Accessing the Single-Particle Structure of the Pygmy Dipole Resonance in 208 Pb
M. Spieker, A. Heusler, B. A. Brown, T. Faestermann, R. Hertenberger, G. Potel, M. Scheck, N. Tsoneva, M. Weinert, H.-F. Wirth, A. Zilges
aa r X i v : . [ nu c l - e x ] S e p Accessing the single-particle structure of the Pygmy Dipole Resonance in Pb M. Spieker, ∗ A. Heusler, B. A. Brown,
3, 4
T. Faestermann, R. Hertenberger, G.Potel, M. Scheck,
8, 9
N. Tsoneva, M. Weinert, H.-F. Wirth, and A. Zilges Department of Physics, Florida State University, Tallahassee, Florida, 32306, USA Niebuhr-Str. 19c, D-10629 Berlin, Germany National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, Michigan 48824, USA Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824, USA Physik Department, Technische Universit¨at M¨unchen, D-85748 Garching, Germany Fakult¨at f¨ur Physik, Ludwig-Maximilians-Universit¨at M¨unchen, D-85748 Garching, Germany Lawrence Livermore National Laboratory, Livermore, California, 94550, USA School of Computing, Engineering, and Physical Sciences,University of the West of Scotland, Paisley, PA1 2BE,UK SUPA, Scottish Universities Physics Alliance, UK Extreme Light Infrastructure (ELI-NP), Horia Hulubei National Institute of Physicsand Nuclear Engineering (IFIN-HH), RO-077125, Bucharest-Mˇagurele, Romania Institut f¨ur Kernphysik, Universit¨at zu K¨oln, Z¨ulpicher Straße 77, D-50937 K¨oln, Germany (Dated: September 3, 2020)New experimental data on the neutron single-particle character of the Pygmy Dipole Resonance(PDR) in
Pb are presented. They were obtained from ( d, p ) and resonant proton scatteringexperiments performed at the Q3D spectrograph of the Maier-Leibnitz Laboratory in Garching,Germany. The new data are compared to the large suite of complementary, experimental dataavailable for
Pb and establish ( d, p ) as an additional, valuable, experimental probe to studythe PDR and its collectivity. Besides the single-particle character of the states, different featuresof the strength distributions are discussed and compared to Large-Scale-Shell-Model (LSSM) andenergy-density functional (EDF) plus Quasiparticle-Phonon Model (QPM) theoretical approachesto elucidate the microscopic structure of the PDR in
Pb.
Atomic nuclei with large proton-neutron asymmetry,like
Pb, form a neutron skin [1]. The neutron-skinthickness, ∆ r np , is directly correlated to properties ofneutron stars [1–5]. This raised the interest of the sci-ence community in determining it experimentally [6–9]. Following the first multi-messenger detection of abinary neutron star merger [10] including gravitationalwaves [11], this interest has been recently reinforced [12].The electric dipole polarizability α D [6, 8, 13–19] isone key observable investigated to obtain constraints on∆ r np . For its precise determination, the low-lying E E S n [20–24]. The PDR strength mightalso correlate more strongly with ∆ r np [18, 25–29] and,thus, provide tighter constraints. However, the possiblystronger correlation has been critically discussed [30–33].In any case, it would be necessary to distinguish the PDRfrom other E e.g. , [17, 34–37]). Ithas been shown that the PDR strength strongly impactsneutron-capture rates in the s and r process [17, 38–41].A precise understanding of its microscopic structure isalso essential to pin down how the PDR contributes tothe γ -ray strength function ( γ SF) often used to calculate( n, γ ) rates [41], i.e. whether there is a dependence of the γ SF’s shape on excitation energy, spin-parity quantumnumber or even specific nuclear structure [42–49]. Depending on the mass region of the nuclear chart, thelow-lying E E Ni [24],which used a fully self-consistent nonrelativistic mean-field approach based on Skyrme Hartree-Fock plus ran-dom phase approximation (HF+RPA), reinforced thatcoherence between several 1p–1h configurations is ratherobserved in the isoscalar than in the isovector channel.Qualitatively comparable results had been obtained for
Sn and
