Theoretical prediction of the Fine and Hyperfine structure of heavy muonic atoms
aa r X i v : . [ phy s i c s . a t o m - ph ] O c t Theoretical prediction of the Fine and Hyperfine structure of heavy muonic atoms
Niklas Michel, ∗ Natalia S. Oreshkina, † and Christoph H. Keitel Max Planck Institute for Nuclear Physics, Saupfercheckweg 1, 69117 Heidelberg, Germany (Dated: October 3, 2017)Precision calculations of the fine and hyperfine structure of muonic atoms are performed in arelativistic approach and results for muonic
Bi,
Sm, and Zr are presented. The hyperfinestructure due to magnetic dipole and electric quadrupole splitting is calculated in first order pertur-bation theory, using extended nuclear charge and current distributions. The leading correction fromquantum electrodynamics, namely vacuum polarization in Uehling approximation, is included as apotential directly in the Dirac equation. Also, an effective screening potential due to the surroundingelectrons is calculated, and the leading relativistic recoil correction is estimated.
PACS numbers: 21.10.Ft, 21.10.Ky, 36.10.Ee, 31.30.jr
I. INTRODUCTION
A muon is a charged elementary particle, which isin many aspects similar to the electron, in particular,it has the same electric charge, but it is ≈
200 timesheavier than the electron [1]. When coming close to anatom, a muon can be captured by the nucleus and forma hydrogen-like muonic ion, which is typically also sur-rounded by the atomic electrons. This atomic system iscommonly referred to as a muonic atom. The lifetimeof the muon is big enough to be considered stable inthe structure calculations of these muonic bound states.Muonic atomic systems feature strong dependence on nu-clear parameters and therefore can provide informationabout atomic nuclei [2]. This triggered interest in preciseknowledge of the level structure of muonic atoms [3, 4].Due to the muon’s high mass, it is located much closer tothe nucleus; and, especially for heavy nuclei, this resultsin big nuclear size effects and a strong dependence of themuon bound state energies on the nuclear charge andcurrent distributions, as well as large relativistic effects.A combination of the knowledge about the level struc-ture and experiments measuring the transition energiesin muonic atoms enabled the determination of nuclear pa-rameters like charge radii [5, 6], quadrupole moments [7],and magnetic HFS constants [8]. One of the most precisemeasurements to date is the determination of the nuclearroot-mean-square radius of
Pb on a 0 .
2% level [9].Recent measurements on muonic hydrogen renewed theinterest in muonic atoms, revealing a disagreement be-tween the values for the proton charge radius extractedfrom muonic and electronic systems [10]. This allowsthe assumption that there can be unidentified effects inmuonic systems, and triggered detailed theoretical inves-tigation of muonic hydrogen und light muonic atoms, e.g.,Refs [11, 12]. Deeper knowledge of the physics of heavymuonic atoms could also contribute to the understandingof the muonic puzzles. In addition, nuclear parameters ∗ [email protected] † [email protected] obtained from muonic x rays would be beneficial for ex-periments on atomic parity violation [13]. For this rea-son, there are upcoming experiments on heavy muonicatoms [14]. The complicated level structure of these sys-tems demands accurate theoretical calculations.We present updated state-of-the-art calculations of thefine and hyperfine structure of heavy muonic atoms andanalyze the individual contributions. In combinationwith experimental data, they can be used for the determi-nation and further improvement of values of nuclear pa-rameters. The fine structure is calculated including finitesize effects and leading order effects of the vacuum polar-ization. Additionally, the screening from the surroundingatomic electrons is considered. The hyperfine structure isthen calculated with extended quadrupole and magneti-zation distributions, including the previously mentionedeffects. Results are presented for muonic Bi,
Sm,and Zr. The dual-kinetic-balance method [15] was ap-plied for the numerical evaluation of the listed contribu-tions.Muonic relativistic units with ~ = c = m µ =1 are used,where m µ is the muon’s mass, and the Heavyside chargeunit with α = e / π , where α is the fine structure con-stant and the electron’s charge is e< II. INTERACTION BETWEEN MUON ANDNUCLEUS
