Gamma strength function and level density of 208 Pb from forward-angle proton scattering at 295 MeV
GGamma strength function and level density of
Pb from forward-angle protonscattering at 295 MeV
S. Bassauer, ∗ P. von Neumann-Cosel, † and A. Tamii Institut f¨ur Kernphysik, Technische Universit¨at Darmstadt, D-64289 Darmstadt, Germany Research Center for Nuclear Physics, Osaka University, Ibaraki, Osaka 567-0047, Japan (Dated: October 24, 2016)
Background:
Gamma strength functions (GSFs) and level densities (LDs) are essential ingredients of statistical nuclearreaction theory with many applications in astrophysics, reactor design, and waste transmutation.
Purpose:
The aim of the present work is a test of systematic parametrizations of the GSF recommended by the RIPL-3data base for the case of
Pb. The upward GSF and LD in
Pb are compared to γ decay data from an Oslo-typeexperiment to examine the validity of the Brink-Axel (BA) hypothesis. Methods:
The E1 and M1 parts of the total GSF are determined from high-resolution forward angle inelastic proton scatteringdata taken at 295 MeV at RCNP, Osaka, Japan. The total LD in
Pb is derived from the 1 − LD extracted with afluctuation analysis in the energy region of the isovector giant dipole resonance.
Results:
The E1 GSF is compared to parametrizations recommended by the RIPL-3 data base showing systematic deficienciesof all models in the energy region around neutron threshold. The new data for the poorly known spinflip M1 resonancecall for a substantial revision of the model suggested in RIPL-3. The total GSF derived from the present data is largerin the PDR energy region than the Oslo data but the strong fluctuations due to the low LD resulting from the doubleshell closure of
Pb prevent a conclusion on a possible violation of the BA hypothesis. Using the parameters suggestedby RIPL-3 for a description of the LD in
Pb with the back-shifted Fermi gas model, remarkable agreement betweenthe two experiments spanning a wide excitation energy range is obtained.
Conclusions:
Systematic parametrizations of the E1 and M1 GSF parts need to be reconsidered at low excitation energies.The good agreement of the LD provides an independent confirmation of the approach underlying the decomposition ofGSF and LD in Oslo-type experiments.
PACS numbers: 25.40.Ep, 21.10.Ma, 21.60.Jz, 27.80.+w
I. INTRODUCTION
Gamma strength functions describe the average γ de-cay behavior of a nucleus. They serve as input for ap-plications of statistical nuclear theory in astrophysics [1],reactor design [2], and waste transmutation [3]. Althoughall electromagnetic multipoles contribute, the GSFs areusually dominated by the E1 component with smallercontributions from M1 strength. Above particle thresh-old it is governed by the isovector giant dipole resonance(IVGDR) but for astrophysical processes the energy re-gion around particle thresholds and at even lower exci-tation energies [4] is more important. There, the situa-tion is more complex: In nuclei with neutron excess oneobserves the formation of the pygmy dipole resonance(PDR) [5] but the low-energy tail of the IVGDR can alsocontribute. Furthermore, the spinflip M1 resonance over-laps with the energy region of the PDR [6]. The impactof the low-energy GSFs on astrophysical reaction ratesand the resulting abundances in the r -process have beendiscussed e.g. in Refs. [7–9].Many applications imply an environment of finite tem-perature, notably in stellar scenarios [10], and thus reac-tions on initially excited states become relevant. Their ∗ Electronic address: [email protected] † Electronic address: [email protected] contributions to the reaction rates are usually estimatedapplying the generalized Brink-Axel (BA) hypothesis[11, 12] which states that the GSF is independent of theproperties of the initial and final states. The validity ofthe BA hypothesis is also implicity assumed in the deriva-tion of the GSFs from many experimental data based onground state photoexcitation. Although historically for-mulated for the IVGDR, where it seems to hold approx-imately for not too