Effect of nuclear magnetization distribution within the Woods-Saxon model: Hyperfine splitting in neutral Tl
aa r X i v : . [ phy s i c s . a t o m - ph ] J a n Nuclear magnetization distribution effect within the Woods-Saxon model: hyperfinesplitting in neutral Tl
S.D. Prosnyak
1, 2, ∗ and L.V. Skripnikov
1, 2, † Petersburg Nuclear Physics Institute named by B.P. Konstantinov of NationalResearch Centre “Kurchatov Institute”, Gatchina, Leningrad District 188300, Russia Saint Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg, 199034 Russia (Dated: 04.01.2021)Three models of the nuclear magnetization distribution are applied to predict the hyperfine struc-ture of the hydrogen-like heavy ions and neutral thallium atoms: the uniformly magnetized ballmodel and single-particle models for the valence nucleon with the uniform distribution and distribu-tion determined by the Woods-Saxon potential. Results for the hydrogen-like ions are in excellentagreement with previous studies. The application of the Woods-Saxon model is now extended to theneutral systems with the explicit treatment of the electron correlation effects within the relativisticcoupled cluster theory using the Dirac-Coulomb Hamiltonian. We estimate the uncertainty for theratio of magnetic anomalies and numerically confirm its near nuclear-model independence. Theratio is used as a theoretical input to predict the nuclear magnetic moments of short-lived thalliumisotopes. We also show that the differential magnetic anomalies are strongly model-dependent. Theaccuracy of the single-particle models significantly surpasses the accuracy of the simplest uniformlymagnetized ball model for the prediction of this quantity. In Ref. [L.V. Skripnikov, J. Chem. Phys. , 114114 (2020)] it has been shown that the Bohr-Weisskopf contribution to the magnetic dipolehyperfine structure constant for an atom or a molecule induced by a heavy nucleus can be factorizedinto the electronic part and the universal nuclear magnetization dependent part. We numericallyconfirm this factorization for the Woods-Saxon single-particle model with an uncertainty less than1%.
INTRODUCTION
The hyperfine splitting in atomic spectra is of great in-terest for many physical applications. From a comparisonof the theoretical and experimental values of the hyper-fine structure (HFS) constants, one can test the accuracyof the electronic structure methods for atoms [1–5] andmolecules [6–14]. Such electronic calculations are neces-sary to extract the value of the electric dipole moment ofthe electron and other fundamental constants and prop-erties from the experimental data [1, 5, 15–18]. Usingthe results of calculations and experimental data, it ispossible to obtain the magnetic moments of short-livednuclei [19–24]. The obtained values can be used for thedevelopment of the nuclear structure theory.In order to reproduce the experimental results for hy-perfine splitting with an uncertainty of an order of 1%,it is necessary to take into account both the finite chargedistribution over the nucleus, the Breit-Rosenthal (BR)effect [25, 26], and the finite nuclear magnetization distri-bution, the Bohr-Weisskopf (BW) effect [27–29]. In stud-ies of neutral atoms, the uniformly magnetized ball modelis widely used to calculate the BW correction [23, 30–33].The only parameter of this model is the radius of theball, R M . Therefore, it can be not equal to the chargeradius to reproduce the experimental value of the hyper-fine splitting [23, 34], which raises questions about thephysical meaning of such a model.In this paper we study a more accurate and more phys-ical single-particle (SP) nuclear magnetization distribu-tion model. In this model it is assumed that the nu- clear magnetic moment is induced by one unpaired nu-cleon, that has both the orbital motion and the spin. Weconsider two approximations for the density of the un-paired nucleon. In the Woods-Saxon (WS) single-particlemodel, the wavefunction of this nucleon