The critical notes to the solutions of the "puzzle" of hyperfine structure in 82+ and 80+ 209 Bi ions
aa r X i v : . [ nu c l - t h ] J a n THE CRITICAL NOTES TO THE SOLUTIONS OF THE “PUZZLE”OF HYPERFINE STRUCTURE IN 82+ AND 80+
BI IONS
F. F. KarpeshinMendeleev All-Russian Research Institute of Metrology190005 Saint-Petersburg, RussiaandM. B. TrzhaskovskayaPetersburg Nuclear Physics Institute of the National Research Center “Kurchatov Institute”198300 Gatchina, Russia
Abstract
Some aspects of description of the Bohr—Weisskopf effect in hyperfine splitting of the H– andLi–like ions of
Bi are considered by application of the surface and volume models of the nuclearcurrents. Extension of these models, used in internal conversion theory, to description of theHFS allows one to successfully describe the effect, without resorting to the specific differences.The latters are shown not to be needed at all. Moreover, they turn out to depend on the nuclearmodel even stronger than the HFS values themselves. Comparison of the calculated HFS valuesto experiment shows a satisfactory agreement. Both models provide equally good descriptionof the effect. However, they result in different values of the retrieved rms radius of the nuclearmagnetization. In this respect, situation resembles the proton radius puzzle. Prospects of futureresearch are discussed. Introduction
A considerable progress during the past decades was archived in investi-gation of few-electron heavy ions. Specifically, this concerns study of theirelectronic structure and its influence on the nuclear processes. Wonderfulexperiments were performed studying the shell effects on the beta decay[1]. There is the comparative study of α decay in H-like and He-like ionson the urgent agenda, with respect to that in neutral atoms [2, 3, 4]. Inspite of that the influence of the electron screening on the α decay is a veryimportant question, in view of many applications in astrophysics and ex-periments with laser-produced plasma, it is only recently that the adequateapproach has been found [2]. It was clearly shown that the frozen-shellapproximation, which was used during half century, exaggerates the effectof the shell at least by an order of magnitude [3, 4]. Moreover, it givesthe wrong sign of the effect [4]. Furthermore, attractive ideas concern pos-sibilities of manipulations by the electromagnetic decay of the nuclei. Away of drastic acceleration of nuclear decay rate by means of resonanceconversion was proposed by [5, 6, 7].A considerable attention was paid to study of the hyperfine structureof heavy ions. As compared to neutral atoms, a few-electron wavefunctioncan be calculated with high accuracy in heavy ions. On the other hand,QED effects give a significant contribution. This gave basis to suggestthat the hyperfine splitting (HFS) can be used to test QED (e.g. [8] andRefs. cited therein). However, it was noted [8] that there is a stumblingstone on the way represented by the Bohr—Weisskopf effect [9]. Thiseffect generates known for decades hyperfine magnetic anomalies in opticalspectra of atoms. The effect is caused by the interaction of atomic electronswith the spatially-distributed magnetic moment of the nucleus. Althoughits contributions to the HFS of the 1 s and 2 s levels in the H-like andLi-like Bi ions comprise approximately 2% and 2.2%, respectively, itsactual contribution depends on the nuclear model. Some attempts were ndertaken, aimed at calculation of this effect ([8] and Refs. cited therein).They showed, however, that there remains a contradiction with theory atthe level of 20 – 30 percent. Such a result should be expected, becausenuclear calculations still cannot be performed ab initio in principle, in viewof absence of an appropriate parameter [10]. At the same time, the Bohr—Weisskopf effect becomes essential for description of experimental data [11].In view of this problem, another, roundabout way was proposed in Ref.