Pb by employing similar theoretical ap-proaches [65, 66]. These theoretical results question theusefulness of studying the PDR’s collectivity based onthe isovector E d, p ) experimental study of the PDR in Pb and com-plement it with available experimental data to discussthe PDR’s microscopic structure and its influence onexperimental observables by comparing to state-of-the-art, theoretical models. The neutron 1p–1h configura-tions contributing to forming the PDR are accessed from( d, p ) data up to the proton-separation energy, S p , and,for a limited number of states, from the results of res-onant proton scattering via isobaric analog resonances(( p, p ′ ) IAR ) [71–76], which probes components that couldnot be populated in the selective one-neutron transfer re-action. An unprecedented access to the theoretical wavefunctions was achieved.When discussing its collectivity within the HF+RPAapproach, Roca-Maza et al. identified the PDR of
Pbabove 7 MeV [65]. Following a comparison of NuclearResonance Fluorescence data and Quasiparticle-PhononModel (QPM) calculations, Ryezayeva et al. had ar-gued that the PDR should indeed correspond to thestrength observed around S n [62]. The lower-lying 1 − states should have a more pure single-particle charac-ter [62]. Poltoratska et al. [35] considered, however, alllow-lying E ∼ O , O ′ γ ) experiment [54],performed to study its isospin character. Based onQPM calculations for Pb, dominantly the neutron1p–1h states below S n were identified to belong to thePDR of the N = 124 Pb isotope [17]. The importanceof including two-particle-two-hole (2p–2h) configurationsto describe the isovector B ( E
1) strength fragmentationwas pointed out in [77] using Large-Scale-Shell-Model(LSSM) calculations [78]. Also the possibility of tetrahe-dral configurations in
Pb was presented and some ofthe lower-lying states, including the 1 − state, were dis-cussed to originate from this exotic type of excitation [79].The new data, presented here, were obtained from aseries of experiments performed to study excited statesin Pb with the high-resolution Q3D spectrograph ofthe Maier-Leibnitz Laboratory (MLL) in Garching, Ger-many [80, 81]. For the ( d, p ) experiments, the deuteronswere accelerated to 22 MeV and impinged onto a 0.11-mg/cm thick, highly-enriched Pb target (99 % enrich-ment) on a Carbon backing. After the reaction, the resid-ual particles were momentum-analyzed with the Q3D anddetected in the focal-plane detection system [82, 83]. Byadjusting the horizontal entrance slits, half ( ± . ◦ ) of theQ3D’s maximum angular acceptance was used and an en-ergy resolution of better than 6 keV (FWHM) achieved.This facilitated the analysis of the dense excitation spec-tra seen in Fig. 1.The ( d, p ) data were analyzed at three scattering an-gles; 20 ◦ , 25 ◦ , and 30 ◦ . This allows to distinguishthe two different transfer configurations through whichthe known 1 − states of Pb [85, 86] can be populatedfrom the J π = 1 / − ground state of Pb; namely(3p / ) − (4s / ) +1 ( l = 0) and (3p / ) − (3d / ) +1 ( l =2). The angular distributions are shown in Fig. 2 along- Figure 1. (a) , (b) Pb( d, p ) Pb spectra taken at θ = 25 ◦ for two different magnetic settings. Only a part of thespectrum is shown in panel (b) . Contamination from the C( d, p ) C reaction is observed (labeled with C) due to theCarbon backing of the target. The kinematic correction withthe Q3D multipole element was applied to the
Pb( d, p )reaction causing the peaks observed from C( d, p ) to be sig-nificantly broader (compare [84]). Known J π = 1 − states of Pb [85–87], which could be resolved, are highlighted withvertical, dashed lines. All other states, seen in the spectra,correspond to excited states of
Pb with a 3p / neutron-hole component in their wave function. Below S n , many ofthem were experimentally observed before [85, 86, 88–90]. side Distorted-Wave-Born-Approximation (DWBA) cal-culations performed with the coupled-channels program chuck3 [91]. The global optical-model parameters(OMP) of [92] were used for the protons and of [93] for thedeuterons with adjustments to the real potential of thevolume Woods-Saxon part from [94]. With the exceptionof using an effective neutron-separation energy for statesabove S n , the same OMP were used for all excited states.As shown in Fig. 2, the measured and DWBA angulardistributions are in excellent agreement. The dominantcontributions of the most strongly excited 1 − states at5292 keV, 5512 keV, and 5947 keV were previously iden-tified [71, 72, 88–90, 95] and confirmed here. Only smalladditional (3p / ) − (4s / ) +1 contributions were neededto explain the experimental angular distributions for the5512-keV and 5947-keV states. In return, this high-lights the sensitivity of the present experiment to suchsmall contributions. The ( p, p ′ ) IAR data on the 3 d / and 4 s / IARs in
