Neutron-hole strength in N = 81 nuclei
A. M. Howard, S. J. Freeman, D. K. Sharp, T. Bloxham, J. A. Clark, C. M. Deibel, B. P. Kay, P. D. Parker, J. P. Schiffer, J. S. Thomas
NNeutron-hole strength in N = 81 nuclei A. M. Howard, ∗ S. J. Freeman, † D. K. Sharp, T. Bloxham,
2, 3
J. A. Clark, C. M. Deibel,
4, 5, ‡ B. P. Kay, P. D. Parker, J. P. Schiffer, and J. S. Thomas Department of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom Physics Department, University of California, Berkeley, California 94720, USA Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA Physics Division, Argonne National Laboratory, Argonne, Illinois 60439, USA Joint Institute for Nuclear Astrophysics, Michigan State University, East Lansing, Michigan 48824, USA A. W. Wright Nuclear Structure Laboratory, Yale University, New Haven, Connecticut 06520, USA (Dated: March 23, 2020)A systematic study of neutron-hole strength in the N = 81 nuclei Ba,
Ce,
Nd and
Smis reported. The single-neutron removal reactions ( p , d ) and ( He, α ) were measured at energiesof 23 and 34 MeV, respectively. Spectroscopic factors were extracted from measured cross sectionsthrough a distorted-wave Born approximation analysis and centroids of single-particle strength havebeen established. The change in these centroid energies as a function of proton number have beencompared to calculations of the monopole shift for the s / and h / orbitals, where the majorityof the strength has been observed. Significant fragmentation of strength was observed for the d and g / orbitals, particularly for the latter orbital which is deeply bound, with summed strengths thatindicate a significant amount lies outside of the measured excitation energy range. I. INTRODUCTION
The description of atomic nuclei in terms of constituentnucleons moving within a mean-field potential is the basisof the shell model, and consequently, much of our under-standing of nuclear structure. Over the past decade or so,evidence has emerged indicating that, when moving awayfrom stability into exotic systems, the ordering of single-particle levels evolves as a function of proton and neutronnumber to the extent that the gaps between levels thatcorrespond to shell and sub-shell closures are found toalter. Significant attention has been paid to these phe-nomena in the literature, which has motivated a care-ful reexamination of how the interaction between valenceprotons and neutrons drives such evolution. On mov-ing through a series of isotopes or isotones, the chang-ing single-particle occupancies of one type of nucleon al-ters the overall effect of interactions with a nucleon ofthe other type, thus changing its effective single-particleenergy. It appears that in some cases both the centraland tensor components of the nucleon-nucleon interac-tion need to be considered carefully in order to reproducethe observed changes in single-particle structure [1–3].It is therefore interesting to carefully reexamine thetrends in single-particle states near the line of β stabil-ity, particularly where changes can be tracked across arange of proton-neutron ratios. Such experimental mea-surements are often easier and tend to yield more detailedinformation compared to studies with radioactive beams, ∗ Current address: FRM-II Heinz Maier-Leibnitz Research Neu-tron Source, Technical University of Munich, Lichtenbergstrasse1 85748 Garching Germany. † Correspondence to: [email protected] ‡ Current address: Department of Physics and Astronomy,Louisiana State University, Baton Rouge, LA 70803, USA. which are performed with inevitably lower beam intensi-ties. In many experiments with stable beams, centroidsof single-particle strength can be constructed from theobservation of several different excited states populatedby transfer of a nucleon to the same orbital and used toestimate its effective single-particle energy. d h d s g FIG. 1. Schematic level diagram of the single-particle or-bitals near stability for the shell between N = 50 and N = 82. Several studies have been performed recently usingconsistent approaches to both experimental and analyti-cal methods that have highlighted the detailed trends insingle-particle orbitals in near stable nuclei. These in-clude studies of high- j proton states outside of stable Sncores [4]; untangling particle-vibration coupling to revealthe underlying neutron orbitals outside N = 82 isotones[5, 6]; single-neutron states in N = 51 nuclei [7]; anda detailed study of the single-particle properties in Niisotopes [8, 9].This paper focusses on a systematic study of hole statesin the N = 82 closed core. The low-lying structure of a r X i v : . [ nu c l - e x ] M a r N = 81 nuclei is largely based on configurations formedvia core coupling with neutron holes in the shell be-tween N = 50 and N = 82 (see, for example, Refer-ence [10]). This shell is composed of 0 g / , 1 d / , 1 d / ,2 s / and 0 h / single-particle orbitals, shown schemat-ically in Figure 1. The even- Z , N = 81 isotopes that canbe studied using stable beams and solid targets rangefrom Ba to
Sm.Light-ion nucleon-transfer reactions are a traditionaltool with which to probe single-particle structure in nu-clei and have been used for many years generating awealth of information in the literature. However, sys-tematic studies across chains of nuclei have been lesscommon in the past and it can be difficult to use iso-lated studies to evaluate systematic trends as differentexperimental conditions and techniques have often beenemployed. In addition, the distorted-wave Born approx-imation (DWBA) calculations required to extract spec-troscopic information have been done with different com-puting codes and different choices of input parameters indifferent studies and were often limited by the compu-tation power available at the time, leading to the use ofmultifarious approximations. Indeed, the researcher try-ing to reassess experiments in the literature with modernreaction approaches is stymied where the original abso-lute cross section data are not available in publicationsand only graphs of relative angular distributions or tablesof spectroscopic factors are reported.Here we describe a series of single-nucleon transfer ex-periments on stable solid N = 82 targets, using a mag-netic spectrometer, that have been used to determine thelocation of single-neutron hole strength in N = 81 sys-tems. These employ both the ( p , d ) and ( He, α ) reactionsto ensure good momentum matching for low- and high- (cid:96) transfers, respectively.There are several published works in the literature onhole strength, but systematic data across the solid sta-ble N = 82 targets using a consistent approach to boththe experimental technique and the DWBA calculationswith each reaction are not available. The ( p , d ) reactionhas been studied previously on Ba,
