Sensitivity of isotope shift to distribution of nuclear charge density
aa r X i v : . [ phy s i c s . a t o m - ph ] J u l Sensitivity of isotope shift to distribution of nuclear charge density
V. V. Flambaum , and V.A. Dzuba School of Physics, University of New South Wales, Sydney 2052, Australia and Helmholtz Institute Mainz, Johannes Gutenberg University, 55099 Mainz, Germany (Dated: July 18, 2019)It is usually assumed that the field isotope shift (FIS) is completely determined by the change ofthe averaged squared values of the nuclear charge radius h r i . Relativistic corrections modify theexpression for FIS, which is actually described by the change of h r γ i , where γ = √ − Z α . Inthe present paper we consider corrections to FIS which are due to the nuclear deformation and dueto the predicted reduced charge density in the middle of the superheavy nuclei produced by a verystrong proton repulsion (hole in the nuclear centre). Specifically, we investigate effects which cannot be completely reduced to the change of h r i or h r γ i . I. INTRODUCTION
Isotope shift (IS) phenomena in heavy atoms are animportant way of probing various scenarios in nuclearphysics and can aid the search for new physics beyond theStandard Model. Nuclear theory predicts the existenceof long-lived isotopes for elements with Z ≥
104 (seee.g. [1, 2]), in particular isotopes with a magic neutronnumber N = 184. However, producing these neutron-rich isotopes in laboratories by colliding lighter atoms iscurrently impossible. The Coulomb repulsion for nucleigrows as Z ; in order to compensate for this with theattractive strong force, the neutron number N must growfaster than Z . Consequently, an isotope from the islandof stability with N = 184 cannot be produced from thecollision of a pair of lighter isotopes with smaller N/Z ratios.In contrast to laboratories, various astrophysical eventssuch as supernovae explosions, neutron stars and neutronstar - black hole/neutron star mergers generate high neu-tron fluxes and may create environments favourable forthe production of neutron-rich heavy elements. For ex-ample, a new mechanism of such a kind due to the cap-ture of the neutron star material by a primordial blackhole has been suggested in [3]. Furthermore, neutron star- neutron star mergers are predicted to generate optimalenvironments for the production of heavy atoms [4, 5].As a consequence, astrophysical data may be the bestplace to observe super-heavy meta-stable elements. It ispossible that optical lines of elements up to Z = 99 havealready been identified in the spectra of Przybylski’s star[6]. These elements include heavy, short-lived isotopeswhich may be products of the decay of long-lifetime nucleinear the island of stability [7].IS calculations for superheavy elements (SHE) can helptrace the hypothetical island of stability in existing as-trophysical data. It may be possible to predict a spectralline of a neutron-rich isotope ν ′ based on the experimen-tal spectrum of a neutron-poor isotope ν and calculationsof IS δν as ν ′ = ν + δν . The results can then be usedto search for the long-lifetime neutron-rich elements incomplicated astrophysical spectra such as that of Przy-bylski’s star. Spectroscopic measurements of IS may also be relevantto the search for strange-matter. Strange nuclei consistof up, down and strange quarks (see [8] and referencestherein). A strange-matter nuclei of charge Z would havea very different radius in comparison to any regular iso-tope. Calculations of IS can be used to predict the effectsof this change in radius on atomic spectra.Calculations of IS allows one to estimate the King-plotnonlinearity of a given element. New long-range forcessuch as Yukawa-type interactions between electrons andnucleus can lead to nonlinearities in a King plot for aseries of isotopes [9]. It is useful to understand otherpossible sources of nonlinearities in the IS in order toconstrain new physics beyond the Standard Model.It should be noted that relativistic corrections pro-duce an important difference in the dependence of thefield shift on the nuclear radius r . The traditional ex-pression for field shift is known as F i δ (cid:10) r (cid:11) where F i isan electronic structure factor and δ (cid:10) r (cid:11) is a nuclear pa-rameter. It is usually assumed that electron factor F i is the same for all isotopes. In fact, relativistic effectsbreak this independence and if the independence on iso-topes is to be kept the field shift should be written as˜ F i δ (cid:10) r γ (cid:11) , where γ = √ − Z α , α is the fine structureconstant. The electronic factor ˜ F i is to be calculated.Analytical estimate of ˜ F i has been done in Ref. [10] (seealso [14, 15, 17]), relativistic many body calculations for Z = 102 −
