Isotopically resolved neutron total cross sections at intermediate energies
C. D. Pruitt, R. J. Charity, L. G. Sobotka, J. M. Elson, D. E. M. Hoff, K. W. Brown, M. C. Atkinson, W. H. Dickhoff, H. Y. Lee, M. Devlin, N. Fotiades, S. Mosby
IIsotopically resolved neutron total cross sections at intermediate energies
C. D. Pruitt, ∗ R. J. Charity, L. G. Sobotka,
1, 2
J. M. Elson, D. E. M. Hoff, † K. W. Brown,
1, 3
M.C. Atkinson, ‡ W.H. Dickhoff, H. Y. Lee, M. Devlin, N. Fotiades, and S. Mosby Department of Chemistry, Washington University, St. Louis, MO 63130 Department of Physics, Washington University, St. Louis, MO 63130 National Superconducting Cyclotron Laboratory, Departments of Physics and Astronomy,Michigan State University, East Lansing, MI 48824, USA Los Alamos National Laboratory, Los Alamos, NM 87545, USA
The neutron total cross sections σ tot of , O, , Ni,
Rh, and , Sn have been measuredat the Los Alamos Neutron Science Center (LANSCE) from low to intermediate energies (3 ≤ E lab ≤
450 MeV) by leveraging waveform-digitizer technology. The σ tot relative differences betweenisotopes are presented, revealing additional information about the isovector components needed foran accurate optical-model (OM) description away from stability. Digitizer-enabled σ tot -measurementtechniques are discussed and a series of uncertainty-quantified dispersive optical model (DOM)analyses using these new data is presented, validating the use of the DOM for modeling lightsystems ( , O) and systems with open neutron shells ( , Ni and , Sn). The valence-nucleonspectroscopic factors extracted for each isotope reaffirm the usefulness of high-energy proton reactioncross sections for characterizing depletion from the mean-field expectation.
INTRODUCTION
Neutron scattering is a direct, Coulomb-insensitivetool for probing the nuclear environment. The simplestneutron-nucleus interaction quantity is the neutron to-tal cross section, σ tot , which provides information aboutnuclear size and the ratio of elastic-to-inelastic compo-nents of nucleon scattering. Additionally, σ tot data arethought to be tightly correlated with a variety of struc-tural nuclear properties of great interest including theneutron skin of neutron-rich nuclei [1] and thus the den-sity dependence of the symmetry energy L , an essentialequation-of-state input for neutron-star structure calcu-lations [2–4].In the crude “strongly-absorbing-sphere” (SAS) ap-proximation, where a target nucleus absorbs incidentneutrons passing within a nuclear radius, σ tot dependssolely on the target nucleus size and the energy of theincident neutron: σ tot ( E ) = 2 π ( R + λ ) , (1)where R = r A and λ is the reduced wavelength of theincident neutron with energy E in the center of mass[5, 6]. While on average , experimental σ tot data com-port with this na¨ıve model, the most prominent featureof experimental σ tot data is the oscillatory behavior cen-tered about the average of Eq. (1), visible in Fig. 1.Peterson [7] interpreted these oscillations as the resultof a phase shift between neutron partial waves passing around the nucleus (thus undergoing no phase shift) and ∗ Corresponding author: [email protected]; Present Address:
Lawrence Livermore National Laboratory, Livermore, CA 94550 † Present Address:
Department of Physics, University ofMassachusetts-Lowell, Lowell, MA USA 01854 ‡ Present Address:
TRIUMF, Vancouver, BC V6T 2A3, Canada
FIG. 1: Experimental σ tot data are shown from 2-500 MeVfor nuclides from A =12 to A =208 [8–12]. Predictions for σ tot given by the “strongly absorbing sphere” (SAS) model [Eq.(1)], are shown as thin dashed lines for each nucleus. Regularoscillations about the SAS model are visible as is the trend forthe oscillation maxima and minima to shift to higher energiesas A is increased. waves passing through the nuclear potential, where theyare refracted and exhibit a retardation of phase (an illus-tration is available in [6]). This explanation was termedthe “nuclear Ramsauer effect” by Carpenter and Wilson[13] based on the analogous effect seen in electron scat-tering on noble gases.Following Angeli and Csikai [14], this explanationcan be incorporated by imbuing the strongly-absorbing-sphere relations with a sinusoidal term: σ tot = 2 π ( R + λ ) (1 − ρ cos( δ )) (2)where ρ = e − Im(∆) and δ = Re(∆), ∆ being the phasedifference between a partial wave traveling around andtraveling through the nucleus. The large amplitude ofthe oscillations suggests that elastic scattering accounts a r X i v : . [ nu c l - e x ] S e p for a significant fraction of the total cross section, in turnimplying a larger mean free path for neutrons through thenucleus than might otherwise be expected in the absenceof Pauli blocking [15, 16]. If we approximate the nucleuswith a real spherical potential of radius R and depth U ,the total phase shift δ is: δ = C (cid:16)(cid:2) E + UE (cid:3) − (cid:17) λ (3)where C = R is the average chord length through thesphere [14]. Rearranging Eq. (3) in terms of A and E and discarding leading constants yields: δ ∝ A × (cid:16) √ E + U − √ E (cid:17) (4)This form reveals an important relation: as A is in-creased, to maintain constant phase δ , E must also in-crease [6, 7]. This is contrary to a typical resonance con-dition where an integer number of wavelengths are fit in-side a potential; in that case, to maintain constant phaseas A is increased, E must be decreased. Thus these σ tot oscillations have been referred to as “anti-resonances” or“echoes” [6, 17]. Other authors [18] have exposed weak-nesses in Angeli and Csikai’s interpretation of Eq. (2)and have provided a more general semi-empirical equa-tion for σ tot . However, Eq. (2) is a valuable startingpoint for connecting σ tot with the depth and shape ofthe nuclear potential as experienced by neutrons.By including additional surface, spin-orbit, and otherterms, OMs have been used to successfully reproduce thegeneral features of all manner of single-nucleon scatter-ing data across the chart of nuclides up to several hun-dred MeV [19–21]. However, despite the excellent agree-ment with experiment, OMs involve the interaction ofmany partial waves with many sometimes-opaque termsin the potential, complicating intuitive understanding ofthe underlying physics at play. In particular, the isovec-tor components of optical potentials are quite difficult toconstrain as they depend on both proton and neutronscattering data, one or both of which are often unavail-able. For example, when Dietrich et al. conducted ananalysis of neutron total cross section differences betweenW isotopes, including standard isovector terms in theiroptical potential worsened the reproduction of experi-mental relative differences, an illustration of how poorlythese isovector components are known [22].With these considerations in mind, our present goalis twofold: first, to provide new isotopically resolved σ tot data useful for identifying the dependence of opti-cal potential terms on nuclear asymmetry; and second,conduct a DOM analysis of these new σ tot data alongwith a large corpus of scattering and bound-state datato extract veiled structural quantities (e.g. neutron skinthicknesses and spectroscopic factors, or SFs) for severalcornerstone, closed-proton-shell nuclei. Key findings ofthis DOM analysis are presented in the companion Let-ter [23]. EXPERIMENTAL CONSIDERATIONS
By scattering secondary radioactive beams off of hy-drogen targets in inverse kinematics, proton-scatteringexperiments are possible even on highly unstable nu-clides. Because neutrons themselves must be generatedas a secondary radioactive beam, neutron-scattering ex-periments are restricted to normal kinematics and σ tot measurements are possible only for relatively stable nu-clides that can be formed into a target. At present, σ tot measurements above the resonance region on nuclideswith short half-lives (shorter than the timescale of days)are technically infeasible for this reason, though a hand-ful have been carried out on samples with half-lives inthe tens to thousands of years [10, 24, 25].Traditionally, σ tot measurements have relied on analog-electronics techniques for recording events, techniquesthat suffer from a large per-event deadtime of up to sev-eral µ s. For a typical analog intermediate-energy σ tot measurement with dozens or hundreds of energy bins,achieving statistical uncertainty at the level of 1% re-quires a thick sample to attenuate a sizable fraction ofthe incident neutron flux. If cross sections are in the 1-10barn range, this means sample masses of tens of grams[8, 12]. Producing an isotopically enriched sample of thissize is often prohibitively expensive. As a result, thereis a dearth of σ tot data on isotopically resolved targetsfrom 1-300 MeV, even for closed-shell isotopes of specialimportance like , He, O, Ni, , Sn, and , Pb(see Fig. 1.3 in [26]).Recent developments in waveform digitizer technologyhave made it possible to reduce the per-event deadtimeby an order of magnitude or more, enabling a correspond-ing reduction in the necessary sample size. In 2008, weembarked on a campaign of σ tot measurements on isotopi-cally enriched samples using these new technical capabil-ities, starting with , Ca from 15 ≤ E lab ≤
