Charge radii of exotic neon and magnesium isotopes
CCharge radii of exotic neon and magnesium isotopes
S. J. Novario,
1, 2
G. Hagen,
2, 1, 3
G. R. Jansen,
4, 2 and T. Papenbrock
1, 2 Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996, USA Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA TRIUMF, 4004 Wesbrook Mall, Vancouver BC, V6T 2A3, Canada National Center for Computational Sciences, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
We compute the charge radii of even-mass neon and magnesium isotopes from neutron number N = 8 to the dripline. Our calculations are based on nucleon-nucleon and three-nucleon potentialsfrom chiral effective field theory that include delta isobars. These potentials yield an accurate satu-ration point and symmetry energy of nuclear matter. We use the coupled-cluster method and startfrom an axially symmetric reference state. Binding energies and two-neutron separation energieslargely agree with data and the dripline in neon is accurate. The computed charge radii have anestimated uncertainty of about 2-3% and are accurate for many isotopes where data exist. Finerdetails such as isotope shifts, however, are not accurately reproduced. Chiral potentials correctlyyield the subshell closure at N = 14 and also a decrease in charge radii at N = 8 (observed inneon and predicted for magnesium). They yield a continued increase of charge radii as neutrons areadded beyond N = 14 yet underestimate the large increase at N = 20 in magnesium. Introduction. —
The radii of atomic nuclei carry in-formation about their structure as isotopic trends reflectchanges in nuclear deformation, shell structure, super-conductivity (pairing), and weak binding. The differencebetween the radii of the neutron and proton distributionsof an atomic nucleus also impact the structure of neutronstars. Matter radii are usually extracted from reactionswith strongly interacting probes, which requires a model-dependent analysis [1, 2]. In contrast, electric chargeradii (and more recently also weak charge radii) can bedetermined using the precisely known electroweak inter-action [3, 4]. Precision measurements of nuclear chargeradii have contributed much to our understanding of sta-ble nuclei and rare isotopes, and they continue to chal-lenge nuclear structure theory [5–8].In the past two decades we have seen a lot of progressin ab initio computations of nuclei, i.e. calculations thatemploy only controlled approximations and are based onHamiltonians that link the nuclear many-body problemto the nucleon-nucleon and few-nucleon systems. Vir-tually exact methods [9–12] scale exponentially with in-creasing mass number and depend on the exponential in-crease of available computational cycles for progress. Agame changer has been combining ideas and soft interac-tions from effective field theory (EFT) [13–18] and therenormalization group [19–21], with approximate (butsystematically improvable) approaches that scale poly-nomially with mass number. Examples of such meth-ods are coupled-cluster theory [22–24], in-medium sim-ilarity renormalization group [25, 26], nuclear latticeEFT [27, 28], and self-consistent Green’s function ap-proaches [29, 30].Nuclei as heavy as
Sn have now been computedwithin this framework [31], and the first survey of nu-clei up to mass 50 or so has appeared [32]. Comput-ing nuclei is much more costly than using, e.g., nu-clear density-functional theory (DFT) [33–35]. However,the ever-increasing availability of computational cyclesmakes these computations both feasible and increasingly
20 22 24 26 28 30 32 34 36 38 40 A R c h ( f m ) MgMg expNNLO GO (394)NNLO GO (450) FIG. 1. (Color online) Charge radii for magnesium iso-topes with even mass numbers computed with the potentials∆NNLO GO (394) (red) and ∆NNLO GO (450) (blue) comparedto data (solid bars) [40]. The model spaces consist of 13 oscil-lator shells with oscillator frequencies (cid:126) ω = 12 and 16 MeV,as indicated by the bands. affordable. The approach based on Hamiltonians offersthe possibility to compute excited states, to perform sym-metry projections, and to treat currents and Hamiltoni-ans consistently. It migh also be possible to link suchHamiltonians back to quantum chromodynamics [36–39].In this paper, we compute the charge radii of neutron-rich isotopes of neon and magnesium. These nuclei areat the center of the island of inversion [41, 42] and atthe focus of current experimental interests. Charge radiiare known for − Ne [43, 44] and − Mg [40] (seeFig. 1), leaving much to explore. Of particular interestis the impact (or lack thereof) of the “magic” neutronnumbers N = 8 ,