[8]. It runs that, instead of calculation of the Bohr—Weisskopf effect, onecan cancel its contribution in the specially constructed linear combination(difference) ∆ ′ E (see Eq. (6) in section 2) of the HFS values of the H- andLi-like ions, being in the 1 s - and 2 s states, respectively. The cancellationsupposedly takes place if the parameter ζ in the combination (see Eq. (7)in section 2) is calculated in such a way that the BOHR—WEISSKOPFcontribution is mutually subtracted in the difference. This combinationwas called specific difference (SD). In the case of Bi ions, the calculatedvalue of ζ = 0.16886 was then listed to fifth decimal [11, 12, 13].The first thing that catches your eye is that consideration of such definedSD is in contradiction with general methods of epistemology, which isfounded on comparison of experimental data with theoretical values of theobservables. But SD, thus defined, cannot be observed experimentally.Moreover, the operation of subtraction, aimed at mutual cancellation ofsmall terms, is incorrect from the viewpoint of mathematics as it leads alsoto a considerable reduction of the main parts. We note in this relation thatuncertainties do not cancel one another in subtraction. Oppositely, theyare summarized in the general case. This makes the result of subtraction,that is SD in our case, less accurate than the 1 s - and 2 s HFS valuesthemselves. We will see this in Section 3 and Table 1.And even more: for the method to work, it is necessary that parameter ζ be model independent. This idea seems to be absolutely incorrect, judgingby experience of application of the theory of internal conversion (IC). It as subject to critical check in Refs. [14, 15]. At first sight, applicationof IC theory may seem unusual. However, this idea is not new. For thefirst time, IC theory was applied in Ref. [16]. Then estimations of theHFS values and the dynamic effect were performed in Refs. [7, 17, 18]. Inthe next section, we will show the relatedness of these two phenomena inmore detail. Meanwhile, based on the methods of IC theory, the method ofthe magnetic moments was developed in Ref. [14] for interpretation of thepenetration effects of the nuclear structure. The method was then appliedin Ref. [15], where it was shown that the conclusion of the “hyperfinepuzzle” [19] was mostly due to underestimation of the model dependenceof the SD. Actually, there was no puzzle at that moment, the authorsshould merely cite the result of paper [14], where encounter with sucha “puzzle” was literally predicted. Herein, we attack the problem in adifferent way as compared to [15]. We study the penetration effect onthe HFS and SD values by means of comparison of the two conventionalmodels, which are known to work well in the IC theory: surface (SC) [20]and volume (VC) [7, 21] nuclear currents. These models are opposite toone another in their physical sense. Due to the latter circumstance, theresults obtained can be considered as quite general.In section 2, we derive shortly the formulas. The results of the calcula-tions are reported in Section 3. Unexpectedly, we arrived at the conclusionthat both of the models work equally well in description of the data at thepresent level of precision. With different parameters, the models give thesame values of the HFS for the 1 s - and 2 s levels up to six decimals. How-ever, the HFS values are, naturally, very sensitive to the only parametersof the models — their radii. The ζ and ∆ ′ E values turn out to be moresensitive, than the HFS values themselves, as expected. The results arediscussed in more detail in the conclusive section. In the same section, weconsider analogy and possible applications of the results for better under-standing of the proton radius puzzle. hen the manuscript was ready, paper [19] was issued. The authorsshow that the value of the magnetic moment of the nucleus µ =4.092 nu-clear magnetons may be more correct than 4.1106 [22] used previously.Such a new value would essentially keep the present results, merely rescal-ing them by ∼ . Bi issue: the data can be fairly explainedwith any of the magnetic moment values.