300 MeV[27]. The data from that measurement were incorporatedinto several DOM analyses [28–30] that yielded protonand neutron SFs, charge radii, and initial estimates ofthe neutron skins [1] for these nuclei. Here we signifi-cantly expand on that effort by providing σ tot results forthe important closed-shell nuclides , O, , Ni, and , Sn. We also present a measurement on a very thinsample of the naturally monoisotopic
Rh to demon-strate that σ tot experiments over a broad energy rangeusing only a few grams of material are feasible. EXPERIMENTAL DETAILS
All σ tot measurements were carried out at the 15Rbeamline at the Weapons Neutron Research (WNR) facil-ity of the Los Alamos Neutron Science Center (LANSCE)during the 2016 and 2017 run cycles. Our experimentwas modeled on previous σ tot measurements at WNR[8, 12, 27]. At WNR, broad-spectrum neutrons up to ≈
700 MeV are generated by impinging proton pulses ontoa water-cooled, 7.5 cm-long tungsten target (Fig. 2). Be-fore the beam enters the experimental area, a permanentmagnet deflects all charged particles generated by theproton pulses, allowing only neutrons and γ rays to reachthe experimental area. At the entrance to the experimen-tal area, the beam was collimated to 0.200 inches usingsteel donuts with a total thickness of 24 inches. In addi-tion, the γ -ray content of the beam was suppressed usinga plug of Hevimet (90% W, 6% Ni, 4% Cu by weight)at the upstream entrance of the collimation stack. Aftercollimation, the beam passed successively through a fluxmonitor, the sample of interest, a veto detector, and fi-nally the time-of-flight (TOF) detector approximately 25meters from the neutron source. All detectors consistedof BC-400 fast scintillating plastic mated with photomul-tiplier tubes (PMTs) and encased in either a plastic oran aluminum housing. The flux monitor and veto de-tector each had scintillator thicknesses of 0.25 inch andthe TOF detector had a scintillator thickness of 1 inch.Signals from all detectors and the target changer were re-layed to a 500-MHz CAEN DT-5730 waveform digitizerrunning custom software. To improve time resolution,the TOF detector used two PMTs (one left, one right)mated to the same plastic scintillator and the PMTs’ sig-nals were summed before digitization.The particular neutron beam structure at WNR dic-tates the energy range achievable for σ tot measurements(Fig. 3). Proton pulse trains, called “macropulses”,are delivered to the tungsten target at 120 Hz. Eachmacropulse consists of ≈
350 individual proton pulses,called “micropulses”, spaced 1.8 µ s apart. Each mi-cropulse consists of a single proton packet that gener-ates γ rays and neutrons within a tight temporal-spatialrange. As neutrons from this micropulse travel alongthe beam path, high-energy neutrons separate in timefrom lower-energy neutrons so that neutron energy canbe determined by standard TOF techniques (see [31] fordetails). Because the γ rays and high-energy neutronsfrom later micropulses can overtake slower neutrons froman earlier micropulse, the distance of the TOF detectorfrom the neutron source determines both the minimumneutron energy that can be unambiguously resolved andthe maximum instantaneous neutron flux, critical to cor-recting for per-event deadtime.A programmable sample changer with six positions wasused to cycle each sample into the beam at a regular in-terval of 150 seconds per sample. Once per macropulse,an analog signal from the sample changer was recordedto indicate its current position. The flux monitor wasused to correct for variations in beam flux betweenmacropulses. The veto detector suppressed events fromcharged-particle production in the samples and in airalong the flight path.Custom digitizer software was used to run the digitizerin two complementary modes, referred to as “DPP mode”and “waveform mode”. In DPP mode, triggers were ini- FIG. 2: Experimental configuration at WNR facility. Samplesare cycled into and out of the beam using a linear actuatorwith a period of 150 seconds. Times-of-flight (TOFs) are de-termined by the TOF detector and used to calculate neutronenergies. tiated by the digitizer’s onboard peak-sensing firmware.For each trigger, several quantities were recorded: thetrigger timestamp, two charge integrals over the detectedpeak with different integration ranges (32 ns for the shortintegral, 100 ns for the long integral), and a 96-ns portionof the raw digitized waveform, referred to as a “wavelet”.DPP mode was used for the vast majority of the ex-periment and accounts for ≈
99% of the total data vol-ume. In waveform mode, the digitizer performs no peak-sensing and was externally triggered. Upon triggering,the trigger timestamp and a very long wavelet (60 µ s)were recorded. While waveform mode data accounts foronly ≈
1% of the total data, the instantaneous data rateis much higher than in DPP mode because hundreds of µ s of consecutive waveform samples are stored. Roughlyonce every three seconds, the digitizer was switched to FIG. 3: Neutron-beam structure at WNR facility.“Macropulses” of protons (d) are delivered to WNR’s tung-sten Target 4, where they generate neutrons by spallation.Each macropulse consists of ≈
350 proton “micropulses” (c).Neutrons from each micropulse (b) disperse in time as theytravel along the flight path so that γ rays and high-energyneutrons catch up to low-energy ones from the previous pulse(a). waveform mode for one macropulse, then switched backto DPP mode as quickly as possible (10-40 ms, dependingon run configuration).Except for the O and Rh samples, all samples were pre-pared as right cylinders 8.25 mm in diameter and rangingfrom 10-27 mm in length (see Table I for sample charac-teristics). For each element studied, a natural-abundancesample was also prepared as were two natural C sam-ples and a natural Pb sample, useful for benchmarkingagainst literature data. The samples were inserted intostyrofoam sleeves and seated in the cradles of the samplechanger. This design minimizes the amount of non-targetmass proximate to the neutron beam path. Our sampleswere generally much smaller than those used in previ-ous measurements; for example, the Ni and Sn samplesused in [8, 12] had areal densities of 1.515 and 0.5475mol/cm , respectively, 12.7 and 6.5 times larger than forour Ni and Sn samples.The O isotopes were prepared as water samples to in-crease the areal density of atoms and for ease of han-dling. Each water sample was contained by a cylindricalbrass vessel with thin brass endcaps (0.002 inches), andan empty brass vessel served as the blank. , O crosssections were calculated by subtracting the well-knownH cross section from the raw H O results. We used H σ tot data sets from Clement et al. [32] and Abfalterer etal. [12], which together cover the range 0 . ≤ E n ≤ σ tot determination, we pre-pared a deuterated water sample, from which the litera-ture σ tot for D could be subtracted, to serve as an addi-tional cross-check. Due to the poor machining propertiesof Rh, the Rh sample was prepared by purchasing and
TABLE I: Physical characteristics of samples used for neutron σ tot measurements. The relevant “sample thickness” for crosssection calculations is the areal density of nuclei ρ A , equal tothe volumetric number density times the length of the sam-ple. For liquid samples H nat2 O, D nat2
O, and H
O, the lengthand diameter given are for the interior of the vessels used tohold the samples and the masses listed are calculated basedon literature values for the density of each sample at 25 C.Isotopic natural abundances (NA) and the abundances in ourenriched samples (SA) are provided for reference.Isotope Length Diam. Mass ρ A NA SA(mm) (mm) (g) (mol/cm ) (%) (%) nat C 13.66(2) 8.260(5) 1.2363 0.1921(1) - - nat
C 27.29(2) 8.260(5) 2.4680 0.3835(2) - -H O 20.00(1) 8.92(1) 1.2461 0.1107(3) - -D O 20.00(1) 8.92(1) 1.3852 0.1107(3) 0.02 99.9H
O 20.00(1) 8.92(1) 1.3844 0.1107(3) 0.20 99.9 Ni 7.97(3) 8.18(2) 3.6438 0.1197(3) 68.1 99.6 nat
Ni 8.00(3) 8.20(2 3.6898 0.1192(3) - - Ni 7.96(2) 8.20(4) 3.9942 0.1192(6) 0.93 92.2
Rh 2.03(1) 10.20(2) 2.8359 0.02426(4) 100 99.9
Sn 13.65(3) 8.245(5) 4.9720 0.08332(5) 0.97 99.9 nat
Sn 13.68(3) 8.245(5) 5.3263 0.08414(5) - -
Sn 13.73(3) 8.245(5) 5.5492 0.08399(5) 5.79 99.9 nat
Pb 10.07(2) 8.27(1) 6.130 0.05508(6) - - stacking a series of thin discs rather than by manufactur-ing a fused cylinder. These discs were held in place by acylindrical plastic case with open ends.