The total Hamiltonian for a muon bound to a nucleuscan be written as a sum of nuclear, muonic, and interac-tion Hamiltonian [4]. Thus, we consider the Hamiltonian H = H N + H (0) µ + H µ − N , (1)with the nuclear Hamiltonian H N , the Dirac Hamiltonian H (0) µ for the free muon, and the interaction Hamiltonian H µ − N . The nucleus is described in the rotational model,i.e. in a state with well defined angular momentum andcharge- and current density in the body fixed nuclearframe [16]. As a next step, the interaction between thebound muon and the atomic nucleus is expanded, whereelectric and magnetic interactions are taken into account.The interaction Hamiltonian is H µ − N = H E + H M (2)where the electric part reads H E = − α Z d V ′ ρ ( ~r ′ ) | ~r µ − ~r ′ | , (3)with the fine structure constant α , the position ~r ′ of thenuclear charge distribution and the position ~r ′ µ of themuon in the nuclear frame. The nuclear charge distribu-tion ρ ( ~r ) is normalized to the nuclear charge Z as Z d V ρ ( ~r ) = Z. (4)Conveniently, the nuclear charge distribution is dividedinto a spherically symmetric part ρ ( r ) and a part ρ ( r )describing the quadrupole distribution in the nuclearframe as [17] ρ ( ~r ′ ) = ρ ( r ′ ) + ρ ( r ′ ) Y ( ϑ ′ , ϕ ′ ) , (5)with the spherical harmonics Y lm ( ϑ, ϕ ). Since an analo-gous part for the dipole distribution would be an oper-ator of odd parity, it would vanish after averaging withmuon wave functions of defined parity [18], and thus it isnot considered here and neither are higher multipoles be-yond the quadrupole term. Correspondingly, the electricinteraction Hamiltonian from (2) can be written as H E = H (0) E + H (2) E , (6)where the spherically symmetric part of the charge dis-tribution gives rise to H (0) E ( r µ ) = − πα Z ∞ d r r ρ ( r ) r > , (7)with r > = max( r, r µ ). This interaction Hamiltonianwill be included in the numerical solution of the Diracequation for the muon as described in Sec. III. Thequadrupole part of the interaction H (2) E causes hyperfinesplitting, which is calculated perturbatively in Sec.VII A.As for the magnetic part, we consider dipole interac-tion. Therefore, the corresponding interaction Hamilto-nian from (2) reads [19] H M = | e | π ~µ · (cid:18) F BW ( r ) ~rr × ~α (cid:19) , (8)with the charge of the muon e = −| e | , the nuclear mag-netic moment ~µ , its distribution function F BW , and theDirac matrices ~α . If the nuclear current density is de-scribed by a normalized scalar function f µ ( r ) as ~j ( r ) = rot ( ~µf µ ( r )) , (9) then the distribution function is given by F BW ( r ) = − r ∂∂r Z d V ′ f µ ( r ′ ) | ~r − ~r ′ | . (10)The difference in the hyperfine splitting between a point-like magnetic moment and a spacial distribution of themagnetization is called the Bohr-Weisskopf effect [20]. InSec. VII A, the matrix elements of the magnetic interac-tion are analyzed, paying special attention to the distri-bution function F BW . We expect the contribution of thehigher-order terms, namely electric octupole, magneticquadrupole, and beyond, to be smaller than the uncer-tainty of the considered terms [4, 21]. Therefore they canbe ignored here.For evaluating these Hamiltonians, the appropriatestates are states of defined total angular momentum.A nuclear state | IM i with nuclear angular momentumquantum number I and projection M on the z axis ofthe laboratory frame and a muonic state | nκm i with to-tal angular momentum j ( κ ) = | κ | − and projection m are coupled to a state | F M F Iκ i with angular momentum F and projection M F as | F M F Iκ i = X M,m C F M F IM jm | IM i | nκm i , (11)where C jmj m j m are the Clebsch-Gordan coefficients [22].Here, n is the principal quantum number of the muon and κ = ( − j + l + ( j + ) with the orbital angular momen-tum quantum number l . III. DIRAC EQUATION WITH FINITE SIZECORRECTIONS
As a basis for further calculations, the Dirac equation( ~α · ~p + β + V ( r µ )) | nκm i = (1 − E nκ ) | nκm i (12)is solved for the muon. Here, ~α and β are the four Diracmatrices, E nκ are the binding energies, and the poten-tial V ( r ) is the spherically symmetric part of the interac-tion with the nucleus, which is the monopole contributionfrom the electric interaction (7) and the Uehling poten-tial from (17). A Fermi type charge distribution [23] isused to model the monopole charge distribution as ρ ( r ) = N r − c ) /a ) , (13)where a is a skin thickness parameter and c the half-density radius. The normalization constant N is cho-sen such that (4) is fulfilled. It has been proven, that a = t/ (4 log3), with t = 2 .
