Nuclear charge densities in spherical and deformed nuclei: towards precise calculations of charge radii
SSpin-orbit contribution to the nuclear charge density
Paul-Gerhard Reinhard and Witold Nazarewicz
2, 3 Institut f¨ur Theoretische Physik, Universit¨at Erlangen, Erlangen, Germany Facility for Rare Isotope Beams, Michigan State University, East Lansing, Michigan 48824, USA Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824, USA
Background:
Precise measurements of atomic transitions affected by electron-nucleus hyperfine interactionsoffer sensitivity to explore basic properties of the atomic nucleus and study fundamental symmetries, includingthe search for new physics beyond the Standard Model of particle physics. In particular, such measurements,augmented by atomic and nuclear calculations, will enable extraction of the higher-order radial moments of thecharge density distribution.
Purpose:
The nuclear charge density is composed of the proton point distribution folded with the nucleoniccharge distributions. The latter induce subtle relativistic corrections due to the coupling of nucleon magneticmoments with the nuclear spin-orbit density. We assess the precision of nuclear charge density calculations bystudying the behavior of relativistic corrections. Special attention has been paid to the magnetic spin-orbit densityassociated with the local variations of the spin-orbit current.
Methods:
The calculations are performed in the framework of self-consistent mean-field theory using Skyrmeenergy density functionals and density-dependent pairing force. We used the general expression for the spin-orbitform factor that is valid for spherical and deformed nuclei.
Results:
We studied the impact of the spin-orbit densities on the charge radii and the fourth radial moments (cid:104) r (cid:105) of spherical and deformed nuclei. The corresponding corrections to charge radii show strong shell fluctuationsand sometimes are quite appreciable when aiming at high-precision predictions. Conclusions:
To establish reliable constraints on the existence of new forces from isotope shift measurements,precise calculations of nuclear charge densities are needed. The proper inclusion of the spin-orbit charge densityis essential when aiming at extraction of tiny effects.
I. INTRODUCTION
High-precision studies of atomic transitions offer com-plementary information on the structure of atomic nu-cleus and fundamental symmetries, including hints ofnew physics beyond the Standard Model of particlephysics [1–5]. In particular, precise measurements oftransition frequencies allow extraction of tiny variationsin the root-mean-square (rms) nuclear charge radii acrosslong isotopic chains of stable and radioactive nuclei [6–12]. This carries the potential to constrain the existenceof new forces and hypothetical particles with unprece-dented sensitivity [2–4, 6, 13–16]. The theoretical find-ings have stimulated considerable developments in high-precision experimental techniques [17–20]. The new un-precedented level of precision offers sensitivity not onlyto explore new physics, but would also provide access tonuclear observables that have so far been elusive, suchas the higher-order radial moment (cid:104) r (cid:105) [15, 21, 22] thatcarries information on nuclear surface properties [23, 24].In order to extract structural information from atomicmeasurements, it is important for nuclear theory to pro-duce reliable predictions of nuclear charge densities andcurrents. Nuclear models usually yield the proton andneutron densities from which the nuclear charge den-sity can be extracted by considering several corrections[25, 26]. The spurious center-of-mass (c.m.) motion iscorrected by an unfolding with the width of the centre-of-mass vibrations. The nucleon structure is taken intoaccount by folding with the intrinsic form factor of thefree nucleons expressed in terms of the Sachs form fac- tors. The leading part comes from the folding with thenucleonic charge form factors. Moreover, there are themagnetic form factors of the nucleons which contributeto the charge density through the coupling to the nuclearspin-orbit density. The latter contributions are called thespin-orbit terms in the following. Together with the rela-tivistic Darwin-Foldy term, they constitute the relativis-tic corrections to the charge density [25, 27–31]In calculations of charge radii based on the self-consistent mean-field theory [26, 32–34], the proton andneutron form factor can be expressed in terms of single-particle Hartree-Fock (HF) or Hartree-Fock-Bogoliubov(HFB) wave functions. The spin-orbit correction is oftenneglected [5, 24, 35, 36] or computed in spherical geom-etry [37–40], which is reasonable for prediction that donot require high accuracy.In this work, we apply the self-consistent mean-fieldtheory to study nuclear charge densities and charge radiiwith emphasis on the intrinsic nucleon form factors andthe relativistic contributions that are essential for the ac-curacy required for precision studies. The paper is orga-nized as follows. The definitions of corrections to chargedensities and charge radii given in Sec. II. Section III de-scribes the theoretical approach used. This is followed bydescription of results in Sec.IV. Finally, Sec. V containsthe conclusions of our study. a r X i v : . [ nu c l - t h ] J a n II. KEY OBSERVABLESA. The charge form factor
The nuclear charge density is uniquely related to thenuclear charge form factor F c : ρ c ( r ) = 1(2 π ) (cid:90) d qe − i q · r F c ( q ) . (1)The latter is the quantity measured by electron scatteringexperiments [27] and used for conveniently including thefolding by the intrinsic nucleon form factors and the c.m.motion. In the following, we recall the relativistic andnon-relativistic expressions for F c .