Feynman graphs of the hyperfine splitting and internal conversion are pre-sented in Figures 1 and 2, respectively. Actually, the both graphs describethe same amplitude, though defined on different areas of the external kine-matical variables of the transition energy and angular momenta. In quan-tum mechanics and theory of field, such values can be related to each otherby making use of the analytical properties of the amplitudes, and the pro-cesses themselves are spoken about as crossing channels. The method ofcomplex transition trajectories by Landau (e. g. [23, 24]), or complexangular momenta by Regge [25] may set examples. In the case of HFS, theanalyticity of the amplitudes in Figs. 1 and 2 means that all the methods,developed in the IC theory, can be directly applied to description of HFSvalues, considered as the diagonal IC matrix elements in the limit of thetransition energy ω → c ( τ, L ) and radiative Γ γ ( τ, L ) transitions: α ( τ, L ) = Γ c ( τ, L ) / Γ γ ( τ, L ) , (1) Feynman graph of hyperfine shift. Nuclear propagator is shown by bold line. Atomic state is defined bythe total momentum F and its projection M , together with I and j — nuclear and electronic spins, respectively. Figure 2:
Feynman graph of internal conversion. I , M , I , M — nuclear quantum numbers (spins and theirprojections on the quantization axes) in the initial and final states, respectively. j , m — electronic quantumnumbers in the initial state. Conversion electron is characterized with the four-vector of its momentum p . where τ, L stand for the type and multipole order of the transition. Weremind that ICC, as well as HFS, would be independent of the nuclearmodel in the limit of the point-like nuclei, where the electron penetrationeffects into the nuclear volume are absent. This approximation comprisesthe “no-penetration” (NP) nuclear model. Calculations within the frame-work of realistic nuclear models generally needs a two-dimensional integra-tion over the electronic and nuclear variables within the nuclear volume e. g. [14]). This results in loss of the factorization, which makes ICC α ( τ, L ) model-dependant, as well as HFS. Some reasonable models, whichtake into account the penetration effects, with also keeping the amplitudesfactorized, were introduced and approved by comparison to experiment.The SC nuclear model served as a basis for a number of tables of ICC(e.g. [26, 27]), highly demanded for the research and application purposes.The VC model is expected to work even better in the case of the valence h / proton orbital in Bi. Furthermore, the VC model was applied fordescription of muonic conversion [7, 21, 28].Generally, in the IC theory, manifestation of the nuclear structure isclassified into two kinds: static and dynamical effects ([20, 26, 27, 29, 30,31, 32] and Refs. cited therein). To the first kind belong the effects thatarise because of change in the electronic wave functions as compared to theDirac wavefunctions for the point-like nucleus. Coulomb wavefunctionsare singular at the origin. Accounting for the finite charge distributionover the nuclear volume makes the functions regular, and brings about acorrection to the internal conversion coefficients (ICC) up to 30 percent inthe case of the M ζ and ∆ ′ E [1]values was checked against variation of the parametersof Fermi charge distribution over the nuclear volume. This just comprisesthe static effect. Therefore, the conclusion of the model independence ofSD [8] is only drawn from investigation of the static effect, which is knownnot to be essential after the main shortcoming — the divergence of theCoulomb wavefunction — is resolved, indeed.Influence of a model, used for the transition nuclear currents on theICC values, is called the dynamical effect of the nuclear structure. It isjust the dynamical effect which is responsible for differences between theexperimental ICC and their table values, which are observed in some casesof forbidden nuclear transitions. These differences are also called anomalies n IC, similar to magnetic anomalies in the hyperfine spectra (e.g., [33, 34,35]). The dynamical effect constitutes up to ∼
10 percent in heavy nucleiin the case of the M W = N w ,w = Z ∞ g ( r ) f ( r ) dr + t ν ≡ w + t ν , (2) N = − I + 1) I ( j + 1) eκµ e ~ M p c . Here g ( r ), f ( r ) are the large and small components of the radial Diracelectronic wavefunction of the i -th level. κ is the relativistic quantumnumber; j , I — the electronic and nuclear spins, respectively, e — theelementary charge, µ — the magnetic moment of the nucleus, and e ~ M p c —the nuclear magneton. w gives the NP value, t ν , which we shall call thepenetration matrix element, bears information on the nuclear structure.Therefore, the t ν value depends on the nuclear model. In the SC and VCnuclear models, t ν = Z R c g ( r ) f ( r ) Y ν ( r ) r dr , (3) ith Y ν ( r ) = rR c − r for ν = SC (4) R c (cid:16) r − r R c (cid:17) − r for ν = VC (5)In the NP model Y ν ( r ) ≡ R c is the model radius of the transitioncurrents. We refer upper index ν to the model, and lower index i — tothe electronic level.It is thus t ν which only bears information about the Bohr—Weisskopfeffect. It was proposed to get rid off it in the linear combination, calledspecific difference: ∆ ′ E = W s − ζ W s . (6)In terms of the penetration matrix elements (2), Eq. (6) has an evidentsolution ζ = t ν s /t ν s . (7)By making use of the last equation, SD (6) can be also expressed in equiv-alent form as follows: ∆ ′ E = N w s − ζ w s = (8)= N w s p s − p s p s , (9)where p νi = t νi /w νi — the relative contribution of the Bohr—Weisskopfeffect to the HFS. In view of that p s > p s , it follows from Eq. (9) that∆ ′ E < Y ( r ) with˜ Y n ( r ) = r n +1 R cn +3 − r . (10) Schematic picture explaining seeming contradiction of the terms of “homogeneous” and “surface-current”for the same nuclear model. Homogeneously distributed inside the nucleus magnetic dipoles are shown by elementarycircular currents. In the bulk, the adjacent currents mutually cancel one another. As a result, only the encirclingeffective current survives, resulting in the surface-current model. It is this surface current which brings about thehyperfine splitting.