14, and 20 on charge radii, the onset ofdeformation past N = 20, and the rotational structureof neutron-rich isotopes as the dripline is approached [45,46]. a r X i v : . [ nu c l - t h ] J u l Among the many available interactions from chiralEFT [15–17, 47–51], NNLO sat [18] and ∆NNLO GO [52–54] stand out through their quality in describing nuclearradii. These interactions contain pion physics, three-nucleon forces, and – in the case of ∆NNLO GO – ef-fects of the ∆ isobars. Both interactions have been con-strained by data on the nucleon-nucleon interaction, andnuclei with mass numbers A = 3 ,
4. While NNLO sat alsowas constrained by binding energies and radii of nucleias heavy as oxygen, ∆NNLO GO was constrained by thebinding energy, density and symmetry energy of nuclearmatter at its saturation point. These potentials use theleading-order three-body forces from chiral EFT [55]. Inthis study we will employ two ∆NNLO GO interactionswhich differ by their respective momentum cutoffs of 394and 450 MeV c − . Theoretical approach.—
Our coupled-cluster calcula-tions start from an axially deformed product state builtfrom natural orbitals. To construct the natural orbitalswe perform a Hartree-Fock calculation that keeps axialsymmetry, parity, and time-reversal symmetry, but is al-lowed to break rotational invariance. Thus, the J z com-ponent of angular momentum is conserved, and single-particle orbitals come in Kramer-degenerate pairs with ± j z . For open-shell nuclei, we fill the partially occupiedneutron and proton shells at the Fermi surface from lowto high values of | j z | ; this creates a prolate Hartree-Fockreference. Following Ref. [56] we use this state to com-pute its density matrix in second-order perturbation the-ory and diagonalize it to obtain the natural orbitals. Asshown in Fig. 2 and discussed below, natural orbitals im-prove the convergence of the ground-state energies withrespect to the number of three-particle–three-hole (3 p –3 h amplitudes in the coupled-cluster wave-function.The natural orbital basis is spanned by up to 13 spher-ical harmonic oscillator shells. We present results for twodifferent oscillator frequencies ( (cid:126) ω = 12 and 16 MeV) togauge the model-space dependence. The three-nucleoninteraction had the additional energy cut of E =16 (cid:126) ω , which is sufficient to converge the energies andradii reported in this work.The breaking of rotational symmetry by the referencestate is consistent with the emergent symmetry breakingand captures the correct structure of the nontrivial vac-uum [57]. However, our approach lacks possible tri-axialdeformation and symmetry restoration, for which severalproposals exist [58–61]. Overcoming these limitations isthus possible but comes at a significant increase in com-putational cost: The loss of symmetries (either by per-mitting tri-axiality or by rotating the Hamiltonian duringprojection) significantly increases the number of non-zeroHamiltonian matrix elements and coupled-cluster ampli-tudes. To estimate the impact of symmetry restorationwe performed projection after variation of the deformedHartree-Fock states for all nuclei considered in this work,and found an energy gain from 3 to 6 MeV. This providesus with an upper limit on the energy that can be gainedthrough symmetry restoration, as we would expect that correlations beyond the mean-field partially restore bro-ken symmetries. We note that tri-axial deformations inthe ground-state are not expected to be significant forthe nuclei we study in this paper [62]. We finally notethat the axially-symmetric coupled-cluster computationsare an order of magnitude more expensive than thosethat keep rotational invariance. Fortunately, the avail-ability of leadership-class computing facilities and the useof graphics processor units (GPUs) now make such com-putations possible.Our calculations start from the “bare” Hamiltonian H = T kin − T CoM + V NN + V NNN (1)based on the ∆NNLO GO nucleon-nucleon and three-nucleon potentials V NN and V NNN , respectively. Here, T kin denotes the kinetic energy, and we subtract the ki-netic energy of the center of mass T CoM to remove thecenter-of-mass from the Hamiltonian. We express thisHamiltonian in terms of operators ˆ a † p and ˆ a q that createand annihilate a nucleon with quantum numbers q and p ,respectively, in the natural orbital basis. The Hamilto-nian H N is normal-ordered with respect to the referencestate, and we only retain up to normal-ordered two-bodyforces; we have H N = F N + V N , where the Fock term F N denotes the normal-ordered one-body part and V N thenormal-ordered two-body terms [63]. E n e r g y [ M e V ] Ne PAV ExperimentHartree-Fock orbitalsNatural orbitals
