Isotope shifts for {}^1S_0-{}^3P_{0,1}^o Yb lines from multi-configuration Dirac-Hartree-Fock calculations
IIsotope shifts for S − P o , Yb lines from multi-configuration Dirac-Hartree-Fockcalculations
Jesse S. Schelfhout ∗ and John J. McFerran † Department of Physics, University of Western Australia, 35 Stirling Highway, 6009 Crawley, Australia (Dated: February 18, 2021)Relativistic multiconfiguration Dirac-Hartree-Fock calculations with configuration interaction arecarried out for the S and P o , states in neutral ytterbium by use of the available grasp2018 package. From the resultant atomic state functions and the ris4 extension, we evaluate the massand field shift parameters for the S − P (clock) and S − P o (intercombination) lines. Wepresent improved estimates of the nuclear charge parameters, λ A,A (cid:48) , and differences in mean-squarecharge radii, δ (cid:104) r (cid:105) A,A (cid:48) , and examine the second-order hyperfine interaction for the P o , states.Isotope shifts for the clock transition have been estimated by three independent means from whichwe predict the unknown clock line frequencies in bosonic Yb isotopes. Knowledge of these linefrequencies has implications for King plot nonlinearity tests and the search for beyond Standard-Model signatures. I. INTRODUCTION
Atomic systems offer a means to test fundamentalphysics at a high level of precision in the search forphenomena beyond the Standard Model of elementaryparticles [1–3]. This may be undertaken by examin-ing King plots that are generated through isotope shiftspectroscopy of at least two transitions in an atomicspecies [4–7]. Nonlinearities in such plots may arise dueto higher-order effects within the Standard Model (SM),such as higher-order mass shifts [8, 9], nuclear deforma-tion [10, 11] or nuclear polarizability [12], or due to phe-nomena beyond the Standard Model [5–7, 12–16]. Accu-rate atomic structure calculations are needed to explorepossible causes for such nonlinearities, as is done by inves-tigating additional contributions to isotope shifts beyondthe simple mass shift and field shift [4, 5, 11, 12, 17] —this can be done by analysing the residuals of a linear fitto a King plot, whereby different nonlinearities are ex-pected to have different signatures in the residuals [4].Such calculations can also be used in the extraction ofinformation about the nuclear structure [18, 19].The recent work of Counts et al. [4], using narrowoptical quadrupole transitions in Yb + , is the only Kingplot to date which demonstrates nonlinearity beyond thelevel of experimental uncertainty. This 3 σ nonlinearityis consistent with interpretations as either higher-orderSM effects or physics beyond the SM [17]. We note thata recent arXiv paper by Allehabi et al. [11] suggests thatnuclear deformation can produce a King plot nonlinear-ity at a level consistent with that found in [4], althoughtheir nuclear calculations neglect spin-orbit corrections ∗ [email protected] † [email protected] which can be important for high-accuracy calculations[20]. A means of exploring the dominant cause of Kingplot nonlinearity is by combining prior Yb + data withisotope shift measurements of the S − P transition inneutral Yb. In this work we provide estimates of these clock transition frequencies for all the bosonic isotopes ofYb i , aiding the experimental search for these lines.Advents in modern computing allow for relativisticatomic structure calculations to be performed with re-sults consistent with experimentally determined valuesto a few parts in 10 [21–24]. Such computations arealso used to determine mass and field shift parametersof isotope shifts for King plot analyses [4, 10, 25–29].Low-lying energy levels in ytterbium have been exploredthrough computational means [30–44], however they donot compute isotopes separately and often avoid the P o state. In this paper, the isotope shifts of the clock andintercombination line (ICL) transitions are computed abinitio and the mass and field shift parameters that aidKing plot analysis are calculated.We describe our computational procedure in Sect. II,where the atomic state function is refined through a re-stricted active set approach using MCDHF computationsfrom a multireference set of configuration state functions.Sect. III summarises the energy level differences and iso-tope shifts resulting from the MCDHF-CI computations.Sect. IV gives a detailed account of the mass and fieldshift parameters that are evaluated with ris S and P o , states and the predictions of the absoluteclock line frequencies, is given in the Appendices. a r X i v : . [ phy s i c s . a t o m - ph ] F e b II. COMPUTATIONAL METHOD
Most ab initio isotope shift computations performcomputations for a single isotope and then calculate themass and field shift parameters, using nuclear charge pa-rameter ( λ A,A (cid:48) ) values derived from experiment to arriveat isotope shifts. In contrast, the computations presentedhere are similar to the “exact” method of [19] and tothose of [45] and [46], in that energies and wavefunctionsare computed for each isotope of interest, and the isotopeshifts are taken as the differences between these energies.It is suggested that this approach can be strongly model-dependent [47], so the more common method of calcu-lating isotope shifts via computed mass and field shiftparameters is also pursued.A two-step approach is used to estimate the isotopeshifts, mass shift and field shift parameters for the clockand intercombination transitions using computationalmethods. First, a multi-configuration Dirac Hartree Fockapproach with configuration interaction (MCDHF-CI)is used to compute the