Model (10) with parameter n = 0, 2 was used in Refs. [36, 37]. In a partic-ular case of n = 0, Eq. (10) coincides with (4). This seeming paradox withthe name has a simple explanation on the physical ground. In fact, it is notthe transition density, but rather the transition current which determinesthe HFS ( e. g. [14]). On the other hand, such a classical picture of “ho-mogenius” density distribution of elementary point-like magnetic dipolesleads to appearance of the effective current along the circle surroundingthe locus, as inside the locus all the elementary currents mutually cancelone another. This directly generates the δ -form nuclear current, as illus- rated in Fig. 3. It is worthy of noting that another such paradox wasshown in Ref. [28]. It was found that muonic conversion from the statesof giant dipole resonance is better described by the VC model, in spite ofthat the transition nuclear density has a sharp maximum on the nuclearsurface.It is worthy of noting some relations concerning the physical sense of themodel parameters. In the VC model, R c , like the equivalent electromag-netic radius, equals the radius of the sphere with the homogeneos sharpedge distribution of the magnetization currents. It is related with the rmsradius R by means of the following expression: R V C = r R V Cc . (11)In the SC model, all the multipole moment radii equal R SCi ≡ R SCc . As it was pointed out in the original paper by Bohr and Weisskopf, theeffects of the nuclear structure can be studied, using as a series expansionof the electronic wave functions within the nucleus [9]. Independently, thesame series was also used in studies of the penetration effects in the casesof anomalous conversion [20, 29]. In this way, the series expansion over themultipole moments R , R . . . of the nuclear magnetization distributionwas developed for HFS in Ref. [14]. The leading term is proportional tothe square of the rms radius R , or the second moment of the distributionof magnetism. Therefore, if two nuclear models have the same R , they willresult in the same HFS in this approximation. This justifies the commonprocedure of studies of variations of R by means of the hyperfine anomaliesalong isotopic chains (e. g. [34, 35, 38]).Herein, we systematize the result in such a way, which keeps minimumthe difference between the results, obtained in the different models. For HFS values for the 1 s and 2 s states (in eV) calculated with various representative model radii (in fm). Theresults are presented for the two values of µ = 4.1106 and 4.092 nuclear magnetons (see text) µ =4.1106 µ =4.092 R V Cc R R SCc W s W s W s W s this purpose, we will fit the SC radius to the value which results in thesame values of w s and w s , as far as possible, to those obtained in the VCmodel, respectively. For this purpose, we note that the both models onlydiffer by the penetration matrix element t i . Given an R V Cc radius, the fitof the R SCc radius was fulfilled by minimization of the following form: χ = (cid:12)(cid:12)(cid:12)(cid:12) t SC s − t V C s t V C s (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) t SC s − t V C s t V C s (cid:12)(cid:12)(cid:12)(cid:12) . (12)Even in such different models as the ones we use, the coincidence ofthe HFS values is achieved up to six decimals. For the purpose of bettercomparison to experiment, we added the latest values of QED correctionsto the calculated HFS values [12]: ∆ E sQED = -0.0268 eV, ∆ E sQED = -0.005eV, and used the contribution from the electron-electron interactions forthe Li-like configuration from Ref. [12], where they were calculated up tothe third order of 1 /Z : ∆ E se − e = -0.030 eV. The results are presented inTable 1 for various representative values of the radii of the models. In thefirst and third columns, R V Cc and related R SCc radii are listed, respectively.The resulting rms radii turn out to be different in the both models. In
