Determination of the ^{60}Zn level density from neutron evaporation spectra
D. Soltesz, M. A. A. Mamun, A. V. Voinov, Z. Meisel, B. A. Brown, C. R. Brune, S. M. Grimes, H. Hadizadeh, M. Hornish, T. N. Massey, J. E. O'Donnell, W. E. Ormand
DDetermination of the Zn Level Density from Neutron Evaporation Spectra
D. Soltesz, ∗ M.A.A. Mamun, A.V. Voinov, Z. Meisel, † B.A. Brown, C.R. Brune, S.M.Grimes, H. Hadizadeh, M.Hornish, T.N. Massey, J.E. O’Donnell, ‡ and W.E. Ormand Institute of Nuclear & Particle Physics, Department of Physics & Astronomy, Ohio University, Athens, Ohio 45701, USA Department of Physics & Astronomy and National Superconducting Cyclotron Laboratory, East Lansing, Michigan 48824, USA Lawrence Livermore National Laboratory, P.O. Box 808, L-235, Livermore, California 94551, USA
Nuclear reactions of interest for astrophysics and applications often rely on statistical model cal-culations for nuclear reaction rates, particularly for nuclei far from β -stability. However, statisticalmodel parameters are often poorly constrained, where experimental constraints are particularlysparse for exotic nuclides. For example, our understanding of the breakout from the NiCu cycle inthe astrophysical rp -process is currently limited by uncertainties in the statistical properties of theproton-rich nucleus Zn. We have determined the nuclear level density of Zn using neutron evapo-ration spectra from Ni( He , n ) measured at the Edwards Accelerator Laboratory. We compare ourresults to a number of theoretical predictions, including phenomenological, microscopic, and shellmodel based approaches. Notably, we find the Zn level density is somewhat lower than expectedfor excitation energies populated in the Cu( p, γ ) Zn reaction under rp -process conditions. Thisincludes a level density plateau from roughly 5–6 MeV excitation energy, which is counter to theusual expectation of exponential growth and all theoretical predictions that we explore. A determi-nation of the spin-distribution at the relevant excitation energies in Zn is needed to confirm thatthe Hauser-Feshbach formalism is appropriate for the Cu( p, γ ) Zn reaction rate at X-ray bursttemperatures.
I. INTRODUCTION
Atomic nuclei have long been known to be well de-scribed by statistical properties at excitation energieswith a relatively dense spacing of nuclear levels [1, 2].The number of levels per excitation energy E ∗ , referred toas the nuclear level density ρ , is of particular importance.For instance, ρ determines the number of states accessi-ble in the decay of a compound nucleus via a particularparticle or γ -emission channel. While ρ has been wellcharacterized for nuclei along the valley of β -stability [3],experimental challenges have made constraints for short-lived nuclides virtually absent. This is troubling consid-ering the key role statistical estimates of nuclear reactionrates play in calculations of astrophysical phenomena andfor nuclear applications [4, 5].The level density enters into the calculationof the nuclear reaction cross section throughthe Hauser-Feshbach (HF) formalism [6, 7] via σ HF ∝ λ ( T form T decay ,i ) / Σ j T decay ,j , where λ is thede Broglie wavelength for the entrance channel and T are the transmission coefficients that describe theprobability for a particle or photon, which defines thechannel, being emitted from (“decay”) or absorbed by(“form”) a nucleus. The T for decay channels in thepreceding equation, T decay ,i and all other open decaychannels included in the sum Σ j T decay ,j , are actually asum over T to individual discrete states added to anintegral over T to levels in a higher-excitation energyregion described by ρ . ∗ [email protected] † [email protected] ‡ Deceased.