Ce,
Nd and
Sm targets, but with worse resolution than the cur-rent work [11–13]. High-resolution measurements of the( He, α ) reaction were studied on Ce,
Nd and
Smtargets in Ref. [14], which also reports measurements ofthe ( d , t ) reaction. However, the helium-induced reactionon a Ba target has not been studied before. In all thisprevious work, a zero-range approximation was used inthe DWBA calculations and it was noted in several casesthat there was sensitivity to some of the associated cor-rections [11, 12]. The calculations were also normalizedby making assumptions about the single-particle purityof the 3 / + ground states in each residual nucleus. Bet-ter approaches can now be employed to both DWBA cal-culations and the determination of their normalization.In addition to these studies, there are also a number ofpublications of reactions on isolated targets [15–20].The current publication is organized in the following manner. Aspects of the experimental methodology willbe discussed first, covering neutron removal with both( p , d ) and ( He, α ) reactions. The approach used to theDWBA calculations and normalization of the calculatedcross sections follows, and the deduced single-neutron en-ergies will then be compared to a simple model basedon a two-body effective interaction between protons andneutrons. II. EXPERIMENTAL DETAILS
Beams of 23-MeV protons and 34-MeV He ions wereprovided by the tandem Van de Graaff accelerator atthe A. W. Wright Nuclear Structure Laboratory of YaleUniversity. These beams were used to bombard targetsof
Ba,
Ce,
Nd and
Sm. Momentum analysisof the ejectile ions was performed using the Yale EngeSplit-Pole Spectrograph. At the focal plane, a multiwiregas proportional counter, backed by a plastic scintillator,was used to measure position, energy loss and residualenergy of the ions passing through it. The ions wereidentified by combining information on magnetic rigid-ity and energy-loss characteristics in the gas detector.The beam dose was measured using a current integratorconnected to a tantalum beam stop positioned behindthe target. A +300 V bias was applied to both the tar-get frame and beam stop to suppress electron sputtering.Beam currents were typically in the range 50 to 100 enAfor each beam species. A 1.5-mm thick silicon detectorwas mounted at 30 ◦ to the beam axis to monitor targetthickness, although the ratio of elastic scattering to beamcurrent varied by less than 3% on individual targets dur-ing the experiment.Given the reactivity of the chemical elements usedas targets, oxygen is an inevitable contaminant and,to avoid complicated vacuum transfer procedures, tar-gets were manufactured by evaporation of isotopically-enriched oxide material onto supporting carbon foils ofthickness 20-40 µ gcm − . Reactions on oxygen and carbondid not overly complicate the analysis since the kinematicproperties of ejectile ions from the contaminant reactionswere sufficiently different from those of interest to be eas-ily identified.To allow the extraction of absolute cross sections, acalibration of the target thickness and spectrograph ac-ceptance was necessary. The product of these two quan-tities was determined for each target by elastic scatteringof 15-MeV α particles into the spectrometer at a labora-tory angle of 20 ◦ . Under these conditions, the cross sec-tion is expected to be within 0.5% of that for Rutherfordscattering. The spectrometer entrance aperture was fixedthroughout the experiment. The systematic uncertaintyin cross sections determined this way was estimated tobe around 5%. Details of the four target foils are givenin Table I, where the thicknesses given assume a nominalacceptance of 2.8 msr, determined by previous calibra-tions using an α source at the target position [21]. TABLE I. Details of the N = 82 target foils.Target Nominal Thickness Isotopic µ g cm − enrichment % Ba 101 99.8(1)
Ce 144 99.9(1)
Nd 150 99.0(1)
Sm 42 93.8(1)
Excitation Energy (MeV) Y i e l d ( a r b i t r a r y un i t s ) Ba(p,d)
Ce(p,d)
Nd(p,d)
Sm(p,d) x5
FIG. 2. Deuteron spectra from the ( p , d ) reaction on targetsof Ba,
Ce,
Nd and
Sm at an angle of 42 ◦ , displayedin terms of the excitation energy of the residual nucleus. Theportions of the data to the right of the dotted line have beenmultiplied by a factor of five for clarity. Representative focal-plane spectra for each target andreaction are shown in Figures 2 and 3. Comparison of the( p , d ) and ( He, α ) data in each case highlight the (cid:96) sensi-tivity of the reaction mechanism; for example, the (cid:96) = 2transitions to the 3/2 + ground states are visibly strongerin the ( p , d ) reactions than the ( He, α ) reactions, whosespectra are dominated by the (cid:96) = 5 population of an ex-cited 11/2 − state at excitation energies ranging from 661to 754 keV across the residual nuclei. These spectra werecalibrated using previously observed states, usefully sum-marized in References [22–25]. The energy resolution wasdetermined to be ∼
25 keV for ( p , d ) data and ∼
85 keV for
Excitation Energy (MeV) Y i e l d ( a r b i t r a r y un i t s ) Ba( He, α ) Ce( He, α ) Nd( He, α ) Sm( He, α ) FIG. 3. α -particle spectra from the ( He, α ) reaction ontargets of Ba,
Ce,
Nd and