Calculated ζ and SD values for the VC and SC models for the same representative values of the model radiias in Table 1. The SD values are presented in meV R V Cc ζ V C ζ SC ∆ ′ E , VC ∆ ′ E , SC9.1214 0.16688 0.16688 -61.11 -61.128.6214 0.16688 0.16689 -61.12 -61.148.1214 0.16689 0.16689 -61.14 -61.157.6214 0.16689 0.16689 -61.16 -61.177.1214 0.16689 0.16690 -61.18 -61.196.6214 0.16690 0.16690 -61.20 -61.216.1214 0.16690 0.16690 -61.22 -61.245.6214 0.16691 0.16691 -61.25 -61.26 order to show this, we list the rms R V C in the second column. In theSC model, R SC ≡ R SCc . For the sake of clarity, the values obtained with µ = 4.092, are also presented in columns 6 and 7. In the both cases,the results are in quite satisfactory agreement with the last experimentalvalues of 5.08503(11) and 0.797645(18) eV, although with different rmsradii. This difference compensates the variation of the magnetic momentof the nucleus.Proceeding with the ζ and ∆ ′ E values, one can see from Eq. (12) thatthe condition of model independence of the ζ value is equivalent to thecondition that both of the t ν values, and therefore, both of the W s and W s HFS’s, might be fitted simultaneously by different models. This conditionis looser than mere proportionality of the 1 s - and 2 s wave functions [8],not speaking on that the proportionality is in fact broken by the e − e interactions, QED effects etc . Naturally, if the equivalence of the modelswere full, the ζ value would coincide in the both models. Differences in the W i values also give rise to the differences in the ζ – and ∆ ′ E values. Allthese consequences are illustrated in Table 2 for the same representativemodel radii as in Table 1. In accordance with what is said previously, Comparison of theoretical results to experiment
Electronic Experiment µ = 4.1106 µ = 4.092state [19] [14] present [12] present [19]1 s s the ζ - and ∆ ′ E values differ from one another much more, than the HFSvalues. ζ ’s coincide up to fifth decimal, and ∆ ′ E [2]values hold up only tothe third one. Both values are very sensitive to the only model parameters R ic .In finer detail, results of the fit of the W i values within the frameworkof the VC model are presented in Table 3, together with the experimentaldata. As one can see, the results are not critical to the µ value: decrease ofthe latter correlates with adequate decrease of the fitted R c value withoutworsening the quality of the fit. A similar fit can be performed, using theSC model. The results obtained in paper [14] within the framework of thetwo-parameter magnetic moment method are also presented. They are ingood agreement with the present calculations. For comparison, the resultsof Refs. [12, 19] are also listed. One can see that the latters are in worseagreement with experiment. In contrast, the authors of [19] are quite satis-fied by their fit. Here is the key point to understanding the Bi hyperfinepuzzle. As a matter of fact, the authors of [19] compare to experiment notthe W i values themselves, but the SD values instead, which are speciallyconstructed by themselves for this purpose. That such a way is misleading,is explained previously. This delusion leads to underestimated values ofthe radii of the nuclear magnetization, as compared to ours. Conclusion