FIG. 2. (Color online) Ground state energy of Ne with re-spect to the number of included 3 p –3 h amplitudes, computedfrom the Hartree-Fock basis (blue circles connected by dashedline), and natural orbitals (red diamonds connected by fullline). For the Hartree-Fock basis we limited the number of 3 p –3 h excitations by the energy cut ˜ E pqr = ˜ e p + ˜ e q + ˜ e r < ˜ E ,where ˜ e p = | N p − N F | is the energy difference between thesingle-particle energies and the Fermi surface N F . The cutin the natural orbital basis is described in the main text.We used the ∆NNLO GO (394) potential and a model-spaceof 11 major spherical oscillator shells with the frequency (cid:126) ω = 16 MeV. The black solid line is the experimental value,while the gray dashed line includes the energy gain from pro-jection after variation of the Hartree-Fock result. The coupled-cluster method [22–24, 64–68] generatesa similarity-transformed Hamiltonian H N ≡ e − ˆ T H N e ˆ T , (2)using the cluster-excitation operatorˆ T = ˆ T + ˆ T + ˆ T · · · = (cid:88) ia t ai ˆ a † a ˆ a i + 14 (cid:88) ijab t abij ˆ a † a ˆ a † b ˆ a j ˆ a i + 136 (cid:88) ijkabc t abcijk ˆ a † a ˆ a † b ˆ a † c ˆ a k ˆ a j ˆ a i + · · · . (3)The operator ˆ T n creates n -particle– n -hole excitations ofthe reference state | ψ (cid:105) ≡ (cid:81) Ai =1 ˆ a † i | (cid:105) . Here and in whatfollows, labels i, j, k refer to single-particle states occu-pied in the reference state, while a, b, c are for unoccupiedstates.We truncate the expansion (3) at the 3 p –3 h level andinclude leading-order triples using the CCSDT-1 approx-imation [69, 70]. In this approximation e T ≈ e T + T + T ,and the amplitudes t ai , t abij , and t abcijk fulfill (cid:104) ψ ai | H N + H N ˆ T | ψ (cid:105) = 0 , (cid:104) ψ abij | H N + H N ˆ T | ψ (cid:105) = 0 , (cid:104) ψ abcijk | (cid:16) F N ˆ T + V N ˆ T (cid:17) con | ψ (cid:105) = 0 . (4)In the first two lines ˆ T = ˆ T + ˆ T enters the similaritytransformation, which gives the commonly used coupled-cluster singles-and-doubles (CCSD) approximation when T = 0. In the last line only the connected terms enter.The correlation energy is then E = (cid:104) ψ | H N | ψ (cid:105) .The CCSD approximation costs o u compute cyclesfor each iteration, with o = A ( u ) being the number of(un)occupied states with respect to the natural-orbitalreference. The cost of CCSDT-1 is o u and thus anorder of magnitude more expensive.Both CCSD and CCSDT-1 are too expensive with-out further optimizations. To overcome this challengewe first take advantage of the block-diagonal structureof the Hamiltonian imposed by axial symmetry, isospin,and parity and only store and process matrix-elementsthat obey these symmetries. Second, we impose a trun-cation on the allowed number of 3 p –3 h amplitudes by acut on the product occupation probabilities n a for threeparticles above the Fermi surface and for three holes be-low the Fermi surface, i.e. we require n a n b n c ≤ ε and(1 − n i )(1 − n j )(1 − n k ) ≤ ε . This cut favors configura-tions with large occupation probabilities near the Fermisurface and – as shown in Fig. 2 – requires only a manage-able number of 3 p –3 h amplitudes to be included. Third,we exploit the internal structure of the three-body sym-metry blocks, which can be expressed as the tensor prod-uct of two- and one-body symmetry blocks, to formulatethe equations as a series of matrix multiplications. Thisallows us to efficiently utilize the supercomputer Summitat the Oak Ridge Leadership Computing Facility, whosecomputational power mainly comes from GPUs. For the computation of observables other than theenergy (the radius in our case), we also need to solvethe left eigenvalue problem as the similarity transformedHamiltonian is non-Hermitian. This is done usingthe equation-of-motion coupled-cluster method (EOM-CCM), see Refs. [24, 67, 70, 71] for details. In this workwe limit the computations of radii to the EOM-CCSDapproximation level. For Mg the inclusion of (com-putationally expensive) triples via the EOM-CCSDT-1approximation [70] increases the radius by less than 1%,consistent with the findings of Refs. [72, 73].In EOM-CCSDT-1 the left ground-state eigenvalueproblem is (cid:104) ψ | (1 + ˆΛ) H N = E (cid:104) ψ | (1 + ˆΛ) . (5)Here Λ is a de-excitation operator with amplitudes Λ ia ,Λ ijab , and Λ ijkabc . We need to solve forˆΛ = ˆΛ + ˆΛ + ˆΛ = (cid:88) ia Λ ia ˆ a † i ˆ a a + 14 (cid:88) ijab Λ ijab ˆ a † i ˆ a † j ˆ a b ˆ a a + 136 (cid:88) ijkabc Λ ijkabc ˆ a † i ˆ a † j ˆ a † k ˆ a c ˆ a b ˆ a a (6)Given H N , Eq. (5) is an eigenvalue problem, and weare only interested in its ground-state solution E = E . In the EOM-CCSDT-1 approximation, the triplesde-excitation part Λ only contributes to the doublesde-excitation part of the matrix-vector product