109 have been done in Refs. [11–13]. Thetraditional formula for the field shift F i δ (cid:10) r (cid:11) still canbe used for neighbouring isotopes where change in F i issmall and can be neglected. The formula is useful forfinding the change in nuclear root mean square (RMS)radius from the IS measurements.Due to the relativistic effects in heavy atoms, the fieldshift of the p / orbital is comparable to that of the s / :the ratio is ∼ (1 − γ ) / (1 + γ ) [10]. The Zα expansiongives the ratio ∼ Z α / Z =137, γ ≈ j > / δV due to theperturbation of the s and p / orbitals by the field-shiftoperator) produces the same dependence of field shift onnuclear radius for all orbitals: ˜ F i δ (cid:10) r γ (cid:11) .The difference between the non-relativistic h r i andrelativistic h r γ i expressions may be explained by the dif-ferent dependence of the non-relativistic and relativisticwave functions near the origin. Another relativistic effectis due the variation of the electron density ρ e inside thenucleus which for the s and p / orbitals is approximatelypresented by the following formula [10]: ρ e ( r ) ≈ ρ e (0) (cid:18) − Z α (cid:16) rc (cid:17) (cid:19) (1)where c is the nuclear radius. The r -dependent term givesus an additional sensitivity of IS to the nuclear chargedistribution beyond the change of h r i .In this work we study the effect of the change in nuclearcharge distribution on the field isotope shift. We considerfour types of charge distribution variation: (a) a hole inthe origin, where nuclear density is small in the originand increases to the periphery; (b) nuclear quadrupoledeformation; (c) change of the skin thickness; and (d)change in nuclear RMS radius. The questions we tryto answer include (a) can isotope measurements be usedto study nuclear structure beyond the change of nuclearRMS radius; (b) what is the best way of using isotopeshift calculations to predict the spectra of neutron-richSHE with the aim to reach the hypothetical island of sta-bility; (c) can nuclear deformation lead to non-linearityof King plot.We choose the E120 + ion for numerical analysis. Itis sufficiently heavy for the relativistic effects to be pro-nounced. On the other hand the ion has relatively sim-ple electron structure (one external electron above closedshells) so that all important points can be illustratedwithout getting into a trouble of complicated many-body calculations. We use the results of nuclear calcu-lations [18] to get the parameters of nuclear deformationand nuclear RMS radius. We consider only even isotopesbecause nuclear calculations for them are more reliable.The work [18] considers a range of nuclear models whichfavour spherical nuclear shape at Z = 120 and N = 172.We use this spherical nucleus as starting point in ourstudy. II. CALCULATIONS
We use an approach similar to one in Ref. [19, 20].Electron potential V for valence orbitals is found bysolving relativistic Hartree-Fock (RHF) equations for aclosed-shell core ( ˆ H HF − ǫ c ) ψ c = 0 , (2)where c numerates states in the core from 1 s to 7 p / and 7 p / . States of valence electron (Brueckner orbitals)are obtained by solving the RHF-like equations for thevalence orbitals( ˆ H HF + λ Σ (2) − ǫ v ) ψ Br v = 0 . (3) Here Σ is the correlation potential responsible for core-valence correlations [21], index ”2” indicates second orderof the many-body perturbation theory. Σ is defined insuch a way that the correlation correction to the energy ǫ v is given by δǫ v = h ψ v | Σ | ψ v i (see, e.g. [21] for details).We calculate Σ ab initio , limiting ourselves to the lowestorder of the perturbation theory. λ is a scaling parameterintroduced to simulate the effect of higher-order correla-tions. Its value ( λ = 0 .
