Nuclear magnetic shielding in heliumlike ions

Ab initio QED calculations of the nuclear magnetic shielding constant in helium-like ions are presented. We combine the nonrelativistic QED approach based on an expansion in powers of the fine-structure constant $\alpha$ and the so-called ``all-order'' QED approach which includes all orders in the parameter $Z\alpha$ but uses a perturbation expansion in the parameter $1/Z$ (where $Z$ is the nuclear charge number). The combination of the two complementary methods makes our treatment applicable both to low-$Z$ and high-$Z$ ions. Our calculations confirm the presence of a rare antiscreening effect for the relativistic shielding correction and demonstrate the importance of the inclusion of the negative-energy part of the Dirac spectrum.


I. INTRODUCTION
Magnetic moments of nuclei are often determined from nuclear magnetic resonance (NMR) measurements.Despite high precision of these experiments, the accuracy of the extracted nuclear moments is severely limited by the restricted knowledge of the magnetic shielding caused by the chemical surrounding.Such effects are difficult to calculate reliably, which often led to significant deficiencies in the literature data on nuclear magnetic moments [1,2].As an example, the so-called "bismuth hyperfine puzzle" [3,4] was recently resolved [5] and traced back to an inaccuracy of the nuclear magnetic moment caused by shortcomings in calculations of the shielding correction.
Much more accurate determinations of nuclear magnetic moments can nowadays be achieved by Penningtrap measurements of the combined Zeeman and hyperfine structure of few-electron atoms or ions.The shielding constants of such systems can be calculated ab initio within the framework of QED, with a detailed analysis of uncertainties due to omitted higher-order effects.Precise determinations of magnetic moments of a number of light nuclei by this method were reported in the last years [6][7][8][9].In particular, the magnetic moment of the proton was accurately measured by the Penning-trap technique in Ref. [10].This technique can in principle be extended to measurements of other nuclei and closed-shell ions.
Highly sophisticated calculations of the nuclear shielding have been recently accomplished for the helium atom [11][12][13], motivated by perspectives of using the hyperpolarized helium NMR probes as a new standard for absolute magnetometry [14][15][16].The calculation of Ref. [11] revealed a rare effect of 'anti-screening' for the relativistic shielding correction, corresponding to the situation when an effect for two correlated (1s) 2 electrons is larger than for two non-interacting 1s electrons.This defies the physically intuitive picture in which each of the correlated electrons should experience a slightly smaller nuclear charge because it is effectively screened by the second electron, the effect commonly known as screening.So far the presence of the anti-screening effect has not been confirmed by an independent calculation.
The goal of the present study is to perform ab ini-tio QED calculations of the nuclear magnetic shielding of helium-like ions for a wide range of nuclear charges Z.This will be achieved by merging together two complementary methods, namely, the nonrelativistic QED (NRQED) approach based on an expansion in powers of the fine-structure constant α and the so-called "allorder" QED approach which includes all orders in the parameter Zα but uses a perturbation expansion in the parameter 1/Z.The NRQED method alone is applicable only to low-Z ions, since the uncalculated higher-order effects scale with high powers of Z.By contrast, the all-order method is effective in the high-Z region, since the 1/Z expansion converges fast there.In this work we unify these two methods, so that the resulting approach becomes applicable for the whole range of Z.For the first time such unified approach was applied by G. Drake for calculating energies and transition rates of helium-like ions in Refs.[17,18].The outline of our calculations is as follows.First, we employ the NRQED approach to calculate the leading shielding contribution of order α 2 as well as the relativistic, nuclear and QED corrections of order α 4 , α 2 m/M , and α 5 ln α.Then we address the higher-order corrections within the all-order method.We calculate the oneelectron shielding contribution, the one-photon exchange, QED, and the nuclear magnetization distribution effects.By analysing the Zα expansion of the individual corrections, we identify the lowest-order contributions already included into the NRQED treatment and remove them, thus avoiding double counting and obtaining the final results for the shielding constant.