A variety of methods can be used to obtain ρ . Pri-mary techniques include a direct determination of levelspacings by level counting or neutron resonance spac-ings [8, 9]; proton scattering at extreme forward an-gles [10]; Ericson fluctuations [11]; β -delayed particlespectrum fluctuations [12]; the regular, inverse kinemat-ics, or β Oslo method [13–16]; and particle evaporationspectra [17–19]. Of the techniques that can be used onexotic nuclides, neutron resonance spacings and β -Oslomeasurements require estimates of the spin distributionof excited states in order to obtain a total level density.Furthermore, the latter technique requires normalizationto neutron resonance spacings, which generally must becalculated for exotic nuclides. Meanwhile, particle evap-oration spectra provide a nearly model-independent leveldensity extraction, albeit presently limited to nuclides afew nucleons from stability.To date, constraints off stability (which do not rely ontheoretical normalizations) are extremely limited. Re-sults for ρ from counting discrete states show hints of areduction in ρ for increasingly exotic nuclides [20, 21].The purpose of this work is to provide a constraint for ρ of a nucleus several nucleons from stability, using amodel-independent measurement of ρ .Here, we focus on ρ for Zn due to its role in theastrophysical rapid proton capture ( rp -) process. The rp -process powers type-I X-ray bursts, thermonuclear ex-plosions that occur on the surfaces of accreting neutronstars [22]. Zn plays a key role in the NiCu cycle ofthe rp -process, where transmutation of Cu via a ( p, α )reaction stalls nuclear energy generation, while a ( p, γ )reaction, which depends on ρ of Zn, enables the rp -process to proceed [23]. X-ray burst model calculationshave shown that the shape of the burst light curve isparticularly sensitive to the Cu( p, γ ) Zn reaction rate, a r X i v : . [ nu c l - e x ] J a n which may impact constraints on the properties of ultra-dense matter [24, 25].We report the first measurement of ρ for Zn, obtainedusing neutron evaporation spectra from Ni( He , n ) Znmeasured at the Edwards Accelerator Laboratory atOhio University [26]. We describe our experiment inSection II and analysis in Section III. In Section IV wepresent our results, followed by a comparison to theoreti-cal level density models and discussion of the implicationsfor astrophysics in Section V. We conclude in Section VI,including some comments on next steps for constrainingthe Cu( p, γ ) reaction rate for the rp -process. II. EXPERIMENT SET-UP
Measurements were performed at the Edwards Accel-erator Laboratory using the beam swinger and neutrontime-of-flight (TOF) tunnel [26, 27]. A beam of He waschopped and bunched into ∼ ± .053 mg/cm Ni target with an average inten-sity of ∼
20 nA. Due to energy loss within the Ni tar-get, the average energy of the He beam at the centerof the target was 9.93 MeV. Incident beam current wascollected by integrating charge deposited in the targetchamber. Neutrons were detected at an angle of 105 ◦ bythree 5.08 cm thick, 12.7 cm diameter NE213 liquid or-ganic scintillators, located at a distance of L path = 5 mfrom the target within the neutron TOF tunnel. TheTOF start was provided by a detection within an NE213detector, with TOF stop provided by charge capacitivelydeposited in a beam pick-off located upstream of the tar-get chamber.To identify contributions from beam-induced back-ground to the Ni( He , n ) spectrum, measurements wereperformed using a thin carbon target and an empty tar-get frame. Additionally, measurements with all targetswere taken at an angle of 115 ◦ to exploit the kinematicshifts of neutron peaks at discrete energies.Detector efficiency calibrations were performed bymeasuring a standard. Here, neutrons from Al( d, n )were measured at 120 ◦ for 7.44 MeV deuterons impingingon a thick Al target [28]. Our neutron detection efficiencyis shown in Figure 1. III. ANALYSISA. Neutron Differential Cross SectionDetermination