is obtained as asolution of the Schr¨odinger equation with the WS poten-tial [35]. In the second single-particle model the uniformdistribution (UD) of the valence nucleon is assumed. Inthe case of zero orbital momentum of the valence nucleon,this model is equivalent to the model of the uniformlymagnetized ball.For a point nuclear model, the ratio of the hyperfinesplittings of two different isotopes 1 and 2 is proportionalto the ratio of the nuclear g -factors of the isotopes. How-ever, this is not the case for the finite-size nucleus modeldue to the BR and BW effects. Corresponding correction, ∆ , is called the nuclear magnetic hyperfine anomaly: ∆ = A g A g − , (1)where A and A are HFS constants (see Eq. (2) below)for a given electronic state, g and g are the nuclear g -factors of considered isotopes and . The ratio ofmagnetic anomalies is a key theoretical input to obtainthe magnetic moments of short-lived isotopes [19–22].In the present paper we apply the WS model to predictthe contribution of the BW effect to the hyperfine struc-ture of the neutral Tl atom in the ground and the firstexcited electronic states. As far as we know, this modelhas not been previously used to calculate the hyperfinestructure of the neutral thallium atom with the explicitand direct treatment of the electron correlation effects.Results are compared with the values obtained within theuniformly magnetized ball model [23]. Next we comparepredictions for the ratio of hyperfine magnetic anomaliesand the differential hyperfine anomaly within differentmodels. For the former we verify its near model inde-pendence and use it deduce the values of the short-livedisotopes of thallium. For the latter we show that the SPmodels give far better results than the simple uniformlymagnetized ball model. Finally, we numerically check thefactorization of the BW contribution into the electronicand nuclear magnetization distribution dependent parts,introduced in Ref. [14] for the WS model. THEORY
In the point magnetic dipole approximation the HFSconstant A (0) for the atomic electronic state Ψ JM J withthe total electronic momentum J and its projection M J on the axis z can be calculated using the following ex-pression: A (0) = µI · M J h Ψ JM J | [ r el × α ] z r el | Ψ JM J i , (2)where µ is the value of the nuclear magnetic dipole mo-ment, I is the nuclear spin, α are Dirac matrices, r el isthe electron radius-vector. The electronic wavefunction Ψ JM J is calculated assuming the finite nuclear chargedistribution. This means that the Breit-Rosenthal effectis considered non-perturbatively and included in A (0) . Inthis case, the expression for the hyperfine splitting con-stant has the following form: A = A (0) − A BW = A (0) (1 − ε ) , (3)where A BW is the Bohr-Weiskopf contribution and ε isthe relative Bohr-Weiskopf correction. One can also addthe quantum electrodynamics contribution A QED to theright side of Eq. (3).In this paper we consider the SP nuclear magnetizationdistribution models in which the nuclear magnetization isgenerated by a single valence nucleon. In the WS modelof the nucleus, the wavefunction of the valence nucleonis determined as a solution of the Schr¨odinger equationwith the Woods-Saxon potential [35, 36]: U ( r ) = V ( r ) + V C ( r ) + V SO ( r ) , (4)where V ( r ) = − V e ( r − R ) /a , (5) V C ( r ) = (cid:26) α ( Z − /r r > R C α ( Z − − r /R C ) / R C r R C , (6) V SO ( r ) = λ ( ℏ m p c ) r ddr V e ( r − R SO ) /a σσσ · lll. (7) R C = p / h r c i / is the nuclear charge radius and h r c i / is the rms charge radius. Parameters of the po-tential R , R SO , a , V , λ are listed in Table I. If thevalence nucleon is the neutron then the Coulomb term V C should be omitted. TABLE I. Parameters of the WS potential from Ref. [36].The radii are calculated as R = r A / and R SO = r SO A / ,where A is the mass number. r (fm) r SO (fm) a (fm) V (MeV) λ proton 1.275 0.932 0.70 58.7 17.8neutron 1.347 1.280 0.70 40.6 31.5 The BW correction