high temperatures [13], the BA hy-pothesis is nowadays commonly used to calculate the low-energy E1 and M1 strength functions. This is questionedby a recent shell-model analysis [14] where it was demon-strated that the strength functions of collective modesbuilt on excited states do show an energy dependenceand this is expected from spectral distribution theory.However, the numerical results for E1 strength functionsshowed an aproximate constancy consistent with the BAhypothesis.Recent work utilizing compound nucleus γ decay withthe so-called Oslo method [15] has demonstrated inde-pendence of the GSF from excitation energies and spins ofinitial and final states in accordance with the BA hypoth-esis once the level densities are sufficiently high to sup-press large intensity fluctuations [16]. However, there area number of experimental results which seem to violatethe BA hypothesis in the low-energy region. For example,the GSFs in heavy deformed nuclei at excitation energiesof 2 − B (M1) strengths are a r X i v : . [ nu c l - e x ] O c t observed between γ emission [18, 19] and absorption [20]experiments. For the low-energy E1 strength the ques-tion is far from clear when comparing results from theOslo method with photoabsorption data. Below parti-cle thresholds much of the information on GSFs stemsfrom nuclear resonance fluorescence (NRF) experiments.A problem of the NRF method are experimentally un-observed branching ratios to excited states which havebeen neglected in many cases [5]. Recent studies of the γ decay after photoabsoprtion indicate that these may besizable [21, 22].Here we present results for Pb from a new methodfor the measurement of E1 and M1 strength distributionsin nuclei (and thus the GSF) from about 5 to 25 MeVbased on relativistic Coulomb excitation in inelastic po-larized proton scattering at energies of a few hundredMeV and scattering angles close to 0 ◦ [23–26]. The E1strength distribution from Coulomb excitation permitsto determine the dipole polarizability which provides im-portant constraints on the neutron skin of nuclei and thepoorly known parameters of the symmetry energy [27].It also allows extraction of the M1 part of the GSF [28]due to spinflip excitations which energetically overlapswith the PDR strength. The high-resolution data alsoprovide information on level densities – another essentialingredient of statistical model cross section calculations– from an analysis of the fine structure of the IVGDR[29].The purpose of the paper is twofold. On one hand,recommended parametrizations of the GSF and LD sum-marized in the RIPL-3 data base [30] are evaluated forthe case of Pb. In particular, we provide new datafor the poorly known M1 part of the GSF. On the otherhand, the comparison with the Oslo experiment providesa test of the BA hypothesis. Moreover, since GSF andLD are independently determined, the decomposition ofboth quantities in the Oslo method, which measures theproduct of GSF and LD [15], can be verified.
II. GAMMA STRENGTH FUNCTION OF Pb In the experiments discussed here, the GSF for an elec-tric or magnetic transition X ∈ { E, M } with multipo-larity λ is related to the photoabsoprtion cross section (cid:10) σ Xλabs (cid:11) f Xλ ( E, J ) = 2 J + 1( π ¯ hc ) (2 J + 1) (cid:10) σ Xλabs (cid:11) E λ − , (1)where E denotes the γ energy and J, J the spins of ex-cited and ground state, respectively [30]. The brackets (cid:104)(cid:105) indicate averaging over an energy interval. In prac-tise, only E1, M1 and E2 provide sizable contributions tothe total GSF. In the following, we discuss the derivationof the GSF for these components from the experimentaldata and compare to parametrizations recommended inRIPL-3. Energy (MeV) − − − − E - G S F ( M e V − ) Pb(p,p’)Sys. GSFSLOMLOEGLO
FIG. 1: (Color online) E1 GSF of
Pb deduced from the( p, p (cid:48) ) data [23, 24] (blue diamonds) in comparison with theSLO (green line), MLO (cyan line), and EGLO (magenta line)models explained in the text. The black circle shows the pre-diction from experimental systematics at the neutron separa-tion threshold [30].