30 fm, is a good approximationfor most of the nuclei [23]. The parameter c is then de-termined by demanding, that the charge radius squared (cid:10) r (cid:11) = R d r r ρ ( r ) R d r r ρ ( r ) (14)agrees with the values from the literature [24]. Since thepotential in (12) is spherically symmetric, the angularpart can be separated and the solution with sphericalspinors Ω κm ( ϑ, ϕ ) can be written as [25] | nκm i = 1 r (cid:18) G nκ ( r ) Ω κm i F nκ ( r ) Ω − κm (cid:19) , (15)and the resulting equations for the radial functions aresolved with the dual-kinetic-balance method [15] to ob-tain G nκ and F nκ , and the corresponding eigenenergiesnumerically.In Table I, the binding energies for muonic Bi,
Sm, and
Zr are shown, both with and without thecorrections from the Uehling potential (17). The finitenuclear size effect is illustrated by also including the bind-ing energies E ( C ) nκ of the pure Coulomb potential − Zα/r µ ,which read [25] E ( C ) nκ = 1 − Zα ) (cid:16) n − | κ | + p κ − ( Zα ) (cid:17) − . (16)The uncertainties include the error in the rms radiusvalue as well as a model error, which is estimated viathe difference of the binding energies with the Fermi po-tential (13) and the potential of a charged sphere with thesame rms radius. For heavy nuclei, the finite nuclear sizecorrection can amount up to 50 %, and thus the bindingenergy is halved. IV. VACUUM POLARIZATION
For atomic electrons, usually the self-energy QED cor-rection is comparable to the vacuum polarization correc-tion [23]. For muons, however, the vacuum polarizationcorrection is much larger due to virtual electron-positronpairs, which are less suppressed due to their low masscompared to the muon’s mass [3]. The spherically sym-metric part of the vacuum polarization to first order in α and Zα is the Uehling potential [19] V Uehl ( r µ ) = − α α π Z ∞ d r ′ πρ ( r ′ ) Z ∞ d t (cid:18) t (cid:19) × √ t − t exp( − m e | r µ − r ′ | t ) − exp( − m e ( r µ + r ′ ) t )4 m e r µ t , (17)where m e is the electron mass and ρ is the sphericallysymmetric part of the charge distribution from (5). Thispotential can be directly added to the Dirac equation(12). In this way, all iterations of the Uehling potentialare included [11]. Results for our calculations can befound in Table I. V. RECOIL CORRECTIONS
Taking into account the finite mass and the resultingmotion of the nucleus leads to recoil corrections to thebound muon energy levels. In nonrelativistic quantummechanics, as in classical mechanics, the problem of de-scribing two interacting particles can be reduced to aone particle problem by using the reduced mass m r ofthe muon-nucleus system [26]. With the mass of the nu-cleus m N , the reduced mass reads in the chosen systemof units as m r = m N m N + 1 , (18) TABLE I. Overview of the binding energies for muonic
Bi,
Sm, and
Zr, obtained by solving the Dirac equation withthe spherically symmetric parts of the muon-nucleus interac-tion. The values for solving the Dirac equation only with theelectric monopole potential, and with the electric monopolepotential and the Uehling potential are presented to show theinfluence of the leading order vacuum polarization. The bind-ing energies (16) for a point like nucleus are shown as well.The reduced mass is used to include the non-relativistic recoilcorrections from Section V. The corrections from section VIare not included in this table. All energies are in keV.state point like finite size (fs) a fs+Uehling b205 Bi 1s / / / / / / / / / Sm 1s / / / / / / / / / Zr 1s / / / / / / / / / a V ( r µ ) = H (0) E ( r µ ) b V ( r µ ) = H (0) E ( r µ ) + V Uehl ( r µ )see eq. (7), (12), and (17) for definitions and the Dirac equation is accordingly modified to( ~α · ~p + β m r + V ( r µ )) | nκm i = ( m r − E nκ ) | nκm i . (19)In relativistic quantum mechanics, this separation is notpossible. We follow an approach used in Refs. [3, 27],which includes the nonrelativistic part of the recoil cor-rection already in the wave functions by using the re-duced mass in the Dirac equation and calculating theleading relativistic corrections perturbatively. If E (fm) nκ denotes the binding energy of (12) with the finite sizepotential (7) but with the reduced mass replaced by thefull muon rest mass, and E (rm) nκ the binding energy in thesame potential but with the reduced mass (18), then theleading relativistic recoil correction ∆ E (rec,rel) nκ accordingto Ref. [3] reads∆ E (rec,rel) nκ = − (cid:16) E (fm) nκ (cid:17) M N + 12 M N D h ( r ) + 2 E (fm) nκ P ( r ) E , (20)where M N is the mass of the nucleus, and the functions h ( r ) and P ( r ) are defined in Eqs. (109) and (111) ofRef. [3], respectively. In Table II, the binding energiesobtained from solving the Dirac equation with the muonrest mass and the reduced mass of the muon-nucleussystem are compared, and the leading relativistic recoilcorrection is shown. The uncertainties include errors inthe rms radius, the model of the charge distribution andfor the relativistic recoil, and a ( m µ /M N ) term due tohigher-order corrections in the mass ratio of muon andnucleus, which dominates the uncertainty for lower Z . VI. ELECTRON SCREENING
The effect of the surrounding electrons on the bindingenergies of the muon was estimated following Ref. [28]by calculating an effective screening potential from thecharge distribution of the electrons as V e ( ~r µ ) = − α Z d V ρ e ( ~r ) | ~r µ − ~r | , (21)and using this potential in the Dirac equation for themuon. The charge distribution of the electrons isobtained by their Dirac wave functions as ρ e ( ~r ) = P i ψ ∗ e i ( ~r ) · ψ e i ( ~r ), where ψ e i ( ~r ) is the four componentspinor of the i -th considered electron. In order to ob-tain the wave functions of the electrons, it has to betaken into account, that the muon essentially screens oneunit of charge from the nucleus. The simplest possibil-ity is to replace the nuclear charge by an effective charge˜ Z = Z − TABLE II. Recoil corrections to the binding energies of themuon. fm (full mass) denotes the finite size binding energy,analogous to the fourth column of Table I, but with the restmass of the muon used in the Dirac equation. ∆ E rec,nr isthe non-relativistic recoil correction, which is the differencebetween the finite size Dirac solutions with reduced mass andfull mass, respectively. ∆ E (rec,rel) nκ is the leading relativisticrecoil correction from Section V. All energies are in keV.state E (fm) ∆ E rec,nr ∆ E (rec,rel) nκ a205 Bi 1s / / / / / / / / / Sm 1s / / / / / / / / / Zr 1s / / / / / / / / / a ∆ E rec,nr := E (red.mass) − E (fm) , see Section V for definitions. required states, adding the screening potential due to thebound muon V µ ( ~r e ) = − α Z d V ψ ∗ µ ( ~r ) · ψ µ ( ~r ) | ~r e − ~r | , (22)analogously to (21). The interaction between the elec-trons is not taken into account here. Finally, the Diracequation for the muon is solved again, now including thenuclear potential and the screening potential (21) duethe atomic electrons from the considered