Bi confirm the dominant structureassignments for the 5292-keV and 5947-keV state [com-pare Fig. 3 (b) ], respectively. Also at higher excitationenergies, superpositions of the two configurations wereoften needed to explain the experimental ( d, p ) data asshown for three examples in Fig. 2. As indicated by the( p, p ′ ) IAR data, other 1p–1h configurations are importantas well and might dominate the structure of the states[compare Fig. 3 (b) ]. In total, 11 out of the 15 ampli-
Figure 2. (color online) Measured ( d, p ) angular distribu-tions (differential cross sections dσ/d
Ω) for selected J π = 1 − states (circles) in comparison to DWBA calculations (lines).Two different 1p–1h configurations, (3p / ) − (4s / ) +1 (blue,longer dashed lines) and (3p / ) − (3d / ) +1 (red, shorterdashed lines), have been assumed to describe the experimentaldistributions. Black, solid lines correspond to superpositionsof these two individual configurations. No multistep transferwas considered. For states above S n , an effective neutron-separation energy of S n = 8 . l = 0 and l = 2 transfers remainunchanged. For the 1 − state at 6264 keV, a Carbon contam-inant prevented a cross-section measurement at θ = 25 ◦ . tudes were studied experimentally [71–76]. More detailson the determination of the relative c LJlj amplitudes forthe different neutron 1p–1h configurations from ( p, p ′ ) IAR are presented in [71–75, 87].The model-indepedent, angle-integrated ( d, p ) crosssections and c LJlj amplitudes from ( p, p ′ ) IAR are shownin Fig. 3 in comparison to a selection of other experimen-tal data on the PDR in
Pb [35, 54]. The ( d, p ) strengthpattern [Fig. 3 (a) ] is dominated by the two strongly pop-ulated 1 − states at 5292 keV and 5947 keV, correspondingto the major fragments of the (3p / ) − (4s / ) +1 [ S =0 . / ) − (3d / ) +1 [ S = 0 . S , are model-dependent but weredetermined consistently, i.e. using the same OMP. Thisis different from the approach chosen in [90, 95], whereOMP were varied depending on the l transfer introduc-ing a stronger model dependency. While the 5292-keVstate shows appreciable B ( E
1) strength [35, 62] and isalso comparably strongly populated in ( O , O ′ γ ) [54],the 5947-keV state is, strikingly, barely excited with theelectromagnetic probe and not at all with the hadronicprobe [compare Figs. 3 (a) , (c) , (d) ]. Remarkably, thegroup of states with excitation energies of 6264 keV, 6314 keV, 6362 keV, and 6486 keV features both gradu-ally decreasing ( d, p ) cross sections and isovector B ( E O , O ′ γ ). For the 6264-keV and 6314-keVstates, mixtures of l = 0 and l = 2 transfers were neededto describe the experimental angular distributions (see,exemplary, the 6264-keV state in Fig. 2). One configu-ration was sufficient for the 6362-keV ( l = 0) and 6486-keV ( l = 2) states. Interestingly, the 6264-keV state isthe only one of the four, which has (2f / ) − (2g / ) +1 and (2f / ) − (2g / ) +1 components in its wave function[compare Fig. 3 (b) ].Figs. 3 (e) - (m) presents the results of LSSM [78] andenergy-density functional (EDF)+QPM [97] calculations.To calculate the differential cross sections dσ/d Ω, pre-dicted spectroscopic factors, i.e. the overlap of the
Pb ground state with excited 1 − states in Pb whenadding a neutron, were combined with the DWBA cal-culations [91], which described the experimental data.The angle-integrated σ ( d,p ) cross sections were also de-termined between θ = 20 ◦ − ◦ .The LSSM calculations [Fig. 3 (e) - (i) ] were introducedin [77, 78]. In addition to the σ ( d,p ) values (see sup-plement for truncation at 1p–1h level [98]), we providethe decomposition of the wave functions into the differ-ent neutron 1p–1h components ( > ψ total [Fig. 3 (f ) ], and the contribu-tions of 1p–1h and 2p–2h components to ψ total [compareFig. 3 (g) ]. The predicted excitation energy, 5226 keV,of the major (3p / ) − (4s / ) +1 [ S LSSM = 0 .
56] frag-ment is very close to the experimental 5292-keV state[ S exp = 0 . S LSSM = 0 .
16 at 5469 keV provides a cen-troid energy of 5280 keV with S LSSM = 0 .