atomic state functions (ASFs)for the S ground state and P o , excited states usingthe Fortran 95 package grasp2018 (General RelativisticAtomic Structure Package) [48]. Isotope shifts are cal-culated as the differences in energy between the groundand excited states for different isotopes. Mass and fieldshift parameters are then extracted using the Fortran90 program ris4 (Relativistic Isotope Shift) [47]. ris4 was written as an addition to the grasp2k package [49];however, we have been able to use it in conjunction with grasp2018 [50]. The computational process is outlinedin Figure 1.A MCDHF-CI approach is used in favour of otherapproaches, e.g. configuration interaction with many-body perturbation theory (CI+MBPT) [10, 51], all-ordermethods [52], and relativistic coupled cluster (RCC) cal-culations [53], due to the recent updates of the grasp and ris packages and their cross-compatibility allowingfor ease of extraction of isotope shift parameters.A restricted active space approach is used to constructthe atomic state functions, whereby a multi-reference set(MR) is chosen, and additional configuration state func-tions (CSFs) are systematically included according toallowed substitution rules. The ground state electronconfiguration for ytterbium is [Xe]4 f s . The multi-reference (MR) set for the S ground state is thus takento be [Xe]4 f { s , d , p } as these are the configu-rations with two valence electrons which can form S terms and are near in energy to the 6 s ground state.This is the same MR set as that of the ‘MCDF IV’ ap-proach used in [54]. The excited states P o , have elec-tron configuration [Xe]4 f s p . Conveniently, these canbe computed simultaneously using the extended optimallevel (EOL) mode of the rmcdhf( mpi) program [48]. Generatenuclear modelComputeMR set (cid:0) { s , p , d } (cid:1) S (cid:0) { s p, d p } (cid:1) P o , rhfs Run ris4
Run ris4grasp2k no yes noyes grasp2018 FIG. 1. Procedure for performing MCDHF-CI computa-tions ( grasp2018 ) and extracting isotope shift information( grasp2k ). Correlation layers are added until n = 12 forground and excited states. Mass and field shift parametersare evaluated with the ris4 package. MR - multireference; rhfs - relativistic hyperfine structure program [48].TABLE I. Multireference configurations for the clock and in-tercombination transition levels in Yb i .Level J Π MR configurations N CSFs s S + [Kr]4 d f s p + { s , p , d } s p P o − [Kr]4 d f s p + { s p, d p } s p P o − [Kr]4 d f s p + { s p, d p } Computing the P o and P o excited states together withthe EOL mode is found to have negligible effect on theclock transition frequency compared with computation ofthe P o state on its own ( ∼ .
3% difference). The MRset for P o , is taken to be [Xe]4 f { s p, d p } . TheMR sets are summarised in Table I, where N CSF s is thenumber of configuration state functions for the MR setwhen using relativistic notation.Correlation orbitals are added layer by layer, where alayer includes orbitals for each angular momentum value(e.g. 7 s, p − , p + , d − , d + , f − , f + , with the subscript ± indicating j = l ± / f ± , 5 s or 5 p ± orbitals, togetherwith unrestricted substitutions from the valence orbitals(6 s, p ± , d ± ). This keeps the computations tractablewhilst allowing a considerable degree of valence-valenceand core-valence correlation. The number of CSFs growsto 30256, 30668 and 84519 for S , P o and P o , respec-tively [57]. The dominant CSFs by percentage contri-bution to the total ASF for each state are tabulated inAppendix B.The atomic state function computed with all the de-sired correlation layers is corrected for higher-order QEDeffects through the rci mpi program. The transversephoton interaction is reduced to the Breit interaction byscaling all transverse photon frequencies by a factor of10 − [48]. Vacuum polarisation effects are accounted for,and self-energy is estimated for orbitals up to n = 6. Thenormal and specific mass shifts due to the nuclear recoilare also included in the CI computations. Ytterbium nu-clei are known to be deformed [58, 59]; however, the nu-clear model used for these computations — see AppendixA — does not account for nuclear deformation.The wavefunction arising from a single CSF is an anti-symmetric product of one electron wavefunctions [60] inthe form of a Slater determinant [61]. The radial func-tions for the 6 p − and 6 p + orbitals resulting from ourMCDHF-CI computations for the S and P o , statesare represented in Fig. 2, where the large-component, P ( r ), and small-component, Q ( r ), radial functions arepresented separately. Less significant deviations betweenthe ground state and excited state radial functions werefound for the 6 s orbital.Where possible, uncertainties in the presented compu-tational results are estimated by direct comparison withexperiment [62]. In other cases, uncertainties are esti-mated by systematically adding correlation layers or in-creasing the size of the core available for correlation, andanalysing the convergence of the desired properties [56].The latter approach may not include systematic uncer-tainties arising from the MCDHF-CI method, and so itis desirable to compare against other computational ap-proaches [63]. We use a combination of these approaches,with quoted uncertainties corresponding to 1 σ unless oth-erwise stated.The efficacy of MCDHF-CI computations has beendemonstrated recently; for example, Zhang et al. calculate energies for sulfur-like tungsten with near-spectroscopic accuracy [64]; Silwal et al. compute isotopeshifts within the uncertainty bounds of experimental re-sults in Mg-like and Al-like systems [45]; and Palmeri et