EXPERIMENTAL ANALYSIS
The quantity of interest, σ tot , is related to the flux lossthrough a sample by: I t = I e − (cid:96)ρ A σ tot (5)or, equivalently, σ tot = − (cid:96)ρ A ln (cid:18) I t I (cid:19) (6)where I is the neutron flux entering the sample, I t isthe neutron flux transmitted through the sample withoutinteraction, ρ A is the number density of nuclei in thesample, and (cid:96) is the sample length. For thin or low-density samples, flux attenuation through the sample willbe small (e.g., 13% for our Ni samples at 100 MeV) anda large number of counts will be required to determinethe cross section to high precision.Two post-processing steps were used to improve TOF-detector timing resolution (see Fig. 4). First, the wave-form for each TOF-detector event was passed througha software constant-fraction discriminator (CFD) logic, FIG. 4: The effects of timing corrections on the γ -ray peak ofa typical run are shown. The uncorrected spectrum is shownin black, the spectrum after correction with our software CFDis shown in blue, and the spectrum after correction with bothour software CFD and γ -averaging is shown in magenta. Forthis run, the final γ -ray peak FWHM after both correctionsis 0.866 ns, comparable to the precision we achieved in ourCa study [27], which also employed γ -averaging. improving precision by a factor of two. Second, a γ -ray-averaging procedure (cf. [27]) was used to improvethe precision of each micropulse start time. The finalcorrected TOF resolution (taken as the FWHM of the γ -ray peak in the TOF spectra) ranged from 0.60-0.90ns over the series of σ tot measurements. This is compa-rable to the resolution from our digitizer-mediated σ tot measurement on Ca isotopes in 2008 [27]. For context,for a 100-MeV neutron and a TOF detector distance of25 meters, a TOF uncertainty of 0.80 ns translates toan energy resolution of ≈
900 keV. For neutrons below ≈
20 MeV, the TOF time resolution worsens because thetraversal time through the 1-inch thickness of the TOFdetector becomes non-negligible. However, because theTOF of these neutrons is already very long (several hun-dred ns or longer) the relative energy resolution ( ∆ EE ) issuperior at low energies. As an example from one of ourruns, a 5 MeV neutron with a 0.82 ns detector-traversaltime and an inherent TOF resolution of 0.80 ns has anenergy uncertainty of 13 keV. These energy uncertaintieshave been propagated through subsequent analysis intoour σ tot results below.Calculating the neutron energy requires knowledge ofthe flight-path distance to high precision. We determinedthis distance by calculating putative σ tot data for nat Cfrom 3-15 MeV from our measurement and comparingthe resonance peaks in this region with high-precisionliterature data sets. From this study, the mean TOFdistance was determined as 2709 ± ± i can becalculated [31]: F i = N − (cid:88) j =0 R ( i − j ) mod N × P j (7)where N is the number of time bins in the micropulse, R x is the rate of detected events per micropulse in bin x , and P j is the probability that the digitizer is still busy from atrigger j bins ago. If the variation in beam flux is signifi-cant, a more advanced formula can be used; however, anexamination of our flux-per-micropulse data showed verylittle flux variance across macropulses, except during thefirst 10% of the micropulses within each macropulse. Inthe final analysis we discarded these first 10% and usedthe simpler Eq. (7) to calculate the dead time fraction.To model the experimentally observed probability-dead, P j , we fitted a logistic function to the observedspectrum for time differences between consecutive events(Fig. 5). For a given bin i , the fraction of time that thedigitizer is dead, F i , is a discrete convolution of the mea-sured TOF spectrum with P j . Note that except for thefirst and last micropulses in a macropulse, all micropulsesare consecutive, so deadtime effects can “wrap around”from the end of one micropulse to the next. For thesewrap-around contributions (that is, j > i ), the (mod N )term ensures that the bin referred to by i − j is non-negative.Because trigger processing is done in firmware onboardthe digitizer, the per-event deadtimes affecting our mea-surement were reduced to between 150-230 ns. After wecalculated the average probability-dead for each time bin,the total number of events detected in that bin, N d [ i ],could be corrected to recover the true number of eventsthat would have been detected in the absence of a per-event deadtime: N t [ i ] = − ln (cid:34) − N d [ i ] M (1 − F i ) (cid:35) × M (8)where M is the total number of micropulse periods. Atlarge TOFs (low energies) the correction is as low as afew percent, but at small TOFs (high energies), the dig-itizer is often still dead from the γ -ray flash and high-energy neutrons. In this regime the correction can bequite large ( ≈
20% for our Ni/Rh runs, and ≈
40% forour Sn/O runs). Still, the corrections needed for ourmeasurement are far smaller than the typical analytic
FIG. 5: The time difference between adjacent TOF-detectorevents for a single run is plotted (black histogram). Below acertain minimum time difference (the “deadtime”), no eventsare recorded. A logistic fit (red line) models the detector’sdeadtime response and is used to generate a deadtime cor-rection. The underlying linearly decreasing count rate (graydashed line) is incorporated into the logistic model. From thefit, a mean deadtime of 228.1 ns was extracted for the Sn andO run configurations (a similar procedure was used to recovera deadtime of 159.7 ns for the Ni and Rh run configurations). deadtime corrections required with the deadtime mitiga-tion scheme of previous analog measurements [8, 12].In addition to analytic deadtime, there is an additionaldeadtime effect associated with digitizer readout to thedata acquisition computer (DAQ). During data collec-tion, each pair of digitizer channels shares a commonbuffer for storing events. After several seconds of ac-quisition, the digitizer begins readout at which time theacquisition is paused and buffer contents are read out tothe DAQ. However, because each buffer is independentlyread out to the DAQ, it is possible that buffers couldbe emptied and readied for new acquisition at slightlydifferent times (10-40 ms apart), and a mismatch coulddevelop between the number of macropulses seen on dif-ferent channels. Such run-time interactions between thefirmware and USB traffic of the DAQ were difficult tocharacterize, but we estimate that they might cause a sys-tematic error of a few tenths of one percent in the num-ber of macropulses seen by different channels, dependingon the user-defined threshold and the buffer size. Thiseffect could contribute to the discrepancy at the high-est energies ( >
100 MeV) between our results and pastanalog-enabled measurements.During analysis, it was noted that occasionally (1 in400 macropulses), one or two adjacent macropulses wouldhave an abnormally small number of events. The fre-quency of these “data dropouts” was similar to the rateof switching between DPP and waveform modes; we sus-pect it is related to edge case behavior right before orafter a mode switch. To mitigate this issue, we threwout any macropulse that had less than 50% of the aver-age event rate in either the flux monitor or TOF detector
FIG. 6: TOF spectra after the analytic deadtime correctionand the veto and integrated charge gating for the blank sam-ple (in red) and the nat
C sample (in blue), from the Ni/Rhexperiment. The γ -ray peak is visible as a sharp spike at 90ns, followed by the highest-energy neutrons at 130 ns. channel.After applying these corrections, the veto and inte-grated charge gates were applied to all events and sur-viving events were populated into TOF spectra (Fig. 6).Next, room background was subtracted (responsible for0.1% to 1% of event rate, depending on energy) and spec-tra were mapped to the energy domain.From these energy spectra, the raw cross sections werecalculated, bin-wise, as follows: σ tot = − (cid:96)ρ A ln (cid:18) I I s × M s M (cid:19) (9)where I / I s is the ratio of counts in the energy spectrabetween the blank and sample, M s / M is the ratio ofcounts in the monitor detector between the sample andblank (for flux normalization).Finally, two isotope-dependent corrections were ap-plied to the raw cross sections. First, because the blanksample contains air and not vacuum, the cross section ofair must be added to each sample’s cross section. Second,the cross section for Ni was corrected for the isotopicenrichment of our sample (92.2%) using our measured nat
Ni cross section. All other isotopes were sufficientlypure such that the impurity correction was negligible.To validate our analysis, we first benchmarked our σ tot measurements of natural samples ( nat C, nat Ni, nat
Sn, and nat
Pb) against the high-precision data sets on naturalsamples from [8] and [12] (Fig. 7). Our natural sampleresults are in excellent agreement with these previous re-sults from 3-100 MeV and show slight deviation above100 MeV (a relative difference of up to 5% at 300 MeV),suggesting a small systematic error at high energies inone or both approaches when the instantaneous neutronflux is highest. As an additional diagnostic, we compared σ tot results from our long and short natural carbon tar-gets and found excellent agreement, within 1% through- FIG. 7: (a) A comparison of literature data (taken with analogtechniques) and our results (signals processed with a digitizer,or “DSP”) for natural C, Ni, Sn, and Pb. The absolute crosssections are shown from 3-500 MeV. (b) Relative differencesbetween the literature data and our data are shown in per-cent. From 3-100 MeV, our data are fully consistent with theliterature but above 100 MeV, a difference arises, peaking at ≈
5% at 300 MeV. out the measured energy domain.Extracting the , O σ tot required subtraction of thewell-measured σ tot for H. To better characterize the ad-ditional systematic uncertainty associated with this sub-tractive analysis, we subtracted our measured values for O neutron σ tot from our raw D O and H O data andcalculated the D-to-H relative difference. A comparisonof our D-to-H relative difference with that of [33] is shownin Fig. 8. Our results differ systematically from the pre-vious (analog) measurement by 2-3% throughout the en-ergy range, comparable to the 2% systematic differencebetween our final O neutron σ tot results and those of[12]. The size and uniformity of these systematic differ-ences is consistent with a combination of slight ( ≈ σ tot results from our measurement or in theliterature data. FIG. 8: The σ tot relative difference between deuterium andhydrogen, as calculated by subtraction of our O σ tot resultsfrom D O and H O. Data from our measurement are shownas red squares; the data of Abfalterer et al. [33], which weregenerated using CH , C H , and D O targets, are shown asblack circles.
EXPERIMENTAL RESULTS
Our absolute σ tot results for O, Ni, and Sn isotopictargets are shown in Fig. 9. Results for Rh are shownin Fig. 11. Literature isotopic σ tot measurements (wherethey exist) are shown alongside our results for compari-son. Residuals between our data and any existing liter-ature data are also shown. In each figure, the literaturedata sets have been rebinned to match the bin structureof our data to facilitate comparison. In regions with alow density of states where individual resonances are vis-ible (e.g., nat C below 10 MeV), this rebinning washes outthe fine structure of the cross sections.Except for the already well-measured O, our newdata significantly extend knowledge of the neutron σ tot for each sample. In the cases of O, Ni,
Rh, and
Sn, almost no previous data were available above 20MeV. Our new data are in good agreement with the pre-vious measurements where available. In the cases of therare isotopes Ni and
Sn, data were available at onlyone energy, 14.1 MeV, from a study from more than 50years ago [34] and our measurement is in excellent agree-ment, within 2-3%.Our results for relative differences between isotopicpairs , O, , Ni, and , Sn are shown in Fig. 10.For , O [Fig. 10(a)], the purely isoscalar SAS model[Eq. (1)] grossly reproduces the relative difference be-low 100 MeV, but fails completely above 100 MeV. Near200 MeV, the O σ tot crosses over that of O result-ing in a negative relative difference, in keeping with theRamsauer-logic expectation of Eq. (2) that σ tot oscilla-tion minima shift to higher energies as A is increased. Inthe relative difference subfigures for , Ni and , Sn[Fig. 10(b) and 10(c)], the average σ tot values are belowthe SAS model trend ( r ∝ A ), shown by the dashedlines. The well-known r ∝ A trend in Sn isotope-shiftdata [35] is also shown for reference and underpredictsthe relative differences. In the DOM analyses presentedbelow, we fit only absolute σ tot data and did not directlyfit these relative differences. Still, the relative differencesbetween our individual DOM fits for , Ni and , Sn(black dashed-dotted lines) show overall agreement withthe experimental relative differences, especially for theSn relative difference. For the , O relative difference,there is an obvious phase mismatch between the oscil-lations of DOM calculation and the experimental data.This mismatch is symptomatic of a slight DOM over-estimation of the O radius (0.02 fm), which nudgesthe DOM-calculated O σ tot rightward so that the Ocrossover occurs at too low an energy. As was noted byDietrich et al. in their study of σ tot relative differencesin W isotopes, a simultaneous OM analysis along the en-tire isotopic chain, as in [28], may be required to realizethe full isovector-constraining power latent in the relativedifferences.