1. The magnetic contribution to charge density in therelativistic mean-field theory
The relativistic operator for the nuclear charge formfactor ˆ F c is the zeroth component of the charge currentˆ J and reads [25]:ˆ F c ( q ) ≡ ˆ J ( q ) = (cid:88) t ∈{ p,n } f ,t ( q )ˆ γ − f ,t ( q ) (cid:126) mc ˆ α · q , (2)where m is the nucleon mass, ˆ α is the three-vector ofDirac matrices [41], f ,t ( q ) is the intrinsic nucleon chargeform factor, and f ,t ( q ) is the intrinsic nucleon magneticform factor. The charge form factor can be written as: F c ( q ) = (cid:88) t ∈{ p,n } (cid:20) f ,t ( q ) F t ( q ) − f ,t ( q ) F tens ,t ( q ) (cid:126) mc (cid:21) , (3)where the form factors F t ( q ) = (cid:90) d r e i q · r ρ t ( r ) ,F tens ,t ( q ) = (cid:90) d r e i q · r ρ tens ,t ( r ) , (4)can be expressed in terms of relativistic densities ρ t ( r ) = (cid:88) α ∈ t v α ψ α ˆ γ ψ α ,ρ tens ,t ( r ) = − i ∇ · (cid:88) α ∈ t v α ψ α i (cid:18) σσ (cid:19) ψ α , (5)where the four-component eigenstate of the Dirac equa-tion ψ α is the nucleonic single-particle (s.p.) wave func-tion, the v α are the BCS or HFB pairing occupations,and the tensor density ρ tens ,t together with the nucleonmagnetic form factors yield the magnetic contribution tothe charge density.The intrinsic nucleon form factors are usually ex-pressed in terms of the Sachs form factors G E and G M as f ,t ( q ) = G E,t ( q ) + q D µ t G M,t ( q )1 + q D ,f ,t ( q ) = − G E,t ( q ) + µ t G M,t ( q )1 + q D , (6)where D = (cid:126) (2 mc ) (7)and µ t are the magnetic moments of the nucleon: µ p =2 .
79 and µ n = − .