electron config-uration. This procedure can be repeated in the spirit ofHartree’s method [29] until the electrons and the muonare self-consistent in the fields of each other, but ourstudies show that one iteration is usually enough sincethe overlap of muon and electron wave functions in heavymuonic atoms is small. It is important to note, that herethe screening potential depends to a small extent on thestate of the muon, since the muon wave function is usedin the calculation for the electron wave function. The TABLE III. Electron screening corrections to the bound muonenergy levels. ∆ E (1)S , eff and ∆ E (1+2)S , eff are the screening cor-rections with the effective nuclear charge method, whereas∆ E (1)S , and ∆ E (1+2)S , use the 3 step calculation, both de-scribed in Section VI. For the superscript (1), only the 1selectrons are considered, while for (1+2), all electrons fromthe first and second shell are considered. All energies are inkeV. µ -state ∆ E (1)S , eff ∆ E (1+2)S , eff ∆ E (1)S , ∆ E (1+2)S , Bi 1s / / / / / / / / / Sm 1s / / / / / / / / / Zr 1s / / / / / / / / / atomic electrons primarily behave like a charged shellaround the muon and the nucleus; thus every muon levelis mainly shifted by a constant term, which is not ob-servable in muonic transitions. The screening correction∆ E S is defined as the difference of the binding energywithout screening potential and with screening potential,therefore a positive value indicates that the muon is lessbound due to the screening effect. The main contributionto the nonconstant part of the screening potential comesfrom the 1 s electrons, since their wave functions have thebiggest overlap with the muon; therefore the exact elec-tron configuration has only a minor effect on transitionenergies [28]. In Table III, results for the screening cor-rection are shown for both mentioned methods and fordifferent electron configurations. Values of the screeningcorrection for different electron configurations show thata 10% error for the non-constant part is a reasonableestimate. VII. HYPERFINE INTERACTIONSA. Electric quadrupole splitting
Since for heavy nuclei the nuclear radius is compa-rable to the muon’s Compton wavelength [1, 24], themuonic wavefunction overlaps strongly with the nucleusand the muon is sensitive to nuclear shape corrections,which results in hyperfine splitting of the energy levels.The quadrupole part of the electric interaction (6) canbe rewritten as [16] H (2) E = − α Q F QD ( r µ )2 r µ X m = − C m ( ϑ N , ϕ N ) C ∗ m ( ϑ µ , ϕ µ ) , (23)where C lm ( ϑ, ϕ ) = p π/ (2 l + 1) Y lm ( ϑ, ϕ ) and angleswith a subscript µ ( N ) describe the position of the muon( z axis of the nuclear frame) in the laboratory frame.Here, the nuclear intrinsic quadrupole moment is definedvia the charge distribution (5) as Q = 2 r π Z ∞ r ρ ( r ) d r, (24)and the distribution of the quadrupole moment is de-scribed by the function f ( r µ ), where in the point-likelimit f ( r µ ) = 1 /r − µ . For the shell model, where thequadrupole distribution is concentrated around the nu-clear rms radius R N , the divergence for r µ = 0 is re-moved, and the corresponding quadrupole distributionfunction is F QD ( r µ ) = ( ( r µ / R N ) r µ ≤ R N r µ > R N . (25)Formally, this corresponds to a charge distribution with ρ ( r µ ) = Q R N r π δ ( r µ − R N ) . (26)The matrix elements of the quadrupole interaction (23)in the states (11) read [30] h F M F Iκ | H (2) E | F M F Iκ i = − α ( − j + I + F (27) × h I || Q b C ( ϑ N , ϕ N ) || I i h nκ || F QD ( r µ ) r µ b C ( ϑ µ , ϕ µ ) || nκ i . The reduced matrix element in the nuclear coordi-nates can be expressed with the spectroscopic nuclearquadrupole moment Q as h I || Q b C ( ϑ N , ϕ N ) || I i = Q s (2 I + 3)(2 I + 1)( I + 1)4 I (2 I − , TABLE IV. Results for the electric quadrupole and magnetic dipole hyperfine splitting for a selection of hyperfine states ofmuonic