ε can be written as follows [27, 28,37]: ε = g S g I (cid:20) I h K S i + (2 I − I ( I + 1) h K S − K L i (cid:21) + g L g I (cid:20) (2 I − I h K L i + (2 I + 1)4 I ( I + 1) h φ SO r K L i (cid:21) (8)for I = L + 1 / , and ε = g S g I (cid:20) − I + 1) h K S i − (2 I + 3)8 I ( I + 1) h K S − K L i (cid:21) + g L g I (cid:20) (2 I + 3)2( I + 1) h K L i − (2 I + 1)4 I ( I + 1) h φ SO r K L i (cid:21) (9)for I = L − / . Here φ SO is the radial part of thespin–orbit interaction V SO = φ SO σσσ · lll , g I is the g -factor ofconsidered nucleus. For the valence proton we set g L = 1 ,for the valence neutron we set g L = 0 . g S is obtainedfrom the following equations: µµ N = 12 g S + (cid:20) I −
12 + 2 I + 14( I + 1) m p h h φ SO r i (cid:21) g L (10)for I = L + 1 / and µµ N = − I I + 1) g S + (cid:20) I (2 I + 3)2( I + 1) − I + 14( I + 1) m p h h φ SO r i (cid:21) g L (11)for I = L − / . h K S i and h K L i are obtained by aver-aging functions K S ( r ) and K L ( r ) over the density of thevalence nucleon | u ( r ) | : h K S,L i = Z ∞ K S,L ( r ) | u ( r ) | r dr. (12)Functions K S ( r ) and K L ( r ) in the case of a hydrogen-like ion have the following form: K S ( r ) = Z r f g dr el Z ∞ f g dr el , (13) K L ( r ) = Z r (1 − r el /r ) f g dr el Z ∞ f g dr el , (14)where g and f are the radial parts of the Dirac wave-function of the electron. For the s ground state of thehydrogen-like ion, the following approximate expressionscan be used [27, 38] K S ( r ) = b (cid:20) a (cid:16) rR C (cid:17) + a (cid:16) rR C (cid:17) + a (cid:16) rR C (cid:17) (cid:21) , (15) K L ( r ) = 3 b (cid:20) a (cid:16) rR C (cid:17) + a (cid:16) rR C (cid:17) + a (cid:16) rR C (cid:17) (cid:21) . (16)The expansion coefficients b and a i can be found inRef. [38].In the approximation of a uniformly distributed va-lence nucleon, the density of the valence nucleon has thefollowing form: | u ( r ) | = 3 R C θ ( R C − r ) , (17)where θ ( R C − r ) is the Heaviside step function: θ ( R C − r ) = ( , if r < R C ;0 , if r > R C . (18)Note, that for this model the terms with the spin-orbitinteraction in Eqs. (8), (9) should be omitted.Hyperfine magnetic anomalies (1) can be used to deter-mine the magnetic moments of short-lived isotopes [20–22, 24]. We denote stable and short-lived isotopes by 1and 2, respectively. Using the experimentally obtainedHFS constants A and A for a given electronic state b , the magnetic moment of the stable isotope, µ , andhyperfine magnetic anomaly one can determine the mag-netic moment µ of the short-lived isotope: µ = µ · A [ b ] A [ b ] · I I · (1 + ∆ [ b ]) . (19)A direct calculation of the anomaly ∆ [ b ] is quite dif-ficult due to a strong dependence of the result on thechoice of the nuclear model. However, the ratio of theanomalies k [ a, b ] = ∆ [ a ] / ∆ [ b ] (20)for two electronic states a and b turns out to be fairlystable, which we verify below. Using this fact, it is pos-sible to extract the desired nuclear magnetic moment ofa short-lived isotope. For this, it is necessary to knowthe magnetic moment of a stable isotope, as well as thehyperfine constants A , [ a ] and A , [ b ] for the electronic states a and b of the nuclei under consideration. For con-venience, we introduce the so-called differential hyperfinemagnetic anomaly θ [ a, b ] [19, 22]: θ [ a, b ] = A [ a ] A [ b ] A [ a ] A [ b ] − ∆ [ a ]1 + ∆ [ b ] − . (21)The important feature of θ [ a, b ] is that it is independenton the magnetic moments and spins of the nuclei underconsideration. As it can be seen from Eq. (21), θ [ a, b ] can be determined using only the experimental values ofthe hyperfine constants. Substituting the ratio of hyper-fine magnetic anomalies into Eq. (21), we find [20–22]: ∆ [ b ] = θ [ a, b ] k [ a, b ] − θ [ a, b ] − . (22)One can put ∆ [ b ] into Eq. (19) to finally obtain thedesired nuclear magnetic moment. Below we explore themodel dependence of both the ratio of hyperfine magneticanomalies k [ a, b ] and the differential magnetic anomaly θ [ a, b ] . CALCULATION DETAILS