al. produce isotope shifts in reasonable agreement FIG. 2. Large, P ( r ), and small, Q ( r ), component radial wave-functions for the S (solid blue) and P o , (dashed orange)states computed using grasp2018 . The abscissa is √ r (inatomic units). P ( r ) is plotted in the left column and Q ( r ) inthe right column. The first (second) row displays the func-tions for the 6 p − (6 p + ) orbital. with experimental results for osmium [65]. Froese Fis-cher and Senchuk note that good accuracy is generallyachieved for light elements or highly-ionised heavy el-ements, but suggest neutral heavy elements with opencore sub-shells can be subject to problems [66]. Theseproblems are not expected to influence the results ofthis paper, due to the simple closed-shell electron con-figuration of neutral ytterbium, in particular the closed4 f sub-shell. Further, neglecting core-core correlationshere is justified: The inclusion of core-core correlationin MCDHF computations of neutral lithium and sodium,which have closed core sub-shells, was found to decreasethe agreement with experiment compared to computa-tions with only valence and core-valence correlation [67].The agreement between computational and experimentalvalues for oscillator strength in singly-ionised thallium, aheavier system than ytterbium with electron configura-tion [Xe]4 f d s , was also found to be better in theabsence of core-core correlation [68].The bulk of the computations were performed atthe University of Western Australia High PerformanceComputing Centre on Kaya [69], one of their high-performance computing machines [70]. Kaya is com-prised of two Dell PowerEdge R740 nodes, each with 2Intel Xeon Gold 6254 processors with 18 cores, 768 GBof RAM and dual 1.6 TB NVMe devices. 34 cores wereutilised for the computations. III. RESULTS: ISOTOPE SHIFTS
The computed energy level differences for the clocktransition are displayed in Table II, and in Table III forthe intercombination line. Energies computed in atomicunits (E h ) are converted into frequencies in Hz via multi-plication by 2 cR ∞ = 6 . × Hz E h − [71]. The computed energy level differences are 0 . .
7% larger for the intercombination line.
TABLE II. Computed energy level separations and isotopeshifts for S ground state and P o excited state for stableisotopes of Yb. Isotope shifts are relative to Yb.Isotope Energy separation (MHz) Isotope shift (MHz)168 522 679 368 − − − − − − S ground state and P o excited state for stableisotopes of Yb. Isotope shifts are relative to Yb.Isotope Energy separation (MHz) Isotope shift (MHz)168 543 180 934 − − − − − − For the clock transition, the isotope shift between
Yb and
Yb is calculated as −
615 MHz and between
Yb and
Yb is calculated as − − . − . .
5% and 10 . .
5% when compared with measuredvalues from [72]. This difference may reduce with the in-clusion of deeper core-valence correlations and possiblycore-core correlations [28, 48, 68]. Variation in nucleardeformation between the isotopes, not accounted for inthese computations, may also contribute to the differ- ences between the experimental and computational iso-tope shifts. While these differences are a concern, they donot prevent us from making viable predictions for clocktransition frequencies in the bosonic isotopes (discussedbelow).
IV. MASS AND FIELD SHIFT PARAMETERS
The differences in nuclear mass and charge distribu-tions between isotopes of the same element give rise tosmall variations in the energy eigenvalues for the atomicsystem, known as isotope shifts. By convention, the iso-tope shift for a pair of isotopes is calculated by sub-tracting the energy of the lighter isotope from that ofthe heavier isotope [73], i.e., for isotopes A and A (cid:48) with m A > m A (cid:48) , the isotope shift is given by δν A,A (cid:48) = ν A − ν A (cid:48) . (1)To a very good approximation, an isotope shift maybe split into a mass shift and a field shift, arising fromdifferences in the nuclear recoil and nuclear charge dis-tribution, respectively, between the isotopes [73]. Underthe approximation that the electronic wavefunction for aparticular state is invariant between isotopes, the massand field shifts for an atomic state i factor into electronicand nuclear components δν A,A (cid:48) i = K i µ A,A (cid:48) + F i λ A,A (cid:48) , (2)where K i ( F i ) is the electronic mass (field) shift factor, µ A,A (cid:48) = 1 m A − m A (cid:48) = m A (cid:48) − m A m A m A (cid:48) (3)is the nuclear mass parameter, and λ A,A (cid:48) = λ A − λ A (cid:48) = (cid:88) n ≥ C n δ (cid:104) r n (cid:105) A,A (cid:48) (4)is the nuclear charge parameter, where C n are Seltzer’scoefficients [43, 74, 75]. For a transition between an up-per state j and a lower state i , the isotope shift is givenby δν A,A (cid:48) = δν A,A (cid:48) j − δν A,A (cid:48) i = Kµ A,A (cid:48) + F λ
A,A (cid:48) , (5)where K = K j − K i and F = F j − F i .The field shift factor, F , has been evaluated for eachisotope for both the clock and intercombination tran-sitions with the ris4 program following grasp2018 .We present the values in Table IV, where we see someisotope-dependence. The mean values across all sevenstable isotopes are F clock = − . − and F ICL = − . − ; we comment on the un-certainties below. For the clock transition, a previouslyreported value of F clock was calculated via amb i t [51], us-ing configuration interaction only (without MBPT) anda very similar correlation model to this work [17]. Forthe intercombination line, the mean value is comparedwith previous evaluations of F ICL at the base of the ta-ble. Our value lies approximately central to the range ofprevious estimations, but with higher precision.