2. The magnetic contribution to charge density innon-relativistic mean-field theory
The corresponding expression for the form factor innon-relativistic models is obtained by the expansion inpowers of
D ∝ m − up to first order [27]. In this non-relativistic limit, the charge form factor reads [25]: F c ( q ) = (cid:88) t ∈{ p,n } (cid:104) G E,t ( q ) (cid:0) − q D (cid:1) F t ( q ) − D [2 µ t G M ( q ) − G E,t ( q )] F (cid:96)s,t ( q ) (cid:105) . (8)The form factors F t ( q ) = (cid:90) d r e i q · r ρ t ( r ) ,F (cid:96)s,t ( q ) = (cid:90) d r e i q · r ∇ · J t ( r ) (9)are given in terms of the local particle densities ρ t ( r ) andspin-orbit currents J t ( r ): ρ t ( r ) = (cid:88) α | ϕ tα ( r ) | , J t ( r ) = i (cid:88) α ϕ ∗ tα ( r )( σ × ∇ ) ϕ tα ( r ) , (10)with ϕ tα ( r ) being the canonical HFB (or BCS) wavefunctions and v α the corresponding pairing occupationcoefficients. Note that the above derivation does notmake assumptions about spatial symmetries. Conse-quently, the expressions can be applied in 3D HFB codesas well as in 2D axial or 1D spherical HFB calculations.There is a subtle difference in the interpretation ofthe relativistic and non-relativistic expressions. In therelativistic form factor, the magnetic contributions areassociated simply with the tensor density. In the non-relativistic case, this becomes the spin-orbit density andit turns out to be of the same order D as the relativis-tic Darwin term G E D . It is customary, to consider thepurely electric contribution G E,t F t as the leading termand everything else as relativistic correction. B. The charge radius
The squared charge radius is obtained from the chargeform factor F c ( q ) as (cid:104) r c (cid:105) = − ∇ F c (q) (cid:12)(cid:12)(cid:12) q =0 F c (0) . (11)For the reflection-symmetric nuclei, all form factors F (q)in Eq. (8) fulfill the condition: ∇ F (q) = 0. The productrule with ∇ then yields only terms with zeroth or secondderivative. We abbreviate ∇ f | q =0 = f (cid:48)(cid:48) for each factorin the form factor and insert the values in zeroth order G E,p (0) = 1, G E,n (0) = 0, G M (0) = 1, F p (0) = Z , F n (0) = N , and F (cid:96)s,t ( q )(0) = 0. This yields at q = 0: F c = Z, (12) F (cid:48)(cid:48) c = F (cid:48)(cid:48) p + ZG (cid:48)(cid:48) E,p − Z D + N G (cid:48)(cid:48)
E,n − (2 µ p − D F (cid:48)(cid:48) (cid:96)s,p − µ n D F (cid:48)(cid:48) (cid:96)s,n . (13)The second derivatives can be written as F (cid:48)(cid:48) p = (cid:90) d r r ρ p ( r ) ≡ Z (cid:104) r (cid:105) pp , (14a) F (cid:48)(cid:48) (cid:96)s,p = (cid:90) d r r ∇ · J p ( r )= − (cid:90) d r r · J p ( r ) ≡ − Z (cid:104) ˆ σ · ˆ l (cid:105) p , (14b) F (cid:48)(cid:48) (cid:96)s,n = (cid:90) d r r ∇ · J n ( r ) = − N (cid:104) ˆ σ · ˆ l (cid:105) n (14c)and similarly G (cid:48)(cid:48) E,p = (cid:104) r (cid:105) (intr) p , G (cid:48)(cid:48) E,n = (cid:104) r (cid:105) (intr) n . Inthe above expression, (cid:104) r (cid:105) pp indicates the point-protonradius as it emerges directly from the mean-field calcu-lation. By combining all contributions, we obtain theexpression or the average squared charge radius: (cid:104) r c (cid:105) = (cid:104) r pp (cid:105) + (cid:104) r p (cid:105) (intr) + NZ (cid:104) r n (cid:105) (intr) + (cid:104) r (cid:105) (rel) , (15)where (cid:104) r (cid:105) (rel) = 3 D + ( µ p − ) D(cid:104) ˆ σ · ˆ l (cid:105) p + µ n NZ D(cid:104) ˆ σ · ˆ l (cid:105) n . (16)is the relativistic contribution to the charge radius. Asdiscussed above, it consists of the Darwin-Foldy (DF)term 3 D and the spin-orbit corrections.The form of the spin-orbit terms in Eq. (14) that in-volves ∇ · J is valid for arbitrary mean-field geometry. Thesecond form, involving (cid:104) ˆ σ · ˆ l (cid:105) t , is particularly useful if thespherical geometry is imposed. In this case, the expecta-tion value of the spin-orbit term becomes independent ofthe radial profile of the wave functions and the expressionreduces (for each nucleon type) to (cid:104) ˆ σ · ˆ l (cid:105) = (cid:80) α v α ( σ(cid:96) ) α where ( σl ) α = j α ( j α + 1) − l α ( l α + 1) − , which is (cid:96) α for j α = (cid:96) α + 1 / − ( (cid:96) α + 1) for j = (cid:96) α − /
