsdonhniticctehdesctireoohuf4srntoentHedosracxiietneniclt+enarthgitteye4aaatmnHoitasttnieireiotttamta+hinyhseneeuessoLorroisffteaeoreimctmfeqothethuntebniecheentafneUtsrclL,hceoVgeiygqwmf,rmutfleotpaaevhauncispanulnetcarl-sordiuetαeriseelsoifelsgdonainamhrenastnttrdi[tde.nte3hrh1atemfe,htooi(e3irennr1v2getaSas,htptl3a)iuiecio3nna=an]gy--,l Schematic representation of the excitation scheme L 107692.522(228)MHzfor the “old”value [34]of the nu- including ionization channels. The dotted line repre- clear charge radius r2 1/2 = 1.673(1)fm and (1S) = sentstheionizationcontinuumthreshold,andεLj are h i L 107693.112(472)MHzfor the “new” value of the charge electronic continuumstates. radius [35], which is r2 1/2 = 1.680(5)fm. (The un- h i certainty estimate for the “old” value has given rise While the UV laser light drives the transition to discussions, see Ref. [31].) For the 2P states, 3/2 1S1/2 (cid:0)mj=+21(cid:1) ⇔ 2P3/2 (cid:0)mj=+23(cid:1), the absorption the Lamb shift is independent of the current uncer- of an additional photon can take place, resulting in tainty in the nuclearradius onthe levelof one kHz and iεoDn5iz/a2t(cid:0)iomnj=th+ro25u(cid:1)g,hwthheerecεhDannareel e2lPec3t/r2o(cid:0)nmicjc=on+t23in(cid:1)uu⇒m rReeafd.s[3L3(])2.P3U/s2i)ng=a2p01ro.1p6e8rMdeHfizni(tsieoenToafbtlheesL3aamnbds4hioftf 4 W.Quint W., B.Nikoobakht B., U.D.Jentschura U.D. Individual contributions to the 1S bound-electron g factor for 4He+. In the labeling of the corrections, we follow the conventions of Ref. [28]. The abbreviations used are as follows: “h.o.” stands for a higher order contribution, “SE” for a self energy correction, “VP-EL” for the electric-loop vacuum-polarization correction, and “VP-ML” for the magnetic-loop vacuum-polarization correction. The value of α−1 =137.035999070(98) is the currently most accurate value from Ref. [29], whereas the value of α−1 =137.03599911(46) is the 2002 CODATArecommended value[30]. g factor contribution (4He+) α−1 =137.035999070(98) α−1 =137.03599911(46) Dirac eigenvalue 1.999 857 988 825 2(2) 1.999 857 988 825 3(9) Finite nuclear size 0.000 000 000 002 3 0.000 000 000 002 3 One-loop QED (Zα)0 0.002 322 819 466 0(17) 0.002 322 819 465 4(76) (Zα)2 0.000 000 082 462 2 0.000 000 082 462 2 (Zα)4 0.000 000 001 976 7 0.000 000 001 976 7 h.o.,SE 0.000 000 000 035 1(2) 0.000 000 000 035 1(2) h.o.,VP–EL 0.000 000 000 002 0 0.000 000 000 002 0 h.o.,VP–ML 0.000 000 000 000 2 0.000 000 000 000 2 two-loop QED (Zα)0 0.000 003 515 096 9(3) 0.000 003 515 096 9(3) ≥ − − (Zα)2 0.000 000 000 124 8 0.000 000 000 124 8 − − (Zα)4 0.000 000 000 002 4(1) 0.000 000 000 002 4(1) Recoil m/M 0.000 000 029 198 5 0.000 000 029 198 5 Radiative recoil (m/M)2 0.000 000 000 025 3 0.000 000 000 025 3 − − Hadronic/weak interaction 0.000 000 000 003 4 0.000 000 000 003 4 Total 2.002 177 406 727 1(17) 2.002 177 406 726 5(77) as given, e.g., in Eq. (10) of Ref. [33], we then obtain of the measurement of the free-electron g factor has re- the transition frequencies as given in Table 1. cently been increased to a level of 7.6 10−13 [38, 29]. × Within our proposed setup, an accuracy on the level of The ground-state g factor can be described natu- j 10−12...10−13 seems to be entirely realistic for bound- rally in an intertwined expansion in the QED loop ex- electron g factors in hydrogen-like ions with a low nu- pansion