TABLE IV. Electronic field shift factor ( F ) for the S − P o clock transition and the S − P o intercombination line(ICL).Isotope F clock (GHz fm − ) F ICL (GHz fm − )168 − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . . − . a Ref. [77] — − . . − . . aa Value is positive in reference (assumed to be absolute value)
The mass shift factors ( K ) experience little changewith isotope; the mean values are K clock = − K ICL = − P states for two-electron spectra is suggested toarise from angular correlation (private communication in[40]). Whilst these negative mass shifts appear to beat odds with the positive value of K ICL = 1 . µ A,A (cid:48) in this work. The same convention is usedin [59, 79]. The convention used in this work is consis- tent with that of [47, 76]. Berengut et al. [17] determine K clock = −
655 GHz u using a CI+MBPT method, imple-mented via amb i t [51], demonstrating the dependence ofthe calculation on the method and supporting its approx-imate magnitude and sign.Uncertainties in K and F for each isotope are esti-mated by systematically increasing the size of the compu-tations. The convergence of the parameters as correlationlayers are added for isotope Yb, and as the set of coreorbitals available for core-valence correlation is extendedfor isotope
Yb, are presented in Appendix C. The un-certainties for the mean values (over isotopes) are takento be the sum by quadrature of (i) the standard deviationof the isotopic data and (ii) the estimated uncertainty foreach isotope. The calculated mass shift factor for the in-tercombination line is consistent with that of [76], andthe field shift factor is consistent with [59, 76].
V. SECOND-ORDER HYPERFINESTRUCTURE
The off-diagonal second-order hyperfine interaction forisotopes with nuclear spin results in a shift of the centroid(center of gravity) of the hyperfine manifold relative tothat of an isotope with no nuclear spin [42, 80]. Correct-ing the experimentally-determined centers of gravity forthese shifts provides a means of comparison between thebosonic and fermionic isotopes (e.g. for King plot anal-ysis). The shift for a state denoted | γJIF m F (cid:105) is givenby∆ E (2) F = (cid:88) γ (cid:48) ,J (cid:48) (cid:54) = γ,J |(cid:104) γJIF m F |H hfs | γ (cid:48) J (cid:48) IF m F (cid:105)| E γ,J − E γ (cid:48) ,J (cid:48) . (6)The matrix element in (6) can be written in terms of theoff-diagonal hyperfine structure constants, A ( J, J (cid:48) ) and B ( J, J (cid:48) ), as (cid:104) γ (cid:48) ( J − IF m F |H hfs | γJIF m F (cid:105) = 12 A ( J, J − (cid:112) ( K + 1)( K − F )( K − I )( K − J + 1)+ B ( J, J −
1) [( F + I + 1)( F − I ) − J + 1] (cid:112) K + 1)( K − F )( K − I )( K − J + 1)2 I (2 I − J ( J − , (7)and (cid:104) γ (cid:48) ( J − IF m F |H hfs | γJIF m F (cid:105) = B ( J, J − , (8)where K = I + J + F .Only isotopes Yb and
Yb have non-zero nuclear spin and thus experience the hyperfine interaction. For J = 1, the off-diagonal hyperfine constant B ( J, J −
1) isvanishing. The hyperfine constants calculated using the rhfs program in the grasp2018 package [48] and arepresented in Table V. Uncertainties are taken to be 4%by comparison of the diagonal hyperfine constants withthe experimental values from Atkinson et al. [72].
TABLE V. Hyperfine interaction constants calculated using rhfs .Isotope A ( P ) (GHz) A ( P , P ) (GHz) B ( P ) (GHz)171 4.07(17) 3.89(16) 0173 -1.12(5) -1.07(5) -0.794(32) Calculation of the centroid shift using (6) makes useof the energy difference between the fine-structure lev-els, P o and P o ; i.e., the value of 21 092 574.882(93)MHz for Yb, based on measurements presented in [72]and [81]. The centroid shifts for the clock transition, tosecond-order in perturbation theory, for the mixing of the P and P states we calculate to be − . − . Yb and
Yb, respectively.For the ICL, the F = I hyperfine level is the only oneinfluenced by mixing with the P o state, and so the shiftto its centroid is smaller. The new centroids for the ICLisotope shifts relative to Yb are − . Yb, and − . Yb ( c.f.
Ref.[72]).The centers of gravity for the intercombination lineisotope shifts presented in [72] are correct to first-orderin perturbation theory; however, the second-order cor-rections due to mixing with the P o state are greaterthan the experimental uncertainty and so are accountedfor here. The effects of mixing with other nearby states( P o , P o ) are estimated to be less than experimentaluncertainty. The centers of gravity determined from themeasured clock transition frequencies for Yb [82, 83]and
Yb [84] must also take into account the higher-order perturbations in order to make comparison withthat of
Yb [81] in a King plot analysis. The resul-tant isotope shifts (between centers of gravity for thefermions) are presented in Table VI. The values are usedlater in Sect. VII.