DOM ANALYSIS
The DOM is a phenomenological Green’s-functionframework enabling a simultaneous and self-consistentanalysis of nuclear structure and reaction data. An es-sential feature of the DOM is the enforcement of a disper-sion relation between the complex components of the self-energy across the entire energy domain, allowing struc-tural data from below the Fermi energy (e.g., charge den-sities, bound levels) to help constrain the potential above,and data from above the Fermi energy (e.g., elastic, re-action, and total cross sections) to help constrain the po-tential below. Using our new σ tot data for , O, , Ni,and , Sn, we performed a simultaneous fit on eachisotopic pair and also revisited , Ca and
Pb. Com-pared to previous DOM analyses [1, 28, 29, 43], we em-ploy an updated version of the DOM that has been gen-eralized for use with any combination of near-sphericaleven-even nuclei. Partial occupation of neutron openshells, as for the neutron d / valence shell in O, is ac-commodated using the level’s energy E and the pairingparameter ∆:∆( N, Z ) ≡
14 [ B ( N − , Z ) − B ( N − , Z )+3 B ( N, Z ) − B ( N + 1 , Z )] , (10)where B ( N, Z ) is the binding energy of the nucleus with N neutrons and Z protons. Occupation for the level issplit into upper ( n + ) and lower ( n − ) components: n ± = 12 (1 ± χ s ) , (11)where χ ≡ E − (cid:15) F , s ≡ ( χ + ∆ ) . Only the lower (oc-cupied) component is included in calculations of bound-state quantities (e.g., total particle number, binding en-ergy). In the appendices, we provide the functional formsused to define the potential (Appendix A), optimizedparameter values with uncertainties (Appendix B), andfigures showing the quality of the DOM reproduction toeach experimental data set (Appendix C). The other ma-jor methodological difference is the use of Markov-ChainMonte Carlo (MCMC) for parameter optimization, dis-cussed below.For additional details on the underlying DOM formal-ism, see [44, 45]. To calculate cross sections from theself-energy, the standard R-matrix approach was used[46]. Except where indicated, experimental data usedfor fitting are the same as in [26]. To situate the reader,we describe the corpus of experimental data and DOMresults for , O in full detail. The experimental dataused and fit quality for , Ca, , Ni, , Sn, and
Pb are similar in quantity and quality and only keydifferences are noted. For systematics of neutron skinsand binding energies, see companion Letter [23]. O experimental data used in DOM analysis
For protons, twenty-eight differential elastic cross sec-tions data sets and twenty analyzing power data setsfrom 10-200 MeV were incorporated. Only three pro-ton reaction cross section data sets, ranging from 20-65MeV, were available. As an added constraint, we usedsystematic trends from the comprehensive proton σ rxn review of Carlson [47] to generate proton σ rxn pseudo-data from 70-200 MeV, which were included in the fit.These pseudo-data are shown as gray open symbols inthe proton σ rxn figures in Appendix C. For neutrons, tendifferential elastic cross section data sets from 10 MeVto 95 MeV, a single neutron reaction cross section datapoint at 14 MeV, and our newly measured σ tot resultsfor O were included. In all, over sixty experimentalnucleon scattering data sets were used to constrain the O parameters.In addition to nucleon scattering data, several sectorsof bound-state data were included in the fit. Neutron(proton) 0p / and 0d / single-particle level energieswere assigned according to the nucleon separation en-ergies of O and O isotopes ( O, F isotopes) [48].Charge density distributions were taken from the compi-lation of [49]. Since the time of that compilation, newexperiments (particularly muonic-atom measurements)have improved the precision of many root-mean-square(rms) charge radii by roughly an order of magnitude [50].To account for these improved data, we rescaled the dis-tributions from [49] to recover the updated rms chargeradii while still conserving particle number. We also fit-ted directly to the updated rms charge radii of [50]. Be-cause the DOM self-energy does not necessarily conserveparticle number, we included the “experimental” protonand neutron numbers of eight as part of the fit. Lastly,the total binding energy of O from [48] was included as
FIG. 9: Neutron σ tot for , O, , Ni, and , Sn: our results and literature data. In the upper three panels, our digitizer-measured isotopic results are shown in red and corresponding analog-measured literature data [8, 34, 36–42] are shown in blue.The data for O have been shifted up by 1 barn for visibility. The lower three panels show residuals between our data and theliterature data shown in the upper panels.FIG. 10: , O, , Ni, , Sn neutron σ tot relative differences from our measurement. In each panel, the colored bandsindicate regions of 1 σ -uncertainty due to target thickness imprecision (blue) and from both target thickness and statistics (red).The gray dashed lines show the prediction for the σ tot relative difference per the strongly absorbing sphere (SAS) model of Eq.(1), which assumes a simple A size scaling for the nuclear radius. The gray dotted lines show the SAS model prediction butwith an A size scaling. The black dash-dotted lines shows the σ tot relative differences from the median parameter values ofthe O, Ni, and Sn DOM analyses performed in this work (detailed in the following section). a constraint. O experimental data used in DOM analysis
Extensive proton elastic scattering data for O wasavailable from the EXFOR database. Twenty-eight pro-ton elastic differential cross sections were included rang-ing from 10-200 MeV. Unfortunately, no proton reactioncross section data were available at all in the relevantrange of 10-200 MeV. As with O, we generated pro-ton reaction cross section pseudo-data from systematictrends in [47] from 70-200 MeV. On the neutron side, two differential elastic cross section data sets were in-cluded, at 14 and 24 MeV, but no analyzing powers wereavailable. One datum for the neutron reaction cross sec-tion, at 14.1 MeV, was incorporated as well. Our σ tot results for O were the sole neutron total cross sectiondata used in the fit. The energies of the proton and neu-tron 0p / and 0d / single-particle levels were assignedaccording to the same procedure used for O.Unlike O, for O, no charge density distribution wasavailable from [49]. To approximate it, we rescaled thecharge density distribution used for O to give the Orms charge radius of [50] while preserving eight units ofcharge. As with O, we also fitted to the experimental0
FIG. 11: Neutron σ tot for Rh: our results and literaturedata. In panel (a), our digitizer-measured results are shownin red and corresponding analog-measured literature data [10]are shown in blue. Panel (b) shows the residuals between ourdata and the literature data, where it exists. rms charge radius directly, to the particle numbers N and Z , and the total binding energy. MCMC analysis
Several aspects of the DOM potential make optimiza-tion challenging. Even with the reduced number of po-tential parameters used in this work (42 for
Pb and43 for all other pairwise fits) compared to past DOMstudies (for example, 60 or more in [1]), we found thatclassical gradient-descent methods were inappropriate forreliably searching the parameter space. A recent study[51] systematically compared Bayesian optical model op-timization techniques to frequentist ones, the type almostuniversally used in previous analyses, and found that tra-ditional algorithms may be overconfident in their param-eter estimation. To avoid these problems, we used theaffine-invariant MCMC library, emcee [52], for optimiza-tion and uncertainty characterization. For an in-depthintroduction to applied MCMC, see [53].In the ensemble-sampling approach, several hundred“walkers” are first randomly initialized in parameterspace for each isotopic system to be fitted. At each sub-sequent step t during the random walk, each walker’sposition is updated from (cid:126)x t → (cid:126)x t +1 either by acceptinga new position (cid:126)x (cid:48) with probability: p ( (cid:126)x → (cid:126)x (cid:48) ) = min(1 , U ( (cid:126)x (cid:48) | D ) U ( (cid:126)x | D ) ) , (12)or by remaining in the same position (cid:126)x with probability1 − p ( (cid:126)x → (cid:126)x (cid:48) ). New positions are proposed according to the stretch-move proposal distribution of [54] (for ourstretch move scaling, we used α = 1 . α = 2 .