75) is chosen to fit the result ofall-oder calculations of Ref. [19, 20].IS is calculated using the so-called random phase ap-proximation (RPA, see e.g. [21]) which can be describedas linear response of self-consistent atomic field to a smallperturbation. In our case the perturbation is the changein nuclear potential δV N due to change in nuclear chargedistribution. The RPA equations are first solved for thecore ( ˆ H HF − ǫ c ) δψ c = − ( δV N + δV core ) , (4)where δψ c is the correction to the core orbitals due tothe effect of δV N , δV core is the correction to the electronpotential of core electrons due to the changes in all coreorbitals. IS for states of a valence electron is found as h ψ Br v | δV N + δV core | ψ Br v i .We use Fermi nuclear charge distribution (solid line onFig. 1) ρ ( r ) f = ρ r − c ) /t , (5)where c is nuclear radius, t is skin thickness, and ρ isnormalisation constant, R ρ ( r ) f dV = Z . Nuclear chargedistribution with a hole in the origin is given by (dashedline on Fig. 1) ρ ( r ) h = ρ ( r ) f (cid:18) k (cid:16) rc (cid:17) (cid:19) . (6)The normalisation constant ρ is adjusted to keep cor-rect normalisation. Nuclear quadrupole deformation isconsidered by replacing constant nuclear radius c in (5)by varying parameter c ( θ ) c ( θ ) = c (1 + βY ( θ )) . (7)and calculating spherical average by integrating over θ . Itis known that this is approximately equivalent to increasein skin thickness [25, 26] t ≈ t + (4 ln 3) (cid:0) / π (cid:1) c β . (8)We also consider the change of nuclear radius. We use the E
120 isotope as a reference one and we take nuclearparameters from nuclear calculations [18].
III. RESULTS
Table I lists isotopes of SHE E120 used in this studywith nuclear parameters taken from [18]. The results
Figure 1. Variations of nuclear density. Solid (black) line- Fermi distribution (5); dashed (blue) line - modified dis-tribution with a hole in the origin, formula (6) with k = 0 . t in (5)) by 14.5 % to simulate the effect ofthe hole; long dashed (red) line - Fermi distribution with in-clreased skin thickness by 30.5 % to simulate the effect ofquadrupole deformation, formula (7) with β = − . v nsms v Figure 2. Dominating contribution to the isotope shift of thesingle-electron states v with total angular momentum j > / p / , d / , d / , etc). Cross stands for δV N , change of nuclearpotential due to change in nuclear charge distribution.Table I. Nuclear parameters for the range of even isotopesfrom E120 to
E120 isotopes taken from [18].
A β p h r i (fm)292 0 . . − .
174 6 . − .
205 6 . − .
218 6 . − .
221 6 . − .
261 6 . − .
290 6 . − .
376 6 . are presented in Tables II and III and Fig. 3. In allcases the IS for s and p / states is dominated by the h φ Br a | δV N | φ Br a i term (see Table II); IS for states with j > / h φ Br a | δV core | φ Br a i . The largest contributions to the CPcomes from the core s states as shown on Fig. 2. There-fore, the effect of change in nuclear charge distribution isvery similar for all states except the p / states. A. Hole in nuclear charge distribution and changeof the nuclear skin thickness
A hole (or, more accurately, central depression) in nu-clear density for E120 was considered in Refs. [22–24]. Itsimportance is related to theoretical prediction of magicnumbers for protons and neutrons. We study the effectof making a hole in nuclear charge distribution by com-paring the energies of the
E120 + ion in which nuclearcharge distribution is pure Fermi distribution (5) to theenergies of the ion in which nuclear density is modifiedaccording to (6) (see also Fig. 1). We use k = 0 . E120 and
E120 calcu-lated with the nuclear parameters from Table I as a ma-trix element h ψ Br a | δV N + δV core | ψ Br a i . The ratio of theenergy shifts due to a hole to the reference IS is about8%. This means that the effect is significant and deservesfurther study.It turns out that a hole in the nuclear charge distri-bution is numerically equivalent to decreasing the valueof the skin thickness (parameter t in (5)). The value k = 0 . t . In both cases the effect is practically the same forall considered states. B. Nuclear quadrupole deformation and change ofnuclear radius
Next we study the effect of nuclear quadrupole defor-mation. We consider a model situation by comparing twonuclei with the same RMS radius but one has no defor-mation, and another has a deformation with β = − . β comes from nuclear calculationsfor the E120 isotope [18]. The effect of quadrupoledeformation is equivalent to increased skin thickness (seeFig. 1). Calculations show that for β = − . t is 30.5% in good agreementwith (8). The shift in energy is significant, ∼ − for s states (see Table II) or ∼
20% of the reference IS forall considered states. This leads to a question whetherIS can be used to study nuclear deformation. There-fore, we check whether nuclear deformation can be dis-tinguished from the change of nuclear RMS radius. Twolast columns of Table II show the effect of the change innuclear RMS radius in which the parameters were cho-
Table II. Isotope shift for specific states of E120 + ( in 10 − cm − ) due to change in nuclear charge distribution. Reference ISis the IS between E120 and
E120 calculated ( h ψ Br a | δV N + δV core | ψ Br a i ) with the nuclear parameters from Table I. ”Hole”is the shift due to the difference between pure Fermi distribution (5) and the distribution with the hole in the origin, formula(6) with k = 0 .