VI. NUCLEAR CHARGE PARAMETER
The nuclear charge parameter can be calculated by re-arranging equation (5) to find, λ A,A (cid:48) = 1 F (cid:16) δν A,A (cid:48) − Kµ A,A (cid:48) (cid:17) . (9)By use of the isotope shifts presented in Table VI, themass shift and field shift parameters calculated in TableIV, and the isotope masses presented in Table X, thenuclear charge parameter λ A,A (cid:48) can be determined fromEq. 9, as presented in Table VII. The uncertainties aredominated by the uncertainty in K , but they are lower TABLE VI. Isotope shifts for the S − P o intercombinationline and S − P o clock line in Yb i . δν A,A (cid:48) = ν A − ν A (cid:48) .The centroid for the hyperfine manifold is used for fermionicisotopes, where the corrections to second order are taken intoaccount. A A (cid:48) δν A,A (cid:48)
ICL (MHz) δν A,A (cid:48) clock (MHz)176 174 − . − . − . − . − . − . − . − . − . − . − . than those of previous estimates by at least a factor offour. King [73] notes that the values from Clark et al. [59] give excessive weight to the muonic and x-ray data intheir combined analysis, which leads to larger values thanour own. Column 5 shows λ A,A (cid:48) values from Clark et al. based on optical data alone, showing better agreementwith our values. Jin et al. [78] assume a specific massshift of zero and use a larger value for the field shiftparameter (12.2 GHz fm − ), leading to their lower valuesfor λ A,A (cid:48) . TABLE VII. Nuclear charge parameters λ A,A (cid:48) determinedfrom the Yb i intercombination line measurements and calcu-lated F parameters − in units of 10 − fm (column 3). Datafrom prior work are presented for comparison. A A (cid:48)
This work Ref. [59] a Ref. [59] b Ref. [78] Ref. [77]176 174 88.86(47) 109(8) 87(13) 79.4(4.0) 86(2)174 172 93.10(48) 114(8) 92(15) 83.3(4.2) 90(2)172 170 119.17(51) 139(8) 116(16) 106.6(5.3) 113(3)170 168 126.86(53) 147(8) 128(19) 113.6(5.7) 120(14) c
173 172 41.46(24) 53(4) 41(10) 37.1(1.9) 40(1)172 171 76.27(27) 85(4) — 68.3(3.4) 71(1)171 170 42.90(25) 54(4) 41(10) 38.3(1.9) 42(1)174 173 51.64(25) 61(4) — 46.2(2.3) 49(1)173 171 117.72(50) — — — 110(2) ca Combined analysis of optical, x-ray & muonic isotope shifts b Optical isotope shifts only c Value calculated using results from Ref. [77]
The nuclear charge parameter, λ A,A (cid:48) can be convertedinto the difference in mean-square nuclear charge radii, δ (cid:104) r (cid:105) A,A (cid:48) , through rescaling [76, 77] or using an itera-tive procedure [79, 85]. The tabulated δ (cid:104) r (cid:105) A,A (cid:48) valuesfor Yb in Angeli and Marinova [79] are calculated us-ing semi-empirical mass shift and field shift parametersof F ICL = − . − and K ICL = − . .
6) THz u[86]. This mass shift parameter is much larger in magni-tude than that calculated in this work and by [17], lead-ing to the tabulated values being larger than those deter-mined in this work. Allehabi et al. [11] also suggest thatthe tabulated δ (cid:104) r (cid:105) A,A (cid:48) values are too large based on theirown nuclear and electronic structure calculations. TableVIII presents the differences in mean-square charge radiiarising from this work. Fricke and Heilig [76] determinethe higher-order moments to contribute − . λ A,A (cid:48) based on experimental data from muonic atoms, so thedifferences in mean-square charge radii are recovered byrescaling via δ (cid:104) r (cid:105) A,A (cid:48) = λ A,A (cid:48) / . VII. KING PLOT AND CLOCK TRANSITIONISOTOPE SHIFTS
A King plot compares the isotopic shifts of one transi-tion, i , against that of another, j . By scaling the isotopeshift with the reciprocal of the nuclear mass parameter,one defines the modified isotope shift, ξ A,A (cid:48) i = δν A,A (cid:48) i /µ A,A (cid:48) . (10)From Eq. 5 and assuming the nuclear parameters λ A,A (cid:48) and µ A,A (cid:48) are the same for both (all) transitions, onefinds, ξ A,A (cid:48) i = ( F i /F j ) ξ A,A (cid:48) j + ( K i − K j F i /F j ) . (11)A plot of ξ AA (cid:48) i versus ξ AA (cid:48) j should thus, to first or-der, form a straight line with slope F i /F j and intercept K i − K j F i /F j , known as a King plot. A King plot con-structed from the measured isotope shifts for the clockand intercombination lines in Yb i is presented in Fig-ure 3. With only three isotopes having frequency mea-surements for the clock transition, only two independentdata points can be used to create the King plot (the171-174 pairing makes it overdetermined). The gradi-ent and intercept for the linear ‘fit’ are 1.0138(12) and − . F ICL /F clock and the interceptas K ICL − K clock F ICL /F clock . The calculated values in Ta-ble IV produce a gradient of 1.0095(28), and an interceptof 0.01(11) THz u. This gradient and intercept values arenot inconsistent with those obtained from the King plotin Fig. 3 (experimental).The unknown isotope shifts for the S − P o clocktransition can be estimated in three different ways (fur-ther information follows),1. Energy level differences for each isotope are foundthrough MCDHF-CI computations. From these, FIG. 3. King plot for clock and intercombination lines of Yb i . Blue circles represent isotope pairs ( A, A (cid:48) ) with clock tran-sition measurements. Dashed black lines represent isotopepairs (