II. NRQED APPROACH
Within the NRQED approach, the shielding constant σ of low-Z atoms is represented as a double expansion in α and the electron-to-nucleus mass ratio m/M , As is customary in NRQED calculations, we will use the atomic units in the following formulas in this Section.
To the leading order in α and zeroth order in m/M , the shielding constant σ takes the form [19] σ where r a is the distance between the nucleus and a'th electron and the summation runs over all electrons.The relativistic shielding correction of order α 4 was derived in Ref. [11], with the result (2) , where r ≡ r 1 − r 2 , (p i q i ) (2) = p i q j /2+p j q i /2−δ ij p• q/3, and 1/(E − H) ′ is the reduced Green function (with the reference state removed from the sum over the spectrum).
The leading-order nuclear recoil correction was derived in Ref. [20] and later corrected in Ref. [13].The result is where p N = − a p a , g N = (M/m p )(µ/µ N )/(ZI) is the nuclear g-factor; M , I and µ are the nuclear mass, spin and the magnetic moment, respectively; m p is the proton mass and µ N is the nuclear magneton.
The logarithmic part of the leading QED correction of order α 5 was derived in Ref. [11].We write it as  This formula differs from the one from Ref. [11] by ln Z in the second-order correction containing δ( r).In obtaining it, we took into account that the two-electron Lamb shift contains, in addition to ln α, an implicit ln Z term usually hidden in the Araki-Sucher term ∼ r −3 [21].The nonlogarithmic QED contribution of order α 5 was derived and calculated for helium in Ref. [12].Its numerical calculation is rather complicated as it involves perturbations of the so-called Bethe logarithm.For this reason, we will address this and higher-order QED corrections within the 1/Z expansion in the next Section.
Numerical calculations of the NRQED corrections summarized above were carried out with the basis set of exponential functions e −αi r1−βi r2−γi r [22].The method of calculations was developed in our previous investigations, see Ref. [23] for a review.The most computationally intense part was the calculation of σ (4) .While the evaluation of first-order matrix elements in Eq. ( 3) was relatively straightforward, the computation of secondorder matrix elements turned out to be rather demanding.To achieve a high numerical accuracy, we used carefully optimized basis sets for the intermediate electron states.The optimization was carried out for symmetric second-order corrections, using non-uniform distributions of nonlinear basis-set parameters (see Ref. [23] for details), with a typical size of the basis set N = 1200.The obtained basis sets were then used for computation of non-symmetric second-order corrections in Eq. ( 3).
Results of our numerical calculations of σ (2) , σ (2,1) , σ (4) , and σ (5) log for Z ≤ 12 are presented in Table I.The corresponding values for Z > 12 can be readily obtained by using the 1/Z expansion, which is of the form and similarly for other corrections.The leading coefficients c 0 are known analytically from the hydrogen theory, whereas the higher-order coefficients c k were obtained by fitting our numerical results.
For the relativistic correction σ (4) we find a small deviation from the helium result of Ref. [11].The difference comes from the second-order contribution with the 3 D intermediate states, labeled as Q 12 in Ref. [11].The deviation is small and does not influence the final theoretical prediction for the helium shielding constant within its estimated error.
Our results summarized in Table I confirm the previous findings [11] of the presence of the unusual antiscreening effect for σ (4) .Indeed, the absolute values of σ (4) /Z 3 for all nuclear charges are larger than the corresponding limiting value for noninteracting electrons, c 0 = 97 /54 ≈ 1.796.This is in contrast to all other corrections examined in Table I, which exhibit the normal screening effect.It is important to note that the first two 1/Z-expansion coefficients c 0 and c 1 of σ (4) are independently cross-checked by our calculation of the one-photon exchange correction in Sec.III, thus excluding a possibility of a technical mistake in the derivation of σ (4) .The probable explanation of the anti-screening effect is the singlet-triplet mixing.Specifically, the relativistic effects mix the ground 1 1 S 0 state with intermediate n 3 S states.This mixing is quite large and changes the behaviour of the relativistic correction in the case of helium-like atoms as compared to the hydrogen-like case.

