The hyperfine anomaly in heavy atoms and its role in precision atomic searches for new physics
aa r X i v : . [ phy s i c s . a t o m - ph ] J a n The hyperfine anomaly in heavy atoms and its rolein precision atomic searches for new physics
B. M. Roberts ∗ and J. S. M. Ginges † School of Mathematics and Physics, The University of Queensland, Brisbane QLD 4072, Australia (Dated: January 26, 2021)We report on our calculations of differential hyperfine anomalies in the nuclear single-particle modelfor a number of atoms and ions of interest for studies of fundamental symmetries violations. Com-parison with available experimental data allows one to discriminate between different nuclear mag-netization models, and this data supports the use of the nuclear single-particle model over thecommonly-used uniform ball model. Accurate modelling of the nuclear magnetization distributionis important for testing atomic theory through hyperfine comparisons. The magnetization distribu-tion must be adequately understood and modelled, with uncertainties well under the atomic theoryuncertainty, for hyperfine comparisons to be meaningful. This has not been the case for a number ofatoms of particular interest for precision studies, including Cs. Our work demonstrates the validityof the nuclear single-particle model for Cs, and this has implications for the theory analysis of atomicparity violation in this atom.
Investigations of atomic parity violation provide someof the most constraining low-energy tests of electroweaktheory [1–4]. These investigations require exquisitely pre-cise measurements of parity-violating transition ampli-tudes [5–7], and equally precise atomic structure calcu-lations [8–11] for their interpretation. Similarly, mea-surements of time-reversal-violating electric dipole mo-ments (EDMs) in atoms and molecules require atomicand molecular structure theory for their interpretationin terms of fundamental charge-parity (CP) violating pa-rameters [12–15]. While such EDMs have eluded de-tection to date, the experimental programs are rampingup and their measurements clamping down on the sizeof these EDMs [16–24]. The implications for new CP-violating models are profound, demanding increasinglyaccurate theory for meaningful constraints and in antic-ipation of non-zero measurements.The magnetic hyperfine structure, which arises due tothe interaction of atomic electrons with the nuclear mag-netic moment, plays an important role in precision stud-ies of violations of fundamental symmetries. The testingand further development of atomic theory depends oncomparisons between calculated and measured quanti-ties that probe the atomic wavefunctions across all lengthscales of the atom. The quantities used for benchmarkinginclude binding energies, electric dipole matrix elements,and hyperfine structure constants. It is the latter that al-lows unique access to the quality of the wavefunctions inthe nuclear region, where the parity-violating and EDMinteractions take place; see, e.g., Ref. [1].Hyperfine structure calculations depend on the mod-elling of nuclear structure effects. In particular, they aresensitive to the distribution of the nuclear magnetic mo-ment across the nucleus, the so-called Bohr-Weisskopfeffect [25, 26]. It has been recognised only recently [27–29] that for a number of heavy atoms of interest thisdependence is much stronger than has been assumed. The effect is so large that in some cases the hyperfinesplitting shifts by more than the claimed atomic the-ory uncertainty when switching from one nuclear mag-netization model to another. Indeed, for Cs and Fr –of particular interest in atomic parity violation studies,and where the claimed atomic structure uncertainty hasreached 0.5% or better – the hyperfine splittings changeby as much as 0.5% and 1.5%, respectively, when a sim-ple nuclear single-particle model is used in place of thewidely-adopted uniform