via (cid:104) ψ | ˆΛ V N | ψ ijab (cid:105) , while the triples de-excitation part ofthe matrix-vector product is (cid:104) ψ | (ˆΛ + ˆΛ ) V N + (ˆΛ +ˆΛ ) F N | ψ ijkabc (cid:105) . To compute the left ground-state we caneither solve a large-scale linear problem (because weknow the ground-state energy E ), or we use an iterativeArnoldi algorithm for general non-symmetric eigenvalueproblems to compute the ground state of H N . In our ex-perience the latter approach is more stable and requiresfewer iterations. The ground-state expectation value ofan operator ˆ O is (cid:104) ˆ O (cid:105) ≡ (cid:104) ψ | (1 + ˆΛ) O | ψ (cid:105) . (7)Here the similarity-transformation O ≡ e − ˆ T ˆ Oe ˆ T of ˆ O enters.The charge radius squared is R = R p + (cid:104) r p (cid:105) + NZ (cid:104) r n (cid:105) + (cid:104) r (cid:105) + (cid:104) r (cid:105) . (8)Here, R p is the radius squared of the intrinsic point-proton distribution and (cid:104) r (cid:105) is the spin-orbit correc-tions. These two quantities are actually computed withthe coupled-cluster method [4]. The corrections (cid:104) r p (cid:105) =0 .
709 fm , (cid:104) r n (cid:105) = − .
106 fm , and (cid:104) r (cid:105) = 3 / (4 m ) =0 .
033 fm (with m denoting the nucleon mass) are thecharge radius squared of the proton (updated accord-ing to Refs. [74, 75]), the neutron (updated value fromRef. [76]), and the Darwin-Foldy term, respectively. Results.—
Our results for the charge radii of magne-sium isotopes are shown in Fig. 1. Here, each bandreflects model-space uncertainties from varying the os-cillator frequency from 12 to 16 MeV. The results forthe softer interaction with a cutoff of 394 MeV c − areshown in red and exhibit less model-space dependencethan those for the harder interaction with 450 MeV c − shown in blue. The overall uncertainty estimate on theradii, both from model-space uncertainties and system-atic uncertainties of the interactions is then about 2-3%,i.e. the full area covered by (and between) both bands.Overall, the ∆NNLO GO potentials reproduce theprominent pattern of a minimum radius at the sub-shellclosure N = 14, and they agree with data within uncer-tainties for mass numbers 22 ≤ A ≤
30. The computedradii continue to increase beyond N = 14, and they re-flect the absence of the N = 20 shell closure in magne-sium. This is, of course, the beginning of the island ofinversion. However, the theory results do not reproducethe very steep increase from A = 30 to 32. Thus, theyseem to reflect remnants of a shell closure at N = 20that are not seen in the data. Theory predicts increasingcharge radii as the dripline is approached. This is con-sistent with an increase in nuclear deformation as neu-trons are added [45]. We also note that theory predictsa marked shell closure at N = 8 for neutron-deficientmagnesium. This is in contrast to the trend projectedin Ref. [40]. The excited 2 + state in Mg at 1.6 MeVis somewhat higher that the 1.2 MeV observed in Mg,and the question regarding a sub-shell closure at N = 8is thus undecided. It will be interesting to compare thetheoretical results with upcoming laser spectroscopy ex-periments that are at the proposal stage [46].The plot of isotopic variations in the charge radii,shown in Fig. 3, is interesting. Theory is not accurateregarding most isotopes shifts and over-emphasizes shellclosures at N = 14 and N = 20 that are not in the data.This is perhaps a most important result of this study:While state-of-the-art potentials can now describe chargeradii within 2-3% of relative uncertainties, finer detailssuch as isotope shifts still escape the computations.We show the results for binding energies in Fig. 4. Ourcalculations yield the dripline at Mg, with Mg beingabout 1.8 MeV less bound for the ∆NNLO GO (394) po-tential. However, computational limitations prevented usfrom including continuum effects, which can easily yieldan additional binding energy of the order of 1 MeV [77].This prevents us from predicting the unknown driplinein magnesium more precisely [78].Another uncertainty stems from the lack of angular-momentum projection. To estimate the correspondingenergy correction, we performed a projection after vari-ation within the Hartree-Fock computations. These pro-jections lower the Hartree-Fock energy by about 3 to5 MeV, see Fig. 2 for an example. We expect that a pro-jection of the coupled-cluster results would yield slightlyless energy gains (because these calculations already in-clude some of the correlations that are associated with a
20 22 24 26 28 30 32 34 36 38 40 A r , A ( f m ) MgMg expNNLO GO (394)NNLO GO (450) FIG. 3. (Color online) As in Fig. 1 but for the isotope shift,i.e. the charge radii squared relative to Mg.