Sm at an angle of 15 ◦ ,displayed in terms of the excitation energy of the residualnucleus. ( He, α ). Information on the excitation energies of knownstates, along with a width calibration determined fromresolved states, were used to assist the analysis of un-resolved peaks, especially in the ( He, α ) spectra. Weakcontaminant peaks resulting from the small quantities of C and O present in the target foils were readily iden-tifiable by their characteristic kinematic shift with angle,which also ensured that states of interest were affectedby contaminant contributions at no more than one mea-surement angle.Data were collected at laboratory angles of 5 ◦ , 20 ◦ ,35 ◦ and 42 ◦ for the ( p , d ) reaction, chosen to be close tothe first maxima of the expected angular distributionsfor (cid:96) = 0 , , He, α ) reaction tend to be less distinctand more forward peaked, so data were only taken at 5 ◦ and 15 ◦ . An additional angle of 10 ◦ was measured forthe Ba target to assist assignments since the reactionhad not been studied previously.For the majority of the states populated in the residualodd nuclei, angular-momentum quantum numbers havealready been determined by a variety of different meth-ods in the literature [22–25]. Previous assignments werechecked using the following strategy. The angle of thefirst maxima of the angular distribution of the ( p , d ) re-action is generally indicative of the angular momentumtransfer, so the shape of the ( p , d ) distribution was used inmost cases to determine the (cid:96) values - some examples ofangular distributions are shown in Figure 4. The angulardistribution for (cid:96) = 4 transitions to states in the residualsystem were found to be increasingly flat at higher exci-tation energies, behavior that is reproduced by DWBAcalculations, but still distinct from those of (cid:96) = 0 , (cid:96) transfer is deduced from the ( He, α ) reactionrather than from ( p , d ) cross sections, as discussed be-low). To confirm the assignments of high- (cid:96) transitions,the slopes of the ( He, α ) angular distributions, in theform of the ratio of cross sections at 5 ◦ and 15 ◦ , werealso used, as illustrated in Fig. 5 for the Ba target. Acomparison of the two differently-matched reactions hasproved valuable in other work in differentiating betweenhigh- (cid:96) assignments (some examples can be found in Ref-erences [7, 9, 26]); it was found to be less useful here inthat respect, but did help to discriminate between high- (cid:96) and low- (cid:96) transitions. C r o ss S e c t i on ( m b / s r) Laboratory Angle (deg) (p,d) ( He, (cid:95) ) FIG. 4. Examples of angular distributions for the ( p , d ) and( He, α ) reactions compared to the results of DWBA calcu-lations discussed in Section III. The distributions are shownfor states populated in Ba by (cid:96) = 0 (black), (cid:96) = 2 (red), (cid:96) = 4 (green) and (cid:96) = 5 (blue) transitions. Transitions with (cid:96) = 0 are not strongly populated in the ( He, α ) reaction. Theangular distributions are labeled with the excitation energyin the residual system in units of MeV. The (cid:96) values deduced from the current work for thethree heaviest targets are generally consistent with thework on ( d , t ) and ( He, α ) reactions by Berrier et al. [14].There is very good agreement for Nd. We note onlyminor discrepancies with Ref. [14] in
Ce; strength at
Excitation Energy (MeV) R a t i o o f c r o ss s e c t i on s o / o FIG. 5. An example of the ratio of cross section at 5 ◦ andto that at 15 ◦ for the ( He , α ) reaction, here shown for thepopulation of states in Ba for (cid:96) = 4 (green) and (cid:96) = 5(blue) as a function of excitation energy. The solid lines arethe results of DWBA calculations discussed in Section III. both (cid:96) = 2 and 4, but here no evidence for the pres-ence of (cid:96) = 4 is found in the former and conversely, no ev-idence for (cid:96) = 2 in the latter. The population of the stateat 2.018 MeV has been noted by several authors to have anon-standard distribution in neutron-removal reactions,which is confirmed here and no firm assignment could bemade. The current work finds evidence for the presenceof a tentative (cid:96) = 0 contribution at 2.556 MeV, alongwith the stronger (cid:96) = 4 transition. Spectroscopic factorsfor this doublet were determined on the basis that the( p , d ) cross section at forward angles is due to the (cid:96) = 0strength and that this component does not contribute tothe ( He, α ) cross section, which was attributed entirelyto (cid:96) = 4.Assignments in Sm also agree well with Ref. [14].However, at a beam energy of 23 MeV, elastically-scattered protons have a lower kinetic energy and mag-netic rigidity than deuterons arising from the popula-tion of the ground-state groups in the ( p , d ) reaction.Whilst the proton groups are fairly well separated fromdeuterons by energy-loss characteristics, a proton taildoes contaminate the deuteron gating conditions, espe-cially at larger angles. This is the origin of the broad peakabove 3 MeV in the Sm( p , d ) reaction in Figure 2. Sim-ilar groups in data on other targets lie higher in effectiveexcitation energy than was studied here. Previous workhas been performed at higher energies [15], moving theelastic group to higher effective excitation energies, whichcircumvented this issue. The ( He, α ) reaction does notsuffer the same problem with elastic scattering, but with-out the ( p , d ) data, assignments are more difficult. Thetwo states at 3.13 and 3.23 MeV observed in the currentwork with the ( He, α ) reaction are likely to be populatedvia high- (cid:96) transitions, but differentiation between (cid:96) = 4and 5 has not been possible. For the later discussion,unobserved (cid:96) = 5 transitions would be a more criticalissue; Ref. [14] observes no further (cid:96) = 5 population,whereas Ref. [15] isolates two higher-lying (cid:96) = 5 transi-tions. If the states at 3.13 and 3.23 MeV were (cid:96) = 5,it would shift the centroid of that strength in Sm byaround 100 keV, which would not significantly alter theinterpretation presented below.In