0, which improved the typical acceptancefraction from around 5% to 15%). In Eq. (12), the util-ity of a parameter vector conditional on the experimentaldata U ( (cid:126)x | D ) was defined according to Bayes rule (omit-ting the evidence term): U ( (cid:126)x | D ) ∝ L ( D | (cid:126)x ) × P ( (cid:126)x ) , (13)where D is the full set of constraining experimental data.The parameter prior distribution P ( (cid:126)x ) was specified asuniform over a physically reasonable range for each pa-rameter. For example, the diffusenesses of all Woods-Saxon potential geometry terms were restricted to 0.4-1.0fm. Other more sophisticated choices for the prior distri-bution (e.g., broad truncated Gaussians) were tested andhad little impact on the resulting posterior distributions.The likelihood function was defined as a least-squaresfunction over all data sectors d : L ( D | (cid:126)x ) = (cid:88) d N d N d (cid:88) i =1 (cid:32) y calcd,i − y expd,i σ calcd,i + σ expd,i (cid:33) , (14)where • N d is the number of experimental data points in adata sector d , • y calc,expd,i are the calculated and experimental values,respectively, for the i th datum of sector d , • σ calc,expd,i are the assigned model and experimentalerrors, respectively, for the i th datum of sector d .Appendix A shows the parameter definitions and priordistributions used in the present analysis.Due to the choice of functional form and finite modelbasis size, DOM predictions for nuclear observables sufferfrom inherent model error. For example, many previousOM analyses tend to easily reproduce low-angle exper-imental dσd Ω data taken at lower scattering energies butare increasingly discrepant with the data at high ener-gies and at backward angles, where the predicted crosssections may differ from experimental results by an or-der of magnitude or more. This discrepancy indicates adeficiency in the potential form of the OM; ignoring itcan lead to drastic underestimation of variances of ex-tracted quantities. In this investigation, we found thatthe inclusion of reasonable model discrepancy terms inour utility function improved the visual fit to experi-mental data while broadening parameter uncertainties,in keeping with the methodological findings of [55]. TableII shows the model error terms we used for each data sec-tor. We assigned model error for each data set accordingto how well preliminary fits could reproduce differing re-gions of each data sector, the flexibility of the functionalforms, and intuition from the successes and failures ofpast OM analyses. In principle, the form of these model1 TABLE II: Model error terms for each data sector used inthe MCMC utility function. For terms with units of %, themodel error was calculated as a percentage of the experimen-tal data point magnitude. For dσd Ω the model error increasedlinearly with respect to the scattering angle in the center-of-mass frame with units of % per degree. (cid:15) nlj are the single-particle energies for valence nucleons as calculated from sep-aration energies in [48]. r rms is the root-mean-square chargeradius and ρ q is the charge density distribution. dσd Ω A σ tot σ rxn (cid:15) nlj BE/A N, Z r rms ρ q (%/ ° ) (-) (%) (%) (MeV) (%) (-) (fm) (%)0.25 0.10 0.25 0.25 0.10 5.0 0.10 0.005 1.0 error terms could also be treated as random variables tobe sampled over during MCMC, but due to computa-tional limitations and the already-challenging size of theDOM parameter space, we elected to fix the model er-ror terms. After N samples have been taken from theposterior distribution, a subset can be used to estimatethe true parameter distributions, and physics results cal-culated for each sample. Ensuring that this subset isrepresentative of the true posterior is discussed in thenext section.Following [52] we attempted an autocorrelation anal-ysis to test for convergence and estimate the number ofindependent samples we had collected for each nucleus.Because of computational limitations on the number ofwalkers and steps used to approximate the posteriors,posterior estimation involves a finite MCMC samplingerror. The integrated autocorrelation time for a physicsfeature f , denoted τ f , represents the number of steps re-quired for a walker to produce a new, decorrelated poste-rior sample for the feature that is independent of the pre-vious independent sample. In an ideal MCMC analysis, τ f could be accurately computed for each physics quan-tity and the MCMC sampling error could be robustlyestimated. In practice, we found this to be computa-tionally infeasible for the DOM parameter space. Forexample, in preliminary analysis of O, we were able toperform N = 31000 steps for each of 336 walkers (morethan 100,000 CPU-hours in total). Over this domain, wecalculated the integrated autocorrelation time for eachpotential parameter p , denoted τ p , to be roughly 2800steps. Assuming a N > τ p rule-of-thumb conditionfor convergence of the τ p estimate near its true value,the decorrelation time appears to be extremely long. Inother words, from τ p alone, we could not exclude the pos-sibility that the parameters had not yet fully “settled” inthe region of their optimal values and begun independentsampling of the parameter posteriors. We note that thetrue τ f could be considerably smaller than τ p due to thehighly correlated nature of DOM parameter space.To proceed, we applied several commonsense tests tojudge whether our parameter and extracted-quantity es-timates were accurate. First, we sampled as long as pos-sible and used as many parallel walkers as possible, given our computational resources. From time to time duringsampling, we analyzed the mean walker positions andthe mean walker position likelihood as a function of sam-pling step. Encouragingly, for all nuclei walkers quicklyconverged on a common region (within 1000 samples)and their mean parameter values stabilized soon after-ward (within 10000 samples), suggesting that walkerswere sampling a reasonably optimal subspace. At thispoint, we considered the chain tentatively converged. Asan additional test, we re-started sampling from a different(uniformly random) initial position for each nucleus andfound that a similar optimal subspace was reached, againwithin roughly 1000 samples, indicating that our resultsare independent of the initial walker positions. Finally,for a “converged” chain, we calculated extracted physicsquantities (e.g., neutron skins, scattering cross sections)for all walkers at several intervals to confirm that theirmean values were stable. Again using O and O asan example, we found their mean neutron skin valuesvaried by less than 0.001 and 0.01 fm, respectively, overseveral thousand sampling steps late in sampling. Out ofcaution (and given our expectation of very large autocor-relation times) we used only the terminal sample for eachwalker chain to produce the results presented here andin the companion Letter [23]. In the end, we expect thatadditional sampling could slightly reduce the estimatedvariance of each extracted quantity but have a negligibleeffect on the mean values. For all quantities derived fromMCMC analysis, the estimated 16 th , 50 th , and 84 th pos-terior percentile values are denoted as 50 . The rangebetween the 16 th and 84 th percentiles corresponds to a1 σ -uncertainty range if the posteriors are assumed to beGaussian. The median values and ranges for each pa-rameter for each isotope system are listed in AppendixB. Fit results on , O Figure 12 in Appendix C shows the DOM fit of O andexperimental data. The experimental proton σ rxn , neu-tron dσd Ω , σ tot , and σ rxn charge density distribution, bind-ing energy per nucleon, and p / and d / single-particleenergy data are all well-reproduced suggesting that theDOM is effective for modeling nuclei as light as A =16.Almost all experimental proton dσd Ω data are accuratelyreproduced by the DOM calculations with the exceptionof an overprediction of cross sections at backward an-gles and high energies, a regime known to be challengingfrom past OM analyses. In addition, the median DOM-generated rms charge radius, 2.72 fm, slightly exceeds theexperimental value of 2.70 fm. Taken together with the , O relative difference results in panel (a) of Fig. 10,these overestimations indicate that the traditional OMassumption of radial proportionality with A / must betweaked for a better description of O.To reproduce the O proton σ rxn pseudo-data gen-2erated from [47], a larger volume imaginary term wasrequired above 100 MeV, which in turn reduced the spec-troscopic strength for the valence π and ν p / nucleonsby roughly 0.05. We also note the importance of thecharge density distribution for determining the magni-tude of the imaginary strength below the Fermi energy.For example, in test fits where the charge density was notincluded as a constraint, most of the negative imaginarystrength was concentrated in the surface term between − < E < (cid:15) F MeV, and the tail of the charge densitywas overpredicted. With the charge density included asa constraint, the imaginary surface magnitude shrank bya factor of two and the volume term grew to compen-sate, pushing nucleon density deeper in energy space andincreasing the binding energy closer to the experimentalvalue.While all data sectors contributed at least some in-formation not fully captured by any other sector, theproton σ rxn , neutron σ tot , and charge density providedthe most stringent constraints on the self-energy. Theanalyzing powers were the most difficult sector of ex-perimental data to reproduce, with moderate deviationsvisible from 10-15 MeV for both protons and neutronsand above 100 MeV for protons [Figs. 12(b) and 12(d)].Some of the difficulty with the analyzing powers is at-tributable to our neglecting of an imaginary spin-orbitterm in the DOM potential used in this work, a choicemade due to the unreasonable unbounded growth of theimaginary spin-orbit term as (cid:96) grows in the traditional (cid:96) · σ definition used in [21]. In a future analysis we intendto quantitatively investigate the importance of the imag-inary spin-orbit term and to compare different optionsfor its functional form.Figure 13 in Appendix C shows the O experimentaldata and the DOM fit. The paucity of O experimen-tal data presented a challenge for our analysis. To