A. E1 contribution
The E1 contribution of the GSF in
Pb was deter-mined using polarized inelastic proton scattering dataobtained at the Research Center for Nuclear Physics(RCNP) at Osaka, Japan with a beam energy of 295 MeVin an excitation energy region from 5 to 23 MeV [23, 24].In Ref. [24], the B (E1) strength was extracted by meansof the multipole decomposition analysis (MDA) in the en-ergy region from 4.8 to 9 MeV. The B(E1) strength pro-vided in Table I was used to determine the E1 part of theGSF. In the IVGDR region, the E1 GSF was determinedfrom photoabsorption cross sections extracted from the( p, p (cid:48) ) data by means of the virtual photon method [23].The resulting E1 GSF is shown in Fig. 1 in comparisonwith three widely used models and with a GSF value atthe neutron separation threshold deduced from experi-mental systematics over a wide mass range [30].The simplest model to describe the E1 GSF is a stan-dard Lorentzian (SLO) function f SLO ( E ) = σ r Γ r π ¯ hc ) Γ r E ( E − E r ) + (Γ r E ) , (2)where σ r is the peak cross section, E r the centroid energyand Γ r the width of the IVGDR.A more sophisticated model is the enhanced general-ized Lorentzian (EGLO) model f EGLO ( E ) = σ r Γ r π ¯ hc ) (3) × (cid:20) Γ K ( E, T ) E ( E − E r ) + (Γ K ( E, T ) E ) + 0 . K ( E = 0 , T ) E r (cid:21) . The EGLO consists of two terms [31], a Lorentzian withan energy- and temperature-dependent width Γ K ( E, T )and a term describing the shape of the low-energy partof the GSF. The temperature dependence is estimatedwithin Fermi liquid theory [32]Γ K ( E, T ) = χ ( E ) Γ r E r ( E + (2 πT ) ) , (4)where χ ( E ) = κ + (1 − κ ) E − E E r − E (5)is an empirical function with parameters κ and E , where κ is adjusted to reproduce the experimental E1 strengthat a reference energy E [30].The SLO and EGLO models are both parametriza-tions of experimental data. In contrast, the modifiedLorentzian model (MLO) is based on general relationsbetween the GSF and the imaginary part of the nuclearresponse function f MLO ( E ) = σ r Γ r π ¯ hc ) Λ( E, T )Γ(
E, T ) E ( E − E r ) + (Γ( E, T ) E ) , (6)where Λ( E, T ) = 11 − exp( − E/T ) . (7)The function Λ( E, T ) accounts for the enhancementof the GSF with increasing temperature. The widthΓ(
E, T ) within the MLO is calculated with microcanon-ically distributed initial states [33].The resulting predictions are shown in Fig. 1 as green(SLO), magenta (EGLO), and cyan (MLO) curves. Inthe region around the maximum of the IVGDR all mod-els provide a good description. The high-energy tailof the IVGDR is well described by SLO and MLOwhile EGLO overestimates the photoabsorption crosssections. The low-energy tail of the IVGDR exhibitsstrong fluctuations which complicate the comparisonwith smooth strength functions. For excitation energiesdown to about 8 MeV, MLO describes the average be-havior fairly well while SLO(EGLO) are roughly consis-tent with the upper(lower) limits of the fluctuations butover(under)estimate the average cross sections. Between6 and 8 MeV a resonance-like structure dominates theGSF identified as the PDR in
Pb [24]. This low-energyresonance is not included in the models. Finally, the GSFvalue expected at neutron threshold ( S n = 7 .
37 MeV inthe present case) from experimental systematics of neu-tron capture cross sections (black circle) is almost an or-der of magnitude smaller than the experimental strengthsin the PDR. However, this may be an artefact of theunusually low level density in the doubly magic nucleus
Pb with corresponding strong fluctuations of individ-ual strengths at energies close to the neutron threshold(note that the GSF values correspond to energy binsrather than to individual transitions for excitation en-ergies above 7 MeV (cf. Tab. I in Ref. [24]).
Energy (MeV) − − − − M - G S F ( M e V − ) Pb(p,p’)Sys. GSFSLOMLOEGLO
FIG. 2: (Color online) Same as Fig. 1 but for the M1 compo-nent of the GSF.