Ba, assignments up to 2 MeV are in agreementwith those of previous ( p , d ) reactions [12, 13]. The 7 / + peak at 1.252 MeV in the current work, also observed byseveral other techniques [22], has a J π assignment from γ -decay measurements following Coulomb excitation [27].It was missed in both previous ( p , d ) experiments, pre-sumably masked by its more intense (cid:96) = 2 neighbour at1.290 MeV. Ref. [12] also identified tentative assignmentsof the 7/2 + state at 2.230 MeV and the 11/2 − state at2.320 MeV, which are confirmed here and supported bythe ( He, α ) data for the first time. The (cid:96) = 4 transitionsalso found in that work at 2.54 and 2.99 MeV have beenrevised here as (cid:96) = 2 and (cid:96) = 5, respectively. The formerstate is not observed strongly in the ( He, α ) reaction, sothe (cid:96) = 4 assignment of Ref. [12] is not confirmed. Thelatter state has angular distributions in both reactionsthat are more consistent with (cid:96) = 5. The previous (cid:96) = 4assignment in Ref. [12] may have been affected by thestate at 3.03 MeV, which was unresolved from that at2.99 MeV; the states were resolved, but no assignmentwas made, in Ref. [13]. In addition, 11 new assignmentsin Ba are made here, mainly (cid:96) = 2 states at excitationenergies above 2.3 MeV.The energies and (cid:96) assignments of all states observedare summarized in Table II, along with spectroscopic fac-tors determined using the procedures outlined below. De-tailed data on cross sections are available as Supplemen-tal Information [28]. The J π values listed in this tableare taken from other measurements [22–25]; where J π assignments are not available, the subsequent analysistakes a model-dependent assumption that the strengthis from the valence shell. However, in many cases, thereis insufficient information to properly assign spin-parityto (cid:96) = 2 strength.Although the extraction of single-particle strength us-ing DWBA calculations is not discussed until the follow-ing section, it is useful at this point to consider the gen-eral picture of the strength distributions in the residualnuclei, which is illustrated in Figure 6; the comparisonwith particle-vibration coupling calculations will be dis-cussed later. The general pattern of behavior is similar tothat revealed in neutron-removal reactions on , Ba[29] and , Te [30]. The ground state in each caseis a 3 / + state carrying a significant fraction of the ex- pected d / strength, increasing with Z from around 64%in Ba to 85% in
Sm. Older studies have made theassumption that this state carries all of the d / strength[11–13]. At a few 100 keV in excitation energy, there isa 1 / + state with significant s / strength (90% on av-erage and not varying significantly across the isotopes).Beyond that lies a strong 11 / − state with around 80%of the expected h / strength. These correspond to thethree low-lying strong peaks that can be seen in the ( p , d )spectra (see Fig. 2) and the population of the 11 / − statedominates the ( He , α ) spectra (see Fig. 3). At higher ex-citation energies, there is a second strong (cid:96) = 2 transitionabove 1 MeV, obvious in the ( p , d ) reactions on Ce,
Nd and
Sm targets, which has been given a 5 / + assignment in other work, carrying between 35 and 50%of the d / strength. In Ba, the corresponding statehas a lower strength and an additional, relatively strong3 / + state occurs just above in excitation energy.Above ∼ (cid:96) = 2 and (cid:96) = 4 strength, with a few evenweaker isolated (cid:96) = 0 and (cid:96) = 5 transitions. It thereforeappears that most of the strength associated with the s / , d / and h / orbitals are generally contained in alow-lying state with low levels of fragmentation. The low-lying (cid:96) = 4 state apparent around 1.2 MeV in Sm, Nd andCe final nuclei only carries only around 10% of the g / strength, the rest is dispersed in small fragments at highexcitation energies with a significant proportion lying athigher excitation energies than studied here; this 10%fragment does not appear in Ba. Across all the resid-ual nuclei the deeper-lying d / and g / hole strengthsare significantly fragmented over many states extendingto high excitation energies. III. DWBA AND NORMALIZATION
Spectroscopic factors were determined from the mea-sured cross sections by comparison with the results ofcalculations using the distorted-wave Born approxima-tion with the finite-range code PTOLEMY [31]. The ap-proach taken here is same procedure adopted in a recentglobal analysis of quenching of spectroscopic strength[43], which has also been used in a number of recentstudies, for example Refs. [26, 32, 49]. The choices forpotentials associated with the optical models describingthe initial and final reaction channels, and those asso-ciated with the neutron bound states in the light andheavy cores, are the same as those used previously, withone minor exception, and are summarized below.The incoming and outgoing partial waves were de-scribed using the global optical potentials for protons[35], deuterons [36], and helions [37]. The deuteron po-tential used here gave a better reproduction of the angu-lar distributions than more recent global potentials [38]that we have employed in previous cases. The poten-tial of Ref. [36] had been used as the starting point in
Theory Expt s d g h E x c i t a t i o n e n e r g y ( M e V ) −
10 0 10 Sm Theory Expt E x c i t a t i o n e n e r g y ( M e V ) −
10 0 10 Ba Theory Expt E x c i t a t i o n e n e r g y ( M e V ) −
10 0 10 Nd Theory Expt E x c i t a t i o n e n e r g y ( M e V ) −
10 0 10 Ce Theory Expt E x c i t a t i o n e n e r g y ( M e V ) −
10 0 10
Nd Theory Expt E x c i t a t i o n e n e r g y ( M e V ) −
10 0 10
Nd Theory Expt E x c i t a t i o n e n e r g y ( M e V ) −
10 0 10
Nd Theory Expt E x c i t a t i o n e n e r g y ( M e V ) −
10 0 10 Nd Theor yExpt E x c i t a t i o n e n e r g y ( M e V )
024 Spectroscopic Factor −
10 0 10 Sm FIG. 6. Distribution of spectroscopic strength of states populated in ( p , d ) and ( He , α ) reactions for (cid:96) = 0 (black), (cid:96) = 2(red), (cid:96) = 4 (green) and (cid:96) = 5 (blue) transitions as a function of the excitation energy in the residual systems, comparedto particle-vibration coupling calculations from Ref. [10]. The strength of individual states has been obtained from measuredreaction cross sections using procedures described in Section III. the search for new parameters to extend the potential towider energy range in Ref. [38], but the current deuteronenergies are within those used in the former potential. Afixed α -particle potential determined from the A = 90region was used [39].Recent microscopic calculations were used as thesource for the internal wave functions of the light ions inthe reactions. For the deuteron, form factors determinedusing the Argonne v potential were used [33] and thosefor the α particle and He ions were taken from recentGreen’s function Monte-Carlo calculations [34].The wave functions of the transferred neutron in theheavy bound state were generated using a Woods-Saxonpotential with a depth adjusted to match the measuredbinding energy. This used fixed geometric parameters:radius parameter r =1.28 fm and diffuseness a = 0 .