Neutrons were identified using TOF and pulse shapediscrimination, providing n - γ separation down to theNE213 threshold near 1 MeV. A linear time calibrationwas performed using each spectrum’s γ -ray peak andthe neutron peak associated with populating the ground Neutron Energy (MeV) E ff i c i e n cy ( C oun t s / N e u t r on ) FIG. 1. Intrinsic NE213 neutron detection efficiency deter-mined using Al( d, n ). state of O via C( He , n ). The neutron laboratoryframe energy is E n = m n (cid:16)(cid:112) / (1 − β n ) − (cid:17) , where m n is the neutron mass, β n = ( L path / TOF) /c , and c is thespeed of light.The constant time-independent background from ran-dom coincidences triggered by background radiation wasdetermined using the superluminal region of each spec-trum. This average background level was determined in-dividually for each NE213 detector and subtracted fromthe corresponding neutron spectrum. Neutrons from re-actions on target contaminants were removed from thedifferential cross section as described below.The approach of Ref. [28] was used to determine theneutron detection efficiency, carrying a statistical uncer-tainty along with a 5% systematic uncertainty. By com-paring our measured Al( d, n ) spectrum to the stan-dard spectrum determined in that work, we determinethe product of the geometric and intrinsic neutron detec-tion efficiency.Following the time calibration and efficiency correc-tion, the spectra of the three NE213 detectors werecombined. The differential cross section dσ/dEd
Ω = N det / ( N beam n A (cid:15) tot ), where N det is the number of de-tected neutrons within an energy interval around energy E , N beam is the number of beam particles incident on thetarget, n A is the areal atomic density of the target, and (cid:15) tot is the total (geometric × intrinsic) neutron detectionefficiency at energy E . N beam was measured using chargeintegration on the target, carrying a 3% uncertainty. Fig-ure 2 compares our dσ/dE , where we have multiplied themeasured dσ/dEd Ω by 4 π sr, in the center-of-mass frameto calculations performed with the code Talys version 1.8(
TalysV1.8 ) [29] using the default model parameters.The Ni( He , n ) dσ/dE is expected to have rela-tively smooth behavior up to E n ∼ Zn become sparse enough thatthe neutron spectrum will be characterized by discretepeaks. Our data largely follow this trend; however, sev-eral prominent discrete peaks are apparent in the con-tinuum region. We do not attribute these to reactionson Ni favoring particular states in a direct reactionmechanism, as this should be disfavored by the relatively
Neutron Energy (MeV) - - -
10 110 / d E ( m b / M e V ) s d * * * * ExperimentalTalysDirectStatisticalPreequilbirum
FIG. 2. Measured (points) dσ/dE for Ni( He , n ) at 105 ◦ compared to default TalysV1.8 calculations, including abreakdown of the total dσ/dE by reaction mechanism. As-terisks indicate contamination from ( He , n ) on carbon andoxygen. low reaction energy and backward angles for neutron de-tection. Instead, these peaks in the continuum regioncan be understood as background from C( He , n ) and O( He , n ) reactions on contamination in the target andcollimator upstream of the target. We confirmed thisidentification using our measurements on the carbon tar-get and empty target frame, including checking the kine-matic shift of the peaks for the 115 ◦ measurement anglerelative to the 105 ◦ angle, along with calculations us-ing TalysV1.8 . Oxygen and carbon background peakswere removed by fitting each peak in the continuum witha Gaussian summed with a polynomial. The Gaussiancontributions were then subtracted from the measured dσ/dE . B. Level Density Determination
The general approach for our level density determina-tion is briefly summarized here and elaborated upon inthe subsequent subsections. Calculations are performedfor the Ni( He , n ) dσ/dE assuming some theoretical ρ for Zn. The ratio between the measured and calculated dσ/dE is determined and then multiplied by ρ used inthe calculation, taking advantage of the proportionality dσ ( E ) /dE ∝ ρ ( E ∗ ), where E ∗ is the excitation energypopulated by a neutron of energy E [30]. The absolute ρ is determined by performing a normalization to matchthe theoretical dσ/dE populating the low-lying excita-tion energy region of Zn, where all levels are thoughtto be known.The detailed description of this process is as follows. dσ/dE Calculation
The level density extraction begins with a theoreticalestimate for dσ/dE . While the final extracted ρ , ρ expt , isrelatively insensitive to the initial theoretical ρ , ρ th , it isnot as clear that our result will be as insensitive to other assumptions in the model calculation of dσ/dE . In par-ticular, there is considerable uncertainty in the opticalmodel potential (OMP) for He and the pre-equilibriumcontribution to dσ/dE , where models in general struggleto reproduce He-induced reaction data [31, 32]. The HeOMP will impact the overall magnitude of the ( He , n )cross section, but this impact is removed in the ρ renor-malization (described below). Here, the concern with the He OMP is the impact on the populated spin distribu-tion for the compound nucleus, Zn, which will modifythe distribution of centrifugal barriers