con-strain the negative-energy domain of the potential, theonly unambiguous experimental data were the neutronand proton separation energies and the overall bindingenergy. As with O, broad agreement with experimentaldata was achieved for experimental proton and neutron dσd Ω data, the neutron σ tot , rms charge radius, binding en-ergy per nucleon, and p / and d / single-particle energydata. The artificially scaled charge density and proton σ rxn data were also easily reproduced. Due to the deteri-oration of systematic trends from [47] below 70 MeV, wedid not generate proton σ rxn pseudo-data for lower ener-gies, so the positive-energy surface term of the potentialwas largely unconstrained in this important area.In symmetric O, the proton and neutron potentialswere identical except for the Coulomb interaction, so theneutron σ tot data provided information about both theproton and neutron imaginary strength at positive ener-gies. For O, this expectation of symmetric potentialswas inapplicable, making proton σ rxn data essential forfixing the positive-energy imaginary strength for protons.In principle, O proton and neutron differential elas- tic scattering cross sections about 100 MeV could jointlyyield some information about the asymmetry-dependenceof the imaginary strength for O, but no neutron elasticscattering data were available above 24 MeV. For a bet-ter characterization of this nucleus, even a single proton σ rxn datum between 10 and 50 MeV would be valuable. Fit results for , Ca, , Ni, , Sn, and Pb Figures 14-20 in Appendix C show , Ca, Ni, , Sn, and
Pb experimental data and the DOMfits. The availability of single-nucleon scattering data for , Ca, , Ni, , Sn, and
Pb followed the sametrends as that for , O: plentiful proton differential elas-tic scattering data, moderate coverage for neutron dif-ferential elastic cross sections and proton reaction crosssections on abundant isotopes ( Ca, Ni, and
Pb),with little-to-no coverage for neutron scattering or protonreaction cross section data on rare isotopes ( Ca, Ni,
Sn,
Sn). For
Sn and
Sn, however, even pro-ton elastic scattering data sets were sparse and no dataabove 50 MeV were available, making our newly collectedneutron σ tot data especially valuable in constraining thepotential. For Ca and
Pb, experimental proton reac-tion cross section data were available up to 200 MeV; forthe other isotopes, proton reaction cross section pseudo-data (discussed in the , O subsections) were used asa constraint. As for O, no charge-density parameteri-zation was available for
Sn in [49], so we rescaled theavailable
Sn distribution to reproduce the
Sn chargeradius.Generally, all sectors of experimental data were well-reproduced; exceptions include the high-angle (above120 ° ) proton elastic scattering data for Ca and
Pb,where data sets were available up to 200 MeV, andthe single-particle energies for neutron open shells in , Sn (see Figs. 18 and 19), where several levelsare partially filled and clustered near the Fermi surface.Achieving more accurate single-particle energies whilepreserving particle number accuracy may require a moresophisticated treatment of pairing. Our new neutron σ tot data were well-reproduced across the board, typi-cally within 2% of the experimental value, by the DOMfits, suggesting that our Lane-like parameterization ofthe potential’s asymmetry dependence [Eqs. (29-32)] isa promising starting point for extrapolation away fromstability. We note that because Pb was fit on its ownwithout an isotopic partner, initial fits showed that theasymmetry-dependence of the HF radius term was toopoorly constrained to yield reliable neutron skin results;in the final treatment, this term was disabled for
Pb.3
TABLE III: Spectroscopic factors for valence proton ( π ) and neutron ( ν ) levels, extracted from our DOM analysis. The 16 th ,50 th , and 84 th percentile values of the MCMC-generated posterior distributions are reported as 50 .Isotope O O Ca Ca Ni Ni Sn Sn Pb π Level 0p / / / / / / / / / SF 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . ν Level 0p / / / / / / / / / SF 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . Discussion
Table III shows DOM-calculated SFs for valence pro-ton and neutron levels for all nine systems. Significantdepletion from the mean-field expectation appears evenin the light systems , O. In the present study, the ex-tracted proton SFs show only a very weak dependenceon neutron-richness within each isotopic pair, in keepingwith the weak dependence extracted in ( e, e (cid:48) p ) and trans-fer reaction studies and at odds with knockout-reactionanalyses that recover a strong asymmetry-dependence[45, 56]. The recent DOM analyses of [43, 57] iden-tified proton reaction cross sections above roughly 100MeV as important for their successful reproduction of , Ca ( e, e (cid:48) p ) cross sections without arbitrary SF rescal-ing. Compared to the present work, these analyses founda much larger reduction of valence proton SFs in Cawith respect to Ca, indicative of an SF asymmetry de-pendence somewhere between the weak dependence de-duced from transfer reactions and the very strong depen-dence from knockout reactions.To understand the differences between these analy-ses, we conducted several diagnostic runs with artifi-cially scaled Carlson pseudo-data in Ca. These diag-nostic runs confirmed that fitting to appropriate high-energy proton reaction cross sections leads to larger Caproton imaginary strength both far above and far be-low the Fermi energy, an effect already seen in previ-ous DOM work. However, the growth we observed inthe imaginary potential was more modest compared toprevious treatments, potentially explaining the weakerasymmetry-dependent SF reduction. We also note thatin the present work, the high-energy neutron total crosssections and proton reaction cross sections appeared tohave little impact on other extracted quantities such asneutron skins, as had been previously hypothesized forthe neutron skin of Ca [1]. We conclude that the dif-ferent methodological choices, especially the focus of thiswork on simultaneous fitting of isotope pairs, is respon-sible for the differences in these asymmetry-dependentquantities. To further clarify the situation, the potentialsof the present work should be used to generate ( e, e (cid:48) p )cross sections that can be compared to the previous find-ings of [57].Surprisingly, despite the extensive proton and neutronelastic scattering data for O, Ca, and
Pb, the ex-tracted spectroscopic factor distributions and parameter uncertainties for these isotopes are just as wide as forthose systems with barely any available elastic scatteringdata, such as Ni. We tentatively conclude that the elas-tic scattering data we used are very weak constraints onthe all-important imaginary terms of the optical poten-tial, at least for the stable, spherical systems discussedhere. Unfortunately, this suggests that elastic scatter-ing measurements in inverse kinematics on radioactivebeams are of diminishing utility for extrapolating opti-cal potentials away from β -stability. A program of pro-ton reaction cross section and neutron total cross sectionmeasurements on radioactive targets could be useful forunderstanding the potential’s near-Fermi-level asymme-try dependence but is experimentally daunting. Instead,a two-pronged approach may be required. On the ex-perimental side, proton reaction and neutron total crosssection measurements on stable isotopic chains can helpidentify which asymmetry-dependence forms are justifi-able for increasingly asymmetric systems. On the theo-retical side, sensitivity studies are needed to clarify howbound-state data on highly asymmetric systems connectto scattering cross sections.Lastly, a few systematics in optical potential param-eter values are worth mention. For most of the param-eters, there was minimal variation with nuclear size orasymmetry, suggesting that a global DOM treatmentusing the functional forms we have selected is achiev-able. The radial term for the real central potential ( r )and for the positive-energy imaginary volume and sur-face ( r + , r + ) are nearly constant among , Ca, , Ni, , Sn, and
Pb, but the values for , O show mod-erate deviations, another indication that the geometricform of the potential is insufficient for light systems.As a consequence of the limited negative-energy dataavailable for fitting, the negative energy geometric terms( r − , r − , a − , a − ) show large variation. The nonlocalitiesfor the negative imaginary components are systematicallylarger than those for the positive imaginary components.This suggests that while traditional OMs have been ableto successfully reproduce positive-energy scattering datawith strictly local potentials, description of hole prop-erties requires true nonlocal character in the negative-energy potential. In practice, we found it impossibleto simultaneously reproduce charge density distributions,binding energies, and scattering data unless the centralpotential and at least the volume imaginary terms wereequipped with a nonlocality. In the end, for simplic-4ity and generality, each element of the potential (exceptCoulomb) was treated nonlocally, but it is unclear whichparticular data are most important for constraining theseseveral nonlocalities. As one moves further from stabilityto systems with even less (or no) scattering data avail-able, the risk of overfitting will loom until this issue isresolved.In preliminary fits, the imaginary volume magnitude( A − ) component of the potential was shown to bestrongly sensitive to the inclusion of the binding energyas a constraint during fitting. We expect the asymmetry-dependence of this term, ( A − vol , asym ), to impact DOM-based predictions of the Ca, Ni, and Sn neutron driplines(as in [28]), though in this work, this dependence wasvery poorly constrained due to the absence of exper-imental asymmetry-dependent data probing the mostdeeply bound nucleons. Because they encode informationabout how protons and neutrons share energy through-out the nucleus, experimental neutron-skin thicknessescould provide this kind of valuable information. For theCa, Ni, Sn, and Pb fits, the median positive-energy sur-face imaginary magnitude ( A + sur , asymm ) is positive, indi-cating enhancement in proton surface imaginary strengthwith increasing neutron richness, and a corresponding de-crease for neutron surface imaginary strength. Of course,the nuclei under study in the present work are stable; thetrend for nuclei with large asymmetries, relevant for ther-process neutron-capture rate, is unknown. CONCLUSION