A, A (cid:48) ) without clock transition measurements. Thesolid orange line is the King linearity relationship. Error barsare smaller than the marker size. the isotope shifts are evaluated, which, becausethere is a consistent offset from measured valuesin S − P o , can be scaled to match the threeknown experimental isotope shifts.2. The mass shift and field shift parameters calculatedusing ris4 are used with the ICL isotope shifts pre-sented in Table VI to estimate the clock transi-tion isotope shifts. This estimate is based predom-inantly on calculation (theory).3. The modified frequency shifts are extrapolatedfrom a King plot constructed using the clock andintercombination lines, and converted back into iso-tope shifts. This estimate is based predominantlyon experimental measurement.The estimated isotope shifts for the clock transition foreach method are presented in Table IX.(Method-1): The ab initio isotope shifts calculated forthe clock transition using MCDHF-CI computations, pre-sented in Table II, are larger than experimental values by ∼
11% (for all the isotopes). This difference we attributeto a systematic effect in the calculations, which we canaccount for by a scaling factor. Accounting for the dif-ference leads to the estimates given in the ‘Method-1’column of Table IX. The adjusted isotope shift between
Yb and
Yb is −
554 MHz and between
Yb and
Yb is − .
5% and − . TABLE VIII. Differences in mean-square nuclear charge radii δ (cid:104) r (cid:105) A,A (cid:48) , determined from the λ A,A (cid:48) values in Table VII via δ (cid:104) r (cid:105) A,A (cid:48) = λ A,A (cid:48) / .
941 [76], in units of 10 − fm (column 3). Data from other works are presented for comparison. A A (cid:48)
This work Ref. [78] Ref. [77] Ref. [79] a Ref. [76] Ref. [11] b
176 174 94.4(0.5) 84.8(4.6) 90(2) 115.9(0.1) 114(30) c a Ref. [79] presents only statistical errors in the uncertainty — the large uncertainty in the mass shift parameter used in calculation isnot propagated through. Propagating the uncertainty from the mass shift parameter leads to an uncertainty of ∼ × − fm for thefirst row. b Purely computational values (presented without uncertainty) c This is mistakenly presented with the equivalent of A (cid:48) = 172 in Ref. [76] TABLE IX. Clock transition isotope shifts ( δν A,A (cid:48) clock ) in MHzdetermined using three different methods, as outlined in thetext.
A A (cid:48)
Method-1 Method-2 Method-3176 174 − − . . − . . − − . . − . . − − . . − . . − − . . − . . − − . . − . . − − . . − . . − − . . − . . − − . . − . . − − . . − . . (Method-2): Equation 5 applies for both the clock andICL transitions, with the nuclear parameters taken tobe independent of the electronic states. Substituting for λ A,A (cid:48) between these two equations leads to δν A,A (cid:48) clock = (cid:18) K clock − F clock F ICL K ICL (cid:19) µ A,A (cid:48) + F clock F ICL δν A,A (cid:48)
ICL . (12)The ICL isotope shifts presented in Table VI can be usedwith the calculated mass shift and field shift parametersto arrive at the clock transition isotope shifts. This isequivalent to constructing a King plot using the theo-retical mass and field shifts computed using ris4 andnuclear charge parameters presented in Table VII, andleads to the isotope shifts presented in the ‘Method-2’column of Table IX. The uncertainties are again domi-nated by the uncertainties in the K parameters for eachtransition, similarly to those for Table VII.(Method-3): Assuming King linearity holds, the Kingplot in Figure 3 can be extrapolated to arrive at the clocktransition isotope shifts for other pairings. These es-timates are presented in the final column of Table IX.We emphasize that the King plot is based on experimen-tal values and not MCDHF-CI calculations. The onlycomputational component is that of the higher order hy-perfine shifts affecting the centers of gravity. Consistentwith this, the uncertainties for ‘Method-3’ are less thanthose of ‘Method-1’. The values in the final two rows ofthis column provide a consistency check, since these arethe isotopes used to construct the King plot — they agreewithin the uncertainties. For comparison, the experimen-tal values appear in Table VI. The presented uncertain-ties for ‘Method-3’ are calculated using propagation oferrors with the uncertainties from the ICL isotope shifts,nuclear masses, and fit parameters.The isotope shifts for Methods 2 and 3 presented inTable IX provide a region in which experimental searchescan be made for the bosonic clock transitions. A weightedmean of the shifts has been used to estimate the absolute clock transition frequencies, as tabulated in Appendix E. VIII. CONCLUSIONS