for outgoing neu-trons, and thereby influence the ( He , n ) spectrum. Sim-ilarly, the pre-equilibrium contribution can modify theneutron spectrum and any suspected contributions fromthis reaction mechanism must be subtracted from thedata in order to maintain the validity of the proportion-ality dσ ( E ) /dE ∝ ρ ( E ∗ ).The He OMP and pre-equilibrium model parameterswere selected by reproducing measured Ni( He , p ) datawith TalysV1.8 , while checking the influence on the asso-ciated Ni( He , n ) spectrum (see Figure 3). The ( He , p )data, which are the focus of a separate work, are from ameasurement following the approach of Ref. [33]. Briefly,these ( He , p ) spectra were produced using a 10 MeVbeam of He by employing charged-particle time-of-flightand energy loss with a silicon surface barrier detector lo-cated at a flight path of 1 m and angle of 112 ◦ . We notethat these data are consistent with the lower statisticsmeasurement of Ref. [34]. The benefit of including theproton spectrum in the present analysis is that these dataare sensitive to the formation cross section of the Zncompound nucleus, but not the neutron-decay daugh-ter Zn. Since the ( He , p ) channel dominates over the( He , n ) channel at these energies, reproducing the mag-nitude of the former is a strong indication that we arereproducing the He + Ni fusion cross section.We explored a variety of modifications to the He OMPand pre-equilbrium models, focusing in Figure 3 on thesubset that best reproduced the shape of the proton andneutron spectra, while using the constant temperaturelevel density model (see Section IV) and the mean beamenergy in the center of the target. We refer to thesemodifications as Mod-1, Mod-2, and Mod-3, and com-pare to the results from using the default
TalysV1.8 in-puts for context. The default He OMP is essentiallya weighted combination [35] of the proton and neutronOMPs from Ref. [36], where in this case local OMPs areavailable. The default pre-equilibrium model is a two-component exciton model [37] supplemented by contri-butions from nucleon transfer and knock-out [31]. ForMod-1, we use the T He based on the systematics estab-lished by Ref. [38, 39]. For Mod-2, we increase the lead-ing coefficient of the surface absorption term for the HeOMP by a factor of 10. For Mod-3, we use the T He fromsystematics referred to in Mod-1 and a multi-step directand compound pre-equilibrium model [40] decreased bythe scale factor 0.3. We find that Mod-2 most satisfacto-rily describes the ( He , p ) data. The increased absorption Proton Energy (MeV) - -
10 110 / d E ( m b / M e V ) s d ExperimentDefaultMod-1Mod-2Mod-3 (a) 0 2 4 6 8 10
Neutron Energy (MeV) - -
10 110 / d E ( m b / M e V ) s d ExperimentDefaultMod-1Mod-2Mod-3 (b)
FIG. 3. He + Ni proton (a) and neutron (b) spectra compared to results from model calculations we describe as Mod-1,Mod-2, and Mod-3 in the text. Solid and long-dashed lines show the total dσ/dE , while the short-dashed and dot-dash linesshow the contribution from the statistical reaction mechanism to the long-dashed and solid lines,respectively. implied by Mod-2 is consistent with the He + Ni fusioncross section enhancement found by Ref. [34], where near10 MeV Mod-2 results in σ fus . = 250 mb, as comparedto 220 mb in Ref. [34]. Therefore, we use the Mod-2 HeOMP and pre-equilibrium combination for the remain-der of the analysis. The corresponding impact on the( He , n ) spectrum is shown in Figure 3b. ρ ( E ∗ ) Extraction
Determining ρ ( E ∗ ) from a particle evaporation spec-trum is based on the concept that dσ ( E ) /dE will dependon the number of states near excitation energy E ∗ thatwill be populated by a particle evaporating from the com-pound nucleus with an energy near E [41–43].The proportionality dσ ( E ) /dE ∝ ρ ( E ∗ ) is only validfor reaction products created via the statistical reactionmechanism and relies on accurate estimates for T . Theanalysis presented in the prior subsection served to de-termine the non-statistical contributions to our ( He , n )spectrum, as well as T He . T n used here result from theneutron OMP of Ref. [36], though we found little differ-ence when employing the JLM potentials of Refs. [44, 45].Essentially, the experimental level density ρ expt is de-termined by scaling a theoretical level density ρ th used tocalculate dσ/dE | th by the ratio between the experimentaland theoretical dσ/dE : ρ expt ( E ∗ ) ∝ ρ th ( E ∗ ) dσ ( E ) /dE | expt dσ ( E ) /dE | th , (1)where E ∗ and E are related by kinematics and here the dσ/dE refer only to the statistical component [41]. Thedetailed ρ extraction steps are as follows:1. As