20 22 24 26 28 30 32 34 36 38 40 A E ( M e V ) MgMg expNNLO GO (394)NNLO GO (450) FIG. 4. (Color online) As in Fig. 1 but for the ground-stateenergies. projection). Overall, Fig. 4 shows that the ∆NNLO GO potentials accurately describe nuclear binding energiesalso for open-shell nuclei.Binding-energy differences, such as the two-neutronseparation energy, is another observable sensitive to shellstructure and dripline physics. Figure 5 shows thatthe overall pattern in the data is accurately reproducedwithin the uncertainties from the employed interactionsand model spaces. However, the details of the sub-shellclosure at N = 14 escape the theoretical description,i.e. theory predicts a slightly stronger sub-shell than ob-served experimentally.We finally turn to neon isotopes. Here, our compu-tations have been less extensive to manage the avail-able computational cycles. We limited the computa-tions of energies to the ∆NNLO GO (394) potentials in amodel space of 13 harmonic oscillator shells at (cid:126) ω =16 MeV. For the charge radii we also employed the∆NNLO GO (450) potential at (cid:126) ω = 12 MeV. Figure 6shows that the ground-state energies are close to the
22 24 26 28 30 32 34 36 38 40 A S n ( M e V ) MgMg expNNLO GO (394)NNLO GO (450) FIG. 5. (Color online) As in Fig. 1 but for the two-neutronseparation energies data. We estimate theoretical uncertainties to be a bitsmaller than for the magnesium isotopes. We also notethat about 3-5 MeV of energy gain is expected from aprojection of angular momentum (see again Fig. 2). Wefind the dripline at Ne, in agreement with data [79].
18 20 22 24 26 28 30 32 34 A E ( M e V ) NeNe expNNLO GO (394) FIG. 6. (Color online) Ground-state energies for neon iso-topes with even mass numbers computed with the potentials∆NNLO GO (394) shown as a red line. The model spaces con-sist of 13 oscillator shells. Data is shown as black bars. The computed two-neutron separation energies, shownin Fig. 7, confirm this picture. Compared to magnesium,it is interesting that the addition of two protons shiftsthe drip line by about six neutrons. Again we estimatethat theoretical uncertainties are a bit smaller than forthe magnesium isotopes.Finally, we show results for charge radii in Fig. 8, us-ing the ∆NNLO GO (394) and ∆NNLO GO (450) potentials.We only employed one oscillator frequency for each in-teraction. Thus, the theoretical uncertainties are esti-mated to be somewhat larger than the area between thetwo lines (compare with Fig. 1 of the magnesium iso-
20 22 24 26 28 30 32 34 A S n ( M e V ) NeNe expNNLO GO (394) FIG. 7. (Color online) As in Fig. 6 but for the two-neutronseparation energies. topes). Based on these estimates, theoretical results arenot quite accurate below Ne, though they qualitativelyreproduce the overall trend. The results accurately re-flect the known sub-shell closures at N = 14 and N = 8.We see no closure at N = 20 and it will be interesting toconfront this prediction with data.
18 20 22 24 26 28 30 32 34 A R c h ( f m ) NeNe expNNLO GO (394)NNLO GO (450) FIG. 8. (Color online) Charge radii for neon isotopeswith even mass numbers computed with the potentials∆NNLO GO (394) and ∆NNLO GO (450) shown in red and blue,respectively. The model spaces consist of 13 oscillator shells.Data is shown as black bars [43, 44]. Conclusion.—
We computed ground-state energies,two-neutron separation energies, and charge radii forneon and magnesium isotopes. Our computations werebased on nucleon-nucleon and three-nucleon potentialsfrom chiral EFT, and we employed coupled-cluster meth-ods that started from an axially symmetric referencestate. The computed energies and radii are accuratewhen taking expected corrections from angular momen-tum projection into account. Trends in charge radii, andthe minimum and neutron number N = 14 are qual-itatively reproduced. Within our estimated uncertain-ties of about 2-3%, however, quantitative accuracy is notachieved for all isotopes, and isotope shifts still challengetheory. Nevertheless, we predict a continuous increaseas the neutron dripline is approached, and this is consis-tent with a considerable nuclear deformation. Proposedexperiments will soon confront these predictions. ACKNOWLEDGMENTS
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