5. The same IS is produced by reducing the skin thickness t in (5) by 14.5%. ”Deformation” is the shiftdue to quadrupole deformation, formula (7) with β = − .
4. The same IS is produced by increasing the skin thickness t in(5) by 30.5%. Note that while changing the hole parameter k or the deformation parameter β we are also changing nuclearradius parameter c to keep the rms radius unchanged. ”Change of p h r i ” is the IS due to change of nuclear RMS radius inpure Fermi distribution (5) from 6.220 fm to 6.211fm. ”Br” stands for IS given by h ψ Br a | δV N | ψ Br a i ; ”Br+CP” includes corepolarization, h ψ Br a | δV N + δV core | ψ Br a i . Note that corresponding matrix elements may be interpreted as isotope shift correctionsto the ionisation potential for an electron on a given orbital.State Reference Hole Deformation Change of p h r i IS Br Br+CP Br Br+CP Br Br+CP8 s s p / p / -485 ∼ − -38 ∼ − ∼ − d / -1350 ∼ − -106 ∼ − ∼ − d / -606 ∼ − -48 ∼ − ∼ − sen to produce the same IS for the 8 s state as in thecase of quadrupole deformation. We see that the shiftis the same for all states except the 8 p / state. Thedifference for the 8 p / state is 4% or 0.014 cm − . Thisis large enough to be detected in spectroscopic measure-ments. However, this is a model case. Let us now con-sider a more realistic case of isotope shift between twoisotopes in which nuclear parameters are taken from nu-clear calculations [18]. We consider isotope shift for fre-quencies of electric dipole transitions in E120 + for iso-topes in Table I. IS for the a → b transition is given by δν ab = h ψ Br b | δV N + δ core | ψ Br b i − h ψ Br a | δV N + δV core | ψ Br a i .The results are presented as case A in Table III. In case Bwe perform model calculations to check whether IS can bereduced to the change in RMS radius. The answer is neg-ative. We see that if we chose the change in RMS radiusto fit the shift of s and p / states (they behave the sameway, see above) then the shift for the p / state is slightlydifferent leading to different IS in the ns − mp / transi-tions. The difference is ∼ .
003 cm − for the 8 s − p / transition which is probably large enough to be detected.This means that nuclear deformation can be studied bycomparing IS in s − p / and s − p / transitions. Boththese IS cannot be fitted by changing just one nuclearparameter, e.g. RMS radius. Change in nuclear defor-mation ( β ) is also needed. Note that this might be theonly way of study nuclear deformation for even-even iso-topes by means of atomic spectroscopy. In odd isotopesone can also measure electric quadrupole hyperfine struc-ture. Note also that since three types of nuclear defor-mations (hole in the origin, quadrupole deformation andchange of thickness) are numerically equivalent in termsof producing similar IS, what is said above about nucleardeformation is also true about having a hole in nuclearcharge distribution; i.e. it can be studied by comparingIS in s − p / and s − p / transitions. Table III. Isotope shift (in cm − ) for the frequencies of the8 s − p and 9 s − p transitions in E120 + . Case A correspondsto nuclear parameters in Table I. Case B is a model case inwhich β = 0 for both isotopes and change in RMS radius ischosen to fit the shift of s states.Transition A B A-B8 s − p / . . − . s − p / . . . s − p / . . − . s − p / . . . C. Isotope shift for large change of neutronnumbers