The values of the charge radii of the stable nuclei weretaken from Ref. [39]. The charge radii of the short-livedthallium isotopes have been taken from Ref. [40]. Nu-clear magnetic moments listed in Table II were takenfrom Ref. [41] for stable nuclei and Ref. [23] for short-lived thallium isotopes. WS potential parameters were
TABLE II. Employed parameters of the nuclei: valence nu-cleon state, nuclear magnetic dipole moments [23, 41–44] andcharge radii [39, 40]. In the square brackets the values of themagnetic moments used in previous papers are given; theyhave been revisited in recent papers [42–44].Nucleus State µ I /µ N h r c i / (fm) Re a d / Tl m h / Tl m h / Tl 3 s / Tl 3 s / Pb a p / Bi a h / taken from Ref. [36] and are listed in Table I.To obtain the nucleon wavefunction in the WS modelthe radial Schr¨odinger equation has been solved on thegrid using the code developed in the present paper. Cal-culated radial probability densities of a valence nucleonfor different isotopes are shown on Fig. 1. The electronicwavefunction for the hydrogen-like ions have been ob-tained by the numerical solution of the Dirac equationusing the Gaussian-type basis set. This basis set includes s-type functions, with exponential parameters forming ⟩ R a d i a l ⟨ p r o b a b ili t y ⟨ r | u ⟩ r ) | Tl Re Bi Pb FIG. 1. Calculated radial probability densities of the valencenucleon for various nuclei. The densities for
Tl and
Tlisotopes coincide with rather high accuracy and are indicatedby a single line. a geometric progression. The common ratio of this pro-gression is 1.8, and the largest element is · .In HFS calculations of the neutral thallium atom, theQED effects were not taken into account. Atomic orbitalsfor subsequent correlation calculations were obtained us-ing the Dirac-Hartree-Fock (DHF) method, where theFock operator is determined by averaging electronic shellconfigurations over p j =1 / and p j =3 / for P / , P / electronic states. For the S / electronic state the av-eraging has been performed over the s j =1 / configu-ration. The main correlation calculations that includeall 81 electrons have been performed using the coupledcluster method with single, double, and perturbativetriple amplitudes, CCSD(T) [45, 46] within the Dirac-Coulomb Hamiltonian. In these calculations the uncon-tracted Dyall’s AAE4Z basis set [47] augmented with one h − and one i − type functions was used. It includes35s − , 32p − , 22d − , 16f − , 10g − , 5h − and 2i − type func-tions. For the calculation virtual orbitals were truncatedat the energy of 10000 Hartree. The importance of thehigh energy cutoff for properties dependent on the be-havior of the wavefunction close to the heavy-atom nu-cleus has been demonstrated in Refs. [10, 18]. In Ta-bles below we also include corrections on the basis setsize extension, high-order correlation effects beyond theCCSD(T) level and the Gaunt interaction contributionfrom Ref. [23]. The basis set correction has been calcu-lated within the CCSD(T) method using the extendedbasis set that includes s, p, d, f, g, h, i basisfunctions. s − d electrons were excluded from the corre-lation treatment and the virtual orbitals were truncatedat the energy of 150 Hartree in these calculations. Calcu-lations of the contributions of correlation effects beyondthe CCSD(T) model have been performed within the cou-pled cluster with single, double, triple and perturbative quadruple amplitudes, CCSDT(Q), method [48–50]. Inthis calculation we have used the SBas basis set that con-sists of 30 − s, 26p − , 15d − , 9f − type functions and corre-sponds to the Dyall’s CVDZ [51, 52] basis set augmentedby diffuse functions. As in the case of basis set correc-tion calculation, s − d electrons were excluded fromthe correlation treatments. Contribution of the Gauntinteraction has been calculated within the SBas basis setusing the CCSD(T) method. In this calculation, all elec-trons were correlated and all virtual orbitals within agiven basis set were considered. Correlation calculationshave been performed using the finite-field technique. Forrelativistic coupled cluster calculations the dirac15 [53]and mrcc codes [49, 50, 54] were used. The code devel-oped in Ref. [55] was used to calculate the HFS integralsin the approximation of a point magnetic dipole. Thecode for calculating the BW matrix elements in the WSmodel has been developed in the present paper. RESULTS AND DISCUSSION