an initial estimate, the constant temperature(CT) phenomenological level density [46] ρ CT isused for ρ th , where ρ CT ( E ∗ ) = 1 T exp (cid:18) E ∗ − δET (cid:19) , (2) and the nuclear temperature T and energy shift δE are from global parameter systematics of Ref. [3].Using this ρ th , dσ/dE | th (the statistical contri-bution to the theoretical dσ/dE ) is calculated. TalysV1.8 results are reported separately for thecontinuum and discrete level region, where the lat-ter includes by default the ten lowest-lying excitedstates. We apply our experimentally determinedneutron energy resolution to dσ/dE calculated forthe discrete-level region in order to compare to dσ/dE | expt . The neutron energy resolution was cal-culated as∆ E n = (cid:113) ∆ E + ∆ E , (3)where ∆ E loss is the energy loss within the targetcalculated using LISE++ [47]. We ignore energyresolution contributions due to energy straggling inthe target, as ∆ E strag << ∆ E loss . ∆ E TOF is theuncertainty in the neutron energy from the TOF:∆ E TOF = 2 E n (cid:115)(cid:18) L path D (cid:19) + (cid:18) ∆ t TOF (cid:19) , (4)where D and ∆ t are the detector thickness andthe bunch length of the beam packets, respectively.For instance, at E n = 7 .
050 MeV, ∆ E TOF =0 .
293 MeV, and ∆ E n = 0 .
323 MeV.2. The direct and pre-equilibrium contributions to thecalculated dσ/dE , scaled along with the statisticalcontribution to minimize χ between the total cal-culated dσ/dE and the measured dσ/dE , are sub-tracted from the measured total dσ/dE to arrive at dσ/dE | expt (which is the statistical contribution tothe measured dσ/dE ). We use ∆ χ = χ − χ to determine the uncertainty [48] in the scale factorand propagate this through as a contribution to thesystematic uncertainty in ρ expt . In this analysis, χ = 8 .
03, where ∆ χ = 1 .
00 for a 68% confidenceinterval.3. We normalize dσ/dE | th to match the integral of dσ/dE | expt over the region corresponding to E ∗ =0 − Zn, and therefore the absolute ρ , are thought tobe known [3]. This normalization is sensitive touncertainties in dσ/dE | expt where the discrete levelregion is populated. As such, to account for thisnormalization uncertainty, we repeat this step 1,000times, each time perturbing the dσ/dE | expt in thisregion randomly within the one standard deviationuncertainty of each data point. The normalizationfactor F = 0 .
795 and its uncertainty δF = 0 . ρ expt ( E ∗ ) = ρ th ( E ∗ ) dσ ( E ) /dE | expt dσ ( E ) /dE | th F (5)5. The level density from the previous step, ρ expt , isnow used as the the theoretical level density ρ th instep 1, in lieu of ρ CT . The levels are distributed byspin J , ρ ( E ∗ , J ) = P ( J ) ρ ( E ∗ ), assuming the Bethespin-distribution [49], which agrees reasonably wellwith measurements and more sophisticated calcu-lations [50–52], where the probability of a given J is P ( J ) = 2 J + 12 σ exp (cid:18) − ( J + 1 / σ (cid:19) (6)and σ is the spin-cutoff parameter.The E ∗ -dependent σ is determined in Talysv1.8 from a linear interpolation between σ calculatedusing the known J for levels with E ∗ ≤ σ , and the Fermi gas estimate σ evaluatedat the neutron separation energy, S n . For E ∗ ≤ σ is represented by σ = (cid:80) J i ( J i + 1)(2 J i + 1)3 (cid:80) J i + 1 , (7)where the sum runs over all levels in the discretelevel region, and σ ∼ . A / √ E ∗ , (8)where A is the mass number. The exact form isavailable in the Talys manual.Ref. [51] found roughly a factor of two variabilityamongst various predictions and experimental con-straints for σ in our E ∗ and A region (see also Fig-ure 8). Therefore, we also investigated performingthe ρ extraction described in this subsection for afactor of two increase and factor of two decrease in σ . Additionally, we explored using the empirical σ determined by Ref. [9], where σ ∝ ( E ∗ ) . .The results from using alternative σ parameteri-zations were used to assess the uncertainty contri-bution to ρ expt from our choice of σ . Neutron Energy (MeV) - -
10 110 / d E ( m b / M e V ) s d Experiment,TotExperiment,StatStatisticalTotalDirectPreequilibriumStatistical,CT
FIG. 4. Measured Ni( He , n ) differential cross section be-fore (open triangles) and after (filled triangles) subtractingcalculation results for non-statistical contributions. Calcula-tion results for the total differential cross section using ρ expt (solid gray line) are shown, along with the contributions to thetotal from the direct (dotted line), pre-equilibrium (dot-dot-dash line), and statistical (dot-dash line) components. Forcomparison, the statistical component of the differential crosssection calculated using ρ CT is also shown.