parameter α and the electron-nucleus interac- clear charge number. It might be very beneficial if the tion strength Zα [28]. We follow the conventions of extremelyimpressive,ultra-precisenewmeasurementof Ref. [28] and take into account all corrections that are relevant at the 10−12 level of accuracy (see Table 2). the free-electron g factor [29] could be supplemented by a potentially equally accurate measurement of the The entry for the “(Zα)0 one-loop QED” is just the bound-electron g factor in the near future, as an alter- Schwinger term α/(2π) and it carries the largest the- native determination of the fine-structure constant is oretical uncertainty, because of the uncertainty in the urgentlyneededinconjunctionwithanimproveddeter- fine-structure constant α itself [36, 37]. mination of the electron mass. In this note, we attempt to formulate a pro- TheauthorsacknowledgehelpfuldiscussionswithV. posalby whichultra-accurateg factor measurementsin A. Yerokhin and K. Pachucki, and support from DFG hydrogen-like systems with low nuclear charge number (Heisenberg program) as well as from GSI (contract might be accessible to laser spectroscopic techniques. HD–JENT). Within the next decade, it is realistic to assume that the necessaryrequirements for experiments will be pro- videdthatfullyprofitfromboththeelectriccouplingof 1. M. Fischer, N. Kolachevsky, M. Zimmermann, R. the electron (via optical electric-dipole allowed Lyman- Holzwarth, Th. Udem, T. W. Hänsch, M. Abgrall, J. α transitions) and from the magnetic coupling of the Gru¨nert, I. Maksimovic, S. Bize, H. Marion, F. Pereira electron (via spin-flip transitions among the Zeeman Dos Santos, P. Lemonde, G. Santarelli, P. Laurent, A. sublevels of the groundstate, see Fig. 1). The accuracy Clairon, C. Salomon, M. Haas, U. D. Jentschura, and Double–Resonance g Factor Measurements... 5 C. H.Keitel, Phys.Rev. Lett. 92, 230802 (2004). number Z. These are as follows: laser frequency: Z2; 2. W. H. Oskay, S. A. Diddams, E. A. Donley, T. M. Rabi frequency: Z−1 for constant Ilas; lifetime of the Fortier, T. P. Heavner,L. Hollberg, W. M. Itano, S. R. upper state: Z−4; AC Stark shift: Z−4, again for con- Jefferts, M. J. Delaney, K. Kim, F. Levi, T. E. Parker, stant Ilas; ionization rate coefficient: Z−4, and ioniza- andJ.C.Bergquist,Phys.Rev.Lett.97,020801(2006). tion cross section: Z−2. Allvaluesgiven in thetextare for Z =2. 3. H.Häffner,T.Beier,N.Hermanspahn,H.-J.Kluge,W. Quint, J. Verdu´, and G. Werth, Phys. Rev. Lett. 85, 24. K.S.E.Eikema,J.Walz,andT.W.Hänsch,Phys.Rev. 5308 (2000). Lett. 86, 5679 (2001). 4. T. Beier, H. Häffner, N. Hermanspahn, S. G. Karshen- 25. C. Gohle, T. Udem, M. Hermann, J. Rauschenberger, boim,H.-J.Kluge,W.Quint,S.Stahl,J.Verdu´,andG. R. Holzwarth, H. A. Schuessler, F. Krausz, and T. W. Werth, Phys.Rev.Lett. 88, 011603 (2001). Hänsch,Nature (London) 436, 234 (2005). 26. M. Haas, U. D. Jentschura, C. H. Keitel, N. Ko- 5. J. Verdu´, S. Djekić, S. Stahl, T. Valenzuela, M. Vogel, lachevsky, M. Herrmann, P. Fendel, M. Fischer, T. R. G. Werth, T. Beier, H.-J. Kluge, and W. Quint, Phys. Rev. Lett.92, 093002 (2004). Holzwarth,T.W.Hänsch,M.O.Scully,andG.S.Agar- 6. W. Quint,Phys. Scr. T 59, 203 (1995). wal, Phys.Rev.A 73, 052501 (2006). 27. W. Nagourney, J. Sandberg, and H. Dehmelt, Phys. 7. T. Beier, Phys.Rep. 339, 79 (2000). Rev.Lett. 56, 2797 (1986). 