III. ALL-ORDER APPROACH
In order to access the higher-order effects ∼ α 5+ , we will adopt the so-called all-order QED approach.This method includes all orders in the parameter Zα but expands in the electron-electron interaction, with the expansion parameter 1/Z.In order to separate out the higher-order contributions beyond what is already included into the NRQED treatment in Sec.II, we will examine the Zα expansion of the all-order results and remove the double counting by subtracting the leadingorder contributions.The zeroth order in 1/Z is delivered by the independent-particle approximation, which neglects the interaction between the electrons.Further terms of the 1/Z expansion are described by Feynman diagrams containing an exchange by one, two, etc. virtual photons between the electrons.In this Section we will use the relativistic units ( = c = 1).

A. Electron-structure effects
We start with examining the so-called electronstructure effects, which are induced by Feynman diagrams without radiative loops.

One-electron
In the independent-particle approximation, the relativistic shielding constant for the (1s) 2 state of the helium-like ion is, see Ref. [24] for details, where µ a is the momentum projection of the 1s electron, the sum over n runs over the complete spectrum of the Dirac equation, and V g and V h are effective interactions of a Dirac electron with the external magnetic field (V g ) and with the magnetic dipole nuclear field (V h ), For the point nuclear charge, Eq. ( 7) can be calculated analytically [25][26][27][28][29], with the result where γ = 1 − (Zα) 2 .For an extended nuclear charge distribution, Eq. ( 7) can be readily calculated numerically with help of the dual-kinetic balance finite basis set method [30].
The higher-order one-electron relativistic correction is obtained from the above expression by subtracting the first two terms of the Zα expansion, rel,1el = σ rel,1el − 2α(Zα) It should be noted that when performing the summation over the Dirac energy spectrum in Eq. ( 7), the inclusion of the negative-energy part of the spectrum is mandatory, as its contribution is very large, especially for low-Z ions.This is explained by the fact that the nonrelativistic limit of the nuclear shielding constant in atoms is induced solely by the negative-energy part of the Dirac spectrum.So, for Z = 2, the negative-energy states induce 99.9% of the total result.With the increase of Z, the relative contribution of the negative-energy states gradually diminishes but is still very significant, e.g., for Z = 60 it is 37%.This is in sharp contrast to calculations of transition energies, where the negative-energy contribution is suppressed by a factor of (Zα) 3 compared to the leading nonrelativistic result [31].

One-photon exchange
The one-photon exchange correction to the nuclear magnetic shielding can be obtained as a perturbation of the one-photon exchange correction to the energy by two external interactions, V g and V h .For the ground state of a helium-like ion, we obtain where a and b denote the two electrons in the (1s) 2 shell, P is the permutation operator (P aP b = ab or ba), (−1) P is the sign of the permutation, the perturbations of the wave functions are defined by and |δ gh a is obtained from the last equation by interchanging g and h.Furthermore, I ≡ I(r 12 ) denotes the electron-electron interaction operator in the Coulomb gauge for the zero transferred energy, where α the Dirac matrices, r 12 = r 1 − r 2 , and ˆ r = r/r.We note that in obtaining Eq. ( 11) we took into account that the two electrons in the (1s) 2 shell have the same energy, so that the frequency dependence of the electron-electron interaction operator does not play any role in this case.
We calculated Eq. ( 11) numerically with help of the Bspline basis set method [32] for the point-charge nuclear model and with the dual-kinetic-balance method [30] for the extended-charge nuclear model.The typical basis size used in the computation was N = 100 -150.In order to achieve a high numerical accuracy in the low-Z region (required for a high-precision fitting of the d 2 coefficient), we had to use the quadruple-precision (appr.32 decimal digits) in our computation.
The results for the point nuclear model are listed in Table II.In order to remove the double counting with the NRQED results, we analyse the Zα-expansion of the one-photon exchange correction, which is of the form The coefficients d 2k obtained by fitting our numerical results are listed in Table II.The first coefficient d 0 = − 5 /24 corresponds to the 1/Z 1 coefficient of expansion of σ (2) /Z, see Table I.The second coefficient d 2 corresponds to the 1/Z 1 coefficient of σ (4) /Z 3 , see Table I.Other coefficients represent contributions of order α 6 and higher which have not been accounted for in Sec.II.
The higher-order one-photon exchange contribution is obtained as where d 0 and d 2 are listed in Table II.
Analysing contributions induced by the positive-and negative-energy parts of the Dirac spectrum in the onephoton exchange shielding correction, we observe that similarly to the one-electron case, the contribution of the negative-energy states is very significant.For Z = 2, they contribute 99.9% of the total result, whereas for Z = 60, the negative-and positive-energy contributions are of the same magnitude and of the opposite sign.This demonstrates the importance of the proper treatment of the negative-energy part of the Dirac spectrum in calculations of the nuclear shielding.We note that a similar dominance of the negative-energy contribution was recently found for the M 1 polarizability in strontium [33].