It was suggested in Ref. [7] to use isotope shift calcu-lations to predict transition frequencies in SHE from ahypothetical island of stability. These metastable SHEdiffer from isotope-poor SHE produced in laboratoriesby large number of neutrons (large ∆ N ). This should betaken into account in the IS calculations. Calculationsreported above use the RPA method which assumes thatthe change in nuclear potential δV N is a small pertur-bation and ignores non-linear in δV N contributions. InSHE with large ∆ N non-linear in δV N contributions arelikely to be important and should not be thrown away.The most obvious way to do calculations properly is tocalculate energy levels for each isotope and then take thedifference. This does not work for light atoms becausethe IS is small and obtaining it as a difference of large al-most equal numbers leads to numerical instabilities. For-tunately, IS in SHE is sufficiently large to ensure stableresults. Even for neighbouring isotopes taking the differ-ence between two RHF calculations produce result whichare very close to the RPA calculations. For large ∆ N , thecalculations based on the difference between two isotopesare preferable because they include non-linear contribu- Figure 3. Fractional deviation from average value for isotopeshift ratio (black solid line); field shift constant F (blue shortdashed lines) and modified field shift constant ˜ F (red longdashed lines) for spherically symmetric and deformed nuclei.Lines, corresponding to spherically symmetric nuclei marketwith ”o”; lines corresponding to deformed nuclei market with”0”. tions.It is customary to present FIS as a formula in whichelectron and nuclear variables are separated. Standardformula reads FIS = F δ h r i . (9)It is assumed that the electron structure factor F doesnot depend on nuclear variables. This formula worksvery well in light atoms and widely used even for atomsclose to the end of known periodic table (e.g. for No, Z = 102, [11]). It was shown in Ref. [10] that relativisticcorrections lead to a different formulaFIS = ˜ F δ h r γ i , (10)where γ = p − ( αZ ) . New electron structure constant˜ F does not depend on nuclei. The formula was obtainedby considering spherical nuclei with uniform change dis-tribution. Below we study the performance of both for-mulae (9) and (10) for deformed nuclei. We calculate iso-tope shifts for the 8 s − p / and 8 s − p / transitions forall even isotopes of E120 + from A =294 to A =306. Wetake nuclear parameters β and RMS radius from Ref. [18](see Table I). We also consider a model case in which allconsidered nuclei are assumed to be spherically symmet-ric ( β = 0). IS is calculated for pairs of neighbouringisotopes using the RPA method as described above. Theconstants F and ˜ F are found using (9) and (10). Thecalculations repeated for both transitions for seven pairs of isotopes. In the end we have fourteen values of iso-tope shift and fourteen values of F and ˜ F . The resultsare presented on Fig. 3 in terms of fractional deviationsof the considered values from their average values, e.g. δ ( F/F ) i = ( F i − h F i ) / h F i , where h F i = P F i /
7. Wepresent on Fig. 3 the variation of the ratio of isotopeshifts in two transitions and variations of F and ˜ F forboth transitions. However, the variations for two transi-tions are too similar to see the difference on the graph.Fig. 3 shows that the ratio of the isotope shifts remainsconstant to very high precision. However, neither for-mula (9) or (10) works well. The value of F in (9) tendsto drift in one direction leading to large variations forlarge difference in neutron numbers. This is similar forboth cases, symmetrical and deformed nuclei. In con-trast, formula (10) works very well for spherical nuclei,showing only about 0.01% variation for ˜ F in the consid-ered interval. However, the formula does not work so wellfor deformed nuclei. The value for ˜ F jumps up and downby several per cent from one isotope to another. This isprobably because the value of h r γ i depends on two nu-clear parameters, nuclear deformation parameter β andnuclear RMS radius, making its behaviour irregular.Note that the difference in the value of F for neigh-bouring isotopes usually does not exceed 1% for bothspherical and deformed nuclei. With this accuracy for-mula (9) can be used for neighbouring isotopes to extractthe change of nuclear RMS radius from isotope shift mea-surements (see, e.g. [11]). Keeping in mind that the valueof F depends on isotope, the calculations should be per-formed for one of isotopes of interest (or for both, takingthen an average value). IV. CONCLUSIONS