To test the developed approach, the HFS constants ofhydrogen-like ions were calculated. The obtained valuesare given in Table III and compared with the previousstudies [56, 57]. A slight difference between the presentand the previous results can be explained by a differentnuclear charge model. In the present calculations theGaussian charge distribution model [58] has been used,while in the previous the Fermi distribution has beenemployed. The Gaussian charge distribution model iswidely used in the molecular calculations of HFS.Table III contains also the BW correction extractedfrom the experimental values of the HFS constants A exp [59, 60] using the following expression: ε exp = 1 − ( A exp − A QED ) µI · M J h Ψ JM J | [ r el × α ] z r el | Ψ JM J i . (23)For calculation of the denominator, we used the datafrom Ref. [56] and the latest values of the nuclear mag-netic moments. QED contributions A QED were takenfrom Refs. [56, 61, 62]. Note that there is a small depen-dence of the BW correction calculated in the SP modelsdue to the dependence of parameter g S on the magneticmoment value, see Eqs. (8)-(11). Therefore, to be ableto compare with previous calculations of the BW cor-rection for H-like ions we used the same values of themagnetic moments that have been used in the previouspapers. However, to obtain the ε exp values the revisitednuclear magnetic moment values [42–44] have been used(see Table II). One can see from Table III that the sim-plified Eqs. (15) and (16) give very good approximationto more accurate Eqs. (13) and (14).Tables IV and V give the values of calculated HFS con-stants for the neutral Tl atom in the ground electronic
TABLE III. Calculated values of the BW correction ε (in %) using the WS model for various hydrogen-like ions in the groundelectronic state s .Author, reference Re
74+ 203 Tl
80+ 205 Tl
80+ 207 Pb
81+ 209 Bi Shabaev et al. [56], Eqs. (15), (16), without SO 1.20 1.77 1.77 4.19 1.33Shabaev et al. [56], Eqs. (15), (16), with SO 1.22 1.79 1.79 - 1.18Gustavsson et al. [57] 1.18 1.74 1.74 4.29 1.31This work, Eqs. (15), (16), without SO 1.20 1.78 1.78 4.44 1.29This work, Eqs. (15), (16), with SO 1.22 1.80 1.79 4.47 1.17This work, Eqs. (13), (14), without SO 1.30 1.87 1.87 4.43 1.43This work, Eqs. (13), (14), with SO 1.32 1.89 1.89 4.45 1.30Experiment 1.35 2.21 2.23 3.81 1.02 state P / and the first excited state P / , respectively.In the last column, the values of HFS constants with BWcontributions calculated within the WS model of the nu-clear magnetization distribution are given. They wereobtained using Eqs. (13) and (14) for one-electron ma-trix elements. For comparison, we also provide resultsobtained within the point magnetic dipole approxima-tion (the second column) and uniformly magnetized ballmodel from Ref. [23] (the third column). One can seefrom Tables IV and V a reasonable agreement betweenthe HFS constants calculated in the ball model and inthe WS model for Tl. The theoretical uncertainty ofelectronic structure calculation in [23] was estimated as1% for P / and about 10% for P / . One can see verygood agreement of the theoretical prediction of the HFSconstant for the P / state with the experimental value, . MHz. A reasonable agreement between thetheoretical value of the HFS constant for the P / stateand the experimental value, . MHz, is ob-tained. It can be noted that the WS model also predictslarge relative BW correction for this state (see a detaileddiscussion in Ref. [23]).