6. We calculate dσ/dE using the newly extracted ρ expt as the input ρ and compared to the data (inFigure 4) to confirm that using the extracted ρ de-scribes the data. IV. RESULTS
Our final dσ ( E ) /dE for Ni( He , n ) and the asso-ciated ρ expt ( E ∗ ) for Zn are shown in Figures 4 and5, respectively. The contributions to the ρ expt uncer-tainty, presented in Table I, include the statistical andsystematic uncertainties from dσ/dE | expt δρ d σ/ dE , a sys-tematic uncertainty from the initial choice of ρ th (es-timated using ρ BSFG and ρ BSFG − Red . described below) δρ model , a systematic uncertainty from the subtractionof non-statistical contributions to dσ/dE (described in ρ -extraction step 2) δρ χ , a systematic uncertainty fromthe normalization (described in ρ -extraction step 3) δρ F ,and a systematic uncertainty for adopting different σ (described in ρ -extraction step 5) δρ σ . The dominantuncertainty contributions are statistical for low E ∗ (highneutron energy) and the uncertainty from the initialchoice of ρ th for high E ∗ (low neutron energy).In principle, our extracted ρ expt is also sensitive to theproperties of the states in the discrete level region. The J π of several states are uncertain and some states havebeen observed by a single measurement, but not oth-ers [53]. Rather than attempting to assign an associateduncertainty contribution, instead we present our ρ expt and F in Table I. If future measurements modify thestructure of Zn for E ∗ < dσ/dE | th to the re-sult from this work in order to determine the requiredmodification to F . In principle, the correction to themeasured dσ/dE subtracting non-statistical components Excitation Energy (MeV) - -
10 110 ) - L eve l D e n s i t y ( M e V extractedBSFGCTBSFG-Red.Known Discrete FIG. 5. The extracted ρ expt for Zn (markers) comparedto global theoretical ρ models discussed in the text. Thegray-dashed histogram is ρ from known levels up to E ∗ whereall levels are thought to be known, where our experimentalenergy resolution has been applied. would also have to be repeated, but in practice this cor-rection is not significant (see Figure 4).In Figure 5 we compare ρ expt to arguably the twomost common phenomenological ρ : the CT model [46](Eqn. 2 [54]) and the back-shifted Fermi gas (BSFG)model [55]. For the latter, ρ BSFG ( E ∗ ) = exp (cid:16) (cid:112) a ( E ∗ − ∆) (cid:17) √ σa / ( E ∗ − ∆) / , (9)where ∆ is a back-shift to ensure ρ is described at low-lying excitation energies where all discrete levels areknown and a is the E ∗ -dependent level-density param-eter of Ref. [56], based on global fits to ρ . For boththe the CT and BSFG models, we use the parameteriza-tion and default parameters of TalysV1.8 . Based on ananalysis of ρ determined from low-lying excited states,Refs. [20, 21] found that a may be reduced for nuclidesaway from the valley of β -stability. A favored reductionwas the form a red = αA exp (cid:16) γ ( Z − Z ) (cid:17) , (10)where α , γ , and Z were determined empirically and, for Zn, the result is a red = 4 .