72 in almostperfect agreement with the experimental data. Below6.25 MeV, the (3p / ) − (3d / ) +1 strength is much morefragmented than experimentally observed. The strongestfragment is predicted at 6171 keV with S LSSM = 0 . l = 2 strength is found at5912 keV with S LSSM = 0 .
78 ( E x < .
25 MeV), whichagain compares well to the experimental centroid at5904 keV [ S exp = 0 . S n are P σ ( d,p ) exp = 1524(17) µ band P σ ( d,p ) LSSM = 1470 µ b. However, the fragmenta-tion of the LSSM spectroscopic strength between S n and S p is not as observed in experiment. For firm 1 − statesabove S n , P σ ( d,p ) exp is 254(9) µ b while the LSSM pre-dicts only 22 µ b. 13 % of the d / and 9 % of the s / strength are pushed to energies higher than 8.6 MeV inthe LSSM. The data suggest that this strength is locatedbelow S p .Many neutron 1p–1h excitations contribute to ψ total with the strongest component never exceeding 56 %.Given the experimental limitation of only being able todetermine three to four amplitudes when studying one Figure 3. (color online) (a)
Angle-integrated ( d, p ) cross sec-tions σ ( d,p ) , (b) c LJlj amplitudes from ( p, p ′ ) IAR [71, 72, 75,76], (c) isovector B ( E
1) strengths from ( p, p ′ ) [35], and (d) differential cross sections from ( O , O ′ γ ) [54]. The lat-ter probe the isoscalar character of the 1 − states [54]. (e) σ ( d,p ) predicted by combining LSSM spectroscopic factorswith DWBA calcultions. (f) Decomposition of the LSSMwave functions into neutron 1p–1h components relative to thetotal wave function ψ total . (g) ψ total . LSSM isovector B ( E
1) strength predicted when (h) including all or (i) exluding the specified contributions. (j) - (m) same as (e) - (h) but for EDF+QPM. SVS stands for“state-vector structure” [97, 98]. IAR, this seems largely consistent with the ( p, p ′ ) IAR data. As seen from the comparison of Figs. 3 (b) and3 (f ) , most 1 − states cannot be considered as simpleneutron 1p–1h states. The ( d, p ) data prove that al-most all 1 − states have at least small (3p / ) − (4s / ) +1 and (3p / ) − (3d / ) +1 components, many of which werebelow the sensitivity limit in ( p, p ′ ) IAR . With only afew exceptions, the neutron 1p–1h contribution makesup around 80 % of ψ total in the LSSM at lower ener-gies (compare R (1 p − h ) ν in Fig. 3 (f ) for the first ten 1 − states). The strongest neutron 1p–1h component in thewave function of the 1 − is identified as (2f / ) − (2g / ) +1 in both ( p, p ′ ) IAR and the LSSM. The experimental datafor the 1 − support that less than 60 % of ψ total are dueto neutron 1p–1h components [74, 79]. For almost alllower-lying 1 − states, the 2p–2h contribution alreadyexceeds 10 % [compare Fig. 3 (g) ]. A clear structuralchange is observed above 7.5 MeV, where 2p–2h configu-rations begin to dominate the wave functions. Note thatPoltoratska et al. [35] experimentally observed a struc-ture change at ∼ . ∼ . ψ total drops well below 10 % in the LSSM.The LSSM B ( E
1) strength distribution [77] is shownin Fig. 3 (h) . Problematically, the most enhanced B ( E / ) − (4s / ) +1 frag-ment, i.e. the 1 − state in conflict with experiment [com-pare Fig. 3 (c) ]. The major (3p / ) − (3d / ) +1 fragmenthas an experimental B ( E
1) = 13(1) × − e fm . TheLSSM calculations predict B ( E
1) = 45 × − e fm .Also, the large B ( E
1) value from the LSSM state at7.5 MeV is not observed for a specific, experimental state.We note that, in general, there is a large amount of can-cellation between the shell-model components of the E B ( E
1) values. To further highlight this problem, we haveexcluded the (3p / ) − (4s / ) +1 and (3p / ) − (3d / ) +1 contributions to the B ( E
1) strengths in Fig. 3 (i) . Thestrength fragmentation below 7 MeV changes drastically.Missing microscopic configurations or incorrect individ-ual contributions, thus, influence the shape of the γ SF.To study the neutron-skin structure of the 1 − states with (3p / ) − (4s / ) +1 , (3p / ) − (3d / ) +1 andother neutron 1p–1h components, EDF+QRPA andEDF+QPM calculations were performed (see [97] fora review). In contrast to [6, 35, 62], single-particleenergies were neither determined from nor adjusted todata. Instead, they were directly obtained at the mean-field level from the EDF [97]. Fig. 3 (j) presents theQPM+DWBA predictions for σ ( d,p ) . Results obtainedat the QRPA level and further details are given in thesupplement [98]. The QPM also predicts a dominant(3p / ) − (4s / ) +1 [ S .