B. M1 contribution
In addition to the B (E1) strengths measured in the( p, pprime ) experiment, M1 cross sections at Θ = 0 ◦ are provided in Tab. I of Ref. [24]. These are concen-trated between 7 and 9 MeV and represent the spinflipM1 resonance [6]. We note that an additional M1 tran-sition to a 1 + state at 5.844 MeV is known (see Ref. [34]and references therein) but omitted here because it is ofdominant isoscalar nature [35]. Recently a method uti-lizing isospin symmetry has been presented to relate thespinflip M1 cross sections to those of Gamow-Teller ex-citations studied with the ( p, n ) reaction and extract thespin-M1 matrix elements [28]. Assuming dominance ofthe spin-isovector part of the electromagnetic M1 opera-tor reduced B (M1) transition strengths can be extracted.In the resonance region these agree well [28] with studiesusing electromagnetic probes [36, 37]. At excitation ener-gies above 8 MeV, where previous experiments had lim-ited sensitivity [36], additional B (M1) strength is foundin the ( p, p (cid:48) ) data which raises the total strength of thespinflip M1 resonance by about 20%.Figure 2 displays the corresponding M1 GSF in com-parison with SLO, EGLO, and MLO model predictionsfor Pb. The M1 GSF model results are derived fromthe E1 models discussed above in the following way [30] f M ( E ) = f E ( S n ) R Φ M ( E )Φ M ( S n ) (8)and R = f E ( S n ) f M ( S n ) = 0 . A . , (9)where Φ M ( E ) is a SLO parametrization of the spinflipM1 resonance with energy centroid E r = 41 · A − / andΓ r = 4 MeV [38]. The mass dependence of the ratio R in Eq. (9) is valid for nuclei with S n ≈ Pb.
Energy (MeV) − − − E - G S F × E x ( M e V − ) Pb( α , α ’)SLO FIG. 3: (Color online) E2 GSF deduced from Ref. [39] incomparison to the SLO model with parameters from Ref. [40].The GSF was multiplied by E x to make the units comparableto the E1 and M1 GSF. The comparison in Fig. 2 indicates that the theo-retical GSF values near maximum are of magnitudesroughly comparable to the data. However, the assumedresonance properties represent a poor approximation ofthe data. The theoretical maxima are about 500 keVtoo low and the experimental width is grossly overes-timated. As a result, the predicted total strengths ofthe spinflip M1 resonance exceed the experimental value (cid:80) B (M1) = 20 . µ N [28] by factors ranging from two(EGLO) to five (SLO). C. E2 contribution
The E2 contribution to the GSF was estimated us-ing ( α , α (cid:48) ) data obtained at the Texas A&M K500 super-conducting cyclotron, College Station, Texas, USA [39].In this experiment several isotopes including Pb wereinvestigated using alpha particles with an energy of240 MeV. The data was taken in an excitation energyregion of 10 to 55 MeV where isoscalar E0, E1, E2 andE3 strength distributions were extracted with the aid of aMDA. The resulting E2 strength distribution exhausted100 ±
15% of the energy weighted sum rule (EWSR).Using this data the B(E2) strength distribution was ob-tained and converted to the E2 GSF shown in Fig. 3.The solid line shows a global parametrization of the E2giant resonance [40] suggested in earlier RIPL versions.
D. Total GSF and comparison with Oslo data
Figure 4 summarizes the E1, M1 and E2 contributionsto the total GSF. As can be seen, the dominant contribu-tion stems from E1 transitions. The M1 contribution isof the order of a few percent for excitation energies above8 MeV and reaches at most 10-30% in the peak of the
Energy (MeV) − − − − − G S F ( M e V − ) E1-GSFM1-GSFE2-GSF × E FIG. 4: (Color online) Comparison of E1, M1 and E2 contri-butions to the GSF of
Pb. resonance around S n . The E2 contribution is of compa-rable magnitude to M1 but located at higher excitationenergies. Because of the simultaneous strong rise of theE1 part in the IVGDR energy region the E2 contributionto the GSF at the maximum of the E2 resonance is about1% only.The total GSF summing all contributions is displayedin Fig. 5 (blue diamonds) and compared with data de-rived with the Oslo method from a Pb( He, He (cid:48) γ ) ex-periment (red circles) [41]. The data set has been reana-lyzed recently [42]. The main changes are new, updatedresponse functions for the CACTUS detector array andan improved error estimate in the simultaneous extrac-tion of level density and γ strength from the primary γ -ray spectra. The initial excitation energy range usedfor the reanalysis was 4 . ≤ E i ≤ .