73. We therefore exploreadjusting a ( E ∗ ) such that a ( S n ) = a red , referring to thiscalculation as BSFG-Red.In Figure 6 we compare our results to microscopiclevel densities. These were obtained by generating sin-gle particle energy levels using various energy densityfunctionals within the Hartree Fock Bogoliubov frame-work [57] and then determining ρ using statistical me-chanics [58, 59]. We adopted energy density functionalsbased on the SLy4 [60], SkT6 [61], Zs [62], SkM ∗ [63], andSkI3 [64] Skyrme forces as a somewhat arbitrary samplingof the vast array of possible choices. Note that we do notreport microscopic ρ below 2 MeV as the saddle-pointapproximation used in the ρ calculation diverges. Suchdivergence can be corrected with asymptotic expressions,but this was not deemed necessary here. Excitation Energy (MeV) - -
10 110 ) - L eve l D e n s i t y ( M e V SLy4SkT6ZsSkM*SkI3Extracted
FIG. 6. Extracted Zn ρ (markers) compared to microscopictheoretical ρ models discussed in the text. Excitation Energy (MeV) -
10 110 ) - L eve l D e n s i t y ( M e V FIG. 7. Extracted Zn ρ (markers) compared to the shellmodel based theoretical ρ model discussed in the text (ELM).Lines are ρ summed up to a maximum J from J = 0 (lowestline) up through J = 12 (red, upper bold line), where the sumup through J = 3 is the lower bold line, indicated in blue. In Figure 7 we compare our results to calculations per-formed with the extrapolated Lanczos method (ELM) ofRef. [52]. This is a shell-model based ρ , using the pf-shell model space and the GXPF1A Hamiltonian [65].The ELM ρ are summed for all positive-parity statesup through a maximum J . For J =0 −
8, we use 8 mo-ments and 30 iterations for each J , known as ELM(8,30),whereas ELM(8,4) is used for higher J . V. DISCUSSIONA. Comparison to Level Density Models
We compare our results to phenomenological, micro-scopic, and shell-model based theory calculations of ρ ,as each technique has its advantages. Each comparisonalso provides unique context for understanding ρ of Znand likely nearby nuclides, which is valuable for predic-tions of statistical properties of nuclides off-stability forastrophysics and applications.Our comparison to phenomenological ρ in Figure 5demonstrates that we clearly favor the CT model of TalysV1.8 . This adds to the body of evidence that theCT ρ is often favored over the BSFG prediction for nu-clides in this mass region [18, 66] (and perhaps all mass TABLE I. Extracted ρ expt for Zn, total uncertainty δρ tot , and uncertainty contributions in MeV − , where F =0.795.Individual error bars are shown within the parentheses, the lower error bar being listed first, and the upper error bar following.If the lower error bar contains 0, the maximum level density is listed instead of the central value since all physical values belowthe upper limit are within the uncertainty. E ∗ [MeV] ρ expt δρ tot δρ d σ/ dE δρ F δρ χ δρ σ δρ model < . , .
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1) (8 . , .
02) (9 . , .
7) (77 . , . ) regions [67]). The agreement may be explained by theconclusion that the CT model effectively captures thephase transition from ordered nuclear structure to quan-tum chaos with increasing E ∗ [68]. While a reduced a within the BSFG model may improve agreement withour results, we find that the required a -reduction is notconsistent with a red predicted by Refs. [20, 21].The comparison to microscopic ρ in Figure 6 demon-strates that these theoretical ρ strongly depend on theadopted Skyrme force. In this case, Zs is clearly favored.However, we note that normalization has not been per-formed to match ρ in the discrete level region, as is of-ten done for databases of ρ intended for HF calculations.Nonetheless, this may provide guidance for microscopiccalculations of statistical nuclear properties for nuclidesin this region of the nuclear chart.Figure 7 shows that the shell-model based ELM calcu-lations significantly overestimate ρ for Zn, while Fig-ure 8 shows general agreement between the ELM P ( J )and commonly used theoretical estimates. The ELMmethod has previously been demonstrated to success-fully reproduce measured ρ ; however, results are some-what dependent on the adopted Hamiltonian [52]. Whileshell model calculations face the usual limits imposed bya finite model space, this would only artificially reduce ρ at high E ∗ . Alternatively, the disagreement betweenour result and the ELM results could be explained if ourmeasurement failed to probe J >