We performed study of some effects arising in description of the Bohr—Weisskopf effect. As expected, the above results disavow the concept ofthe specific differences as a significant model independent value. Theseexhibit even stronger sensitivity to the models used than the HFS them-selves. On the other hand, they show that there is no fundamental problemin the interpretation of the Bohr—Weisskopf effect: by mere fitting the pa-rameters, the effect can be equally well reproduced by either of the modelswithin six decimals, which is quite enough for the present purposes. At thesame time, we have to conclude that such an equivalence means absenceof physical sense in agreement of either of the models with experiment. Inthe other words, one cannot conclude that the real distribution is surface-or volume-like one, based on agreement with experiment, as the modelsare mutually exclusive.Regarding the calculated values of HFS, they remain in general agree-ment with experiment, as in [14]. This is true with both values of µ n =4.1106 and 4.092, with the corresponding values of the model parameters R V Cc and R SCc .Without going into details, let us draw an analogy of the results obtainedabove with the proton radius puzzle. The proton rms radius extracted fromthe levels in muonic hydrogen turns out to be different from that retrievedby means of electron scattering experiment. But we already saw above thatdescription of the levels does not provide the rms radius unambiguously.Instead, the value retrieved depends on the model used. Cf. also Refs.[39, 40].An alternative way, based on the two-parameter model, was proposedin Ref. [14]. This way allows one to unambiguously retrieve objectivecharacteristics of the distribution of magnetism inside the nucleus, suchas the second and fourth moments. Model independence of the valuesthus obtained has been demonstrated in [14]. Within this method, the ifference obtained above in the rms values, calculated within the SC andVC models, can be attributed as a manifestation of the truncated terms,containing R and higher moments. Analysis of the results presented inTable 3 suggests that it might be impossible to describe the HFS values forthe both levels simultaneously within the framework of an one-parametermodel. The two-parameter method of magnetic moments [14] can. Furtherresearch, both experimental and theoretical, is needed in order to betterunderstand the above peculiarities. Specifically, measuring the 2 p / HFSvalue may be critical to this end [14].The authors would like to express their gratitude to L. F. Vitushkin,D. P. Grechukhin, V. M. Shabaev, I. I. Tupitsin for fruitful discussions ofthe topic and helpful comments. eferences [1] M. Jung, F. Bosch, K. Beckert, et al., Phys. Rev. Lett. , 2164 (1992); F. Bosch, T. Faestermann,J. Friese, et al., Phys. Rev. Lett. , 5190 (1996).[2] F. F. Karpeshin, Phys. Rev. C87 , 054319 (2013).[3] F. F. Karpeshin, M. B. Trzhaskovskaya, in: Exotic Nuclei, Proc. of the First African Symposiumon Exotic Nuclei, Cape Town, South Africa, 2 – 6 December 2013. Ed. E. Cherepanov et al., WorldScientific: New Jersey—London—Singapore, 2014, p. 201.[4] F. F. Karpeshin, M. B. Trzhaskovskaya, Yad. Fiz. , 1055 (2015). ( In Russian. Engl transl.: ) Phys.At. Nucl. , 993 (2015).[5] B.A.Zon, F.F.Karpeshin, Zh. Eksp. i Teor. Fiz., , 401 (1990) [Sov. Phys. — JETP (USA), ,224 (1990)].[6] F.F. Karpeshin, Zhang Jing-Bo and Zhang Wei-Ning, Chinese Physics Letters, , 2391 (2006).[7] F.F.Karpeshin, Nuclear Fission in Muonic Atoms and Resonance Conversion , Saint Petersburg:“Nauka”, 2006.[8] V. M. Shabaev, A. N. Artemyev, V. A. Yerokhin, O. M. Zherebtsov, and G. Soff, Phys. Rev. Lett. , 3959 (2001).[9] A.Bohr, V.F.Weisskopf, Phys. Rev. , 94 (1950).[10] A. B. Migdal. Theory of Finite Fermi Systems:And Applications to Atomic Nuclei. IntersciencePublishers (Wiley), New York, 1967.[11] Johannes Ullmann, Zoran Andelkovic, Carsten Brandau et al. , Nature Communic., , 15484 (2017).DOI: 10.1038/ncomms15484.[12] A.V. Volotka, D. A. Glazov, O.V. Andreev, V. M. Shabaev, I. I. Tupitsyn, and G. Plunien, Phys.Rev. Lett. , 073001 (2012).[13] Matthias Lochmann, Raphael J¨ohren, Christopher Geppert, Zoran Andelkovic et al. , Phys. Rev. A90 , 030501(R) (2014).[14] F. F. Karpeshin, M. B. Trzhaskovskaya, Nucl. Phys.
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