≥ 2 photon exchange
The uncertainty due to two and more photon exchange is estimated basing on the pattern of the available 1/Zexpansion coefficients of the σ (4) correction.Specifically, where 2 is the conservative factor.

B. QED effects
In the independent-particle approximation, the QED contribution for the (1s) 2 state is twice the 1s QED correction calculated for hydrogen-like ions to all orders in Zα in Refs.[24,34].The analytical result for the leading Zα-expansion term was obtained in Refs.[24,34] and later corrected in Ref. [12].Separating out the leadingorder result, we represent the one-electron QED contribution for the (1s) 2 state of helium-like ions as where G QED (Zα) ≈ 2.182 (Zα) + O((Zα) 2 ) is the remainder function containing one-electron contributions of higher orders in Zα.The logarithmic term in the above formula corresponds to the 1/Z 0 term of the expansion of σ log , whereas the other terms have not been included in the NRQED treatment of Sec.II.The remainder function G QED (Zα) was calculated to all orders in Zα in Refs.[24,34].In the present work, we use the data obtained in those works and ascribe to it the relative uncertainty of ±0.3 (Zα) 2 , to account for the uncalculated diagrams with magnetic-loop vacuum-polarization.So, the higher-order QED contribution beyond those included in Sec.II is In order to estimate the effects of the screening of the one-electron QED correction by the second electron, we introduce the screening factor ζ scr basing on known results for the logarithmic QED contribution σ (5) log .Specifically, we define where c 0 = − 32 /9π is the leading 1/Z-expansion coefficient, see Table I.Using this screening factor, we estimate the QED screening contribution as The above estimate of uncertainty is supported by the complete NRQED calculation of σ (5) for helium [12].

C. Nuclear magnetization distribution
Within the independent-particle approximation, the nuclear magnetization distribution correction for the (1s) 2 state is twice the 1s hydrogen-like contribution, derived to the leading order in Zα in Ref. [13].Formulas presented in Ref. [13] include both the finite nuclear size (fns) and the nuclear magnetization distribution (Bohr-Weisskopf, BW) effects.Removing the fns part, we get where r M and r Z are the magnetic and the Zemach radius, respectively and r is the mean nuclear charge radius.Within the Gaussian model for the nuclear charge distribution ρ C (r) = ρ 0 exp −3 r 2 /(2r 2 C ) (and similar to the magnetization distribution), we obtain where r C = r 2 1/2 is the root-mean-square nuclear charge radius.
For light nuclei with Z = 3 and 4, we use Eq. ( 23) with the experimental values of the Zemach radii, r Z ( 7 Li) = 3.33 fm [35] and r Z ( 9 Be) = 4.04 fm [36].For heavier nuclei, the Zemach radius is not readily available from experiment.For some nuclei, the 1s shielding BW correction was calculated in Ref. [24] within the effective single-particle model of the nuclear magnetization distribution.However, this model is not universal and is applicable for some selected nuclei only.In the present work we will use Eq. ( 23) with the magnetic radius expressed in terms of the charge radius by r M = √ 3 r C for nuclear charges Z < 80.We found that with this choice of the magnetic radius, Eq. ( 23) qualitatively reproduces results of the single-particle model calculations of Ref. [24].We estimate the uncertainty of this approximation of the one-electron BW correction to be 50%, which can be compared to the 30% uncertainty estimate of the singleparticle model results in Ref. [24].For Z > 80, Eq. ( 23) is no longer adequate.We thus apply the single-particle nuclear model as described in Ref. [24] to compute the BW correction for several high-Z ions, specifically, with Z = 80, 83, and 91.
The effects of the electron-electron interaction on the one-electron BW corrections are estimated analogously to Eq. (22).