By adopting a digitizer-driven approach, we measured σ tot on the important closed-shell nuclides , O, , Ni,and , Sn across more than two orders of magnitudein energy (3-450 MeV). Except at the highest energies,our results on natural targets are in good agreement withprevious analog-mediated measurements that required10-20 times more target material.Using these new data and a suite of scatteringand bound-state literature data on , O, , Ni, and , Sn, we extracted DOM potentials capable of re-producing a diverse range of scattering and structuraldata for both neutrons and protons, validating the useof the DOM away from doubly closed shells from A =16to A =208, though with indications that the traditional A / radial dependence may require modification for lightsystems. These analyses further indicate that simultane-ous fits of isotopically resolved neutron σ tot , proton σ rxn ,and charge-density distribution data on isotopic partnersprovide a more stringent constraint on the asymmetry-dependence of both real and imaginary components. ACKNOWLEDGEMENTS
This work is supported by the U.S. Department ofEnergy, Office of Science, Office of Nuclear Physics un-der award numbers DE-FG02-87ER-40316, by the U.S.National Science Foundation under grants PHY-1613362and PHY-1912643, and by the National Nuclear Se-curity Administration of the U.S. Department of En-ergy at Los Alamos National Laboratory under Con-tract No. 89233218CNA000001. C.D.P. acknowledgessupport from the U.S. Department of Energy SCGSRProgram (2014 and 2016 solicitations) and the Na-tional Nuclear Security Administration through the Cen-ter for Excellence in Nuclear Training and UniversityBased Research (CENTAUR) under grant number DE-NA0003841. Computations were performed in part us-ing the facilities of the Washington University Center forHigh Performance Computing, which were partially pro-vided through NIH grant S10 OD018091, and in partunder the auspices of the U.S. Department of Energyby Lawrence Livermore National Laboratory under Con-tract DE-AC52-07NA27344. [1] M. H. Mahzoon, M. C. Atkinson, R. J. Char-ity, and W. H. Dickhoff, Phys. Rev. Lett. ,222503 (2017), URL https://link.aps.org/doi/10.1103/PhysRevLett.119.222503 .[2] F. J. Fattoyev and J. Piekarewicz, Phys. Rev. C , 015802 (2012), URL https://link.aps.org/doi/10.1103/PhysRevC.86.015802 .[3] X. Vi˜nas, M. Centelles, X. Roca-Maza, and M. Warda,Eur. J. Phys. A , 27 (2014), URL https://doi.org/10.1140/epja/i2014-14027-8 .[4] B. A. Brown, Phys. Rev. Lett. , 5296 (2000), URL https://link.aps.org/doi/10.1103/PhysRevLett.85.5296 .[5] S. Fernbach, R. Serber, and T. B. Taylor, Phys. Rev. ,1352 (1949), URL https://link.aps.org/doi/10.1103/PhysRev.75.1352 .[6] G. R. Satchler, Introduction to Nuclear Reactions (JohnWiley And Sons, 1980).[7] J. M. Peterson, Phys. Rev. , 955 (1962), URL https://link.aps.org/doi/10.1103/PhysRev.125.955 .[8] R. W. Finlay, W. P. Abfalterer, G. Fink, E. Montei,T. Adami, P. W. Lisowski, G. L. Morgan, and R. C.Haight, Phys. Rev. C , 237 (1993), URL http://dx.doi.org/10.1103/PhysRevC.47.237 .[9] R. B. Schwartz, R. A. Schrack, and H. T. Heaton II, Tech.Rep. 138, National Bureau of Standards (1974).[10] W. P. Poenitz and J. F. Whalen, Tech. Rep. 80, ArgonneNational Laboratory (1983).[11] W. P. Abfalterer, R. W. Finlay, and S. M. Grimes, Phys.Rev. C , 064312 (2000), URL https://link.aps.org/doi/10.1103/PhysRevC.62.064312 .[12] W. P. Abfalterer, F. B. Bateman, F. S. Dietrich, R. W.Finlay, R. C. Haight, and G. L. Morgan, Phys. Rev. C , 044608 (2001), URL http://dx.doi.org/10.1103/PhysRevC.63.044608 . [13] S. G. Carpenter and R. Wilson, Phys. Rev. ,510 (1959), URL http://journals.aps.org/pr/pdf/10.1103/PhysRev.114.510 .[14] I. Angeli and J. Csikai, Nucl. Phys. A , 389(1970), URL .[15] C. B. O. Mohr, Proc. Phys. Soc. A , 340 (1955), URL http://stacks.iop.org/0370-1298/68/i=4/a=410 .[16] H. Feshbach, Ann. Rev. Nucl. Part. Sci. , 49(1958), URL https://doi.org/10.1146/annurev.ns.08.120158.000405 .[17] K. W. McVoy, Ann. Sci. , 91 (1967), URL .[18] I. Ahmad, N. Bano, and A. N. Saharia, Pramana - J.Phys. , 188 (1973), URL https://link.springer.com/article/10.1007/BF02847190 .[19] C. M. Perey and F. G. Perey, Atom. Data Nucl. DataTables (1976).[20] R. L. Varner, W. J. Thompson, T. L. McAbee,E. J. Ludwig, and T. B. Clegg, Phys. Rep. , 57(1991), URL .[21] A. J. Koning and J. P. Delaroche, Nucl. Phys. A , 231(2003), URL .[22] F. S. Dietrich, J. D. Anderson, R. W. Bauer, S. M.Grimes, R. W. Finlay, W. P. Abfalterer, F. B. Bateman,R. C. Haight, G. L. Morgan, E. Bauge, et al., Phys. Rev.C , 044606 (2003), URL https://link.aps.org/doi/10.1103/PhysRevC.67.044606 .[23] C. D. Pruitt, R. J. Charity, L. G. Sobotka, M. C. Atkin-son, and W. H. Dickhoff, Phys. Rev. Lett. , 102501(2020).[24] T. W. Phillips, B. L. Berman, and J. D. Seagrave, Phys.Rev. C , 384 (1980), URL https://link.aps.org/doi/10.1103/PhysRevC.22.384 .[25] D. G. Foster and D. W. Glasgow, Phys. Rev. C ,576 (1971), URL https://link.aps.org/doi/10.1103/PhysRevC.3.576 .[26] C. D. Pruitt, Ph.D. thesis, Washington University in StLouis (2019).[27] R. Shane, R. J. Charity, J. M. Elson, L. G. Sobotka,M. Devlin, N. Fotiades, and J. M. O‘Donnell, Nucl. In-strum. Meth. , 468 (2010), URL http://dx.doi.org/10.1016/j.nima.2010.01.005 .[28] J. M. Mueller, R. J. Charity, R. Shane, L. G. Sobotka,S. J. Waldecker, W. H. Dickhoff, A. S. Crowell, J. H.Esterline, B. Fallin, C. R. Howell, et al., Phys. Rev. C , 064605 (2011), URL https://link.aps.org/doi/10.1103/PhysRevC.83.064605 .[29] M. H. Mahzoon, R. J. Charity, W. H. Dickhoff,H. Dussan, and S. J. Waldecker, Phys. Rev. Lett. , 162503 (2014), URL https://link.aps.org/doi/10.1103/PhysRevLett.112.162503 .[30] M. Mahzoon, Ph.D. thesis, Washington University inSt Louis (2015), URL http://libproxy.wustl.edu/login?url=https://search.proquest.com/docview/1749780826?accountid=15159 .[31] M. S. Moore, Nucl. Instrum. Meth. , 245(1980), URL .[32] J. M. Clement, P. Stoler, C. A. Goulding, and R. W.Fairchild, Nucl. Phys. A , 51 (1972), URL http://dx.doi.org/10.1016/0375-9474(72)90930-X . [33] W. P. Abfalterer, F. B. Bateman, F. S. Dietrich, C. El-ster, R. W. Finlay, W. Gl¨ockle, J. Golak, R. C. Haight,D. H¨uber, G. L. Morgan, et al., Phys. Rev. Lett. (1998).[34] Y. V. Dukarevich, A. N. Dyumin, and D. M. Kaminker,Nucl. Phys. A , 433 (1967), URL http://dx.doi.org/10.1016/0375-9474(67)90228-X .[35] M. Anselment, K. Bekk, A. Hanser, H. Hoeffgen,G. Meisel, S. Goring, H. Rebel, and G. Schatz, Phys. Rev.C , 1052 (1986).[36] F. G. Perey, T. A. Love, and W. E. Kinney, Tech. Rep.4823, Oak Ridge National Lab (1972).[37] F. J. Vaughn, H. A. Grench, W. L. Imhof, J. H. Rowland,and M. Walt, Nucl. Phys. , 336 (1965), URL http://dx.doi.org/10.1016/0029-5582(65)90361-5 .[38] S. R. Salisbury, D. B. Fossan, and F. J. Vaughn, Nucl.Phys. , 343 (1965), URL http://dx.doi.org/10.1016/0029-5582(65)90362-7 .[39] C. M. Perey, F. G. Perey, J. A. Harvey, N. W. Hill,N. M. Larson, R. L. Macklin, and D. C. Larson, Phys.Rev. C , 1143 (1993), URL http://dx.doi.org/10.1103/PhysRevC.47.1143 .[40] R. W. Harper, T. W. Godfrey, and J. L. Weil, Phys. Rev.C , 1432 (1982), URL http://dx.doi.org/10.1103/PhysRevC.26.1432 .[41] V. M. Timokhov, M. V. Bokhovko, A. G. Isakov, L. E.Kazakov, V. N. Kononov, G. N. Manturov, E. D. Poletaev,and V. G. Pronyaev, Yad. Fiz. , 609 (1989).[42] J. Rapaport, M. Mirzaa, M. Hadizadeh, D. E. Bainum,and R. W. Finlay, Nucl. Phys. A , 56 (1980), URL http://dx.doi.org/10.1016/0375-9474(80)90361-9 .[43] M. C. Atkinson, H. P. Blok, L. Lapik´as, R. J.Charity, and W. H. Dickhoff, Phys. Rev. C ,044627 (2018), URL https://link.aps.org/doi/10.1103/PhysRevC.98.044627 .[44] C. Mahaux and R. Sartor, Adv. Nucl. Phys. , 1 (1991).[45] W. H. Dickhoff and R. J. Charity, Prog. Part. Nucl. Phys.(2018).[46] A. M. Lane and R. G. Thomas, Rev. Mod. Phys. ,257 (1958), URL https://link.aps.org/doi/10.1103/RevModPhys.30.257 .[47] R. F. Carlson, Atom. Data Nucl. Data Tables , 93(1996), URL .[48] M. Wang, G. Audi, F. G. Kondev, W. Huang, S. Naimi,and X. Xi, Chin. Phys. C , 030003 (2017).[49] H. D. Vries, C. W. D. Jager, and C. D. Vries,Atom. Data Nucl. Data Tables , 495 (1987), URL .[50] I. Angeli and K. P. Marinova, Atom. Data Nucl. DataTables , 69 (2013), URL .[51] G. B. King, A. E. Lovell, L. Neufcourt, and F. M. Nunes,Phys. Rev. Lett. , 232502 (2019).[52] D. Foreman-Mackey, D. W. Hogg, D. Lang, and J. Good-man, Publ. Astron. Soc. Pac. , 306312 (2013), URL http://dx.doi.org/10.1086/670067 .[53] S. Sharma, Ann. Rev. Astron. Astrophys. , 213 (2017).[54] J. Goodman and J. Weare, Commun. Appl. Math. Com-put. Sci. , 65 (2010), URL https://doi.org/10.2140/camcos.2010.5.65 .[55] J. Brynjarsdttir and A. O’Hagan, Inverse Problems (2014).[56] J. A. Tostevin and A. Gade, Phys. Rev. C ,057602 (2014), URL https://link.aps.org/doi/10. .[57] M. C. Atkinson and W. H. Dickhoff, Phys. Lett. B , 135027 (2019), URL .[58] F. Perey and B. Buck, Nucl. Phys. , 353(1962), URL .[59] R. J. Charity, J. M. Mueller, L. G. Sobotka, and W. H.Dickhoff, Phys. Rev. C , 044314 (2007), URL https://link.aps.org/doi/10.1103/PhysRevC.76.044314 . Appendix A: Definition of DOM PotentialFunctional Forms
Before giving the full parameterization, we identify afew standard functional forms. Radial dependences aredefined by a Woods-Saxon shape or a derivative: f vol ( r ; r , a ) = −