TABLE IV. Calculated values of the HFS constant of the P / state of Tl (in MHz) using different levels of elec-tronic theory and nuclear models. The numbers in the firstline in columns 2 and 3 indicate the ratio of the model mag-netic radius and the charge radius R M /R C . The values of theBW contributions, − A BW , are given in ()-brackets.Method 0 [23] 1.0 [23] WSDHF 18805 18681 18696(-124) (-109)CCSD 21965 21807 21826(-158) (-139)CCSD(T) 21524 21372 21390(-152) (-134)+Basis corr. -21 – –+CCSDT-CCSD(T) +73 – –+ CCSDT(Q)-CCSDT -5 – –+Gaunt -83 – –Total a a Instead of missing corrections, the contributions calculatedfor the point magnetic dipole moment model given in thefirst column were used. TABLE V. Calculated values of the HFS constant of the P / state of Tl (in MHz) using different levels of electronictheory and nuclear models. The numbers in the first linein columns 2 and 3 indicate the ratio of the model magneticradius and the charge radius R M /R C . The values of the BWcontributions, − A BW , are given in ()-brackets.Method 0 [23] 1.0 [23] WSDHF 1415 1415 1415CCSD 6 40 36(+34) (+30)CCSD(T) 244 273 269(+29) (+25)+Basis corr. +4 – –+CCSDT-CCSD(T) -49 – –+ CCSDT(Q)-CCSDT +14 – –+Gaunt +1 – –Total a
214 243 239 a Instead of missing corrections, the contributions calculatedfor the point magnetic dipole moment model given in thefirst column were used.
Table VI presents the values of calculated ratios of hy-perfine magnetic anomalies k x [7 S / , P / ] , where x is Tl, Tl m or Tl m . Results are given at differentlevels of the electronic structure theory for three modelsof the magnetization distribution: the uniformly magne-tized ball model [23], UD and WS single particle mod-els. In the former model the ball radius is equal to thecharge radius. The obtained values are in fairly goodagreement. This numerically justifies the assumed nearmodel-independence of such ratio. Thus, a theoreticalcalculation of the ratio of hyperfine magnetic anomaliesfor a pair of electronic states can be used to determine themagnetic moments of short-lived isotopes. It should benoted that for stable isotopes, the charge and magnetiza-tion distribution effects give comparable contributions tothe anomalies, and hence to their ratio. However, for thecase of isotopes having different states of the valence nu-cleon, the main contribution to the anomaly comes fromthe BW effect.Table VII gives the values of the differential hyper-fine magnetic anomalies θ x [7 S / , P / ] defined byEq. (21), where x is Tl, Tl m or Tl m . As in theprevious case, three nuclear magnetization distribution TABLE VI. The ratio of magnetic hyperfine anomalies k x [7 S / , P / ] , where x is Tl, Tl m or Tl m . Forthe Ball and UD models the magnetic rms radius was set tobe equal the experimental rms charge radius.Nucleus Method Ball [23] UD WSDHF 3.77 3.77 3.85 Tl CCSD 3.38 3.38 3.44CCSD(T) 3.47 3.47 3.54DHF 3.73 3.55 3.54 Tl m CCSD 3.36 3.23 3.22CCSD(T) 3.45 3.32 3.31DHF 3.74 3.55 3.54 Tl m CCSD 3.36 3.23 3.22CCSD(T) 3.46 3.32 3.31 models have been used: the simplest uniformly magne-tized ball model and two single particle models: UD andWS. The obtained values of the differential anomaly for Tl m and Tl m isotopes are slightly smaller than theestimate θ x ( I =9 / [7 S / , P / ] = − . · − fromRef. [22]. This can be explained by the fact that inRef. [22] the effective value of the orbital g-factor of thevalence nucleon g L = 1 . from paper [63] has been used.In our calculations, the value g L = 1 . has been used. Forcomparison, we have also performed calculations at theDHF level using the effective value of g L from paper [63]and corresponding value g S derived using Eqs. (10), (11).We estimate: θ x ( I =9 / [7 S / , P / ] = − . · − and θ x ( I =9 / [7 S / , P / ] = − . · − for the UDand WS models of magnetization distribution, respec-tively. TABLE VII. The differential magnetic hyperfine anomalies θ x [7 S / , P / ] , where x is Tl, Tl m or Tl m , − .For the Ball and UD models the magnetic rms radius was setto be equal the experimental rms charge radius. The experi-mental values [22, 64, 65] are given in the last column.Nucleus Method Ball [23] UD b WS ExperimentDHF -1.09 -1.09 -0.86 Tl CCSD -1.05 -1.05 -0.83 -1.9(8)CCSD(T) -1.06 -1.06 -0.84DHF -5.14 -93 -69 Tl m CCSD -4.92 -90 -66 -129(62)CCSD(T) -4.98 -91 -67DHF -6.06 -96 -72 Tl m CCSD -5.80 -92 -69 -154(60)CCSD(T) -5.87 -93 -70