We studied the effects of nuclear deformations on thefield isotope shift in SHE. We demonstrated that mak-ing a hole in nuclear charge distribution and havingquadrupole deformation can be reduced to changing nu-clear skin thickness. On the other hand, changing in skinthickness is not totally equivalent to change of nuclearRMS radius. There is small difference in energy shift ofthe p / states compared to states of other symmetries.With sufficiently accurate measurements of the IS thisdifference can probably be used to study nuclear defor-mations in even nuclei. The total effect of the nuclearhole on the isotope shift is up to ∼ ∼ V. ACKNOWLEDGEMENTS
This work was supported by the Australian Re-search Council and the Gutenberg Fellowship. The au-thors are grateful to Anatoli Afanasjev, Jie Meng, Bao- hua Sun, Jorge Piekarewicz, Shashi K. Dhiman, JacekDobaczewski, Peter Ring, Stephane Goriely, ZhongzhouRen, Ning Wang, Yibin Qian, Witold Nazarewicz, Paul-Gerhard Reinhard, Peter Schwerdtfeger and Bryce Lack-enby for valuable discussions. [1] Y. T. Oganessian, V. K. Utyonkov, Y. V. Lobanov,F. S. Abdullin, A. N. Polyakov, I. V. Shirokovsky, Y. S.Tsyganov, G. G. Gulbekian, S. L. Bogomolov, B. N.Gikal, et al., Nuclear Physics A , 109 (2004).[2] J. H. Hamilton, S. Hofmann, and Y. T. Oganessian, An-nual Review of Nuclear and Particle Science , 383(2013).[3] G. M. Fuller, A. Kusenko, and V. Takhistov, PhysicalReview Letters , 061101 (2017).[4] S. Goriely, A. Bauswein, and H.-T. Janka, The Astro-physical Journal Letters , L32 (2011).[5] A. Frebel and T. C. Beers, Physics Today , 30 (2018),ISSN 0031-9228.[6] V. F. Gopka, A. V. Yushchenko, V. A. Yushchenko, I. V.Panov, and C. Kim, Kinematics and Physics of CelestialBodies , 89 (2008).[7] V. A. Dzuba, V. V. Flambaum, and J. K. Webb, PhysicalReview A , 062515 (2017).[8] E. Witten, Physical Review D , 272 (1984).[9] J. C. Berengut, D. Budker, C. Delaunay, V. V. Flam-baum, C. Frugiuele, E. Fuchs, C. Grojean, R. Harnik,R. Ozeri, G. Perez, et al., arXiv:1704.05068 (2017), (ac-cepted by Phys. Rev. Lett.).[10] Isotope shift, nonlinearity of King plots, and the searchfor new particles V. V. Flambaum, A. J. Geddes, and A.V. Viatkina Phys. Rev. A 97, 032510 (2018).[11] S. Raeder, D. Ackermann, H. Backe, et al , Phys. Rev.Lett. , 232503 (2018).[12] B. G. C. Lackenby, V. A. Dzuba, and V. V. Flambaum,Phys. Rev. A , 022518 (2018). [13] B. G. C. Lackenby, V. A. Dzuba, and V. V. Flambaum,Phys. Rev. A , 042509 (2019).[14] G. Racah, Nature , 723 (1932).[15] J. E. Rosenthal and G. Breit, Physical Review , 459(1932).[16] W. H. King, Isotope Shifts in Atomic Spectra (SpringerScience & Business Media, 2013).[17] V. M. Shabaev, Journal of Physics B: Atomic, Molecularand Optical Physics , 1103 (1993).[18] S. E. Agbemava, A. V. Afanasjev, T. Nakatsukasa, andP. Ring, Phys. Rev. C , 054310 (2015).[19] T. H. Dinh, V. A. Dzuba, V. V. Flambaum and J. S. M.Ginges, Phys. Rev. A , 022507 (2008).[20] V. A. Dzuba, Phys. Rev. A , 042516 (2013).[21] V. A. Dzuba, V. V. Flambaum, P. G. Silvestrov, O.P. Sushkov, J. Phys. B: At. Mol. Phys. , , 1399-1412(1987).[22] M. Bender, K. Rutz, P.-G. Reinhard, J. A. Maruhn, andW. Greiner, Phys. Rev. C , 034304 (1999).[23] J. Decharge, J.-F. Berger, K. Dietrich, and M. S. Weiss,Phys. Lett. B , 275 (1999).[24] A. V. Afanasjev and S. Frauendorf, Phys. Rev. C ,024308 (2005).[25] D. L. Clark, M. E. Cage, D. A. Lewis, and G. W. Green-lees, Phys. Rev. A , 239 (1979).[26] J. H. Heisenberg, J. S. McCarthy, I. Sick, and M. R.Yearian, Nucl. Phys. A164