Ab initio computations of the isotope shifts for theclock transition and its partnering intercombination line( S − P o ) have been performed separately for each sta-ble isotope using a MCDHF-CI method implemented bythe grasp2018 [48] package. Absolute transition fre-quency measurements agree with experimental resultsto less than 1% error, with isotope shifts differing fromexperimental values by 11%. Using these same compu-tations, the hyperfine interaction constants for the P o state have been calculated to within 4% of correspondingexperimental values. Corrections of the centroids of thehyperfine manifolds for the second-order hyperfine inter-action in the fermionic isotopes have also been made.The electronic mass shift and field shift parameters arecomputed with the program ris4 [47] using the resultsof the MCDHF-CI computations. The corrected isotopeshifts for the intercombination line together with theseelectronic mass shift and field shift parameters enablecomputation of the nuclear charge parameters, λ A,A (cid:48) ,consistent with previous results, but with an estimatedorder of magnitude reduction in uncertainties. The dif-ferences in mean-square charge radii, δ (cid:104) r (cid:105) A,A (cid:48) , are calcu-lated and found to be significantly smaller than tabulatedvalues in Angeli and Marinova [79].Experimental isotope shifts for the clock and intercom-bination lines, corrected for the second-order hyperfineinteraction, have been used to construct a King plot withtwo data points. This King plot is used to estimate theisotope shifts for the clock transition for the undiscoveredbosonic isotopes. These estimates are found to be rea-sonably consistent with estimates based on the calculatedmass shift and field shift parameters.The computations may be increased in size by includ-ing deeper core-valence correlation, and by extending theactive set of orbitals beyond a principal quantum num-ber of 12, given sufficient computational resources. Theinclusion of deeper core-valence correlation is expectedto reduce the 11% discrepancy between the computedand experimental isotope shifts [28, 48]. Different nu-clear models, including models accounting for the knowndeformation of Yb nuclei, may also be explored to inves-tigate their potential systematic effects on the computedresults.With suggestions to combine the results of Counts et al. [4] with isotope shift measurements of a clocktransition in neutral ytterbium [4, 17], the undiscoveredbosonic-isotope clock transitions should be sought usingthe isotope shift estimates presented in this work (e.g.with cold Yb atoms in an optical lattice trap and a DC0magnetic field applied [88]). Once the clock isotope shiftsare identified, King plots can be constructed with otherhigh-precision isotope shift measurements in neutral andionised ytterbium in order to investigate King nonlinear-ity and identify or constrain physics beyond the StandardModel.
ACKNOWLEDGMENTS
We are grateful for the assistance provided by Christo-pher Bording and Hayden Walker from the UWA HighPerformance Computing Team. J. S. acknowledgessupport from the University of Western Australia’sWinthrop Scholarship, and St Catherine’s College. Thisresearch was undertaken with the assistance of resourcesfrom the University of Western Australia High Perfor-mance Computing Team, an University IT enabled ca-pability.
APPENDIX A: NUCLEAR MODEL
The nuclear charge distribution is modelled as a two-component Fermi distribution [89, 90] ρ ( r ) = ρ e ( r − c ) /a , (A1)where c is the half-density radius, a is related to thenuclear skin thickness t by t = (4 ln 3) a , and ρ is a nor-malisation factor such that (cid:90) ∞ πr ρ ( r ) dr = Z. (A2)For all isotopes the atomic number is Z = 70 and thenuclear skin thickness is taken to be t = 2 . t = 2 . t = 2 . t = 2 . , Yb.
Yb has a nuclear spin of I = 1 / (cid:126) and a magneticdipole moment of µ = 0 . µ N [92]. Yb has anuclear spin of I = 5 / (cid:126) , a magnetic dipole moment of µ = − . µ N [92], and nuclear electric quadrupolemoment of Q = 2 . TABLE X. Isotope-dependent parameters for the Yb nuclearmodel. A , mass number; R , rms nuclear charge radius; m ,atomic mass. R values obtained from [79], and mass valuesare obtained from [93] except for Yb [94].