2. It isimmediately seen that if both sub-shells of the spin-orbit doublet are occupied with the same weight, their con-tribution to the (cid:96)s term in (15) vanishes (spin-saturatedcase). The maximal spin-orbit contribution is attainedwhen the lower-energy member of the spin-orbit dou-blet is fully occupied and the upper-energy member with j = (cid:96) − / III. COMPUTATIONAL FRAMEWORK
The examples presented here were computed with non-relativistic nuclear density-functional theory (DFT) us-ing the well known Skyrme energy-density functional, fora detailed review see [33]. In our applications, we em-ploy the Skyrme parametrization SV-bas from Ref. [42]which has been optimized to a large experimental cali-bration dataset including information on several exoticnuclei. This is appropriate for the present study, whichcovers long isotopic and isotonic chains. We have re-peated calculations presented in this work with otherSkyrme parametrizations and obtained results that arevery similar to those with SV-bas.To cover deformed nuclei, we use the recently publishedcode
SkyAx which allows for deformed axially symmetricshapes [43]. A word is in order about the treatment ofpairing. The code
SkyAx implements pairing at the BCSlevel using a soft cutoff in pairing space with the cutoffprofile as used in Ref. [44] w α = [1 + exp (( ε α − ( (cid:15) F ,q α + (cid:15) cut )) / ∆ (cid:15) )] − (17)where ε α are the s.p. energies, (cid:15) cut marks the cutoffband, and ∆ (cid:15) = (cid:15) cut /
10 is its width. We use a dynamicalsetting of the pairing band where (cid:15) cut is adjusted suchthat a fixed number of nucleons N q + η cut N / q is includedin the sum (cid:80) α ∈ q w α [45], here with η cut = 1 .
65 as wasdone in the fits of SV-bas in Ref. [42].It is to be noted that mere BCS is not always appropri-ate for nuclei at the edges of stability [9, 46, 47], for whichone should use, in principle, the full HFB framework. Inthis study, however, we limit the selection generally tonuclei whose proton and neutron Fermi energies are suf-ficiently bound so the unphysical particle gas effects areavoided.The intrinsic form factors of the nucleons were com-puted as in Ref. [34] with the Sachs form factors takenfrom Refs. [48, 49].It is to be noted that we consider here subtle ef-fects stemming from the relativistic corrections that placegreat demands on the accuracy of underlying calcula-tions. In order to compute charge radii with precisionbetter than 0.001 fm, the calculations were carried outwith enhanced demands on grid spacing, box size, Fouriertransform, and HF+BCS termination criteria. c h a r g e d e n s it y (f m - ) r (fm) -6 -6
9 10 11 1201∙10 -6
9 10 11 12 Ca Ca ρ c ρ p FIG. 1. Comparison of charge and proton densities for Caand Ca computed with SV-bas. The density dependence atlarge distances is shown in the insets.
IV. RESULTS
We shall start from a pedagogical Fig. 1 showing thecharge density (1) predicted with SV-bas for Ca and Ca. It is seen that at large distances the neutron chargedistribution and, to a lesser extent, the neutron spin-orbitdensity produce a negative contribution to the chargedensity in Ca, while the effect of correction terms to theproton density in Ca is less pronounced. The resultingnegative contribution to the charge radius helps bringingthe charge radius of Ca very close to the value in Ca[6, 38, 39, 50, 51].Figure 2 shows the predicted rms proton and chargeradii along selected isotopic chains which cover sphericaland deformed nuclei. The nucleonic and relativistic cor-rections are of the order of 0.05 fm. This suggests thatin many practical applications one can safely use the ap-proximate relation [25] (cid:104) r c (cid:105) ≈ (cid:104) r pp (cid:105) + (cid:104) r p (cid:105) (intr) + ( N/Z ) (cid:104) r n (cid:105) (intr) (18)with the constant proton and neutron charge radii: (cid:104) r p (cid:105) (intr) = 0 .
848 fm [52] and (cid:104) r n (cid:105) (intr) = − . [53], which is the radius correction (15) without the rel-ativistic term (cid:104) r (cid:105) (rel) . (Note that the previous imple-mentation of the proton form factor in Refs. [34, 48, 49]implies the older value of the proton radius (cid:104) r p (cid:105) (intr) =0 .