11 + e ( r − R ) /a ,f sur ( r ; r , a ) = 1 r ddr f vol ( r ; r , a ) . (15) R is the nuclear radius, calculated as R = r A . Thesign of the potential is such that the Woods-Saxon formprovides an attractive interaction. For nonlocalities, weuse a Gaussian nonlocality first proposed by [58]: N ( r, r (cid:48) ; β ) = 1 π β e − ( r − r (cid:48) ) /β , (16)where β sets the Gaussian width. The energy-dependences of the imaginary components is based onthe functional form of [59]: ω n ( E ; A, B, C ) = Θ( X ) A X n X n + B n (17)where X = | E − (cid:15) F | − C and Θ( X ) is the Heaviside step function.For symmetric nuclei, the same potential was used forprotons and neutrons, excepting Coulomb. For asym-metric nuclei, we introduced five asymmetry-dependentterms. For all energy dependences, the energy domainwas (cid:15) F -300 MeV to (cid:15) F +200 MeV.The irreducible self-energy (optical potential) used inthis work is definedΣ ∗ ( α, β ; E ) = Σ ∗ s ( α, β ) + Σ ∗ im ( α, β ; E ) + Σ ∗ d ( α, β ; E ) . (18)The energy-independent real part Σ s ( α, β ) and energy-dependent imaginary part Σ ∗ im ( α, β ) parameterizationsare given in the following two subsections. The dispersivecorrection term Σ ∗ d ∗ α, β ; E ) is completely determinedby an integral over the imaginary part [Eq. (3) of [29]].All free parameters that are fit via MCMC sampling aretypeset in bold . Real Part
The energy-independent real part of the self-energyconsists of a nonlocal Hartree-Fock and a spin-orbit com-ponent (plus a local Coulomb term if the nucleon in ques-tion is a proton):Σ s ( r, r (cid:48) ) = Σ HF ( r, r (cid:48) ) + V so ( r, r (cid:48) ) + V C ( r ) δ ( r − r (cid:48) ) . (19) The Coulomb potential is calculated using the same ex-perimentally derived charge density distributions (see[49]) used in fitting. The Hartree-Fock component V HF has two subcomponents:Σ HF ( r, r (cid:48) ) = V vol ( r, r (cid:48) ) + V wb ( r ) , (20)where the nonlocal Hartree-Fock volume term V vol ( r, r (cid:48) ),is defined as a Woods-Saxon form coupled to a Gaussiannonlocality: V vol ( r, r (cid:48) ) = − V × f vol ( r ; r , a ) × N ( r, r (cid:48) ; β ) . (21)The local Hartree-Fock wine-bottle term V wb , named forresemblance to the dimple at the bottom of a wine bottle,is defined as a Gaussian centered at the nuclear origin, V wb ( r ) = V × e r / σ . (22)The real spin-orbit component V so is defined using aderivative-Woods-Saxon shape in keeping with the ex-pectation that the spin-orbit coupling is strongest nearthe nuclear surface: V so ( r, r (cid:48) ) = (cid:18) (cid:126) m π c (cid:19) V × r f sur ( r ; r , a ) × N ( r, r (cid:48) ; β ) × ( (cid:96) · σ ) . (23)The leading constant (cid:16) (cid:126) m π c (cid:17) is taken to be 2.0 fm [30].In total, there are ten free parameters for the symmetricreal part of the potential. Imaginary Part
The imaginary part of the potential is comprised ofindependent surface and volume terms both above andbelow the Fermi surface:Σ ∗ im ( r, r (cid:48) , E ) = Σ ± vol ( r, r (cid:48) , E ) + Σ ± sur ( r, r (cid:48) , E ) , (24)where the volume and surface components are defined:Σ ± vol ( r, r (cid:48) , E ) = W ± vol ( E ) × f vol ( r ; r ± , a ± ) × N ( r, r (cid:48) ; β ± ) , Σ ± sur ( r, r (cid:48) , E ) = 4 a W ± sur ( E ) × f sur ( r ; r ± , a ± ) × N ( r, r (cid:48) ; β ± ) . (25)The terms labeled with + determine the potential above (cid:15) F , and the terms labeled with − determine the poten-tial below (cid:15) F . The energy dependence of the imaginaryvolume terms read: W ± vol ( E ) = A ± (cid:20) ( E ∆ ) ( E ∆ ) + ( B ± ) + W ± NM ( E ) (cid:21) , (26)8where E ∆ = | E − (cid:15) F | and W + NM ( E ) = α (cid:34) √ E + ( (cid:15) F + E + ) E − (cid:113) (cid:15) F + E + (cid:35) ,W − NM ( E ) = ( (cid:15) F − E − E − ) ( (cid:15) F − E − E − ) + ( E − ) . (27)The terms W ± NM are asymmetric above and below theFermi surface and are modeled after nuclear-matter cal-culations. They account for the decreasing phase space atnegative energies and the increasing phase space at pos-itive energies. The energy-dependence of the imaginarysurface terms read: W ± sur ( E ) = ω ( E, A ± , B ± , − ω ( E, A ± , B (cid:48) ± , C ± )(28)In total, there are thirteen free parameters for the sym-metric imaginary volume terms of the potential and four-teen free parameters for the symmetric imaginary surfaceterms of the potential. Thus for symmetric nuclei, thirty-seven real and imaginary parameters were used. Parameterization of Asymmetry Dependence
For asymmetric nuclei, the parametric forms must bemodified to account for the different potential experi-enced by protons and neutrons. For the real central po-tential, the depth V and radius r from Eq. (21) wereallowed to vary linearly with asymmetry: V ⇒ (cid:40) V + V asym × N − ZA for protons V − V asym × N − ZA for neutrons , (29) r ⇒ (cid:40) r + r asym × N − ZA for protons r − r asym × N − ZA for neutrons . (30)The magnitude of the energy-dependence for the imag-inary surface and volume potentials, A ± and A ± fromEqs. (26) and (28), were also allowed to vary with lin-early with asymmetry: A ± ⇒ (cid:40) A ± + A ± vol , asym × N − ZA for protons A ± − A ± vol , asym × N − ZA for neutrons , (31) A ± ⇒ (cid:40) A ± + A ± sur , asym × N − ZA for protons A ± − A ± sur , asym × N − ZA for neutrons . (32)There should be no confusion between A ± , , A (the to-tal number of nucleons), and the analyzing power. Withthese six additional asymmetry-dependent terms, the to-tal number of free parameters used for fitting asymmetricnuclei in the present work totals forty-three.9 Appendix B: Parameter Values for DOM Potential
Parameter labels correspond to those in the equations of Appendix A. For each parameter, the prior distributionwas defined to be uniform with minimum and maximum values listed in columns 2 and 3 of each table. For eachnucleus, the 16 th , 50 th , and 84 th percentile values for each estimated parameter distribution are listed. The format is50 . For Pb, the asymmetry-dependent HF radius term ( r asym ) was disabled during fitting. TABLE IV: Real central potential parameters
Par. Min Max Units Eq. , O , Ca , Ni , Sn PbV
50 150 MeV 19 112 . . . . . . . . . . . . . . . V asym -100 200 MeV 27 − . . − . . . . − . . − . . . . . . . r . . . . . . . . . . . . . . . r asym -1.0 1.0 fm 28 0 . . − . − . . − . . . . − . . − . - a . . . . . . . . . . . . . . . β . . . . . . . . . . . . . . . V . . . . . . . . . . . . . . . σ . . . . . . . . . . . . . . . TABLE V: Imaginary central potential parameters
Par. Min Max Units Eq. , O , Ca , Ni , Sn PbA + . . . . . . . . . . . . . . . B + . . . . . . . . . . . . . . . r + . . . . . . . . . . . . . . . a + . . . . . . . . . . . . . . . β + . . . . . . . . . . . . . . . A − . . . . . . . . . . . . . . . B − . . . . . . . . . . . . . . . r − . . . . . . . . . . . . . . . a − . . . . . . . . . . . . . . . β − . . . . . . . . . . . . . . . α . . . . . . . . . . . . . . . E +
50 200 MeV 25 109 . . . . . . . . . . . . . . . E −
50 200 MeV 25 104 . . . . . . . . . . . . . . . A + vol , asym -100 200 MeV 29 37 . . . . . − . . . − . . . − . . . . A − vol , asym -100 200 MeV 29 131 . . . . . − . − . . − . − . . − . − . . − . TABLE VI: Imaginary surface potential parameters
Par. Min Max Units Eq. , O , Ca , Ni , Sn PbA + . . . . . . . . . . . . . . . B + . . . . . . . . . . . . . . . B (cid:48) + . . . . . . . . . . . . . . . C + . . . . . . . . . . . . . . . r + . . . . . . . . . . . . . . . a + . . . . . . . . . . . . . . . β + . . . . . . . . . . . . . . . A − . . . . . . . . . . . . . . . B − . . . . . . . . . . . . . . . B (cid:48) − . . . . . . . . . . . . . . . C − . . . . . . . . . . . . . . . r − . . . . . . . . . . . . . . . a − . . . . . . . . . . . . . . . β − . . . . . . . . . . . . . . . A + sur , asym -100 200 MeV 30 − . . − . . . . . . − . . . . . . . A − sur , asym -100 200 MeV 30 48 . . − . − . . − . . . − . . . − . − . . − . TABLE VII: Spin-orbit parameters
Par. Min Max Units Eq. , O , Ca , Ni , Sn PbV . . . . . . . . . . . . . . . r . . . . . . . . . . . . . . . a . . . . . . . . . . . . . . . β . . . . . . . . . . . . . . . Appendix C: DOM Fit Comparison to Experimental Data
Figures 12-20 show the data sectors used to constrain the DOM potential. Experimental scattering cross sectionsare shown as points with associated experimental error bars in panels (a) through (f) of each figure. Experimentalbound-state data are shown as bands in panels (g) through (j). DOM calculations for each data sector are plotted as1 σ and 2 σ uncertainty bands. References for each data set are provided in Appendix B of [26].Panels (a) and (c) show proton dσd Ω and analyzing powers from 10-200 MeV. Panels (b) and (d) show neutron dσd Ω and analyzing powers from 10-200 MeV. For visibility, data sets at different energies are offset vertically and coloredaccording to the scattering energy. Panels (e) show proton σ rxn data. Experimental data are plotted as black pointsand pseudo-data generated from [47] are plotted as gray open circles. Panels (f) show the neutron σ tot and σ rxn . Thecharge distributions of panels (g) are derived from the compilation of [49] (see comments in DOM Analysis section),and are displayed with an arbitrary 1% uncertainty band in black. In panels (h), single-particle energies (cid:15) nlj areshown as horizontal lines. In the “calc” column, DOM-calculated single-particle energies are plotted; the height ofeach rectangle spans the 1 σ calculated uncertainty for that level. Panels (i) show DOM-calculated charge radii; theexperimental charge radius is displayed using dark gray and light gray bands representing 1 σ and 2 σ uncertainties,respectively. Panels (j) show the DOM-calculated binding energy per nucleon; the experimental value is shown witha thin gray band.1 FIG. 12: O: constraining experimental data and DOM fit. See introduction of Appendix C for description. No O neutronanalyzing powerswere availableFIG. 13: O: constraining experimental data and DOM fit. See introduction of Appendix C for description. FIG. 14: Ca: constraining experimental data and DOM fit. See introduction of Appendix C for description. No Ca neutronanalyzing powerswere availableFIG. 15: Ca: constraining experimental data and DOM fit. See introduction of Appendix C for description. FIG. 16: Ni: constraining experimental data and DOM fit. See introduction of Appendix C for description. No Ni neutronanalyzing powerswere availableFIG. 17: Ni: constraining experimental data and DOM fit. See introduction of Appendix C for description. No Sn protonanalyzing powerswere available No
Sn neutronanalyzing powerswere availableFIG. 18:
Sn: constraining experimental data and DOM fit. See introduction of Appendix C for description. No Sn neutronanalyzing powerswere availableFIG. 19:
Sn: constraining experimental data and DOM fit. See introduction of Appendix C for description. FIG. 20:208