95 MeV, and theapplied low- E γ threshold was 2.65 MeV. For consistencywith the previous work, the level density has been nor-malized to the p -wave resonance data of RIPL-2, see Tab.I in Ref. [41]. Further, the γ strength has been normal-ized to recent ( γ ,n) data by Kondo et al. [43] and alsocompared to older data [44, 45]. This is considered as thelow-limit estimate of the γ strength from the He-inducedreaction.There are overlapping results from both experimentsin the energy region between 5 and 8 MeV (see inlet ofFig. 5). The GSF derived from the ( p, p (cid:48) ) data is system-atically higher in the PDR region although they seem stillcompatible within error bars in the peak region aroundthe neutron threshold. Between 6 and 7 MeV consistentresults are found while below 6 MeV the strong transi-tions observed in Ref. [24] exceed the average γ strengthin the Oslo data by factors 4 to 5. However, one shouldbe aware that single transitions are analyzed for excita-tion energies E x < − states excited from the ground state is probably toolow to discuss an average behavior in the PDR region.Rather the upward GSF is dominated by Porter-Thomasintensity fluctuations. Energy (MeV) − − − − − − T o t a l G S F ( M e V − ) Pb(p,p’)
Pb( He, He’ γ )Total GSF (SLO)Total GSF (EGLO) − − − FIG. 5: (Color online) Total GSF of
Pb from the ( p, p (cid:48) )data [23, 24] in comparison to the reanalyzed [42] resultsfromm the Oslo experiment [41]. The inlet shows and ex-panded view of the low-energy region 5 − III. LEVEL DENSITY OF PB Level density of 1 − states in the excitation energy re-gion from 9 to 12.5 MeV was determined from ( p, p (cid:48) ) data[29] using a fluctuation analysis [46]. However, the LDsfor Pb derived from the Oslo method represent a dif-ferent spin window depending on the specific reaction.Thus all results are converted to total level densities us-ing Fermi gas models [15]. This can be achieved usingthe following equation ρ tot ( E ) = 2 ρ ( E, J, Π) f ( J ) , (10)where ρ tot ( E ) is the total level density at energy E and ρ ( E, J,
Π) is the level density for transitions with spin J and parity Π. The function f ( J ) is the so-called spindistribution function defined as f ( J ) = 2 J + 12 σ exp (cid:20) − ( J + ) σ (cid:21) , (11)where σ is the spin cutoff parameter. Since the spin cut-off depends on the parameters of the Fermi gas model onehas to investigate the model dependence. For this pur-pose, we considered three parameter sets derived withinthe backshifted Fermi gas model (BSFG) approach [47].These include the one used in the original analysis ofthe Oslo experiment [41], a global set recommended inRIPL-3 [30], and the parametrization of Ref. [48] devel-oped for s -process reaction network calculations, whichhas been shown to provide a good description of LD formany nuclei near the valley of stability [29, 49–51].Figure 6 shows the spin distribution functions from thethree different parametrizations at excitation energies of8 and 15 MeV, which show significant differences. Thevalues for J = 1 are indicated by the vertical dashedlines. S p i n D i s t r i b u t i o n F u n c t i o n f(J) E x = 8 MeV BSFGM (RIPL-3)BSFGM (Rauscher et al.)BSFGM (Syed et al.)
Spin J E x = 15 MeV BSFGM (RIPL-3)BSFGM (Rauscher et al.)BSFGM (Syed et al.)
FIG. 6: (Color online) Spin distribution functions of LDs in
Pb at excitation energies of 8 and 15 MeV from BSFGmodel predictions with the parameters of Ref. [41] (magentadashed-dotted lines), Ref. [30] (green solid lines), and Ref. [48](cyan dashed lines). The vertical dashed lines indicate thevalues for spin J = 1. Figure 7 presents the resulting total LDs in
Pbfrom the three models for an excitation energy range E x = 9 − . − LD discussed inRef. [29].The comparison with the Oslo results (red squares) is
Energy (MeV) T o t a l L e v e l D e n s i t y ( M e V − ) Pb(p,p’) (Mean)
Pb(p,p’) (RIPL-3)
Pb(p,p’) (Rauscher et al.)
Pb(p,p’) (Syed et al.)