65 fm.The derivative of a Woods-Saxon potential with ra-dius r so =1.10 fm, diffuseness a so = 0 .
65 fm and depth V so =6 MeV was used to model the spin-orbit component.The approximations involved in the DWBA approachare best satisfied where there is a large probability of a di-rect reaction mechanism. Spectroscopic factors are there-fore extracted using experimental cross sections mea-sured as close as possible to the angle of the first max-imum of the angular distribution of the most appropri-ately matched reaction. The ( p , d ) reaction was used todetermine the spectroscopic strength for (cid:96) = 0 and 2 fromdata at 5 ◦ and 20 ◦ , respectively, whereas that for (cid:96) = 4and 5 was extracted from the ( He , α ) reaction at 5 ◦ .The DWBA calculations carry an overall uncertainty in absolute normalization. Consistent results have beenobtained by adopting systematic approaches (for exam-ple, Ref. [8, 9]) using the Macfarlane-French sum rules[40] which associate the summed spectroscopic strengthsto the occupancies and vacancies of single-nucleon or-bitals. If a normalization factor is chosen such that thetotal observed strength is equal to the full single-particlevalue, the degree to which that factor deviates from unityis related to quenching of single-particle strength. Suchquenching has been observed in other reactions, suchas ( e, e (cid:48) p) [41, 42], where the total low-lying strengthaccounts for approximately half that expected by theindependent-particle model. A recent large-scale analysisof transfer data has found normalization factors that arequantitatively consistent with previous studies of suchquenching [43] and here we follow the same procedure.The total spectroscopic strength was required to repro-duce the number of expected neutrons in the correspond-ing orbital in the target nucleus. On the assumption ofthe closed neutron shell at N = 82, this corresponds tothe degeneracy of the orbital. This assumption can betested by probing the vacancy of the orbitals below theshell closure by looking for population of the relevant (cid:96) transfer in ( d , p ) reactions on N = 82 targets. Sev-eral such studies exist in the literature, but evidence forpopulation of orbitals with the quantum numbers of thenominally-filled neutron orbitals is sparse and any suchstates are populated very weakly. As examples, Ref. [44]observes an (cid:96) = 0 transition at 3.351 MeV and three ten-tative (cid:96) = 2 transitions above 2.2 MeV, with strengths of T A B L E II . Su mm a r y o f s t a t e s p o pu l a t e d i nn e u t r o n - r e m o v a l r e a c t i o n s o n N = t a r g e t s ,i n c l ud i n g e x c i t a t i o n e n e r g y E , o r b i t a l a n g u l a r m o m e n t u m t r a n s f e r (cid:96) , s p i n - p a r i t y J π , a ndn o r m a li s e d s p e c t r o s c o p i c f a c t o r C S . E x c i t a t i o n e n e r g i e s a r e g i v e n i n M e V a nd a r ee s t i m a t e d t o c a rr y a nun c e r t a i n t y o f ∼ k e V , r i s i n g t o ∼ k e V i n t h e c a s e o f B aa t h i g h e r e x c i t a t i o n e n e r g i e s . T h e s p e c t r o s c o p i c f a c t o r s a r e d e du c e d f r o m t h e ( p , d ) r e a c t i o n f o r (cid:96) = nd t r a n s f e r s a nd f r o m t h e ( H e , α ) r e a c t i o n f o r (cid:96) = nd , a ndh a v e b ee nn o r m a li z e du s i n g t h e m e t h o dd e s c r i b e d i n t h e t e x t . T h ee rr o r s i n t h e n o r m a li s e d v a l u e s a r e t y p i c a ll y % du e t o v a r i a t i o n o f D W B A w i t h d i ff e r e n t i npu t p a r a m e t e r s , bu t f o r w e a k e r t r a n s i t i o n s t h e s e r i s e w h e r e s t a t i s t i c a l e rr o r s b e c o m e m o r e s i g n i fi c a n t( m o r e i n f o r m a t i o n i s a v a il a b l e i n t h e Supp l e m e n t a l M a t e r i a l[ ] ) . J π a r e t a k e n f r o m t h e li t e r a t u r e [ ]; w h e r e a J π v a l u e i s n o t li s t e d , a m o d e l - d e p e nd e n t a ss u m p t i o n w a s m a d e t h a tt h e s i n g l e - p a r t i c l e o r b i t a l s i s i n t h e v a l e n c e s h e ll. B a C e N d S m E (cid:96) J π C S E (cid:96) J π C S E (cid:96) J π C S E (cid:96) J π C S . + . . + . . + . . + .
40 0 . + . . + . . + . . + .
84 0 . − . . − . . − . . − .
36 1 . + . . + . . + . . + .
54 1 . + . . + . . + . . + .
02 1 . + . . ( ) + . . ( ) + . . ( ) + .
17 1 . + . . + . . ( , ) + . . ( ) + .
40 1 . + . . + . . . . .
07 2 . ( ) + . . + . . . . .
11 2 . . . ( ) + . . + . . .
47 2 . + . . . ( + , + ) . . + .
01 2 . ( + , ) . . . . + , + . . + . . . . . . . . .
52 2 . . . ( )( + ) . . ( ) − . . .