854 fm which amounts to a constant reduction of about0.001 fm, with no effect on trends.) On the other hand,the relativistic correction must be included in precisioncalculations (which aims at average uncertainties as low
CaCr Sr Sn YbPb Uneutron number protoncharge r m s r a d i u s (f m ) FIG. 2. The rms point-proton (blue) and charge radii (red)for isotopic chains of magic (Ca, Sn, Pb) and open-shell (Cr,Sr, Yb, U) nuclei computed with SV-bas. as 0.015 fm) and studies of small local variations of chargeradii such as the discontinuities across shell closures,which requires accuracy on charge radius prediction wellbelow 0.01 fm [10]. neutron number fullno-rel (f m ) CaCr Sr Sn Yb Pb U r c r pp FIG. 3. The difference (cid:104) r c (cid:105) − (cid:104) r pp (cid:105) for several isotopic chains.For comparison, the results without relativistic contribution(16) are shown. Magic numbers are indicated by verticaldashed lines. Positions of unique-parity shells are marked. To show the effect of spin-obit correction on chargeradii in detail, Fig. 3 displays the difference (cid:104) r c (cid:105) − (cid:104) r pp (cid:105) for the isotopic chains of Fig. 2. For each chain, theresults without the spin-orbit term exhibit a smooth de-crease with neutron number that is consistent with thebehavior of the intrinsic neutron charge distribution termin Eq. (18). As expected, the spin-orbit contributionstrongly fluctuates with N . In the regions correspond-ing to the gradual occupation of high- j unique-parityshells the spin-orbit correction rapidly decreases due tothe negative value of µ n . The local increasing trends canbe associated with the gradual occupation of the upperspin-orbit partner.The most dramatic local variation of relativistic contri-butions is predicted between Ca and Ca (due to thepopulation of 1 f / and 2 p / neutron shells) and in theSr chain around N = 50 (due to the population of 1 g / and 2 d / neutron shells). In heavy nuclei the variationstend to be more gradual due to the fragmentation of thespin-orbit strength and the smoothing effect of pairing.
20 28 50 82 126 proton number fullno-rel (f m ) r c r pp FIG. 4. Similar to Fig. 3 but for several isotonic chains ofsemi-magic nuclei.
Figure 4 illustrates the behavior of (cid:104) r c (cid:105) − (cid:104) r pp (cid:105) alongthe isotonic chains of semi-magic nuclei. Here, due to thepositive value of ( µ p − / Z in the regions in which high- j shell areoccupied. The largest shell effect is predicted for N =28; it is see in the rapid rise of the spin-orbit correctionbetween Ca and Ni. Appreciable kinks in (cid:104) r c (cid:105) areexpected at Z = 50 and 82 where the j = (cid:96) + 1 / j = (cid:96) − / (cid:104) r c (cid:105) − (cid:104) r pp (cid:105) alongthe Yb chain. We note that the deformed Yb isotopesare of particular interest in the context of ongoing experi-mental searches of new physics [20]. The intrinsic protoncontribution, DF, and c.m. terms do not vary with N .The intrinsic neutron contribution shows the trivial linear N/Z dependence. Note that in the deformed region thespin-orbit contributions change gradually as the single-particle spin-orbit strength becomes highly fragmentedby deformation and pairing. The prolate-to oblate shapetransitions seen in the extremely proton-rich and ex- tremely neutron-rich isotopes result in noticeable vari-ations of spin-orbit contributions. -0.200.2 90 100 110 120 d e f o r m a ti on β neutron number -0.200.20.40.6 c on t r i bu ti on t o (f m ) total p r o t on s . o . n e u t r on s . o . DF c.m. r c r pp r p (intr) p r o t on s . o . r n (intr) N/Z