FIG. 7: (Color online) Total level density of
Pb between 9and 12.5 MeV obtained with the spin distribution functionsof Ref. [41] (rightpointing magenta triangles), Ref. [30] (greenupward triangles), and Ref. [48] (cyan downward triangles).Mean values averaged over the three models are shown as bluediamonds.
Energy (MeV) − T o t a l L e v e l D e n s i t y ( M e V − ) Pb(p,p’)
Pb( He, He’ γ )Discrete LDBSFGM (RIPL-3)BSFGM (Rauscher et al.)BSFGM (Syed et al.) FIG. 8: (Color online) Total LD from the ( p, p (cid:48) ) data [23, 24]in comparison to the reanalyzed [42] results from the Osloexperiment [41]. The black downward triangles are resultsfrom from counting the levels identified in Ref. [52] in 200keV bins. The magenta dashed-dotted, green solid, and cyandashed lines are BSFG model predictions with the parametersof Ref. [41], Ref. [30], and Ref. [48], respectively. finally presented in Fig. 8. The value at neutron thresh-old (black circle) is deduced from p -wave resonance neu-tron capture converted to a total LD with the aid of theRIPL-3 BSFG parametrization. The black downward tri-angles denote the results from level counting from a re-cent study claiming essentially complete spectrocopy upto 6.2 MeV [52]. Indeed, the LD agrees well with theOslo result up to about 5 MeV. For E x > S n with factors 1.15, 2.18, and 0.52. However, absolutevalues for the RIPL-3 parametrization are obtained bynormalizing to s -wave neutron capture resonance spac-ings. As pointed out in Ref. [41], the data are ratherpoor in Pb and one should rather normalize to the p wave spacings, i.e., the solid curve is absolute. Theenergy dependence of the BFSG model shows differencesover the wide energy range spanned by the two data sets.Remarkably, the RIPL-3 parameter set, whose predicionsof the total LD are closest to the mean value (cf. Fig. 7),provides a consistent description of all data. IV. SUMMARY AND CONCLUSIONS
The main aim of this work was to determine the E1,M1 and total GSF of
Pb for tests of models recom-mended in the RIPL-3 data base as well as to study the BA hypothesis by comparison with decay data obtainedwith the Oslo method. It is shown that the E1 GSF canbe described well by the SLO and MLO models in theGDR region. In the low-energy region strong fluctuationsoccur, so that no particular model can be favored. Theaverage behavior of the low-energy tail of the IVGDR isprobably best described by the MLO model. However,none of models includes the PDR and thus the predictivepower at low excitation energies is generally limited.The presently recommended parametrization of thespinflip M1 resonance provides only a poor descriptionof the
Pb data. Although the absolute magnitude ofthe resonance maximum is reproduced within a factorof 2 to 3, the width of the M1 GSF is strongly overes-timated. As a result the B (M1) strengths is predictedby the empirical models are too large by factors 2 to 5.Since the excitation energy ranges of the spinflip M1 res-onance and the PDR overlap in heavy nnuclei, this has astrong impact on attempts to extract model parametersfor the PDR contribution in decay experiments. Clearly,more data are needed to establish the systematics of thepoorly known spinflip M1 resonance in heavy nuclei. Themethod presented in Ref. [28] promises experimental in-formation from the ( p, p (cid:48) ) data on spherical [25, 53] aswell as deformed [54, 55] nuclei.The comparison of the present GSF derived fromground-state absorption with the Oslo results showslarger values in the PDR energy region, where both datasets overlap. However, the fluctuations of the GSF arevery strong due to the anomalously small level densitiesin the closed-shell nucleus Pb, which prevents conclu-sions on a possible violation of the BA hypothesis in thePDR energy region. Here, tests in open-shell nuclei withhigher level densities are required and a correspondingstudy [54] is underway.Total level densities for
Pb were derived from fluc-tuations of the high-resolution ( p, p (cid:48) ) cross sections inthe IVGDR energy region and compared to those fromthe Oslo method covering much lower energies. Using theBSFG model parameters suggested by RIPL-3 to convertthe experimental partial-spin results to total level densi-ties and to describe their energy dependence, remarkableagreement between the two results is obtained. This pro-vides an independent confirmation of the approach [15]to separate GSF and LD in Oslo-type data.
Acknowledgments
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