distribution. The ability to testthe validity of these models is therefore critically impor-tant to the field.The most precise atomic parity violation measurementhas been performed for Cs [5], and there are new exper-iments underway in Cs [7] and Fr [30, 31] and interestin studying Ba + , Ra + , and Rb [32–41]. Measurements ofatomic parity violation across a chain of Yb isotopes haverecently been performed [42], and while the consideredratios of measured values do not rely on atomic struc-ture for their interpretation, they strongly depend on theneutron distribution. Systems under recent and ongoingexperimental investigation for detection of EDMs includethe paramagnetic atoms Tl [16], Fr [43], and moleculesYbF [17], BaF [21], and the diamagnetic systems Hg [19],Ra [18], and TlF [24].In this work, we calculate the Bohr-Weisskopf effectand differential hyperfine anomalies for systems of inter-est for precision atomic studies that may be treated assingle-valence-electron atoms or ions and for which thereis experimental data to compare – Rb, Cs, Ba + , Yb + ,Hg + , and Tl. We studied these effects in Fr isotopesin our recent paper in which improved nuclear magneticmoments were deduced [29], and they were studied morerecently in thallium isotopes [44]. The differential hyper-fine anomaly gives the difference in the hyperfine struc-ture for different isotopes of the same atom that arisesdue to finite nuclear size effects. We show that availableexperimental data allows one to distinguish between thedifferent nuclear magnetization models. This data sup-ports the use of the nuclear single-particle model [45–49],rather than the uniformly magnetized ball, for modellingthe Bohr-Weisskopf effect.The relativistic operator for the electron interactionwith the nuclear magnetic moment is h hfs = α µ · ( r × α ) F ( r ) /r , (1)where α is a Dirac matrix, µ = µ I /I is the nuclear mag-netic moment, I is the nuclear spin, and F ( r ) describesthe nuclear magnetization distribution [ F ( r ) = 1 for apointlike nucleus], and α ≈ /
137 is the fine-structureconstant (we use atomic units ~ = | e | = m e = 1, c = 1 /α ).The expectation value of the operator (1) may be ex-pressed as Ah I · J i , where J is the electron angular mo-mentum, and A is the magnetic dipole hyperfine con-stant.The Bohr-Weisskopf (BW) effect arises from the finitenuclear magnetization distribution and gives a significantcontribution to the hyperfine structure [25]. For heavyatoms, it has been standard to model the nucleus as aball of uniform magnetization, such that F Ball ( r ) = ( ( r/r m ) r < r m r ≥ r m , (2)with the nuclear magnetic radius typically taken as r m = p / r rms , where r rms is the root-mean-square(rms) charge radius.A more sophisticated modelling of the magnetizationdistribution, that takes into account the nuclear angularmomenta and configuration, may be given by the sim-ple nuclear single-particle (SP) model [45, 46, 48]. Forodd isotopes, we take the distribution as presented inRef. [48], F I ( r ) = F Ball ( r ) (cid:2) − δF I ln( r/r m ) Θ( r m − r ) (cid:3) , (3)where Θ is the Heaviside step function and δF I = I − I + 1) 4( I + 1) g L − g S g I I I = L + 1 / I + 3)8( I + 1) 4 Ig L + g S g I I I = L − / . (4)Here, I , L , and S are respectively the total, orbital,and spin angular momenta for the unpaired nucleon [48], g L = 1(0) for a proton(neutron), and g I = µ/ ( µ N I ) isthe nuclear g -factor with µ N the nuclear magneton. Thespin g -factor, g S , is chosen so that the experimental valuefor g I is reproduced using the Land´e g -factor expression.Formulae (3), (4) are found by taking the radial part ofthe probability density of the nucleon to be constantacross the nucleus. The model may be improved, e.g., byfinding the nucleon wavefunction in a Woods-Saxon po-tential and including the spin-orbit interaction [49]. The effect of accounting for these has been shown to be small( . Rb,
Cs,and