87 2 . . . − . . + , ( + ) . . . . . . . . . . ( ) .
64 2 . ( ) . . . . + . . . . . ( ) . . . . . . & ( ) . & . . ( ) . . ( ) . . ( ) . . ( ) . . . . ( ) . . ( ) .
05 2 . . . + . . ( ) .
07 3 . . . − , − . . . . . . . . . . ( ) . . . . . . . . ( ) . . . . . . . . ( ) . . . . . . . . . . . around 1% in Ce. Ref. [45] reports an (cid:96) = 0 transitionat 1.616 MeV in
Nd with a similar intensity. Suchweak transitions are also likely to be subject to highercontributions from indirect processes. There appears tobe no evidence for the relevant (cid:96) transfer in
Ba or
Sm. The assumption of a closed shell looks reason-able, at least compared to other uncertainties.Initially normalization was performed separately foreach (cid:96) value in the appropriately matched reaction andthe results are shown in in Table III.
TABLE III. Normalization factors for DWBA calculationswith the associated mean and standard deviation acrossthe four targets studied. Asterisks indicate cases that areaffected by significant unobserved strength.( p, d ) ( He, α ) (cid:96) = 0 (cid:96) = 2 ∗ (cid:96) = 4 ∗ (cid:96) = 5 Ba 0.58 0.40 0.22 0.58
Ce 0.55 0.40 0.40 0.52
Nd 0.51 0.42 0.23 0.54
Sm 0.53 0.44 0.31 0.59Mean 0.54 0.41 0.27 0.56St Dev 0.03 0.02 0.06 0.04
The mean normalization factors for the (cid:96) = 0 and (cid:96) = 5are 0.54 and 0.56, respectively, with a variation of around0.03 across the targets. These values compare favourablywith a recent systematic analysis of transfer data on tar-gets from O to
Pb for a variety of different protonand neutron transfer reactions over a range of (cid:96) values,which deduced a quenching with respect to independent-particle models of 0.55 [43]. The mean quenching factorsdeduced in that work for low (cid:96) transitions in ( d , p ) and( p , d ) reactions was 0.53; the excellent correspondencewith the current normalization for (cid:96) = 0 is particularlyencouraging. It relieves a potential concern that, givenmeasurements at 0 ◦ are not possible, (cid:96) = 0 spectroscopicfactors cannot be obtained as close to the first maxi-mum of the angular distribution as other (cid:96) values and,by necessity, are extracted in a region of a rather stronglysloping angular distribution.However, the average values for (cid:96) = 2 and (cid:96) = 4, at0.41 and 0.27, respectively, are significantly lower. Thissuggests that the experiment is missing some of the low-lying strength associated with the corresponding orbitals.This finding is not inconsistent with the observed distri-bution of high-lying, dispersed and fragmented strengthfor (cid:96) = 2 and 4 (see Fig. 6) where the risk of missingstrength is high, either in the form of transitions lyingoutside the measured excitation range or in the form ofsmall unresolved fragments of strength in the measuredspectra. We therefore adopt the values of 0.54 and 0.56for the DWBA normalizations for the ( p, d ) and ( He, α )reactions, respectively.The choice of potentials used in the DWBA calcula-tion has a significant effect on the absolute magnitude of the raw unnormalised spectroscopic factors; calculationswere repeated with a number of other physically reason-able potentials and a variation of ∼
20% in the calculatedabsolute cross sections was found.
Normalised spectro-scopic factors, determined using the procedures outlinedabove, are far less sensitive to choices of optical modelsand were found to vary by around ∼ T ± / T is the target isospin. The states corre-sponding to the higher isospin coupling T > lie at excita-tion energies higher than those accessed here experimen-tally. In principle, the Macfarlane and French sum rulesused in the normalization procedure for neutron-removalreactions need to include the T > strength. This can bedone on the basis of isospin symmetry, using spectro-scopic factors C S for analogous states in proton-removalreactions and applying the appropriate isospin Clebsch-Gordan coefficients to deduce the spectroscopic factor as-sociated with the higher isospin [46].The nuclei studied here are near the beginning of the Z = 50 −
82 shell and protons are known to occupy mainlythe g / and d / orbitals [47]; the spectroscopic factorsfor proton removal from the (cid:96) = 0 and 5 orbitals relevantfor the normalisation are consequently small (see Fig-ure 7). Moreover, the ratios of isospin Clebsch-Gordancoefficients that are required to convert these into thespectroscopic factors for the higher isospin states in neu-tron removal are also small. The overall correction for thenon-observation of the upper isospin component is lessthan a 1% effect for these orbitals and is smaller thanother uncertainties. The correction has therefore beenneglected in the normalization procedure here. Largercorrections would apply to the summed strengths for g / and d / , which have significant population of protonsand large proton removal strengths, but these are notused to determine the normalization. IV. DISCUSSION
Spectroscopic factors, extracted using the procedureoutlined in the previous section, were used to determinethe centroids of observed single-neutron hole strengthsfor the T < isospin components. These centroids and theassociated summed strength are summarized in Table IVand shown as a function of atomic number in Figure 8.In some previous studies, it has been assumed thatthe 3/2 + ground state exhausted the d / strength, buthere it is found that the associated spectroscopic fac-tor increases from Ba to
Sm. In addition to thetotal (cid:96) = 2 strength, Table IV also shows values asso-ciated with (cid:96) = 2 transitions populating states with a g d s h d Proton Occupancies P r o t o n o cc u p a n cy FIG. 7. Occupancy of single-proton orbitals in N = 82 nu-clei as a function of proton number, taken from from Ref. [47]for Ce, Nd and Sm and Ref. [49] for Xe and Ba. No pro-ton strength was observed for the s / orbital in Ref.[47] forCe and an upper limit of 0.2 was placed on the associatedoccupancy. firm or tentative 3/2 + spin assignment and the centroidof these are shown in Fig. 8. The associated summedstrengths are not as consistent across the isotopes as forthe other (cid:96) values, indicating that in some cases there ismissing d / strength and in others that there are likelysome mis-assignments of j values. The remaining (cid:96) = 2strength is likely attributable to the d / orbital, but itvaries between 50% and 76% of the full strength acrossthe isotopes. Fragmentation is high and a significant por-tion of the strength lies at excitation energies higher thanmeasured here.In the case of the g / strength, there is significantmissing strength and the current work only observed be-tween 40 and 61%, depending on the isotope. The truesingle-particle centroid lies higher than the observed cen-troid quoted in Table IV; we estimate that the true cen-troid lies at least Ba,