Bi ( I = ), Sm ( I = ), and Zr ( I = ). h H (2) E i are the values of the electric quadrupole splitting. h H hom M i is themagnetic dipole splitting from (29) using a homogeneous nuclear current distribution and h H sp M i using the nuclear magnetizationdistribution in the single particle model. See Section VII for definitions. All energies are in keV.nucleus state h H (2) E i h H hom M i h H sp M i F = I − F = I + F = I − F = I + F = I − F = I + Bi 1s / / / / -175.(24.) 175.(24.) -0.55(2) 0.010(4) -0.554(22) 0.098(4)3s / / / -48.9(8.0) 48.9(8.0) -0.160(7) 0.028(1) -0.163(7) 0.029(1)3d / -25.4(1.3) 25.4(1.3) -0.161(6) 0.028(1) -0.163(6) 0.029(1)3d / Sm 1s / / / / -32.8(3.2) 32.8(3.2) 0.066(8) -0.004(1) 0.058(8) -0.004(1)3s / / / -9.4(1.1) 9.4(1.1) 0.020(3) -0.001 0.017(3) -0.0013d / -3.2(0.1) 3.2(0.1) 0.015(1) 0.000 0.014(1) 0.0003d / Zr 1s / / / / / / / / / -1.1(0.4) 1.1(0.4) 0.003 0.000 0.003 0.000 and the reduced matrix elements in the muonic coordi-nates are h nκ || f ( r µ ) b C ( ϑ µ , ϕ µ ) || nκ i = (28) − s (2 j + 3)(2 j + 1)(2 j − j ( j + 1) × Z ∞ (cid:0) G nκ ( r µ ) + F nκ ( r µ ) (cid:1) F QD ( r µ ) r µ d r µ . The values for the nuclear rms-radii R N and the spectro-scopic quadrupole moments Q are taken from Refs. [24,31]. In Table IV, results for the electric quadrupole hy-perfine splitting for the nuclei Bi,
Sm, and
Zr areshown for a selection of hyperfine states, including un-certainties stemming from the error in the quadrupolemoment and an estimation of the modeling uncertainty.
B. Magnetic dipole splitting
In addition, the hyperfine splitting arises from theinteraction of the nuclear magnetic moment with the muon’s magnetic moment, which is also sensitive to thespatial distribution of the nuclear currents. Since themagnetic moment of the muon is inversely proportionalto its mass, the magnetic hyperfine splitting in muonicatoms is less important than in electronic atoms. Thematrix elements of the corresponding Hamiltonian (8) inthe state (11) are [30] h F M F Iκ | H M | F M F Iκ i = (29)[ F ( F + 1) − I ( I + 1) − j ( j + 1)] × α m p µµ N κIj ( j + 1) Z ∞ G nκ ( r µ ) F nκ ( r µ ) F BW ( r µ ) r µ d r µ , where m p is the proton mass, and the ratio of the ob-served magnetic moment µ := h II | ( ~µ ) z | II i and the nu-clear magneton µ N can be found in the literature [31].For the simple model of a homogeneous nuclear currentdistribution the distribution function (10) of the Bohr-Weisskopf effect reads F BW ( r µ ) = ( ( r µ / R N ) r µ ≤ R N r µ > R N . (30)Furthermore, an additional method is used to obtain thedistribution function F BW from the nuclear single par-ticle model, where the nuclear magnetic moment is as-signed to the odd nucleon and the Schr¨odinger equationfor this nucleon is solved in the Woods-Saxon potentialof the other nucleons [19]. In Table IV, results for themagnetic dipole hyperfine splitting for the nuclei Bi,
Sm, and
Zr are presented for a selection of hyper-fine states, using both methods for obtaining F BW , wherethe model error is estimated by the difference of these twomethods and the uncertainty in the magnetic moment isalso taken into account. VIII. CONCLUSION