32 MeV = 0 .
92] fragment but ex-
Figure 4. (color online) Summed transition densities for thefirst five 1 − QRPA states, which contain the (3p / ) − (4s / ) +1 and (3p / ) − (3d / ) +1 components. All five states are dom-inant neutron 1p–1h states (compare [98]). For comparison,the summed transition densities for the GDR are shown. pects, different from the LSSM and in agreement with ex-periment, the (3p / ) − (3d / ) +1 strength to be mainlyconcentrated in one state [ S .
12 MeV = 0 . P σ ( d,p ) QPM = 1676 µ b below S n . However, alsothe QPM does not fragment the l = 0 and l = 2 strengthsufficiently to describe the strength above S n . While theQPM reproduces the experimental B ( E
1) strength dis-tribution around and above S n , i.e. where the states’structure becomes more complex [Figs. 3 (k) , (l) ], it doesnot generate sufficiently enhanced strength at lower en-ergies [Fig. 3 (m) ]. Due to the doubly magic nature of Pb, the 1p–1h structure of the QRPA phonons dom-inates the configuration mixing and polarization contri-butions (compare [98]). In order to improve the compar-ison with experiment, dynamic effects beyond the staticmean field would need to be implemented. As 2p–2hcontributions in the LSSM, multiphonon contributionsare small below 8 MeV. Interestingly, the 1 − ,QP M stateseems to correspond to the LSSM and experimental 1 − state. It has a significant 2-phonon admixture [compareFig. 3 (l) ]. Fig. 4 presents the summed transition densi-ties for the first five QRPA 1 − phonons, which containthe (3p / ) − (4s / ) +1 and (3p / ) − (3d / ) +1 spectro-scopic strengths. The summed transition densities showfeatures which are compatible with the oscillation of theneutron skin [26] and clearly different from the GDR.In summary, we performed the first extensive study ofthe single-particle structure of the PDR in Pb basedon experimental data. The LSSM and EDF+QPM cal-culations were able to account for the main features ofthe ( d, p ) data. However, both models do not generateenough spectroscopic strengths above S n . Such short-comings could have significant influence on ( n, γ ) rateswhen determined via surrogate methods using theoreti-cal nuclear-structure input [41, 99, 100]. The extendedcomparison, including the ( p, p ′ ) IAR data, suggests thatthe LSSM wave functions might be slightly too complex.At lower energies, the QRPA 1 − phonons might not besufficiently admixed to several QPM 1 − states. Most1 − states can, however, not be considered as simple neu-tron 1p–1h states as many neutron 1p–1h excitations con-tribute to their respective wave function. We pointed out the big cancellation effects between individual E B ( E
1) strengths. Enhanced strength is observedbelow S n . In contrast to previous claims, the transi-tion densities for the low-lying 1 − states with dominantneutron 1p–1h character clearly resemble features of adipole-type neutron-skin oscillation. The present workproves the value of complementary, experimental data onand the theoretical analysis of the PDR’s 1p–1h structureto access the microscopic wave functions. Similar stud-ies will help to further understand the microscopic originof the low-lying isovector and isoscalar B ( E
1) strengths.High-resolution, one-nucleon transfer experiments on sta-ble nuclides in different mass regions, where the changeof the underlying single-particle structure can be trackedas both proton and neutron number change, are planned.Further developments at next-generation exotic beam fa-cilities might allow access to the PDR with one-nucleontransfer in inverse kinematics using, e.g. , solenoidal spec-trometers [101–105].B.A.B. acknowledges support by the National ScienceFoundation under Grant No. PHY-1811855. M.Sch.acknowledges financial support by the UK-STFC. N.T.was supported by Extreme Light Infrastructure NuclearPhysics (ELI-NP) Phase II, a project co-financed by theRomanian Government and the European Union throughthe European Regional Development Fund “the Compet-itiveness Operational Programme” (1/07.07.2016, COP,ID 1334). A.Z. acknowledges support by the DeutscheForschungsgemeinschaft under grant ZI 510/9-1. M.Sp.wants to thank K. Kemper and J. 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