Ce,
Ndand
Sm, respectively, and because of this large uncer-tainty, we make no further discussion of (cid:96) = 4 strengthhere.In the cases where most of the low-lying strength hasbeen captured ( (cid:96) = 0 and 5), the centroid across both T < and T > isospin components would reflect the underlyingsingle-neutron energy. As discussed above, only the T < strength is observed in the current work. The locationand strength of the T > component were estimated usingCoulomb displacement energies and data from proton-removal reactions [47] using isospin symmetry. It wasfound that the difference between the full centroid andthat for the T < component of the (cid:96) = 0 and 5 strengthincreases with Z from around 20 to 90 keV across theisotopes. This is relatively small since the associatedorbitals have low proton occupancy. The correction ismuch larger for (cid:96) = 2 and 4 strength, but these are thesame orbitals where significant strength remains unob-served in the current experiment and the interpretationof the measured centroids is difficult. We therefore usethe variation in the measured centroids of (cid:96) = 0 and 5strength as an estimate for the changes in the underlyingsingle-neutron energies across the isotones studied.Changes in orbital energies across chains of nuclideshave been interpreted in terms of the effect of va-lence proton-neutron interactions as the nucleon num-ber varies. Here we follow the approach of Reference [2]where changes in the effective single-neutron energieswere compared to calculations using a two-body centralplus tensor force between neutrons and valence protons,taking information on proton occupancy from proton-transfer experiments in the literature.The occupancies of single-proton orbitals are availablefrom previous measurements of proton removal using the( d , He) reaction. Reference [47], which reports reactionson N = 82 nuclei from Xe through to Sm, is broadlyin agreement with a contemporaneous study on Ba, Ceand Nd [48]. A more recent study has been made of Xeand Ba nuclei [49] with higher precision. Here we adoptthe Ba occupancies from Ref. [49] and those for
Ce,
Nd and
Sm from Ref. [47].The pattern of proton occupancies is illustrated in Fig-ure 7, showing significant occupation of the g / and d / orbitals. The occupancy of the g / orbital increases un-til Z = 58, beyond which the changes in occupancy aremainly in the d / orbital. Other orbitals are filled to lessthan 10%. The h / orbital gradually increases in pop-ulation across the isotopes, but remains small. Evidencefor a low level of occupancy of the s / orbital by protonshas been found in all nuclei, except for Ce where onlyan upper limit is available. The proton occupancy of the d / orbital begins to be observable in the two heaviestsystems. Although the population of low- (cid:96) single-protonstates are small, they can have a significant effect on theenergies of certain neutrons where the orbital overlap islarge.Calculations of the changes in effective single-neutronenergies presented here were performed using the effec-tive two-body force from Reference [53] (labelled here asHKT) which was deduced from a G-matrix treatmentof the Paris nucleon-nucleon interaction. The results ob-tained with that force are very similar to those done usingthe phenomenological Schiffer and True [50] interaction.0 TABLE IV. Observed summed hole strengths and the associated centroid excitation energies for the T < components. Thesummed strength is deduced from spectroscopic factors that were normalized using the method described in the text. Theerrors quoted on the summed strength are on the basis of the variations due to choices of potentials in the DWBA (see textfor details). The errors on the centroid in the table are statistical. Values are given for the sum of d / and d / orbitalsdeduced for the (cid:96) = 2 transitions and also separately for states populated by (cid:96) = 2 transitions with a spin-3/2 assignment inthe literature. Asterisks indicate cases that are affected by significant unobserved strength, which gives rise to a significantsystematic uncertainty in the true single-particle centroid.Orbital Summed Strength Centroid Energy (MeV) Ba Ce Nd Sm Expected Ba Ce Nd Sm s / d ∗ d / g ∗ / h / s d g h Neutron-Hole Centroids E x c i t a t i o n En e r g y ( M e V ) FIG. 8. Variation in the excitation energy of the centroidof observed single-particle strength for the T < component asa function of proton number. Statistical errors are of theorder ∼
10 keV. The open circles and dotted lines indicateinstances where the full single-particle strength has not beenobserved. The centroid for the d / orbital uses states thathave a 3 / + spin-parity in the literature. The data for the g / orbital suffers from significant unobserved strength outsideof the excitation-energy range measured and the true single-particle centroid will lie significantly higher than the observedcentroid (see text for details). Both used single-particle wave functions from infiniteoscillator potentials. Individual matrix elements werecalculated using the computer code of Reference [54], proton-neutron monopole shifts were constructed (theseare available as part of the Supplemental Information[28]) and the changes in neutron single-particle energyacross the N = 81 nuclei were obtained using the protonoccupancies described above.To study the effect of the proton occupancy on therelative changes in neutron binding as a function of pro-ton number across the isotopes studied, the experimen-tal data (solid dots) are plotted in Figure 9. A smoothincrease in the binding energy of the neutron s / and h / orbitals is found when adding protons, due to thetrends in proton occupancy shown in Figure 7, and thefact that many of the monopole terms have a similar am-plitude. Consequently, the effective energy follows that ofan averaged global trend of an attractive proton-neutroninteraction. Since some of the two-body interactions aredifferent, the change in binding was calculated using themonopole shifts with the HKT interaction and the ex-perimental proton occupancies. Since only the variationwith A is meaningful, the