Fr [27], as well as in isotopes of Tl [44], and largerin Ba + and Ra + [27]. The single-particle modelmay be extended in a simple way to describe the mag-netization distribution of doubly-odd (odd proton, oddneutron) isotopes [29, 47, 50].The BW effect may be parameterized as [51] A = A (1 + ǫ ) , (5)where A is the hyperfine constant with a pointlikemagnetization distribution ( F = 1). Here, A includesthe Breit-Rosenthal correction, δ , due to the finite nu-clear charge distribution, which is taken into accountby solving the electron wavefunctions in the field of afinite nucleus (we use a Fermi distribution with rmscharge radii from Ref. [52]). This may be expressed as A = A (1 + δ ), where A is the hyperfine constantwith pointlike nuclear magnetic and charge distributions.Since the nuclear charge distribution is known with rel-atively high accuracy, errors associated with the Breit-Rosenthal correction are typically negligible [29, 53, 54].Note that radiative quantum electrodynamics (QED)corrections contribute to the hyperfine structure withcomparable size to ǫ [27, 49, 55], though they are largelyindependent of the isotope and therefore mostly cancelin the differential hyperfine anomaly considered below.Therefore, we don’t consider QED contributions further.We calculate A using the relativistic Hartree-Fockmethod, including the important core-polarization con-tribution by means of the time-dependent Hartree-Fock(TDHF) method [59, 60], equivalent to the randomphase approximation with exchange (RPA). We consideratoms with a single valence electron above a closed-shellcore, for which the valence wavefunction is found in theHartree-Fock potential due to the ( N −
1) core electrons( N = Z for neutral atoms). The set of TDHF equations,( H − ε c ) δψ c = − ( h hfs + δV − δε c ) ψ c , (6)is then solved self-consistently for each electron in thecore. Here, H , ψ c , and ǫ c are the relativistic Hartree-Fock Hamiltonian, core electron orbitals, and core elec-tron binding energies, respectively, and δψ c and δε c arehyperfine-induced corrections for core orbitals and en-ergies. The resulting hyperfine-induced correction to theHartree-Fock potential is given by δV . Since h hfs can mixstates with different angular momenta, δψ c is not an an-gular momentum eigenstate and contains contributionsfrom states with j = j c , j c ± j c is the angular momen-tum of single-electron state c ). Then, matrix elementsfor valence states, v , are calculated as h v | h hfs + δV | v i ,which includes the core-polarization effects to all-ordersin the Coulomb interaction [60]. Correlation correctionsto hyperfine structure were studied recently by us in de-tail [61], and those beyond core polarization were shown TABLE I. Bohr-Weisskopf corrections, ǫ , and hyperfine anomalies, ∆ , calculated in the ball and single-particle (SP) nuclearmagnetization models for the lowest states of several atoms of interest, and comparison with experimental differential anomalies. A is the atomic mass number for the isotope, I π is the nuclear spin and parity.Isotope 1 Isotope 2 Differential anomaly ∆ (%) A I π ǫ Ball (%) ǫ SP (%) A I π ǫ Ball (%) ǫ SP (%) Ball SP Expt. [51] Rb 5 s /
85 5/2 − − .
306 0 .
044 87 3/2 − − . − . − .
001 0 .
323 0 . − − . − .
139 0 .
000 0 .
183 0 . Ag 5 s /
107 1/2 − − . − .
20 103 7/2 + − . − . − . − . − . − − . − .
78 0 . − . − . Cs 6 s /
133 7/2 + − . − .
209 131 5/2 + − . − . − .
001 0 .
389 0 . a
135 7/2 + − . − .
247 0 .
002 0 .
039 0 . b
134 4 + − . − .
371 0 .
000 0 .
163 0 . Ba + s /
135 3/2 + − . − .
03 137 3/2 + − . − .
03 0 .
001 0 . − . Yb + s /
171 1/2 − − . − .
41 173 5/2 − − . − .
79 0 . − . − . Au 6 s /
197 3/2 + − .
97 15 . + − .
97 7 .
47 0 .
013 7 .
48 3 . Hg + s /
199 1/2 − − . − .
57 201 3/2 − − . − .
20 0 . − . − . Tl 7 s /
203 1/2 + − . − .
13 205 1/2 + − . − .
13 0 .
015 0 .
015 0 . c81 Tl 6 p /
203 1/2 + − . − .
780 205 1/2 + − . − .
781 0 .
005 0 .