As one can see from Table VII, the dependence of thedifferential magnetic anomaly on the level of the includedelectronic correlation effects is slightly smaller than inthe case of the ratio of the magnetic anomalies. In caseof the differential magnetic anomaly θ [7 S / , P / ] theoretical and experimental values are of the same orderof magnitude. However, for short-lived isotopes Tl m and Tl m , SP models give much more accurate results than the model of a uniformly magnetized ball. Thiscan be explained by the fact that the Tl and
Tlthallium isotopes have the same valence nucleon state s / with zero orbital momentum (see Table II). In thiscase the uniformly magnetized ball model reduces to thesingle particle UD model. This is not the case for short-lived isotopes Tl m and Tl m with the valence nu-cleon state having nonzero orbital momentum. Thus, itfollows from Table VII that it is important to use of morecomplex nuclear magnetization distribution models thanthe simplest uniformly magnetized ball model.It has been shown in Ref. [14] that the BW contribu-tion to the hyperfine structure constant of an atom ora molecule induced by a heavy nucleus can be factorizedinto the electronic part, E , and the nuclear magnetizationdistribution dependent part, N , with very high accuracy,see Eq. (29) in Ref. [14]. The electronic part dependsonly on the considered electronic state. The nuclear mag-netization distribution dependent part does not dependon the actual electronic state. In Ref. [14] the nuclearpart corresponds to the matrix element of the BW cor-rection operator over the s function of the correspond-ing hydrogen-like ion, B s . In particular, it means thatwithin a given level of the electronic structure theory theratio of two BW corrections calculated using two differ-ent models of the nuclear magnetization distribution isequal to the ratio of the nuclear parts and should not bedependent on the level of the considered electronic struc-ture theory. Moreover, it should not be dependent on theactual electronic and charge state of the considered open-shell system (we do not consider here situations when theHFS constant is determined exclusively by an electron inthe electronic state with j ≥ / ). Tables IV and V givethe BW contributions, − A BW , calculated within the uni-formly magnetized ball model and the single-particle WSmodel for different levels of electronic structure theory,see the numbers in brackets. According to our findings,the ratio of these BW contributions is indeed practically(with the uncertainty less than 1%) independent on thelevel of the electronic structure theory as well as on theconsidered electronic and charge state: P / , P / ofthe neutral Tl and S / of the hydrogen-like Tl.The theory formulated in Ref. [14] can be also used toillustrate the dependence of the ratio of magnetic anoma-lies and the differential anomalies on the model of thenuclear magnetization distribution. For convenience ofconsideration, we rewrite Eq. (3) by separating furtherthe Breit-Rosenthal correction δ : A = A (0) (1 − ε ) = A ( p.n. ) (1 − δ )(1 − ε ) , (24)where A ( p.n. ) is the HFS constant corresponding to thepoint nucleus. In this case, in the leading order, themagnetic anomaly is determined by the magnetic andcharge distribution contributions: ∆ ≈ ∆ m + ∆ c = ε − ε + δ − δ . (25)For isotopes with different valence nucleon states themain contribution to the anomaly comes from the mag-netiс distribution term, while the charge distributionterm, ( δ − δ ) , can be neglected for a qualitative treat-ment, i.e ∆ ≈ ε − ε . Using the factorization of theBW corrections [14] we obtain the following expression: ∆ [ a ] ≈ ε [ a ] − ε [ a ] = E [ a ]( N − N ) . (26)As one can see, the ratio of anomalies for two electronicstates depends on the ratio of electronic parts: k [ a, b ] = ∆ [ a ] ∆ [ b ] ≈ E [ a ] E [ b ] . (27)A slight deviation from this equality can be due theneglected charge distribution contribution. Thus, forthis case, it is reasonable to suggest that the uncer-tainty of the ratio of magnetic anomalies is mainly dueto the uncertainty of the electronic structure calculation.For example, according to Table VI, below we assume k x ( I =9 / [7 S / , P / ] = 3 . . At the same time,a differential anomaly depends on both the electronic andnuclear parts: θ [ a, b ] ≈ ∆ [ a ] − ∆ [ b ] = ( E a − E b )( N − N ) . (28)Table VIII gives the values of magnetic momentsfor short-lived thallium nuclei calculated according