absolute value of these calcu-lations along the vertical axis in the figure was shifted tofit the experimental points. These calculations, includ-ing the experimental uncertainties in the proton occupan-cies, are represented by the shaded areas. (Additionally,the two-body matrix elements themselves are subject tosome uncertainty. This is rather difficult to estimate, butis likely of the order of 10%).The monopole shifts for neutron states are particularlysensitive to uncertainties in the occupancy of the corre-sponding proton orbital due to their large overlap. This iscompounded in the case of Ce where only an upper limiton the s / proton occupancy had been determined. In-deed, the case of s / may be more complicated if some ofthe weak unassigned strength in the proton-removal reac-tions is in reality (cid:96) = 0; for example, there is unassignedstrength in the Ba( d , He) reaction that amounts toaround 0.1 protons (see Table VIII in Ref. [49]).The trend in the energy of the neutron h / orbitalappears reasonably well reproduced by the calculations,as shown in Figure 9, but the slope of the neutron s / orbital is less well predicted in the calculations using1monopole shifts from the HKT interaction with harmonicoscillator wave functions. The difference in slope in Fig-ure 9 between the data and the monopole-shift calcu-lations for the neutron s / orbital suggests that othereffects are playing a role for that single-particle state.The two-body matrix elements yielding the monopoleshifts were calculated using single-particle wave functionsin an infinite harmonic oscillator potential where the or-dering of the different states is fixed. However, any po-tential with finite binding is subject to geometric effectssuch that the single-particle states behave somewhat dif-ferently depending on their binding energy relative to theheight of the binding potential including the centrifugalterm (and Coulomb effects where relevant). Such effectsare known; for instance, they were demonstrated in Fig2.30 of Ref. [51] where different neutron orbitals in the50-82 shell have different behaviors as a function of A ,notably the s / state, and this was discussed in moredetail in Ref. [52].The mean field is a sum of two-body interactions, butit is not easy to separate effects that depend on angularmomentum (such as the tensor interaction) from thosecaused by geometric effects from finite binding. It istherefore instructive to also compare the data to Woods-Saxon calculations, where geometric effects are included,but the angular-momentum dependence from the two-body interaction is not. Fig. 9 shows the results of suchcalculations with standard radius and asymmetry terms,with parameters fixed to the binding energy of the 11/2 − state in Ba. Such calculations do appear to better re-produce the slope of the s / data.Given these limitations, the level of agreement betweendata and monopole-shift calculations displayed in Fig. 9is probably reasonable, and constitutes a check on howwell the changes in binding energies across the isotopescan be reproduced by the effect of microscopic interac-tions.The interpretation of experimental centroids in termsof monopole-shift calculations presented above is a coarsecomparison and it would be useful to understand the frag-mentation of single-neutron hole strength across states inthe populated nucleus. The general distribution of trans-fer strength revealed here is reasonably well reproducedby particle-vibration coupling calculations performed anumber of years ago [10], given the limitations of themodel used (see Fig. 6). The strong low-lying (cid:96) = 0 , (cid:96) = 4strength is predicted to be higher-lying and fragmented,as observed, but any state-to-state correspondence be-tween the experimental data and calculated strength isdifficult due to the extent of the fragmentation seen inthe experiment.It would be interesting to compare the strength distri-butions with the results from modern large-scale shell-model calculations. However, the dimensions of the Exp s
Exp h
HKT s
HKT h
Woods Saxon s
Woods Saxon h B i n d i n g e n e r g y ( M e V ) − − − − FIG. 9. Experimental single-particle binding energies for theneutron s / (black) and h / (blue) orbitals, deduced fromthe centroids of hole excitation energies. Calculations usedthe effective two-body interaction (HKT) of Ref. [53] and pro-ton valence occupancies from Refs. [47, 49]. These are shownas bands reflecting the uncertainties in the proton occupanciesand the absolute value of these calculations along the verticalaxis in the figure was shifted to fit the experimental points(see text for more details). The solid lines are Woods-Saxoncalculations with standard radius and asymmetry terms withparameters fitted to the 11 / − state in Ba. model space in such a large shell are currently ratherdifficult to manipulate, making such calculations tricky.Some shell-model calculations have been made around A = 130 nuclei [55], which includes Ba as one of theheaviest systems considered. Pair-truncated shell-modelcalculations have been discussed for
Ba and
Ce [56].The results in both cases have so far only been comparedto level energies and electromagnetic moments; predic-tions of spectroscopic factors are not readily available inthe literature. We hope that the current data will informlarge-scale calculations as they become available in thefuture.In summary, neutron-hole strength in the N = 81 nu-clei Ba,
Ce,
Nd and
Sm has been studied inthe ( p , d ) and ( He, α ) neutron-removal reactions at en-ergies of 23 and 34 MeV, respectively. Relative spec-troscopic factors extracted through a DWBA analysisand centroids of single-particle strength have been estab-lished. The majority of the strength has been observedfor the s / and h / orbitals. Strong fragmentation ofstrength was observed for the g / orbital, which is moredeeply bound and significant strength lies outside of themeasured excitation energy range. It proved difficult toproperly disentangle d / and d / strength; the com-bined (cid:96) = 2 strength distribution is broad and also seemsto suffer from unobserved, presumably d / , fragments.Changes in the effect of monopole shifts of neutron ener-gies due to changes in proton occupancy appear to repro-duce the trends in the effective single-particle energies ofthe s / and h / orbital, at least given the influence of2a number of other effects on the former orbital. ACKNOWLEDGMENTS
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