3. However, this wouldrequire significantly reduced T n for large angular momen-tum (cid:96) transfer neutron emissions.It is interesting to note that our ρ expt exhibits a plateaufrom E ∗ ∼ − ρ with increasing E ∗ and is not seen in any of the ρ models that we com-pare to. At present, we do not have an explanation forthis feature, though we note something similar is seenat E ∗ ∼ − Fe [43]. We note that thoughexponential ρ expt growth for E ∗ ∼ − ρ expt trend is primarily dueto statistical uncertainties, which are much smaller thanthe total error bars. B. Implications for Astrophysics
The importance of Zn to the rp -process and Type-I X-ray bursts is discussed in Section I. In calculating Cu( p, γ ) Zn via the HF-formalism, it should be notedthat the uncertainties in the Cu + p proton opticalpotential (at low temperature) and the Zn γ -strengthfunction (at high temperature) are far more significantcontributors to the astrophysical reaction rate uncer-tainty than the Zn ρ [69]. However, ρ can help answerthe key question as to whether the HF approach is likelya valid approximation for this reaction. In particular, weare interested in whether the E ∗ region populated in the Spin (J) P e r c en t age o f T o t a l L D ( % ) XRB Populated J
ELMGorielyHilaireD1Mvon EgidyRigid BodyRigid Body, s · Rigid Body, s · FIG. 8. Calculated P ( J ) of Zn at E ∗ = 5 MeV using thenominal σ (“Rigid Body”), σ increased or decreased bya factor of two, or σ from Ref. [9] (“von Egidy”), comparedto ELM P ( J ) and microscopic P ( J ) available in TalysV1.8 .The green band indicates the relevant J for Cu( p, γ ) Zn ina Type-I X-ray burst (XRB) environment.
X-ray burst environment has a sufficiently high ρ .For the relevant temperature range, 0 . − . Cu( p, γ ) will populate E ∗ ≈ . − . Zn [70].This happens to correspond to the near-plateau presentin our data in Figure 5 and Table I, where ρ maintains ∼
20 MeV − , contrary to the exponential growth presentin theory calculation results. While this is above the ρ ∼
10 levels-per-astrophysical-window (here ∼
12 window − )heuristic for the applicability of HF results [71], it is lowenough that in practice HF-applicability will depend on σ . This is because, in the X-ray burst environment,the only relevant states in Zn are those populated byproton-captures involving low (cid:96) transfer. Assuming only (cid:96) = 0 and (cid:96) = 1 proton-captures are relevant, thenroughly half of the total ρ is encompassed for the nominalchoice of σ (See Figure 8). However, if σ is doubled,following the systematics of Ref. [51], only around a thirdof all levels are within the relevant J range. Therefore,a measurement of σ is needed to confirm the applica-bility of HF calculations for the Cu( p, γ ) Zn reactionrate in X-ray burst conditions. In this regard, the par-ity ratio [21] may also be of interest, though we do notexamine its impact here.
VI. CONCLUSION
Using neutron evaporation spectra from Ni( He , n ),we determined ρ for Zn in the E ∗ range of astrophys-ical interest and compared our results to several theo-retical predictions, including phenomenological, micro-scopic, and shell-model based calculations. We find thatthe CT model of TalysV1.8 is in best agreement withour data amongst phenomenological models and thatthere is no evidence for a ρ reduction moving away fromthe valley of β -stability. Of our microscopic ρ calcula-tions, the results from using the Zs Skyrme force mostclosely resemble our experimental results. Our ELM ρ results are substantially larger than our experimental re-sults, indicating either a limitation of the shell modelapproach here, or that, contrary to expectations, our ex-periment failed to probe high- J levels. For E ∗ relevantfor the Cu( p, γ ) Zn reaction rate at X-ray burst tem-peratures, we find a plateau in ρ , where the exponentialgrowth with increasing E ∗ is temporarily halted. De-pending on the adopted σ , it is possible that the HFformalism may not be applicable for this reaction rate.To resolve this issue, as well as the discrepancy with ELMresults, we suggest measurements of σ and perhaps high-resolution spectroscopy for E ∗ ∼ Zn.
ACKNOWLEDGMENTS
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