005 0 . a Ref. [56], Ref. b [57], Ref. c [58]. not to be important for the relative BW effect (see alsoRef. [27, 28, 62]). The insensitivity of the relative BWeffect to correlations is due to the short-range nature ofthe effect, with account of correlations affecting the nor-malization of the wave functions which largely factors outin the relative correction.The differential hyperfine anomaly, ∆ , is defined viathe ratio of the hyperfine constants for different isotopesof the same atom (see, e.g., [51]): A (1) A (2) = g (1) I g (2) I (cid:0) ∆ (cid:1) . (7)This quantity, which is a measure of the deviation ofthe hyperfine structure from the case of a pointlike nu-cleus, may be found with high accuracy from experiment,provided the nuclear magnetic moments are known welland determined independently of the hyperfine measure-ments [51]. As for the theoretical determination of ∆ ,the correlation corrections beyond RPA that contributeto the hyperfine constants A (1) and A (2) cancel in the ra-tio [61, 62], making the electronic structure calculationsrobust at the level of RPA and of high accuracy. Notethat there is a strong cancellation of the Breit-Rosenthalcorrections, δ (1) − δ (2) , in the differential anomaly, andtypically the differential hyperfine anomaly is stronglydominated by the differential Bohr-Weisskopf effect [51], ∆ ≈ ǫ (1) − ǫ (2) . (8)Therefore, comparison of calculated and measured hy-perfine anomalies presents a powerful test of the validityof nuclear magnetization models.In Table I we present our results for the BW effectsand differential hyperfine anomalies obtained using the ball (2) and single-particle (3) models. The numericalaccuracy for the BW calculations is better than 1%, wellbelow the model uncertainty. We present results for thelowest states of systems of interest for atomic parity vi-olation and electric dipole moment studies, and we alsopresent results for Ag and Au, which may be treated assingle valence electron systems and for which the BW ef-fects and hyperfine anomalies are particularly large. Theanomaly is calculated using Eq. (7) rather than Eq. (8),which means that the small differential Breit-Rosenthaleffect is included. The calculated values for ∆ are com-pared against available experimental data [51]. Note thatthe uncertainties in the measured values are dominatedby uncertainties in the nuclear magnetic moments [63].We draw attention to several points. Firstly, the BWcorrection is a significant effect, typically entering at thelevel of several 0 .
1% to several 1% for the considered sys-tems. For Ag and Au the effect is even larger, contribut-ing at around 10% for Au. Secondly, the ball and single-particle models often lead to substantially different BWeffects. For Rb and
Cs, the difference is as large as0 .
4% and 0 . Yband
Hg, this difference is 1.0% and 1.5%. On the otherhand, for
Tl and
Tl, the two models give the sameBW effect due to their nuclear spin-parity being 1 / + .Finally, from a comparison of the calculated and mea-sured hyperfine anomalies in Table I, it is seen that thenuclear SP model leads to substantially better agreementwith experiment for the majority of cases. The agree-ment is particularly good for isotopes of Rb and Cs. Forexample, for Cs the SP model gives ∆ = 0 . ∆ = 0 . ∆ = 0 . . . . + and Au, including reproducingthe (atypical) sign of the effect for the latter. Our calcu-lations for Au are in good agreement with previous cal-culations [64, 65], though we use a simplified model (seealso [66–68]). In the ball model, the only difference in theBW effect between isotopes comes from changing the nu-clear radius, similarly to the differential Breit-Rosenthaleffect, so the calculated anomaly is always small and themodel generally cannot produce the observed anomalies.Recently, we considered the case of Fr in detail, andwe demonstrated using “double” differential hyperfineanomalies [69] that the single-particle model works verywell for both odd and doubly-odd isotopes between 207–213 [29] (see also Refs. [70–72]). This allowed us to ex-tract nuclear magnetic moments for these isotopes withsignificantly higher precision than was previously possi-ble. The BW effect is particularly large for these Fr iso-topes ( ∼ Pb and
Bi: the calculated and experimentalBW results for Pb are − .
55% and − . Bi are − .
07% and − . Ra + , which was calculated in Ref. [27]. Gen-erally, the size of atomic structure uncertainties precludesthe direct extraction of the BW effect from hyperfinecomparisons in many-electron systems. For this system,however, the effect is so large that direct extraction ispossible. This was done in Ref. [73] where the value ǫ = − .
7% was obtained, which may be compared to thesimple SP (and ball) result ǫ = − .
8% and to the moresophisticated SP result with the nucleon wave functionfound in the Woods-Saxon potential and with spin-orbitinteraction included, ǫ = − .