IV. RESULTS AND DISCUSSION
In this work we performed numerical calculations of the nuclear shielding correction for the ground state of helium-like ions for a wide range of nuclear charges.The nuclear parameters were taken from Ref. [2] (magnetic moments), Ref. [37] (charge radii), and Ref. [38] (masses).Individual shielding contributions for selected ions are presented in Table III.We observe that for the lightest ions, the dominant theoretical uncertainty comes from the QED screening effect.This uncertainty can be improved further by a calculation of the nonlogarithmic α 5 QED correction, as accomplished for helium in Ref. [12].For heavier ions, the largest theoretical uncertainty comes from the extended distribution of the nuclear magnetic moment (the BW effect).This uncertainty can in principle be improved by dedicated calculations of the BW correction for specific nuclei with a microscopic nuclear model [39].An even better way is to use experimental values of the effective Zemach radius rZ [35] obtained from the hyperfine-splitting measurements.One can then use rZ instead of r Z in Eq. ( 23) and compute r M from Eq. ( 24).An additional benefit is that this automatically accounts for some higher-order nuclear effects included into rZ .
Table IV lists our theoretical predictions of the nuclear shielding constant for helium-like ions.We do not present results for neutral helium since more complete calculations are available in this case [12,13].The absolute accuracy of theoretical predictions for the shielding constant σ varies from 2 × 10 −10 for Z = 3 to 1 × 10 −4 for Z = 91.This accuracy demonstrates the precision possible for determination of nuclear magnetic moments from helium-like ions.
Summarizing, we performed calculations of the nuclear magnetic shielding for helium-like ions in the ground state.By combining two complementary approaches, we obtained results for a wide region of nuclear charges.Our calculations confirmed the presence of a rare antiscreening effect for the relativistic shielding correction.They also demonstrated the importance of inclusion of the negative-energy part of the Dirac spectrum in calculations of the nuclear shielding, especially for low-Z ions.In future, the developed approach can be extended to calculations of nuclear shielding in Li-like ions, which are of immediate experimental interest [8,9,40].

σ
rel,1ph = α P (−1) P P aP b|I|δ hg a b + P aP b|I|δ gh a b + P aP b|I|δ h a δ g b + δ h P aP b|I|δ g a b + δ h P aP b|I|a δ g b − P aP b|I|ab δ h a|δ g a − a|V g |a P aP b|I| δ h a b − a|V h |a P aP b|I| δ g a b ,

TABLE I :
NRQED shielding corrections for different nuclear charges Z and their 1/Z-expansion coefficients c k . [11]f.[11]

TABLE II :
One-photon exchange shielding correction σ rel,1ph for different nuclear charges and coefficients of its Zα expansion, for the point nuclear model.

TABLE III :
Individual contributions to the shielding constant in He-like ions, units are 10 −6 .

TABLE IV :
Nuclear magnetic shielding constant σ for the ground state of helium-like ions with the nuclear charge number Z and mass number A.