FIG. 5. Top: various corrections to (cid:104) r c (cid:105)−(cid:104) r pp (cid:105) along the chainof Yb isotopes. Bottom: dimensionless quadrupole shape de-formation parameter β = 4 π (cid:104) Q (cid:105) / (5 AR ), where Q is themass quadrupole moment and R the diffraction radius. As demonstrated recently [23], the fourth radial mo-ment (cid:104) r (cid:105) can be directly related to the surface thickness σ of nuclear density. (See also discussion in Ref. [39].)Precise knowledge of (cid:104) r (cid:105) is essential to establish reli-able constraints on new physics. We compute (cid:104) r (cid:105) , σ ,and the diffraction radius R consistently from the chargeform factor F c . The R and σ are deduced from firstzero and first maximum of F c [43]. The fourth radialmoment (cid:104) r (cid:105) is computed from the charge density as ob-tained from F c by the inverse Fourier transform (whichwe find the simplest and most robust procedure). Inorder to demonstrate the sensitivity of (cid:104) r c (cid:105) and bulk nu-clear surface properties on the spin-orbit charge densitiesFigs. 6 and 7 illustrate the impact of relativistic correc-tions on (cid:104) r (cid:105) , surface thickness σ , and diffraction radii R (see Ref. [23] for definitions). It is seen that the shellfluctuations of relativistic corrections to these quantitiesare appreciable for (cid:104) r (cid:105) and σ while R is less sensitive. V. CONCLUSIONS
In this study, we investigated the nucleonic correctionsto the nuclear charge density and charge radii. The basicnucleonic corrections stem from folding with the nucle- -0.0100.01 20 40 60 80 100 120 140 1600.10.110.120.132030405060 neutron number R c − R pp ( f m ) σ c − σ pp ( f m ) r c r pp ( f m ) fullno-rel Ca CrSr Sn YbPb U
FIG. 6. The corrections to the fourth radial moment (cid:104) r (cid:105) ,surface thickness σ , and diffraction radius R along several iso-topic chains. For comparison, the results without relativisticcontribution are also shown. ons intrinsic charge distribution. At a more detailed levelcome the relativistic corrections covering the Darwin-Foldy term and the coupling of nucleon magnetic mo-ments to the spin-orbit fields. The calculations were per-formed in the framework of self-consistent mean-field the-ory using Skyrme energy density functionals and density-dependent pairing force. We used the general expressionfor the spin-orbit form factor that is valid for deformednuclei. The main conclusions and results of our studycan be summarized as follows:(i) The nucleonic corrections are of the order of0.05 fm. While the basic nucleonic corrections tocharge radii do not depend on shell structure andcan be simply accounted for, the spin-orbit correc-tions strongly vary with particle number and re-quire careful modelling. Although the basic nucle-onic corrections constitute the largest contribution,relativistic corrections can amount up to 0.01 fmand need to be accounted for in precision studies.(ii) Relativistic corrections with their pronounced shelleffect become crucial if one is after small local vari- -0.02-0.0100.010.020.03 10 20 30 40 50 60 70 80 90 proton number R c − R pp ( f m ) σ c − σ pp ( f m ) r c r pp ( f m )
20 28 50 82 126
FIG. 7. Similar to Fig. 6 but for several isotonic chains ofsemi-magic nuclei. ations in charge radii. The required accuracy onpredicted values should then be well below 0.01 fmand both corrections must be considered.(iii) The discontinuities in charge radii across shell clo-sures results in kinks, which are well below 0.01 fm,[10]. Since some of the nuclei of interest are open-shell systems, see, e.g., Ref. [54], contributions fromdeformed spin-orbit densities can be appreciable.(iv) The spin-orbit correction to charge radii exhibitsappreciable shell fluctuations; it is important toconsider them when aiming at extraction of tinyeffects due to new physics.(v) Deformation and pairing give rise to the fragmen-tation of the spin-orbit strength. This results ina smoothing of the relativistic correction to chargeradii.(vi) It will be interesting to investigate experimentallythe charge radii along the isotonic chains of semi-magic nuclei. Here, our calculations predict a largeshell effect for N = 28 that is characterized in therapid rise of the spin-orbit correction between Caand Ni. Also, appreciable kinks in (cid:104) r c (cid:105) are ex-pected at Z = 50 and 82 due to the closing of pro-ton 1 g / and 1 h / intruder shells and filling the1 g / and 1 h / spin-orbit partner shells ACKNOWLEDGMENTS
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