Improved calculations for the fine and hyperfine struc-ture of heavy muonic atoms were presented. In this work, finite-size corrections, leading-order vacuum po-larization, electron screening, and nonrelativistic recoilcorrections are already included in the solution of theDirac equation. Thus, all further calculations of the hy-perfine structure also contain these corrections via usingthe corrected wave functions. The electric quadrupoleand magnetic dipole hyperfine structure was calculatedto first order, using extended charge and current distri-butions. The detailed shape of these distributions repre-sent a source of uncertainty for the predicted values, andthus motivates the comparison with experimental data,especially for nuclei with to date unknown charge distri-butions.The presented usage of modified wave functions for thecalculation of hyperfine effects can be extended to otherphenomena in muonic atoms, for example, the dynamichyperfine structure with highly deformed nuclei. [1] P. J. Mohr, D. B. Newell, and B. N. Taylor, Rev. Mod.Phys. , 035009 (2016).[2] J. A. Wheeler, Rev. Mod. Phys. , 133 (1949).[3] E. Borie and G. A. Rinker, Rev. Mod. Phys. , 67(1982).[4] S. Devons and I. Duerdoth, “Muonic atoms,” in Advancesin Nuclear Physics: Volume 2 , edited by M. Barangerand E. Vogt (Springer US, Boston, MA, 1995) pp. 295–423.[5] C. Piller, C. Gugler, R. Jacot-Guillarmod, L. A. Schaller,L. Schellenberg, H. Schneuwly, G. Fricke, T. Hennemann,and J. Herberz, Phys. Rev. C , 182 (1990).[6] L. Schaller, L. Schellenberg, A. Ruetschi, andH. Schneuwly, Nuclear Physics A , 333 (1980).[7] W. Dey, P. Ebersold, H. Leisi, F. Scheck, H. Walter, andA. Zehnder, Nuclear Physics A , 418 (1979).[8] A. R¨uetschi, L. Schellenberg, T. Phan, G. Piller,L. Schaller, and H. Schneuwly, Nuclear Physics A ,461 (1984).[9] P. Bergem, G. Piller, A. Rueetschi, L. A. Schaller,L. Schellenberg, and H. Schneuwly, Phys. Rev. C ,2821 (1988).[10] R. Pohl et al. , Nature , 213 (2010).[11] P. Indelicato, Phys. Rev. A , 022501 (2013).[12] K. Pachucki and A. Wienczek, Phys. Rev. A , 040503(2015).[13] L. W. Wansbeek, B. K. Sahoo, R. G. E. Timmermans,K. Jungmann, B. P. Das, and D. Mukherjee, Phys. Rev.A , 050501 (2008).[14] K. Kirch, ArXiv e-prints (2016), arXiv:1607.07042 [hep-ph].[15] V. M. Shabaev, I. I. Tupitsyn, V. A. Yerokhin, G. Plu-nien, and G. Soff, Phys. Rev. Lett. , 130405 (2004). [16] Y. S. Kozhedub, O. V. Andreev, V. M. Shabaev, I. I.Tupitsyn, C. Brandau, C. Kozhuharov, G. Plunien, andT. St¨ohlker, Phys. Rev. A , 032501 (2008).[17] D. Hitlin, S. Bernow, S. Devons, I. Duerdoth, J. W. Kast,E. R. Macagno, J. Rainwater, C. S. Wu, and R. C. Bar-rett, Phys. Rev. C , 1184 (1970).[18] W. R. Johnson, Atomic Structure Theory , 1st ed.(Springer, Berlin Heidelberg, 2007).[19] A. A. Elizarov, V. M. Shabaev, N. S. Oreshkina, and I. I.Tupitsyn, Nucl. Instrum. Methods Phys. Res. B , 65(2005).[20] A. Bohr and V. F. Weisskopf, Phys. Rev. , 94 (1950).[21] R. M. Steffen, Hyperfine Interactions , 223 (1985).[22] D. A. Varshalovich, A. N. Moskalev, and V. K. Kher-sonskii, Quantum Theory of Angular Momentum (WorldScientific, Singapore, 1988).[23] T. Beier, Physics Reports , 79 (2000).[24] I. Angeli and K. Marinova, Atomic Data and NuclearData Tables , 69 (2013).[25] W. Greiner, Relativistic Quantum Mechanics , 3rd ed.(Springer-Verlag, Berlin Heidelberg, 2000).[26] L. D. Landau and L. M. Lifshitz,
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