8. S. G. Karshenboim, Phys.Rep. 422, 1 (2005). 28. K.Pachucki,A.Czarnecki,U.D.Jentschura,andV.A. 9. S.V.Andreev,V.I.Balykin,V.S.Letokhov,andV.G. Yerokhin,Phys.Rev.A 72, 022108 (2005). Minogin, JETP Lett. 34, 442 (1981); D. J. Wineland, 29. B. Odom et al., Phys. Rev. Lett. 97, 030801 (2006); H. Dehmelt, Bull. Am. Phys.Soc. 20, 637 (1975). G.Gabrielse et al.,Phys.Rev.Lett.97,030802 (2006); 10. V.I.Balykin,V.S.Letokhov,andA.I.Sidorov,JETP G.Gabrielseetal.,Phys.Rev.Lett.99,0399902(2007); Lett. 40, 1026 (1984). T. Aoyama et al.,e-print hep-ph/0706.3496. 11. W.M.ItanoandD.J.Wineland,Phys.Rev.A24,1364 30. P. J. Mohr and B. N. Taylor, Rev. Mod. Phys. 77, 1 (1981). (2005). 12. D.J.Wineland,J.J.Bollinger,andW.M.Itano,Phys. 31. A. van Wijngaarden, F. Holuj, and G. W. F. Drake, Rev. Lett.50, 628 (1983). Phys.Rev.A 63, 012505 (2000). 13. G.Marx,G.Tommaseo,andG.Werth,Eur.Phys.J.D 32. U.D.JentschuraandG.W.F.Drake,Can.J.Phys.82, 4, 279 (1998). 103 (2004). 14. G. Breit, Nature (London) 122, 649 (1928). 33. U. D. Jentschura and M. Haas, Can. J. Phys. 85, 531 15. N.Hermanspahn,H.Häffner,H.J.Kluge,W.Quint,S. (2007). Stahl,J.Verdu´,andG.Werth,Phys.Rev.Lett.84,427 34. E.BorieandG.A.Rinker,Phys.Rev.A18,324(1978). (2000). 35. I. Sick, talk given at the PSAS–2006 conference (preci- 16. V. M. Shabaev, D. A. Glazov, N. S. Oreshkina, A. V. sion physics of simple atomic systems), Venice (June Volotka, G. Plunien, H. J. Kluge, and W. Quint, Phys. 2006) and private communication (2007). A further Rev. Lett.96, 253002 (2006). reevaluationofscatteringdata,accordingtoI.Sick,has 17. H. Dehmelt, Rev.Mod. Phys. 62, 525 (1990). meanwhilerevealedthattheradiusuncertaintycouldbe 18. L. S. Brown and G. Gabrielse, Phys. Rev. A 25, 2423 decreased further to hr2i1/2 =1.681(4)fm. However, in ordertoremain fully consistent with thetables3and 4 (1982). ofRef.[33],weusethemoreconservativeuncertaintyes- 19. S.Rainville,J.K.Thompson,andD.E.Pritchard,Sci- timate of hr2i1/2 =1.680(5)fm as given at PSAS–2006 ence 303, 334 (2004). byI.Sick.Notethattheprecisevalueofthenuclear-size 20. R. S. van Dyck, Jr., S. L. Zafonte, S. Van Liew, and correction to the transition frequency is important for P. B. S. D. B. Pinegar, Phys. Rev. Lett. 92, 220802 ourproposalonlyinsofarastheUVlaserhastobetuned (2004). to thetransition 1S1/2 `mj=+21´⇔2P3/2 `mj=+23´. 21. W. Shi, M. Redshaw, and E. G. Myers, Phys. Rev. A 36. This observation illustrates that if two g factors could 72, 022510 (2005). j be measured to sufficient accuracy in thelow-Z region, 22. M.I.Eides,H.Grotch,andV.A.Shelyuto,Phys.Rep. thenthefine-structureconstantcouldbeinferredinad- 342, 63 (2001). dition to the electron mass. Namely, we would have 23. The discussed excitation schemeis in principleapplica- two equations of type (2) and two unknowns: the fine- ble to any low-Z hydrogen-like ion with a spinless nu- structureconstant and theelectron mass. Seealso [37]. cleus(forhigherZ,itappearsthatthelaserrequiredto 37. U.D.Jentschura,A.Czarnecki,K.Pachucki,andV.A. drivethe1S–2P3/2transitionisstillfarfromexperimen- Yerokhin,Int.J. Mass Spectrometry 251, 102 (2006). tal possibilities in the foreseeable future). Therefore, it 38. R. S. van Dyck, Jr., P. B. Schwinberg, and H. G. is adequatetoindicate thescaling oftherelevantphys- Dehmelt, Phys.Rev.Lett. 59, 26 (1987). ical quantities for the scheme with the nuclear charge