A R (fm) m (u)168 5.2702(56) [79] 167.93389132(10) [94]170 5.2853(56) [79] 169.934767246(11) [93]171 5.2906(57) [79] 170.936331517(14) [93]172 5.2995(58) [79] 171.936386659(15) [93]173 5.3046(59) [79] 172.938216215(12) [93]174 5.3108(60) [79] 173.938867548(12) [93]176 5.3215(62) [79] 175.942574709(16) [93] APPENDIX B: STATE COMPOSITIONS
The atomic state functions determined using theMCDHF-CI method consist of weighted combinations ofmany configuration state functions (CSFs). The percent-age contributions of the most significant CSFs are listedfor the S ground state and the P o , excited states inTable XI. Our values are consistent with those reportedby Migdalek and Baylis [39], where their calculation ex-tended only to our MR set. TABLE XI. The highest contributing CSFs in the composi-tions of three Yb i atomic states.CSF Percentage S s p +2 p − s s d +2 d − P o s p − d − p + p − s P o s p − s p + d − p + d − p − p − s APPENDIX C: UNCERTAINTY ESTIMATESFOR ISOTOPE SHIFT PARAMETERS
Systematic expansions of the active space and corre-lation model have been undertaken in order to estimatethe uncertainties for the isotope shift parameters, K and1 F . The error introduced by truncating the active spaceat 12 spdf is estimated by analysing the K and F val-ues after adding each new correlation layer. This analy-sis was performed using Yb with core-valence correla-tions restricted to single excitations from 5 s, p, f andunrestricted valence-valence correlations. The results arepresented in Table XII. Based on these results, the uncer-tainty in the final K and F values due to the truncatedactive space is estimated to be the absolute differencebetween the 12 sp d f and 11 sp d f layers, as thesewere the largest two correlation layers added with an or-bital of each symmetry. TABLE XII. Sequences of isotope shift parameters upon addi-tion of correlation layers. The final row constitutes estimatesfor the uncertainty in each of the parameters for each isotope.The units for K are GHz u and the units for F are GHz fm − .Layer K clock F clock K ICL F ICL sp d f -192.67 -10.4060 -176.48 -10.51518 sp d f -204.07 -10.0544 -189.54 -10.16979 sp d f -282.05 -10.9553 -273.16 -11.069510 sp d f -268.73 -10.9639 -257.58 -11.071011 sp d f -290.47 -10.8480 -281.57 -10.954812 sp d f -288.15 -10.8386 -279.59 -10.941812 spdf -288.07 -10.8393 -279.55 -10.9425Uncertainty estimate 2.32 0.0094 1.98 0.0130 The error introduced by restricting the core-valencecorrelation to single excitations from 5 s, p, f is esti-mated similarly, by analysing the K and F values withincreasingly more core orbitals available for excitation.This analysis was performed using Yb with the activespace up to 12 spdf and unrestricted valence-valence cor-relation. The results are presented in Table XIII. Basedon these results, this uncertainty is estimated to be twicethe absolute difference between this core and the nextlargest available core of 5 s, d, p, f . TABLE XIII. Sequences of isotope shift parameters upon in-clusion of deeper core-valence correlation. The final row con-stitutes estimates for the uncertainty in each of the param-eters for each isotope. The units for K are GHz u and theunits for F are GHz fm − .Available core K clock F clock K ICL F ICL f p, f -265.41 -10.5830 -255.65 -10.66815 s, p, f -288.05 -10.8326 -279.53 -10.93585 s, d, p, f -325.22 -10.8255 -315.26 -10.9297Uncertainty estimate 74.34 0.0142 71.46 0.0122 APPENDIX D: ALTERNATIVE PRESENTATIONOF DIFFERENCES IN MEAN-SQUARECHARGE RADII
The differences in mean-square charge radii are pre-sented in Table VIII for pairs of isotopes. Alternatively, a single reference isotope may be chosen and differences inmean-square charge radii given relative to this referenceisotope. For ytterbium, this reference isotope is com-monly chosen to be
Yb. Differences in mean-squarenuclear charge radii of this type are presented in TableXIV, with the reference isotope of
Yb.
TABLE XIV. Differences in mean-square nuclear charge radiirelative to
Yb, δ (cid:104) r (cid:105) ,A (cid:48) , in units of 10 − fm (column 2).Data from other works are presented for comparison. A (cid:48) This work Ref. [77] Ref. [79] a
174 94.4(0.5) 90(2) 115.9(0.1)173 149.3(0.8) 142(3) 181.0(0.1)172 193.4(1.0) 184(5) 236.6(0.1)171 274.4(1.3) 259(6) 327.3(0.1)170 320.0(1.6) 303(7) 384.5(0.1)168 454.8(2.1) 428(13) 540.6(0.3) a Ref. [79] presents only statistical errors in the uncertainty —the large uncertainty in the mass shift parameter used incalculation is not propagated through. Propagating theuncertainty from the mass shift parameter leads to anuncertainty of ∼ × − fm for the first row. APPENDIX E: CLOCK TRANSITIONFREQUENCIES
Table XV lists our estimates for the absolute S − P transition frequencies in neutral ytterbium for isotopeswhere it is yet to be measured, together with the knownfrequencies. Our estimates and their uncertainties arebased on the weighted mean of the isotope shift valuespresented in Table IX using Methods 2 and 3, and theexisting absolute transition frequency measurements. TABLE XV. Estimated (this work) and previously measuredclock transition frequencies in Yb i .Isotope Transition frequency (MHz)168 518 297 652.3(3.5)170 518 296 294.7(1.4)171 518 295 836.59086361(13) [83]171 518 295 836.59086371(11) [82]172 518 295 019.7(1.9)173 518 294 576.845268(10) [84]174 518 294 025.3092178(9) [81]176 518 293 076.4(2.7) [1] M. S. Safronova, D. Budker, D. DeMille, D. F. J. Kimball,A. Derevianko, and C. W. Clark, Search for new physicswith atoms and molecules, Rev. Mod. Phys. , 025008(2018).[2] V. A. Dzuba, V. V. Flambaum, and S. 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