3% [27].The simple nuclear single-particle model does not al-ways work well. This may be seen from Table I for Ba + ,Hg + , and Tl. For the considered isotopes of Hg + , the SPmodel produces a differential hyperfine anomaly that issignificantly larger than the observed value. For isotopesof Ba + and Tl, the nuclear states are the same, and theSP model produces very similar BW effects which cancelstrongly in the anomaly (8). In this case, the neglectednuclear many-body contributions will be more importantfor the differential anomaly than for the BW effect. An-other reason for the discrepancy that appears in the hy-perfine anomaly may arise due to the magnetic radius. Inour calculations we have taken the magnetic rms radius tobe the same as the charge rms radius, though there is no reason for them to be the same. Indeed, there are indica-tions that the magnetic radii for Tl and
Tl are dif-ferent from one another and from the charge radius [74–76]. It was shown very recently that the nuclear single-particle model outperforms the ball model for several Tlisotopes with different nuclear states [44], and the BWeffects extracted from experiments with , Tl arein good agreement with nuclear SP calculations [49, 74].For Ba + , we can look to the Cs differential anomaliesinvolving Cs, with the same neutron configuration as Ba + ; coincidence of the SP and experimental results(Table I) lends support to the validity of the model, whichdescribes the magnetization distribution of the unpairedneutron. While the nuclear single-particle model doesnot always give differential anomalies in good agreementwith experiment, it generally performs better than theball model across the board and is expected to producemore accurate values for the BW effect.We now consider the implications for atomic parity vi-olation studies. The dominant parity violating effects inatoms arise due to the exchange of neutral weak bosonsbetween atomic electrons and the nucleus, leading tothe mixing of atomic states of opposite parity; see, e.g.,Ref. [1]. Such interactions are localized on the nucleus,and the theoretical evaluation of the relevant matrix el-ements, e.g., h s / | ˆ H PV | p / i , therefore depends on pre-cise knowledge of the wavefunctions in this region. Thesematrix elements cannot be directly compared to exper-iment, and information about the accuracy of the wavefunctions and the matrix elements is found from a sur-vey of the deviations between theory and experiment forthe hyperfine constants of the relevant states and for thecombination √A s A p , which is considered to give a morereliable indication of the accuracy for the off-diagonalmatrix elements [1, 8].Separating out the Bohr-Weisskopf contribution, therelevant quantity becomes √A s A p ≈ q A s A p h ǫ s + ǫ p ) / i . (9)The claimed accuracy of the most precise atomic parityviolation calculations for Cs was based in part on de-viations of the hyperfine constants and the quantity (9)from experiment, where the nuclear magnetization distri-bution was treated as uniformly distributed (ball model).Correcting the nuclear magnetization model (to single-particle) leads to a significant change in the hyperfineconstants for s states by +0 . p / states, the BWeffect is an order of magnitude smaller and the hyper-fine constants change by only 0 . . s stateand the associated quantity (9) in Ref. [10], while theresults of the work [8] are hardly changed when account-ing also for QED contributions, which is consistent withthe 0.5% claimed uncertainty for the atomic parity vio-lation calculation in that work [8, 9]; see the analysis inRef. [28]. This illustrates the importance of understand-ing and controlling the nuclear magnetic structure, forboth reliable benchmarking and continued developmentof precision atomic theory, and for assigning atomic the-ory uncertainty which has ramifications for constraintson new physics.In summary, hyperfine comparisons form an importantpart of the atomic theory analysis in atomic tests of fun-damental physics, most notably atomic parity violationand EDM studies in atoms and molecules. These com-parisons are only reliable if the nuclear magnetizationdistribution is adequately understood and modelled, withuncertainties well under the atomic theory uncertainty.This has not been the case for a number of atoms of par-ticular interest for precision studies, including Cs. In thiswork, we point out that sufficient experimental data ex-ists for many isotopes of interest to be able to test nuclearmagnetization models using hyperfine anomalies. Froma study of the hyperfine anomalies, we demonstrate thatthe single-particle model generally outperforms the nearuniversally-used ball model. It is simple enough to in-clude into atomic structure codes without the need forany sophisticated nuclear calculations, and we advocateits use in future studies. These investigations into hyper-fine anomalies open a new window for probing nuclearstructure, including the neutron distribution. Acknowledgments
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