toEqs. (19)-(22) using the calculated ratio of anomaliesfrom Table VI and the experimental values of HFS con-stants from Ref. [22]. For Tl m and Tl m isotopes,this ratio is the same within a given uncertainty. There-fore, the same value, 3.31(10), has been used for otherisotopes in Table VIII, all of which also have one va-lence proton in the h / state. Following [22], we usedthe mean weighted value of the experimental differentialanomaly θ x ( I =9 / [7 S / , P / ] = − . · − forthe isotopes under consideration. The magnetic momentsobtained with this value are given in the third columnof Table VIII. As one can see, the obtained values arein good agreement with the results of Ref. [22]. Theirdifference is mainly due to the different values of the ra-tio of the magnetic anomalies. In the present paper theWS model has been used while in Ref. [22] the singleparticle model with a uniform valence nucleon distribu-tion model from Ref. [57] has been used. Alternatively,the differential anomaly can be determined for each iso-tope separately using Eq. (21). For this, the experimen-tal values of HFS constants A [7 S / ] = 12296 . from [64], A [6 P / ] = 21310 . from [65], as wellas the hyperfine constants for short-lived thallium iso-topes from [22] were used. The obtained results aregiven in the last column of Table VIII. The determinedmagnetic moments are in good agreement with the val-ues µ ( Tl m ) = 3 . µ N and µ ( Tl m ) = 3 . µ N from [23], where the same approach was used. The main source of the magnetic moments uncertainty is the exper-imental uncertainty of the HFS constants of the short-lived isotopes. TABLE VIII. Magnetic moments µ ( µ N ) for short-lived thal-lium isotopes with I = 9 / . The values in column 3 wereobtained using the averaged value of the differential anomaly,while the values in column 4 were obtained using the individ-ual experimental values of the differential anomalies. In thelast two columns, the first uncertainty corresponds to the ex-periment, and the second to the theoretical value of the ratioof magnetic anomalies.Nucleus Ref. [22] This work This work Tl m Tl m Tl m Tl m CONCLUSION
In the present paper, we have developed the approachto treat the nuclear magnetization distribution contri-bution to the hyperfine structure constants in many-electron atoms, that can be used in the calculations withthe explicit treatment of the electronic correlation effects.The approach can be further generalized to the molecularcase.Using the approach, we have numerically verified thatthe ratio of the magnetic hyperfine anomalies for a pairof electronic states is rather stable with respect to thechoice of the nuclear magnetization distribution model.The obtained uncertainty can be taken into account whenone uses the ratio for determining the magnetic momentsof short-lived nuclei.It has been demonstrated, that the order of magnitudeof the differential hyperfine anomaly for Tl isotopes hav-ing the s / valence nucleon state can be calсulated usingthe model of the uniformly magnetized ball and singleparticle models. However, the uniformly magnetized ballmodel cannot be used for isotopes with different nuclearconfigurations. It gives a wrong order of magnitude forthe differential hyperfine anomaly. At the same time, thesingle particle models with a uniform or Woods-Saxondistribution of the valence nucleon give reasonable result. ACKNOWLEDGMENTS
We are grateful to A.V. Oleichnienko, M.G. Kozlov,A.E. Barzakh, V.M. Shabaev and Yu.A. Demidov forhelpful discussions. Electronic structure calculations inthe paper were carried out using resources of the collec-tive usage center “Modeling and predicting properties ofmaterials” at NRC “Kurchatov Institute” - PNPI.The research (except for calculation of the point mag-netic dipole HFS constants and Gaunt interaction inte-grals) has been supported by the Russian Science Foun-dation Grant No. 19-72-10019. Calculations of the pointmagnetic dipole HFS constants have been supported bythe foundation for the advancement of theoretical physicsand mathematics “BASIS” grant according to the re-search project No. 18-1-3-55-1. Calculation of the Gauntcontribution has been supported by RFBR grant No. 20-32-70177. ∗ [email protected], [email protected] † [email protected], [email protected][1] M. S. Safronova, D. Budker, D. DeMille, D. F. J. Kimball,A. Derevianko, and C. W. Clark, Rev. Mod. 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